AE3-302
Flight Dynamics Lecture Notes
March 7, 2007
J.A. Mulder / W.H.J.J. van Staveren / J.C. van der Vaart / E. de Weerdt
Flight Dynamics Lecture Notes J.A. Mulder / W.H.J.J. van Staveren / J.C. van der Vaart / E. de Weerdt March 7, 2007
Faculty of Aerospace Engineering
·
Delft University of Technology
Delft University of Technology
c Control and Simulation division, Delft University of Technology Copyright All rights reserved.
Table of Contents
Nomenclature
xxiii
1 Introduction 1-1 Introduction to flight dynamics and control . . . . . . . . . . . . . . . . . . . .
I
1 2
1-1-1
Flight control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1-1-2
Flight control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1-2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1-3 Book outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Equations of Motion
19
2 Reference Frames 2-1 Overview of reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1-1 Inertial reference frame, FI . . . . . . . . . . . . . . . . . . . . . . . . .
21 22 22
2-1-2
Earth-centered, Earth-fixed reference frame (FC ) . . . . . . . . . . . . .
24
2-1-3 2-1-4 2-1-5
Normal Earth-fixed reference frame, FE . . . . . . . . . . . . . . . . . . Vehicle carried normal Earth reference frame, FO . . . . . . . . . . . . . Body-fixed reference frame, Fb . . . . . . . . . . . . . . . . . . . . . . .
25 26 27
2-1-6
Aerodynamic (air-path) reference frame, Fa . . . . . . . . . . . . . . . .
30
2-1-7
Kinematic (flight-path) reference frame, Fk . . . . . . . . . . . . . . . .
30
2-1-8 Vehicle reference frame, Fr . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Transformations between reference frames . . . . . . . . . . . . . . . . . . . . . 2-2-1 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32 33 33
2-2-2 2-2-3 2-2-4
Transformation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation from FI to FE . . . . . . . . . . . . . . . . . . . . . . . Transformation from FI to FO . . . . . . . . . . . . . . . . . . . . . . .
38 42 43
Flight Dynamics
iv
Table of Contents 2-2-5
Transformation from FE to FO . . . . . . . . . . . . . . . . . . . . . . .
45
2-2-6
Transformation from FO to Fb . . . . . . . . . . . . . . . . . . . . . . .
46
2-2-7
Transformation from FO to Fa . . . . . . . . . . . . . . . . . . . . . . .
49
2-2-8
Transformation from FO to Fk . . . . . . . . . . . . . . . . . . . . . . .
50
2-2-9
Transformation form Fb to Fa . . . . . . . . . . . . . . . . . . . . . . .
52
2-2-10 Transformation form Fb to Fk . . . . . . . . . . . . . . . . . . . . . . .
53
2-2-11 Transformation from Fk to Fa . . . . . . . . . . . . . . . . . . . . . . .
55
2-3 Rotating reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2-3-1
Derivation of the angular velocity vector . . . . . . . . . . . . . . . . . .
63
2-3-2
Derivation of ΩE EI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
2-3-3
Derivation of ΩO OI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
2-3-4
Derivation of ΩO OE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2-3-5
Derivation of ΩbbO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
2-3-6
Derivation of ΩaaO and ΩkkO . . . . . . . . . . . . . . . . . . . . . . . .
68
2-3-7
Derivation of Ωaab and Ωkkb . . . . . . . . . . . . . . . . . . . . . . . . .
69
2-4 Quaternions, a brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2-4-1
Euler angle singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2-4-2
Quaternion transformation matrix . . . . . . . . . . . . . . . . . . . . .
72
2-4-3
Quaternion propagation through time . . . . . . . . . . . . . . . . . . .
77
3 Derivation of the Equations of Motion
83
3-1 Newton’s laws of physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3-2 Derivation of body translational acceleration . . . . . . . . . . . . . . . . . . . .
91
3-3 Derivation of body angular momentum derivative . . . . . . . . . . . . . . . . .
95
3-4 External forces and moments . . . . 3-4-1 External forces . . . . . . . . 3-4-2 External moments . . . . . . 3-5 Composition of equations of motion
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99 99 102 102
3-5-1
Force equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
3-5-2
Moment equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
3-5-3 Kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Rotating masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105 106
3-7 Simplifications of the equations of motion . . . . . . . . . . . . . . . . . . . . .
111
3-7-1
Non-rotating Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
3-7-2 3-7-3
Flat Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat and Non-rotating Earth . . . . . . . . . . . . . . . . . . . . . . . .
112 113
Table of Contents
v
4 Linearized Equations of Motion
117
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame 119
II
4-1-1
Linearization of states . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
4-1-2
Linearization of forces and moments . . . . . . . . . . . . . . . . . . . .
122
4-1-3
Linearization of kinematic relations . . . . . . . . . . . . . . . . . . . . .
128
4-2 Linearization about steady, straight, symmetric flight condition . . . . . . . . . .
130
4-2-1
Linearized set of equations . . . . . . . . . . . . . . . . . . . . . . . . .
131
4-2-2
Moments and products of inertia . . . . . . . . . . . . . . . . . . . . . .
132
4-3 Equations of motion in non-dimensional form . . . . . . . . . . . . . . . . . . .
134
4-3-1
Symmetric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
4-3-2
Asymmetric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
4-4 Symmetric equations of motion in state-space Form . . . . . . . . . . . . . . . .
148
4-5 Asymmetric equations of motion in state-space form . . . . . . . . . . . . . . .
151
Static Stability Analysis
153
5 Analysis of Steady Symmetric Flight
155
5-1 Aerodynamic center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
5-1-1
Aerodynamic forces and moments acting on a wing . . . . . . . . . . . .
156
5-1-2
A compact description for the aerodynamic moment characteristics
. . .
169
5-1-3
The role of the aerodynamic center and Cmac in static stability . . . . . .
175
5-1-4
The position of the aerodynamic center and Cmac of a wing . . . . . . .
184
5-1-5
Influence of wing shape on the aerodynamic center and Cmac . . . . . . .
195
5-1-6
Characteristics of wing-fuselage-nacelle configurations . . . . . . . . . . .
206
5-1-7
Aerodynamic effects of nacelles . . . . . . . . . . . . . . . . . . . . . . .
219
5-2 Equilibrium in steady, straight, symmetric Flight . . . . . . . . . . . . . . . . . .
220
5-2-1
Conditions for equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . .
226
5-2-2
Normal force on the horizontal tailplane . . . . . . . . . . . . . . . . . .
233
5-2-3
Hinge moment of the elevator . . . . . . . . . . . . . . . . . . . . . . .
238
5-2-4
Flow direction and dynamic pressure at the horizontal tailplane . . . . . .
241
5-2-5
Effect of airspeed and center of gravity on tail load . . . . . . . . . . . .
255
5-2-6
Elevator deflection required for moment equilibrium . . . . . . . . . . . .
258
5-2-7
Stick forces in steady flight . . . . . . . . . . . . . . . . . . . . . . . . .
262
Flight Dynamics
vi
Table of Contents
6 Longitudinal Stability and Control Derivatives
267
6-1 Aerodynamic forces in the nominal flight condition . . . . . . . . . . . . . . . .
268
6-2 Derivatives with respect to airspeed . . . . . . . . . . . . . . . . . . . . . . . .
270
6-2-1
Stability derivative CXu . . . . . . . . . . . . . . . . . . . . . . . . . . .
270
6-2-2
Stability derivative CZu . . . . . . . . . . . . . . . . . . . . . . . . . . .
271
6-2-3
Stability derivative Cmu . . . . . . . . . . . . . . . . . . . . . . . . . . .
272
6-3 Derivatives with respect to angle of attack . . . . . . . . . . . . . . . . . . . . .
276
6-3-1
Stability derivative CXα . . . . . . . . . . . . . . . . . . . . . . . . . . .
276
6-3-2
Stability derivative CZα . . . . . . . . . . . . . . . . . . . . . . . . . . .
278
6-3-3
Stability derivative Cmα . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
6-4 Derivatives with respect to pitching velocity . . . . . . . . . . . . . . . . . . . .
280
6-5 Derivatives with respect to the acceleration along the top axis . . . . . . . . . .
291
6-6 Derivatives with respect to the elevator angle . . . . . . . . . . . . . . . . . . .
295
6-6-1
Control derivative CXδe . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
6-6-2
Control derivative CZδe . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
6-6-3
Control derivative Cmδe . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
6-7 Symmetric inertial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
296
7 Lateral Stability and Control Derivatives
299
7-1 Aerodynamic force and moments due to side slipping, rolling and yawing . . . . .
299
7-2 Stability derivatives with respect to the sideslip angle . . . . . . . . . . . . . . .
303
7-2-1
Stability derivative CYβ . . . . . . . . . . . . . . . . . . . . . . . . . . .
303
7-2-2
Stability derivative Cℓβ . . . . . . . . . . . . . . . . . . . . . . . . . . .
307
7-2-3
Stability derivative Cnβ . . . . . . . . . . . . . . . . . . . . . . . . . . .
312
7-2-4
Stability derivative Cnβ˙ . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
7-2-5
Stability derivative Cmβ 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
328
7-3 Stability derivatives with respect to roll rate . . . . . . . . . . . . . . . . . . . .
329
7-3-1
Stability derivative CYp . . . . . . . . . . . . . . . . . . . . . . . . . . .
329
7-3-2
Stability derivative Cℓp . . . . . . . . . . . . . . . . . . . . . . . . . . .
331
7-3-3
Stability derivative Cnp . . . . . . . . . . . . . . . . . . . . . . . . . . .
336
7-4 Stability derivatives with respect to yaw rate . . . . . . . . . . . . . . . . . . . .
339
7-4-1
Stability derivative CYr . . . . . . . . . . . . . . . . . . . . . . . . . . .
340
7-4-2
Stability derivative Cℓr . . . . . . . . . . . . . . . . . . . . . . . . . . .
343
7-4-3
Stability derivative Cnr . . . . . . . . . . . . . . . . . . . . . . . . . . .
345
7-5 The forces and moments due to aileron, rudder and spoiler deflections . . . . . .
347
7-6 Aileron control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6-1 Control derivative CYδa . . . . . . . . . . . . . . . . . . . . . . . . . . .
348 348
Table of Contents
vii
7-6-2
Control derivative Cℓδa . . . . . . . . . . . . . . . . . . . . . . . . . . .
350
7-6-3
Control derivative Cnδa . . . . . . . . . . . . . . . . . . . . . . . . . . .
351
7-7 Spoiler control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
352
7-8 Rudder control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8-1 Control derivative CYδr . . . . . . . . . . . . . . . . . . . . . . . . . . .
355 355
7-8-2
Control derivative Cℓδr . . . . . . . . . . . . . . . . . . . . . . . . . . .
355
7-8-3
Control derivative Cnδr . . . . . . . . . . . . . . . . . . . . . . . . . . .
357
8 Longitudinal Stability and Control in Steady Flight
359
8-1 Stick Fixed Static Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . .
359
8-1-1
Stick fixed static longitudinal stability in gliding flight . . . . . . . . . . .
360
8-1-2
Neutral point, stick fixed . . . . . . . . . . . . . . . . . . . . . . . . . .
365
8-1-3
Elevator trim curve and elevator trim stability . . . . . . . . . . . . . . .
368
8-1-4
Influence of various parameters on elevator trim curve . . . . . . . . . . .
377
8-1-5
Static longitudinal stability of tailless aircraft . . . . . . . . . . . . . . .
379
8-1-6
Determination Cmδe from measurements in flight . . . . . . . . . . . . .
381
8-2 Stick Free Static Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . . .
385
8-2-1
Stick free static longitudinal stability in gliding flight . . . . . . . . . . .
388
8-2-2
Neutral point, stick free . . . . . . . . . . . . . . . . . . . . . . . . . . .
390
8-2-3
Elevator stick force curves and elevator stick force Stability . . . . . . . .
392
8-2-4
Effect of center of gravity and mass unbalance on stick force stability . .
403
8-2-5
Influence of design variables on control forces . . . . . . . . . . . . . . .
408
8-2-6
Control forces a pilot can exert . . . . . . . . . . . . . . . . . . . . . . .
410
8-2-7
Airworthiness requirements for steady straight symmetric flight . . . . . .
411
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns . . . . . . . . . .
412
8-3-1
Characteristics of longitudinal control in turning flight . . . . . . . . . . .
417
8-3-2
The stick displacement per g . . . . . . . . . . . . . . . . . . . . . . . .
420
8-3-3
The manoeuvre point, stick fixed . . . . . . . . . . . . . . . . . . . . . .
426
8-3-4
The stick force per g . . . . . . . . . . . . . . . . . . . . . . . . . . . .
431
8-3-5
The manoeuvre point, stick free . . . . . . . . . . . . . . . . . . . . . .
437
8-3-6
Non-aerodynamic means to influence the stick force Per g . . . . . . . .
442
9 Lateral Stability and Control in Steady Flight
445
9-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Equations of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
445 445
9-3 Steady Horizontal Turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
448
9-3-1
Turns Using the Ailerons Only, δr = 0 . . . . . . . . . . . . . . . . . . .
448
9-3-2
Turns Using the Rudder Only, δa = 0 . . . . . . . . . . . . . . . . . . . .
450
9-3-3
Coordinated Turns, β = 0 . . . . . . . . . . . . . . . . . . . . . . . . . .
450
Flight Dynamics
viii
Table of Contents 9-3-4 Flat Turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-4 Steady, Straight, Sideslipping Flight . . . . . . . . . . . . . . . . . . . . . . . .
453 453
9-5 Steady Straight Flight with One or More Engines Inoperative . . . . . . . . . . .
458
9-6 Steady Rolling Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
463
9-7 Control Forces and Hinge Moments for Lateral Control . . . . . . . . . . . . . .
470
9-7-1 9-7-2
III
Roll Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudder Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic Stability Analysis
10 Analysis of Symmetric Equations of Motion
470 472
473 475
10-1 Solution of the equations of motion . . . . . . . . . . . . . . . . . . . . . . . .
477
10-2 Stability criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
489
10-3 The complete solution of the equations of motion . . . . . . . . . . . . . . . . .
490
10-4 Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
494
11 Analysis of Asymmetric Equations of Motion
503
11-1 Solution of the equations of motion . . . . . . . . . . . . . . . . . . . . . . . .
503
11-2 General character of the asymmetric motions . . . . . . . . . . . . . . . . . . .
507
11-3 Routh-Hurwitz stability criteria for the asymmetric motions . . . . . . . . . . . .
513
11-4 Spiral and Dutch roll mode, the lateral stability diagram . . . . . . . . . . . . .
517
11-5 Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
519
A List of Symbols
535
B Aircraft Parameters 547 B-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 B-2 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 B-3 LTI-System Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
548
C MATLAB Files 559 C-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 C-2 Root Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 C-3 Eigenvalue Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
560
C-4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
560
List of Figures
1
Definition of the mean aerodynamic cord and related parameters . . . . . . . . .
xxviii
2
Parameters defining the geometry of the wing . . . . . . . . . . . . . . . . . . .
xxix
3
Examples of definitions of the vertical tailplane area (see also NACA TN 775)
.
xxx
1-1 The Wright Flyer I [136] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1-2 Boeing 767 3D-view [113] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1-3 Concorde 3D-view [113] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1-4 X-29 in flight [35] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1-5 737-400 Flight and ground spoiler deployment on landing [32] . . . . . . . . . .
7
1-6 Boeing 737 slats [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1-7 Mechanical flight control system example [132] . . . . . . . . . . . . . . . . . .
9
1-8 KC-135/707 Aileron Balance Panel Installation [132] . . . . . . . . . . . . . . .
10
1-9 Boeing 707 Flight Control Surfaces [132] . . . . . . . . . . . . . . . . . . . . . .
11
1-10 Boeing 707 Stabilizer-Actuated Tab Mechanism [132] . . . . . . . . . . . . . . .
12
1-11 Boeing 727 Aileron Control and Trim Systems [132] . . . . . . . . . . . . . . . .
14
1-12 Boeing 747 Elevator Control System [132] . . . . . . . . . . . . . . . . . . . . .
15
2-1 Aircraft velocity vector decomposition . . . . . . . . . . . . . . . . . . . . . . .
21
2-2 Definition of ecliptic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2-3 Definition of earth-centered inertial reference frame . . . . . . . . . . . . . . . . 2-4 Polar motion - precession track . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 24
2-5 Definition of earth-centered, earth-fixed reference frame . . . . . . . . . . . . . . 2-6 Earth geoid and latitude definitions . . . . . . . . . . . . . . . . . . . . . . . . .
25 26
2-7 Normal earth-fixed reference frame for spherical earth . . . . . . . . . . . . . . .
27
2-8 Vehicle carried normal earth reference frame for spherical earth . . . . . . . . . .
28
Flight Dynamics
x
List of Figures 2-9 Body-fixed Reference Frame
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2-10 Stability Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2-11 Aerodynamic Reference Frame in relation to Body-fixed Reference Frame . . . .
31
2-12 Left-handed (!) Aircraft Reference Frame Fr
. . . . . . . . . . . . . . . . . . .
32
2-13 Euler angles define the attitude of Fb irrespectively of translation . . . . . . . . .
33
2-14 Euler angles tutorial: initial and end body orientation . . . . . . . . . . . . . . .
36
2-15 Euler angles tutorial: rotation sequence ϕz → ϕy → ϕx . . . . . . . . . . . . . .
37
2-17 Simultaneous vector decomposition in two reference frames
. . . . . . . . . . .
38
2-18 Vector decomposition for rotation about X-axis . . . . . . . . . . . . . . . . . .
40
2-19 Vector decomposition for rotation about Y -axis . . . . . . . . . . . . . . . . . .
40
2-20 Vector decomposition for rotation about Z-axis . . . . . . . . . . . . . . . . . .
41
2-21 Transformation from inertial reference frame to normal earth-fixed reference frame 2-22 Transformation from vehicle carried normal earth reference frame FO to the bodyfixed reference frame Fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2-16 Euler angles tutorial: rotation sequence ϕx → ϕy → ϕz . . . . . . . . . . . . . .
37
48
2-23 Transformation from vehicle carried normal earth reference frame FO to the aerodynamic reference frame Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2-24 Transformation from the body-fixed reference frame Fb to the aerodynamic reference frame Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2-25 General situation for vector differentiation . . . . . . . . . . . . . . . . . . . . . 2-26 Change of a vector due to rotation vector Ω . . . . . . . . . . . . . . . . . . . .
57 59
2-27 Example - derivation of pilot velocity with respect to FO . . . . . . . . . . . . .
61
2-28 Angular velocity vector
ΩE EI
explanation . . . . . . . . . . . . . . . . . . . . . .
64
2-29 Euler angle singularity: Aircraft in vertical climb . . . . . . . . . . . . . . . . . .
71
2-30 Direction cosines equality for Euler axis rotations . . . . . . . . . . . . . . . . .
73
2-31 Rotating a reference frame about the Euler axis . . . . . . . . . . . . . . . . . .
73
2-32 Rotating an arbitrary vector about the Euler axis . . . . . . . . . . . . . . . . .
74
3-1 Vector definition for point mass P in a body. . . . . . . . . . . . . . . . . . . .
85
3-2 Moment about center of mass due to force on point mass . . . . . . . . . . . . .
86
3-3 Earth gravity vector definition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
3-4 Engine rotation axis definition
. . . . . . . . . . . . . . . . . . . . . . . . . . .
107
4-1 Linearization about a single point and a trajectory . . . . . . . . . . . . . . . . .
120
4-2 Example of a time-function: u(.) . . . . . . . . . . . . . . . . . . . . . . . . . .
123
4-3 Airflow tutorial: example of time-dependent aerodynamic characteristics . . . . .
125
4-4 The impossibility of a force in the XB -direction arising from a variation in velocity dv along the YB -axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
4-5 The angles between the principal inertial axes and the aircraft reference axes . .
133
4-6 Relation between moment of inertia Iy and the mass . . . . . . . . . . . . . . .
141
List of Figures
xi
5-1 The aerodynamic forces and the moment acting on a wing in symmetric flight
.
156
5-2 CN , CL and CT , CD as functions of α for the case of a Fokker F-27 wing (from reference [149]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
5-3 CN as a function of CT for a Fokker F-27 wing (from reference [149])
159
. . . . .
5-4 Rectangular wing configuration with zero-twist, λ = 13 , as used for linear potential flow simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 Simulated CN , CL and CT , CD as functions of α for the wing configuration depicted in figure 5-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159 160
5-6 Simulated CN as a function of CT and CL as a function of CD for the wing configuration depicted in figure 5-4 . . . . . . . . . . . . . . . . . . . . . . . .
161
5-7 The variation of the moment with changes in the reference point
162
. . . . . . . .
5-8 Location of point (x, z) with respect to the position of the mean aerodynamic chord 162 5-9 Moment curves for various positions of the reference point for the Fokker F-27 wing (from reference [149]) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164
5-10 Simulated moment curves for various positions of the reference point for the wing configuration of figure 5-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165
5-11 Definition of the angle χ1
166
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-12 Definition of the center of pressure relative on the mean aerodynamic chord (mac) 167 5-13 The magnitude of C R and the position of the line of action of C R as a function of the angle of attack for Fokker F-27 wing (from reference [149]) . . . . . . . .
167
5-14 The position of the center of pressure as a function of the angle of attack for Fokker F-27 wing (from reference [149]) . . . . . . . . . . . . . . . . . . . . . .
168
5-15 The aerodynamic forces and moment about an arbitrary point at two adjacent values of the angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169
5-16 The line of action of the difference between aerodynamic force vectors at adjacent values of the angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . .
170
5-17 The position of the aerodynamic center . . . . . . . . . . . . . . . . . . . . . .
172
5-18 The aerodynamic forces and moment, using the ac as the moment reference point 174 5-19 The moment about an arbitrary point (x, z) if the position of the ac and the Cmac are known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Equilibrium and stability of a wing, suspension point ahead of the ac, Cmac negative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21 Equilibrium and stability of a wing, suspension point ahead of the ac, Cmac positive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 Equilibrium and stability of a wing, suspension point in the aerodynamic center, Cmac equals 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23 Equilibrium and stability of a wing, suspension point in the aerodynamic center, Cmac negative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176 178 179 180 181
5-24 Equilibrium and stability of a wing, suspension point behind the ac, Cmac negative 182 5-25 Equilibrium and stability of a wing, suspension point behind the ac, Cmac positive 183 5-26 The basic, additional and total lift distribution of a wing with negative twist
. .
185
Flight Dynamics
xii
List of Figures 5-27 Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wingtwist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 0o , λ = 1, αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panelmethod, linearized potential flow. . . . . . . . . . . . . . . . . . . . . . . . . .
187
5-28 Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wingtwist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 37o , λ = 1, αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panelmethod, linearized potential flow. . . . . . . . . . . . . . . . . . . . . . . . . .
188
5-29 Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wingtwist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 0o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panelmethod, linearized potential flow. . . . . . . . . . . . . . . . . . . . . . . . . .
189
5-30 Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wingtwist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 37o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panelmethod, linearized potential flow. . . . . . . . . . . . . . . . . . . . . . . . . .
190
5-31 Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wingtwist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 5o , Λ = 0o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panelmethod, linearized potential flow. . . . . . . . . . . . . . . . . . . . . . . . . .
191
5-32 Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wingtwist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 5o , Λ = 37o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panelmethod, linearized potential flow. . . . . . . . . . . . . . . . . . . . . . . . . .
192
5-33 The calculation of the position of the ac and Cmac for a wing with arbitrary geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193
5-34 The influence of camber on the moment curves (from references [119, 118])
196
. .
5-35 The variation of Cmac with wing sweep Λ and angle of twist ε (from reference [49])197 5-36 The positions of the local ac’s of swept wings and delta wings . . . . . . . . . .
198
5-37 The influence of wing sweep on the moment curves of wings of aspect ratio A = 4, λ = 0.6, c¯t = 0.06, M = 0.40 (from reference [169]) . . . . . . . . . . . . . . .
199
45o ,
5-38 The influence of taper ratio on the moment curves of swept wings, A = 4, Λ = t c¯ = 0.06, M = 0.40 (from reference [87]) . . . . . . . . . . . . . . . . . . . . .
201
5-39 The position of the ac of delta wings and concept wings as a function of aspect ratio, Λ = 45o (from reference [159]) . . . . . . . . . . . . . . . . . . . . . . .
202
List of Figures
xiii
5-40 The influence of aspect ratio on the moment curve of swept wings, Λ = 45o , λ = 0.6, c¯t = 0.06, M = 0.40 (from reference [97]) . . . . . . . . . . . . . . . .
203
5-41 The moment curve of a straight wing and a swept wing, A = 5, λ = 1, Re = 1·106 (DUT measurements) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204
5-42 Boundaries for the combinations of aspect ratio and sweep angle for which destabilizing changes in the moment curves may be expected (from references [58, 75]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-43 Pressure distribution over a fuselage in inviscid flow . . . . . . . . . . . . . . .
205 208
5-44 Numerical simulation of particle lines (top) and pressure distribution over a fuselage in inviscid flow, α = 10o , linearized potential flow . . . . . . . . . . . . . . . . . 5-45 The variation of the angle of attack and the normal force along the fuselage axis in a wing induced flow field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Wing-fuselage configuration as used in the flow field simulations depicted in figures 5-47 and 5-48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47 Numerical simulation of the velocity field of a wing-fuselage configuration, α = 10o 5-48 Numerical simulation of the velocity field of a wing-fuselage configuration, α = 10o , detail of figure 5-47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209 211 212 .212 213
dα
5-49 The influence of aspect ratio on the variation of dαf along the fuselage axis (Λ = 0) (from reference [143]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
dα
5-50 The influence of sweep angle of wings with A = ∞ on dαf along the fuselage () (from reference [143]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
5-51 Fuselage induced variations of the local angle of attack along the wing span . . 215 5-52 Numerical simulation of the fuselage induced upwash along the wing span, α = 5o 215 5-53 The cℓ · c-distribution along the span of a wing with and without a fuselage (from reference [54]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
216
5-54 The change in wing moment due to loss in lift over the wing center part
. . . .
217
5-55 Shift of ac due to fuselage effects, as a function of wing sweep angle(from reference [141]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
218
15o ,
5-56 Flow simulations over a nacelle, α = computed with a source/doublet singularity panel-method, linearized potential flow . . . . . . . . . . . . . . . . . . .
219
5-57 Generic Large Transport Aircraft (GLTA), no nacelles . . . . . . . . . . . . . . .
220
5-58 Particle lines in an XB OZB -plane (top) and non-dimensional sectional pressure distribution Cp (bottom) for the left wing of a generic transport aircraft model, no nacelles, α = 10o , computed with a source/doublet singularity panel-method, linearized potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
221
5-59 Particle lines in an XB OZB -plane (top) and non-dimensional sectional pressure distribution Cp (bottom) for the left wing of a twin-engined transport aircraft , α = 10o , computed with a source/doublet singularity panel-method, linearized potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-60 Spanwise lift distribution cℓ c at an angle-of-attack α = 10o twin-engined transport aircraft configuration , source/doublet singularity panel-method, linearized potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Particle lines in an XB OZB -plane and non-dimensional sectional pressure distribution coefficients Cp for the left wing of a four-engined transport aircraft , α = 10o , source/doublet singularity panel-method, linearized potential flow . . . . . . . .
222
223
224
Flight Dynamics
xiv
List of Figures 5-62 Four-engined transport aircraft with spanwise lift distribution cℓ c at an angle-ofattack α = 10o , source/doublet singularity panel-method, linearized potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-63 The equilibrium in steady, symmetric flight . . . . . . . . . . . . . . . . . . . . 5-64 The forces and moments in steady, straight, symmetric flight . . . . . . . . . . 5-65 Geometry parameters of the horizontal tailplane, elevator and trim tab, and their positive deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-66 Simplified version of the equilibrium of forces and moments . . . . . . . . . . . 5-67 Pressure distributions over the tailplane chord due to αh , δe and δte . . . . . . . 5-68 The normal force coefficient CNh as a function of αh , δe and δte for the tailplane of the Fokker F-27 (Wind tunnel measurements from reference [17]) . . . . . . CNh
5-69 The quotient
CNh
reference [21])
δ
as a function of
α
c¯e c¯h
225 227 229 231 234 236 237
for various trailing edge angles τ (from
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
238
5-70 The hinge moment coefficient Che as a function of αh , δe and δte for a tailplane of the Fokker F-27 (wind tunnel measurements from reference [17]) . . . . . . .
239
5-71 Overbalance with respect to angle of attack and elevator angle. . . . . . . . . . 5-72 The angle of attack αh of the horizontal tailplane, αh = α − ε + ih . . . . . . . 5-73 The contribution of the lifting and free vortices to the induced vertical velocities in front of and behind a wing . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-74 The position of the vortex plane behind the wing . . . . . . . . . . . . . . . . . 5-75 The vertical displacement of the wake and the tailplane relative to the flow field x Rh due to an increase in angle of attack ∆α; ∆zh = ∆ α dx = ∆α (xh − xw ), ∆zwake =
x Rh
242 243 244 244
xw
∆ ε(x) dx, ∆zwake < ∆zh as ∆ε(x) is smaller than ∆α. With
xw
increasing angle of attack the horizontal stabilizer moves downwards through the wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-76 Numerical simulation of the deformation and wake roll-up of the vortex sheet behind a wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dε 5-77 Computed values of dα behind a wing of elliptical lift distribution, as a function of aspect ratio and distance behind the wing . . . . . . . . . . . . . . . . . . . . .
245 246 247
cℓ ·c 2b
5-78 Numerical simulation of for wings with varying taper ratio λ (CL = 1.0, Λ = 0o , A = 6), computed with a source/doublet singularity method, also see reference [143] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
248
ℓ ·c 5-79 Numerical simulation of c2b for wings with varying sweep angle Λ (CL = 1.0, λ = 1, A = 6), computed with a source/doublet singularity method, also see reference [143] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
249
5-80 Numerical simulation of downwash angles in the plane of symmetry behind a wing (α = 13o , NACA 23014 airfoil, A = 6, λ = 0.5 and CL = 1.1615, , computed with a source/doublet singularity method, see also reference [146]) . . . . . . .
250
5-81 Cessna Ce550 Citation II configuration with and without nacelles . . . . . . . . .
252
dε 5-82 The effect of nacelles on dα (at X = 5.0 m, Z = 1.5 m) aft of the wing of a Cessna Ce550 Citation II, computed with a source/doublet singularity panelmethod, linearized potential flow . . . . . . . . . . . . . . . . . . . . . . . . . .
253
List of Figures 5-83 Calculation of ground effect
xv . . . . . . . . . . . . . . . . . . . . . . . . . . . .
254
5-84 The ground effect on the downwash angle ε and the location of the wake relative to the horizontal tailplane of a Siebel 204 D-1 aircraft . . . . . . . . . . . . . .
256
5-85 Simplified picture of the equilibrium of moments
. . . . . . . . . . . . . . . . .
257
5-86 The variation of the tail toad with airspeed at different c.g. positions and values of Cmac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
5-87 The tail load as a function of airspeed at two c.g. positions for De Havilland ‘Mosquito II F’ (from reference [26]), W = 6800 kg, S = 41.8 m2 , lh = 8.0 m and c¯ = 2.81 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-88 The positive direction of control deflections, control forces, control surface deflections and hinge moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
259
5-89 The relation between the control force and the hinge moment . . . . . . . . . .
263
5-90 Contributions to the hinge moment
. . . . . . . . . . . . . . . . . . . . . . . .
264
6-1 The attitude of the stability reference frame relative to the Xr - and Zr -axis, after a disturbance from the equilibrium flight condition . . . . . . . . . . . . . . . .
268
6-2 The definition of aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylon’, ‘nacelles’, ‘vertical fin’ and ‘fuselage’ for the Cessna Ce550 ‘Citation II’ model (starting from the left top figure, in clockwise direction) . . . . . . . . . . . . . . . . . . . . .
277
6-3 Calculated contributions of various aircraft parts to the force curve CX versus α for the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . . 6-4 Calculated contributions of various aircraft parts to the force curve CZ versus α for the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 Calculated contributions of various aircraft parts to the moment curve Cm versus α for the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . 6-6 The pure q-motion, with ∆α =
x−xc.g. R
=
x−xc.g. q¯ c c¯ V
263
278 280 281
. . . . . . . . . . . . . . .
282
6-7 Harmonic q-motion, α = constant, θ = γ . . . . . . . . . . . . . . . . . . . . .
283
6-8 Harmonic α-motion, q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
284
6-9 Combined harmonic q- and α-motion, q = α, ˙ γ = constant
. . . . . . . . . . .
285
. . . . . . . . . . . . . . . aircraft frame of reference, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
286
6-10 Aircraft in a curved flow field . . . . . . . . . . . . 6-11 Equivalent curved aircraft in a straight flow field with Fr . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Equivalent curved aircraft in a straight flow field . .
6-13 Equivalent curved aircraft in a straight flow field, 3-dimensional view
. . . . . . q¯ c V
6-14 Calculated contributions of various aircraft parts to the force curve CX versus for the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . . q¯ c V
6-15 Calculated contributions of various aircraft parts to the force curve CZ versus for the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . . 6-16 Calculated contributions of various aircraft parts to the moment curve Cm versus q¯ c V for the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . 7-1 The six degrees of freedom of a rigid aircraft
286 287 287 289 291 292
. . . . . . . . . . . . . . . . . . .
300
7-2 Definition of the angle of attack α and sideslip angle β . . . . . . . . . . . . . .
301
7-3 Asymmetric force and moments
302
. . . . . . . . . . . . . . . . . . . . . . . . . .
Flight Dynamics
xvi
List of Figures 7-4 CY as a function of β measured on a model of the Fokker F-27 in gliding flight (from references [148, 150, 24]) . . . . . . . . . . . . . . . . . . . . . . . . . .
304
7-5 Relation between the sideslip angle β, the sidewash angle σ and the angle of attack αv at the vertical tailplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
306
7-6 Aerodynamic force coefficient CY as a function of β for the Cessna Ce550 ‘Citation II’, α = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7 The origin of the difference in angles of attack at the left and right wing for a wing with dihedral in sideslipping flight . . . . . . . . . . . . . . . . . . . . . . . . .
309
7-8 The origin of the difference in the velocities over the left and right wing for a swept wing in sideslipping flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
310
7-9 The variation of Cℓβ with CL for swept wings . . . . . . . . . . . . . . . . . . .
310
7-10 High-wing-fuselage configuration with the calculated velocity field in sideslipping flight (β = 10o , α = 0o ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
7-11 Low-wing-fuselage configuration with the calculated velocity field in sideslipping flight (β = 10o , α = 0o ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
314
7-12 Calculated particle traces around the bottom half of the Ce550 ‘Citation II’, sideslipping flight β = 10o , α = 0o related to the magnitude of the airflow’s velocity along traces) . . . . . . . . . . . . . . . . . . . . . . . . . . .
fuselage of the Cessna (note: the speedbar is the calculated particle . . . . . . . . . . . . .
315
7-13 Origin of a rolling moment caused by wing-fuselage interactions in sideslipping flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316
7-14 Aerodynamic moment coefficient Cℓ as a function of β for the Cessna Ce550 ‘Citation II’, α = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316
7-15 The position of the point of action of (CYv )v relative to the X- and Z-axis in the stability reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
317
7-16 The contribution of the slipstream to Cℓβ . . . . . . . . . . . . . . . . . . . . .
319
7-17 Cn as a function of β measured on a model of the Fokker F-27 in gliding flight (from references [148, 150, 24]) . . . . . . . . . . . . . . . . . . . . . . . . . .
320
7-18 The sidewash induced by the fuselage at the vertical tailplane
. . . . . . . . . .
321
7-19 Calculated particle traces around the fuselage of the Cessna Ce550 ‘Citation II’, sideslipping flight β = 10o , α = 0o (note: the speedbar is related to the magnitude of the airflow’s velocity along the calculated particle traces) . . . . . . . . . . .
322
7-20 Calculated particle traces around the fuselage of a large 4-engined transport aircraft, sideslipping flight β = 10o , α = 0o (note: the speedbar is related to the magnitude of the airflow’s velocity along the calculated particle traces) . . . . .
323
7-21 The effect of wing-fuselage interactions on the sidewash at the vertical tailplane of a low-wing aircraft in sideslipping flight . . . . . . . . . . . . . . . . . . . . .
325
7-22 The effect of wing-fuselage interactions on the sidewash at the tailplanes and the derivative Cnβ (from reference [142]) . . . . . . . . . . . . . . . . . . . . . . .
326
7-23 The contribution of the sideforce on the propeller to Cnβ
. . . . . . . . . . . .
327
7-24 Aerodynamic moment coefficient Cn as a function of β for the Cessna Ce550 ‘Citation II’, α = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
327
7-25 The pitching moment as a function of sideslip angle (from reference [126]) . . .
328
7-26 The influence of the vertical position of the horizontal tailplane on the variation of Cm in sideslipping flight for a fuselage with tailplanes (from reference [126]) . .
330
307
List of Figures
xvii
7-27 The variation of the local geometric angle of attack at the wing and the tailplane of a rolling aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pb 2V
7-28 Aerodynamic force coefficient CY as a function of for the Cessna Ce550 ‘Citation II’, α = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-29 The variation of the local geometric and effective angle of attack along the span of a rolling wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-30 The roll-damping as a function of aspect ratio and sweep for various wing taper ratio’s (from reference [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-31 The flow at the tailplanes of a rolling aircraft . . . . . . . . . . . . . . . . . . . pb 7-32 Aerodynamic moment coefficient Cℓ as a function of 2V for the Cessna Ce550 o ‘Citation II’, α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-33 The origin of a negative yawing moment about the Z-axis of the stability reference frame for a wing having a positive rate of roll (attached flow) . . . . . . . . . .
7-34 The side force and yawing moment on a rolling, swept back wing
. . . . . . . .
pb 7-35 Aerodynamic moment coefficient Cn as a function of 2V for the Cessna Ce550 o ‘Citation II’, α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-36 The pure ‘r- motion’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-37 The change in geometric flow direction at an arbitrary point of the aircraft due to an ‘r-motion’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-38 The variation in airspeed in spanwise direction due to an r-motion . . . . . . . 7-39 The side forces on the vertical tailplane and the propeller due to an r-motion. . rb 7-40 Aerodynamic force coefficient CY as a function of 2V for the Cessna Ce550 ‘Citao tion II’, α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rb 2V
7-41 Aerodynamic moment coefficient Cℓ as a function of for the ‘Citation II’, α = 0o . . . . . . . . . . . . . . . . . . . . . . . . 7-42 The effects of the size of the vertical tailplane and the taillength reference [88]) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cessna Ce550 . . . . . . . . on Cnr (from . . . . . . . .
rb 7-43 Aerodynamic moment coefficient Cn as a function of 2V for the Cessna Ce550 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘Citation II’, α = 0 7-44 The positive direction of control deflections, control forces, control surface deflections and hinge moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-45 The effect of an aileron deflection on the lift distribution in spanwise direction . 7-46 The Frise aileron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-47 Spoiler types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-48 The local wing twist of a wing cross-section due to elastic deformation caused by an aileron deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-49 The side force and yawing moment due to a rudder deflection . . . . . . . . . . 7-50 Asymmetric stability and control derivatives . . . . . . . . . . . . . . . . . . . .
8-1 The contribution of various parts of the aircraft to the moment curve Cm − α . 8-2 Measured contributions of various aircraft parts to the moment curve of the Fokker F- 27, reference point at 0.346 c¯ (from reference [147]) . . . . . . . . . . . . . 8-3 Calculated contributions of various aircraft parts to the moment curve of the Cessna Ce550 ‘Citation II’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
330 331 332 333 335 336 337 338 339 341 342 342 343 344 345 346 347 349 350 351 353 354 356 358 362 363 364
Flight Dynamics
xviii
List of Figures
8-4 Influence of the tailplane incidence on the moment curve of the Fokker F-27, reference point at 0.346 c¯ (from reference [147]) . . . . . . . . . . . . . . . . .
365
8-5 Influence of the position of the reference point (center of gravity) on the moment curves of the Fokker F-27, (from reference [147]) . . . . . . . . . . . . . . . . .
367
8-6 Change in the moment dCm due to a change in the angle of attack ∆α
. . . .
368
. . . . . . . . . . . . . . . . . .
370
8-8 Initial control surface deflections . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9 Initial and ultimate control displacement and control surface deflection for the transition to a lower airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . .
372 373
8-10 Aircraft responses to a control deflection
. . . . . . . . . . . . . . . . . . . . .
374
8-11 Influence of cg position on the trim curve . . . . . . . . . . . . . . . . . . . . .
375
8-12 Influence of the tailplane angle of incidence on the trim curve (aircraft has control position stability) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
378
8-13 A positive Cma.c is obtained by combining sweepback and negative wing twist
.
380
8-14 The determination of elevator efficiency from flight tests . . . . . . . . . . . . .
382
8-15 Trim curves for two cg positions of the Fokker F-27 (from reference [51]) . . . .
384
8-16 Some stability characteristics derived from the measured trim curves shown in figure 8-15 (Fokker F-27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
386
8-17 The equilibrium of the free elevator
. . . . . . . . . . . . . . . . . . . . . . . .
387
8-18 The change in the moment dCmf ree due to a change in angle of attack ∆α . . .
390
8-19 The behaviour of the free elevator after a change of the angle of attack . . . . .
392
8-20 Schematic form of the elevator control force curve, Fe as a function of (a) dynamic pressure 12 ρV 2 and (b) Fe as a function of equivalent airspeed Ve . . . . . . . .
394
8-21 Measured trim curves and elevator control force curves for the De Havilland D.H.98 ‘Mosquito’ M II F in gliding flight (from reference [27]) . . . . . . . . . . . . .
397
8-7 Moment curves and corresponding trim curves
8-22 Schematic representation of the concept of ‘positive feel’,
dFe dse
. . . . . . . . . .
8-23 Uncertainty in trim speed due to friction in the control mechanism
398
. . . . . . .
399
8-24 Measured elevator control force curves for the North American ‘Harvard’ II B in gliding flight (from reference [15]) . . . . . . . . . . . . . . . . . . . . . . . . .
404
8-25 Trim curves, hinge moment coefficients and required trim tab angles as functions of CL for the Siebel 204-D-1 aircraft (from reference [16]) . . . . . . . . . . . .
406
8-26 The influence of a bobweight in the control mechanism
. . . . . . . . . . . . .
407
8-27 The influence of a spring in the control mechanism . . . . . . . . . . . . . . . .
407
8-28 The influence of a spring or bobweight on the elevator control force curve . . . .
408
8-29 Maximum control forces as a function of the duration (from reference [140])
411
. .
8-30 Response curves due to an elevator step deflection, Lockheed 1049 C ‘Super Constellation’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31 Response curves due to an elevator step deflection, Auster J-5B ‘Autocar’ (from reference [23]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-32 Description of the character of the control feel for various ratio’s of the control force per g to the control displacement per g (from reference [170]). . . . . . .
415 416 419
List of Figures
xix
8-33 The influence of the stick displacement per g and the stick force per g on the pilot’s opinion of the control characteristics at constant airspeed (from reference [86]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
419
8-34 The forces and angular velocity q in the idealized pull-up manoeuvre
. . . . . .
423
8-35 The forces and angular velocity in a steady horizontal turn . . . . . . . . . . . .
424
8-36 The incremental elevator deflection ∆δe as a function of the incremental load factor ∆n in pull-up manoeuvres, Auster J-5B ‘Autocar’ (from reference [23]). .
427
8-37 The incremental elevator angle ∆δe as a function of the incremental load factor in turns, Auster J-5B ‘Autocar’ (from reference [23]) . . . . . . . . . . . . . . . .
428
8-38 Calculated positions of the manoeuvre point, stick fixed, and the stick displacement per g of the Fokker F-27 ‘Friendship’ . . . . . . . . . . . . . . . . . . . . . . .
430
8-39 The variation of the angle of attack in an arbitrary point of the aircraft caused by a pure q-motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
432
8-40 The incremental control force ∆Fe as a function of the incremental load factor ∆n in pull-up manoeuvres, Auster J-5B ‘Autocar’ (from reference [23]) . . . . .
438
8-41 The incremental control force ∆Fe as a function of the incremental load factor ∆n in turns, Auster J-5B ‘Autocar’ (from reference [23]) . . . . . . . . . . . . .
438
8-42 Calculated positions of the manoeuvre point, stick free, and the stick force per g of the Fokker F-27 ‘Friendship’ . . . . . . . . . . . . . . . . . . . . . . . . . . .
440
8-43 Influence of altitude and magnitude and sign of Chα on the position of the manoeuvre point, stick free, and the stick force per g . . . . . . . . . . . . . . . .
441
8-44 Increasing the range of permissible cg positions by decreasing |Chδ | . . . . . . .
8-45 The incremental hinge moment due to a bobweight in the control mechanism in a pull-up manoeuvre at a load factor n . . . . . . . . . . . . . . . . . . . . . . .
443 444
9-1 The forces along the YB -axis of an aircraft in steady, horizontal, asymmetric flight 446 9-2 Steady turns using ailerons only, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) . . . . . . . . .
449
9-3 Steady turns using rudder only, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) . . . . . . . . .
451
9-4 Steady coordinated turns, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) . . . . . . . . . . . . . . .
454
9-5 Steady flat turns, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) . . . . . . . . . . . . . . . . . . .
456
9-6 Steady, straight, sideslipping flight, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) . . . . . . . . .
457
9-7 The magnitude of Cne due to an engine failure, as a function of the location and the sense of rotation of the propeller and the vertical position of the wing, measured on a model of a twin-engined propeller-driven aircraft (from reference [105]). . .
459
9-8 The magnitude of Cne due to an engine failure, as a function of the location and the sense of rotation of the propeller and the vertical position of the wing, measured on a model of a twin-engined propeller-driven aircraft (from reference [105]). . .
461
9-9 Steady, straight, single-engined flight at ϕ = 0
. . . . . . . . . . . . . . . . . .
462
9-10 Steady, straight, single-engined flight at β = 0
. . . . . . . . . . . . . . . . . .
463
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
464
9-11 Minimum control speed
Flight Dynamics
xx
List of Figures 9-12 Steady, straight, sideslipping flight with the right propeller feathered, Fokker F-27, h = 1850 m, CL = 1.92, xc.g. = 0.28 c¯, V = 46.3 m/sec (from reference [51]) .
465
9-13 The limitation in the maximum attainable roll-rate due to elastic wing deformation 467 9-14 Variation of the wing twist angle ϕ along the wing span due to aileron deflection for two locations of the ailerons . . . . . . . . . . . . . . . . . . . . . . . . . . 9-15 The average value of ∆α along the aileron span in steady, rolling flight . . . . .
468 469
9-16 Roll-rate p as a function of airspeed V for various values of δa and Fa
. . . . .
470
10-1 Aperiodic motions corresponding to a real eigenvalue λ . . . . . . . . . . . . . .
483
10-2 Periodic motion corresponding to a pair of complex, conjugate eigenvalues λ1,2 = ξ±j η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
486
10-3 The location of the eigenvalues for the symmetric motions . . . . . . . . . . . .
493
10-4 Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, phugoid response . . . . . . . . . . . . . . . . . . . . . . . .
495
10-4 (Continued) Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, phugoid response . . . . . . . . . . . . . . . .
496
10-5 Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, magnification of figure 10-4, short period response . . . . . . .
497
10-5 (Continued) Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, magnification of figure 10-4, short period response 498 11-1 The location of the eigenvalues λ = λb Vb for the asymmetric motions . . . . . .
507
11-2 Response curves for a pulse-shaped rudder deflection (∆δr = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’ . . . . . . . . . . . . . . . . . . . . .
508
11-2 (Continued) Response curves for a pulse-shaped rudder deflection (∆δr = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’ . . . . . . . . . . . . . .
509
11-2 (Continued) Response curves for a pulse-shaped rudder deflection (∆δr = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’ . . . . . . . . . . . . . .
510
11-3 Response curves for a pulse-shaped aileron deflection (∆δa = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’ . . . . . . . . . . . . . . . . . . . . .
511
11-3 (Continued) Response curves for a pulse-shaped aileron deflection (∆δa = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’ . . . . . . . . . . . . . .
512
11-3 (Continued) Response curves for a pulse-shaped aileron deflection (∆δa = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’ . . . . . . . . . . . . . .
513
11-4 The spiral motion of an aircraft
. . . . . . . . . . . . . . . . . . . . . . . . . .
514
11-5 Characteristics of the Dutch Roll motion . . . . . . . . . . . . . . . . . . . . . 11-6 Roll- and yaw-rate characteristics of the Dutch Roll motion . . . . . . . . . . .
515 516
11-7 Lateral stability diagram
518
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables
4-1 Non-dimensional parameters in the equations of motion, symmetric motions
. .
142
4-2 Non-dimensional parameters in the equations of motion, asymmetric motions . .
143
4-2 (Continued) Non-dimensional parameters in the equations of motion, asymmetric motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Stability derivatives, symmetric motions . . . . . . . . . . . . . . . . . . . . . .
144 144
4-4 Control derivatives, symmetric motions . . . . . . . . . . . . . . . . . . . . . . .
145
4-5 Moments and products of inertia, symmetric motions . . . . . . . . . . . . . . .
145
4-6 Stability derivatives, asymmetric motions
. . . . . . . . . . . . . . . . . . . . .
146
4-7 Control derivatives, asymmetric motions . . . . . . . . . . . . . . . . . . . . . .
147
4-8 Moments and products of inertia, asymmetric motions . . . . . . . . . . . . . .
147
4-9 Symbols appearing in the general state-space representation of equation (4-46) .
150
4-10 Symbols appearing in the general state-space representation of equation (4-50) .
152
5-1 Shorthand notation for the normal force derivatives on a tailplane . . . . . . . .
235
5-2 Abbreviated notation for the hinge moment derivatives . . . . . . . . . . . . . .
240
6-1 Simplified formulae for the calculation of stability and control derivatives and coefficients in the initial, steady flight condition for the symmetric motions (without the effects of propellers and jets, aeroelasticity and compressibility of the air). .
297
6-1 (Continued) Simplified formulae for the calculation of stability and control derivatives and coefficients in the initial, steady flight condition for the symmetric motions (without the effects of propellers and jets, aeroelasticity and compressibility of the air). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
298
6-2 Influence of the center of gravity position on some stability derivatives
. . . . .
298
8-1 Maximum permissable values of the required control forces, according to the U.S. and British Civil Airworthiness Regulations . . . . . . . . . . . . . . . . . . . .
413
8-2 Permissable values of the stick force per g in kg
418
. . . . . . . . . . . . . . . . .
Flight Dynamics
xxii
List of Tables 10-1 Symmetric stability and control derivatives for the Cessna Ce500 ‘Citation’ . . .
492
11-1 Asymmetric stability and control derivatives for the Cessna Ce500 ‘Citation’ . . .
506
B-1 Symmetric and asymmetric stability and control derivatives for the Cessna Ce500 ‘Citation’, Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Symmetric and asymmetric stability and control derivatives for the Fokker F-27 ‘Friendship’, Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
550 551
B-3 Symmetric stability and control derivatives for the Cessna Ce-172 ‘Skyhawk’, Cruise 551 B-4 Symmetric stability and control derivatives for the Learjet I, Approach . . . . . .
552
B-5 Symmetric stability and control derivatives for the Beechcraft M99, Cruise
. . .
552
B-6 Symmetric stability and control derivatives for the Boeing 747-100, Approach . .
553
B-7 Symmetric stability and control derivatives for the Boeing 747-100, Holding, flaps up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
553
B-8 Symmetric stability and control derivatives for the Boeing 747-100, Approach, flaps 33o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-9 Symmetric stability and control derivatives for the Boeing 747-100, Landing . .
554 554
B-10 Asymmetric stability and control derivatives for the Lockheed L1049C stellation’, Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11 Asymmetric stability and control derivatives for the Lockheed L1049C stellation’, Approach . . . . . . . . . . . . . . . . . . . . . . . . .
‘Super Con. . . . . . . ‘Super Con. . . . . . .
555 555
B-12 Asymmetric stability and control derivatives for the BAC-Aerospatiale ‘Concorde’, Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
556
B-13 Asymmetric stability and control derivatives for the North-American X-15 experimental aircraft, Cruise (unspecified) . . . . . . . . . . . . . . . . . . . . . . . .
556
B-14 Asymmetric stability and control derivatives for the De Havilland Canada DHC-2 ‘Beaver’, Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
557
Nomenclature
Before reading the main chapters of this book, we advise the reader to start with this chapter. We will introduce several key notations and characters which will be used throughout the book. Knowing these notations and characters will facilitate the reading of the book and will improve the understanding of the text. We have tried to keep the reasoning behind the notations the same throughout the book. Only one exception exists and is with respect to the use of subscripts (see page xxv).
Vectors All vectors in this book are in three dimensional space, i.e. ∈ R3 , unless defined otherwise. All vectors are in boldface. The superscript denotes the reference frame in which the vector is expressed while the subscript indicates the particular parameter in question. For example: Vkb is the kinematic (subscript k) velocity (vector V) expressed in the body-fixed reference frame (superscript b). The legend for the subscripts en superscripts are given in the section on page xxv. Sometimes a second subscript is added. In that case the subscript denotes the point of origin of the vector or to which point or body the vector properties belong. For example: b Vk,G
is the kinematic velocity of point G expressed in the body-fixed reference frame. Four types of vectors have fixed component notations. Position vectors have components x, y, z while velocity vectors have components u, v, w. Acceleration components are denoted by ax , ay , az . Finally, the rotational velocity vector has components p, q, r. Other vectors do not have a fixed components notation. The same ’rules’ regarding the superscript Flight Dynamics
xxiv
Nomenclature
and the subscript notation apply for the components of the vectors. A vector can be differentiated with respect to time which is always coupled to a reference frame (without a reference frame time differentiation only has a pure theoretical meaning). The time derivative of vector with respect to reference frame Fb is written as: dX dt b
Note that the derivative of a vector is again a vector. Thus the previous expression can be appended by a superscript denoting the reference frame in which the time derivative is expressed. In cases where the meaning is obvious, the superscript and/or the subscripts are sometimes dropped, cleaning the text of any unnecessary notations which would only cloud the text. If any notation is simplified, it is explicitly mentioned so in the text.
Reference frames A crucial element in this book is the definition of reference frames. All reference frames are right-handed orthogonal unless stated otherwise and are defined by their origin and orientation of the axes. A reference frame is denoted by F and its axes by X, Y, Z. The subscript appended to these letters indicates the reference frame we are talking about. For example: Fb −→ (GXb Yb Zb ) is the body-fixed reference frame which has the origin in the vehicle center of mass G and has the axes Xb , Yb , Zb . A list of all reference frames used in this book is given next.
FI FE FO Fb Fs Fp Fz Fa Fk Fr
Inertial reference frame Normal Earth-fixed reference frame Vehicle carried normal Earth reference frame Body-fixed reference frame Stability reference frame Principle axis reference frame Zero-lift body axis reference frame Aerodynamic reference frame Kinematic reference frame Vehicle reference frame
(section (section (section (section (section (section (section (section (section (section
2-1-1) 2-1-3) 2-1-4) 2-1-5) 2-1-5) 2-1-5) 2-1-5) 2-1-6) 2-1-7) 2-1-8)
After chapter 2 the notations Fx , Fy , Fz are used to denote the total aerodynamic forces along the X, Y, Z-axis respectively. Although the context in which these parameters are used is different, care should be taken with the interpretation of the parameter F . Transformation matrix The orientation between reference frames is defined by a maximum of three Euler angles. The
Nomenclature
xxv
sequence of rotation combined with the set of angles enables us to transform any coordinate from one reference frame to another. A transformation matrix T is used to quickly transform a complete vector. The subscripts indicate the reference frames involved in the transformation. For example: Xb = Tba Xa where Xb is the vector X expressed in reference frame Fb , Tba is the matrix for the transformation from frame Fa to Fb , and Xa is the vector X expressed in reference frame Fa . Angular velocity vectors When the orientation between reference frames is time-variant, one can define a rotation vector describing this change through time. The variable defining a rotation vector is Ω. The subscripts indicate which reference frames are involved while the superscript denotes the reference frame in which the vector is expressed. For example: Ωbba is the rotation vector describing the angular velocity of reference frame Fb with respect to reference frame Fa (subscripts) expressed in frame Fb (superscript). One should be careful not to mix-up the interpretation of the subscripts for the angular velocity vectors and for the transformation matrices.
Superscripts and subscripts At this point a distinction must be made between chapters 1 to 3 and the remainder of the book. Chapters 1 to 3 For the first three chapters the main subject is the derivation of the equations of motion for the most general case of a spherical, rotating Earth. In those chapters the superscripts appended to a vector indicate the reference frame in which the vector is expressed. The possible superscripts are:
I E O b s p z a k r
Inertial reference frame Normal Earth-fixed reference frame Vehicle carried normal Earth reference frame Body-fixed reference frame Stability reference frame Principle axis reference frame Zero-lift body axis reference frame Aerodynamic reference frame Kinematic reference frame Vehicle reference frame Flight Dynamics
xxvi
Nomenclature
The subscripts in chapters 1 to 3 denote either reference frames (in case of transformation matrices and angular velocity vectors) or indicate a particular form of the vector. The subscript for the reference frames are the same as for the superscripts given above. The other subscripts are: a k
aerodynamic kinematic
Remaining chapters In the remainder of the book superscripts are not used often. One exception which occurs in chapter 4, is the superscript ’ indicating the derivative with respect to parameter i, e.g. f ′ = df di . The function of the subscript changes considerably. A subscript in chapter 4 and subsequent chapters indicates the partial derivative of the function with respect to the parameter indicated by the subscript. For example: Xu =
∂X ∂u
Geometric of aircraft parameters The geometric aircraft parameters are used to determine the aerodynamic force and moments and to make the equations of motion dimensionless. Figure 2 illustrates the various parameters defining the geometry of the wing. The reference axes used in this figure are those of the Vehicle reference frame (see section 2-1-8). • Wing area, S The area of the wing projection on the Xr OYr -plane. Often the wing is partially covered by the fuselage and the engine nacelles. The wing area is then calculated using straight line extensions of the wing leading and trailing edges through the fuselage and the nacelles. The wing area can be expressed as, b
S=
Z+ 2
c dy
− 2b
where y is the coordinate in the Yr -direction. • Wingspan, b The distance in Yr -direction between the wing tips. • Mean aerodynamic chord, c¯ (mac) is defined as,
Nomenclature
xxvii
b
1 c¯ = S
Z+ 2
c2 dy
− 2b
One can define the location and orientation of the mean aerodynamic cord within the OXr Zr plane (see figure 1). Four coordinates, x ¯o , x ¯e , z¯0 , z¯e ,determine the location and orientation of the mac. There definition is similar that of the mean aerodynamic cord: x ¯o = z¯o = ε¯ =
1 S
1 S
1 S
+Rb/2
xo (y) c (y) dy x ¯e =
−b/2 +b/2
R
zo (y) c (y) dy
R
ε (y) c (y) dy
z¯e =
−b/2 +b/2
1 S
1 S
+Rb/2
xe (y) c (y) dy
−b/2 +b/2
R
ze (y) c (y) dy
−b/2
−b/2
• Mean or geometric chord, cm Is very often used in the literature and is defined as: cm =
S b
• Taper ratio, λ A measure of the variation in chord length along the span. It is expressed by, λ=
ct cr
A=
b2 S
• Aspect ratio, A Defined as,
• Wing sweep, Λ The angle between the Yr -axis and the projection of the 1/4-chord line on the Xr OYr plane (figure 2). In some cases such as delta wings, only the angle between the Yr -axis and the projection of the wing’s leading edge on the Xr OYr -plane is given. • Dihedral, Γ The dihedral of a wing is the angle between the Yr -axis and the projection of the 1/4chord line on the Yr OZr -plane. • Washout or wing-twist, ε Expresses the variation in direction of the local wing chord relative to the direction of the chord line at the wing root. Neither the gradient of the washout ( dd yε ) nor the magnitude of wing sweep or dihedral need to be constant along the span. If necessary, these parameters are given as functions of the coordinate in span direction. Flight Dynamics
xxviii
Nomenclature
Zr c¯ z¯o z¯e
ε¯
Xr
x ¯e
x ¯o
c(y) y(< 0) xo (y)
xe (y) Xr
x ¯o
c¯
x ¯e
Yr Figure 1: Definition of the mean aerodynamic cord and related parameters
Nomenclature
xxix
Yr
O
1 4 -chord
Λ
ε
cr
line
c dy
ct
y Xr b Zr
Γ Yr O
Figure 2: Parameters defining the geometry of the wing
Flight Dynamics
xxx
Nomenclature
Figure 3: Examples of definitions of the vertical tailplane area (see also NACA TN 775)
• Wing airfoil The shape of the cross section of the wing parallel to the plane of symmetry. The above geometric parameters apply not only to wings, but to tailplanes as well. Definitions of the geometric parameters for the elevator, or control surfaces in general, are given in chapter 7. It proves to be difficult to define the geometry of vertical tailplanes in a way applicable to all aircraft. In particular the distinction between fuselage and vertical tailplane is often hard to make. In many instances the division between vertical tailplane and dorsal fin is also more or less arbitrary. Usually the surface of the dorsal fin is not considered to be part of the vertical tailplane. Figure 3 shows examples of definitions of vertical tailplanes, see reference [11]. Similar and other definitions are given in references [8] and [101].
Aircraft Configurations The flight dynamic characteristics are dependent on the aircraft configuration. A full description of the aircraft configuration gives the aircraft’s weight or mass, center of gravity position and internal as well as external loading, undercarriage position, control surface deflections, flap angle, airbrake and spoiler deflections. A description of the engine operating condition, such as throttle position, engine speed etcetera, is also required.
Nomenclature
xxxi
Some definitions as used in U.S. military requirements are briefly described below. • CR (Cruising Flight) Engine thrust or power for level flight at cruising speed, flaps in the position for cruising flight, undercarriage retracted. • L (Landing) Throttle closed, undercarriage down, flaps in the position for landing. • PA (Powered Approach) Undercarriage down, flaps and airbrakes in the normal position for the powered approach, engine thrust or power for level flight at 1.15 VSL or the normal airspeed in the powered approach, if the latter is lower. NOTE: VSL is the stall speed (in the aircraft configuration for landing).
Flight Conditions By specifying a specific flight condition, we specify (a part of) the state of the aircraft, i.e. we indicate which aircraft states are varying. This information can be used to deduce the equations of motion for that flight condition. The following definitions of flight conditions will be used throughout this book. • Steady flight An aircraft is in steady flight if the components of the aerodynamic velocity vector VG (u, v, w) and the components of the body rotation vector (p, q, r) in the body-fixed reference frame Fb are constant. In this flight condition the aerodynamics force vector components in Fb and the aircraft’s pitch and roll attitude (θ, ϕ) are constant too. • Straight flight An aircraft is in a straight flight condition, if the flightpath vector is straight. • Symmetric flight An aircraft is in symmetric flight, if the velocity vector of any point of the aircraft is parallel to the plane of symmetry (roll angle ϕ = 0). • Slipping flight An aircraft is in slipping flight, if the velocity vector of the aircraft’s center of gravity is not parallel to the plane of symmetry of the aircraft.
Flight Dynamics
xxxii
Nomenclature
Chapter 1 Introduction
This book discusses the theory of stability and control of aircraft at subsonic airspeeds. This book handles the theory of flight dynamics which describes the aircraft/spacecraft velocities (translational and rotational), position, and orientation in four dimensional space. Velocities, position, and orientation through time can be computed from the accelerations in time. The analysis of the flight dynamics is based on Newton’s Second law: F = ma. If we know the forces acting on the aircraft/spacecraft and the mass distributions we can determine the accelerations. By integrating the accelerations with respect to time and by knowing the initial velocities, the velocities at each moment in time can be calculated. Likewise, by integrating the velocities with respect to time and knowing the initial position and orientation, the position and orientation at each time instant can be determined. The dynamics, i.e. the ’behavior’ through time, of the aircraft or spacecraft can be influenced by using controls, e.g. aerodynamic surfaces or thrusters. There are two types of flight dynamics analysis. The first concerns the stability. How does the aircraft react to movements of the controls or other types of disturbances? Does it have inherent stability? This type of analysis concerns the characteristics of the aircraft without the pilot aspect in it. The second type is about the aircraft/pilot connection. It is the analysis of the dynamics of the aircraft or spacecraft with respect to the ability of the pilot to control the craft. This is the field of ’handling qualities’. There are specific regulations that state the required ’handling qualities’ for each type of aircraft. These regulations are usually defined in terms of responses of the aircraft to control inputs, like elevator, rudder or throttle inputs. For aviation these regulations have matured trough the years and are described in detail in regulation documents, such as the Joint Aviation Regulations (JAR) set up by the Joint Aviation Authorities (Europe) and the Federal Aviation Regulations set up by the Federal Aviation Administration (USA). The handling qualities of an aircraft are related to the control surfaces and control mechanism. There is a rich history on the development of control surfaces and control mechanism. The following section gives an overview. After that a list of terms and definitions used in flight dynamics is given. The setup of the book is to keep everything as general as possible and to simplify things through the use of assumptions Flight Dynamics
2
Introduction
where necessary. The assumptions which are used throughout the book are listed in section 1-2. Finally, the book outline will be given in section 1-3.
1-1
Introduction to flight dynamics and control
The Wright Flyer was the first powered piloted aircraft which had full attitude control (see figure 1-1). It had a double rudder to control yaw, a double elevator to control pitch and used warping of the wings to control roll. The control technique of using mechanical links connected to aerodynamic control surfaces was the founding of modern controlled flight. Modern aircraft still use the idea of pitch, roll, and yaw control through means of deflectable control surfaces. (Warping of wings requires flexible wings therefore limiting its field of application. Soon rigid wings with control surfaces where used.) However, over the years, many additional (control) surfaces have been developed. The location, shape, and purpose of each surface varies. Also the mechanisms to move the flight control surfaces, i.e. the flight control systems, have changed over the years. To fully comprehend the influences these development have on the field of flight dynamics and control, a short introduction is given in this section. First the control surfaces which have been developed throughout the years are addressed (section 1-1-1). Thereafter, in section 1-1-2, the mechanisms of translating pilot commands to the control surface deflections are discussed.
1-1-1
Flight control surfaces
Flight control surfaces have been developed throughout the years and many variants exist. A classification has been made for control surfaces. Two types are defined: Primary flight control surfaces Flight-critical control surfaces. If control of these surfaces is lost, control over the aircraft is (partially) lost and the aircraft is likely to crash. Secondary flight control surfaces Non-flight-critical control surfaces. If control of these surfaces is lost, complete control over aircraft is still possible. The most obvious examples of primary flight control surfaces are the elevator, aileron, and rudder. These basic control surfaces are needed to control the aircraft about all three axis of the aircraft. Without any redundancy of these flight control surfaces, malfunction of one of these control surfaces would make the aircraft hard to control. Stable flight may still be possible if the pilot has thorough knowledge of the flight dynamics of the aircraft. Examples of secondary flight control surfaces are: speed brakes, lift dumpers, slats, flaps, and trim surfaces. Control over the orientation of the aircraft is still possible when these surfaces malfunction. As said, many control surfaces have been developed. The difference between surfaces are, apart from their dimensions, the location on the aircraft and their purpose. In the following, a list is presented in which different types of control surfaces are explained briefly.
1-1 Introduction to flight dynamics and control
3
(a) First flight (Kittyhawk England, 10:35 AM, 17 Dec 1903)
Elevator
Rudder (b) Profile view
Figure 1-1: The Wright Flyer I [136]
Ailerons Purpose: provide roll control. Location: on the trailing edge of the main wing. Extra information: ailerons are placed either near the tip of the wing (Out-board ailerons) or near the root of the wing (In-board ailerons). Out-board ailerons are only active at low speeds. In-board ailerons are active at all speeds. Elevators Purpose: provide pitch control. Location: trailing edge of the horizontal stabilizer. Rudders Purpose: provide yaw control. Location: trailing edge of the vertical stabilizer. Canards Purpose: provide pitch control. Location: in front of the main wing. Canards can consist of a fixed surface with trailing edge control surface or the whole canard surface can be rotated and controlled. Elevons Flight Dynamics
4
Introduction Purpose: provide pitch and roll control. Location: trailing edge of the main wing. Extra information: to provide pitch control, the elevons are extended symmetrically. For roll control the elevons are extended asymmetrically. Elevons are typically used on tailless delta-wing aircraft.
Flaperons Purpose: provide roll control and additional lift. Location: trailing edge of main wing. Extra information: the flaperons is a combination of the ailerons and trailing edge flaps. The flap function is obtained by symmetrical extension of the flaperons. Roll control is obtained through asymmetric extension. Flaps Purpose: increase lift. Location: either on the leading-edge of the wing or the trailing edge of the wing. Extra information: the leading-edge and trailing-edge flaps are used to deform the shape of the wing cross-section (extending the chamber line). The increase in wing surface causes an increase in lift. Therefore slower take-off and landing speeds are possible. Slots Purpose: increase lift (at higher angles of attack). Location: trailing edge of the wing. Extra information: a slot is an alternative to the trailing-edge flap. Slots outperform trailing-edge flaps by letting air flow through the wing such that, at high angles of attack, the collapse of airflow on the upper surface of an airfoil is reduced, thus maintaining maximum lift at high angles of attack. Slats Purpose: increase lift (at higher angles of attack). Location: leading edge of the wing. Extra information: the slat is actually a leading-edge slot. It delays wing stall at higher angles of attack. Spoilers Purpose: spoil lift, increase drag. Location: on main wing surface. Extra information: Spoiler have two main effects. By symmetric extension on both wings in flight, the spoilers act as speed brakes. During landing, symmetric extension will lead to a smaller roll-out distance. Asymmetric extension in flight will provide additional roll control without inducing wing twist to cause roll-reversal. There are two types of spoiler known: ground spoilers and flight spoilers. Ground spoilers are used during landing and extend further than flight spoilers (which are used in flight). Speed brakes Purpose: add drag to decelerate aircraft. Location: on wing surface. Extra information: speed brakes are aerodynamic surfaces which pop out of the wing at a near perpendicular angle to the airflow. This increases drag to the maximal possible extent. Stabilators, Stabilons, and Tailerons (names are synonymous) Purpose: provide pitch and roll control. Location: These surfaces are the two halves (left and right) of the horizontal stabilizer. Extra information: pitch control is achieved by rotating the horizontal stabilizer symmetrically. Roll control is achieved through asymmetric rotation. Stabilators are used to augment the surfaces which provide pitch (elevators, elevons, canards) and roll (ailerons, flaperons, spoilers) control.
1-1 Introduction to flight dynamics and control
5
Trim surfaces Purpose: remove required control force exerted by the pilot. Location: trailing-edge of the vertical stabilizer (or part of the rudder) and/or trailing-edge of the horizontal stabilizer (or part of the elevator). Extra information: trim surfaces are used to balance the aerodynamic moment on control surfaces (during steady flight) such that the loads on the controls are eliminated . The pilot can then fly hands-free. Two main possibilities exist for the horizontal trim surface. Either the entire horizontal stabilizer acts as a trim surface (so-called variable- incidence horizontal stabilizer) or a smaller surface is used (trailing-edge surface of part of the elevator, creating the so-called variable camber horizontal stabilizer).
Each aircraft can have a different configuration regarding its control surfaces. In figures 1-2 to 1-6, several aircraft are depicted. Each figure shows the location of several specific control surfaces.
Rudder
Aileron
Elevator
Outboard trailingedge flap Flight spoiler Slats
Stabilizer Inboard trailingedge flap Ground spoiler
Leadingedge flap
Figure 1-2: Boeing 767 3D-view [113]
Flight Dynamics
6
Introduction
Elevons
Figure 1-3: Concorde 3D-view [113]
Carnard
Flaperon
Elevator
Figure 1-4: X-29 in flight [35]
1-1-2
Flight control systems
How do you move the control surfaces of an aircraft and what does a pilot feel when moving a control stick or wheel? Why have the flight control systems changed so much through time? These are just a few questions which one can ask when looking at the history of flight control systems. In the development of control systems their have been two main drives. The first is the desire to build larger aircraft. Larger aircraft means larger aerodynamic surfaces which in turn means larger aerodynamic forces and thus larger control forces. The very first control systems where purely mechanical consisting of wires, springs, and wheels, which relied on the pilot strength to move them. The control forces a pilot can generate (especially for larger periods of time) is limited, so when the aircraft size increased, something had to be done to the controls.
1-1 Introduction to flight dynamics and control
7
Aileron - Ground spoiler Outboard t railing-edge flap
- Flight spoiler
Inboard t railing-edge flap - Ground spoiler
Figure 1-5: 737-400 Flight and ground spoiler deployment on landing [32]
Figure 1-6: Boeing 737 slats [32]
Flight Dynamics
8
Introduction
The second ’drive’ in the development of control systems is the invention of the autopilot. With the development of new technical systems, like computers and electric actuators, it became possible to control the aircraft electrically. Looking at the past, several types of flight control systems can distinguished: • Mechanical human powered system (reversible) • Mechanical hydraulic powered system (irreversible/reversible) • Fly-by-wire system (irreversible) These different types will be briefly discussed in the next three sections. Mechanical human powered flight control system The first powered aircraft, the Wright Flyer, used a simple form of a mechanical human powered control systems. The pilot lay head-forward on the lower wing. With his left hand the pilot controlled the elevator pitch by moving a lever attached to the elevator. By using a cable system attached to a saddle, in which the pilot would lay, the pilot could control the warping of the wings by moving his hips sideways. This cable system was also connected to the rudder, thus creating two control surface movements with one movement of the pilot. This enabled the pilot to balance the aircraft and provided directional steering. Although advanced in its time, the control system of the Wright Flyer was very simple indeed. An example of a more complicated system is the mechanical flight control system given in figure 1-7. One characteristic of a mechanical control system is that a direct linkage exists between the pilot and the control surfaces. Another characteristic is that it is reversible, i.e. the amount of control force needed to move an aerodynamic surface is directly felt by the pilot. Mechanical flight control systems are found in small aircraft where control loads are not excessive. For flying larger aircraft, flight control systems had to be changed. Two mechanical solutions were developed. The first one is an attempt to extract the maximum mechanical advantage through levers and pulleys. The maximum reduction in forces in such a system is limited by the strength of the cables, levers, and pulleys. The second solution is the introduction of so-called control tabs (also known as servo tabs). These are small surfaces at the end of the control surface which reduce the required control force exerted by the pilot by using aerodynamic properties of the aircraft, see figure 1-8. In this case the pilot controls are directly linked to the control tabs. The control tab generates an aerodynamic force which in turn moves the aerodynamic surface. The Boeing 707 used the concept of control tabs in its flight control system (see figure 1-9 and 1-10). The ideas of extracting maximum mechanical advantage and using control tabs worked well for middle sized aircraft. However, when the aircraft became even larger, neither of the two solutions did suffice. This lead to the use of hydraulic power.
1-1 Introduction to flight dynamics and control
9
(a) Elevator control - fighter aircraft
(b) Aileron control - commercial aircraft
Figure 1-7: Mechanical flight control system example [132]
Flight Dynamics
10
Introduction
Figure 1-8: KC-135/707 Aileron Balance Panel Installation [132]
1-1 Introduction to flight dynamics and control
11
Figure 1-9: Boeing 707 Flight Control Surfaces [132]
Flight Dynamics
12
Introduction
Figure 1-10: Boeing 707 Stabilizer-Actuated Tab Mechanism [132]
1-1 Introduction to flight dynamics and control
13
Mechanical hydraulic powered flight control systems The development of large aircraft lead to the need for hydraulic powered flight control systems. The pilot could no longer generate the required control forces by using purely mechanical links. Hydraulic-power became the source of the control forces. The mechanical hydraulic powered flight control system consists of two parts: 1. A mechanical part (the same as for fully mechanical FCS) 2. A hydraulic part The mechanical part transports the pilot commands to the hydraulic system. This is done by opening the appropriate set of servo valves of the hydraulic system. The hydraulic system then generates the forces for the actuators which move the aerodynamic surfaces. The Boeing 727 and 737 used such a flight control system (see figure 1-11). They both have a manual, mechanical linkage backup, in case of any failure of the primary flight control system. The backup system is called manual reversion since the control is done by the pilot and the force is directly felt by the pilot (thus noted as a reversible system). The mechanical hydraulic powered FCS is an irreversible system because the forces are not felt by the pilot. The feedback of forces tells the pilot a great deal about the current airspeed. In order to still keep the feeling of controlling the aircraft, artificial feel devices were installed that generate forces on the mechanical part of the FCS. The amount of force exerted on the mechanical system is made dependent on airspeed. The Boeing 747 was the first aircraft in the Boeing series to have a fully powered actuation system. The manual reversion system was omitted from the FCS because the required control forces for any flight condition would have been too large to be generate by the pilot. The schematic layout of part of the flight control system of the Boeing 747 can be seen in figure 1-12. The benefits of hydraulic-powered control surfaces with respect to human-powered control surfaces are two-fold. Firstly, the drag on the control surfaces is reduced be eliminating the need for control tabs. Also the control surface effectiveness is increased by eliminating control tabs. Secondly, the control surface flutter characteristics have improved since the hydraulic system introduces high mechanical stiffness. The main drawback of hydraulic-powered control surfaces is that it is an irreversible system.
Flight Dynamics
14
Introduction
Figure 1-11: Boeing 727 Aileron Control and Trim Systems [132]
1-1 Introduction to flight dynamics and control
15
Figure 1-12: Boeing 747 Elevator Control System [132]
Flight Dynamics
16
Introduction
Fly-by-wire flight control system With the invention of the (analog) computer it became possible to control an aircraft electronically. Initially the layout of the flight control system did not change very much. The mechanical part of the hydraulic powered control system was replaced by an electronic circuit and the cockpit controls now operate signal transducers which generate the appropriate commands. The only change to the hydraulic part of the system was the replacement of mechanical servo valves with electronic servo valves (the valves were operated by an electronic controller). This initial setup of the Fly-by-wire (FBW) FCS is known as the analog FBW FCS. The force feedback to the pilot is controlled by the electronic controller which drives an electric feel device. The main advantage of the (initial) analog fly-by-wire FCS was that there was no longer a need for a complex, fragile, and heavy mechanical linkage between the pilot and the hydraulic system. The advantages of analog FBW FCS further increased when analog computers were developed. By replacing the electronic controller with analog computers the designer of the FCS has more flexibility with regard to setting the flight control characteristics. The pilot commands no longer needed to be interpreted as control surface deflection but more as ’intentions’, like flying straight (column center) or commanding a constant pitch rate. The analog computer would then translate the pilot commands to actual control surface deflections. This concept opened the way for aircraft which where inherently unstable but could still be controlled as stable aircraft, the so-called relaxed stability concept. The development of the digital computer expanded the application of the FBW FCS considerably. The analog computers had limited capabilities, but the possibilities with the digital computers were vast. Now, the flying and handling characteristics of an aircraft could be set precisely. More and more exotic (inherently unstable) aircraft could be flown with the help of a digital FBW FCS. The next step in de development of flight control systems was to replace the heavy and high maintenance hydraulic mechanism. They were replaced with electrical power circuits. These circuits power electrical or self-contained electro-hydraulic actuators that are controlled by the digital flight control computers. This type of flight control system is also known as Power-bywire FCS. Other developments are Fly-by-optics FCS which use light pulses instead of electric pulses to transport pilot commands. Even more futuristic developments are the designs of intelligent FCS which can reconfigure themselves when the aircraft sustains damage. The FCS can choose the best set of control surfaces to perform the desired/commanded motion.
1-2
Assumptions
In this book the derivation for the equations of motion will be kept as general as possible. The equations of motion are derived in chapter 3. Two assumptions are made during that derivation. These are: Assumption 1: Spherical Earth In reality the Earth is an ellipsoid. Assuming that the Earth is spherical will simplify
1-2 Assumptions
17
the definition of latitude (see section 2-1-3). This is turn simplifies the transformation between reference frames which simplifies the equations of motion. Assumption 2: Vehicle is a rigid body of constant mass It is assumed that the vehicle under consideration does not deform under normal flight conditions. When this is true and the mass of the vehicle is constant then the matrix of inertia will be constant. This assumption implies no fuel consumption and no vehicle elastic modes. After the general equations of motion are derived, other assumptions are used to simplify the equations. This reduced set of equations will be used to clarify certain aspects of flight dynamics in chapter 5 till 11. Whenever an assumption is used it will be clearly stated in the text. The following assumptions will be used in this book (after Chapter 3): Assumption 3: Flat Earth When the duration of the motion under consideration is short, the influence of the Earth’s curvature is negligible. In such a case we can assume that the Earth is flat. This will make the vehicle carried normal Earth reference frame coincide with the normal Earth fixed reference frame. Assumption 4: Non-rotating Earth By neglecting the angular velocity of the Earth one neglects the influence of two types of acceleration, i.e. coriolis acceleration and centripetal acceleration. If real-life measured accelerations are used as input for simulations, one should remove the coriolis and centripetal components. Large errors are introduced if the time-span of the motion is large (in the order of hours). Assumption 5: The vehicle has a plane of symmetry If this is true then the orientation of the body-fixed reference frame can aligned with the principle axis of the vehicle in the symmetry plane. This will cause Ixy and Iyz to be zero. Assumption 6: Aircraft has a conventional configuration An aircraft with conventional configuration has one main wing, a horizontal and vertical stabilizor, ailerons, elevators, and one rudder. Such an aircraft is best suited for the explanation of flight dynamics principles. The reader can deduce the influence of other control systems (like the use of flaperons, elevons, etc.) by themselves. The basic principle of pitch, roll, and yaw control, remains. The only difference is that (stronger) coupling effects may be introduced. Assumption 7: Zero wind velocity No wind means that the undisturbed air is a rest relative to the Earth’s surface. This means that the kinematic velocity is equal to the aerodynamic velocity. Assumption 8: Resultant thrust lies in the symmetry plane The means that the aerodynamic force caused by the engine(s), i.e. thrust, only influences the symmetric aerodynamic forces X, Z and the symmetric aerodynamic moment M. Flight Dynamics
18
1-3
Introduction
Book outline
In part I, the derivation of the equations of motion for aircraft is given. Before the derivation is performed the reference frames which will be used are defined (see section 2-1). The building blocks of the equations of motion are the orientation and angular velocity of one reference frames to another. These block will be given in the remainder of chapter 2. The general equations of motion are derived in chapter 3. Simplification of the equations is performed thereafter to define the set of equations which will be used in the remainder of the book. Linearization of this set is performed in chapter 4. In part II, static stability and control is considered. First several aspects of aircraft aerodynamics, like the aerodynamic center and the process of deriving an state of equilibrium, are discussed (chapter 5). The aircraft characteristics for symmetric flight are explained and analyzed (chapter 6 and 8). Stability and Control derivatives will be derived analytically. An analysis of the aircraft characteristics is performed for symmetric, steady flight, followed by an analysis on the characteristic modes for symmetric flight from the view of static stability. The same analysis and derivations as for symmetric flight will be done for asymmetric flight (chapter 7 and 9). In Part III we will perform simulations with the symmetric (chapter 10) and asymmetric (chapter 11) linearized equations of motion.
Part I
Equations of Motion
Flight Dynamics
Chapter 2 Reference Frames
This chapter defines the reference frames used in flight dynamics. First of all one needs at least one frame of reference to describe any motion or position. The need for multiple reference frames arises from two considerations. The first is that the definition of a vector makes more sense for one particular reference frame than for another. The aircraft’s velocity vector expressed with respect to the Earth surface has a direct physical meaning while another decomposition will have no direct meaning (see figure 2-1, where VN is the velocity component due North, VE is the velocity component due East, and VZ is the velocity component perpendicular the to Earth surface). The other consideration is that the use of additional reference frames will make the derivation of the equations of motion easier. Certain motions are more easily described in a particular reference frame before a translation is made to the reference frame for which the equations of motion are derived.
North
VN
North
V
V VX
Versus VE Earth surface
VZ
VY Earth surface
VZ
Figure 2-1: Aircraft velocity vector decomposition Flight Dynamics
22
Reference Frames
Several reference frames will be used in subsequent chapters to derive the equations of motion. The orientation and state of the a vehicle (aircraft or spacecraft) can be defined in these reference frames. An important aspect when using multiple reference frames is the transformation of vector coordinates from one frame to another. A set of rotation angles is defined for each transformation. The rotation matrices given in this chapter are all defined using so-called Euler angles. The Euler angles are useful in many (normal-flight) cases. However, when certain situations occur such as in acrobatic flight, the Euler angle transformations will fail. A solution to this problem is the use of Quaternions. In the last part of this chapter the subject of vector differentiation is discussed. In particular the cases where the set of reference frames used during the derivation of the equations of motion are rotating relative to one another. The reference frames will be described in section 2-1. In section 2-2, the transformations between the reference frames will be given using Euler angles. Transformation matrices will be established for every transformation between reference frames. The effect of changes in orientation between reference frames over time is an important aspect of deriving the equations of motion. This effect will be described in section 2-3. Finally, a short introduction to Quaternions is given in section 2-4, p. 70.
2-1
Overview of reference frames
In this section a total of nine reference frames will be defined. The reason for defining multiple reference frames is to make the derivation of the equations of motion easier and more meaningful. In principle only one reference frame (an inertial reference frame) would suffice, but this would not make the understanding of flight dynamics easier.
2-1-1
Inertial reference frame, FI
The inertial reference frame (AXI YI ZI ), also known as the Earth-Centered Inertial (ECI) reference frame, is a right-handed orthogonal axis-system. The origin is at the center of mass of the earth. The ZI -axis is directed to the north along the spin-axis of the earth which goes through the poles of the earth. The XI -axis passes through the equator at the location where the ecliptic and the equator cross: the vernal equinox (see figure 2-2, p. 23). 1 The YI passes through the equatorial plane perpendicular to the XI end ZI to complete the coordinate axes system. The spin-axis of earth does not remain in the same direction through time. Due to gravitational influences of the moon and other celestial bodies, the spin-axis ’wobbles’ about a fictive mean spin-axis. This motion is known as polar motion (see figure 2-4, p. 24). The definition of the inertial reference frame still holds although the choice of spin-axis orientation can differ per country. For purpose of clarity an International Celestial Reference System (or 1
The ecliptic is defined as the great circle formed by the intersection of the plane of the earth’s orbit with the celestial sphere (it can be seen as the apparent annual path of the sun in the heavens).
2-1 Overview of reference frames
23
Conventional Celestial Reference System) has been defined. The coordinate system is defined by the mean equator and equinox of J2000.0. (The standard epoch J2000.0 is 12hr on January 1, 2000.) The mean equator and equinox are the fictitious equator and equinox, derived after removing the effects of nutation. For more information about polar motion the reader is directed to Ref. [93].
Figure 2-2: Definition of ecliptic plane
ZI North
Equatorial plane
A
Noval equinox
YI
XI Ecliptic plane
South
Ω Figure 2-3: Definition of earth-centered inertial reference frame
Flight Dynamics
24
Reference Frames
Figure 2-4: Polar motion - precession track
2-1-2
Earth-centered, Earth-fixed reference frame (FC )
The Earth-centered, Earth-fixed reference frame (ECEF) (AXC YC ZC ) is a right-handed coordinate axis-system which is very similar to the Earth-centered inertial reference frame. The difference between the two reference frames are the locations where the X-axis and Y -axis pass through the equatorial plane. The XC -axis crosses the Greenwich meridian (see figure 2-5, p. 25) while the XI -axis passes through the vernal equinox (see figure 2-3, p. 23). The ZC -axis is directed along the spin-axis of the earth, equal to the ZI -axis. The YC passes through the equatorial plane perpendicular to the XC end ZC to complete the coordinate axes system. Since the XC -axis is fixed to the Greenwich meridian, FC is fixed to the Earth and will thus rotate about its ZC -axis with the angular velocity of the Earth. This in contrast with the inertial reference frame which has no angular velocity. Similar to the definition of the International Celestial Reference System, there is the definition of the International Terrestrial Reference System (or Conventional Terrestrial Reference System). It takes the spin-axis as the true celestial pole during the period between 1900-1906 (this period has been chosen for historical reasons). The Greenwich meridian remains the same. This reference frame is widely used for navigational purposes (like GPS and GLONASS).
2-1 Overview of reference frames
25 ZC Greenwich meridian
North
YC
Equatorial plane
A Ω∆t XC
Noval equinox
Ecliptic plane
South
Ω
Figure 2-5: Definition of earth-centered, earth-fixed reference frame
2-1-3
Normal Earth-fixed reference frame, FE
The normal Earth-fixed frame of reference (OXE YE ZE ) is a right-handed orthogonal axissystem with the origin at an arbitrary location in the Inertial reference frame. The reference frame is fixed to the earth, meaning that it rotates with the same velocity as the earth itself (with respect to the Inertial reference frame). The XE YE plane is tangent to the earth surface. The average earth surface, the earth geoid2 , is taken as a reference since the earth surface changes constantly (see figure 2-6, p. 26). The XE -axis is directed to the north and the YE -axis is directed 90 degrees to the right of the XE axis. When the earth is considered as a sphere, the ZE -axis point to the center of the earth (see figure 2-7). The earth is actually an ellipsoid such that the gravitational attraction vector gr is no longer directed to the center of the earth. Due to the rotational acceleration generated by the earths spin velocity Ω, the apparent gravitational pull is always tangent to the earth geoid. A specific latitude can be defined for each vector in figure 2-6 on page 26: Φgc Geocentric latitude 2
Geoid: the particular equipotential surface that coincides with mean sea level and that may be imagined to extend through the continents. This surface is everywhere perpendicular to the force of gravity. Flight Dynamics
26
Reference Frames Defined as angle between AO and the equatorial plane. It denotes the direction of the center of the earth.
Φgr Gravitation latitude Indicates the direction of the gravitational attraction g r . Φgd Geodesic latitude Indicates the direction of apparent gravity (which is a combination of the gravitational attraction and the centripetal acceleration due to earth rotational speed Ω). The direction of the apparent gravity is always perpendicular to the earth geoid. However, in the remainder of this book it is assumed that the earth is a perfect sphere such that latitudes all become equal to each other and thus only one latitude angle exists. The location of the origin of the normal Earth-Fixed reference frame is arbitrary but often chosen as the aircraft’s center of gravity at the beginning of the considered motion. Another common choice is the location of the aircraft position projected along the ZE -axis on the earth geoid (see figure 2-7). The advantage of the latter choice of origin is that the altitude of the origin is always zero. In the remainder of this book the origin of the Normal Earth-Fixed reference frame will always be placed at the earth geoid. Polar axis (North) Earth surface
p
b
o
gr A
Φgc
Φgr
Φgd
Ω × (Ω × R)
g
Equatorial plane
a
Earth geoid
Ω Figure 2-6: Earth geoid and latitude definitions
2-1-4
Vehicle carried normal Earth reference frame, FO
The vehicle carried normal Earth reference frame is similar to the normal earth-fixed reference frame. It has the same origin O as the normal earth-fixed reference frame, but the ZO -axis points in the direction of the local gravity vector seen by the aircraft center of gravity G. Also, the XO -axis points to the direction of north as seen from the aircraft center of gravity, thus from point G instead of O. As a result the XO - and XE -axis are mostly not equal.
2-1 Overview of reference frames
27 ZI North
XE Equatorial plane
O
YE
ZE Noval equinox
Lt0
A
Rt
YI
ωt0 XI South
Ω
Figure 2-7: Normal earth-fixed reference frame for spherical earth
The same holds for the ZO - and ZE -axis and thus also for the YO and YE -axis. This can clearly be seen in figure 2-8 (p. 28). The FO and Fo′ reference frames are parallel to each other. This frame is used to demonstrate which terms will disappear in the equations of motion when a flat earth is assumed.
2-1-5
Body-fixed reference frame, Fb
The body-fixed reference frame (GXb Yb Zb ) is a right-handed orthogonal axis-system with the origin at the aircraft’s reference point. In general the center of mass of the aircraft is chosen as the aircraft’s reference point. If the gravity field is constant than this point coincides with the center of gravity of the aircraft. The reference frame remains fixed to the aircraft even in perturbed motion. The choice of the direction of the axes is rather arbitrary, the most commonly used orientation of the axes-system is depicted in figure 2-9 (p. 29). The Xb -axis is in the symmetry plane of the aircraft 3 and points forward. The actual direction is still arbitrary which influences the definition of the angle of attack (which will be explained in future sections). The Zb -axis also lies in the symmetry plane and points downward. Finally the Yb -axis is directed to the right, perpendicular to the symmetry plane. Because the X-axis direction can be chosen arbitrarily, an infinite number of possible body-fixed reference frames exist. The most commonly used body-fixed reference frames definitions are: 3
This is only possible under the assumption that such a plane exists. Flight Dynamics
28
Reference Frames ZI
G
North
h Xo′ o′
Yo′
Zo′
Equatorial plane
LtG
XO ZE
Rt
Noval equinox
ZO
A ωtO ωtG
XE
YE
o
YO
LtO
YI
LgG
XI South
Ω
Figure 2-8: Vehicle carried normal earth reference frame for spherical earth
Principle axes, Fp When a symmetry plane exist, the direction of the reference frame axis can be chosen such that most cross-products of inertia are zero. For a symmetric aircraft, two axes, i.e. Xp and Zp , will lie in the symmetry plane. The direction of the axes is chosen such that they lie parallel to the axes defined by the geometry of the aircraft such that the difference between moments of inertia are largest. This usually means that the Xp axis is pointed towards the nose of the aircraft. Zero-lift body axes, FZ Taking a look at the relative wind in the symmetry plane, there is a wind direction for which the total lift is zero. The XZ -axis lies in this direction. The remaining two axis can be chosen arbitrarily. Stability reference frames, FS For this reference frame to be defined, a reference flight condition must be chosen which is in most cases a condition of steady flight. For this condition the relative wind can be projected onto the plane of symmetry. The XS -axis is chosen in this direction. The ZS -axis is also in the plane of symmetry but perpendicular to the XS -axis pointing downward, The YS -axis completes the reference frame. Once the orientation of the reference frame relative to the aircraft is chosen, it will remain fixed thereafter. A
2-1 Overview of reference frames
29
graphical description is given in figure 2-10 (p. 29). The stability reference frame will be used in subsequent sections of this book. The origin is chosen in the center of mass of the aircraft.
Aircraft center of gravity: G
Yb
Xb
Zb
Figure 2-9: Body-fixed Reference Frame
Plane of symmetry
G
V
YS Va∞
′
XS
ZS
Figure 2-10: Stability Reference Frame
Flight Dynamics
30
Reference Frames
2-1-6
Aerodynamic (air-path) reference frame, Fa
The aerodynamic (air-path) reference frame is coupled to the aerodynamic velocity V a (also known as air-path velocity of ’air velocity’). The aerodynamic velocity is defined as the velocity of the center of mass G relative to the undisturbed air. The origin of the reference frame is in general the same as that of the body-fixed reference frame Fb . The Xa -axis is in the direction of the aerodynamic velocity V a . The Za -axis is in the symmetry plane of the aircraft. The Ya -axis is perpendicular to the GXb Zb plane to complete the (right-orthogonal) axis system (see figure 2-11, p. 31). The angles αa and βa are the aerodynamic angle of attack and the aerodynamic sideslip angle respectively. They denote the orientation of the aerodynamic reference frame with respect to the body-fixed reference frame. The aerodynamic velocity expressed in the aerodynamic reference frame is denoted as: a Va ua Vaa = vaa = 0 (2-1) a wa 0
Common practice is to express the aerodynamic velocity vector in the body-fixed reference frame: b u ua (2-2) Vab = vab = v wab w
The parameters (u, v, w) are commonly used to denote the components of any velocity vector along the axis of any reference frame. If the parameters are stated without any superscript then they are equal to the components of the aerodynamic velocity (in this book).
2-1-7
Kinematic (flight-path) reference frame, Fk
The kinematic (flight-path) reference frame is coupled to the kinematic velocity V k (also known as flight-path velocity or ’ground velocity’, i.e. the velocity relative to the ground4 ). The kinematic velocity is defined as the derivative of the position of the center of mass OG relative to the normal earth-fixed reference frame FE : dOG Vk,G = Vk = dt E The origin of the reference frame is in general the same as that of the body-fixed reference frame Fb . The Xk -axis is in the direction of the kinematic velocity V k . The Zk -axis is in the symmetry plane of the aircraft. The Yk -axis is perpendicular to the GXk Zk plane to complete the (right-orthogonal) axis system. The angles αk and βk are the kinematic angle of attack and the kinematic sideslip angle respectively. They are used to describe the orientation of the kinematic reference frame with respect to the body-fixed reference frame. The graphical representation of the kinematic reference frame is the same as for the aerodynamic reference frame (figure 2-11, p. 31). If there is no wind, there is no difference between these two 4
The ground velocity is always tangent to the ground track, i.e. the projection of the aircraft trajectory on the ground
2-1 Overview of reference frames
31
αa βa Ya
Wing plane
βa Yb
αa
Va
Xb
βa Xa Aircraft symmetry plane αa Za
Zb
Figure 2-11: Aerodynamic Reference Frame in relation to Body-fixed Reference Frame
reference frames. Further clarification on this matter will be given in section 2-2-11 (p. 55). The kinematic velocity expressed in the kinematic reference frame is denoted as:
ukk Vk Vkk = vkk = 0 wkk 0
(2-3)
Common practice is to express the kinematic velocity vector in the vehicle carried normal earth reference frame: O uk VN VkO = vkO = VE (2-4) wkO −VZ Where VN is the velocity due local North, i.e. the North seen from point G, VE is the velocity due local East, and −VZ is the velocity in local vertical direction. Flight Dynamics
32
Reference Frames
2-1-8
Vehicle reference frame, Fr
The vehicle frame of reference (Or Xr Yr Zr ) is a left-handed orthogonal axis-system with the origin at an arbitrary, yet fixed and invariable, position. The reference frame is fixed to the vehicle. The usual definitions, in case of an aircraft, for the axes’ directions are:
• Xr -axis Parallel to the plane of symmetry, in a direction fixed and invariable relative to the aircraft. The positive Xr -axis points to the rear of the aircraft. • Yr -axis Perpendicular to the plane of symmetry. The positive Yr -axis points to the left. • Zr -axis Perpendicular to the Or Xr Yr -plane. The positive Zr -axis points upward in normal, upright, flight. This frame of reference is used to prescribe locations of important parts of the aircraft, e.g. elevator hinge lines, location of aerodynamic centers, dimensions and positions of control surfaces, etcetera. An example of the aircraft frame of reference is given in figure 2-12 (p. 32).
Zr
Yr
Xr Or
Figure 2-12: Left-handed (!) Aircraft Reference Frame Fr
2-2 Transformations between reference frames ZI
33 ZI
Yb Xb Yb
O r Zb A XI
YI
Xb
=
YI
A, O XI
(a)
Zb (b)
Figure 2-13: Euler angles define the attitude of Fb irrespectively of translation
2-2
Transformations between reference frames
The reference frames defined in the section 2-1 will be used to define position parameters (longitude, latitude, etc.) and vector parameters (airspeed, forces, moments, etc.). They are needed to express the motion of the aircraft (or spacecraft) by differential equations. To make equations of motion more readable, different motions will be expressed in different reference frames. To couple the motions in each reference frame to get the overall aircraft motion through time in one particular reference frame, a transformation between reference frames is necessary. The transformation between any two reference frames at any point in time can be described by three angles, known as Euler angles. Mathematically, the rotation using the three angles results in a particular transformation matrix. The Euler angles and the relevant transformation matrices will be explained in section 2-2-1 and 2-2-2 (p. 38) respectively. Thereafter the most important transformations between reference frames and the angles (including the rotation sequence) which are needed to accomplish the transformation will be discussed (section 2-2-3 till 2-2-11, p. 42 till p. 56).
2-2-1
Euler angles
The orientation of a body in Euclidean space relative to an inertial reference frame can be described by three successive rotations through three body-referenced Euler angles5 . Therefore we need a body-fixed reference frame is necessary6 . Note that only the orientation of a reference frame is described by Euler angles, not the translation of one reference frame relative to another. This means that, in terms of Euler angles, position (a) in figure 2-13 (p. 33) is equivalent to position (b). In principle the transition from the orientation of an inertial reference frame FI to the 5 Named after Leonhard Euler (April 15, 1707 till September 18, 1783), a Swiss mathematician and physicist. He is considered to be on of the greatest mathematician of all times. 6 The rotation method given here is valid for any set of reference frames. To make the explanation easier to understand a body-fixed (non-inertial) and inertial reference frame are taken here.
Flight Dynamics
34
Reference Frames
orientation of an body-fixed reference frame Fb can be performed in 12 different ways. When starting from the orientation of the inertial reference frame, three consecutive rotations are performed. Each rotation is performed about one of the axes of the body-fixed reference frame. A maximum of three rotations are needed to arrive from the attitude of the inertial reference frame at the orientation of the body-fixed reference frame. The only condition required to guarantee that transition to any attitude of the body-fixed reference frame is that no two consecutive rotations are about the same axis. Bearing this in mind one can set up six ’symmetric’ sets of Euler angles:
ϕx → ϕy → ϕx , ϕx → ϕz → ϕx ,
ϕy → ϕx → ϕy , ϕy → ϕz → ϕy ,
ϕz → ϕx → ϕz ϕz → ϕy → ϕz
and six sets of ’asymmetric’ Euler angles:
ϕx → ϕy → ϕz , ϕx → ϕz → ϕy ,
ϕy → ϕx → ϕz , ϕy → ϕz → ϕx ,
ϕz → ϕx → ϕy ϕz → ϕy → ϕx
Where ϕx , ϕy , ϕz denote rotations around the X-axis, Y -axis, and Z-axis. Any of these sets of Euler rotations allows transition to any attitude of Fb . One should bear in mind, however, that different sets of Euler angle rotation sequences will result in different magnitudes of the corresponding Euler angles. An example is given in the mini-tutorial on page 35. Transformations using Euler angles are very often used because one can easily visualize the orientation changes per Euler angle. However, a major drawback exists, the so-called ’Euler angle singularity’ or ’gimbal lock’. The exact meaning of this drawback will not be explained until section 2-4, but it suffices to say that the singularity occurs at an pitch angle of 90◦ and −90◦ . During most (normal) aircraft manoeuvres these pitch angles will not be reached such that the singularity is of no concern.
2-2 Transformations between reference frames
35
Mini-tutorial: Rotation sequence using Euler angles This tutorial shows why the specification of the rotation sequence is important. It will show that when a different rotation sequence is applied using a fixed set of angles will lead to a different orientation. Suppose one wants to rotate a body about the origin of an inertial reference frame such that a transition is made from the initial orientation (figure 2-14a, p. 36) to the final orientation (figure 2-14b). This transition will be done using Euler angles of 30◦ , −30◦ , and 60◦ for ϕx , ϕy , ϕz , respectively. The correct rotation sequence belonging to this set of Euler angles is the a-symmetric sequence: ϕz → ϕy → ϕx , the so-called 3-2-1 rotation. This sequence is the most common sequence in the aerospace industry. The rotations are taken about the axis of the body-referenced axis system. After the first and the second rotation an intermediate reference frame is defined. The complete rotation sequence is stated as: ϕz
ϕy
ϕx
FI −→ FI ′ −→ FI ′′ −→ Fb The rotations are visualized in figure 2-15a till c (p. 37). Now, what will happen if the Euler angles stay the same but the rotation sequence is altered? To show the consequences a second rotation sequence is applied: ϕx
ϕy
ϕz
FI −→ FI ′ −→ FI ′′ −→ Fb The rotations are visualized in figure 2-16a till c (p. 37). As one can see, the end orientation of the body clearly differs from the previous case. The same end orientation can be reached using the second rotation scheme, but it would require a different set of Euler angles! In subsequent sections the rotation sequences and the names of the Euler angles for each transition between specific reference frames are given. Only the transition between reference frames, mentioned in the previous section, will be discussed. Using the Euler angles and the specific rotation sequences, one can set up so-called transformation matrices. These matrices can be used to transform vector coordinates in one reference frame to vector coordinates in another reference frame. The method of deriving such a transformation matrix will be discussed in the next section. In subsequent sections, the transformation matrices for each transition between reference frames will be specified.
Flight Dynamics
36
Reference Frames
ZI
YI
XI (a) Initial orientation of body
ZI Zb
Yb Xb
YI
XI (b) Final orientation of body
Figure 2-14: Euler angles tutorial: initial and end body orientation
2-2 Transformations between reference frames
37
ZI ′ , ZI
ZI ′
ϕx
ZI
+
YI ′ ϕx
YI ′
+ XI
XI ′
ϕz
ϕz
YI
YI
XI ′ , XI
(a) ϕz rotation in FI (60◦ )
(a) ϕx rotation in FI (30◦ )
ZI ′′ ϕy ZI ′ , ZI
ZI ′
ϕy
ZI
ZI ′′ XI ′′
YI ′′ , YI ′
XI ′′ YI ′′ , YI ′
ϕy
+ ϕy
+ XI
XI ′
YI
(b) ϕy rotation in FI ′ (−30◦ )
Zb
ϕx
ZI ′′
YI
XI ′ , XI
(b) ϕy rotation in FI ′ (−30◦ ) ZI
ZI Zb , ZI ′′ ϕz
Yb
+ Xb
Xb , XI ′′ ϕx
+
YI ′′
XI ′′
ϕz
YI ′′ YI
XI
(c) ϕx rotation in FI ′′ (30◦ )
YI
XI
Yb
(c) ϕz rotation in FI ′′ (60◦ )
Figure 2-15: Euler angles tutorial: rotation sequence Figure 2-16: Euler angles tutorial: rotation sequence ϕx → ϕy → ϕz ϕz → ϕy → ϕx
Flight Dynamics
38
2-2-2
Reference Frames
Transformation matrices
The transition between the orientation of one reference frame to another can be described in terms of Euler angles. These angles can be used to set up a so-called transformation matrix T. Any vector which is expressed in one reference frame can be expressed in a second reference frame (see figure 2-17, p. 38). Z2
Z1
v X2 Y2 O
Y1
X1 (a) Vector definition
Z1
Z2
Z2
Z1
vz1 v X2
X2 Y2
O
X1
vx1
v
vz2
vx2
vy2
Y2
O
vy1
Y1
(b) Vector decomposition F1
X1
Y1
(c) Vector decomposition F2
Figure 2-17: Simultaneous vector decomposition in two reference frames
The transition matrix contains the information necessary to calculate the coordinates of vector v in the second reference frame F2 when knowing the coordinates of vector v in the first reference frame F1 :
1 vx2 vx vy2 = T21 vy1 vz2 vz1
(2-5)
The transformation matrix T2,1 must thus contain information regarding the orientation of reference frame F1 with respect to reference frame F2 . It does not contain any information about a possible translation between reference frame origins. When using Euler angles (see previous section) the transformation matrix can be defined as a multiplication of three (sub) transformation matrices. Each (sub-)transformation matrix describes one rotation in the
2-2 Transformations between reference frames
39
entire rotation sequence used in combination with the Euler angles. Thus the complete transformation matrix can be defined as: 1 vx vx2 vy2 = T21′′ T1′′ 1′ T1′ ,1 vy1 vz2 vz1
(2-6)
where T1′ 1 is the transformation matrix for the first rotation in the overall sequence expressing ′ the vector v1 in terms of the axes of the intermediate reference frame F1′ , i.e. v1 . T1′′ 1′ is ′ the matrix for the second rotation expressing the vector v1 , which is expressed in F1′ , into ′′ to the second intermediate reference frame F1′′ (v1 ). Finally the transition from F1′′ to ′′ F2 is described by the transition matrix T21′′ (v1 → v2 ). Equation (2-6) is obtained by substitution: ′
v1 ′′ v1 v2
= T1′ 1 v1 ′ v2 = T21′′ T1′′ 1′ T1′ ,1 v1 = T1′′ 1′ v1 ′′ 1 = T21′′ v
(2-7)
The information on a single rotation depends on the particular axis around which the rotation is performed. If the axis of rotation is the X-axis of the reference frame in which the vector is currently decomposed (F1 ), then the coordinates of vector v in F2 are given by (see figure 2-18, p. 40)): +0.vy1 +0.vz1 vx2 = 1.vx1 vy2 = 0.vx1 + cos ϕx .vy1 + sin ϕx .vz1 vz2 = 0.vx1 − sin ϕx .vy1 + cos ϕx .vz1
(2-8)
The transformation matrix is then defined as:
T21
1 0 0 = 0 cos ϕx sin ϕx 0 − sin ϕx cos ϕx
(2-9)
If the axis of rotation is the Y -axis of the reference frame in which the vector in currently decomposed (F1 ), then the coordinates of vector v in F2 are given by (see figure 2-19, p. 40)): vx2 = cos ϕy .vx1 +0.vy1 − sin ϕy .vz1 vy2 = 0.vx1 +1.vy1 +0.vz1 2 1 1 vz = sin ϕy .vx +0.vy + cos ϕy .vz1
(2-10)
The transformation matrix is then defined as:
T21
cos ϕy 0 − sin ϕy = 0 1 0 sin ϕy 0 cos ϕy
(2-11)
Flight Dynamics
40
Reference Frames
a = Z1 sin ϕx
A-A
Z1
Z1
ϕx
b = Y1 cos ϕx
Z2
Z2
Y2
Y2 ϕx
ϕx
Z1 cos ϕx
Y1
O
X1 , X2
+
a
ϕx
Y1
O
b
Y1 sin ϕx
A-A (a) 3D view of rotation
(b) 2D view of rotation
Figure 2-18: Vector decomposition for rotation about X-axis
ϕy
B-B
Z1
a = X1 sin ϕy
Z1
b = Z1 cos ϕy
Z2
Z2
b
Z1 sin ϕy
+
O
X1 Y1 , Y2
X1 ϕy
a ϕy
ϕy
O
B-B
X1 cos ϕy
X2
X2 (a) 3D view of rotation
(b) 2D view of rotation
Figure 2-19: Vector decomposition for rotation about Y -axis
If the axis of rotation is the Z-axis of the reference frame in which the vector in currently decomposed (F1 ), then the coordinates of vector v in F2 are given by (see figure 2-20, p. 41):
vx2 = cos ϕz .vx1 + sin ϕz .vy1 +0.vZ1 vy2 = − sin ϕz .vx1 + cos ϕz .vy1 +0.vz1 vz2 = 0.vx1 +0.vy1 +1.vz1
(2-12)
2-2 Transformations between reference frames
41
The transformation matrix is then defined as:
T21
cos ϕz sin ϕz 0 = − sin ϕz cos ϕz 0 0 0 1
(2-13)
a = Y1 sin ϕz b = X1 cos ϕz
C-C
Y1
+
C-C
Za , Zb
Y2
X2
Y1 cos ϕz
ϕz
b a
ϕz Y2 O X1
ϕz
ϕz Y1
O
X1
X1 sin ϕz
X2
(a) 3D view of rotation
(b) 2D view of rotation
Figure 2-20: Vector decomposition for rotation about Z-axis
If the rotation sequence is ϕz → ϕy → ϕx the transformation for the entire rotation is described by: v2 = T21 v1 = T21′′ T1′′ 1′ T1′ 1 v1 1 0 0 cos ϕy 0 − sin ϕy = 0 cos ϕx sin ϕx 0 1 0 0 − sin ϕx cos ϕx sin ϕy 0 cos ϕy cos ϕz sin ϕz 0 − sin ϕz cos ϕz 0 v1 0 0 1 cos ϕy cos ϕz cos ϕy sin ϕz − sin ϕy ! ! sin ϕx sin ϕy cos ϕz sin ϕx sin ϕy sin ϕz sin ϕx cos ϕy − cos ϕ sin ϕ + cos ϕx cos ϕz = x z ! ! cos ϕx sin ϕy sin ϕz cos ϕx sin ϕy cos ϕz cos ϕx cos ϕy + sin ϕx sin ϕz − sin ϕx cos ϕz
(2-14)
v1
Transformation matrix T21 depend on the selected set of Euler rotations. This follows easily from the algebraic multiplication of the transformation matrices for a different sequence, for Flight Dynamics
42
Reference Frames
instance ϕx → ϕy → ϕz , resulting in a different matrix then given by equation (2-14). Transformation matrices have special characteristics. These characteristics stem from two criteria. First of all, one wishes to apply a orthogonal transformation, i.e. a rotation from one orthogonal reference frame to another, thus the transformation matrix must also be orthogonal. Secondly, after the rotation of a reference frame, a vector must still have its original length. This implies that the determinant of the transformation matrix must be equal to 1. This leads to the following criteria for transformation matrices (see Ref. [166]): 1. The eigenvalues of the transformation matrix must satisfy one of the following (orthogonality property): • All eigenvalues are 1.
• One eigenvalue is 1 and the other two are -1.
• One eigenvalue is 1 and the other two are complex conjugates of the form eiθ and e−iθ . 2. The determinant of the transformation matrix must be: det (T) ≡ 1 When these criteria are met for a transformation matrix T two important properties are guaranteed: t 1. T−1 ab = Tab = Tba
2. Ttab Tab = T−1 ab Tab = I Proof: v2 = T21 v1 v1
= T12
(taking the inverse)
→
(orthogonality) → v2
2 v1 = T−1 21 v v1 = Tt21 v2
t T−1 21 = T21
(2-15)
From property one follows property two. These properties will be used during the derivation of the equations of motion and kinematic relations. In the following sections the transformation matrices will be derived which are frequently used in practice. As explained above, the particular form of each matrix strictly depends on the specific order of the rotation sequence. Each matrix belongs to one particular transformation between two reference frames and is thus only valid for the rotation sequence as stated per transformation.
2-2-3
Transformation from FI to FE
The transformation from the inertial reference frame FI to the normal earth-fixed reference frame is performed using two consecutive rotations (see figure 2-8 p. 28 and 2-21 p. 44). For
2-2 Transformations between reference frames
43
the rotation sequence one intermediate reference frame will be defined, FI ′ : FI → FI ′ → FE The two rotations in sequence are: 1. Rotation −ωtO about the ZI -axis. 2. Rotation −LtO −
π 2
about the YE -axis (= YI ′ -axis).
The first rotation brings to YI -axis to the YE -axis. The rotation is equal to: −ωtO = Ωt tO
(2-16)
Where Ωt is the rotational speed of the earth and tO is known as the stellar time of point O. The second rotation consists of two parts. First the reference frame is rotated by 90◦ = − π2 [rad] about the YE -axis to bring the XE -axis in northern orientation, i.e. in ZI -direction. Thereafter a rotation of −LtO is performed to point the XE -axis to the geographical north (seen from point O on the earth surface). Angles sign and limits ωtO is positive if point O is west of the XI ZI -plane. LtO is positive if point O is on the northern hemisphere. The ranges of both angles are: −π < ωtO < π − π2 < LtO < π2 Transformation matrix XE = TEI XI TEI
2-2-4
= TEI ′ TI ′ I − sin LtO 0 cos LtO cos ωtO − sin ωtO 0 sin ωtO cos ωtO 0 0 1 0 = − cos LtO 0 − sin LtO 0 0 1 − sin LtO cos ωtO sin LtO cos ωtO cos LtO sin ωtO cos ωtO 0 = − cos LtO cos ωtO cos LtO sin ωtO − sin LtO
(2-17)
(2-18)
Transformation from FI to FO
The transformation from the inertial reference frame FI to the vehicle carried normal earth reference frame FO is similar to the transformation from FI to the normal earth-fixed reference frame FE (described in the previous section, see also figure 2-8, p. 28). Two consecutive rotations are required: Flight Dynamics
44
Reference Frames
ZI ZI XE
North Equatorial plane
xe ye
ze
A Noval equinox
LtO
o
YE YI
LtO
o
ωtO XI
A ZE
South Ω
YI
ωtO
XI Ω
(a) Location and orientation of FI and FE
(b) Origin of FE shifted to A
ZI
ZI XE
ZI ′
YI ′
YE
A −ωtO XI
A
YI
o XI ′
o
−LtO −
π 2
YI
ZE XI
Ω
(c) 1st rotation
Ω
(d) 2nd rotation
Figure 2-21: Transformation from inertial reference frame to normal earth-fixed reference frame
2-2 Transformations between reference frames
45
1. Rotation −ωtG about the ZI -axis. 2. Rotation −LtG −
π 2
about the YO -axis.
The first rotation brings the YI -axis to the YO -axis. The rotation is equal to: −ωtG = Ωt tG
(2-19)
Where Ωt is the rotational speed of the earth and tG is known as the stellar time of point G. The second rotation consists of two parts. First the reference frame is a rotation of 90◦ = − π2 [rad] about the YG -axis to bring the XG -axis in northern orientation, i.e. in ZI direction. Thereafter a rotation of −LtG is performed to point the XG -axis to the geographical north (seen from point G, see figure 2-8, p. 28).
Angles sign and limits ωtG is positive if point G is west of the XI ZI -plane. LtG is positive if point G is on the northern hemisphere. The ranges of both angles are: −π < ωtG < π − π2 < LtG < π2 Transformation matrix XO = TOI XI
TOI
2-2-5
= TOI ′ TI ′ I − sin LtG 0 cos LtG cos ωtG − sin ωtG 0 sin ωtG cos ωtG 0 = 0 1 0 − cos LtG 0 − sin LtG 0 0 1 − sin LtG cos ωtG sin LtG cos ωtG cos LtG = sin ωtG cos ωtG 0 − cos LtG cos ωtG cos LtG sin ωtG − sin LtG
(2-20)
(2-21)
Transformation from FE to FO
The transformation matrix for the case of the normal earth-fixed reference frame FE to the vehicle carried normal earth reference frame FO can be easily deduced using the results from the previous two sections: XE
= TEO XO = TEI TIO XO = TEI TtOI XO
(2-22)
Flight Dynamics
46
Reference Frames
Performing the multiplication using equation (2-18) and (2-21) and by using the summation rules: sin(A − B) = sin A cos B − cos A sin B cos(A − B) = cos A cos B + sin A sin B
(2-23)
with LgG = ωtG − ωtO , one will come to the following transformation matrix: T = EO sin LtO sin LtG cos LgG sin LtO cos LtG cos LgG − sin LtO sin LgG + cos LtO cos LtG − cos LtO sin LtG cos LgG cos LtG sin LgG sin LtG sin LgG cos LtO sin LtG cos LgG sin LtO cos LtG cos LgG − cos LtO sin LgG − sin LtO cos LtG + sin LtO sin LtG (2-24)
2-2-6
Transformation from FO to Fb
The transformation from the vehicle carried normal earth reference frame to the body-fixed reference frame is done in three consecutive rotations. For the rotation sequence two intermediate reference systems will be defined, i.e. FO′ and FO′′ : FO → FO′ → FO′′ → Fb The three rotations in sequence are (see figure 2-22, p. 48): • Rotation ψ yaw angle about the ZO -axis • Rotation θ pitch angle about the YO′ -axis • Rotation ϕ roll angle about the Xb -axis (= XO′′ -axis) In the aerospace industry the most commonly used rotation sequence is that of ψ → θ → ϕ. Other sequences are also possible (see section 2-2-1, p. 33). The yaw, pitch, and roll angle are the rotations about the Z, Y , and X-axis respectively. This set of angles is commonly confused with another set of angles, namely: the azimuth, climb, and bank angle. However the definitions of these angles are: • azimuth angle: angle between Xb -axis and its projection on a predefined (local) vertical plane. • climb angle: angle between Xb -axis and its projection on the (local) horizontal plane. • bank angle: angle between Yb -axis and its projection on the (local) horizontal plane.
2-2 Transformations between reference frames
47
For the current rotation sequence these two sets of angles are related in the following way: azimuth angle = yaw angle climb angle = pitch angle bank angle = roll angle x cos (pitch angle) Two definitions for the azimuth angle can be stated depending on the orientation of the XO -axis: • XO is the direction of the geographic North: ψv true heading • XO is the direction of the magnetic North: ψm magnetic heading The difference between the two headings is the magnetic declination dm, which is positive when the magnetic North is east of the geographic North. Angles sign and limits The positive rotation direction of the angles are in accordance with the definition of right-hand reference systems. In figure 2-22 (p. 48) a negative yaw angle (−ψ) is taken to enhance the graphical representation. The ranges of the angles according to convention are: −π < ψ < π − π2 < θ < π2 −π < ϕ < π Transformation matrix Xb = TbO XO TbO = TbO′′ TO′′ O′ TO′ O 1 0 0 = 0 cos ϕ sin ϕ 0 − sin ϕ cos ϕ cos ψ sin ψ 0 − sin ψ cos ψ 0 0 0 1 cos θ cos ψ ! sin ϕ sin θ cos ψ = − cos ϕ sin ψ ! cos ϕ sin θ cos ψ + sin ϕ sin ψ
(2-25)
cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ (2-26) cos θ sin ψ !
− sin θ
sin ϕ cos θ + cos ϕ cos ψ ! cos ϕ sin θ sin ψ cos ϕ cos θ − sin ϕ cos ψ sin ϕ sin θ sin ψ
Flight Dynamics
48
Reference Frames
GYb Zb -plane
Vertical
Xb
plane
YO
XO Horizontal
Zb
plane
Yb ZO
(a) Aircraft orientation
Horizontal plane
YO −ψ
XO
YO ′
XO′ −ψ
ZO ,ZO′
(b) 1st rotation: ψ
GYb Zb -plane XO′′ Vertical plane θ YO −ψ
YO′′
XO
−ψ Horizontal plane
Xb θ YO −ψ
XO ϕ
Zb ϕ
Yb
Vertical plane
−ψ Horizontal plane
ZO θ ZO′′
ZO θ
(c) 2nd rotation: θ
(d) 3rd rotation: ϕ
Figure 2-22: Transformation from vehicle carried normal earth reference frame FO to the body-fixed reference frame Fb
2-2 Transformations between reference frames
2-2-7
49
Transformation from FO to Fa
The transformation from the vehicle carried normal earth reference frame to the aerodynamic reference system is done in three consecutive rotations. For the rotation sequence two intermediate reference systems will be defined, FO′ and FO′′ : FO → FO′ → FO′′ → Fa The three rotations in sequence are (see figure 2-22, p. 48): • Rotation χa aerodynamic yaw angle about the ZO -axis • Rotation γa aerodynamic pitch angle about the YO′ -axis • Rotation µa aerodynamic roll angle about the Xa -axis (=XO′′ -axis) The first two rotations, χa and γa , are used to ’bring’ the Xa -axis parallel with the aerodynamic velocity vector. The third rotation, µa , is used to place the Za -axis into the symmetry plane of the aircraft. The rotation sequence is similar to that of the previous transformation FO → Fb . The complete transformation can be seen in figure 2-23 , p. 50 (similar to figure 2-22, p. 48). For the current rotation sequence the relation between yaw, pitch, roll and azimuth, climb, bank angle is: azimuth angle = yaw angle climb angle = pitch angle bank angle = roll angle x cos (pitch angle) The aerodynamic azimuth angle, climb angle, and bank angle, are sometimes called the airpath azimuth or air-path track angle, air-path climb or air-path inclination angle, and air-path bank angle respectively.
Angles sign and limits The positive rotation direction of the angles are in accordance with the definition of right-hand reference systems. In figure 2-23 (p. 50) a negative azimuth angle (−χa ) is taken to enhance the graphical representation. The ranges of the angles according to convention are: −π < χa < π − π2 < γa < π2 −π < µa < π Transformation matrix Xa = TaO XO
(2-27) Flight Dynamics
50
Reference Frames
GYa Za -plane symmetry plane
Xa Va
γa
Vertical plane
YO XO
−χa µa
Za
Ya
−χa Horizontal plane
µa ZO
γa
Figure 2-23: Transformation from vehicle carried normal earth reference frame FO to the aerodynamic reference frame Fa
TaO = TaO′′ TO′′ O′ TO′ O 1 0 0 cos γa 0 − sin γa = 0 cos µa sin µa 0 1 0 0 − sin µa cos µa sin γa 0 cos γa cos χa sin χa 0 − sin χa cos χa 0 0 0 1 cos γa cos χa cos γa sin χa − sin γa ! ! sin µa sin γa cos χa sin µa sin γa sin χa sin µa cos γa − cos µ sin χ + cos µa cos χa = a a ! ! cos µa sin γa sin χa cos µa sin γa cos χa cos µa cos γa + sin µa sin χa − sin µa cos χa
2-2-8
(2-28)
Transformation from FO to Fk
The transformation from the vehicle carried normal earth reference frame FO to the kinematic reference frame Fk is similar to the transformation from FO to the aerodynamic reference frame Fa in the previous section. The rotation sequence is the same, only the magnitudes and names of the angles are different. For the rotation sequence two intermediate reference systems will be defined, FO′ and FO′′ :
2-2 Transformations between reference frames
51
FO → FO′ → FO′′ → Fk The three rotations in sequence are: • Rotation χk kinematic yaw angle about the ZO -axis • Rotation γk kinematic pitch angle about the YO′ -axis • Rotation µk kinematic roll angle about the Xk -axis (=XO′′ -axis) The first two rotations, χk and γk , are used to bring the Xk -axis parallel with the kinematic velocity vector. The third rotation, µk , brings the Zk -axis into the symmetrical plane of the aircraft. For the current rotation sequence the relation between yaw, pitch, roll and azimuth, climb, bank angle is: azimuth angle = yaw angle climb angle = pitch angle bank angle = roll angle x cos (pitch angle) The kinematic azimuth angle, climb angle, and bank angle, are sometimes called the flightpath azimuth or flight-path track angle, flight-path climb or flight-path inclination angle, and flight-path bank angle respectively. Angles sign and limits The positive rotation direction of the angles are in accordance with the definition of right-hand reference systems. The ranges of the angles according to convention are: −π < χk < π − π2 < γk < π2 −π < µk < π Transformation matrix Xk = TkO XO
(2-29)
Analogue to equation (2-28) (p. 51): TkO = TkO′′ TO′′ O′ TO′ O cos γk cos χk ! sin µk sin γk cos χk = − cos µk sin χk ! cos µk sin γk cos χk + sin µk sin χk
cos γk sin χk !
− sin γk
sin µk cos γk + cos µk cos χk ! cos µk sin γk sin χk cos µk cos γk − sin µk cos χk sin µk sin γk sin χk
(2-30)
Flight Dynamics
52
Reference Frames
2-2-9
Transformation form Fb to Fa
The transformation from the body-fixed reference frame Fb to the aerodynamic reference frame Fa consists of two consecutive rotations. For the rotation sequence one intermediate reference frame will be defined, Fb′ : Fb → Fb′ → Fa The two rotations in sequence are (see figure 2-24, p. 53): • Rotation −αa aerodynamic angle of attack about the Yb -axis. • Rotation βa aerodynamic sideslip angle about the Za -axis (= Zb′ − axis). The first rotation brings the Xb -axis to the Xb′ -axis and the Zb -axis to the Zb′ of the intermediate reference frame Fb′ . Since the second and last rotation is about the Zb′ -axis, the Zb′ -axis is equal to the Za -axis. The second rotation bring the Xb′ -axis to the Xa -axis, thus into the direction of the aerodynamic velocity.
Angles sign and limits The positive rotation direction of the angles are in accordance with the definition of right-hand reference systems. The angle of attack is positive if the body axis Xb lies above the Xa , thus if the nose of the aircraft is pointed above the aerodynamic velocity vector. The sideslip angle is positive when the body axis Xb lies left from the aerodynamic velocity vector (seen from the origin of the reference frame). The ranges of the angles according to convention are: −π < αa < π − π2 < βa < π2 Transformation matrix Xa = Tab Xb
Tab = Tab′ Tb′ b cos βa sin βa 0 = − sin βa cos βa 0 0 0 1 cos βa cos αa sin βa = − sin βa cos αa cos βa − sin αa 0
(2-31)
cos αa 0 sin αa 0 1 0 − sin αa 0 cos αa cos βa sin αa − sin βa sin αa cos αa
(2-32)
2-2 Transformations between reference frames
53
Symmetry plane
Wing plane Ya Xb
βa Yb , Yb′
Va
αa βa
Xa
Za , Zb′
αa
Xb′
Zb
Figure 2-24: Transformation from the body-fixed reference frame Fb to the aerodynamic reference frame Fa
Useful relations The aerodynamic velocity vector can be related to the aerodynamic angle of attack and aerodynamic angle of sideslip. The aerodynamic velocity vector can be expressed in the body-fixed reference frame as stated in equation (2-2). Using this relation the magnitude of the aerodynamic velocity vector can be defined as: q Va = uba2 + vab2 + wab2 The aerodynamic angle of attack is defined as: αa = tan
−1
wab uba
Finally the aerodynamic angle of sideslip is defined as: b va βa = sin−1 Va
2-2-10
Transformation form Fb to Fk
The transformation from the body-fixed reference frame Fb to the kinematic reference frame Fk consists of two consecutive rotations, similar to those for the transformation of Fb to the aerodynamic reference frame Fa (see previous section): Flight Dynamics
54
Reference Frames • Rotation −αk kinematic angle of attack about the Yb -axis. • Rotation βk kinematic sideslip angle about the Zk -axis (= Zb′ − axis).
The first rotation brings the Xb -axis to the Xb′ -axis and the Zb -axis to the Zb′ of the intermediate reference frame Fb′ . Since the second and last rotation is about the Zb′ -axis, the Zb′ -axis is equal to the Zk -axis. The second rotation bring the Xb′ -axis to the Xk -axis, thus into the direction of the kinematic velocity. Angles sign and limits The positive rotation direction of the angles are in accordance with the definition of right-hand reference systems. The kinematic angle of attack is positive if the body axis Xb lies above the Xk , thus if the nose of the aircraft is pointed above the kinematic velocity vector. The sideslip angle is positive when the body axis Xb lies left from the kinematic velocity vector (seen from the origin of the body-fixed reference frame). The ranges of the angles according to convention are: −π < αk < π − π2 < βk < π2 Transformation matrix Xk = Tkb Xb
Tkb = Tkb′ Tb′ b cos βk sin βk = − sin βk cos βk 0 0 cos βk cos αk = − sin βk cos αk − sin αk
0 cos αk 0 sin αk 0 0 1 0 1 − sin αk 0 cos αk sin βk cos βk sin αk cos βk − sin βk sin αk 0 cos αk
(2-33)
(2-34)
Useful relations The kinematic velocity vector can be related to the kinematic angle of attack and kinematic angle of sideslip. The kinematic velocity vector can be expressed in the body-fixed reference frame as: b uk Vk = vkb wkb
Using this relation the magnitude of the kinematic velocity vector can be defined as: q 2 2 2 Vk = ubk + vkb + wkb
2-2 Transformations between reference frames
55
The aerodynamic angle of attack is defined as: αk = tan
−1
wkb ubk
Finally the aerodynamic angle of sideslip is defined as: b vk −1 βk = sin Vk
2-2-11
Transformation from Fk to Fa
The transformation from the kinematic reference frame Fk to the aerodynamic reference frame Fa can be done several ways. The first one is to use the Tab and Tkb to derive the transformation matrix Tak : Tak = Tab Tbk = Tab Ttkb Tak = Tab Tbk = Tab Ttkb cos βa cos αa sin βa cos βa sin αa = − sin βa cos αa cos βa − sin βa sin αa − sin αa 0 cos αa cos βk cos αk − sin βk cos αk − sin αk sin βk cos βk 0 cos βk sin αk − sin βk sin αk cos αk
The second method is to go straight from the first to the second reference frame using three consecutive rotations. For the rotation sequence two intermediate reference frames will be defined, Fk′ and Fk′ : Fk → Fk′ → Fk′′ → Fa The three rotations in sequence are: • Rotation −αw wind angle of attack about the Yk -axis. • Rotation βw wind sideslip angle about the Zk′ -axis. • Rotation µw wind roll angle about the Xk′′ -axis (= Xa -axis) The need for a third rotation can be explained by saying that the first rotation about the Yk -axis will move the Zk -axis to the Zk′ -axis which lies outside the vehicle symmetry-plane. The second rotation is about the Zk′ -axis such that the Zk′′ -axis is also outside the vehicle symmetry-plane. Since the Za -axis is in the vehicle symmetry-plane, a third rotation is necessary to move the Zk′′ -axis to the Za -axis. Flight Dynamics
56
Reference Frames
Wind relations The difference between the aerodynamic velocity vector and the kinematic velocity vector is due to the wind velocity vector. The relation between the three velocity vectors can be written as: Vk = Va + Vw The wind velocity vector is defined as the velocity of an undisturbed particle W in the center of gravity G expressed in the vehicle carried normal earth reference frame. Thus: uw dOW = vw Vw = VW,G = dt O ww The magnitude of the wind velocity vector can be defined as: q 2 O O2 O2 Vw = uO w + vw + ww The aerodynamic angle of attack is defined as:
αw = tan−1
b ww ubw
Finally the aerodynamic angle of sideslip is defined as: b vw βw = sin−1 Vw
2-3
Rotating reference frames
At any given moment in time we can express any vector in any reference frame. Vector expressions can be related through the relation r2 = T21 r1
(2-35)
But what will happen if we look at this relation through time? What additional relations will we have? In this section we will incorporate the aspect of time by looking at the time derivatives of vectors. We will see that the properties of the reference frames, i.e. location, orientation, and velocities, with respect to each other must all be used to correctly specify vector derivative relations. We will start this discussion by giving a general situation specified in figure 2-25. Imagine that we have the inertial reference frame FI (O, I, J, K) and an body fixed reference frame Fb (G, i, j, k) attached to an aircraft. As we can see, we have the following relation RP = R + r
(2-36)
We can express this relation in either reference frame. Expressing it in the inertial reference frame we obtain RP
= RI .I + RJ .J + RK .K + rI .I + rJ .J + rK .K = RI .I + RJ .J + RK .K + TIb (ri .i + rj .j + rk .k) ˆ = RI .I + RJ .J + RK .K + ri .ˆi + rj .ˆj + rk .k
(2-37)
2-3 Rotating reference frames
57
j −K
P
G
i RP
O
r
R
k
I
J
Figure 2-25: General situation for vector differentiation
ˆ are the unit vectors of reference frame Fb expressed in the inertial reference frame where ˆi, ˆj, k ˆi = TIb i = TIb 1 0 0 T ˆj = TIb j = TIb 0 1 0 T ˆ = TIb k = TIb 0 0 1 T k
(2-38)
The derivative of the vector RP , using the chain rule, is given by dRP d ˆ ˆ ˆ = .I + R .J + R .K + r . i + r . j + r . k R I J K i j k dt dt d ˆ = R˙ I .I + R˙ J .J + R˙ K .K + dt ri .ˆi + rj .ˆj + rk .k ˆ + ri . dˆi + rj . dˆj + rk . dkˆ = R˙ I .I + R˙ J .J + R˙ K .K + r˙i .ˆi + r˙j .ˆj + r˙k .k dt dt dt
(2-39)
dJ dK since dI dt = dt = dt = 0. This derivative is the derivative with respect to the inertial reference frame since the vector RP is expressed in FI during the derivation (and is expressed in FI after the derivation). This fact is incorporated in the notation as follows I I I I ˆ ˆ ˆ dRP ˙ I .I + R˙ J .J + R˙ K .K + r˙i .ˆi + r˙j .ˆj + r˙k .k ˆ + ri . di + rj . dj + rk . dk = R dt I dt dt dt I
I
I
(2-40) The subscript next to the sign | denotes the reference frame in which the derivative is taken. The previous relation is a very important one and we will analyze it piece by piece. The first three terms on the right hand side denote the translation of the origin of the body fixed reference frame. We can also write these terms as dR I I ˙ ˙ ˙ RI .I + RJ .J + RK .K = = VG (2-41) dt I The second group of terms, denote the relative time derivative of r with respect to Fb . It is the change in vector r as we would see it if we stand at the origin of Fb . We will write these
Flight Dynamics
58
Reference Frames
terms as
dr I ∂r ˆ ˆ ˆ r˙i .i + r˙j .j + r˙k .k = = dt b ∂t
(2-42)
∂ where ∂t denotes the derivative with respect to the body fixed reference frame. It is called the relative derivative. The last group of terms contain information regarding the rotation of Fb as we will see later on.
Moving, non-rotating reference frame As we can already guess, the relation for the time derivative when the body fixed reference frame is non-rotating is given by I dRP ˙ I .I + R˙ J .J + R˙ K .K + r˙i .ˆi + r˙j .ˆj + r˙k .k ˆ = R dt I (2-43) I I = dR + dr dt I
dt b
since the orientations (and the lengths) of the unit vectors i, j, k do not change.
Non-moving, rotating reference frame Now let us look at the situation where the body fixed reference frame is non-moving but rotating ( dR dt = 0, angular velocity vector = Ω). It that case we have I I I ˆi ˆj ˆ dRP I ˆ d d d k ˆ + ri . + rj . + rk . = r˙i .i + r˙j .ˆj + r˙k .k (2-44) dt I dt dt dt I
I
I
We need the expressions of the unit vector time derivatives. To derive these expressions we will take a closer at the change of a vector b (of constant length) caused by a single rotation about the vector Ω (see figure 2-26). In the case that the rotation angle α goes to zero we have the following relation (α is replaced by the infinitely small angle dα) lim {|db| = dα |QA| = dα |b| sin θ}
α→0
(2-45)
We also have the following relation by definition dα = |Ω| → dα = |Ω| dt dt
(2-46)
Combining relations we obtain db lim {|db| = |Ω| |b| sin θdt} → = |Ω| |b| sin θ α→0 dt
(2-47)
The latter equation is a scalar equation defining the relations between vector magnitudes and angles. The direction of db dt is perpendicular to the plane spanned by the vectors Ω and QA, thus we have − sin φ db db = cos φ (2-48) dt dt 0
2-3 Rotating reference frames
59
k B
α
db
Q
A
b′
θ
b
ω
j G
ψ
i (a) 3D view
Q
j B α db
ψ A i (b) Top view
Figure 2-26: Change of a vector due to rotation vector Ω
Flight Dynamics
60
Reference Frames
Substituting the obtained relation for db dt gives
− sin φ − |Ω| |b| sin θ sin φ db = |Ω| |b| sin θ cos φ = |Ω| |b| sin θ cos φ dt 0 0
(2-49)
This relation is given exactly by the cross product of Ω and b
0 sin θ cos φ − |Ω| |b| sin θ sin φ Ω × b = |Ω| 0 × |b| sin θ sin φ = |Ω| |b| sin θ cos φ 1 0 0
(2-50)
Thus we have the following important relation for the time derivative of a vector (of constant length) due to a rotation vector Ω db =Ω×b (2-51) dt Using this result in equation 2-44 yields I dRP dt I
I I dˆj dˆi ˆ ˆ ˆ = r˙i .i + r˙j .j + r˙k .k + ri . dt + rj . dt + rk . I I I + ri .ΩI × ˆi + rj .ΩI × ˆj + rk .ΩI × k ˆ = dr bI bI bI dt b dr I I ˆ = dt b + ΩbI × ri .ˆi + rj .ˆj + rk .k I I I = dr dt b + ΩbI × r
ˆ I dk dt I
(2-52)
The latter result is an important relation which is often used during the derivation of the equations of motion. Some remarks are in order: • The time derivative dr dt b is the time derivative of the vector r relative to the moving, rotating reference frame! It is the derivative of r as seen by an observer standing in the origin of Fb . • The rotation vector ΩbI denotes the relative angular velocity of reference frame Fb with respect to FI . This angular velocity vector can be expressed in any reference frame as it is a vector. As usual, the superscript denotes in which reference frame it is expressed. • The rotation vector has some useful properties: – For any set of reference frames we have Ω212 = −Ω221
(2-53)
Ω220 = Ω221 + Ω210
(2-54)
– and
For mathematical proof of these two properties see [30], p. 228 and the mini tutorial 2-3 on page 63.
2-3 Rotating reference frames
61 P
q r
G
R
i θ
k RP I
O
K
Figure 2-27: Example - derivation of pilot velocity with respect to FO
Moving, rotating reference frame If we combine the two relations obtained in the previous two sections, we get dR dr dRP = + + ΩbI × r dt I dt I dt b
(2-55)
This relation is valid for any two reference frames. Take one reference frame as the fixed (inertial) reference frame and let the other frame move with respect to the first. The most important aspect of this relation is that all terms should be expressed in the same reference frame otherwise the relation is not valid.
Example 2.1 Consider an aircraft with pitch angle θ (10 deg) and a pitch up rate q (5 deg/s) as depicted in figure 2-27. The aircraft is flying with a speed of 200 m/s in the direction of I at an altitude T of 11 km at a distance of 5 km from point O, i.e. RO = 5.103 , 0, −11.103 . The location of T the pilot (point P ) given with respect to the body fixed reference frame is rb = [10, 0, −5] . The vehicle carried normal earth reference frame FO is given by the quadruple (O, I, J, K) and is fixed. The moving, rotating reference frame Fb is fixed to the aircraft in the center of gravity (G, i, j, k). We will derive the velocity of the pilot with respect to FO . The location of the pilot expressed in FO is given by RO P
= RO + rO = RI .I + RJ .J + RK .K + rI .I + rJ .J + rK .K = RI .I + RJ .J + RK .K + TIb (ri .i + rj .j + rk .k) ˆ = RI .I + RJ .J + RK .K + ri .ˆi + rj .ˆj + rk .k
(2-56)
Flight Dynamics
62
Reference Frames This relation is given in inertial coordinates. The unit vectors of FI are off course given by I J K
= 1 = 0 = 0
T 0 0 T 1 0 T 0 1
(2-57)
Since we have a single rotation, the transformation matrix is simply (see section 2-2-6, ψ = φ = 0) cos θ 0 sin θ 0 1 0 TOb = TtbO = (2-58) − sin θ 0 cos θ We can now express the unit vectors defining Fb in terms of Fo ˆi = TOb i = TOb 1 0 ˆj = TOb j = TOb 0 1 ˆ = TOb k = TOb 0 0 k
T 0 T 0 T 1
T = cos θ 0 − sin θ T = 0 1 0 T = sin θ 0 cos θ
(2-59)
Using previous results, the location of the pilot expressed in FO is RO P
= RO + rO = RO + TIb rb T 5.103 cos 10◦ 0 + 0 0 1 = 3 ◦ −11.10 − sin 10 0 5.103 + 10 cos 10◦ − 5 sin 10◦ 0 = 3 −11.10 − 10 sin 10◦ − 5 cos 10◦ 5008.98 0 ≈ 11006.66
sin 10◦ 10 0 0 ◦ cos 10 −5
(2-60)
The angular velocity of Fb with respect to FO expressed in FO is given by: ΩO bO
0 = q 0
(2-61)
We now have all the information required to determine the velocity of point P with respect to FO and expressed in FO : VPO
O O O dr O O P = dR = dR bO ×r dt O dt O+ dt b + Ω 200 0 0 10 cos 10◦ − 5 sin 10◦ 5π 0 = 0 + 0 + 180 × ◦ ◦ 0 0 0 −10 sin 10 − 5 cos 10 5π 200 + 180 (−10 sin 10◦ − 5 cos 10◦ ) 0 = 5π − 180 (10 cos 10◦ − 5 sin 10◦ )
(2-62)
2-3 Rotating reference frames
63
Mini-tutorial: angular velocity vector Ω interpretation The goal of this mini-tutorial is to demonstrate the principles of the angular velocity vector. Consider the situation of the rotating earth. Two reference frames are considered, i.e. the inertial reference frame FI and the normal earth-fixed reference frame FE at longitude ωtO and latitude LtO = 0◦ , i.e. point O is on the equator (see figure 2-28). The goal is to determine the angular velocity vector ΩE EI , i.e. to determine the angular velocity of reference frame FE with respect to reference frame FI expressed in frame FE . Since point O is fixed on earth, the only rotation between the reference frames is the rotation of the earth. The rotation can be described in the inertial reference frame by: 0 ΩIEI = 0 Ωt The angular velocity vector can be transformed to another reference frame through multiplication with a transformation matrix (since it is simply a vector). The desired angular velocity vector thus becomes: ΩE EI
I =T EI ΩEI − sin LtO cos ωtO sin LtO cos ωtO cos LtO 0 0 = sin ωtO cos ωtO 0 − ωtO cos LtO sin ωtO − sin LtO Ωt cos LtO cos cos LtO = Ωt 0 − sin LtO
For the simpler case considered here where LtO = 0 the angular velocity vector t becomes: ΩE EI = [1 0 0] , which states that the angular velocity vector is in the direction of the XE -axis. Looking at figure 2-28 (p. 64) one can see that this is also the case: the FE frame is rotation counterclockwise about the XE -axis with respect to the inertial reference frame.
2-3-1
Derivation of the angular velocity vector
Two methods exist to derive the angular velocity vector. The first method is to directly perform the mathematics when considering: Ωaab = Tab
dTba dTba = Ttba dt dt
This method works fine but can lead to complex expressions which must be simplified by hand. The second method is more simple and comprehensable. Imagine a set of reference frames consisting of two reference frames Fa and Fb . The transformation in orientation from Fa to Fb is done according to the rotation sequence ϕz → ϕy → ϕx . Consider the angular Flight Dynamics
64
Reference Frames
Ωt A-A North
ZI
Equatorial plane
XE
p3
p2 A
Noval equinox p4
O, p1
ωt0
Time: tp1
A-A
p3
Time: tp4 p3
p2
p2
YI
ZE
YI
ZE
YE
p4
YE
South
XI
A-A
YI
ZE
p4
p1
p1
YE
XI
A-A
XI
A-A
Time: tp2
Time: tp3
YE
p3
p3 p2
YE
p2
ZE
ZE YI p4
p1 XI
YI p4
p1 XI
Figure 2-28: Angular velocity vector ΩE EI explanation
2-3 Rotating reference frames
65
velocity vector Ωbba , i.e. the angular velocity of Fb with respect to Fa expressed in Fb . During the rotation sequence two intermediate reference systems are encountered:
Fa → Fa′ → Fa′′ → Fb According to equation (2-54), the angular velocity vector in reference frame Fb can be described by the summation of the angular velocities per rotation step considered in reference frame Fb : Ωbba = Ωbba′′ + Ωba′′ a′ + Ωba′ a The angular velocities between each of the reference frames is described by: t ′′ Ωbba′′ = Tba′′ Ωaba′′ = Tba′ ϕ˙ x 0 0 t ′ Ωba′′ a′ = Tba′ Ωaa′′ a′ = Tba′ 0 ϕ˙ y 0 t Ωba′ a = Tba Ωaa′ a = Tba 0 0 ϕ˙ z
(2-63)
These relations all use the following simple principles:
1. Consider the angular velocity of the next reference frame with respect to the current reference frame. 2. Express the angular velocity vector in the current reference frame. 3. Transform the obtained angular velocity vector to Fb using the proper transformation matrix. If the transformation matrices are considered to be composed of sub-transformation matrices belonging to each single rotation: Tba′′ Tba′ Tba equation (2-63) can be rewritten as: Ωbba
= T ϕx = T ϕx T ϕy = T ϕx T ϕy T ϕz
0 0 ϕ˙ x = Tϕx Tϕy Tϕz 0 + Tϕx Tϕy ϕ˙ y + Tϕx 0 ϕ˙ z 0 0
(2-64)
Equation (2-64) is a very useful relation for the determination of any angular velocity vector. An interesting point is that equation (2-64) can be simplified to: 0 0 ϕ˙ x Ωbba = Tϕx Tϕy 0 + Tϕx ϕ˙ y + 0 (2-65) ϕ˙ z 0 0
This is because the multiplication of Tϕi ϕ˙ i is always equal to ϕ˙ i for any ϕi since the rotation axis for ϕi is the same as the angular velocity axis, i.e. ϕ˙ i -axis. This also means that the ′ relation Ωaa′ a = Ωaa′ a is valid for any single rotation transition from reference frame Fa to frame Fa′ . In the following sections, relation (2-65) will be used to derive several angular velocity vectors concerning particular combinations of reference frames. Flight Dynamics
66
Reference Frames
2-3-2
Derivation of ΩE EI
ΩE EI is the angular velocity of the normal earth-fixed reference frame FE with respect to the inertial reference frame FI , expressed in FE . The transformation from FI to FE is performed using two consecutive rotations (see section 2-2-3, p. 42): 1. Rotation −ωtO = Ωt tO about the ZI -axis. 2. Rotation −LtO −
π 2
about the YE -axis.
The derivative of these angles are: d dt d dt
= ΩII ′ I z ′ π −LtO − 2 = ΩIEI ′
(−ωtO )
= −ωt ˙ O = Ωt ˙ O≡0 = −Lt
y
The derivative of the latitude is zero since the reference frame FE is earth-fixed. Using equation (2-65) one can derive the following expression for the angular velocity: 0 0 ΩE EI = TEI ′ −ωt ˙ O 0 − sin LtO 0 cos LtO 0 = 0 1 0 −ωt ˙ O − cos LtO 0 − sin LtO cos LtO = −ωt ˙ O 0 − sin LtO cos LtO = Ωt 0 − sin LtO
2-3-3
Derivation of ΩO OI
ΩO OI is the angular velocity of the vehicle carried normal earth reference frame FO with respect to the inertial reference frame FI , expressed in FO . The transformation from FI to FO is performed using two consecutive rotations (see section 2-2-4, p. 43): 1. Rotation −ωtG = Ωt tG about the ZI -axis. 2. Rotation −LtG −
π 2
about the YE -axis.
The derivative of these angles are: d dt d dt
= ΩII ′ I z ′ −LtG − π2 = ΩIOI ′
(−ωtG )
y
= −ωt ˙ G ˙ G = −Lt
2-3 Rotating reference frames
67
Using equation (2-65) one can derive the following 0 0 O ˙ G ΩOI = TOI ′ 0 + −Lt −ωt ˙ G 0 − sin LtG 0 cos LtG 0 1 0 = − cos LtG 0 − sin LtG −ωt ˙ G cos LtG ˙ G = −Lt ωt ˙ G sin LtG
2-3-4
expression for the angular velocity:
0 0 ˙ G 0 + −Lt −ωt ˙ G 0
Derivation of ΩO OE
ΩO OE is the angular velocity of the vehicle carried normal earth reference frame FO with respect to the normal earth-fixed reference frame FE , expressed in FE . The determination of ΩO OE can be performed using the following relation: O O O O ΩO OE = ΩOI + ΩIE = ΩOI − ΩEI I = ΩO OI − TOI ΩEI
Using the results from the previous two section one can form the following relation: −ωt ˙ G cos LtG − sin LtG 0 cos LtG 0 ˙ G − 0 ΩO 0 1 0 −Lt OE = Ωt − cos LtG 0 − sin LtG ωt ˙ G sin LtG −ωt ˙ G cos LtG Ωt cos LtG ˙ G − = 0 −Lt −Ωt sin LtG ωt ˙ G sin LtG − (ωt ˙ G + Ωt ) cos LtG ˙ G = −Lt (ωt ˙ G + Ωt ) sin LtG
The relations between FO and FE can be described by stating the latitude LtG and longitude LgG of point G (aircraft center of mass) with respect an observer in point O (the origin of FE ). These relations are: LgG = ωtG − ωtO LtG = ∆Lt + LtO From this follows:
˙ G + ωt ˙ G − Ωt ωt ˙ G = Lg ˙ O = Lg ˙ G = ∆Lt ˙ Lt
Thus: ΩO OE
˙ G cos LtG −Lg ˙ G = −Lt ˙ LgG sin LtG Flight Dynamics
68
Reference Frames
2-3-5
Derivation of ΩbbO
ΩbbO is the angular velocity of the body reference frame Fb with respect to the vehicle carried normal earth reference frame FO , expressed in FB . The transformation from FO to Fb is performed using three consecutive rotations (see section 2-2-6, p. 46): 1. Rotation ψ azimuth angle about the ZO -axis 2. Rotation θ inclination angle about the YO′ -axis 3. Rotation ϕ bank angle about the Xb -axis The derivative of these angles are: (ψ) = ΩO = ψ˙ O ′ O z ′ O (θ) = ΩO′′ O′ = θ˙ ′′ y d O = ϕ˙ dt (ϕ) = ΩbO ′′ d dt d dt
x
Using equation (2-65) one can derive the following expression for the angular velocity: ′
′′
O O ΩbbO = TbO′ ΩO O ′ O + TbO ′′ ΩO ′′ O ′ + ΩbO ′′ cos θ 0 − sin θ = sin ϕ sin θ cos ϕ sin ϕ cos θ cos ϕ sin θ − sin ϕ cos ϕ cos θ 0 1 0 0 0 cos ϕ sin ϕ θ˙ + 0 − sin ϕ cos ϕ 0 ϕ˙ − ψ˙ sin θ = θ˙ cos ϕ + ψ˙ sin ϕ cos θ −θ˙ sin ϕ + ψ˙ cos ϕ cos θ
2-3-6
0 0 + ψ˙ ϕ˙ 0 0
Derivation of ΩaaO and ΩkkO
ΩO aO is the angular velocity of the aerodynamic reference frame FX with respect to the vehicle carried normal earth reference frame FO , expressed in Fa . The transformation from FO to Fa is performed using three consecutive rotations (see section 2-2-7, p. 49): 1. Rotation χa aerodynamic azimuth angle about the ZO -axis 2. Rotation γa aerodynamic climb angle about the YO′ -axis 3. Rotation µa aerodynamic bank angle about the Xa -axis
2-3 Rotating reference frames
69
The derivative of these angles are: (χa ) = ΩO O′ O z ′ d (γ ) = ΩO a O ′′ O ′ dt d dt
d dt
′′ (µa ) = ΩO aO ′′
x
χ˙ a y
γ˙ a µ˙ a
Using equation (2-65) one can derive the following expression for the angular velocity: ′
′
′′
O O ΩaaO = TaO′ ΩO O ′ O + TaO ′′ ΩO ′′ O ′ + ΩaO ′′ 0 − sin γa cos γa 0 = sin µa sin γa cos µa sin µa cos γa 0 cos µa sin γa − sin µa cos µa cos γa χ˙ a 0 µ˙ a 1 0 0 0 cos µa sin µa γ˙ a + 0 0 − sin µa cos µa 0 0 µ˙ a − χ˙ a sin γa = γ˙ a cos µa + χ˙ a sin µa cos γa −γ˙ a sin µa + χ˙ a cos µa cos γa
The derivation of the angular velocity vector ΩkkO is analogue to the derivation of ΩaaO . The final expression is given by: ΩkkO
2-3-7
µ˙ k − χ˙ k sin γk = γ˙ k cos µk + χ˙ k sin µk cos γk −γ˙ k sin µk + χ˙ k cos µk cos γk
Derivation of Ωaab and Ωkkb
Ωaab is the angular velocity of the aerodynamic reference frame Fa with respect to the bodyfixed reference frame Fb , expressed in Fa . The transformation from Fa to Fb is performed using two consecutive rotations (see section 2-2-9, p. 52): 1. Rotation −αa aerodynamic angle of attack about the Yb -axis. 2. Rotation βa aerodynamic sideslip angle about the Za -axis (= Zb′ − axis). The derivative of these angles are: d dt d dt
(−αa ) = Ωbb′ b y ′ (βa ) = Ωbab′
z
= −α˙ a = β˙a Flight Dynamics
70
Reference Frames
Using equation (2-65) one can derive the following expression for the angular velocity: ′
Ωaab = Tab′ Ωbb′ b + Ωbab′ cos βa sin βa 0 0 = − sin βa cos βa 0 −α˙ a + 0 0 1 0 −α˙ a sin βa = −α˙ a cos βa β˙a
0 0 ˙ βa
The derivation of the angular velocity vector Ωkkb is analogue to the derivation of Ωaab . The final expression is given by: −α˙ k sin βk Ωkkb = −α˙ k cos βk β˙k
2-4
Quaternions, a brief introduction
In this chapter the principle of Euler angle rotations has been explained as it will be used throughout the book. The great drawback of using Euler angle rotations is the chance of getting ’stuck’ in the singularity (as mentioned in section 2-2-1). Luckily, the singularity is positioned at +90◦ and −90◦ pitch angle, a flight condition which is normally not reached for non-acrobatic aircraft. For simulating aircraft motion one can thus use Euler angles. However, for spacecraft the condition of the singularity will almost certainly be reached, i.e. there is no real ’up’ and ’down’ in space. The existence of the singularity will be explained in section 2-4-1. To remove the singularity we will introduce the concept of quaternions in section 2-4-2. The transformation between two reference frames expressed in the transformation matrix will be derived in the same section. When using quaternions during simulations, one integrates the time derivatives of the quaternions. How to determine the derivative is explained in section 2-4-3.
2-4-1
Euler angle singularity
The presence of a singularity at θ = ±90◦ is best explained by looking at the Euler angle values for the 3-2-1 rotation sequence, i.e. ψ → θ → ϕ. Imagine that the aircraft is in a vertical climb, see figure 2-29. If we try to determine the set of Euler angles for the particular case of vertical climb, it become clear that infinitely many sets satisfy this flight condition! One set is: −90◦ → 90◦ → 0◦ . Another set is: 0◦ → 90◦ → 90◦ . The yaw and roll angle are now both defined about the same axis vertical axis. The singularity at θ = ±90◦ can be avoided by choosing a different rotation sequence. However, another singularity will still exist for that new rotation sequence! For example, when the rotation sequence is ϕy → ϕx → ϕz the singularity will lie at ϕx = ±90◦ .
2-4 Quaternions, a brief introduction
71
Xb
Zb
Yb
North XO YO Earth surface ZO
Figure 2-29: Euler angle singularity: Aircraft in vertical climb
For a-symmetrical rotation sequences (see 2-2-1) the singularity is always at ϕ2 = ±90◦ , i.e. ±90◦ of the second rotation. Due to this rotation the axis for the third rotation will be aligned with the axis of the first rotation. It will therefore not matter how much the body is rotated in the first and third rotation, as long as the subtraction (for ϕ2 = +90◦ ) or summation (for ϕ2 = −90◦ ) of the two angles remains fixed! This can be shown mathematically by looking at the transformation matrix for the 3-2-1 rotation sequence (similar to TbO , section 2-2-6): T321 = Tϕx Tϕy Tϕz 1 0 0 cos ϕy 0 − sin ϕy = 0 cos ϕx sin ϕx 0 1 0 0 − sin ϕx cos ϕx sin ϕy 0 cos ϕy cos ϕz sin ϕz 0 − sin ϕz cos ϕz 0 0 0 1 cos ϕy cos ϕz cos ϕy sin ϕz − sin ϕy ! ! sin ϕx sin ϕy cos ϕz sin ϕx sin ϕy sin ϕz sin ϕx cos ϕy − cos ϕ sin ϕ + cos ϕx cos ϕz = x z ! ! cos ϕx sin ϕy sin ϕz cos ϕx sin ϕy cos ϕz cos ϕx cos ϕy + sin ϕx sin ϕz − sin ϕx cos ϕz
(2-66) Flight Dynamics
72
Reference Frames
Substituting ϕy = 90◦ will lead to: 0 0 −1 sin ϕx cos ϕz sin ϕx sin ϕz 0 T321 = − cos ϕx sin ϕz + cos ϕx cos ϕz cos ϕx sin ϕz sin ϕx cos ϕz 0 − cos ϕx sin ϕz − sin ϕx cos ϕz Using the trigonometric rules of:
sin (ϕx − ϕz ) = sin ϕx cos ϕz − cos ϕx sin ϕz sin (ϕz − ϕx ) = cos ϕx sin ϕz − sin ϕx cos ϕz cos (ϕx − ϕz ) = sin ϕx sin ϕz + cos ϕx cos ϕz one can rewrite the transformation matrix like: 0 0 −1 0 0 −1 T321 = sin (ϕx − ϕz ) cos (ϕx − ϕz ) 0 = sin A cos A 0 cos A − sin A 0 cos (ϕx − ϕz ) − sin (ϕx − ϕz ) 0
This means that the transformation matrix will be similar for a constant A, meaning that the entire collection of ϕx , ϕz satisfying A = ϕx − ϕz can describe the same rotation. For ϕy = −90◦ , the collection of ϕx , ϕz satisfying A = ϕx + ϕz is obtained. The possibilities are infinite, therefore indicating the singularity. For symmetric rotation sequences the singularity is at 0◦ and 180◦ for the second rotation. In both cases the rotation axis for the third rotation is aligned with of the first rotation, creating a singularity.
2-4-2
Quaternion transformation matrix
The solution to the problem is to use one rotation instead of three. This single rotation is taken about the so-called Euler axis or eigenaxis. Such an axis of rotation is fixed to the reference frame which is rotated and has a fixed orientation to this frame and to the inertial reference frame during the rotation. Consider two reference frames Fa and Fb respectively. The orientation of Fb with respect Fa can be characterized by a single rotation θ about the unit vector e along the Euler axis, e = e1 Xa + e2 Ya + e3 Za = e1 Xb + e2 Yb + e3 Zb
(2-67)
where (e1 , e2 , e3 ) are the direction cosines of the Euler axis relative to both Fa and Fb (see figure 2-30). The fact that the Euler axis is stationary to the body-fixed reference frame is shown clearly in equation (2-67), i.e. relation between Fb and e is equal to the Fa , e relation. It is graphically presented in figure 2-31 Let us take a look at the rotation about the Euler axis regarding an arbitrary vector v (see figure 2-32). We want to find the relation between the vector expressed in Fa and Fb . From vector algebra one knows that: vb = OA + AB + BC
2-4 Quaternions, a brief introduction
73
Za
Zb
e
e
θ3
θ3
e3 = cos θ3 θ2
θ1
θ1
Ya
e3 = cos θ3
θ2
e1 = cos θ1 Xb
e2 = cos θ2
e2 = cos θ2
e1 = cos θ1
Yb
Xa
(a) Direction cosines for Fa
(b) Direction cosines for Fb
Figure 2-30: Direction cosines equality for Euler axis rotations
Za
θ Zb e
Xb
θ
Ya Yb
θ
Xa
Figure 2-31: Rotating a reference frame about the Euler axis
Flight Dynamics
74
Reference Frames
e A
A
D θ
vA × e
B
C
va
O
B
D vb
C
θ
Figure 2-32: Rotating an arbitrary vector about the Euler axis
The first term can be expressed in the Euler axis unity vector e and the vector v (see figure 2-32): OA = e (e · va ) The second term can be defined as:
AB = AD cos θ = [va − OA] cos θ
= [va − e (e · va )] cos θ
(2-68)
Finally, third term is expressed as:
BC = [va × e] sin θ where [va × e] denotes the direction of BC with magnitude equal to the radius of the rotation circle while sin θ will scale the magnitude to the correct value. Combining the relations will lead to:
v2 = e (e · va ) + [va − e (e · va )] cos θ + [va × e] sin θ = va cos θ + e (e · va ) (1 − cos θ) + [va × e] sin θ
(2-69)
2-4 Quaternions, a brief introduction
75
Elaborating equation (2-69): b a a vx vx vx e1 vyb = vya cos θ + e2 e1 e2 e3 vya (1 − cos θ) e3 vzb vza vza a vx e1 a vy × e2 sin θ + e3 vza a e1 e1 vxa + e2 vya + c3 vza (1 − cos θ) vx cos θ = vya cos θ + e2 e1 vxa + e2 vya + c3 vza (1 − cos θ) vza cos θ e3 e1 vxa + e2 vya + c3 vza (1 − cos θ) e3 vya − e2 vza sin θ + (e1 vza − e3 vxa ) sin θ e2 vxa − e1 vya sin θ
One can group the terms such that the relation can be characterized by a transformation matrix: vb = Tba va If this relation holds for an arbitrary vector v, then it will also hold for the unity axes of any reference frame. Therefore one can also write: Fb = Tba Fa After grouping the transformation becomes: e3 sin θ+ −e3 sin θ+ cos θ+ 2 e1 (1 − cos θ) e1 e2 (1 − cos θ) e1 e3 (1 − cos θ) −e3 sin θ+ cos θ+ e1 sin θ+ Tba = 2 e1 e2 (1 − cos θ) e2 (1 − cos θ) e2 e3 (1 − cos θ) e2 sin θ+ −e1 sin θ+ cos θ+ e1 e3 (1 − cos θ) e2 e3 (1 − cos θ) e23 (1 − cos θ)
(2-70)
Now that we have defined the transformation matrix between reference Fa and Fb in terms of the Euler axis e and the rotation angle θ it is time to introduce the Euler parameters, also called Quaternions. Euler parameter set consists of four parameters: θ (2-71) 2 θ q2 = e2 sin (2-72) 2 θ q3 = e3 sin (2-73) 2 θ q4 = cos (2-74) 2 The set is usually described as q q4 with the vector q defined as [q1 , q2 , q3 ]T . The Euler parameters are not independent of each other. Taking the square of each quaternion and q1 = e1 sin
Flight Dynamics
76
Reference Frames
summing all four yields: θ θ q12 + q22 + q32 + q42 = e21 + e22 + e23 sin2 + cos2 2 2 θ θ = sin2 + cos2 2 2 =1
(2-75)
where use is made that the length of the rotation axis is unity, i.e. e21 + e22 + e23 = 1. A short-hand notation of the condition is: qT q + q42 = 1
(2-76)
The transition matrix in equation (2-70) can be described in terms of the Euler parameters. When one uses the following trigonometric relations: θ θ cos 2 2 θ 2 θ cos θ= cos − sin2 2 2 θ = 2 cos2 − 1 2 θ = 1 − 2 sin2 2 sin θ= 2 sin
one will come to the following transition 1 − 2 q22 + q32 Tba = 2 (q1 q2 − q3 q4 ) 2 (q1 q3 + q2 q4 )
(2-77) (2-78) (2-79) (2-80)
matrix: 2 (q1 q2 + q3 q4 ) 2 (q1 q3 − q2 q4 ) 2 (q2 q3 + q1 q4 ) 1 − 2 q12 + q32 2 (q2 q3 − q1 q4 ) 1 − 2 q12 + q22
(2-81)
To clarify this result two examples are given. The relation between matrix in (2-81) and the one in equation (2-70) will be determined for the first and second entry one the first row using the given relations7 . Substituting the expressions for the quaternions into the first entry yields: 2 2 2 2 θ 2 2 θ 1 − 2 q2 + q3 = 1 − 2 e2 sin + e3 sin 2 2 2 θ 2 2 θ = 1 − 2 1 − cos − e1 sin 2 2 2 θ 2 2 2 θ 2 = 2 cos − 1 + e1 + 2e1 sin − e1 2 2 = cos θ + e21 (1 − cos θ)
where use has been made in the first line of the relation,
7
θ θ e21 + e22 + e23 sin2 + cos2 = 1 2 2
The other derivations are similar.
(2-82)
2-4 Quaternions, a brief introduction
77
and where e21 − e21 has been added in the third line. Substituting the expressions for the quaternions into the second entry of the first row in (2-81) yields: θ θ θ θ 2 (q1 q2 + q3 q4 ) = 2e1 sin e2 sin + 2e3 sin cos 2 2 2 2 θ = e3 sin θ + e1 e2 2 sin2 − e1 e2 + e1 e2 2 = e3 sin θ + e1 e2 (1 − cos θ)
2-4-3
(2-83)
Quaternion propagation through time
What is the new set of quaternions for two reference frames when those frame are rotating with respect to another? We could determine the set of quaternions at each time step by determining the transformation matrix using Euler angle and then use that matrix to determine the values of the quaternions. However, we are using quaternions such that we do not have to use Euler angles! Another method of deriving the new set of quaternions is to integrate its derivatives. To derive the expression for the derivative a quaternion, we will have to enter the domain of vector algebra. Up till now we have defined vectors as components based parameters, i.e.: a vx a v = vya vza
where vxa , vya , and vza are the components of vector v expressed along the X, Y, and Z-axis of reference frame Fa respectively. Such a notation is used in the field of linear algebra. Linear algebra is a sub-domain of vector algebra. The previous relation can also be expressed as: va = vx aax + vy aay + vz aaz where ax , ay , and az are the unity vectors defining the orientation of the Xa , Ya , and Za -axis respectively. Expressing these vector in reference frame Fa will lead to:
1 0 0 vx va = vx 0 + vy 1 + vz 0 = vy 0 0 1 vz which is thus equal to the vector expressed as before in case of linear algebra. In the field of vector algebra the unity vectors defining the reference frame axes are denoted as simply ax , ay , az or more commonly a1 , a2 , a3 where the letter denotes which frame the vectors span and in which frame the vectors are expressed (thus a1 is used instead of aa1 ). Another important aspect of vector algebra is the fact that vector expressions are combined in one expression: a v1b = Tba v1a v1b v1 b a b v2 = Tba v2 v2 = Tba v2a b a v3a v3 = Tba v3 v3b Flight Dynamics
78
Reference Frames
Note that the entries in the column vectors are vectors themselves and not components (scalars) of one vector! Using this notation one can write the angular velocity vector Ωbba in terms of the unity vectors of reference frame Fb : Ωbba = Ω1 b1 + Ω2 b2 + Ω3 b3
(2-84)
where (Ω1 , Ω2 , Ω3 ) are the vector components of Ωbba . Now, the derivation of the quaternion derivatives starts with the following relation:
a1 b1 a2 = Ttba b2 a3 b3 Taking the derivative with respect to reference frame Fa will lead to the expression of: a a a1 b1 o d d Tt b2 a2 = o = dt dt ba a3 b3 o a a
Using the chain rule one obtains:
b b1 o b1 d Ttba d o = b2 + Ttba b2 dt dt o b3 b3 a a
(2-85)
As for the case of linear algebra the vector differentiation rules also applies for vector algebra, i.e.: b b1 b1 b1 d d b2 + Ωbba × b2 b2 = dt dt b3 a b3 b b3
where Ωba is a vector based angular velocity vector (as defined in equation (2-84)). The first term in the previous relation is equal to zero since the vectors b1 , b2 , b3 span frame Fb , therefore fixating them to Fb . Collecting terms, relation (2-85) can be written as: d Ttba 0= dt
b1 b1 b2 + Ttba Ωbba × b2 b3 b3 a
(2-86)
The cross product can be written in terms of the components of the angular velocity vector.
2-4 Quaternions, a brief introduction
79
The exact formulation of the cross-product term is:
b1 b1 Ωbba × b2 = (Ω1 b1 + Ω2 b2 + Ω3 b3 ) × b2 b3 b3 Ω1 b1 × b1 + Ω2 b2 × b1 + Ω3 b3 × b1 = Ω1 b1 × b2 + Ω2 b2 × b2 + Ω3 b3 × b2 Ω1 b1 × b3 + Ω2 b2 × b3 + Ω3 b3 × b3 Ω2 b2 × b1 + Ω3 b3 × b1 = Ω1 b1 × b2 + Ω3 b3 × b2 Ω1 b1 × b3 + Ω2 b2 × b3 −Ω2 b3 + Ω3 b2 = Ω1 b3 − Ω3 b1 −Ω1 b2 + Ω2 b1
(2-87)
where the following relations where used: b1 = b2 × b3 b2 = b3 × b1 b3 = b1 × b2 Relation (2-87) can be rewritten in terms of the skew-symmetric matrix MΩ :
b1 0 −Ω3 Ω2 b1 b1 0 −Ω1 b2 = −MΩb b2 Ωbba × b2 = − Ω3 ba b3 −Ω2 Ω1 0 b3 b3 Using the skew-symmetric matrix relation (2-86) can be written as: b1 0 = T˙ tba − Ttba MΩb b2 ba b3 Since the vectors b1 , b2 , b3 are nonzero the term between brackets must be zero, i.e.: T˙ tba − Ttba MΩb = 0 ba
Taking the transpose of this relation and using the property of the angular velocity vector MtΩb = −MΩb one can rewrite the relation as: ba
ba
T˙ ba + MΩb Tba = 0 ba
(2-88)
Equation (2-89) is called the kinematic differential equation for the transition matrix Tab . It is valid for any transition matrix provided that the correct angular velocity skew matrix is taken. The differential equation for each element of an arbitrary T (with appropriate M) can Flight Dynamics
80
Reference Frames
be written as: T˙ 11 = Ω3 T21 − Ω2 T31 T˙ 12 = Ω3 T22 − Ω2 T32 T˙ 13 = Ω3 T23 − Ω2 T33 T˙ 21 = Ω1 T31 − Ω3 T11 T˙ 22 = Ω1 T32 − Ω3 T12 T˙ 23 = Ω1 T33 − Ω3 T13 T˙ 31 = Ω2 T11 − Ω1 T21 T˙ 32 = Ω2 T12 − Ω1 T22 T˙ 33 = Ω2 T13 − Ω1 T23 These terms can be rewritten to constitute relations which express the angular velocity components in terms of transition matrix entries: Ω1 = T˙ 21 T31 + T˙ 22 T32 + T˙ 23 T33 Ω2 = T˙ 31 T11 + T˙ 32 T12 + T˙ 33 T13
(2-89)
Ω3 = T˙ 11 T21 + T˙ 12 T22 + T˙ 13 T23
(2-91)
(2-90)
By substituting the transformation matrix entries in relation (2-81) into the previous expression one obtains: Ω1 = 2 (q˙1 q4 + q˙2 q3 − q˙3 q2 − q˙4 q1 ) Ω2 = 2 (q˙2 q4 + q˙3 q1 − q˙1 q3 − q˙4 q2 ) (2-92) Ω3 = 2 (q˙3 q4 + q˙1 q2 − q˙2 q1 − q˙4 q3 ) Differentiating equation (2-76) yields the forth equation: 0 = 2 (q˙1 q1 + q˙2 q2 + q˙3 q3 + q˙4 q4 ) These four relations can be q˙1 q˙2 q˙3 q˙4
rewritten in matrix form as: 0 Ω3 −Ω2 1 −Ω3 0 Ω1 = 2 Ω2 −Ω1 0 −Ω1 −Ω2 −Ω3
Ω1 q1 q2 Ω2 Ω3 q3 0 q4
(2-93)
(2-94)
Relation (2-94) can be used to determine the derivatives of the quaternions provide that one know the angular velocity components and the previous set of quaternions. By numeric integration one can determine the quaternion set for the next time step. In practice one would only need to determine the set of quaternions at t = 0. This set is usually derived using the relations between the quaternions and the direction cosine or transformation matrix, i.e. using equation (2-81). The initial transformation matrix can be determined using Euler angles. As for the angular velocity components, one can typically measure angular acceleration onboard the vehicle. Integrating the angular acceleration will yield the angular velocity.
2-4 Quaternions, a brief introduction
81
The conclusion is that there is no longer any need for Euler angles (except at t = 0). The complete equations of motion could be derived using quaternions. Since quaternions have no physical interpretation, the Euler angles will still be presented to the pilot. As for the singularity encountered with Euler angles, one simply creates a small offset when the orientation of the vehicle is close to the singularity which avoids any numerical problems. This book still uses Euler angles instead of quaternions because they have physical meaning! The aspects of flight dynamics can be better explained when the reader can make the mental coupling between expressions in Euler angles and the physical world.
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82
Reference Frames
Chapter 3 Derivation of the Equations of Motion
To describe the dynamics of a vehicle we need a model of that vehicle. The model we will use in subsequent chapters is governed by differential equations also known as the ”Equations of Motion” (EOM). These EOM will be derived in this chapter. During the derivation the effects of rotating masses is not taken into account. It is assumed that the body is rigid. The effects will be appended to the EOM in section 3-6. The chapter will begin with a review of Newton’s law in section 3-1. Two fundamental equations, which are the base on which the EOM are build, are derived in the same section. The four components in these equations will be investigated in section 3-2, 3-3, and 3-4. In section 3-2 and 3-3, an explicit expression will be derived for the translational acceleration and for the rotational acceleration respectively. Thereafter, in section 3-4, de external forces and moments are identified. After this section all necessary components for constructing the EOM are available. The EOM are constructed in section 3-5. A total of 12 equations in 3 sets of equations constitute the EOM: 3 force equations, 3 moment equations, and 6 kinematic relations. Having defined the EOM in the most general manner, simplifications are applied in section 3-7. At the end of section 3-7 the EOM for flat, non-rotating Earth are derived. This section of EOM will be used in subsequent chapter to explain aircraft dynamics. A simple set of EOM is used to ensure a clear explanation.
3-1
Newton’s laws of physics
The equations of motion are based on Newton’s laws and are stated as follows (see [112] and [153]): Newton’s 1st law: A body of constant mass is at rest or moves at a constant velocity, unless a force acts on the body. Flight Dynamics
84
Derivation of the Equations of Motion
Newton’s 2nd law: If a force acts on a body, the force is equal to the time derivative of the momentum, i.e. the product of mass and velocity of the body. Newton’s 3rd law: If two bodies at rest, or moving at constant velocity, exert forces upon one another, the force on the first body is equal in magnitude but opposite in direction to the force of the second body. Newton’s laws are only valid when the motion of a body is expressed in an inertial reference frame, i.e. a frame with absolute translational velocity and absolute angular velocity equal to zero, or when it is expressed in a Galilean frame. A Galilean frame is one which is in rectilinear translation with respect to an inertial reference frame. This means that the angular velocity of such a frame is equal to zero and the translational velocity of the origin is constant. Normally, a reference frame based on stellar references is chosen as inertial reference frame. In this book, however, it is assumed that the absolute translational velocity of the Earth centered inertial reference frame FI (see section 2-1-1) is negligible such that FI can be seen as an inertial reference frame. The equations of motion will be derived with respect to the inertial reference frame FI , but as explained in the previous chapter they can be expressed in any reference frame. In this section we will only discuss derivatives with respect the inertial reference frame. We will therefore drop the derivative notation and we will write da da instead of dt dt I
Only once will we use another reference frame (equation (3-3)). We will then use the official notation.
Translational acceleration Consider a moving body. This body can be seen as a collection of point masses. If a point mass P with mass dm moves with time varying inertial velocity VP under the influence of a force dFP , Newton’s second law is expressed by:
dFP =
d (VP dm) dt
(3-1)
The motion of the complete collection of point masses is then characterized by the forces acting on each point mass and subsequent accelerations of each point mass. When the body is rigid, the motions of the masses in the body are coherent, i.e. the body will not deform. In this book it is assumed that the vehicle under consideration is indeed a rigid body. Since the accelerations of the point masses in a rigid body are coherent, one can state, using Newton’s third law, that the internal forces the masses exert on each other within the body cancel. This means that the motion of such a body is only influenced by external forces. The summation of the external forces exerted on the body is equal to the integration of the momentum of
3-1 Newton’s laws of physics
85
P r G
rP rG
ZI
YI
A
XI
Figure 3-1: Vector definition for point mass P in a body.
each point mass in the body, so: X
dFextP
P
Z d = VP dm dt
(3-2)
m
We can write the velocity vector of point mass P can be expressed as (see figure 3-1): drP drG dr dr = + −→ VP = VG + dt dt dt dt
(3-3)
So, equation (3-2) can be written as: X
dFextP =
P
d dt
Z
VG dm +
m
The integral can be spilt up into two parts: Z Z dr VG dm + dm dt m
dr dm dt
(3-4)
(3-5)
m
Using the relation for continuous, differential functions, Z Z d d (. . .) = (. . .) dt dt the second part can be rewritten as: Z Z dr d dm = r dm = 0 dt dt m
(3-6)
m
Flight Dynamics
86
Derivation of the Equations of Motion dFP dm
dMG,P
r
G
Figure 3-2: Moment about center of mass due to force on point mass
since the definition of the center of mass is that the integral of the distance between mass points and the center of mass is zero. The first part of equation (3-5) can be written as, Z Z VG dm = VG dm = VG m (3-7) m
m
where m is the total mass of the body. So equation (3-2) can be rewritten as: X
dFextP =
P
d (VG m) dt
(3-8)
When the rigid body is assumed to have a constant mass, the previous relation can be rewritten as: dVG = m AG (3-9) dt which states that the translational acceleration of the center of mass with respect to the inertial space AI,G times the total mass m is equal to the resultant external force Fext . Fext = m
Angular acceleration Next to a translational acceleration, a body can also have an angular acceleration. From equation (3-1) (Newton’s second law) follows that the moment of a force about a reference point is equal to the cross product of the force vector and the relative distance vector (see figure 3-2): d (VP dm) (3-10) dt The point G is chosen in the center of mass in order to simplify the following derivations. Equation (3-10) can be rewritten in terms of the angular momentum about G which is defined for a mass element in point P as: dMG,P = r × dFP = r ×
dBG,P = r × (VP dm)
(3-11)
3-1 Newton’s laws of physics
87
Taking the time derivative of the angular momentum with respect to the inertial reference frame yields: d (dBG,P ) d (r × (VP dm)) = dt dt d (VP dm) dr = × (VP dm) + r × dt dt
(3-12) (3-13)
The vector r can be written as (see figure 3-1): r = rP − rG where A is the center of the spherical Earth and by definition the origin of the inertial reference frame. Substituting this relation in equation 3-13 will lead to: d (dBG,P ) = dt
drP drG − dt dt
× VP dm
= (VP − VG ) × VP dm = −VG × VP dm
d (VP dm) dt d (VP dm) +r × dt d (VP dm) +r × dt +r ×
(3-14)
Combining relation (3-14) with equation (3-10) leads to: dMG,P =
d (dBG,P ) + VG × VP dm dt
(3-15)
When the body is assumed to be rigid the forces between the point masses in the body cancel. So, only the external applied forces will affect the motion of the body, in this case the angular motion. The total external moment is equal to the integration of all external moments on the body, i.e.: Z
dMG,P =
m
Z
m
d (dBG,P ) + VG × VP dm dt
(3-16)
Which is equal to:
MG,ext
Z Z d = dBG,P + VG × VP dm dt m m Z dBG = + VG × VP dm dt
(3-17)
m
Since VG is independent of mass element P the second term on the right hand side in equation Flight Dynamics
88
Derivation of the Equations of Motion
(3-17) can be written as: Z Z VG × VP dm= VG × VI,P dm m
m
= VG × = VG ×
Z
m
dr VG + dt
Z
m
dm
Z d VG dm + rdm dt
(3-18)
m
Again, for the center of mass it holds that the integral of the distance between mass points in the body and the center of mass is zero: Z r dm = 0 (3-19) m
So: Z
m
Z d VG × VP dm= VG × VG dm + VG × r dm dt m m Z = VG × VG dm Z
m
=0
(3-20)
Using this result, we state that the total external moment about the center of mass is equal to the time derivative of the angular momentum of the body about the center of mass with respect tot the inertial reference frame: MG,ext =
dBG dt
(3-21)
The relation given by equation (3-21) can be reformulated by rewriting the term for the angular momentum BG . The definition of the angular momentum about point G is: Z BG = r × VP dm (3-22) m
The velocity vector of the mass dm in point P can be written as: dr VP = VG + dt I dr = VG + + ΩbI × r dt
(3-23)
b
Since the body is rigid the distance between the point of reference G and the location P of the mass does not vary over time the second term on the right hand side is equal to zero. This means we have: VP = VG + ΩbI × r
(3-24)
3-1 Newton’s laws of physics
89
Substituting this result into equation (3-22) gives: Z BG = r × (VG + ΩbI × r) dm m
=
Z
m
r × VG dm +
Z
m
r × (ΩbI × r) dm
(3-25)
Since the point of reference G is chosen in the center of mass and because VG is independent of mass element P , the first term on the right hand side of the previous equation is equal to zero: Z Z r × VG dm = − VG × r dm (3-26) m
m
= −VG ×
Z
r dm
(3-27)
m
=0
(3-28)
Thus the following relation remains: BG =
Z
m
r × (ΩbI × r) dm
(3-29)
With the cross product rule: a × (b × c) = b (a · c) − c (a · b)
(3-30)
equation (3-29) can be rewritten as: Z BG = ΩbI (r · r) − r (r · ΩbI ) dm m
Z = m Z = m Z = m
Z = m
= IG ΩbI
ΩbIX xr ΩbIY (xr )2 + (yr )2 + (zr )2 − yr (xr ΩbIX + yr ΩbIY + zr ΩbIZ ) dm ΩbIZ zr ΩbIX (yr )2 + (zr )2 − ΩbIY xr yr − ΩbIZ xr zr 2 2 ΩbIY (xr ) + (zr ) − ΩbIX xr yr − ΩbIZ yr zr dm ΩbIZ (xr )2 + (yr )2 − ΩbIX xr zr − ΩbIY yr zr (yr )2 + (zr )2 −xr yr −xr zr ΩbIX ΩbIY dm −xr yr (xr )2 + (zr )2 −yr zr 2 2 ΩbIZ −xr zr −yr zr (xr ) + (yr ) (yr )2 + (zr )2 −xr yr −xr zr ΩbIX dm ΩbIY −xr yr (xr )2 + (zr )2 −yr zr 2 2 ΩbIZ −xr zr −yr zr (xr ) + (yr )
(3-31)
Flight Dynamics
90
Derivation of the Equations of Motion
Where IG is the matrix of inertia of the body about point G. A general definition of the matrix of inertia is: Ixx −Jxy −Jxz I = −Jxy Iyy −Jyz (3-32) −Jxz −Jyz Izz with the moments of inertia defined as:
Ixx =
Z
m
Iyy =
Z
m
Izz =
Z
m
y 2 + z 2 dm
(3-33)
x2 + z 2 dm
(3-34)
x2 + y 2 dm
(3-35)
and the products of inertia as: Jyz
=
Z
yz dm
(3-36)
xz dm
(3-37)
xy dm
(3-38)
m
Jxz =
Z
m
Jxy =
Z
m
This means that equation (3-21) can be rewritten as: dBG d (IG ΩbI ) = (3-39) dt dt which states that the resultant external moment (MG,ext ) about point G is equal to the derivative of the angular velocity (ΩbI ) of the body times the matrix of inertia about point G, i.e the derivative of body angular momentum about point G. MG,ext =
Formal notation In the previous paragraphs, the formal notation has been dropped to make the text clearer. We will now give the formal notation for the two most important relations: G Fext = m dV = m AG dt I d(IG ΩbI ) G MG,ext = dB = dt dt I
I
Equations of Motion derivation The force equation (3-9) and the moment equation (3-39) form the basis for the derivation of the equations of motion. There are three main parts to be considered: 1. Derivation of body translational acceleration
3-2 Derivation of body translational acceleration
91
2. Derivation of body angular momentum 3. Resultant external force and moment on the body In the next three sections these parts will be discussed individually.
3-2
Derivation of body translational acceleration
The goal in this section is to find the expression for the inertial translational acceleration of a body in case of a spherical, rotating Earth. The body is again assumed to be a rigid so that all mass points have the same translational acceleration. Our task is now to find the inertial translational acceleration of the center of mass, i.e. AG . The derivation starts with the notion that the inertial acceleration is equal to the derivative of the inertial velocity which in turn is equal to the derivative of the inertial position: dVG d drG AG = = (3-40) dt I dt dt I I Using the rule for vector differentiation (see section 2-3) we can also write dVG dVG AG = = + ΩOI × VG dt I dt O
(3-41)
The inertial velocity of the body VI,G can be expressed as the derivative of the inertial position of the body center of mass: drG VG = (3-42) dt I Which is equivalent to:
drG VG = + ΩEI × rG dt E
(3-43)
The position vector of the center of mass rG can be composed of two other vectors: the position vector for the origin of FO , AO, plus the position vector for the center of mass seen from the origin of FO , OG. Using this in the previous equation: dAO dOG VG = + + ΩEI × rG (3-44) dt E dt E
The position vector AO expressed in the normal Earth fixed reference frame is always directed E along the ZE -axis and stays constant through time. The derivative dOG is by definition equal dt to the kinematic velocity Vk,G . This leads to following expression for VG : VG = Vk,G + ΩEI × rG
(3-45) Flight Dynamics
92
Derivation of the Equations of Motion
Substituting this relation into equation (3-41) will lead to: AG
dVG = + ΩOI × VG dt O d (Vk,G + ΩEI × rG ) = + ΩOI × (Vk,G + ΩEI × rG ) dt O dVk,G dΩEI drG = + × rG + ΩEI × + dt O dt O dt O ΩOI × Vk,G + ΩOI × (ΩEI × rG )
(3-46)
Relation (3-46) is the general expression of the inertial translational acceleration of a body, it is valid in any reference frame. An alternative equation can be defined to be able to indicate what consequences particular assumptions have on the formulation of the inertial translational acceleration. The alternative equation is: AG = Ar + AS + AR
(3-47)
Where Ar is the relative acceleration of the aircraft with respect to the vehicle carried normal Earth reference frame. It is defined as: dVk,G Ar = (3-48) dt O
AS is the acceleration due to the sphericity of the Earth (note that ΩOI = ΩOE + ΩEI , used to split the term ΩOI × Vk,G in equation 3-46). It is defined as: AS = ΩOE × Vk,G
(3-49)
Finally, AR is the acceleration due to the rotation of the Earth. It is defined as: dΩEI drG AR = × rG + ΩEI × + ΩEI × Vk,G + ΩOI × (ΩEI × rG ) dt O dt O
(3-50)
For now, the inertial translational acceleration of the body will be expressed in the vehicle carried normal Earth reference frame FO . By selecting this reference frame the notation of the final expression for the inertial translation acceleration will be simpler then for the case of any other previously defined reference frame. Each term in equation (3-46), expressed in FO , will be expanded before grouping the results. During the expansion of each term two relations will be used. They are given by: ˙ G= Lt
VN Rt + h
˙ G − Ωt = − ωt ˙ G = Lg
(3-51)
VE + Ωt (Rt + h) cos LtG
(3-52)
Starting with the first term, the derivative of the kinematic velocity expressed in FO . The kinematic velocity expressed in FO has the components: velocity due local North VN , velocity
3-2 Derivation of body translational acceleration
93
due local East VE , and velocity in local vertical direction −VZ (see section 2-1-7). The derivative then simply becomes:
˙ V dVk,G O ˙N = VE dt O −V˙ Z
(3-53)
Moving on to the second term on the right in equation (3-46):
I Ω d T dΩEI O OI EI × rO = × rO G G dt O dt O cos LtG 0 d × Ωt 0 = 0 dt − (Rt + h) − sin LtG − sin LtG 0 ˙ G × 0 0 = Ωt Lt − cos LtG − (Rt + h) 0 ˙ G − (Rt + h) sin LtG = Ωt Lt 0 = −Ωt VN sin LtG
(3-54)
The third term becomes:
ΩO EI
O cos LtG 0 drG × d 0 × = Ωt 0 dt O dt − (Rt + h) − sin LtG 0 cos LtG × 0 = Ωt 0 − sin LtG −h˙ 0 = Ωt h˙ cos LtG 0 0 = Ωt VZ cos LtG 0
(3-55)
Flight Dynamics
94
Derivation of the Equations of Motion
The fourth term becomes: ΩO OI
−ωt ˙ G cos LtG VN ˙ G × VE × VkO = −Lt −VZ ωt ˙ G sin LtG VE VN (Rt +h) + Ωt cos LtG − (RVt N = VE × +h) E −VZ − (RVt +h) tan LtG − Ωt sin LtG VN VZ +VE2 tan LtG + Ωt VE sin LtG Rt +h N tan LtG = VE VZ −VREtV+h + Ωt VZ cos LtG − Ωt VN sin LtG VE2 +VN2 Rt +h
(3-56)
+ Ωt VE cos LtG
And the fifth and final term becomes: O O ΩO OI × ΩEI × rG O O O O O = ΩO EI ΩOI · rG − rG ΩOI · ΩEI T −ωt ˙ G cos LtG 0 Ωt cos LtG ˙ 0 0 = − Lt G −Ωt sin LtG −(Rt + h) ωt ˙ G sin LtG T −ωt ˙ G cos LtG Ωt cos LtG 0 ˙ − 0 0 −LtG − (Rt + h) −Ωt sin LtG ωt ˙ G sin LtG
Ωt cos LtG (VE tan LtG + Ωt (Rt + h) sin LtG ) = 0 −Ωt sin LtG 0 VE − 0 Ωt cos LtG + Ω2t cos2 LtG Rt + h − (Rt + h) VE + Ωt tan LtG sin LtG + Ω2t sin2 LtG Rt + h VE Ωt sin LtG + Ω2t (Rt + h) sin LtG cos LtG = 0 2 2 VE Ωt cos LtG + Ωt (Rt + h) cos LtG
(3-57)
Now that all the terms from equation (3-46) are known, they can be combined to form the complete expression. After grouping the terms the relation for the translational acceleration of a body expressed in the vehicle carried normal Earth reference frame is given by: V V +V 2 tan LtG V˙ N + N Z Rt E+h + 2Ωt VE sin LtG + Ω2t (Rt + h) sin LtG cos LtG VE VZ −VE VN tan LtG ˙ AO = V + + 2Ω V cos Lt − 2Ω V sin Lt (3-58) E t Z G t N G G Rt +h 2 2 VE +VN −V˙ Z + R + 2Ωt VE cos LtG + Ω2t (Rt + h) cos2 LtG t +h
3-3 Derivation of body angular momentum derivative
95
The inertial translation acceleration is built up by three terms, see equation (3-47). Expressed in the vehicle carried normal Earth reference frame they are: ˙ O VN dVk AO = V˙ E r = dt O −V˙ Z
(3-59)
VN VZ + VE2 tan LtG 1 O O VE VZ − VE VN tan LtG AO S = ΩOE × Vk = Rt + h VE2 + VN2 2VE sin LtG + Ωt (Rt + h) sin LtG cos LtG 2VZ cos LtG − 2VN sin LtG AO R = Ωt 2VE cos LtG + Ωt (Rt +
3-3
h) cos2 Lt
(3-60)
(3-61)
G
Derivation of body angular momentum derivative
The goal in this section is to find an expression for the derivative of the body angular momentum. The body angular momentum is equal to the multiplication of the matrix of inertia IG and the angular velocity of the body-fixed reference system with respect to the inertial reference system ΩbI . It is common practice to define the matrix of inertia using the bodyfixed reference system, i.e. IbG . In most cases, a specific orientation of the Xb and Zb -axis can be chosen such that some components in the matrix of inertia become zero. If it is assumed that the vehicle mass does not change over time and that the body is rigid, the derivative of the matrix of inertia with respect to the body-fixed reference frame will be equal to zero. This is a very useful aspect which simplifies the equations of motion. The derivation of the expression for the derivative of the body angular momentum starts with the relation given by equation (3-62). dBG MG,ext = (3-62) dt I To use the properties of the matrix of inertia, equation (3-62) can be written as: dBG dBG = + ΩbI × BbG dt I dt b dBG = + ΩbE × BbG + ΩEI × BbG dt b
(3-63)
BG = IG ΩbI = IG ΩbE + IG ΩEI
(3-64)
When using the following relation,
Flight Dynamics
96
Derivation of the Equations of Motion
equation (3-63) can be written as: dBG d (IG ΩbE ) = + ΩbE × (IG ΩbE ) + ΩEI × (IG ΩbE ) + dt I dt b d (IG ΩEI ) + ΩbE × (IG ΩEI ) + ΩEI × (IG ΩEI ) dt b dΩbE b b =IG + Ω × I Ω + Ω × I Ω G EI G bE bE bE + dt b dΩEI b b + Ω × I Ω + Ω × I Ω IG G EI G bE EI EI dt
(3-65)
b
Equation (3-65) is a general expression for the derivative of the angular momentum about point G. The expression is valid in any reference frame. Equation (3-65) can be divided into three groups, the first group containing all terms consisting of only ΩbbE and Ω˙ bbE terms, the second group only consisting of ΩbEI and Ω˙ bEI terms, and the third group consisting of multiplications of terms from ΩbbE and ΩbEI . Equation (3-65) can be written as: dBG = ∆1 MG + ∆2 MG + ∆3 MG (3-66) dt I With:
dΩbE b b ∆1 MG = IG + Ω × I Ω G bE bE dt b ∆2 MG = ΩbEI × IG ΩbEI dΩEI b b b b ∆3 MG = IG + Ω × I Ω + Ω × I Ω G EI G bE EI bE dt b
(3-67) (3-68) (3-69)
Since the matrix of inertia is commonly defined in the body-fixed reference frame, the derivative of the body angular momentum will also be expressed in the body-fixed reference frame. Before expanding equation (3-65), several subcomponents are formulated explicitly. The first is the matrix of inertia expressed in the body-fixed reference system: Ixx −Jxy −Jxz A −F −E IbG = −Jxy Iyy −Jyz = −F B −D (3-70) −Jxz −Jyz Izz −E −D C The second component is the angular velocity of the body-fixed reference frame with respect to the normal Earth-fixed reference frame expressed in the body-fixed reference frame: p ΩbbE = q = ΩbbO + TbO ΩO OE r ˙ ϕ˙ −LgG cos LtG 1 0 − sin θ ˙ G = 0 cos ϕ cos θ sin ϕ θ˙ + TbO (3-71) −Lt ˙ ˙ 0 − sin ϕ cos θ cos ϕ ψ LgG sin LtG
3-3 Derivation of body angular momentum derivative
97
The third and last component is the angular velocity of the normal Earth-fixed reference frame with respect to the inertial reference frame expressed in the body-fixed reference frame:
ΩbEI
pbt pt = qtb = qt = TbO ΩO EI rtb rt cos θ cos ψ cos θ sin ψ − sin θ sin ϕ sin θ sin ψ Ωt cos LtG sin ϕ sin θ cos ψ sin ϕ cos θ = − cos ϕ sin ψ + cos ϕ cos ψ 0 cos ϕ sin θ cos ψ −Ωt sin LtG cos ϕ sin θ sin ψ cos ϕ cos θ + sin ϕ sin ψ − sin ϕ cos ψ cos LtG cos θ cos ψ + sin LtG sin θ sin ϕ sin θ cos ψ − sin LtG sin ϕ cos θ cos LtG − cos ϕ sin ψ = Ωt (3-72) cos ϕ sin θ cos ψ cos LtG − sin LtG cos ϕ cos θ + sin ϕ sin ψ
Each term in equation (3-65) will be expanded in the subsequent text. Thereafter the terms will be grouped where possible. The first term can be written as:
IbG
dΩbbE dt
b
A −F −E p˙ = −F B −D q˙ −E −D C r˙ Ap˙ − F q˙ − E r˙ = B q˙ − F p˙ − Dr˙ C r˙ − E p˙ − Dq˙
(3-73)
The second term can be written as:
ΩbbE
p A −F −E p × IbG ΩbbE = q × −F B −D q r −E −D C r p Ap − F q − Er = q × Bq − F p − Dr r Cr − Ep − Dq (C − B) qr − Epq + F pr + D r 2 − q 2 = (A − C) pr − F qr + Dpq + E p2 − r 2 (B − A) pq − Dpr + Eqr + F q 2 − p2
(3-74)
Flight Dynamics
98
Derivation of the Equations of Motion
The third term can be written as: A −F −E p pt ΩbEI × IbG ΩbbE = qt × −F B −D q rt −E −D C r Ap − F q − Er pt = qt × Bq − F p − Dr Cr − Ep − Dq rt Crqt − Epqt − Bqrt + F prt + D (rrt − qqt ) = Aprt − F qrt − Crpt + Dqpt + E (ppt − rrt ) Bqpt − Drpt − Apqt + Erqt + F (qqt − ppt ) The fourth term can be written as: ! b dΩEI b b dΩEI b b b IG = IG + ΩEb × ΩEI dt b dt E b b dΩEI b b b + I Ω × Ω = IG G EI Eb dt E = IbG −ΩbbE × ΩbEI A −F −E p pt = −F B −D − q × qt −E −D C r rt A (rqt − qrt ) − F (prt − rpt ) − E (qpt − pqt ) = −F (rqt − qrt ) + B (prt − rpt) − D (qpt − pqt) −E (rqt − qrt ) − D (prt − rpt ) + C (qpt − pqt ) The fifth term is similar to the third term: p A −F −E pt ΩbbE × IbG ΩbEI = q × −F B −D qt r −E −D C rt Cqrt − Eqpt − Brqt + F rpt + D (rrt − qqt ) = Arpt − F rqt − Cprt + Dpqt + E (ppt − rrt ) Bpqt − Dprt − Aqpt + Eqrt + F (qqt − ppt ) And finally, the sixth term is equal to the second term: pt A −F −E pt ΩbEI × IbG ΩbEI = qt × −F B −D qt rt −E −D C rt (C − B) qt rt − Ept qt + F pt rt + D = (A − C) pt rt − F qt rt + Dpt qt + E (B − A) pt qt − Dpt rt + Eqt rt + F
(3-75)
(3-76)
(3-77)
rt2 − qt2 p2t − rt2 qt2 − p2t
(3-78)
These six terms can be grouped together to form the general expression for the derivative of angular momentum about point G expressed in the body-fixed reference frame. The grouping
3-4 External forces and moments
99
is done according to equations (3-66) through (3-69) (expressed in the body-fixed reference frame): dBG b = ∆1 MbG + ∆2 MbG + ∆3 MbG dt I
With:
(3-79)
Ap˙ + (C − B) qr − E (pq + r) ˙ + F (pr − q) ˙ + D r2 − q2 ∆1 MbG = B q˙ + (A − C) pr − F (qr + p) ˙ + D (pq − r) ˙ + E p2 − r 2 C r˙ + (B − A) pq − D (pr + q) ˙ + E (qr − p) ˙ + F q 2 − p2 (C − B) qt rt − Ept qt + F pt rt + D rt2 − qt2 ∆2 MbG = (A − C) pt rt − F qt rt + Dpt qt + E p2t − rt2 (B − A) pt qt − Dpt rt + Eqt rt + F qt2 − p2t (C − A − B) qrt + (C + A − B) rqt ∆3 MbG = (A + B − C) prt + (A − B − C) rpt (B + C − A) qpt + (B − C − A) pqt pt (F r − Eq) + D (rrt − qqt ) + 2 qt (Dp − F r) + E (ppt − rrt ) rt (Eq − Dp) + F (qqt − ppt )
3-4
(3-80)
(3-81)
(3-82)
External forces and moments
One could define two types of forces acting on an aircraft: 1. Gravity 2. Aerodynamic forces (includes thrust) These forces are by nature distributed forces. To make the equations of motion more easy to read, these distributed forces are ’replaced’ by point forces acting at particular points on the body. Summation of all point forces will result in the total resultant force. In general these forces will generate a moment about the point of reference, which on the aircraft is chosen at the center of gravity. This choice is made for two reasons. Firstly, the force due to gravity will not cause a moment since it acts at the center of gravity. Secondly, the center of gravity is also the origin of the body-fixed reference system such that no additional reference frames are needed. External forces will be discussed first , next to be followed by the external moments.
3-4-1
External forces
We first consider the gravity vector. Flight Dynamics
100
Derivation of the Equations of Motion
Gravity Gravity is composed of two parts: the gravitational pull towards the center of the Earth gr,G and the centripetal acceleration due to the spinning of the Earth AI,G (see figure 3-3). In mathematical terms the relation is expressed as: mgG = m (gr,G − AI,G )
(3-83)
However, the situation in figure 3-3 is not the one assumed in this book. The hypothesis of a spherical Earth will lead to an erroneous gravity vector if the centripetal force is taken into account. Because the gravitation vector gr,G points to the center of the Earth it will be perpendicular to the Earth surface when a spherical Earth is assumed. When the centripetal force is subtracted from the gravitation vector the resulting gravity vector gG is no longer perpendicular to the Earth surface. To keep the gravity vector perpendicular to the Earth surface, within the framework of the assumption of a spherical Earth, the centripetal acceleration is omitted, i.e.: (3-84) mgG = mgr,G Using the assumption of a spherical Earth, the gravity vector is expressed in the vehicle carried normal Earth reference frame FO as: 0 O O 0 gG = gr,G = (3-85) G mt (Rt +h)2
Where:
G : the constant of gravitation mt : Earth total mass Rt : mean Earth radius
6.664 10 - 11 SI 5.983 1024 kg 6.368 106 m
The gravity vector can be rewritten as a function of the gravity at zero altitude gr , 0: Gmt Rt2 gr,0 gr = 2 1 + Rht
gr,0 =
(3-86) (3-87)
When investigating equation (3-87) one can see that the gravity does not vary much with altitude. To have a difference of 1% in gravity one would have to travel 32 km in height: gr,2 − gr,0 1 = −0.01 = 2 − 1 gr,0 h2 1 + Rt 2 h2 1 1+ = Rt 0.99 ! r 1 h2 = Rt −1 0.99 ≈ 32 km
(3-88)
(3-89) (3-90) (3-91)
3-4 External forces and moments
101
Spherical Earth
Ellipsoid Earth G
G
−AI,G
gr,G
gr,G gG
−AI,G
gG
A
A
Ωt
Ωt
Figure 3-3: Earth gravity vector definition
This means that for normal aircraft the gravity vector magnitude can be assumed to be constant. For spaceflight, however, the distances traveled in height are much larger creating larger changes in gravity, and the variation of the magnitude of gravity can no longer be neglected. Another issue is the fact that the gravity experienced by a body traveling over the Earth surface changes with latitude. This effect is caused by the flattening of the Earth. A simple model for the gravity at zero altitude as a function of latitude was developed by Somigliana in 1929. The numerical version of that model is given by equation (3-92) (Ref. [30]). gr,0 = 9.8703 1.00335 + 0.00195 sin 2 LtG (3-92) where gr,0 is the magnitude of the gravity.
To determine the gravity force one has to multiply the gravity acceleration with the total mass of the body. This force is also known as the weight of the body, i.e. WG = mgG
(3-93)
Aerodynamic forces The flow of particles around a body creates pressure differences that cause the body to experience a force. For flight dynamics the considered particles are air particles, hence the forces are called aerodynamic forces. The overall resultant aerodynamic force which acts on the body can be composed by a summation of aerodynamic forces which act on each part of the aircraft. This includes the forces generated by the aerodynamic surfaces and by the engines, i.e. thrust. The thrust can be seen as an aerodynamic force since an engine creates disturbances in the airflow such that the aircraft is pulled forward. The resultant aerodynamic force is usually decomposed into either the body-fixed reference frame or the aerodynamic reference frame. The decomposition into the body-fixed reference frame is denoted as: b X b Faero = Y b (3-94) Zb Flight Dynamics
102
Derivation of the Equations of Motion
The decomposition into the aerodynamic reference frame Fa is denoted as: a X Faaero = Y a Za
(3-95)
The two decompositions of the resultant aerodynamic force are related as follows: a b F aerob = T ab Faero a X cos βa cos αa sin βa cos βa sin αa X Y b = − sin βa cos αa cos βa − sin βa sin αa Y a Zb − sin αa 0 cos αa Za
3-4-2
(3-96)
External moments
The external moments are usually defined about the center of gravity G. Of course the point where the external forces act on the body can differ from the center of gravity, and they will then cause a moment about the center of gravity. In this section the external forces are reviewed to find out whether they produce additional moments about the center of gravity. Gravity Since the gravity force mgG acts in the center of gravity, it will not cause a moment about the center of gravity. Aerodynamic forces The resultant aerodynamic force typically does not act in the center of gravity. It therefore causes a moment about the center of gravity. The resultant aerodynamic moment is defined in the body-fixed reference frame as: b L Mbaero,G = M b (3-97) Nb The resultant aerodynamic moment is defined in the aerodynamic reference frame as: a L Maaero,G = M a (3-98) a N Further details about the aerodynamic moment about the center of gravity will be given in chapter 4.
3-5
Composition of equations of motion
The components necessary to ‘build’ the equation of motion have been defined in the previous sections. The complete set of equations of motions consists of fifteen scalar equations: three
3-5 Composition of equations of motion
103
force equations, three moment equations, and nine kinematic relations1 . The equations of motion will be expressed in the body-fixed reference frame with the origin in the center of mass. Three assumptions are made:
• Constant gravity field, i.e. center of mass is equal to the center of gravity (= G) • Vehicle is a rigid body • Vehicle has constant mass
First the force equations will be derived (section 3-5-1), then the moment equations (in section 3-5-2). The linear and angular velocities used will be stated in the kinematics section, section 3-5-3. It is important to note that the effects of rotating masses on the rotational accelerations is not taken into account here, i.e. the moment equations are not complete. These effects will be appended to the equations of motion in section 3-6.
3-5-1
Force equations
The derivation of the force equations starts with the basic formula of Newton’s second law for rigid, constant mass body: mAbI,G = Fbext
(3-99)
Gathering the information obtained in previous sections, the force equation can be written as: b mAbr + mAbS + mAbR = mgG + Fbaero
(3-100)
Which is equal to: O O O b mTbO AO r + AS + AR = mTbO gG + Faero
(3-101)
1
In literature one often considers twelve scalar equations. In this book, however, three additional (rotational) kinematic relations are added to make the equations of motion easier to read. Flight Dynamics
104
Derivation of the Equations of Motion
with:
cos θ cos ψ
cos θ sin ψ
− sin θ
sin ϕ sin θ sin ψ sin ϕ sin θ cos ψ sin ϕ cos θ + cos ϕ cos ψ TbO = − cos ϕ sin ψ cos ϕ sin θ cos ψ cos ϕ sin θ sin ψ cos ϕ cos θ + sin ϕ sin ψ − sin ϕ cos ψ ˙ VN V˙ E AO r = −V˙ Z VN VZ + VE2 tan LtG 1 VE VZ − VE VN tan LtG AO S = Rt + h VE2 + VN2 2VE sin LtG + Ωt (Rt + h) sin LtG cos LtG AO 2VZ cos LtG − 2VN sin LtG R =Ωt
O gG =
Fbaero
0 0
gr,O
2VE cos LtG + Ωt (Rt + h) cos2 LtG
Xb = Y b Yb
The definition of the aerodynamic forces expressed in the body-fixed reference frame will be given in chapter 4.
3-5-2
Moment equations
The derivation of the moment equations starts with the modified form of the basic formula of Newton’s second law for rigid, constant mass body: dBG b = MbG,ext dt I
(3-102)
Gathering the information obtained in previous sections, the force equation can be written as: ∆1 MbG + ∆2 MbG + ∆3 MbG = Mbaero,G
(3-103)
3-5 Composition of equations of motion
105
With: Ap˙ + (C − B) qr − E (pq + r) ˙ + F (pr − q) ˙ + D r2 − q2 ˙ + D (pq − r) ˙ + E p2 − r 2 ∆1 MbG = B q˙ + (A − C) pr − F (qr + p) C r˙ + (B − A) pq − D (pr + q) ˙ + E (qr − p) ˙ + F q 2 − p2 (C − B) qt rt − Ept qt + F pt rt + D rt2 − qt2 ∆2 MbG = (A − C) pt rt − F qt rt + Dpt qt + E p2t − rt2 (B − A) pt qt − Dpt rt + Eqt rt + F qt2 − p2t (C − A − B) qrt + (C + A − B) rqt ∆3 MbG = (A + B − C) prt + (A − B − C) rpt (B + C − A) qpt + (B − C − A) pqt pt (F r − Eq) + D (rrt − qqt ) + 2 qt (Dp − F r) + E (ppt − rrt ) rt (Eq − Dp) + F (qqt − ppt ) b L Mbaero,G = M b Nb
3-5-3
Kinematic relations
The linear velocity scalars are related to the changes in latitude and longitude in the following manner: drG Vk,G = + ΩOE × rG (3-104) dt O
Expressed in the vehicle carried normal Earth reference frame, this relation can be written as: ˙ 0 −LgG cos LtG 0 VN ˙ × VE = 0 + 0 −∆Lt ˙ ˙ − (Rt + h) −VZ −h LgG sin LtG ˙ (Rt + h) ∆Lt ˙ = −LgG cos LtG (Rt + h) (3-105) ˙ −h
Where:
˙ G =ωt ˙ G − ωt ˙ O = ωt ˙ G − Ωt Lg ˙ =Lt ˙ G − Lt ˙ O = Lt ˙ G ∆Lt
(3-106) (3-107)
The angular velocity scalars are related to the changes in latitude and longitude and the changes in vehicle orientation with respect to the Earth surface, i.e. with respect to the vehicle carried normal Earth reference frame. The first set of angular velocity scalars can be represented by: ΩbE = ΩbO + ΩOE
(3-108) Flight Dynamics
106
Derivation of the Equations of Motion
Expressed in the body-fixed reference frame, this relation can be written as: p ΩbbE = q = ΩbbO + TbO ΩO OE r ˙ ϕ˙ −LgG cos LtG 1 0 − sin θ ˙ G = 0 cos ϕ cos θ sin ϕ θ˙ + TbO −Lt ˙ G sin LtG 0 − sin ϕ cos θ cos ϕ ψ˙ Lg
Or, likewise: ˙ ϕ˙ 1 sin ϕ tan θ cos ϕ tan θ −LgG cos LtG p ˙ G θ˙ = 0 cos ϕ − sin ϕ q − TbO −Lt sin ϕ cos ϕ ˙ G sin LtG r 0 ψ˙ Lg cos θ cos θ
(3-109)
(3-110)
The second set of angular velocity scalars denote the angular relation between the normal Earth fixed reference frame and the inertial reference frame expressed in the body-fixed reference frame: b pt pt ΩbEI = qtb = qt = TbO ΩO EI rtb rt cos LtG cos θ cos ψ + sin LtG sin θ sin ϕ sin θ cos ψ − sin LtG sin ϕ cos θ cos LtG = Ωt − cos ϕ sin ψ (3-111) cos ϕ sin θ cos ψ cos LtG − sin LtG cos ϕ cos θ + sin ϕ sin ψ
3-6
Rotating masses
When deriving the angular momentum of the aircraft, the assumption was made that the coordinates xb , yb and zb of all mass elements are constant as the aircraft was considered to be a rigid body. In many cases the aircraft has one or more rotating masses, such as propellers or rotors, either helicopters rotors or the rotating masses of jet engines. Due to their own rotation, these rotors may provide an appreciable contribution to the angular momentum. In the following, this contribution is derived. The derivation starts with the assumption that the axis of rotation of the rotating mass does not change its attitude relative to the body-fixed reference frame. In the following the case of an engine containing rotors is considered2 . The angular velocity Ωr about this axis of rotation is then dependent on the commanded thrust level. The situation to be considered is shown in figure 3-4. There are two rotations which transform the Xb -axis to the rotation-axis orientation. The first rotation is about the Yb -axis, rotation angle −αm . The second rotation is about the Zb′ (intermediate axis), rotation angle βm . 2
The derivation stated in this section is valid for any rotating mass.
3-6 Rotating masses
107 G
Yb Xb
rgp rgq
P
rpq
Zb
Q
VP
αm
// to Xb
βm
Ωr
Rotation axis
Figure 3-4: Engine rotation axis definition
The position of an arbitrary mass element dm, P , with respect to the center of mass G of the aircraft is, GP = GQ + QP
(3-112)
where Q is the center of mass of the rotating mass (in this case an engine rotor). The velocity of dm now is: V P = V P0 + Ωr × QP
(3-113)
where V P0 , is the velocity of dm if the engine rotor does not rotate at angular velocity Ωr . The contribution of dm to the total angular momentum of the aircraft about its center of mass is:
dB G = dm GP × V P
= dm GP × V P0 + dm GP × (Ωr × QP ) = dB 1 + dB 2
(3-114)
Here dB 1 is the contribution to the angular momentum, as derived in section 3-3. The additional part, due to the angular velocity of the engine rotor, is:
dB 2 = dm GP × (Ωr × QP )
= dm GQ × (Ωr × QP ) + dm QP × (Ωr × QP )
(3-115) Flight Dynamics
108
Derivation of the Equations of Motion
In the integration across the entire rotor GQ and Ωr in the first term at the right hand of the latter expression are constants, hence,
Z
mrotor
dm GQ × (Ωr × QP ) = GQ × Ωr ×
Z
mrotor
dm QP = 0
(3-116)
This integral is equal to zero, as QP denotes the position of the mass element dm relative to the center of mass Q of the engine rotor. As a result, dB 2 may now be written as,
dB 2 = dm QP × (Ωr × QP )
= dm {Ωr (QP · QP ) − QP (Ωr · QP )} = dm Ωr Rx2 + Ry2 + Rz2 − QP Ωrx · Rx + Ωry · Ry + Ωrz · Rz
(3-117)
where Rx , Ry , Rz are the components of the vector QP along the X, Y , and Z-axis of the chosen reference frame respectively. The required addition to the angular momentum of the aircraft is,
∆B G =
Z
dB 2
(3-118)
mrotor
Expansion results in the components of ∆B G , ∆Bx =
Ω rx
R
mrotor R
∆By = −Ωrx ∆Bz = −Ωrx
mrotor R mrotor
R R Ry2 + Rz2 dm − Ωry Rx Ry dm − Ωrz Rx Rz dm mrotor R mrotor2 R Rx Ry dm + Ωry Rx + Rz2 dm − Ωrz Ry Rz dm mrotor R R mrotor Rx2 + Ry2 dm Rx Rz dm − Ωry Ry Rz dm + Ωrz mrotor
mrotor
In these expressions the inertial parameters of the rotor are taken about axes parallel to the body-fixed reference frame axes, as it applies to the inertial parameters of the aircraft itself. The moments of inertia are,
Ixxr
=
Z
mrotor
Iyyr
=
Z
mrotor
Izzr
=
Z
mrotor
Ry2 + Rz2 dm
(3-119)
Rx2 + Rz2 dm
(3-120)
Rx2 + Ry2 dm
(3-121)
3-6 Rotating masses
109
and the products of inertia,
Jyzr
=
Z
Ry Rz dm
(3-122)
Z
Rx Rz dm
(3-123)
Z
Rx Ry dm
(3-124)
mrotor
Jxzr
=
mrotor
Jxyr
=
mrotor
(3-125) and hence the addition to the angular momentum of the rotor expressed in the body-fixed reference frame can be defined as: ∆BbG = Ibrotor Ωbr
(3-126)
↓
∆Bxb
Ixxr
b ∆By = −Jxyr b −Jxzr ∆Bz
−Jxyr
−Jxzr
Iyyr
−Jyzr
−Jyzr
Izzr
Ωbrx
b Ωr y Ωbrz
(3-127)
The rotor angular velocity expressed in the body-fixed reference frame is defined as: Ωbr = Tbr Ωrr
(3-128)
where Tbr is the transition matrix from the reference frame of the engine3 to the body-fixed reference frame. The relation can be written as:
Ωbrx cos βm cos αm − sin βm cos αm − sin αm Ωr Ωbr = 0 sin βm cos βm 0 y b cos βm sin αm − sin βm sin αm cos αm 0 Ω rz cos βm cos αm = Ωr sin βm cos βm sin αm pr = qr rr
(3-129)
∆B G provides an extra contribution in the moment equation: 3
Which is defined as, Xr -axis along rotor angular velocity vector, Zr -axis is aircraft symmetry plane, and Yr -axis to complement the right-handed reference frame Flight Dynamics
110
Derivation of the Equations of Motion
∆4 MbG
d∆B G b = dt I d∆B G b = + ΩbbI × ∆B bG dt b dIrotor Ωr b b b b b = + ΩbE × ∆B G + ΩEI × ∆B G dt b
(3-130)
It is assumed that the rotor will stay intact during the operation of the engine, such that the moment of inertia of the rotor will stay constant with respect to the body-fixed reference frame, i.e.: ∆4 MbG
=Ibrotor
dΩr b + ΩbbE × ∆B bG + ΩbEI × ∆B bG dt b
(3-131)
The derivative of the rotor angular velocity vector with respect to the body-fixed reference frame is simply: b cos β cos α p ˙ m m r dΩr ˙r = q˙r =Ω sin βm dt b sin βm sin αm r˙r
(3-132)
The definitions for the rotation vectors ΩbEI and ΩbbE are (see section 3-3, p. 96):
ΩbbE
p pt = q , ΩbEI = qt rt r
(3-133)
The expression for the additional moment due to rotating masses expressed in the body-fixed reference frame can be written as: ∆4 MGb = ∆4a MGb + ∆4b MGb + ∆4c MGb
(3-134)
with: dΩr b = dt b ∆4b MGb = ΩbbE × Ibrotor Ωbr ∆4c MGb = ΩbEI × Ibrotor Ωbr
∆4a MGb
Ibrotor
Analogous to the derivation of ∆3 (M)bG on page 96 till 99 the expressions for the three
3-7 Simplifications of the equations of motion
111
components are:
p˙r Ar −Fr −Er ∆4a MGb = −Fr Br −Dr q˙r −Er −Dr Cr r˙r Ar p˙r − Fr q˙r − Er r˙r = Br q˙r − Fr p˙r − Dr r˙r Cr r˙r − Er p˙r − Dr q˙r pr p Ar −Fr −Er ∆4b MGb = q × −Fr Br −Dr qr −Er −Dr Cr rr r Cqrr − Eqpr − Brqr + F rpr + D (rrr − qqr ) = Arpr − F rqr − Cprr + Dpqr + E (ppr − rrr ) Bpqr − Dprr − Aqpr + Eqrr + F (qqr − ppr ) pt Ar −Fr −Er pr ∆4c MGb = qt × −Fr Br −Dr qr rt −Er −Dr Cr rr Cqt rr − Eqt pr − Brt qr + F rt pr + D (rt rr − qt qr ) = Art pr − F rt qr − Cpt rr + Dpt qr + E (pt pr − rt rr ) Bpt qr − Dpt rr − Aqt pr + Eqt rr + F (qt qr − pt pr )
(3-135)
(3-136)
(3-137)
where the subscript r denotes that the moments of inertia (A, B, C) and the products of inertia (D, E, F ) are the properties of the rotor. The total moment equation defined in section 3-5-2 (equation (3-103), page 104) has to be extended to the form: ∆1 MbG + ∆2 MbG + ∆3 MbG + ∆4 MbG = Mbaero,G
(3-138)
with ∆4 MbG consisting of the three components defined previously.
3-7
Simplifications of the equations of motion
The set of equations of motion derived in section 3-6 are for a spherical, rotating Earth considering a rigid vehicle with constant mass moving in a constant gravity field. In this section, the influence of two commonly made assumptions on the equations of motion will be investigated. They are: Non-rotating Earth This assumption greatly reduces the complexity of the equations of motion. However, depending on the study case, it can introduce large errors during computations. Flat Earth If the distance traveled during the study case is small, then the curvature of the Earth will not play an important role in the changes of vehicle state. One could therefore assume that the Earth is flat, which simplifies the equations of motion. For larger distances, this assumption will cause significant errors. Flight Dynamics
112
3-7-1
Derivation of the Equations of Motion
Non-rotating Earth
If the Earth is assumed to be non-rotating then the angular velocity of the normal Earth fixed reference frame with respect to the inertial reference frame will be zero, i.e. Ωt = 0. This will simplify the equations of motions since in the force equations the term due to the rotation of the Earth ∆AR will be zero. The force equations thus become (see equation (3-101), p. 103): O O b mTbO AO r + AS = mTbO gG + Faero
(3-139)
∆1 MbG + ∆4a MbG + ∆4b MbG = Mbaero,G
(3-140)
In the moment equations the terms ∆2 MG , ∆3 MG , and ∆4c MG will be zero (since ΩEI = 0). The moment equations thus become (see equation (3-138), p. 111):
The kinematic relations which contain terms with Ωt will become zero.
3-7-2
Flat Earth
If one assumes that the Earth is flat, then the normal Earth fixed reference frame will coincide with the vehicle carried normal Earth reference frame. Therefore the angular velocity between those two reference frames will be zero, i.e. ΩOE = 0. Using this in equation (3-49), one can see that the term ∆AS in the force equations will become zero. Also, the term ∆AR changes. The relation given by equation (3-50) will become: dΩEI drG AR = × rG + ΩEI × + ΩEI × Vk,G dt O dt O
+ (ΩOE + ΩEI ) × (ΩEI × rG ) dΩEI drG = × rG + ΩEI × + ΩEI × Vk,G dt dt O
O
+ ΩEI × (ΩEI × rG )
(3-141)
Only the last term of the relation has changed. Expressing the last term in the vehicle carried normal Earth reference frame will lead to: O O ΩO EI × ΩEI × rG O O O O O = ΩO EI ΩEI · rG − rG ΩEI · ΩEI T Ωt cos LtG Ωt cos LtG 0 = 0 0 0 −Ωt sin LtG −Ωt sin LtG −(Rt + h) T 0 Ωt cos LtG Ωt cos LtG − 0 0 0 − (Rt + h) −Ωt sin LtG −Ωt sin LtG
(3-142)
3-7 Simplifications of the equations of motion
=
−
=
Ωt cos LtG (Ωt (Rt + h) sin LtG ) 0 −Ωt sin LtG 0 Ω2t cos2 LtG + Ω2t sin2 LtG 0 − (Rt + h) Ω2t (Rt + h) sin LtG cos LtG 0 2 2 Ωt (Rt + h) cos LtG
113
(3-143)
The relations given by the moment equation remain the same. Only the definition of the term ΩbbE will change: ΩbbE
p b = q = ΩbbO + TbO ΩO OE = ΩbO r ϕ˙ 1 0 − sin θ = 0 cos ϕ cos θ sin ϕ θ˙ 0 − sin ϕ cos θ cos ϕ ψ˙
(3-144)
The consequences for the kinematic relations is that all terms which contain ΩOE will become zero.
3-7-3
Flat and Non-rotating Earth
Combining the assumptions of a flat Earth and a non-rotating Earth will result in the following force equations: O b mTbO AO r = mTbO gG + Faero
(3-145)
and the following moment equations: ∆1 MbG + ∆4a MbG + ∆4b MbG = Mbaero,G
(3-146)
with: ΩbbE
ϕ˙ p 1 0 − sin θ = q = 0 cos ϕ cos θ sin ϕ θ˙ r 0 − sin ϕ cos θ cos ϕ ψ˙
The relations given in this section can directly be used to obtain the scalar form of the equations of motion for a flat, non-rotating Earth expressed in the body-fixed reference frame. To show that the relations in this section are correct, another derivation is performed. The following tutorial will show this second derivation of the equations of motion for a flat, nonrotating Earth and no wind. The derivation is similar to one used to obtain the general equations of motion but it is much more simpler. The simplified set of equations will be used in subsequent chapters to demonstrate the principles of flight dynamics. Flight Dynamics
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Derivation of the Equations of Motion
Mini-tutorial: Derivation of the equations of motion for a flat, non-rotating Earth, no wind and no rotating masses. A simplified set of equations of motion is derived that will be used in the remainder of this book. The assumptions made here are: • Vehicle is a rigid body. • Vehicle mass is constant. • Earth is flat. • Earth is non-rotating. • Body-fixed reference frame is chosen such that Jxy and Jyz are zero. • Effects of rotating masses are neglected. Since the Earth is assumed to be flat the vehicle carried normal earth reference frame and the normal Earth fixed reference frame coincide. Because the Earth is assumed to be non-rotating, those two reference frames can also be seen as inertial reference frames, so: AE,G = AO,G = AG VE,G = VO,G = VG rE,G = rO,G = rG Here the normal Earth fixed reference frame will be selected as inertial reference frame. The equations of motion will be expressed in the body-fixed reference frame (with origin in the center of gravity G). The derivation starts with the force equations: b mAbG = Fext
(3-147)
The external forces on the vehicle remain the same as given in section 3-4 (keeping in mind that TbE = TbO in case of a flat earth): b 0 X b Fext = TbO 0 + Yb Zb mgr,0 − sin θ X = mgr,0 sin ϕ cos θ + Y (3-148) cos ϕ cos θ Z
The inertial acceleration in the vehicle carried normal Earth reference frame can be rewritten as: dVG b b AG = dt E (3-149)
3-7 Simplifications of the equations of motion
115
dVG b b = + ΩbbE × VG dt b b b p uk u˙ k = v˙ kb + q × vkb w˙ kb r wkb
(3-150)
Since there is no wind, the kinematic velocity is equal to the aerodynamic velocity. We can therefore drop the sub-scripts and define the vector [u, v, w] as the inertial velocity vector expressed in the body-fixed reference frame, i.e.: b b uk ua u v b = vab = v (3-151) k wkb wab w
Substituting relation (3-150) into equation (3-150) and substituting the result in equation (3-147) along with equation (3-148) will lead to the following set of force equations: − sin θ X u˙ + qw − rv (3-152) m v˙ + ru − pw = mgr,0 sin ϕ cos θ + Y cos ϕ cos θ w˙ + pv − qu Z
Similar to the force equations, one can write for the moment equations: dBG dBG dBG = = = Mext dt I dt E dt O
(3-153)
When neglecting the effect of rotating masses we can write the derivative of the inertial angular momentum as: dBG b dBG b = + ΩbbO × BbG dt E dt b b b dΩbE b b b = IG + Ω × I Ω G bE bE dt
(3-154)
(3-155)
b
By substituting:
p ΩbbE = q r A −F −E Ixx 0 −Jxz IbG = −F B −D = 0 Iyy 0 −E −D C −Jxz 0 Izz
(3-156)
Flight Dynamics
116
Derivation of the Equations of Motion
into equation (3-154) and substituting the result into equation (3-153) will lead to the following moment equations: Ixx p˙ + (Izz − Iyy ) qr − Jxz (pq + r) ˙ L Iyy q˙ + (Ixx − Izz ) pr + Jxz p2 − r 2 = M (3-157) Izz r˙ + (Iyy − Ixx ) pq + Jxz (qr − p) ˙ N
The kinematic relations describe the relation of angular velocities between reference frames and the relation of translational velocities between reference frames. First the angular velocities. Only two reference frames are used, the body-fixed reference frame and the normal earth-fixed reference frame. The angular velocity expressed in Fb is related to the angular velocity expressed in FE by the following expression: ϕ˙ p 1 0 − sin θ q = 0 cos ϕ cos θ sin ϕ θ˙ (3-158) ˙ r 0 − sin ϕ cos θ cos ϕ ψ For the relation between velocities, the following expression holds: VN u u u VE = TEb v = TOb v = TtbO v w w w −VZ sin ϕ sin θ cos ψ cos ϕ sin θ cos ψ cos θ cos ψ − cos ϕ sin ψ + sin ϕ sin ψ u v = sin ϕ sin θ sin ψ cos ϕ sin θ sin ψ cos θ sin ψ w + cos ϕ cos ψ − sin ϕ cos ψ − sin θ sin ϕ cos θ cos ϕ cos θ [u cos θ + (v sin ϕ + w cos ϕ) sin θ] cos ψ − (v cos ϕ − w sin ϕ) sin ψ [u cos θ + (v sin ϕ + w cos ϕ) sin θ] sin ψ (3-159) = + (v cos ϕ − w sin ϕ) cos ψ u sin θ − (v sin ϕ + w cos ϕ) cos θ
The displacement of the aircraft’s center of gravity relative to the earth is obtained by integrating the three components of the ground speed with respect to time, hence, x(t) =
Zt
VN dt
(3-160)
y(t) =
Zt
VE dt
(3-161)
h(t) =
Zt
−VZ dt
(3-162)
0
0
0
Chapter 4 Linearized Equations of Motion
The equations of motion of an aircraft may be used in several different ways. Most commonly, a disturbance function is given, such as a time-dependent control surface deflection or a gust velocity, and the response of the aircraft is then calculated. An inverse situation of this problem also exists, when a certain non-steady motion of the aircraft is given and the control surface deflection as a function of time causing this motion is to be found. In a third application of the equations of motions both the disturbing control surface deflection and the resulting aircraft motion have been measured in flight. The purpose is then to find expressions for the aerodynamic forces and moments in the equations of motions, as functions of the components of the motion and the disturbances, or stated differently, to dins a mathematical model of the aircraft. The problems described above can be solved by a variety of numerical algorithms. c For example the numerical integration routines provided by MatLab may readily be applied to solve the set of nonlinear ordinary differential equations derived in chapter 3, even in real time as needed for flight simulation. The present chapter focusses on the linearization of these nonlinear differential equations, resulting in a set of linear differential equations describing small deviations from a chosen reference condition. Performing simulations with a set of linearized equations will always lead to less accurate results. So why should we derive linearized versions of the equations of motion? The answer is that these linearized equations of motion enable us to: • study the so-called ’characteristic modes of aircraft’, • derive important handling quality and stability characteristic parameters • design manual or automatic flight control systems using well established linear control theories. The goal of this book is to provide insight into the characteristics of aircraft, presented in two ways: through determination of the characteristics modes on the one hand and of Flight Dynamics
118
Linearized Equations of Motion
stability and control derivatives on the other. Characteristic modes are described by the impulse response of the aircraft and are completely determined by the eigenvalues of the system. Stability and control derivatives describe the relation between the variation of the aircraft state due to variation of that state and due to variation of control input respectively. The knowledge about the characteristics of the aircraft can be used to design control systems, such as autonomous control systems, like an autopilot, or performance increasing control systems to be used during manual control. The set of equations of motion used in this chapter is obtained by applying the following assumptions: • Vehicle is a rigid body. • Vehicle’s mass is constant. • Earth is flat. • Earth is non-rotating. • Gravity field is constant. • Aircraft has a plane of symmetry and the body-fixed reference frame is chosen such that Jxy and Jyz are zero. • Effects of rotating masses are neglected. • The resultant thrust vector lies in the symmetry plane and thus only affects the aerodynamic forces X, Z and the aerodynamic moment M . Moreover we assume that the aircraft has a normal configuration with respect to control surfaces, i.e. it has ailerons (deflection δa ), an elevator (deflection δe ), a rudder (deflection δr ), and engines (setting δt ). The complete set of equations can be described by (see section 3-7-3): Fx = Fy =
−W sin θ + X
+W cos θ sin ϕ + Y
=m (u˙ + qw − rv)
Fz =
+W cos θ cos ϕ + Z
=m (w˙ + pv − qu)
Mx =
L
Mz =
N
My =
where W = mgr,0 has been used. are: ϕ˙ = θ˙ = ψ˙ =
M
=m (v˙ + ru − pw)
=Ix p˙ + (Iz − Iy ) qr − Jxz (r˙ + pq)
=Iy q˙ + (Ix − Iz ) rp + Jxz p2 − r 2 =Iz r˙ + (Iy − Ix ) pq − Jxz (p˙ − rq)
(4-1)
The kinematic relations accompanying this set of equations p + q sin ϕ tan θ + r cos ϕ tan θ q cos ϕ − r sin ϕ ϕ cos ϕ q sin cos θ + r cos θ
Linearization of the equations of motion can be performed about any given flight condition. The results generated with the linearized equations are valid for that flight condition and
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame
119
for the conditions close to that particular point (see figure 4-1). This means that there is a certain ’validity’-region about the initial state, i.e. the state about which the equations are linearized. It depends on the chosen flight condition and on the desired equation of motion output accuracy whether or not the aircraft moves quickly out of this region or not. If, for instance, a steady state flight condition is chosen, the aircraft will remain close to this condition if no changes are made to the inputs. On the other hand, if we select a instable or non-steady flight condition (with the same desired accuracy), the aircraft will deviate quickly from this point leaving us with a set of equations incapable of describing the motion accurately. If we want to simulate the response of the aircraft along a particular trajectory which consists of multiple flight conditions, we could linearize the equations for multiple conditions and switch between sets as we move along the trajectory (see figure 4-1). Especially when we want to design a control system for the entire trajectory (for instance, to keep the aircraft on that trajectory) we can design a controller for each flight condition and switch between controllers. Also, linearizing the equations for multiple flight conditions gives us insight into the variation of the aircraft characteristics.
Linearization about one flight condition or about an entire flight trajectory boils down to the same thing. The technique of linearization about an arbitrary flight condition is described in section 4-1. No specific body-fixed reference frame is selected. In practice a steady state flight condition is usually chosen for the linearization. A steady state, straight, symmetric flight condition is chosen in section 4-2 for the linearization. At first no choices are made regarding the type of body-fixed reference frame. Thereafter we do make a choice and derive the linearized equation set for the same flight condition. The non-dimensional form the latter set of equations is derived in section 4-3. The set is made non-dimensional to decouple the aerodynamic properties of the aircraft from its dimensions and mass, creating the opportunity to compare different aircraft types.
4-1
Linearization about arbitrary flight condition in arbitrary bodyfixed reference frame
Linearization can be performed on any function which is differentiable. Consider a general function: Y = f (X) which states that the output Y is a function of the state X which can have any dimension. This function can also be written in a Taylor expansion form. If we consider the state to be one dimensional and the function to be real then the Taylor expansion about state x0 is given by: ′
f (x0 ) y =f (x0 ) + f (x0 ) (x − x0 ) + (x − x0 )2 2! f (3) (x0 ) f (n) (x0 ) + (x − x0 )3 + . . . + (x − x0 )n + . . . 3! n! ′
(4-2) Flight Dynamics
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Linearized Equations of Motion
x2 confidence region x20
x10
x1
(a) ’Point-linearization’
x2
x2
x1
x1
(b) ’Trajectory-linearization’
Figure 4-1: Linearization about a single point and a trajectory
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame ′
121
′
where f denotes the first derivative of function f with respect to x, and f denotes the second derivative etc. If the state X is two dimensional, e.g. X = (x, u), the Taylor expansion about state (x0 , u0 ) of a real function can be written as: y =f (x0 , u0 ) + [fx (x0 , u0 ) ∆x + fu (x0 , u0 ) ∆u] 1 fxx (x0 , u0 ) ∆x2 + 2fxu (x0 , u0 ) ∆x∆u + fuu (x0 , u0 ) ∆u2 + 2! 1 + fxxx (x0 , u0 ) ∆x3 + 3fxxu (x0 , u0 ) ∆x2 ∆u 3! + 3fxuu (x0 , u0 ) ∆x∆u2 + fuuu (x0 , u0 ) ∆u3 + ...
(4-3)
where fx denotes the partial derivative of f with respect to x and fu denotes the partial derivative of f with respect to u, etc. Of course the Taylor expansion can become rather large when the dimension of the state grows, however, the principle remains the same. Linearization of a function about a point (known as the initial point of linearization or simply the initial point) is defined as expanding the function in a Taylor series and taking only the first term and the terms with a first (partial) derivative from that expansion. In other words disregarding terms which contain (partial) derivatives of order higher than 1. For a function with an n-dimensional state the linearized function about point X0 becomes: Y = f (X0 ) + fx1 (X0 ) ∆x1 + fx2 (X0 ) ∆x2 + . . . + fxn (X0 ) ∆xn n X = f (X0 ) + fxi (X0 ) ∆xi
(4-4)
i=1
The function can be replaced by the relations given by the equations of motion. In the next section the linearization of the states, i.e. the right hand side of the equations, is performed while in section 4-1-2 the forces and moments, i.e. the left hand side of the equations, are linearized. In section 4-1-3 the kinematic relations are linearized.
4-1-1
Linearization of states
The equations of motion can be written as: Fx =f (u, ˙ v, w, q, r) Fy =f (v, ˙ u, w, p, r) Fz =f (w, ˙ u, v, p, q) Mx =f (p, ˙ r, ˙ p, q, r) My =f (q, ˙ p, r)
Mz =f (p, ˙ r, ˙ p, q, r)
(4-5) Flight Dynamics
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Linearized Equations of Motion
Applying the Taylor expansion and neglecting the higher order terms will give: Fx =f (X0 ) + fu˙ (X0 ) ∆u˙ + fv (X0 ) ∆v + fw (X0 ) ∆w + fq (X0 ) ∆q + fr (X0 ) ∆r Fy =f (X0 ) + fv˙ (X0 ) ∆v˙ + fu (X0 ) ∆u + fw (X0 ) ∆w + fp (X0 ) ∆p + fr (X0 ) ∆r Fz =f (X0 ) + fw˙ (X0 ) ∆w˙ + fu (X0 ) ∆u + fv (X0 ) ∆v + fp (X0 ) ∆p + fq (X0 ) ∆q Mx =f (X0 ) + fp˙ (X0 ) ∆p˙ + fr˙ (X0 ) ∆r˙ + fp (X0 ) ∆p + fq (X0 ) ∆q + fr (X0 ) ∆r My =f (X0 ) + fq˙ (X0 ) ∆q˙ + fp (X0 ) ∆p + fr (X0 ) ∆r
Mz =f (X0 ) + fp˙ (X0 ) ∆p˙ + fr˙ (X0 ) ∆r˙ + fp (X0 ) ∆p + fq (X0 ) ∆q + fr (X0 ) ∆r
(4-6)
where X0 denotes the state about which the linearization is performed. Taking the specific partial derivatives of the function using the equations in (4-1) and substituting the initial state for linearization results in: Fx = Fy = Fz = Mx =
m (u˙ 0 + q0 w0 − r0 v0 ) + m (∆u˙ − r0 ∆v + q0 ∆w + w0 ∆q − v0 ∆r) m (v˙ 0 + r0 u0 − p0 w0 ) + m (∆v˙ + r0 ∆u − p0 ∆w − w0 ∆p + u0 ∆r) m (w˙ 0 + p0 v0 − q0 u0 ) + m (∆w˙ − q0 ∆u + p0 ∆v + v0 ∆p − u0 ∆q) ( ) Ix p˙0 + (Iz − Iy ) q0 r0 − Jxz (r˙0 + p0 q0 ) + Ix ∆p˙ − Jxz ∆r˙
My =
(
Mz =
4-1-2
−Jxz q0 ∆p + [(Iz − Iy ) r0 − Jxz p0 ] ∆q + (Iz − Iy ) q0 ∆r Iy q˙0 + (Ix − Iz ) p0 r0 + Jxz p20 + r02
+Iy ∆q˙ + [(Ix − Iz ) r0 + Jxz p0 ] ∆p + [(Ix − Iz ) p0 + Jxz r0 ] ∆r ( ) Ix r˙0 + (Iy − Ix ) p0 q0 − Jxz (p˙0 − q0 r0 ) − Jxz ∆p˙ + Ix ∆r˙
)
(4-7)
+ (Iy − Ix ) q0 ∆p + [(Iy − Ix ) p0 + Jxz r0 ] ∆q + Jxz q0 ∆r
Linearization of forces and moments
The left hand side of the equations of motion can be written as: Fx = f (θ, X) Fy = f (θ, ϕ, Y ) Fz = f (θ, ϕ, Z) Mx = f (L) f (M ) My = Mz = f (N )
(4-8)
where X, Y, Z are the aerodynamic forces and L, M, N are the aerodynamic moments, they are all dependent on the entire history of the aircraft state in time! The question is which states influence which forces and moments. The most general framework to be chosen is to assume that the aerodynamic forces and moments are functions of the components of the motion and control surface deflections, i.e.: X, Y, Z, L, M, N −→ f (u(.), v(.), w(.), p(.), q(.), r(.), δa (.), δe (.), δr (.), δt (.))
(4-9)
where the notation (.) denote the dependence on the time-history of the parameters. The (.) function contains information regarding the entire time history of a specific parameter. The aerodynamic forces and moments are thus a function of time-functions.
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame
123
u(.)
u
u(t)
t
−∞
Figure 4-2: Example of a time-function: u(.)
Let us take a look at the time-functions, taking for example u(.), i.e. the time-function of the velocity along the Xb -axis (see figure 4-2). The information regarding the entire time history must now be molded into a form fit for the equations of motion. The method adopted here is to replace the time-function u(.) by the following Taylor series: u (.) = u (t) + u˙ (t) ∆t +
1 1 ... u ¨ (t) ∆t2 + u (t) ∆t3 + . . . 2! 3!
∞ X 1 di u i = u (t) + ∆t i! dti
(4-10)
i=1
It is assumed that if we take into account all possible derivatives of u(t), that we have captured all information regarding the parameter history. In theory all the derivatives have an influence on the value of u(.) on time t. However, it has been observed in practice that for most parameters the influences of the derivatives is limited and can be neglected in the relation (i.e. neglecting the summation term in equation (4-10)). A few exceptions are the derivatives ∂Y ∂N ∂Z ∂ v˙ and ∂ v˙ with respect to the acceleration v˙ along the YB -axis and the derivatives ∂ w˙ and ∂M ˙ along the ZB -axis. Of these four partial derivatives, ∂ w˙ with respect to the acceleration w ∂N ∂M the two relating the moments ∂ v˙ and ∂M ∂ w˙ (and of these two again ∂ w˙ ) are by far the most important. In the following, all derivatives (with the exception of the previously mentioned four derivatives) will be neglected. By neglecting the influence of the derivatives we are in fact neglecting the influence of parameter variation through time. These influences arise from non-stationary wing-fuselage and tail interference. An example is given in the next mini-tutorial.
Flight Dynamics
124
Linearized Equations of Motion
Mini-tutorial: Influence of non-stationary wing-fuselage and tail interference on aerodynamic forces and moments. We look here at the wing-fuselage and tail interference in the case of cross-wind, i.e. v 6= 0. The discussion in this tutorial is hypothetical and is not proven here, it is only meant to present some insight into why the aerodynamic forces and moments are dependent on the history of all motion parameters. Consider an aircraft flying at a velocity V which experiences at a certain moment a sudden cross-wind at time t (see figure 4-3(a)). Let us focus momentarily on the airflow over the wing and fuselage due to the cross-wind only (see figure 4-3(b)). At the root of the wing the airflow encounters the fuselage. Since the airflow cannot penetrate the fuselage it needs to follow a path around the fuselage. The airflow will be split in two, one part will travel over the fuselage and one will go underneath. The airflow underneath the fuselage will travel around the wing downwards, causing a decrease in the angle of attack for that section of the wing which leads to a decrease in lift. On the other side of the fuselage the opposite happens. So the lift on one side decreases and on the other side increases. This will cause a moment about the Xb -axisa Now consider the lift generating mechanism. The magnitude of lift is directly related with the strength of the vortices generated by the wings. If the lift decreases then the vortex strength will also decrease and vice versa. So, in our situation, we have a local decrease in vortex strength on one side and an increase on the other. This can be represented by a placing a imaginary vortex with its centerline on the Xb -axis (see figure 4-3(c)). After time ∆t the vortex hits the tail, therefore the tail will experienced a different airflow. The forces generated by this change in airflow will cause a rolling and yawing moment. This example shows that events on time t will still have effect at time t + ∆t, showing the time-dependency of the aerodynamic force and moments on the motion parameters. a
Under the assumption that the Xb -axis is aligned with the fuselage.
By neglecting the time-dependency, the relations given by equation (4-9) can thus be redefined as:
X, Y, Z, L, M, N −→ f (u(t), v(t), v(t), ˙ w(t), w(t), ˙ p(t), q(t), r(t), δa (t), δe (t), δr (t), δt (t)) (4-11) When dropping the time notation (t), the relations given equation (4-8) can be rewritten as:
Fx Fy Fz Mx My Mz
= = = = = =
f (θ, u, v, w, p, q, r, δa , δe , δr , δt ) f (θ, ϕ, u, v, w, v, ˙ p, q, r, δa , δe , δr ) f (θ, ϕ, u, v, w, w, ˙ p, q, r, δa , δe , δr , δt ) f (u, v, w, p, q, r, δa , δe , δr ) f (u, v, w, w, ˙ p, q, r, δa , δe , δr , δt ) f (u, v, w, v, ˙ p, q, r, δa , δe , δr )
(4-12)
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame
125
Airflow direction
(a) Cross-wind relative to the aircraft
Line of stagnation points
Streamlines
Increased Angle of attack Decreased Angle of attack Airflow direction
(b) Streamlines over wing and fuselage due to cross-wind
∆F
∆F
∆F
∆t
(c) Additional forces on tail due to time-delayed vortices
Figure 4-3: Airflow tutorial: example of time-dependent aerodynamic characteristics
Flight Dynamics
126
Linearized Equations of Motion
Linearizing these equations leads to: Fx Fy Fz Mx My Mz
= = = = = =
Fx (X0 ) + Fx (∆X) Fy (X0 ) + Fy (∆X) Fz (X0 ) + Fz (∆X) Mx (X0 ) + Mx (∆X) My (X0 ) + My (∆X) Mz (X0 ) + Mz (∆X)
(4-13)
with Fx (X0 ) = Fy (X0 ) = Fz (X0 ) = Mx (X0 ) = My (X0 ) = Mz (X0 ) =
−W sin θ0 + X0 W cos θ0 sin ϕ0 + Y0 W cos θ0 cos ϕ0 + Z0 L0 M0 N0
(4-14)
where X0 , Y0 , and Z0 are the aerodynamic forces and L0 , M0 , and N0 are the aerodynamic moments for the initial state of linearization, i.e. for X0 = [u0 , v0 , w0 , p0 , q0 , r0 , δa0 , δe0 , δr0 ]. The second part of equation 4-13 can be written as: ( ) −W cos θ0 ∆θ + Xu ∆u + Xv ∆v + Xw ∆w + Xp ∆p Fx (∆X) = +Xq ∆q + Xr ∆r + Xδa ∆δa + Xδe ∆δe + Xδr ∆δr + Xδt ∆δt −W sin θ0 sin ϕ0 ∆θ + W cos θ0 cos ϕ0 ∆ϕ Fy (∆X) = +Yu ∆u + Yv ∆v + Yw ∆w + Yv˙ ∆v˙ + Yp ∆p + Yq ∆q + Yr ∆r +Yδa ∆δa + Yδe ∆δe + Yδr ∆δr −W sin θ0 cos ϕ0 ∆θ − W cos θ0 sin ϕ0 ∆ϕ Fz (∆X) = +Zu ∆u + Zv ∆v + Zw ∆w + Zw˙ ∆w˙ + Zp ∆p + Zq ∆q + Zr ∆r (4-15) +Zδa ∆δa + Zδe ∆δe + Zδr ∆δr + Zδt ∆δt ( ) Lu ∆u + Lv ∆v + Lw ∆w Mx (∆X) = +Lp ∆p + Lq ∆q + Lr ∆r + Lδa ∆δa + Lδe ∆δe + Lδr ∆δr ( ) Mu ∆u + Mv ∆v + Mw ∆w + Mw˙ ∆w˙ + Mp ∆p + Mq ∆q My (∆X) = +Mr ∆r + Mδa ∆δa + Mδe ∆δe + Mδr ∆δr + Mδt ∆δt ( ) Nu ∆u + Nv ∆v + Nw ∆w + Nv˙ ∆v˙ Mz (∆X) = +Np ∆p + Nq ∆q + Nr ∆r + Nδa ∆δa + Nδe ∆δe + Nδr ∆δr As a consequence of the assumed symmetry of the aircraft and the linearization of the equations of motion a further simplification is possible. Suppose, a change in velocity +∆v along the YB -axis causes an extra force ∆Xdv = Xv ∆v = dX. Then, since the derivative is a constant, a change in velocity −∆v will generate an extra force ∆X−dv = Xv − ∆v = −dX. Reflecting the situation with −∆v in the plane of symmetry must result in the situation with +∆v, see figure 4-4. This means, however, that dX must be equal to zero and, as a consequence, X cannot vary linearly with ∆v. Only a variation with ∆v 2 , ∆v 4 , ∆v 6 etcetera is possible. Such terms of higher degree however, have already been neglected during the
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame
127
XB
+ dX due to +dv - dv
+ dv
YB
- dX due to −dv - dX = + dX = 0
Figure 4-4: The impossibility of a force in the XB -direction arising from a variation in velocity dv along the YB -axis
linearization. This leads to the conclusion that the asymmetric deviation ∆v has no influence on the symmetric aerodynamic force X. A similar argument can be given for any of other combinations of symmetric and asymmetric variables. Evidently, the following rules apply,
• Small asymmetric deviations and disturbances have no influence on the symmetric forces X and Z or on the symmetric moment M • Small symmetric deviations and disturbances have no influence on the asymmetric force Y or on the asymmetric moments L and N
Expressed in a different way,
No aerodynamic coupling exists between the symmetric and the asymmetric degrees of freedom, as long as the deviations and disturbances remain small! 2
2 to dM , which has in many A single exception to this rule is the contribution ∂∂vM 2 · ∆v practical situations a non-negligible magnitude at values of dv occurring in flight conditions where no other significant non-linearities are noticeable. In the following, however, this contribution of dv 2 to dM will not be further considered.
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128
Linearized Equations of Motion
Under the previous assumptions, the linearized set of aerodynamic forces and moments and the weight can be written as follows: Fx = Fx (X0 ) − ( W cos θ0 ∆θ + Xu ∆u + Xw ∆w + Xq ∆q + Xδe ∆δe + X) δt ∆δt −W sin θ0 sin ϕ0 ∆θ + W cos θ0 cos ϕ0 ∆ϕ Fy = Fz (X0 ) + +Yv ∆v + Yv˙ ∆v˙ + Yp ∆p + Yr ∆r + Yδa ∆δa + Yδr ∆δr ( ) −W sin θ0 cos ϕ0 ∆θ − W cos θ0 sin ϕ0 ∆ϕ Fz = Fz (X0 ) + +Zu ∆u + Zw ∆w + Zw˙ ∆w˙ + Zq ∆q + Zδe ∆δe + Zδt ∆δt
(4-16)
Mx = Mx (X0 ) + Lv ∆v + Lp ∆p + Lr ∆r + Lδa ∆δa + Lδr ∆δr My = My (X0 ) + Mu ∆u + Mw ∆w + Mw˙ ∆w˙ + Mq ∆q + Mδe ∆δe + Mδt ∆δt Mz = Mz (X0 ) + Nv ∆v + Nv˙ ∆v˙ + Np ∆p + Nr ∆r + Nδa ∆δa + Nδr ∆δr where Fx (X0 ) , ..., Mx (X0 ) , ... remain the same (see equation 4-14).
4-1-3
Linearization of kinematic relations
The kinematic relations for a non-rotating flat Earth are functions of the Euler angles ϕ, θ and of the angular velocities p, q, r: ϕ˙ = f (ϕ, θ, p, q, r) θ˙ = f (ϕ, θ, q, r) ψ˙ = f (ϕ, θ, q, r)
(4-17)
Linearization of these relations results in:
with
and
ϕ˙ = ϕ˙ (X0 ) + ϕ˙ (∆X) θ˙ = θ˙ (X0 ) + θ˙ (∆X) ψ˙ = ψ˙ (X0 ) + ψ˙ (∆X)
(4-18)
ϕ˙ (X0 ) = p0 + q0 sin ϕ0 tan θ0 + r0 cos ϕ0 tan θ0 θ˙ (X0 ) = q0 cos ϕ0 − r0 sin ϕ0 ϕ0 cos ϕ0 ψ˙ (X0 ) = q0 sin cos θ0 + r cos θ0
(4-19)
∆p + sin ϕ0 tan θ0 ∆q + q0 cos ϕ0 tan θ0 ∆ϕ sin ϕ0 +q ∆θ + cos ϕ tan θ ∆r 0 0 0 ϕ˙ (∆X) = 2 cos θ0 cos ϕ0 −r0 sin ϕ0 tan θ0 ∆ϕ + r0 2 ∆θ cos θ0 θ˙ (∆X) = cos ϕ0 ∆q − q0 sin ϕ0 ∆ϕ − sin ϕ0 ∆r − r0 cos ϕ0 ∆ϕ sin ϕ cos ϕ0 sin ϕ0 0 ∆q + q0 ∆ϕ + q0 tan θ0 ∆θ+ cos θ0 cos θ0 cos θ0 ψ˙ (∆X) = cos ϕ0 ∆r − r0 sin ϕ0 ∆ϕ + r0 cos ϕ0 tan θ0 ∆θ cos θ0 cos θ0 cos θ0
(4-20)
Since the parameters of the linear velocity, i.e. VN , VE , VZ , are not included in the state vector, we do not have to linearize the equations relating these parameters with the state
4-1 Linearization about arbitrary flight condition in arbitrary body-fixed reference frame
129
parameters u, v, w, ϕ, θ, ψ. After we known the values for the state vector we can determine VN , VE , VZ using the relation:
VN u VE = TtbO v w −VZ sin ϕ sin θ cos ψ cos ϕ sin θ cos ψ cos θ cos ψ u − cos ϕ sin ψ + sin ϕ sin ψ v = sin ϕ sin θ sin ψ cos ϕ sin θ sin ψ cos θ sin ψ w + cos ϕ cos ψ − sin ϕ cos ψ − sin θ sin ϕ cos θ cos ϕ cos θ At this point we have linearized the equations of motion and the kinematic relations. With this set of equations we are able to simulate any motion about the initial state of linearization X0 . During simulation one uses a state vector consisting of nine parameters: x = [u, v, w, p, q, r, ϕ, θ, ψ] T and an input vector containing three input parameters: u = [δa , δe , δr , δt ]T After rearranging the terms in the linearized equations of motion and kinematic relations one can write the relations in the state-space format: x˙ 0 + ∆x˙ = A0 x0 + A∆x + B0 u0 + B∆u where A0 , A, B0 , B are constant matrices and the vector x˙ 0 is also constant. The simulation is initialized at the initial state of linearization X0 . Bear in mind that this state includes information about the initial state vector and the initial input vector. If the initial state of linearization is a steady state, then the term x˙ 0 will be zero meaning that the terms A0 x0 and B0 u0 cancel and the state-space format becomes: ∆x˙ = A∆x + B∆u The matrix A contains the stability derivatives and inertial properties and the B contains the control derivatives. The derivatives will be defined in section 4-3. The previous relation is used during simulations to determine the changes in state due to input changes. The most common flying condition is a steady-state flight. In particular the steady, straight, symmetric flight conditions are often encountered. Therefore most analysis of the characteristics of an aircraft are performed for such flight conditions. We will derive the linearized set of equations about a steady, straight, symmetric flight condition in the next section. Flight Dynamics
130
4-2
Linearized Equations of Motion
Linearization about steady, straight, symmetric flight condition
A steady, straight (wing level), symmetric flight condition has the following properties: u 6= 0 u˙ = 0 v = 0 v˙ = 0 w 6= 0 w˙ = 0
p=0 q=0 r=0
p˙ = 0 q˙ = 0 r˙ = 0
ϕ = 0 ϕ˙ = 0 θ 6= 0 θ˙ = 0 ψ 6= 0 ψ˙ = 0
X 6= 0 X˙ = 0 Y = 0 Y˙ = 0 Z 6= 0 Z˙ = 0
which in principle means that the state of the aircraft remains constant, i.e. all derivatives, the lateral velocity, and the angular velocities are zero. The conditions that u, w, X, Z are √ unequal to zero means that the aircraft has a velocity equal to u2 + w2 and that the resultant aerodynamic force R = X + Z is equal in magnitude but opposite in sign with respect to the gravity vector W . The set of linearized equations of motion is obtained by taking the flight condition described above as initial state for linearization. By substituting the properties in the general linearized equations of motion (equation (4-21)) we get: Fx = m (∆u˙ + w0 ∆q) Fy = m (∆v˙ − w0 ∆p + u0 ∆r) Fz = m (∆w˙ − u0 ∆q)
Mx = Ix ∆p˙ − Jxz ∆r˙ My = Iy ∆q˙ Mz = Ix ∆r˙ − Jxz ∆p˙
(4-21)
The linearized forces and moments equation becomes: Fx Fy Fz Mx My Mz
= = = = = =
−W cos θ0 ∆θ + Xu ∆u + Xw ∆w + Xq ∆q + Xδe ∆δe + Xδt ∆δt +W cos θ0 ∆ϕ + Yv ∆v + Yv˙ ∆v˙ + Yp ∆p + Yr ∆r + Yδa ∆δa + Yδr ∆δr −W sin θ0 ∆θ + Zu ∆u + Zw ∆w + Zw˙ ∆w˙ + Zq ∆q + Zδe ∆δe + Zδt ∆δt Lv ∆v + Lp ∆p + Lr ∆r + Lδa ∆δa + Lδr ∆δr Mu ∆u + Mw ∆w + Mw˙ ∆w˙ + Mq ∆q + Mδe ∆δe + Mδt ∆δt Nv ∆v + Nv˙ ∆v˙ + Np ∆p + Nr ∆r + Nδa ∆δa + Nδr ∆δr
(4-22)
here, we applied the knowledge that the aerodynamic forces are canceled by the weight terms for the initial state. Therefore they are removed from the equations. The linearized kinematic relations become (see equation 4-18 to 4-20): ϕ˙ = ∆p + tan θ0 ∆r θ˙ = ∆q ψ˙ = cos1θ0 ∆r
(4-23)
The aircraft body axes (Fb ) used in the foregoing to describe the motions, have not yet been completely defined. The reference frame is fixed relative to the aircraft, but the direction of
4-2 Linearization about steady, straight, symmetric flight condition
131
the Xb -axis in the plane of symmetry has not yet been defined. Often, the choice depends on the type of motion to be studied. If the aircraft motion about a steady, straight, symmetric flight are to be determined, for instance to investigate the stability of the steady flight conditions, it is advantageous to use the so-called stability reference frame. This is a system of body axes, as previously defined, of which the XS -axis is parallel to the direction of motion of the center of gravity in the steady initial flight condition. But during the disturbed motion of the aircraft the reference frame is fixed to the aircraft. This particular choice of the XS -axis results in the equations of motion in, u0 = V w0 = 0 In addition, the angle of attack in the steady reference condition, measured relative to the XS -axis, is zero, α0 = 0 (4-24) θ0 = γ0 A disadvantage of the stability axes becomes apparent, if the disturbed motions about several different equilibrium conditions have to be studied in succession. During each of these disturbed motions the XS -axis and the ZS -axis have a different direction relative to the aircraft. This implies, that the aerodynamic variables, the components of the motion and also the moments and products of inertia in each new case refer to a different attitude of the reference frame relative to the aircraft, although that attitude remains invariant during each of the disturbed motions. In the following section the linearized set of equations for steady, straight, symmetric flight expressed in the stability reference frame is given. Thereafter the equations needed to determine the moment and products of inertia about the stability reference frame are given.
4-2-1
Linearized set of equations
In order to simplify the notation in the remainder of this chapter the changes in the variable angles relative to the reference condition are indicated without the letter ∆, e.g. θ rather than ∆θ, δe instead of ∆δe 1 . The magnitude of the relevant variable in steady flight is indicated by the subscript 0 ( 0 ). Since the aerodynamics are considered to be uncoupled, the set of equations can be split up into two parts. One describes the motion in the symmetry plane of the aircraft, i.e. the symmetric motion, and the other part describes the asymmetric motion. If the stability reference frame is used the equations of motion can now be written as follows,
1
During simulations one should bear in mind that u does not represent the true velocity along the Xb -axis. One should add the initial condition u0 to obtain the true value for that velocity! Flight Dynamics
132
Linearized Equations of Motion
I Symmetric motions −W cos θ0 θ −W sin θ0 θ
+Xu u + Xw w + Xq q + Xδe δe + Xδt δt = mu˙ +Zu u + Zw w + Zw˙ w˙ + Zq q + Zδe δe + Zδt δt = m (w˙ − q V )
(4-25)
Mu u + Mw w + Mw˙ w˙ + Mq q + Mδe δe + Mδt δt = Iy q˙ θ˙ = q II Asymmetric motions W cos θ0 ϕ
+Yv v + Yv˙ v˙ + Yp p + Yr r + Yδa δa + Yδr δr = m (v˙ + r V ) Lv v + Lp p + Lr r + Lδa δa + Lδr δr = Ix p˙ − Jxz r˙ Nv v + Nv˙ v˙ + Np p + Nr r + Nδa δa + Nδr δr = Iz r˙ − Jxz p˙ ψ˙ =
(4-26)
r cos θ0
ϕ˙ = p + r tg θ0 The kinematic relations remain the same, i.e.: ϕ˙ = p + tan θ0 r θ˙ = q ψ˙ = cos1θ0 r
4-2-2
(4-27)
Moments and products of inertia
The moments and products of inertia of the aircraft depend not only on the mass distribution of the aircraft, but also on the chosen directions of the axes of the reference frame relative to the aircraft (stability frame of reference FS versus body-fixed frame of reference Fb ). Due to the symmetry of the aircraft two of the three products of inertia are zero, Z Jyz = yz dm = 0 m
Jxy =
Z
xy dm = 0
m
This result does not depend on the directions of the two sets of body axes (stability and body-fixed frame of reference) in the plane of symmetry. Suppose now, that for an arbitrary direction of the Xb -axis the magnitudes of the three moments of inertia and the remaining product of inertia are given. The direction of the Xb -axis is given by the angle ε + η1 between the Xb -axis and the negative Xr -axis, see
4-2 Linearization about steady, straight, symmetric flight condition
133
Zr
X0a
E n1 X1a n2
X2a
Z2
Z1 X2
P
O
X1
Xr
Figure 4-5: The angles between the principal inertial axes and the aircraft reference axes
figure 4-5. This fixes also the direction of the Zb -axis. If the Xb -axis has a different angle ε + η2 with the negative Xr -axis, see figure 4-5, and the first and second reference frame are indicated by the subscripts 1 and 2 respectively, the following relations hold,
x2 = x1 cos (η2 − η1 ) + z1 sin (η2 − η1 ) y2 = y1
z2 = −x1 sin (η2 − η1 ) + z1 cos (η2 − η1 ) This leads to the following moments and products of inertia, relative to the new reference frame, Ix2
= Ix1 cos2 (η2 − η1 ) + Iz1 sin2 (η2 − η1 ) − Jxz1 sin2 (η2 − η1 )
Iy2
=
Iz2
= Ix1 sin2 (η2 − η1 ) + Iz1 cos2 (η2 − η1 ) + Jxz1 sin2 (η2 − η1 )
Jxz2
=
Iy1 (4-28)
1 2
(Ix1 − Iz1 ) sin (2 (η2 − η1 )) + Jxz1 cos (2 (η2 − η1 ))
As could be expected, the moment of inertia about the Yb -axis does not depend on the directions of the body axes in the plane of symmetry. For one particular direction of the Flight Dynamics
134
Linearized Equations of Motion
Xb -axis, and correspondingly of the Zb -axis, the product of inertia Jxz is zero. The axes having these directions are called the principal inertial axes in the plane of symmetry. They are indicated as X0 - and Z0 -axes and have an angle ε with the negative Xr - and Zr -axis, see figure 4-5. As the products of inertia Jxy and Jyz are always zero, the Yb -axis is always a principal inertial axis. This follows also directly from the fact that the Yb -axis is perpendicular to the plane of symmetry. If for an arbitrary direction ε + η of the Xb -axis the inertial parameters are known the angle η of these axes with the principal inertial axes follow from equation (4-28) by equating Jxz2 and η2 to zero, tan 2η =
2 Jxz Ix − Iz
Using the principal moments of inertia Ix0 and Iz0 , which can be calculated from equation (4-28) for a given value η by letting η2 = 0, the inertial parameters for an arbitrary direction η of the body axes are, Ix = Ix0 cos2 η + Iz0 sin2 η Iy = Iy0 Iz = Ix0 sin2 η + Iz0 cos2 η Jxz =
4-3
1 (Ix0 − Iz0 ) sin 2η 2
Equations of motion in non-dimensional form
The differential equations derived in the foregoing can be applied directly to the calculation of the symmetric and asymmetric aircraft motions about a given condition of steady, straight, symmetric flight as a response to a given disturbance. Very often, however, the equations are used in a non-dimensional form. The reason for this practice is that the aerodynamic forces and moments in the equations are usually expressed in non-dimensional coefficients2 . This leads to writing the entire equations in non-dimensional form. To this end, also the inertial characteristics and the time scale are made non-dimensional. One of the several ways in which this can be done, is discussed in the following. As a starting point, three independent units are chosen, i.e. length having the dimension [ℓ], velocity with dimension [ℓ][t]−1 and mass with dimension [m]. The actual values of these units are, Symmetric 2
Asymmetric
With the non-dimensional form of the aerodynamic forces and moments, we can compare aircraft independent of aircraft dimensions and weight.
4-3 Equations of motion in non-dimensional form
135
motions :
motions :
length
c¯
b
velocity
V
V
mass
ρS¯ c
ρSb
Evidently different units of length are used for the symmetric and the asymmetric motions, the mean aerodynamic chord c¯ and the wing span b respectively. These are also the customary lengths used to make the symmetric moment M and the asymmetric moments L and N , respectively, non-dimensional. The unit of mass chosen for both types of motion is the mass of a certain volume of air surrounding the aircraft equal to the wing area multiplied by the respective unit of length. With these units all variables in the equations of motion can be made non-dimensional, according to the schemes in tables 4-1 and 4-2. Using these schemes, the way in which the equations of motion are made non-dimensional can now be discussed more in detail.
4-3-1
Symmetric Motions
For the symmetric motions the two force equations, in which each of the terms has the dimension of a force, i.e. X and Z, are made non-dimensional by a division by 12 ρV 2 S. The moment equation, in which each of the terms has the dimension of a moment, i.e. M , is divided by 12 ρV 2 S¯ c. Starting from equations (4-25) the result is, Xu u + Xw w + Xq q + Xδe δe + Xδt δt m · u˙ W cosθ0 ·θ+ = 1 2 1 1 2 2 2 ρV S 2 ρV S 2 ρV S Zu u + Zw w + Zw˙ w˙ + Zq q + Zδe δe + Zδt δt W sinθ0 m (w˙ − q · V ) − 1 2 ·θ+ = 1 1 2 2 2 ρV S 2 ρV S 2 ρV S Iy q˙ Mu u + Mw w + Mw˙ w˙ + Mq q + Mδe δe + Mδt δt = 1 2 1 2 c c 2 ρV S¯ 2 ρV S¯ ˙c θ¯ q¯ c = V V
(4-29)
−
(4-30) (4-31) (4-32)
In elaborating the two force equations, use is made of the two equilibrium conditions, W sin θ0 = X0 W cos θ0 = −Z0
Using the previous scheme of table 4-1, these conditions are written as follows in equation (4-29), −W cos θ0 = CZ0 1 2 2 ρV S
(4-33) Flight Dynamics
136
Linearized Equations of Motion
and in equation (4-30), −W sin θ0 = −CX0 1 2 2 ρV S
(4-34)
Next, the partial derivatives of the aerodynamic forces and the moment are further elaborated. As an example the contribution Xu · u is considered. It is written as, Xu · u 1 2 2 ρV S
=
Xu 1 2 ρV S
·
u Xu = 1 ·u ˆ V 2 ρV S
As both the left and right hand sides of this latter expression are non-dimensional, as u ˆ is, Xu the quotient 1 ρV S must also be non-dimensional. This quotient is called a stability derivative 2
written as CXu , CXu =
Xu 1 2 ρV S
and, Xu · u 1 2 2 ρV S
ˆ = CXu · u
It should be emphasized that for the stability derivative CXu in particular, but also for CZu and Cmu , the order of differentiation with respect to airspeed and non-dimensionalizing using the factor 12 ρV 2 S in which the airspeed figures as well, is relevant for the resulting values of CXu , CZu and Cmu , CXu =
1 1 2 ρV
S
·
∂X ∂CX 6= ∂u ∂u ˆ
(4-35)
In a similar manner it follows, Xw w Xw · w = 1 · = CXα α 1 2 V 2 ρV S 2 ρV S where the stability derivative CXα is, CXα =
Xw 1 2 ρV S
=
∂CX ∂α
Now the order of differentiation with respect to w and non-dimensionalizing using the factor 1 1 2 2 2 ρV S may be changed, as 2 ρV S does not contain the factor w. The expressions for the different stability derivatives for symmetric motions are collected in table 4-3. The non-dimensional control derivatives are given in table 4-4. See also table 4-5 for the definition of the non-dimensional moments and products of inertia.
4-3 Equations of motion in non-dimensional form
137
The non-dimensional equations for the symmetric motions now become, CZ0 θ + CXu u ˆ + CXα α + CXq Dc θ + CXδe δe + CXδt δt = 2µc Dc u ˆ ˆ + CZα α + CZα˙ Dc α + CZq Dc θ + CZδe δe + CZδt δt = 2µc (Dc α − Dc θ) −CX0 θ + CZu u q¯ c Cmu u ˆ + Cmα α + Cmα˙ Dc α + Cmq Dc θ + Cmδe δe + Cmδt δt = 2µc KY2 Dc V ˙θ¯ c q¯ c = (4-36) V V The four equations (4-36) are now rewritten in an ordered form. In this way, the following c differential equations in the variables u ˆ, α, θ and q¯ V result in, q¯ c (CXu − 2µc Dc ) u ˆ + CXα α + CZ0 θ + CXq + CXδe δe + CXδt δt V q¯ c ˆ + [CZα + (CZα˙ + 2µc ) Dc ] α − CX0 θ + CZq + 2µc + CZδe δe + CZδt δt CZu u V q¯ c −Dc θ + V q¯ c Cmu u ˆ + (Cmα + Cmα˙ Dc ) α + Cmq − 2µc KY2 Dc + Cmδe δe + Cmδt δt V
=0 =0 =0 =0
(4-37)
or using matrix notation,
CXu − 2µc Dc u ˆ CXα CZ0 CXq α C C + (C − 2µ ) D −C C + 2µ Zu Zα Zα˙ c c X0 Zq c θ 0 0 −Dc 1 q¯ c 2 Cmu Cmα + Cmα˙ Dc 0 Cmq − 2µc KY Dc V
and with,
−CXδe −CZδ e = 0 −Cmδe
−CXδt −CZδt δe δt 0 −Cmδt
(4-38)
θ = α+γ the left hand side of equation (4-38) can also be expressed in the variables u ˆ, γ, θ and q¯ c T equation (4-38) is written as P x = Q δe with x = u ˆ, γ, θ, V , matix P is equal to,
CXu − 2µc Dc −CXα CXα + CZ0 C −C − (C − 2µ ) D Zu Zα Zα˙ c c CZα − CX0 0 0 −Dc Cmu −Cmα − Cmα˙ Dc Cmα
Cmα˙
q¯ c V .
If
CXq CZα˙ + CZq 1 2 + Cmq − 2µc KY Dc
The right hand side of equation (4-38), i.e. Q [δe , δt ]T , remains unchanged with matrix Q equal to, Flight Dynamics
138
Linearized Equations of Motion
−CXδe −CZδ e Q= 0 −Cmδe
4-3-2
−CXδt −CZδt 0 −Cmδt
Asymmetric Motions
The equations for the asymmetric motions are made non-dimensional in close analogy with the foregoing. The force equation is again divided by 12 ρV 2 S and the two moment equations by 12 ρV 2 Sb. The result is,
W cos θ0 · ϕ Yv v + Yv˙ v˙ + Yp p + Yr r + Yδa δa + Yδr δr + 1 1 2 2 2 ρV S 2 ρV S
=
m (v˙ + r · V ) 1 2 2 ρV S
Lv v + Lp p + Lr r + Lδa δa + Lδr δr 1 2 2 ρV Sb
=
Ix p˙ − Jxz r˙ 1 2 2 ρV Sb
Nv v + Nv˙ v˙ + Np p + Nr r + Nδa δa + Nδr δr 1 2 2 ρV Sb
=
Iz r˙ − Jxz p˙ 1 2 2 ρV Sb
˙ ψb V /
=
b r V cos θ0
ϕb ˙ V
=
pb rb + tg θ0 V V
The component of the weight along the ZS -axis is now written as,
W cos θ0 W cos γ0 = 1 2 = CL 1 2 2 ρV S 2 ρV S Next, the partial derivatives of the aerodynamic force and the two moments are expressed using the non-dimensional stability derivatives according to the scheme of table 4-6. See also table 4-7 and 4-8 for the definition of the non-dimensional control derivatives and the moments and products of inertia respectively. If in addition θ0 in the kinematic relations is replaced by γ0 , using equation (4-24), the non-dimensional equations for the asymmetric motions are,
4-3 Equations of motion in non-dimensional form
139
pb rb CL ϕ + CYβ β + CYβ˙ Db β + CYp 2V + CYr 2V + CYδa δa + CYδr δr = 2µb Db β + 2
rb 2V
pb rb 2 D pb Cℓβ β + Cℓp 2V + Cℓr 2V + Cℓδa δa + Cℓδr δr = 4µb KX b 2V rb −KXZ Db 2V
pb rb rb + Cnr 2V + Cnδa δa + Cnδr δr = 4µb KZ2 Db 2V Cnβ β + Cnβ˙ Db β + Cnp 2V pb −KXZ Db 2V 1 2 Db ψ
=
1 rb cos γ0 2V
1 2 Db ϕ
=
pb 2V
+
rb 2V
tg γ0 (4-39)
In level flight (γ0 = 0) the kinematic relations are reduced to, 1 2 Db ψ
=
rb 2V
1 2 Db ϕ
=
pb 2V
By adding the latter of these two relations, 1 pb − Db ϕ + =0 2 2V the equations of motion (4-39) are changed into a set of four first order differential equations pb rb in β, ϕ, 2V and 2V , h “ ” i pb CYβ + CYβ˙ − 2µb Db β + CL ϕ + CYp 2V + (CYr − 4µb )
rb 2V
+ CYδa δa + CYδr δr
=
0
pb 2V
=
0
+ Cℓδa δa + Cℓδr δr
=
0
+ Cnδa δa + Cnδr δr
=
0
− 21 Db ϕ + “ ” 2 D Cℓβ β + Cℓp − 4µb KX b
pb 2V
+ (Cℓr + 4µb KXZ Db )
“ ” ` ´ Cnβ + Cnβ˙ Db β + Cnp + 4µb KXZ Db
pb 2V
` ´ 2D + Cnr − 4µb KZ b
rb 2V
rb 2V
(4-40) Using matrix notation, equations (4-40) are written as,
CYβ + CYβ˙ − 2µb Db Cnβ
0 Cℓβ + Cnβ˙ Db
CL
CYp
− 12 Db 1 2D 0 Cℓp − 4µb KX b 0 Cnp + 4µb KXZ Db
β ϕ 0 pb Cℓr + 4µb KXZ Db 2V rb Cnr − 4µb KZ2 Db 2V CYr − 4µb
Flight Dynamics
140
Linearized Equations of Motion
−CYδa 0 = −Cℓ δa −Cnδa
−CYδr δa 0 −Cℓδr δr −Cnδr
(4-41)
At the end of this section it should be noted that,
• Often, especially in the British literature, the equations of motion are made nondimensional using a unit of time not equal to Vc¯ , but τc µc Vc¯ seconds for the symmetric motions and τb µb Vb seconds rather than Vb seconds for the asymmetric motions. The particular way of making the equations non-dimensional is determined in the first place by the habits of the user. None of the various methods offers clear advantages or disadvantages in comparison with other possible methods. This explains why only one method has been described in detail here. • In chapters 6 and 8 attention is given to the methods used to obtain the various stability and control derivatives for the symmetric and asymmetric motions respectively. • Reliable quantitative data on moments and products of inertia of aircraft are rare. Quantitative calculations can be time-consuming and may produce rather inaccurate results. The most reliable inertial parameters of aircraft are those obtained experimentally, usually by oscillating the suspended aircraft on the ground. Details on this experimental technique may be found in references [40, 95, 164, 168]. Figure 4-6 shows the relation between the moment of inertia Iy and the mass for multiple types of aircraft. One can make educated guesses for the inertia properties of new types of aircraft using such figures. • Appendix D lists symmetrical and asymmetrical inertial data, as well as stability and control derivatives for both the symmetric and asymmetric motions.
4-3 Equations of motion in non-dimensional form
141
[Moment of Inertia Iy] 1/5 vs [mass]1/3, inertia2002,12−Jan−2006 50 Best Fit border 50% of data
45 40
IY(1/5) [kgm2]
35 30 25 20 15 10 5 0
0
10
20
30
40
50 (1/3)
Mass
60
70
80
90
[kg]
Figure 4-6: Relation between moment of inertia Iy and the mass
Flight Dynamics
142
Linearized Equations of Motion
Dimensional parameter
Dimension
Divisor
Non-dimensional parameter
t
[t]
c¯ V
sc =
d dt
[t]−1
V c¯
Dc =
c¯ d V dt
d2 dt2
[t]−2
V2 c¯2
Dc2 =
c¯2 d2 V 2 dt2
u
[ℓ][t]−1
V
u ˆ=
u V
w
[ℓ][t]−1
V
α=
w V
q
[t]−1
V c¯
q¯ c V
u˙
[ℓ][t]−2
V2 c¯
Dc u ˆ=
u˙ c¯ V V
=
u ˆ˙ c¯ V
w˙
[ℓ][t]−2
V2 c¯
Dc α =
w˙ c¯ V V
=
α¯ ˙c V
q˙
[t]−2
V2 c¯2
c c¯ Dc q¯ V = q˙ V 2
m
[m]
ρS¯ c
µc =
IY
[m][ℓ]2
ρS¯ c · c¯2
µc KY2 =
ky
[ℓ]
c¯
KY =
X
[m][ℓ][t]−2
1 c· 2 ρS¯
c¯ ·
V2 c¯2
CX =
Z
[m][ℓ][t]−2
1 c· 2 ρS¯
c¯ ·
V2 c¯2
CZ =
M
[m][ℓ]2 [t]−2
1 c2 2 ρS¯
· c¯ ·
V c¯
t d dsc
=
=
d2 ds2c
2
V2 c¯2
m ρS¯ c
Cm =
IY ρS¯ c3
ky c¯ X
1 ρV 2 S 2
Z
1 ρV 2 S 2
M
1 ρV 2 S¯ c 2
Table 4-1: Non-dimensional parameters in the equations of motion, symmetric motions
4-3 Equations of motion in non-dimensional form
143
Dimensional parameter
Dimension
Divisor
Non-dimensional parameter
t
[t]
b V
sb =
d dt
[t]−1
V b
Db =
b d V dt
d2 dt2
[t]−2
V2 b2
Db2 =
b2 d2 V 2 dt2
v
[ℓ][t]−1
V
β=
p
[ℓ][t]−1
2V b
pb 2V
r
[t]−1
2V b
rb 2V
v˙
[ℓ][t]−2
V2 b
Db β =
p˙
[t]−2
2V 2 b2
pb Db 2V =
pb ˙ 2 2V 2
r˙
[t]−2
2V 2 b2
rb Db 2V =
rb ˙ 2 2V 2
m
[m]
ρSb
µb =
IX
[m][ℓ]2
ρSb · b2
2 = µb KX
IX ρSb3
IZ
[m][ℓ]2
ρSb · b2
µb KZ2 =
IZ ρSb3
JXZ
[m][ℓ]2
ρSb · b2
µb KXZ =
kx
[ℓ]
b
KX =
kx b
kz
[ℓ]
b
KZ =
kz b
V b
t d dsb
=
=
d2 ds2b
v V
v˙ b V V
=
˙ βb V
m ρSb
JXZ ρSb3
Table 4-2: Non-dimensional parameters in the equations of motion, asymmetric motions
Flight Dynamics
144
Linearized Equations of Motion
Dimensional parameter
Dimension
Divisor
Non-dimensional parameter
Y
[m][ℓ][t]−2
1 2 ρSb
L
[m][ℓ]2 [t]−2
1 2 2 ρSb
·b·
V2 b2
Cℓ =
N
[m][ℓ]2 [t]−2
1 2 2 ρSb
·b·
V2 b2
Cn =
·b·
V2 b2
CY =
Y
1 ρV 2 S 2
L
1 ρV 2 Sb 2
N
1 ρV 2 Sb 2
Table 4-2: (Continued) Non-dimensional parameters in the equations of motion, asymmetric motions
Definition
∂CX ∂u ˆ
S
·
∂Z ∂u
6=
∂CZ ∂u ˆ
S¯ c
·
∂M ∂u
S
·
∂X ∂w
=
∂CX ∂α
1 ρV S 2
·
∂Z ∂w
=
∂CZ ∂α
·
∂M ∂w
1
1 ρV 2
1
1 ρV 2
CZα =
1
1
Cmα =
1 ρV 2
S¯ c
∂Cm ∂u ˆ
6=
=
∂Cm ∂α
∂CZ ˙c ∂ α¯ V
1
·
∂Z ∂ w˙
1
·
∂M ∂ w˙
=
∂Cm ˙c ∂ α¯ V
S¯ c
·
∂X ∂q
=
∂CX c ∂ q¯ V
S¯ c
·
∂Z ∂q
=
∂CZ c ∂ q¯ V
1 ρV S¯ c2 2
·
∂M ∂q
CZα˙ =
Cmq =
6=
1
CXα =
CZq =
∂X ∂u
1 ρV 2
Cmu =
CXq =
·
1 ρV 2
CZu =
Cmα˙ =
S
1
CXu =
1 ρS¯ c 2
1 ρS¯ c2 2
1
1 ρV 2
1
1 ρV 2
1
=
=
∂Cm c ∂ q¯ V
Table 4-3: Stability derivatives, symmetric motions
4-3 Equations of motion in non-dimensional form
145
Definition
CXδe = CZδe = Cmδe = CXδt = CZδt = Cmδt =
1
·
∂X ∂δe
=
∂CX ∂δe
1
·
∂Z ∂δe
=
∂CZ ∂δe
1
·
∂M ∂δe
1
·
∂X ∂δt
=
∂CX ∂δt
1
·
∂X ∂δt
=
∂CZ ∂δt
1
·
∂X ∂δt
=
∂Cm ∂δt
1 ρV 2 S 2
1 ρV 2 S 2
1 ρV 2 S¯ c 2
1 ρV 2 S 2
1 ρV 2 S 2
1 ρV 2 S 2
=
∂Cm ∂δe
Table 4-4: Control derivatives, symmetric motions
Definition
µc =
m ρS¯ c
KY2 =
IY m¯ c2
Table 4-5: Moments and products of inertia, symmetric motions
Flight Dynamics
146
Linearized Equations of Motion
Definition
CYβ = Cℓβ = Cnβ = CYβ˙ = Cnβ˙ = CYp = Cℓp = Cnp = CYr = Cℓr = Cnr =
1
1 ρV 2
S
∂Y ∂v
·
=
∂CY ∂β
·
∂L ∂v
=
∂Cℓ ∂β
Sb
·
∂N ∂v
=
∂Cn ∂β
Sb
·
∂Y ∂ v˙
=
∂CY ˙ ∂ βb V
1 ρSb2 2
·
∂N ∂ v˙
=
∂Cn ˙ ∂ βb V
Sb
·
∂Y ∂p
=
∂CY pb ∂ 2V
2 Sb2
·
∂L ∂p
=
∂Cℓ pb ∂ 2V
2 Sb2
·
∂N ∂p
=
∂Cn pb ∂ 2V
2 Sb
·
∂Y ∂r
=
∂CY rb ∂ 2V
2 Sb2
·
∂L ∂r
=
∂Cℓ rb ∂ 2V
2 Sb2
·
∂N ∂r
=
∂Cn rb ∂ 2V
1
1 ρV Sb 2
1
1 ρV 2
1
1 ρV 2
1
2
1 ρV 2
1 ρV 2
1 ρV 2
1 ρV 2
1 ρV 2
1 ρV 2
Table 4-6: Stability derivatives, asymmetric motions
4-3 Equations of motion in non-dimensional form
147
Definition
CYδa = Cℓδa = Cnδa = CYδr = Cℓδr = Cnδr =
1
1 ρV 2 S 2
∂Y ∂δa
·
=
∂CY ∂δa
1
·
∂L ∂δa
=
∂Cℓ ∂δa
1
·
∂N ∂δa
=
∂Cn ∂δa
1 ρV 2 Sb 2
1 ρV 2 Sb 2
1
1 ρV 2 S 2
∂Y ∂δr
·
=
∂CY ∂δr
1
·
∂L ∂δr
=
∂Cℓ ∂δr
1
·
∂N ∂δr
=
∂Cn ∂δr
1 ρV 2 Sb 2
1 ρV 2 Sb 2
Table 4-7: Control derivatives, asymmetric motions
Definition
µb =
m ρSb
2 = KX
IX mb2
KZ2 =
IZ mb2
JXZ =
IXZ mb2
Table 4-8: Moments and products of inertia, asymmetric motions
Flight Dynamics
148
Linearized Equations of Motion
4-4
Symmetric equations of motion in state-space Form
The linearized ‘deviation equations’ for the symmetric motions, where CXq = 0, are, see equation (4-38),
CXα CZ0 0 CXu − 2µc Dc u ˆ α C C + (C − 2µ ) D −C C + 2µ Zu Zα Zα˙ c c X0 Zq c θ 0 0 −Dc 1 q¯ c 2 Cmu Cmα + Cmα˙ Dc 0 Cmq − 2µc KY Dc V
−CXδe −CZδ e = 0 −Cmδe
=
−CXδt −CZδt δe δt 0 −Cmδt
Linear Time Invariant (LTI) systems are given in the so called state-space form, x˙ = A x + B u
(4-42)
y =Cx+D u
(4-43)
With A (n × n) the state-matrix, B (n × m) the input-matrix, C (r × n) the output matrix and D (r × m) the direct matrix. Furthermore, x (n × 1) and y (r × 1) are the state and output vector respectively, while u (m × 1) is the input vector. Since the system is LTI, the matrices A, B, C, and D have constant elements. This form can be obtained by rearranging equation (4-38). First, all terms without the differential operator Dc are placed on the right hand side of equation (4-38). Furtherd . The result is, more, the differential operator Dc is replaced by Vc¯ dt
d −2µc Vc¯ dt 0 0 0
−CXu −CZu 0 −Cmu
0 d (CZα˙ − 2µc ) Vc¯ dt 0 d Cmα˙ Vc¯ dt
−CXα −CZ0 −CZα CX0 0 0 −Cmα 0
d − Vc¯ dt 0
0 0 0 −2µc KY2
0 u ˆ α − CZq + 2µc θ −1 q¯ c −Cmq V
This equation can be written in the form, Px˙ = P with,
0 0
c¯ d V dt
u ˆ α θ =
−CXδe −CZδ e + 0 −Cmδe
dx =Qx+Ru dt
q¯ c V
−CXδt −CZδt δe δt 0 −Cmδt
(4-44)
4-4 Symmetric equations of motion in state-space Form
−2µc Vc¯ 0 P= 0 0
−CXu −CZu Q= 0 −Cmu
0 (CZα˙ − 2µc ) Vc¯ 0 Cmα˙ Vc¯ −CXα −CZ0 −CZα CX0 0 0 −Cmα 0
−CXδe −CZδ e R= 0 −Cmδe
149
0 0 − Vc¯ 0
0 0 0 −2µc KY2
c¯ V
0 − CZq + 2µc −1 −Cmq
−CXδt −CZδt 0 −Cmδt
c T and x = u ˆ α θ q¯ , while u = δe . The final state-space form is obtained by premultiplying V equation (4-44) by the inverse of matrix P, x˙ = P−1 Q x + P−1 R u = A x + B u
(4-45)
In flight dynamics, equation (4-45) is usually written as, ˙ u ˆ α˙ θ˙
q¯ ˙c V
xu zu = 0 mu
xα zα 0 mα
xθ 0 u ˆ α zθ zq V θ 0 c¯ q¯ c mθ mq V
xδe zδe + 0 mδe
xδt zδt δe 0 δt mδt
(4-46)
The definition of the newly introduced symbols are recapitulated in table 4-9. Dimensional eigenvalues λ are obtained using MATLAB’s routine eig.m (for the state matrix A).
Flight Dynamics
150
Linearized Equations of Motion
x···
z···
m···
u
V CXu c¯ 2µc
CZu V c¯ 2µc −CZα˙
V c¯
α
V CXα c¯ 2µc
CZα V c¯ 2µc −CZα˙
V c¯
θ
V CZ0 c¯ 2µc
CX0 − Vc¯ 2µc −C Zα ˙
q
V CXq c¯ 2µc
V 2µc +CZq c¯ 2µc −CZα˙
δe
V CXδe c¯ 2µc
CZδ V e c¯ 2µc −CZα˙
V c¯
δt
C V Xδt c¯ 2µc
CZδ V t c¯ 2µc −CZα˙
V c¯
Cm α ˙ c −CZ
Cmu +CZu 2µ 2µc KY2
Cm α ˙ c −CZ
Cmα +CZα 2µ 2µc KY2
α ˙
Cm α ˙ c −CZ α ˙ 2 2µc KY
CX0 2µ
− Vc¯ V c¯
α ˙
2µc +CZ q c −CZ α ˙ 2µc KY2
Cmq +Cmα˙ 2µ
Cm α ˙ e 2µc −CZα ˙ 2µc KY2
Cmδ +CZδ e
Cm α ˙ t 2µc −CZα ˙ 2 2µc KY
Cmδ +CZδ t
Table 4-9: Symbols appearing in the general state-space representation of equation (4-46)
4-5 Asymmetric equations of motion in state-space form
4-5
151
Asymmetric equations of motion in state-space form
Similar to section 4-4 the linearized ‘deviation equations’ for the asymmetric motions, are, see equation (4-41),
CYβ
+ CYβ˙ − 2µb Db Cnβ
0 Cℓβ + Cnβ˙ Db
CL
CYp
− 12 Db 1 2D 0 Cℓp − 4µb KX b 0 Cnp + 4µb KXZ Db
−CYδa 0 −Cℓ δa −Cnδa
β ϕ 0 = pb Cℓr + 4µb KXZ Db 2V rb Cnr − 4µb KZ2 Db 2V CYr − 4µb
−CYδr δa 0 −Cℓδr δr −Cnδr
These equations of motion are transformed to the state-space form as in, x˙ = A x + B u
(4-47)
This form can be obtained by rearranging equation (4-41). First, all terms without the differential operator Db are placed on the right hand side of equation (4-41). Furthermore, d the differential operator Db is replaced by Vb dt . The result is, CYβ˙ − 2µb Vb 0 0 d Cnβ˙ Vb dt
−CYβ 0 −Cℓ β −Cnβ
d dt
−CL −CYp 0 −1 0 −Cℓp 0 −Cnp
0 − 12 Vb 0 0
0 d dt
0
0 2 b d −4µb KX V dt d 4µb KXZ Vb dt
0 4µb KXZ Vb −4µb KZ2 Vb
d dt d dt
β − (CYr − 4µb ) −CYδa ϕ 0 0 pb + −Cℓr −Cℓδa 2V rb −C −Cnr nδa 2V
Again, this equation can be written in the form, Px˙ = P
β ϕ = pb 2V rb 2V
−CYδr δa 0 −Cℓδr δr −Cnδr
dx =Qx+Ru dt
(4-48)
with in this case, CYβ˙ − 2µb Vb 0 P= 0 Cnβ˙ Vb
0
0
− 12 Vb 0 0
0 2 b −4µb KX V 4µb KXZ Vb
0
0 4µb KXZ Vb −4µb KZ2 Vb Flight Dynamics
152
Linearized Equations of Motion
y···
l···
n···
β
V CYβ b 2µb
2 V Cℓβ KZ +Cnβ KXZ b 4µb (K 2 K 2 −K 2 ) X Z XZ
2 V Cℓβ KXZ +Cnβ KX b 4µb (K 2 K 2 −K 2 ) X Z XZ
ϕ
V CL b 2µb
0
0
p
V CYp b 2µb
2 V Cℓp KZ +Cnp KXZ b 4µb (K 2 K 2 −K 2 ) X Z XZ
2 V Cℓp KXZ +Cnp KX b 4µb (K 2 K 2 −K 2 ) X Z XZ
r
V CYr −4µb b 2µb
2 V Cℓr KZ +Cnr KXZ b 4µb (K 2 K 2 −K 2 ) X Z XZ
2 V Cℓr KXZ +Cnr KX b 4µb (K 2 K 2 −K 2 ) X Z XZ
δa
V CYδa b 2µb
2 V Cℓδa KZ +Cnδa KXZ b 4µb (K 2 K 2 −K 2 ) X Z XZ
2 V Cℓδa KXZ +Cnδa KX b 4µb (K 2 K 2 −K 2 ) X Z XZ
δr
V CYδr b 2µb
2 V Cℓδr KZ +Cnδr KXZ b 4µb (K 2 K 2 −K 2 ) X Z XZ
2 V Cℓδr KXZ +Cnδr KX b 4µb (K 2 K 2 −K 2 ) X Z XZ
Table 4-10: Symbols appearing in the general state-space representation of equation (4-50)
−CYβ 0 Q= −Cℓ β −Cnβ
−CL −CYp 0 −1 0 −Cℓp 0 −Cnp
−CYδa 0 R= −Cℓ δa −Cnδa
− (CYr − 4µb ) 0 −Cℓr −Cnr
−CYδr 0 −Cℓδr −Cnδr
h iT pb rb and x = β ϕ 2V , while u = [δa δr ]T . The final state-space form is obtained by 2V pre-multiplying equation (4-48) by the inverse of matrix P, x˙ = P−1 Q x + P−1 R u = A x + B u
(4-49)
In flight dynamics, equation (4-49) is usually written as,
β˙ yβ ϕ˙ 0 pb ˙ = lβ 2V rb ˙ nβ 2V
yϕ 0 0 0
yp 2 Vb lp np
β yr 0 0 ϕ 0 pb + lδa lr 2V rb nδa nr 2V
yδr 0 δa lδr δr nδr
(4-50)
The definition of the newly introduced symbols are recapitulated in table 4-10. Dimensional eigenvalues λ are obtained using MATLAB’s routine eig.m (for the state-matrix A).
Part II
Static Stability Analysis
Flight Dynamics
Chapter 5 Analysis of Steady Symmetric Flight
The analysis of both static and dynamic stability and control characteristics of aircraft requires a mathematical model describing the effects of aerodynamic flow parameters as airspeed, angle of attack, angle of sideslip and body rotation rates on the aerodynamic forces and aerodynamic moments acting on the aircraft, see also chapters 3 and 4. Current practice is to extensively rely on tools from Computational Fluid Dynamics (CFD) to derive these characteristics from. It was and and still is considered good practice, however, to augment and verify these results from CFD by using results from steady or rotary balance wind tunnel experiments with models of the complete aircraft or of isolated components. During the flight test campaign of the prototype aircraft these models can subsequently be improved by including flight test results using aerodynamic model identification techniques, see for example [94]. The disadvantage, however, of just using data from either CFD, wind tunnel experiments or flight tests on the complete model or actual aircraft is that it will not necessarily contribute much to the physical understanding of the resulting aerodynamic mechanisms leading to the observed aerodynamic force and moments characteristics. Also, there is still the problem of incorporating and interpreting the wind tunnel experimental results on isolated aircraft components. This is the rationale behind the approach taken in the present chapter to first look at the characteristics of the isolated wing to which subsequently a fuselage and nacelles on the wing or fuselage are added. The final step is to add the effects of horizontal and vertical tail planes to this wing-body-nacelles composite. The present chapter starts with an analysis of the aerodynamic forces and the aerodynamic moment acting on a symmetrical wing at some angle of attack. It is shown that the introduction of the concept of aerodynamic center leads to a surprisingly simple model for the force and moment characteristics. The concept of static stability is introduced and the necessary conditions to allow steady, stable flight. Next, it is shown how to derive a model of the force and moment characteristics of wings of arbitrary shape and twist from wing profile characteristics by applying classical 2-dimensional ’strip theory’. It is shown in particular how to compute the location of the wing aerodynamic center as well as the aerodynamic moment Flight Dynamics
156
Analysis of Steady Symmetric Flight CN CL
CR α
XB Cm
α
CD
V
CT
ZB Figure 5-1: The aerodynamic forces and the moment acting on a wing in symmetric flight
about it. Next, the isolated fuselage is discussed and the effect the fuselage has on the aerodynamic characteristics of the wing, and vice versa, the so called interference effects. In a similar way the effects of wing nacelles are discussed. The contribution of the horizontal tailplane to the aerodynamic forces and moment is discussed next. In classical aircraft configurations the horizontal tail plane is located somewhere behind the wing in the flow field induced by the wing-body-nacelles composite. This has a significant effect on the contribution of the tail plane to the aerodynamic forces in the plane of symmetry and on the pitching moment. The chapter proceeds with a general discussion on steady state flight conditions and is concluded with a discussion on the aerodynamic hinge moments of control surfaces.
5-1 5-1-1
Aerodynamic center Aerodynamic forces and moments acting on a wing
A. Aerodynamic moment as a function of angle of attack Figure 5-1 shows the components CL and CD , and CN and CT , of the dimensionless total aerodynamic force vector C R and the dimensionless aerodynamic moment Cm about a given arbitrary reference point, acting on a wing in symmetric flight, with
CN = CL cos α + CD sin α CT = CD cos α − CL sin α Figure 5-2 presents the familiar picture of CL and CD of a typical wing as a function of α, as measured in the wind tunnel. In addition the corresponding CN -α and CT -α curves are
5-1 Aerodynamic center
157
shown. From these figures it follows that the CL and CN curves are almost identical while the CD and CT are much more different. While CL and CN vary almost linearly with α over a large range of angles of attack, the relations betweenCD and CT and α is clearly non-linear. Figure 5-3 shows the relation between CN and CT of this wing. Note that CT turns negative as α increases! Similar results are obtained using linear-potential flow simulations. The configuration used for these simulations is depicted in figure 5-4 while the CN , CL versus α, CT , CD versus α, CN versus CT and CL versus CD curves are plotted in figures 5-5 and 5-6. The aerodynamic moment coefficient Cm not only depends on α but also on the position of the selected reference point. This holds true also for the slope of the Cm versus α curve which, as we will show later is of crucial importance for longitudinal stability. It is easy to compute Cm for an arbitrary reference point from wind tunnel measurements about a fixed model suspension point. Given the measured force and moment coefficients CN and CT and Cm about the suspension point (x1 , z1 ) for some angle of attack, the moment about an arbitrary reference point (x2 , z2 ) in the plane of symmetry can be computed as, see figure 5-7, Cm(x2 ,z2 ) = Cm(x1 ,z1 ) + CN
x2 − x1 z2 − z1 − CT c¯ c¯
(5-1)
where x and z are coordinates in the aircraft reference frame Fr . Figure 5-9 shows the resulting curves of Cm as a function of α for various positions of the reference point on the mac again for the case of a Fokker F-27 wing. From this figure it appears that, 1. The slope of the Cm − α curve strongly depends on the position of the reference point in the Xr -direction 2. I a large interval Cm varies almost linearly with α for reference points close to the mac Figure 5-9 also shows the influence on the Cm −α curve of a change in position of the reference point parallel to the Zr -axis. Now, the effect on the moment is much smaller. This is easily explained by equation (5-1), where CT is small relative to CN , see also figure 5-3. It appears that when the reference point is situated far below or above the mac, the moment is no longer a linear function of α which is due the non-linear variation of CT with α, see figure 5-2. Similar results from linear potential flow simulations are given in figure 5-10. For this case use has been made of the wing configuration as depicted in figure 5-4 (the data apply to the same wing as used in figures 5-5 and 5-6. The dependence of the Cm − α curve the selected reference point is a general phenomenon which does not apply to only wings but also to complete aircraft configurations. The relevant reference point in that case is the aircraft center of gravity, as we will see later.
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Analysis of Steady Symmetric Flight
CL
CN
CL
1.0
CN 0.8
0.6
0.4
0.2
-4
-2
0
2
4
6
8
10
α [o ]
-0.2
(A) CD
CT 0.08
CD 0.04
-4
-2
0
2
4
6
8
10
α [o ]
-0.04
-0.08
CT -0.12
-0.14
(B) Figure 5-2: CN , CL and CT , CD as functions of α for the case of a Fokker F-27 wing (from reference [149])
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159
CN 1.0
0.8
0.6
0.4
0.2
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0
-0.2
CT
Figure 5-3: CN as a function of CT for a Fokker F-27 wing (from reference [149])
Figure 5-4: Rectangular wing configuration with zero-twist, λ = simulations
1 3,
as used for linear potential flow
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Analysis of Steady Symmetric Flight
1
CN , CL
0.5
CN CL
0
−0.5 −5
0
α
5
10
0.04 0.02 0
CT CD
CT , CD
−0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 −5
0
α
5
10
Figure 5-5: Simulated CN , CL and CT , CD as functions of α for the wing configuration depicted in figure 5-4
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161
1
CN
0.5
0
−0.5 −0.14
−0.12
−0.1
−0.08
−0.06 CT
−0.04
−0.02
0
0.02
0.005
0.01
0.015
0.02 CD
0.025
0.03
0.035
0.04
1
CL
0.5
0
−0.5 0
Figure 5-6: Simulated CN as a function of CT and CL as a function of CD for the wing configuration depicted in figure 5-4
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Analysis of Steady Symmetric Flight
Zr
CN
CN
CR
CR
z2
z1 2
1
CT
CT Cm(x2 ,z2 )
x1
Cm(x1 ,z1 ) x2 Xr
1 1 Cm(x2 ,z2 ) = Cm(x1 ,z1 ) + CN x2 −x − CT z2 −z c¯ c¯
Figure 5-7: The variation of the moment with changes in the reference point
z
mac x
c¯ Figure 5-8: Location of point (x, z) with respect to the position of the mean aerodynamic chord
5-1 Aerodynamic center
163
It is possible to interpret the bundle of Cm − α curves in terms of some new concepts: the line of action of C R , the center of pressure, the aerodynamic center (ac) and the aerodynamic moment about the ac. B. The line of action of C R and the center of pressure The aerodynamic forces as well as the aerodynamic moment acting on the wing may also be obtained from the line of action along of the resulting aerodynamic force vector C R . The magnitude of the vector C R is, CR =
q
2 = CL2 + CD
q
2 + C2 CN T
The direction of C R is follows from the angle χ1 , see figure 5-11, of the vector with the CN -axis, χ1 = arctan
CT CN
The position of the line of action of C R in the plane of symmetry, may be defined by its intersection with the (extended line through the ) mac. This point is called the center of pressure. The position of the center of pressure (xd , z0 ) for an arbitrary moment reference point (x1 , z1 ) can be derived with equation (5-1) from Cm(x1 ,z1 ) , Cm(xd ,z0 ) = 0 = Cm(x1 ,z1 ) + CN
xd − x1 z0 − z1 − CT c¯ c¯
(5-2)
or simply 0 = Cm(x0 ,z0 ) + CN
xd − x0 c¯
if the leading edge of the mac (x0 , z0 ) is taken as the moment reference point. Now the center of pressure follows as, Cm(x0 ,z0 ) e xd − x0 = =− c¯ c¯ CN
(5-3)
It was shown above that for moment reference points on the mac, the wing aerodynamic moment Cm varies more or less linearly with the angle of attack, and thus also with CN . The relation between Cm(x0 ,z0 ) and CN can then be written as, Cm(x0 ,z0 ) = Cm0 +
dCm(x0 ,z0 ) dCN
CN
(5-4)
Cm0 denoting the value of Cm at CN = 0. We note that all moment lines intercept at one single point at αCN =0 with one value for Cm0 for all moment reference points. We note also Flight Dynamics
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Analysis of Steady Symmetric Flight
Cm 0.8
z/¯ c=0 x/¯ c = 1.00
0.6
0.4
x/¯ c = 0.75 0.2
x/¯ c = 0.50
αCN =0 -4
-2
Cm0 = Cmac
0
2
4
6
8
x/¯ c = 0.25
x/¯ c = 0.00
-0.4
x c¯
α [o ]
-0.2
Cmx0 ,z0 (A)
10
variable
Cm 0.2
z/¯ c = 1.00
x/¯ c = 0.25 -4
-2
0
-0.2
2
4
6
z/¯ c = 0.50 8
10
z/¯ c = 0.00 α [o ] z/¯ c = −0.50 z/¯ c = −1.00
-0.4
(B)
z c¯
variable
Figure 5-9: Moment curves for various positions of the reference point for the Fokker F-27 wing (from reference [149])
5-1 Aerodynamic center
165
1
x/¯ c = 1.00 0.5
Cm
x/¯ c = 0.75 x/¯ c = 0.50 x/¯ c = 0.25
0
x/¯ c = 0.00
−0.5 −4
−2
0
2
α
4
6
(a) Moment curve for varying
x z , ¯ c ¯ c
8
10
=0
0.4
0.3
0.2
z/¯ c = 1.00
0.1
Cm
z/¯ c = 0.50 z/¯ c = 0.00
0
z/¯ c = −0.50
−0.1
z/¯ c = −1.00
−0.2
−0.3
−0.4 −4
−2
0
2
α
4
(b) Moment curve for varying
6 z x , ¯ c ¯ c
8
10
= 0.25
Figure 5-10: Simulated moment curves for various positions of the reference point for the wing configuration of figure 5-4
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Analysis of Steady Symmetric Flight CN
χ1
CR
O
CT
Figure 5-11: Definition of the angle χ1
that this and all other moment lines are close to straight up to an angle of attack of almost 10 ◦ The center of pressure may now be written as, dCm(x0 ,z0 ) e Cm0 =− − c¯ CN dCN
(5-5)
dCm
(x0 ,z0 ) is negative, the center of pressure lies behind the leading edge of the mac for As dCN positive values of CN .
Figure 5-13 shows the total aerodynamic force vector on the Fokker F-27 wing for a range of angles of attack. Figure 5-14 gives the corresponding position of the center of pressure. The different ways to describe the aerodynamic moment characteristics of the wing, i.e. the Cm − α-curve and the line of action of C R with the center of pressure, both have their disadvantages. The Cm − α-curve for example is not unique but depends on the location of the reference point. In the other case of the line of action and the center of pressure both C R as well as the position of the center of pressure depend on α. As a consequence the resulting ’picture’ of the aerodynamic moment character is not simple to interpret, and easily leads to mistakes. There, however, is an alternative and more attractive way to describe the aerodynamic moments which is based on the concept of the so called aerodynamic center. The aerodynamic center (ac) is a very special moment reference point with the important property that the aerodynamic moment about it is invariant, i.e. it does not change with the angle of attack! In many cases of practical interest also the position of the aerodynamic center (ac) is inde-
5-1 Aerodynamic center
167
Zr
CN line of action of C R
(x0 , z0 )
(xd , z0 )
V
mac
CT Cm(x0 ,z0 )
e Cm(xd ,z0 ) = 0
Xr
Figure 5-12: Definition of the center of pressure relative on the mean aerodynamic chord (mac)
α = 18.95o α = 8.97o α = 12.94o
α = 6.87o α = 2.65o α = −0.56o
mac
α = −1.62o α = −4.81o
Figure 5-13: The magnitude of C R and the position of the line of action of C R as a function of the angle of attack for Fokker F-27 wing (from reference [149])
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Analysis of Steady Symmetric Flight
16
α[o ]
12
8
4
-1.2
-0.8
-0.4
0
0.4
αCN =0
0.8 e c¯
-4
Figure 5-14: The position of the center of pressure as a function of the angle of attack for Fokker F-27 wing (from reference [149])
5-1 Aerodynamic center
169 CN + ∆CN C R + ∆C R CN ∆CN
CR
∆C R ∆CT α CT ∆α CT + ∆CT Cm + ∆Cm
Cm
Figure 5-15: The aerodynamic forces and moment about an arbitrary point at two adjacent values of the angle of attack
pendent of the angle of attack. In the next section the aerodynamic center will be discussed in more detail.
5-1-2
A compact description for the aerodynamic moment characteristics
The following discussion it is shown that a moment reference point called the aerodynamic center (ac), about which aerodynamic moment (Cmac ) is constant, does indeed exist. In this section the simplifying assumption is made that also the position of the ac does not vary with angle of attack.
A. The first metacenter, the neutral line and the neutral point Figure 5-15 shows the forces and the moment about an arbitrary reference point at two adjacent angles of attack, α and α + ∆α. Now the line of action of ∆C R in Figure 5-16, may readily be constructed as the vector difference of the aerodynamic forces corresponding to α and α + ∆α. About any point on the lines of action of C R and C R + ∆C R the aerodynamic moment at the respective angle of attack, i.e. α and α + ∆α, is zero. Consequently, the moment about the point of intersection of the two lines, M1 , equals zero both at α and α + ∆α. In the limiting case, where ∆α → 0, this point of intersection of the two adjacent lines of action is well known as the first metacenter, M1 , in naval architecture. Flight Dynamics
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Analysis of Steady Symmetric Flight
Zr centers of pressure ∆C R (x0 , z0 )
C R + ∆C R
∆CN
∆CT χ2 (< 0)
α
CR mac
∆α
Xr neutral point (xn , z0 )
∆C R
M1 , first metacenter
neutral line Figure 5-16: The line of action of the difference between aerodynamic force vectors at adjacent values of the angle of attack
5-1 Aerodynamic center
171
We may conclude that in the first metacenter the following two conditions hold, Cm
= 0 (5-6)
dCm dα
= 0
The physical meaning of the line of action of ∆C R is that for each reference point on the line the change of aerodynamic moment will be zero. Consequently, for infinitely small change of alpha the analytical expression for all points on the line of action of dC R is, dCm =0 dα
(5-7)
The line of action of dC R is called the neutral line. The first metacenter turns out to be a special point on the neutral line where Cm = 0 as it is the intersection of the line of action of C R and the neutral line. If the aerodynamic forces and the moment about the leading edge of the mac (x0 , z0 ) are given the expression for the neutral line can be extracted from equations (5-1) and (5-7), dCm(x0 ,z0 ) dα
+
dCN x − x0 dCT z − z0 − =0 dα c¯ dα c¯
(5-8)
The intersection of the neutral line with the mac, where z = z0 , is called the neutral point of the wing. The distance of this neutral point (xn , z0 ) behind the leading edge of the mac follows from equation (5-8), dCm(x0 ,z0 ) xn − x0 =− c¯ dCN
(5-9)
with, zn = z0 The orientation of the neutral line follows from (see figure 5-16), χ2 = arctan
dCT dCN
(5-10)
B. The second metacenter and the aerodynamic center Next, we consider two vectors dC R at two values of the angle of attack, α1 = α and α2 = α + ∆α. It follows that now, we have two neutral lines, one for α1 and for α2 . The point where these two neutral lines intersect satisfies the following conditions: Flight Dynamics
172
Analysis of Steady Symmetric Flight neutral lines
∆C R2
∆C R1
α
∆α M2
(∆α → 0)
second metacenter = aerodynamic center Figure 5-17: The position of the aerodynamic center
dCm dα dCm dα
=0
α1
=0
α2
see figure 5-17. In the limiting case where ∆α → 0 the point of intersection follows from, dCm =0 dα d2 Cm =0 dα2
(5-11) (5-12)
The point of intersection of two neutral lines belonging to two adjacent angles of attack is called the second metacenter, M2 in naval architecture. In aeronautical engineering this point is referred to as the aerodynamic center (ac). In theoretical aerodynamics it is possible to prove that thin airfoils in a potential flow have an ac which is exactly invariant with the angle of attack. According to this thin airfoil theory, the ac of all thin airfoils is located at 25 % of the airfoil chord. In experimental aerodynamics, measurements on two-dimensional wing airfoils show that in the range of angles of attack where no flow separation occurs there is indeed an ac position which is very nearly independent of the angle of attack. Also in the case of a complete wing in three-dimensional flow the ac turns out to have a constant position over a wide range of angles of attack in many cases for moderate to large aspect ratios and small to moderate sweep angles.
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173
The . The location of the ac with coordinates xac and zac with respect to the leading edge of the mac follows from the conditions (5-11, 5-12), dCmac =0 dα or, dCm(x0 ,z0 ) dα
+
dCN xac − x0 dCT zac − z0 − =0 dα c¯ dα c¯
(5-13)
and,
d2 Cmac dα2 2 d Cm(x0 ,z0 ) dα2
= 0 +
(5-14)
d2 CN xac − x0 d2 CT zac − z0 − =0 dα2 c¯ dα2 c¯
(5-15)
in which it was assumed that xac and zac are independent of α. After computing the two coordinates xac and zac of the ac by solving these equations, the aerodynamic moment about the ac, Cmac , follows by substitution into (5-1). This leads to the following expression for Cmac : Cmac = Cm(x0 ,z0 ) + CN
xac − x0 zac − z0 − CT c¯ c¯
(5-16)
If CN , CT and Cm(x0 ,z0 ) are known from measurements as functions of the angle of attack, it is possible in principle to obtain the coordinates xac and zac using the equations above (5-13). This would require, however, to differentiate the experimental data on CN , CT and Cm two times with respect to alpha, which maybe difficult to do. An alternative way is to use the expressions for the neutral line (5-7), dCmac =0 dα for two adjacent angles of attack. The coordinates xac and zac follow from their intersection .
With the concept of the aerodynamic center, the model of the aerodynamic forces and moment acting on a wing can be written in a very simple way. Each neutral line for any angle of attack α must pass through the ac . As by definition the moment Cmac is constant, the model of the aerodynamic forces and moment acting on the wing thus consists of just a constant Cmac about the ac of the wing and the total aerodynamic force C R through the ac for all angles of attack (see figure 5-18). Flight Dynamics
174
Analysis of Steady Symmetric Flight Zr
C R + ∆C R ∆C R
zac
aerodynamic center α mac
xac ∆α
Cmac
Xr
Figure 5-18: The aerodynamic forces and moment, using the ac as the moment reference point
C. Replacing the ac by the neutral point We have seen that the aerodynamic center is the moment reference point for which Cm is constant. It follows from practice that the position of the ac is indeed invariant for a lange of angles of attack. Furthermore, it appears that the aerodynamic center is usually located very close to the wing’s mac. This follows also from equation 5-15 if we take account of the virtually perfect linear relationships between Cm(x0 ,z0 ) and alpha, and CN and alpha. So, their second derivatives will vanish, resulting in zac = z0 , see again figure 5-9. By assuming that the ac is located exactly on the mac, the ac coincides with the neutral point. The following shows also that this simplification is usually quite acceptable. As Cmac is constant: dCmac =0 dα it follows from equation (5-1) that, dCm(x0 ,z0 ) dCmac dCN xac − x0 dCT zac − z0 = + − dα dα dα c¯ dα c¯
(5-17)
dCN T In this expression dC dα is relatively small compared to dα . Moreover, the distance from the ac to the mac is small (a few percent of c¯ at most). This means that the last term in equation eq:eq2p14 may be neglected and the x-coordinate of the ac xc¯ac , follows from equation (5-17) as,
dCm(x0 ,z0 ) xac − x0 =− c¯ dCN
(5-18)
5-1 Aerodynamic center
175
This result is identical to the expression for the location of the neutral point, equation (5-9), meaning that if the contribution of the tangential force is neglected in (5-17), the ac will indeed coincide with the neutral point. With zac = z0 , Cmac follows, with (5-1), from, Cmac = Cm(x0 ,z0 ) + CN
xac − x0 c¯
(5-19)
= Cm0
(5-20)
and, clearly, Cmac = Cm(x0 ,z0 )C
N =0
As has been shown, the ac and Cmac can be obtained simply from experimental data on the aerodynamic moment about the leading edge of the mac. Next, using equation (5-18), xac is obtained from the slope of the Cm − CN -curve. According to equation (5-20), Cmac is equal to Cm0 at CN = 0. In many cases in the literature this simplified and approximated position of the ac is used. See for instance reference [49]. Finally if xc¯ac and Cmac are known, the aerodynamic moment coefficient about an arbitrary reference point on the mac is given by the fo0llowing very simple expression: Cm (x) = Cmac + CN
x − xac c¯
The description of Cm using the ac and Cmac derives its practical value from the fact that Cmac is constant and the position of the ac is independent of the angle of attack . However, for wings with a low aspect ratios and/or a large sweep angle the variation of the ac with angle of attack may not always be negligible, not even at those angles of attack occurring in normal flight. In order to enlarge the applicability of the ac and Cmac , a more general definition may be used in which Cmac is kept constant but the ac is allowed to shift position depending on alpha.
5-1-3
The role of the aerodynamic center and Cmac in static stability
If the Cmac and the position of the ac (xac , zac ) of a wing are known, the moment about an arbitrary point (x.z) is given by, see figure 5-19, Cm (x, z) = Cmac + CN
x − xac z − zac − CT c¯ c¯
(5-21)
For qualitative discussions it usually is permissable to simplify equation (5-21), as was already indicated in section 5-1-2, by neglecting the contribution of the tangential force, Cm = Cmac + CN
x − xac c¯
(5-22) Flight Dynamics
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Analysis of Steady Symmetric Flight Zr
CN xac
zac
x
Cm(x,z) CT
Or z
aerodynamic center
Cmac Xr
Figure 5-19: The moment about an arbitrary point (x, z) if the position of the ac and the Cmac are known
This simplified expression for Cm will often be used in the following discussions. Suppose a model of a wing has been mounted in a windtunnel so that the model is free to rotate about an axis through the moment reference point, serving now as the suspension point of the model. Furthermore, it is supposed that the center of gravity of the model has been made to coincide with this axis of rotation. Now a necessary condition for a steady equilibrium condition is Cm = 0. Small disturbances will induce model attitude (i.e. angle of attack) deviations from the nominal equilibrium condition. If such a deviation causes an aerodynamic moment acting against the change in angle of attack the original equilibrium situation was a stable one. This means that an increase in angle of attack must give rise to a negative, nose-down, change in aerodynamic moment. The condition for stability may thus be written as: dCm < 0 at Cm = 0 dα
(5-23)
m It is easy to see that if dC dα > 0 at Cm = 0, the equilibrium is unstable. If a change in angle of attack causes no change in the aerodynamic moment, the equilibrium is said to be indifferent or neutrally stable. In that case Cm = 0 at all angles of attack and there will be equilibrium at any value of α. The importance of the position of the suspension point relative to the ac of the wing is illustrated by considering the equilibrium of the moment and the stability of the equilibrium of the wing for different positions of the suspension point. From equation (5-22) follows for the equilibrium (Cm = 0),
Cm = Cmac + CN
x − xac =0 c¯
(5-24)
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177
and for the change of the moment with angle of attack, dCm dCN x − xac = dα dα c¯
(5-25)
N where dC dα in the range of linear variation of CN with α is always positive. Three cases are considered,
1. suspension point ahead of the ac,
x−xac c¯
<0
According to equation (5-24), equilibrium is possible for either positive or negative values of CN depending on the magnitude and the sign of Cmac , see figures 5-20 and m 5-21. From equation (5-25) follows that dC dα < 0. This means that the equilibrium is stable, as can be seen from figures 5-20 and 5-21. 2. suspension point in the ac,
x−xac c¯
=0
In this situation, equilibrium is possible only if Cmac = 0, see equation (5-24). If this is the case, equilibrium exists at any value of CN , or α (see figure 5-22). The equilibrium m is indifferent ( dC dα = 0). If Cmac 6= 0 no equilibrium situation is possible and as a consequence there can be no discussion of stability either (see figure 5-23). 3. suspension point behind the ac,
x−xac c¯
>0
if Cmac < 0 equilibrium will exist at a certain positive CN and for Cmac > 0 at a negative value of CN (see figures 5-24 and 5-25). In both cases the equilibrium is m unstable, dC dα > 0.
In summary it can be said that in the general case, i.e. where Cmac 6= 0, a condition of equilibrium (Cm = 0) is possible for any position of the suspension point ahead or behind the ac. The value of α or CN for which Cm = 0 does depend on Cmac , see equation (5-24), but has by itself nothing to do with the stability of the equilibrium. At a given CNα the stability follows only from the position of the suspension point relative to the ac, see equation (5-25). Summarizing, m • If the suspension point is situated ahead of the ac the model is always stable, dC dα < 0. m If the suspension point moves back, the model becomes less stable as dC dα becomes less negative.
• If the suspension point coincides with the ac, the slope of the moment curve is zero. In that case equilibrium is possible only if Cmac = 0. This equilibrium is said to be indifferent or neutrally stable. • The equilibrium is always unstable if the suspension point lies behind the ac as then dCm dα > 0. Flight Dynamics
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Analysis of Steady Symmetric Flight
CN Cm
CN
equilibrium at α1 ∆α α1
α2 ∆Cm
Cmac
α Cmac < 0; Cm
dCm dCN
<0
stable equilibrium
(a)
V
Cmac
α1
Cm = 0
ac suspension point
CN1
(b)
∆Cm α2 ∆α
∆CN
V
Cmac CN2
suspension point
ac
∆Cm < 0
CN1
(c) Figure 5-20: Equilibrium and stability of a wing, suspension point ahead of the ac, Cmac negative
5-1 Aerodynamic center
179
CN Cm
CN
equilibrium at α1
∆α
Cmac
α1
α2 ∆Cm
α Cmac > 0; Cm
dCm dCN
<0
stable equilibrium
(a) CN1
α1
Cmac
V ac
Cm = 0
suspension point (b)
CN2 CN1 ∆Cm
∆α α2
∆CN Cmac
V ac suspension point
∆Cm < 0
(c) Figure 5-21: Equilibrium and stability of a wing, suspension point ahead of the ac, Cmac positive
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Analysis of Steady Symmetric Flight
CN Cm
CN
equilibrium at any α
∆α α1
α2
Cm
Cmac = 0
α Cmac = 0;
dCm dCN
=0
neutrally stable equilibrium (a) CN1
α1
V suspension point
ac
Cm = 0
CN2
(b)
CN1 ∆CN
∆α α2
V suspension point
ac ∆Cm = 0
(c) Figure 5-22: Equilibrium and stability of a wing, suspension point in the aerodynamic center, Cmac equals 0
5-1 Aerodynamic center
181
CN Cm
CN
∆α α1
α2 Cm
Cmac Cm1
α
Cm2 Cmac < 0;
dCm dCN
=0
no equilibrium (a) CN1 Cmac
α1
V ac
suspension point
Cm < 0
CN2
(b)
CN1 ∆CN
∆α α2
Cmac
V suspension point
ac ∆Cm = 0, Cm < 0
(c) Figure 5-23: Equilibrium and stability of a wing, suspension point in the aerodynamic center, Cmac negative
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Analysis of Steady Symmetric Flight
CN Cm
CN
equilibrium at α1 Cm ∆Cm α2
α1 ∆α
Cmac
α Cmac < 0;
dCm dCN
>0
unstable equilibrium (a) CN1 Cmac
α1
V ac
(b)
suspension point
Cm = 0
CN2 CN1 ∆CN
∆α α2
Cmac ∆Cm
V ac
suspension point ∆Cm > 0
(c) Figure 5-24: Equilibrium and stability of a wing, suspension point behind the ac, Cmac negative
5-1 Aerodynamic center
183
CN
CN
Cm Cm
equilibrium at α1 ∆α Cmac
∆Cm α1
α2 α Cmac > 0;
dCm dCN
>0
unstable equilibrium (a)
Cmac α1
V suspension point Cm = 0
ac CN1 (b) ∆CN Cmac α2 ∆α
∆Cm
V ac
suspension point CN2 CN1 ∆Cm > 0
(c) Figure 5-25: Equilibrium and stability of a wing, suspension point behind the ac, Cmac positive
Flight Dynamics
184
Analysis of Steady Symmetric Flight
In the foregoing the influence of a change in the x-coordinate of the moment reference point (the suspension point and rotation axis) was considered. In a similar way the influence of a change in the position of the ac at a constant position of the suspension point can be studied. Such a shift in ac location may be caused for instance by the influence of the fuselage on Cm as will be discussed later. From equation (5-25) it follows, that a forward shift of the ac (xac decreases) always has a destabilizing influence and a rearward shift of the ac a stabilizing effect. In chapter 8 it is shown that the discussion given here for a wing, in a more extended form applies to the complete aircraft as well. The center of gravity of an aircraft has in this respect the same role as the axis of rotation of the wing. The slope of the moment curve (Cm − α), with the center of gravity as the suspension point, is a measure of the static stability of the aircraft, i.e. its tendency to return the original angle of attack following a hypothetical disturbance.
5-1-4
The position of the aerodynamic center and Cmac of a wing
In the foregoing sections an elegant and compact model was derived for the aerodynamic moment acting on a wing. Section 5-1-3 discussed how to use that model for analysis of static stability. The present section describes a method to derive such a model for a wing of arbitrary shape and twist from ’strip theory’ rather than from wind tunnel measurements. The main motive to discus this theory here is to allow the reader to understand the relationships between the parameters defining wing shape and the aerodynamic center ac and Cmac The basic idea behind strip theory is to derive the aerodynamic characteristics of wings of arbitrary geometry, dihedral and twist from the aerodynamic characteristics of two dimensional wing profiles. A ’two-dimensional’ dimensionless lift coefficient may be defined as cℓ =
ℓ
1 ρV 2 c 2
, in which
ℓ denotes lift force per unit of length. In a similar way the two dimensional normal force coefficient cn , drag coefficient cd , tangential force coefficient ct and moment coefficient cm may be defined. Just for simplicity, below, the difference between the two dimensional lift and normal force is neglected, as well as the contributions of the drag or tangential force to the overall aerodynamic moment. The latter implies that effects of dihedral are neglected too. All formula’s, however, may easily be extended to include these effects. The simple model of the aerodynamic forces and moments developed in part c of section 5-1-2 is now used as the model for the aerodynamic forces and moment acting on a wing strip at a given span-wise location. The lift distribution across the span of a wing can be divided into two parts. One part is the ’basic lift distribution’ cℓb (y), which is the lift distribution at αCL =0 . This lift distribution depends on only on the geometry of the wing and the wing profiles applied. The other part is
5-1 Aerodynamic center
185
cℓb c
− 2b
+ 2b
y
(A) Basic lift distribution cℓa c
α increasing
− 2b
+ 2b
y
(B) Additional lift distribution cℓ c
cℓa c − 2b
+ 2b
y
(C) Total lift distribution Figure 5-26: The basic, additional and total lift distribution of a wing with negative twist
Flight Dynamics
186
Analysis of Steady Symmetric Flight
the ’additional lift’ distribution cℓa (y) which depends on the wing geometry and wing profiles as well but in addition depends on the angle of attack, see figure 5-26. Several examples of basic, additional and total lift distributions using a source/doublet singularity panel method (based on linearized potential flow) are given in figures 5-27 to 5-32. The figures clearly illustrate the effects of wing-twist, taper ratio and wing sweep angle. We note that Cmac is the aerodynamic moment when CL = 0. This must imply that Cmac is caused by the torque resulting from the basic lift distribution and the cmac ’s of the wing strip airfoils. From the characteristics of the applied wing airfoils as well as the shape of the wing we now continue by computing the position of the ac, the magnitude of Cmac and lift gradient CL α. The lift acting on a wing strip dy is, dL = dLa + dLb = (cℓa + cℓb )
1 2 ρV c dy 2
This local lift force is assumed to act through the local ac, the constant moment cmac must be added, see figure 5-33. The moment due to the additional lift distribution about x0 , the leading edge of the mac, follows as, b
Ma = −2
Z2
cℓa
0
1 2 ρV c (x − x0 ) cos α dy 2
The resultant force of the additional lift distribution of the wing may be assumed to act through xac the ac of the wing on the mac. So, the moment about the leading edge x0 of the mac due to the additional lift distribution may be written as, Ma = −La (xac − x0 ) Combining the above two expressions results in, b
La (xac − x0 ) = 2
Z2
cℓa
0
1 2 ρV c (x − x0 ) cos α dy 2
Using, L = La = CL
1 2 ρV S 2
and assuming cos α ≈ 1, results in the required position of the ac of the wing, b
xac − x0 1 2 = c¯ CL S¯ c
Z2 0
cℓa c (x − x0 ) dy
(5-26)
Flight Dynamics
187
5
Z [m]
Z [m]
5
0 5
−5 −2
0
4
6
−5
8
5
−5 −2
0 2
0
0
0 2
Y [m]
X [m]
4
6
without wing-twist
including wing-twist 1.4
cℓ c cℓa c cℓb c
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
y/( 2b )
0.8
cℓ c cℓa c cℓb c
1.2
cℓ c
cℓ c
1.2
5-1 Aerodynamic center
Y [m]
X [m]
1.4
−0.2
−5
8
1
−0.2
0
0.2
0.4
0.6
y/( 2b )
0.8
1
Figure 5-27: Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wing-twist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 0o , λ = 1, αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panel-method, linearized potential flow.
5
Z [m]
Z [m]
Analysis of Steady Symmetric Flight
5
0 5
−5 −2
0
4
6
−5
8
5
−5 −2
0 2
0
0
0 2
Y [m]
X [m]
4
6
without wing-twist
including wing-twist 1.4
cℓ c cℓa c cℓb c
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
y/( 2b )
0.8
cℓ c cℓa c cℓb c
1.2
cℓ c
cℓ c
1.2
188
Y [m]
X [m]
1.4
−0.2
−5
8
1
−0.2
0
0.2
0.4
0.6
y/( 2b )
0.8
1
Figure 5-28: Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wing-twist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 37o , λ = 1, αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panel-method, linearized potential flow.
Flight Dynamics
189
5
Z [m]
Z [m]
5
0 5
−5 −2
0
4
6
−5
8
5
−5 −2
0 2
0
0
0 2
Y [m]
X [m]
4
6
without wing-twist
including wing-twist 1.4
cℓ c cℓa c cℓb c
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
y/( 2b )
0.8
cℓ c cℓa c cℓb c
1.2
cℓ c
cℓ c
1.2
5-1 Aerodynamic center
Y [m]
X [m]
1.4
−0.2
−5
8
1
−0.2
0
0.2
0.4
0.6
y/( 2b )
0.8
1
Figure 5-29: Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wing-twist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 0o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panel-method, linearized potential flow.
5
Z [m]
Z [m]
Analysis of Steady Symmetric Flight
5
0 5
−5 −2
0
4
6
−5
8
5
−5 −2
0 2
0
0
0 2
Y [m]
X [m]
4
6
without wing-twist
including wing-twist 1.4
cℓ c cℓa c cℓb c
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
y/( 2b )
0.8
cℓ c cℓa c cℓb c
1.2
cℓ c
cℓ c
1.2
190
Y [m]
X [m]
1.4
−0.2
−5
8
1
−0.2
0
0.2
0.4
0.6
y/( 2b )
0.8
1
Figure 5-30: Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wing-twist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 0o , Λ = 37o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panel-method, linearized potential flow.
Flight Dynamics
191
5
Z [m]
Z [m]
5
0 5
−5 −2
0
4
6
−5
8
5
−5 −2
0 2
0
0
0 2
Y [m]
X [m]
4
6
without wing-twist
including wing-twist 1.4
cℓ c cℓa c cℓb c
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
y/( 2b )
0.8
cℓ c cℓa c cℓb c
1.2
cℓ c
cℓ c
1.2
5-1 Aerodynamic center
Y [m]
X [m]
1.4
−0.2
−5
8
1
−0.2
0
0.2
0.4
0.6
y/( 2b )
0.8
1
Figure 5-31: Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wing-twist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 5o , Λ = 0o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panel-method, linearized potential flow.
5
Z [m]
Z [m]
Analysis of Steady Symmetric Flight
5
0 5
−5 −2
0
4
6
−5
8
5
−5 −2
0 2
0
0
0 2
Y [m]
X [m]
4
6
without wing-twist
including wing-twist 1.4
cℓ c cℓa c cℓb c
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
y/( 2b )
0.8
cℓ c cℓa c cℓb c
1.2
cℓ c
cℓ c
1.2
192
Y [m]
X [m]
1.4
−0.2
−5
8
1
−0.2
0
0.2
0.4
0.6
y/( 2b )
0.8
1
Figure 5-32: Numerical simulation of the basic lift distribution (cℓb c at α = αCL =0 ), additional lift distribution (cℓa c) and total lift distribution (cℓ c) for a wing without wing-twist (left, ε = 0o ) and a wing including wing-twist (right, ε = −3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c¯ = 2 m, Γ = 5o , Λ = 37o , λ = 13 , αr = 0o , ε = −3o , α = 5o . Results from a source/doublet singularity panel-method, linearized potential flow.
5-1 Aerodynamic center
193
y
Yr x0
x line of local ac’s
mac
c
dy
Xr
Zr (cℓa + cℓb ) 12 ρV 2 cdy
α
cmac 12 ρV 2 c2 dy x0 x ac Xr
Figure 5-33: The calculation of the position of the ac and Cmac for a wing with arbitrary geometry
Flight Dynamics
194
Analysis of Steady Symmetric Flight
The contribution of a wing strip dy to the total moment about x0 , the leading edge of the mac, follows from figure 5-33, 1 dM = {cmac c − cℓ (x − x0 ) cos α} ρV 2 c dy 2 The non-dimensional moment of the entire wing about x0 , if again cos α ≈ 1, is obtained by integration along the wing span and division by 12 ρV 2 S¯ c
Cm
b b Z2 Z2 2 2 = cmac c dy − cℓ c (x − x0 ) dy S¯ c 0
or since,
(5-27)
0
cℓ = cℓa + cℓb equation (5-27) may be written as,
Cm
b b b Z2 Z2 Z2 2 2 = cmac c dy − cℓb c (x − x0 ) dy − cℓa c (x − x0 ) dy S¯ c 0
0
(5-28)
0
According to equation (5-22), Cm about x0 ), the leading edge of the mac, assuming again CN ≈ CL , can be written as, Cm = Cmac + CL
x0 − xac c¯
(5-29)
Substituting equation (5-26) in equation (5-29) and equating equation (5-28) to equation (5-29) results in the expression for Cmac ,
Cmac
b b Z2 Z2 2 2 = cmac c dy − cℓb c (x − x0 ) dy S¯ c 0
(5-30)
0
We now have expressed the position of the ac of the wing as in equation (5-26) and the value of Cmac in equation (5-30). These results are subject to the same simplifying assumptions as were made in part c of section 5-1-2, 1. The contribution of the tangential force to the moment is neglected and as a consequence the ac is located on the mac 2. The position of the ac is assumed to be independent of α Reference [5] provides for a large number of wing shapes the values of xac according to equation (5-26), in addition to the magnitudes of the first and second integral in equation (5-30). Data on the position of the ac and of Cmac can be found in references [74, 152, 172].
5-1 Aerodynamic center
5-1-5
195
Influence of wing shape on the aerodynamic center and Cmac
It is quite difficult to give general rules for the influence of the various wing geometry parameters describing the wing shape, such as aspect ratio and wing sweep on the aerodynamic moment curve of wings. The reason is, that the influence of one parameter strongly depends on the value of the others. Not too many experimental (wind tunnel) results are available in which those parameters were systematically varied. Most of these experimental results apply to wing-fuselage combinations, so variations of the influence of the fuselage with wing sweep, wing taper ratio, etcetera, are included in the experimental results. For these reasons it is not feasible to aim for completeness and a description of only a few of the more general effects of wing shape on the aerodynamic moment has to suffice. We have seen that the aerodynamic moment characteristics of a wing can be expressed in terms of a constant value of Cmac and an approximated position of the ac on the wing mac. Relative to an arbitrary reference point (x1 , z0 ) on the mac the aerodynamic moment may then be written as see equation (5-21), Cm(x1 ,z0 ) = Cmac + CN
x1 − xac c¯
This equation is valid as far as linear wing strip theory is valid, so up to only ’moderate’ values of the angle of attack, wing sweep angle and slender wings with moderate to large wing aspect ratios. For such wings, Cm and CN vary more or less linearly with α if the same holds true for the individual wing strip profiles. The Cmac of a complete wing could be written as as, see equation (5-30),
Cmac
b b Z2 Z2 2 2 = cmac c dy − cℓb c (x − x0 ) dy S¯ c 0
0
From this expression we see that Cmac is partly determined by the cmac ’s in the first integral, i.e. by the camber of the individual wing airfoils. However, from the second integral in equation (5-30) it appears that Cmac also depends on the wing twist in case the wings are swept , due to the contribution made of the basic lift distribution. Straight wings have a constant factor (x − x0 ) in the second integral in equation (5-30), reducing it to zero, and Cmac now only results from the contribution of the camber of the wing airfoils to the local cmac . Increasing the camber leads to a more negative cmac , see for example the wind tunnel results of figure 5-34. If a straight wing has a constant cmac along the wing span, the result is Cmac = cmac thanks to the definition of the mac. For sweptback wings having a negative wing twist the positive cℓb at the wing root and the negative cℓb at the tip will result in a positive (’tail-heavy’) contribution to Cmac , see figure 5-35a. This contribution increases with wing sweep and aspect ratio, see figure 5-35b. Decreasing the taper ratio (λ = ccrt ) decreases again this effect.
Flight Dynamics
196
Analysis of Steady Symmetric Flight
Cm0.25
0.1
NACA Airfoil 63A010 -0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
63A210
-0.1
63A410
(A) λ = 1, Λ = 0o , A = 4, M = 0.6, Cm0.25
CL
t c¯
= 0.10
0.1
NACA Airfoil 63A010 -0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
CL
63A210 63A410 -0.1
(B) λ = 1, Λ = 0o , A = 2, M = 0.6, Cm0.25
t c¯
= 0.10
0.1
NACA Airfoil 63A004 -0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
CL
63A204 63A404 -0.1
(C) λ = 1, Λ = 0o , A = 4, M = 0.6, Cm0.25
t c¯
= 0.04
0.1
NACA Airfoil 63A004 -0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
63A204 63A404 -0.1
(D) λ = 1, Λ = 0o , A = 2, M = 0.6,
t c¯
= 0.04
Figure 5-34: The influence of camber on the moment curves (from references [119, 118])
CL
5-1 Aerodynamic center
197
YB cℓb c XB
ZB (A) The moment due to basic lift distribution (left wing)
∆Cmac (per o ε)
0.008
A = 6.0 A = 3.5
0.004
A = 1.5 -60
-40
-20
0
20
40
60
Λ (o )
-0.004
(B) ∆Cmac as a function of A and Λ, λ = 0.5 Figure 5-35: The variation of Cmac with wing sweep Λ and angle of twist ε (from reference [49])
Flight Dynamics
198
Analysis of Steady Symmetric Flight ac’s of the local airfoils
local ac’s ac 1 ¯ 4c
1 ¯ 4c
ac
Figure 5-36: The positions of the local ac’s of swept wings and delta wings
The deflection of landing flaps may cause a significant increase of the camber of the wing cross-section over a part of the wing span. For straight wings the result is a more negative Cmac for the entire wing. Flap deflection also causes a change of the basic lift distribution similar to the effect an increased negative twist at the wing tips. If the wing has positive (backward) sweep back, this change in the basic lift distribution induces itself a positive change in Cmac . As a consequence, the total change in Cmac due to flap deflection may even be positive. Calculations of this effect have been made for instance in references [121, 47]. The position of the ac is, (5-26), b
xac − x0 2 = c¯ CL S¯ c
Z2 0
cℓa c (x − x0 ) dy
The position of local ac’s of wing airfoils of straight wings with moderate to large aspect ratios correspond are close to the ac’s in two-dimensional flow. As a consequence, the ac of such wings will lie around the 25 % position on the mac of the wing. Straight wings with low aspect ratio (A < 3), however, show an appreciable forward shift with decreasing aspect ratio. For these straight wings, flap deflection causes no significant shift of the ac. The lift distribution in chordwise direction of swept wings and delta wings is strongly influenced by wing sweep. The result is that the local ac’s of the wing no longer agree with the ac’s of the local wing airfoils in two-dimensional flow, see figure 5-36. The local ac’s at the tips of swept back wings lie ahead of the ac’s of the corresponding wing
5-1 Aerodynamic center
199
Cm0.25
Λ = 60o 0.28
Λ = 60o
0.24
Λ = 45o
0.20
0.16
0.12
Λ = 32.6o
Λ = 45o 0.08
Λ = 32.6o
0.04
-0.2
0
0.2
0.4
0.6
0.8
CL -0.04
Λ = 3.6o
Λ = 3.6o
Figure 5-37: The influence of wing sweep on the moment curves of wings of aspect ratio A = 4, λ = 0.6, t c¯ = 0.06, M = 0.40 (from reference [169])
Flight Dynamics
200
Analysis of Steady Symmetric Flight
airfoils. Since for swept back wings the additional lift distribution is concentrated more near the wing tips, the ac of the complete wing will be situated ahead of the 25 % mac position. This forward shift increases with increasing sweepback angle and we know that this has a destabilizing influence, see also [5]. However, the combination of a swept back wing with a fuselage often shows a rearward shift of the ac with increasing sweepback angle, see the linear part of the moment curves up to CL ≈ 0.4 in figure 5-37. This is due to the influence of wing sweepback on the wing-fuselage interference, see also section 5-1-6. If the taper ratio (λ = ccrt ) decreases, the contribution of the wing tips to the aerodynamic moment decreases. As the ac at the wing tip is shifted forward due to wing sweep, see 5-36, decreasing the taper-ratio of swept back wings moves the ac backwards, thus has a stabilizing effect, see the linear part of the moment curves up to CL ≈ 0.4 in figure 5-38. Wing aspect ratio as such only has a relatively minor influence on the ac. Figure 5-39 for example shows the position of the ac as a function of aspect ratio for wings with 45o of sweep back angle . Delta wings have local ac’s situated behind the quarter-chord position over a large part of the wing span, see figure 5-36. In addition, the lift distribution of delta wings shows a concentration more towards the center parts of the wing. This causes the wing ac to be situated behind the 25 % mac position. Wing aspect ratio and sweep back of delta wings cannot be varied independently. Increasing sweepback decreases aspect ratio, leading to a rearward shift of the ac, see figure 5-39. Similar to the case of swept wings, a fuselage added to a delta wing has a stabilizing effect (i.e. rearward shift of the ac), which increases with decreasing wing aspect ratio, and thus with increasing sweepback angle. At sufficiently large angles of attack in particular the effects of flow separation result in nonlinear moment curves for straight wings, wings with sweep back angle as well as for delta wings. Now, for these large angles of attack, the concepts of ac and Cmac start loosing their significance as in order to match the slope of the moment curve, not only the location of the ac, but also the value of the Cmac must be adapted. As also argued in reference [1] the characteristics of wings in such conditions may best be judged from aerodynamic moment curves for some suitable reference point as stability characteristics may still be deduced from the slope of the moment curves, as discussed earlier. dCm m Straight wings usually exhibit large negative changes in dC dCL ≈ dCN at lift coefficients near CLmax due to flow separation. Figures 5-41 and 5-34 give examples of this phenomenon. As in flight such a negative (nose-down) change in Cm would help to decrease the angle of attack this is a favourable characteristic of straight wings.
Swept wings,however, with moderate to large aspect ratios usually suffer from a large positive slope change increasing angle of attack. The change may start already at relatively small
5-1 Aerodynamic center
201
Cm0.25
0.28
0.24
λ = 1.0
0.20
0.16
λ = 1.0 0.12
λ = 0.6 0.08
0.04
λ = 0.6 -0.2
0
0.2
0.4
0.6
0.8
CL
λ = 0.3
λ = 0.3 -0.04
Figure 5-38: The influence of taper ratio on the moment curves of swept wings, A = 4, Λ = 45o , t c¯ = 0.06, M = 0.40 (from reference [87])
Flight Dynamics
202
Analysis of Steady Symmetric Flight xac c¯ 0.37
0.33
Delta wings
xac
1 ¯ 4c
c¯
ac 0.29
0.25
1
2
3
4
5
A xac
0.21
0.17
ac 1 ¯ 4c
c¯
Swept wings
Figure 5-39: The position of the ac of delta wings and concept wings as a function of aspect ratio, Λ = 45o (from reference [159])
values of α. Figures 5-37, 5-38, 5-40 and 5-41 give examples. These changes are caused by premature flow separation at the outer wings, predominantly caused by the high tip loading and the cross flow towards the wing tips. The result is an increasing thickness of the boundary layer and early flow separation. This effect occurs at lower CL values, and the effect on Cm is stronger as sweep back angle and aspect ratio are larger. Figure 5-42 gives an indication of the combinations of aspect ratio and wing sweep angle for which destabilizing changes in the slope of the moment curves of the wing may be expected. This ’sudden’ change of slope of the moment curve is called ‘pitch up’. If it would occur in flight its overwhelming effect is to increase the angle of attack which evidently is highly undesirable and potentially very dangerous. A horizontal tailplane has an important effect on the aerodynamic pitching moment and may be designed as to compensate for the adverse effect of ‘pitch up’ as we will see later. However, even then ‘pitch up’ remains a very undesirable wing characteristic which should best be avoided through design. To avoid too much lift concentration near the wing tips (and the pitch up phenomenon) negative wing twist is often used. Camber and other shape parameters of the wing airfoils may be adjusted to obtain desired values of local cℓmax without undue drag increases in cruising flight. To prevent cross-flow towards the wing tips (inducing early flow separation), several means may be applied such as boundary layer fences or a ‘saw tooth’ leading edge. Reference [68] provides an extensive compilation of the various methods and tools which may be employed for this purpose.
5-1 Aerodynamic center
203
Cm0.25
0.28
A=6 0.24
A=6
0.20
0.16
0.12
A=4 0.08
A=4 0.04
-0.2
0
0.2
0.4
0.6
0.8
CL -0.04
A=2
A=2
Figure 5-40: The influence of aspect ratio on the moment curve of swept wings, Λ = 45o , λ = 0.6, t c¯ = 0.06, M = 0.40 (from reference [97])
Flight Dynamics
204
Analysis of Steady Symmetric Flight
Cm0.25 0.12
0.08
Swept wing Λ = 45o 0.04
Straight wing Λ = 0o
0
4
8
12
16
20
α [o ]
-0.04
-0.08
Figure 5-41: The moment curve of a straight wing and a swept wing, A = 5, λ = 1, Re = 1 · 106 (DUT measurements)
5-1 Aerodynamic center
205
A 14
12
λ=1
Ref. [58]
10
λ=0 8
Ref. [75] unstable 6
4
stable 2
0
10
20
30
40
50
60
70
Λ 1 c [o ] 4
Figure 5-42: Boundaries for the combinations of aspect ratio and sweep angle for which destabilizing changes in the moment curves may be expected (from references [58, 75])
Flight Dynamics
206
5-1-6
Analysis of Steady Symmetric Flight
Characteristics of wing-fuselage-nacelle configurations
The model of the aerodynamic moment of the combination of a wing with fuselage and nacelles attached to the wing is in principle the same as for the isolated wing. At small angles of attack Cm varies linearly with CN , as it does for the isolated wing. As a consequence, changes in the longitudinal moment caused by the fuselage and the nacelles are commonly expressed as a shift in the position of the ac and a change in Cmac . Neglecting again the contribution of the tangential force, the moment coefficient of the wing was written as, see equation (5-22), Cmw = Cmacw + CN
x − xacw c¯
In a similar way the moment of the combination of the wing with fuselage and nacelles can be expressed as, Cmw+f +n = Cmacw+f +n + CN
x − xacw+f +n c¯
At constant CN the change in the moment is, ∆Cm = Cmw+f +n − Cmw = ∆Cmac − CN
∆xac c¯
(5-31)
Similar to Cmacw , also Cmacw+f +n is defined as constant and independent of the angle of attack. This means that ∆Cmac is independent of α as well. ∆Cmac is the change of Cm at CN = 0 due to the addition of a fuselage (and nacelles). The shift in the ac position then follows from equation (5-31), d (∆Cm ) ∆xac 1 d (∆Cm ) =− =− c¯ dCN CNα dα
(5-32)
∆Cmac = (∆Cm )CN =0
(5-33)
and the change in Cmac is,
The change in the longitudinal moment due to the presence of the fuselage and the wing’s nacelles can be thought of as consisting of the contributions of the fuselage and the nacelles separately, the contribution of the wing-fuselage interference and the contribution of the wing-nacelle interference. Nacelles mounted at the rear fuselage are commonly regarded as part of the fuselage. Wing mounted nacelles exert an influence on the longitudinal moment similar to the fuselage. For these reasons only the effect of adding a fuselage to the wing is discussed in the following. The change in the moment due to the presence of the fuselage, ∆Cm , can be written in the following expression, Cmw+f = Cmw + ∆Cm
5-1 Aerodynamic center
207
where, ∆Cm = Cmf + Cmi Here Cmf denotes the longitudinal moment of the free fuselage in undisturbed flow while Cmi denotes the wing-fuselage interference. From windtunnel tests the total ∆Cm can be simply obtained by comparing the moment curves of the isolated wing and the wing-fuselage combination at equal values of CN . If, in addition, measurements have been made on the isolated fuselage, i.e. to determine Cmf , Cmi can be found as well. For theoretical studies it appears to be useful to divide the total interference effect Cmi into two parts, 1. ∆Cmf.i. ; the change in the fuselage moment caused by situating the fuselage in the field of flow around the wing. 2. ∆Cmw.i. ; the change in the wing moment caused by placing the wing in the field of flow around the fuselage. So, ∆Cm = Cmf + ∆Cmf.i. + ∆Cmw.i. Usually, the total moment Cmf.i. contributed by the fuselage in the field of flow around the wing is calculated as one effect, Cmf.i. = Cmf + ∆Cmf.i. This results in, ∆Cm = Cmf.i. + ∆Cmw.i. In the following the latter two contributions to the change in Cm and the subsequent changes in the position of the ac and the Cmac are discussed. A. The moment on the fuselage positioned in the field of flow induced by the wing, the magnitude of Cmf.i. For a fuselage placed under a non-zero angle of attack in an inviscid and incompressible flow it is possible to prove that it will experience just an aerodynamic moment while the aerodynamic force will be zero. The mechanism behind this moment is illustrated in figure 5-43 (see also figure 5-44 for a numerical simulation of particle lines and pressure distribution over a slender fuselage, linearized potential flow). The magnitude of this moment acting on Flight Dynamics
208
Analysis of Steady Symmetric Flight - +
+
++
- -
+
α +
+ + + + - - - - -
Figure 5-43: Pressure distribution over a fuselage in inviscid flow
a slender fuselage of circular crosssection in these conditions of incompressible, inviscid flow has been derived by Munk (reference [115]), Mf = ρV 2 · αf · (Volume of Fuselage) From this expression follows,
Cmf
παf = 2S¯ c
Zlf
b2f (x) dx
(5-34)
0
Where, bf (x) lf αf S, c¯
local fuselage width fuselage length fuselage angle of attack, [Rad.] characteristic parameters of the wing to which Cmf is referred
According to equation (5-34) the moment is positive (tail-heavy) at positive angles of attack and it increases with increasing angles of attack. The implication is that the idealized isolated fuselage is statically unstable at αf = 0o and cannot be in equilibrium at αf 6= 0o . If the wing and the fuselage are combined, the fuselage will experience the effect of air velocity variations as induced by the wing. This causes the angle of attack along the fuselage axis to be different from that of an isolated fuselage. As a consequence, also the pressure distribution over the fuselage is different from that of an isolated fuselage. The wing induced variation of the local angle of attack and the corresponding local normal force along the fuselage axis is shown schematically in figure 5-45. Ahead of the wing, for positive values of CL , the upwash causes an increase of αf (x) up to the wing leading edge. Across the wing chord the airflow is ’guided’ parallel to the wing chord resulting in αf (x) being zero. Behind the wing the local angle of attack along the fuselage is reduced due to the downwash induced by the wing. Results of some numerical simulations of the flowfield of the wing-fuselage combination of figure 5-46 are depicted in figures 5-47 and 5-48. The result of the induced upwash in front of the wing is an increase in tail-down moment with increasing angle of attack, which as we know, is destabilizing. The wing induced downwash has the opposite effect, however, usually not enough to compensate fully the destabilizing
5-1 Aerodynamic center
209
Pressure distribution Cp
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
Figure 5-44: Numerical simulation of particle lines (top) and pressure distribution over a fuselage in inviscid flow, α = 10o , linearized potential flow
Flight Dynamics
210
Analysis of Steady Symmetric Flight
effect of the wing induced flow field. Now it is not difficult to see that a long fuselage in front of the wing, as needed for transport aircraft with rear mounted engines, amplifies this effect. As shown in the classical reference [114], the aerodynamic moment acting on a slender fuselage in an inviscid, incompressible flow field with induced velocities can be written as,
Cmf.i.
π = 2S¯ c
Zlf
b2f (x) αf (x) dx
(5-35)
0
Note that by substituting αf (x) for the case of an isolated fuselage in (5-35 this equation should yield the same result as equation (5-34) ! αf (x) can be expressed in terms of the wing angle of attack α as, αf (x) = αf0 +
dαf (x) (α − α0 ) dα
(5-36)
where α0 denotes αCN =0 and αf0 is the fuselage angle of attack at α = α0 . With equation (5-36) follows for Cmf.i. ,
Cmf.i.
παf0 = 2S¯ c
Zlf
π (α − α0 ) b2f (x) dx + 2S¯ c
0
Zlf
b2f (x)
d αf (x) dx dα
(5-37)
0
in which the first term on the right hand side represents the fuselage moment at CN = 0, and so ∆Cmac due to the fuselage is,
∆Cmac = (∆Cm )CN =0
παf0 = 2S¯ c
Zlf
b2f (x) dx
(5-38)
0
With equation (5-32) the shift in ac position due to the fuselage is,
∆xac c¯
f.i.
1 dCmf.i. 1 π =− =− CNα dα CNα 2S¯ c
Zlf
b2f (x)
d αf (x) dx dα
(5-39)
0
Reference [66] presents some classical semi-empirical correction formulae for ∆Cmac and ∆xac accounting for friction effects of fuselages of finite length. c¯ f.i. dα
Computing Cmf.i. with equation (5-37)is possible if the variation of d αf along the fuselage axis is known. Some classical references are [114], [165] and [160]. Nowadays one would no doubt rather rely on some numerical method as used to compute the numerical simulations in the figures above. Some nice qualitative results are shown in Figures 5-49 and 5-50. It turns out that both dα a reduction of aspect ratio and an increase of sweepback angles result in decreasing d αf in
5-1 Aerodynamic center
211
α X
αf (x)
dαf dα
upwash downwash 1
1−
0
αf = 0 and
dαf dα
=0
dε dα
=
dαh dα
X
dNf dx
X Figure 5-45: The variation of the angle of attack and the normal force along the fuselage axis in a wing induced flow field .
Flight Dynamics
212
Analysis of Steady Symmetric Flight
Figure 5-46: Wing-fuselage configuration as used in the flow field simulations depicted in figures 5-47 and 5-48 .
Figure 5-47: Numerical simulation of the velocity field of a wing-fuselage configuration, α = 10o .
5-1 Aerodynamic center
213
Figure 5-48: Numerical simulation of the velocity field of a wing-fuselage configuration, α = 10o , detail of figure 5-47 . d Cm
front of the wing. This means that the contribution of the fuselage nose to d αf.i. becomes less stabilizing, see equation (5-39). The downwash angle behind the wing increases with dα decreasing wing aspect ratio and increasing sweepback angle, dd αε increases and d αf is closer to zero. In this way the destabilizing contribution to Cmf.i. of the rear part of the fuselage behind the wing decreases as well. We conclude that both decreasing the wing aspect ratio and increasing the sweepback angle reduce the destabilizing effect of the fuselage to Cm . B. The influence of the fuselage on the wing moment, the magnitude of ∆Cmw.i. The influence of the fuselage on the characteristics of the wing itself consists of two parts, 1. The extra upwash induced by the presence of the fuselage over the wing root next to the fuselage, see figures 5-47, 5-48 and 5-51. Results of numerical simulations are shown in figure 5-52. These simulations were performed using the wing-fuselage configuration as of figure 5-46. 2. The change in lift distribution over that part of the wing covered by the fuselage, see figure 5-53. A further look at the fuselage induced upwash learns that its effect on the wing moment is relatively small and can be usually be neglected, see reference [143]. The induced upwash of wing-mounted nacelles, however, may have a non-negligible effect, see reference [114].
Flight Dynamics
214
Analysis of Steady Symmetric Flight dαf dα
A=∞
3
A=9
increasing A 2
A=3 increasing A
A=0
1
A=∞ A=9 A=3
Γ
A=0
0 -2
-1
0
1
2
3 x cr
Figure 5-49: The influence of aspect ratio on the variation of reference [143])
dαf dα
along the fuselage axis (Λ = 0) (from
dαf dα
Λ = −45o
3
Λ = 0o increasing Λ 2
Λ = 45o increasing Λ 1
Λ = −45o Λ = 0o Λ = 45o
Γ 0 -2
-1
0
1
2
3 x cr
Figure 5-50: The influence of sweep angle of wings with A = ∞ on reference [143])
dαf dα
along the fuselage () (from
5-1 Aerodynamic center
215 Z
Y
(A) Flow around the fuselage influence of the fuselage α(y)
y (B) The variation of the angle of attack along the wingspan Figure 5-51: Fuselage induced variations of the local angle of attack along the wing span
Left wing
Figure 5-52: Numerical simulation of the fuselage induced upwash along the wing span, α = 5o Flight Dynamics
216
Analysis of Steady Symmetric Flight cℓ · c 0.8
wing 0.6
0.4
wing with fuselage 0.2 1 2 bf
0 0
0.2
0.4
0.6
0.8
1.0
y/ 2b Figure 5-53: The cℓ · c-distribution along the span of a wing with and without a fuselage (from reference [54])
Figure 5-53 shows measurements of the lift distribution across the span of a wing with and without fuselage. It appears that the fuselage affects lift only close to the fuselage. For straight wings this fuselage effect on the Cmac of the wing can safely be neglected. For swept back wings, and also for delta wings, the ac of the entire wing lies behind the local ac’s of the center wing profiles , see figure 5-54. This means that the lift reduction ∆CLw of the center part of the wing will induce in a nose-down change in moment, i.e. ∆Cmw.i. is negative. As ∆CLw is more or less proportional to CL this effect is stabilizing , shifting ac of the wing rearward. For swept forward wings, however, the same effect causes a tail-down change in the wing moment and is destabilizing ,shifting the ac forward . Reference [8] allows one to estimate ∆Cmw.i. with a simple method. The influence of the sweep angle on the ac as computed with,
∆xac c¯
w.i.
=−
1 d∆Cmw.i. CNα dα
(5-40)
is shown in figure 5-55 for a set of idealized wing-body configurations. The figure shows the contribution to ∆xac caused by the fuselage in the wing induced field of flow Cmf.i. as derived from equation (5-37), the effect of the fuselage on aerodynamic moment of the wing ∆Cmw.i. and the resulting shift of ac position caused by the fuselage, both calculated and measured.
5-1 Aerodynamic center
217
line of ac’s
ac of the wing center part
ac of the wing
ac of the wing center part
ac of the wing Figure 5-54: The change in wing moment due to loss in lift over the wing center part
Flight Dynamics
218
Analysis of Steady Symmetric Flight ∆xac c¯
-0.18
measurements -0.12
fuselage in wing induced field of flow Cmf.i. -0.06
total fuselage influence -30
-20
-10
0
0.06
10
20
30
40
Λ [o ]
effect of fuselage on the wing ∆Cmw.i.
A = 5, λ = 0.2
Λ
Xr Figure 5-55: Shift of ac due to fuselage effects, as a function of wing sweep angle(from reference [141])
5-1 Aerodynamic center
219
Figure 5-56: Flow simulations over a nacelle, α = 15o , computed with a source/doublet singularity panelmethod, linearized potential flow
From this figure it appears once again that the fuselage has a destabilizing influence, whereas a wing with positive sweepback angle has a stabilizing effect. As has already been noted in 5-1-5 this stabilizing influence of wing sweepback may compensate the destabilizing effect of the fuselage. The resulting shift of the ac may turn out to be stabilizing.
5-1-7
Aerodynamic effects of nacelles
Analysis of the aerodynamic effects of wing- or fuselage-mounted nacelles on the aerodynamic characteristics of a wing-fuselage configuration starts with the isolated nacelle . In figure 5-56 a generic configuration of a engine nacelle including simulated particle lines is given for the case of an ’empty’ nacelle (no engine thrust effects ). Considering this nacelle at an angle-of-attack of α = 15o it is obvious that the local airflow will conform to the exterior of the nacelle, the nacelle can be seen as a ring shaped wing .
Example: aircraft with wing-mounted nacelles We first consider a transport aircraft model with swept back wings without its nacelles, see figure 5-57. The local airflow over the wing at the nacelle position is shown in figure 5-58. In figure 5-58 simulated particle lines are shown for an angle-of-attack α = 10o as well as the non-dimensional pressure distribution coefficient Cp . The aerodynamic effect of the sweep Flight Dynamics
220
Analysis of Steady Symmetric Flight
Figure 5-57: Generic Large Transport Aircraft (GLTA), no nacelles
angle on the non-dimensional pressure distribution Cp of the wing sections is evident from the increasing pressures moving to the wing tip. Next we add nacelles to the wing . In figure 5-59 results are shown for the case of one nacelle per wing including simulated particle traces and non-dimensional sectional pressure distribution coefficients Cp . It is clear that the nacelle has a significant effect on the local pressure distribution resulting in a reduction of the local angle-of-attack as the airflow conforms to the external shape of the nacelle. This is confirmed by figure 5-60 showing the spanwise lift distribution expressed in terms of cℓ · c for an angle-of-attack α = 10o with and without nacelles. Similar results are shown for the case of a four-engined model in figures 5-61 and 5-62. The numerical examples shown above on the example case of a transport aircraft configuration show the potential benefits of numerical flow simulation tools as the source/doublet singularity panel-method, linearized potential flow applied in our case. We may use these methods in several ways to compute flow characteristics including pressure distributions and lift distributions, to analyze the effect of aircraft configuration changes including the effect of components as nacelles and, by computing results for a range of angle of attach values, computing the location of the aerodynamic center xac and the Cmac , and their variations for different configurations ∆xac and ∆Cmac . This includes the computation of the corresponding important effects on the downwash characteristics behind the wing, as we will see below.
5-2
Equilibrium in steady, straight, symmetric Flight
The present section discusses steady, symmetric, straight flight conditions. The stability of these conditions is discussed in subsequent chapters.
5-2 Equilibrium in steady, straight, symmetric Flight
221
Figure 5-58: Particle lines in an XB OZB -plane (top) and non-dimensional sectional pressure distribution Cp (bottom) for the left wing of a generic transport aircraft model, no nacelles, α = 10o , computed with a source/doublet singularity panel-method, linearized potential flow Flight Dynamics
222
Analysis of Steady Symmetric Flight
Figure 5-59: Particle lines in an XB OZB -plane (top) and non-dimensional sectional pressure distribution Cp (bottom) for the left wing of a twin-engined transport aircraft , α = 10o , computed with a source/doublet singularity panel-method, linearized potential flow
5-2 Equilibrium in steady, straight, symmetric Flight
223
3 No nacelles 1 nacelle per wing
2.5
cℓ c
2
1.5
1
0.5
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y/( 2b ) Figure 5-60: Spanwise lift distribution cℓ c at an angle-of-attack α = 10o twin-engined transport aircraft configuration , source/doublet singularity panel-method, linearized potential flow
Flight Dynamics
224
Analysis of Steady Symmetric Flight
Figure 5-61: Particle lines in an XB OZB -plane and non-dimensional sectional pressure distribution coefficients Cp for the left wing of a four-engined transport aircraft , α = 10o , source/doublet singularity panel-method, linearized potential flow
5-2 Equilibrium in steady, straight, symmetric Flight
225
3 No nacelles 2 nacelles per wing
2.5
cℓ c
2
1.5
1
0.5
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y/( 2b ) Figure 5-62: Four-engined transport aircraft with spanwise lift distribution cℓ c at an angle-of-attack α = 10o , source/doublet singularity panel-method, linearized potential flow
Flight Dynamics
226
Analysis of Steady Symmetric Flight
For an aircraft it should be possible to obtain equilibrium of the (longitudinal) aerodynamic moment at any angle of attack and center of gravity position at which equilibrium of the forces is possible, i.e. at all feasible flight conditions, see reference [123]. To achieve this a horizontal tailplane with an elevator is used. The horizontal tailplane serves a dual purpose of generation a longitudinal moment about the center of gravity such that the resulting aerodynamic moment is zero and of changing the aerodynamic moment characteristics in a stable sense, as we will discuss in more detail later. The following sections discuss the equilibrium of the forces and moments in steady, symmetric, straight flight conditions. In particular the contribution of the horizontal tailplane and the elevator to the longitudinal equilibrium of moments is considered. The stability of these conditions of equilibrium is the subject of next chapters.
5-2-1
Conditions for equilibrium
Since only steady flight is considered here, forces and moments due to mass and rotational inertia of the aircraft are omitted. The aircraft weight and the aerodynamic forces and moments are in equilibrium so,
W +R = 0 M = 0 The forces and the moment are resolved in components along the aircraft body axes. In this way the conditions for equilibrium may be written as,
Wx + X = 0 Wy + Y
= 0
Wz + Z = 0 L = 0 M
= 0
N
= 0
In the following, it is always assumed that the aircraft is symmetric and that in the considered flight conditions the flow around the aircraft has a plane of symmetry coinciding with that of the aircraft. In these symmetric flight conditions the resultant aerodynamic force R and the weight vector W lie in the plane of symmetry of the aircraft, so,
5-2 Equilibrium in steady, straight, symmetric Flight
227 R
Z
XB
θ
W sin θ
X
horizontal
c.g. W cos θ W ZB
Figure 5-63: The equilibrium in steady, symmetric flight
Wy
=
Y
= 0
L
=
N
= 0
The three conditions for equilibrium of these symmetric flight conditions are , see figure 5-63,
−W sin θ + X = 0 W cos θ + Z = 0 M
= 0
θ denoting the pitch angle. These equations may also be written in terms of normal and tangential force N and T , T = −W sin θ
(5-41)
N = W cos θ
(5-42)
M =0
(5-43)
with T = X and N = Z. For simplicity , the aircraft body axes are chosen parallel to the aircraft reference axes, origin of the body axes remaining in the aircraft center of gravity. In the following, these conditions are further analyzed.
Flight Dynamics
228
Analysis of Steady Symmetric Flight
The resultant aerodynamic force R is built up from the contributions of the different parts of the aircraft. These parts also provide the elements of the resultant aerodynamic moment M . Three parts of the aircraft may be distinguished. • Wing with fuselage and nacelles The contributions of these parts of the aircraft usually are taken as one entity, the index w refering to the combination of the wing with fuselage and nacelles. The model of the forces acting on this part of the aircraft is given by the components Nw and Tw of the total aerodynamic force Rw , acting in the aerodynamic center (ac) of the wing with fuselage and nacelles (acw ), and the aerodynamic moment Macw acting about the acw . • Horizontal tailplane Just as for the wing with fuselage and nacelles, the model of the forces acting on the horizontal tailplane is given by the components Nh and Th , acting in the ac of the tailplane (ach ), and the moment Mach . • The propulsion unit The contribution of the propeller, or the jet engine, consists of the thrust Tp along the propeller or engine-axis and the normal force Np acting in the plane of the propeller or in the plane of the engine inlet depending on the local angle of attack . Figure 5-64 depicts the contributions of the various parts of the aircraft. The symmetric conditions of equilibrium, equations (5-41), (5-42) and (5-43) can now be written as follows, • Forces along the XB -axis T = +Tw + Th − Tp cos ip + Np sin ip = −W sin θ
(5-44)
• Forces along the ZB -axis N = +Nw + Nh + Np cos ip + Tp sin ip = +W cos θ
(5-45)
• Moments about the YB -axis M
= +Macw + Nw (xc.g. − xw ) − Tw (zc.g. − zw ) + +Mach + Nh (xc.g. − xh ) − Th (zc.g. − zh ) + + (Np cos ip + Tp sin ip ) (xc.g. − xp ) + + (Tp cos ip − Np sin ip ) (zc.g. − zp ) = 0
(5-46)
5-2 Equilibrium in steady, straight, symmetric Flight
229
Zr
Nw
in plane of propeller ip
Tp
Np ach acw
ih
c.g.
iw
Nh Th
Tw
zp Mac
Mach zc.g.
zw
zh
W xp
θ θ
xw
Xr
xc.g. xh
Figure 5-64: The forces and moments in steady, straight, symmetric flight
Flight Dynamics
230
Analysis of Steady Symmetric Flight
Before bringing the expressions (5-44), (5-45) and (5-46) in a non-dimensional form, some new non-dimensional coefficients have to be defined. The non-dimensional coefficients of normal force and tangential force on the tailplan, and the moment about the ac of the tailplane are now defined by, Nh
= CNh 12 ρVh2 Sh
Th
= CTh 12 ρVh2 Sh
Mach = Cmach
(5-47)
1 2 ¯h 2 ρVh Sh c
The non-dimensional coefficients CNh , CTh and Cmach are thus referred to the surface area Sh and the mac ch of the horizontal tailplane (see figure 5-65a) and the average local dynamic pressure 12 ρVh2 at the horizontal tailplane. In general, this dynamic pressure is different from that of the undisturbed flow (at infinity). This is further discussed in section 5-2-4. The thrust of the propeller and the normal force in the plane of the propeller may be written as, Tp = Tc ρV 2 D2 Np = CNp
1 2 ρV Sp 2
(5-48) (5-49)
where D is the propeller diameter and Sp = π4 D2 is the area of the propeller disc. The equations of equilibrium are made non-dimensional in the usual way by dividing equations (5-44) and (5-45) by 12 ρV 2 S and equation (5-46) by 12 ρV 2 S¯ c. Using , −Tp cos ip + Np sin ip ≈ −Tp the result is,
CT = +CTw + CTh
CN = +CNw + CNh
Vh V
2
Vh V
2
Sh 2D2 W − Tc = − 1 2 sin θ S S 2 ρV S
Sp Sh 2D2 W + Tc sin ip + CNp = 1 2 cos θ S S S 2 ρV S
2 xc.g. − xw zc.g. − zw Vh Sh c¯h − CTw + Cmach + c¯ c¯ V S c¯ 2 2 Vh Sh xc.g. − xh Vh Sh zc.g. − zh +CNh − CTh + V S c¯ V S c¯
Cm = +Cmacw + CNw
(5-50)
(5-51)
5-2 Equilibrium in steady, straight, symmetric Flight
231
Sh
Se
c¯e
c¯h
St
hinge line of elevator
hinge line of tab (A) Geometry parameters of the horizontal tailplane, elevator and trim tab
αh Vh δe δte
(B) Positive deflections of αh , δe and δte Figure 5-65: Geometry parameters of the horizontal tailplane, elevator and trim tab, and their positive deflections
Flight Dynamics
232
Analysis of Steady Symmetric Flight
Sp xc.g. − xp 2D2 2D2 zc.g. − zp + Tc sin ip + CNp + Tc =0 S S c¯ S c¯
(5-52)
These equilibrium equations may be used for different purposes such as to explore the set of feasible steady state flight conditions. In the following, however, the purpose is an analysis of stability and flying qualities parameters. In that case it is permissable and even desirable to simplify the equations of equilibrium in such a way that only the most important contributions remain, resulting in the following simplifications , 1. Propulsion effects, i.e. the terms with Tc and CNp are neglected. 2. The contribution of CTh to the force in XB -direction in equation (5-50) and to the moment in equation (5-52) is neglected. This simplification is nearly always permissable. 3. The contribution of CTw to the moment in equation (5-52) is omitted. For large angles of attack, or in cases when the center of gravity is situated far above or below the acw , this simplification may not always be permissable. 4. The contribution of Cmach is relatively small, even zero in case of symmetric airfoils, and so Cmach in equation (5-52) is neglected too. The index w of Cmacw may therefore be omitted as well without raising confusion. The resulting equations of equilibrium now read , W sin θ 2 2 ρV S
CT = CTw = − 1 CN = CNw + CNh
Cm = Cmac
Vh V
2
(5-53)
Sh W = 1 2 cos θ S 2 ρV S
xc.g. − xw + CNw + CNh c¯
Vh V
2
Sh xc.g. − xh =0 S c¯
(5-54)
(5-55)
The assumptions made in the model of the equilibrium lead to a very simple model, see figure 5-66. The distance between the ac of the horizontal tailplane and the ac of the wing with fuselage and nacelles, (xh − xw ), is called the tail length lh . For conventional aircraft, lh = xh − xw ≈ xh − xcg
(5-56)
With (5-56) equation (5-55) may be written as, Cm = Cmac
xc.g. − xw + CNw − CNh c¯
Vh V
2
Sh lh =0 S¯ c
(5-57)
5-2 Equilibrium in steady, straight, symmetric Flight
233
or, Cm = Cmw + Cmh in which
Sh lh S¯ c
is called the ’tailplane volume’.
The tailplane serves to compensate for Cmw through an appropriate deflection of the elevator or by adjusting the angle of incidence of the entire horizontal tailplane. In this way the contribution Cmh from the horizontal tailplane to the moment as generated by CNh , is equal and opposite to the contribution Cmw of the wing with its fuselage and nacelles. In the following sections the characteristics of the horizontal tailplane and the effects of the elevator are discussed in more detail. Also, expressions will be derived for the elevator angle and the pilot’s control force necessary to maintain equilibrium for given flight conditions. The equations of equilibrium (5-53), (5-54) and (5-55) apply to the conventional aircraft configurations where the horizontal tailplane is placed behind the wing. In the given form the expressions are valid also for aircraft where the horizontal ’tailplane’ is located in front of the wing. In that case lh in equation (5-57) is negative , see also equation (5-56). For tailless aircraft the contribution of the horizontal tailplane obviously does not exist. In that case equation (5-57) reduces to, Cm = Cmac + CNw
xc.g. − xw =0 c¯
(5-58)
see also figure 5-21. The ’elevator’ of tailless aircraft is situated at the trailing edge of the wing. Deflection of the elevator of such an aircraft causes a change of Cmac . The elevator deflection also has an influence on CN which may be non-negligible in many cases. For tailless aircraft, equilibrium for some value of CN is possible if Cmac in equation (5-58) is made equal and opposite in sign to the moment of CN about the center of gravity.
5-2-2
Normal force on the horizontal tailplane
The horizontal tailplane usually consists of a stabilizer and a control surface. In many cases a small auxiliary control surface is added to the control surface. This may be a trim tab, a balance tab or a servo tab. The non-dimensional pressure distribution over the horizontal tailplane is completely determined by three deflection angles, 1. The angle of attack of the horizontal tailplane, αh 2. The control deflection angle, δe (index e stands for elevator) Flight Dynamics
234
Analysis of Steady Symmetric Flight
CNh Zr
Vh V
2
Sh S
CNw ach
c.g. Cmac
W sinθ 1 ρV 2 S 2
acw CTw W cosθ 1 ρV 2 S 2
Xr
xw xc.g. lh xh
Figure 5-66: Simplified version of the equilibrium of forces and moments
5-2 Equilibrium in steady, straight, symmetric Flight
235
Normal force derivatives
Derivative
∂CNh ∂αh
∂CNh ∂δe
∂CNh ∂δte
Shorthand notation
CNhα
CNhδ
CNhδ
t
Table 5-1: Shorthand notation for the normal force derivatives on a tailplane
3. The tab angle, δte (index t stands for tab) Figure 5-65b shows the positive sense of the three angles αh , δe and δte . Figure 5-67 shows the changes in the chordwise pressure distribution caused by changes in the local angle of attack, the control surface deflection and the tab angle deflection. The non-dimensional normal force CNh as defined in equation (5-47) is determined by the local angle of attack and two deflection angles , CNh = CNh (αh , δe , δte ) In figure 5-68 results are given of measurements of CNh as a function of αh , δe and δte . It follows that in selected small intervals of αh , δe and δte , CNh may be assumed to vary approximately linearly with these parameters. In the range of those small angle excursions a reasonable approximation of CNh is obtained by, CNh = CNh0 +
∂CNh ∂CNh ∂CNh αh + δe + δte ∂αh ∂δe ∂δte
(5-59)
A convenient shorthand notations is used for the aerodynamic partial derivatives introduced in equation (5-59), see table 5-1. The expression (5-59) can be simplified a little. First we note that the influence of δte on CNh is usually small, see for example figure 5-68 and may be neglected. Furthermore, many tailplanes have a symmetric or close to symmetric airfoils, so CNh0 = 0 . Using the shorthand notation (which corresponds to the American notation) this results in, CNh = CNhα αh + CNhδ δe
(5-60)
which serves as an approximation of the much more complex nonlinear relation, Flight Dynamics
236
Analysis of Steady Symmetric Flight
∆p q
δe = δte = 0 αh2 αh1 x/¯ c
αh1 αh2 ∆p q
αh = δte = 0
δe2 δe1 x/¯ c δe1
∆p q
δe2
αh = δe = 0
δte2 δte1 x/¯ c δte1
δte2
Figure 5-67: Pressure distributions over the tailplane chord due to αh , δe and δte
5-2 Equilibrium in steady, straight, symmetric Flight
δte =
0o
CNh
δe [o ]
237
20 15
δe = 0o
10
CNh
δte [o ]
5 0 0.8 -5
15
0.8
-10
0
-15 -20 0.4
0.4
-15 -25 -16
-8
0
αh
8
-16
-8
0
-0.4
-0.4
-0.8
-0.8
8
αh
Figure 5-68: The normal force coefficient CNh as a function of αh , δe and δte for the tailplane of the Fokker F-27 (Wind tunnel measurements from reference [17])
CNh = CNh (αh , δe , δte ) as may be obtained from windtunnel measurements or flight test. Both positive changes in the angle of attack αh and the control deflection δe cause positive changes in CNh which means, CNhα > 0
and
CNhδ > 0
With references [71, 157, 4, 101, 61, 21], based on empirical data, it is possible to estimate CNhα and CNhδ . The normal force gradient CNhα can be obtained in principle using one of the available wing theories. The presence of the fuselage and the existence of an open gap between the stabilizer and the control surface must be taken into account, reducing the value of CNhα . From measurements on two- and three-dimensional models it was shown that CNhδ depends almost linearly on CNhα . This is reflected in figure 5-69 showing c¯e c¯h
CNh
CNh
δ
to depend on the
α
ratio of the control surface chord to the tailplane chord, and the trailing edge angle τ of the airfoil. This result holds for aerodynamically unbalanced control surfaces extending over the entire span of the horizontal tailplane with a closed gap between the horizontal tailplane (’stabilizer’) and the control surface. However, CNhδ is affected by several other factors such as cut-outs in the control surface at the fuselage and the vertical tailplane, a gap between Flight Dynamics
238
Analysis of Steady Symmetric Flight 0o
CNh CNh
δ
τ
α
10o 20o
0.7
0.6
0.5
0.4
0.3
0.2
Closed gap Full span elevator No Aerodynamic balance
0.1
0
Figure 5-69: The quotient
0.1
CNh
δ
CNh
0.2
as a function of
α
c¯e c¯h
0.3
0.4
0.5
c¯e c¯h
for various trailing edge angles τ (from reference
[21])
the stabilizer and the aerodynamic shape of the nose of control surfaces (used to ’balance’ or compensate hinge moments), see references [101, 61].
5-2-3
Hinge moment of the elevator
The aerodynamic hinge moment about the elevator hinge line, He , is determined by the pressure distribution over the elevator. This hinge moment is expressed by a non-dimensional coefficient, Che , Che =
He 1 2 ¯ e 2 ρVh Se c
(5-61)
The hinge moment coefficient is thus referred to the average local dynamic pressure at the horizontal tailplane and to the surface Se and the mac ce of the part of the control surface
5-2 Equilibrium in steady, straight, symmetric Flight Che δte =
239
αh = 0o
0o
Che
δe [o ] -12
0.08
-9 -6
δe
0.04
0.04
[o ]
-15
-3
-10 0
-5 -16
-8
0
0
αh
8
-16
-8
0
-0.04 -0.04
8
δe [o ]
3
5 6
δte [o ]
-0.08 9
-0.08
12
Figure 5-70: The hinge moment coefficient Che as a function of αh , δe and δte for a tailplane of the Fokker F-27 (wind tunnel measurements from reference [17])
behind the hinge line, see figure 5-65a. The hinge moment is defined positive in the same direction as the control surface deflection. Similar to CNh , also Che depends on the three angles αh , δe and δte , see the pressure distributions in figure 5-67, Che = Che (αh , δe , δte ) Measurements of Che as a function of αh , δe and δte are shown in figure 5-70. We now note that δte has an important effect on Che . Again for a limited interval of αh , δe and δte variations a linear approximation for Che may be written as, Che = Ch0 +
∂Che ∂Che ∂Che αh + δe + δt ∂αh ∂δe ∂δte e
(5-62)
If a symmetric airfoil is used for the tailplane, Ch0 = 0. Using in addition the shorthand notation of table 5-2, Che can be written as, Che = Chα αh + Chδ δe + Chδt δte
(5-63)
Note that in equation (5-63), Chδt is the derivative of the moment about the elevator hinge axis with respect to the tab deflection. Table 5-2 also defines the derivatives Chtα , Chtδ and Chtδ which refer to the hinge moment about the tab hinge axis. These latter derivatives are t made non-dimensional using St and ct of the tab. Flight Dynamics
240
Analysis of Steady Symmetric Flight
Hinge moment derivatives
Derivative
∂Che ∂αh
∂Che ∂δe
∂Che ∂δte
∂Chte ∂αh
∂Chte ∂δe
∂Chte ∂δte
American notation
Chα
Chδ
Chδt
Chtα
Chtδ
Chtδ
t
Table 5-2: Abbreviated notation for the hinge moment derivatives
Computing the hinge moment derivatives of a control surface of given shape and dimensions is not always possible with sufficient accuracy. The shape of the tailplane, the airfoil and minute details of the form of the gap between stabilizer and control surface have an effect on the derivatives. The main problem, however, is the relatively large influence of viscous effects which are difficult to account for quantitatively. Best is to obtain data on Che from windtunnel measurements, using as large a model of the tailplane as possible. Depending on the location of the hinge line of the control surface the change in hinge moment, with increasing angle of attack and control surface deflection, will be negative in most cases, see also the pressure distributions given in figure 5-67. In that case both derivatives are negative,
Chα < 0
and
Chδ < 0
If the hinge line would be moved backwards, however, in most cases it is the derivative Chδ which first changes sign to positive, see again 5-67, and would result in unacceptable flying qualities as discussed later. It follows from equation (5-61) that the hinge moment varies with the square of the airspeed and the cube of the dimensions of the aircraft. Fast and large aircraft would soon be confronted with very large hinge moments which would be difficult for pilots to handle by conventional mechanical control systems . This led early mechanical flight control system designers to carefully aerodynamically balance control surfaces such that Chα and Chδ stay small, while keeping Chδ negative. As long as both Chα and Chδ are negative, the control surface is called aerodynamically underbalanced. Shifting the hinge-line rearward will result in the control surface to become overbalanced with respect to the angle of attack, or with respect to the control deflection, or both. Chα and Chδ will then be positive see figure 5-71. One of the means to change Chα and Chδ in the positive sense, is to employ a rearward shift of the hinge axis, or a forward extension of the nose of the control surface in front of the hinge line. As long as both Chα and Chδ are negative, the control surface is called aerodynamically underbalanced.
5-2 Equilibrium in steady, straight, symmetric Flight
5-2-4
241
Flow direction and dynamic pressure at the horizontal tailplane
A horizontal tailplane located behind the wing experiences the field of flow disturbed by the wing, the fuselage, the nacelles and propellers. Mainly due to the downwash behind the wing, the average angle of attack αh of the horizontal tailplane is decreased. If the horizontal tailplane lies in the wake produced by the wing, or in the boundary layer of the fuselage, the local average dynamic pressure at the horizontal tailplane is reduced. This is expressed in a 2 value of VVh < 1 in equation (5-57). Both effects cause a change in the contribution of the horizontal tailplane to the moment about the aircraft’s lateral axis. The slipstream of propeller driven aircraft and the exhaust 2 stream of jet-propelled aircraft may also cause changes in αh and VVh . If the average downwash angle at the location of the horizontal tailplane is designated ε, and the tailplane incidence setting relative to the Xr -axis is ih , the average angle of attack of the horizontal tailplane follows from (see figure 5-72), αh = α − ε + ih
(5-64)
The downwash angle is approximately proportional to CL , in the range where CL varies linearly with α, ε = (α − α0 ) where α0 = αCL =0 and
dε dα
dε dα
is approximately constant. Thus αh can be written as, dε αh = (α − α0 ) 1 − + (α0 + ih ) dα
(5-65)
From (5-65) the derivative of αh with respect to α is, dαh = dα
dε 1− dα
(5-66)
We will see below that this derivative is always smaller than one, resulting in a reduction of the positive contribution of the horizontal tailplane to the static stability of conventional aircraft with rear mounted tail planes. 2 The downwash angle ε and the factor VVh depend in the first place on the shape of the wing and on the location of the horizontal tailplane relative to the wing. The downwash behind the wing may be thought of as to be induced by a system of nested U shaped vortices. Part of each vortex is bound to the wing, inducing an upwash in front of the wing and a downwash behind the wing. The free, remaining parts of each vortex (the two antisymmetrical ’tails’) leave the wing at the trailing edge of the wing in a vortex sheet of free vortices, reinforcing the downwash and reduce the upwash, see figure 5-73. The vertical velocities induced by these vortices result in the vortex sheet itself being pushed downwards, assuming the form Flight Dynamics
242
Analysis of Steady Symmetric Flight
∆p q
due to ∆αh
x/¯ c
∆αh
∆Che > 0
(A)
∆Che ∆αh
> 0 → Chα > 0
∆p q
due to ∆δe
∆Che > 0
∆δe (B)
∆Che ∆δe
> 0 → Chδ > 0
Figure 5-71: Overbalance with respect to angle of attack and elevator angle.
x/¯ c
5-2 Equilibrium in steady, straight, symmetric Flight
243
α ε
V
αh
ih
Vh k Xr Xr αh = α − ε + ih Figure 5-72: The angle of attack αh of the horizontal tailplane, αh = α − ε + ih
as depicted in figure 5-74. While the inclination of the vortex sheet conforms to the upper and lower surfaces of the wing at the trailing edge (Joukowski condition), downstream the inclination decreases and the vortex sheet is ’bent’ towards the direction of the undisturbed flow. Upwash, downwash and the vertical displacement of the vortex sheet are approximately proportional to CL for a given wing geometry. The boundary layers over the wing result in a ’wake’ of reduced dynamic pressure which is assumed to coincide with the the vortex sheet. The vertical displacement of the wing wake and vortex sheet increases with angle of attack but at a smaller rate than that of the horizontal tailplane , see figure 5-75. It is good practice to keep the horizontal tailplane out of the wing wake at all flight conditions. We conclude that this can indeed be assured by mounting it either high, keeping it above, or mounting it low, keeping it below the wake of the wing. The tendency of a vortex plane is to roll-up, changing its shape until it is transformed into two tip vortices, see figure 5-76. This process depends on the lift distribution over the wing and it is intensified as the lift coefficients are larger and the wing aspect ratio smaller, see references [151, 171]. For wings with a large aspect ratio and a small sweep angle the roll-up of the vortex plane is of little importance in the range of practical angles of attack. For these wings we may compute the downwash at the horizontal tailplane by assuming a flat vortex plane, only slightly curled at the edges. For small aspect ratio wings , or wings with a large sweep angle Λ, or both, however, this would be too rough an approximation. One would have to take account in these cases of the intricate deformed shape of the vortex sheet. The same holds true in cases where at some values of the angle of attack local flow separation occurs, often resulting in the ’functional’ form of the wing to be different from what one would expect by considering just its geometry. The main effect of flow separation is to reduce downwash rather than to affect the shape (rolling-up) of the vortex sheet, as discussed below. According to reference [117], the influence of the deformation and the rolling-up of the vortex sheet on the downwash close to the wing plane of symmetry may as well be neglected at low values of CL . dε Figure 5-77 gives the theoretical value for dα in the core of the vortex plane, as a function of aspect ratio and distance behind the wing, for wings having an elliptical lift distribution and Flight Dynamics
244
Analysis of Steady Symmetric Flight
w
Γ
x
due to free vortex
total induced vertical velocity
x
due to lifting vortex
Figure 5-73: The contribution of the lifting and free vortices to the induced vertical velocities in front of and behind a wing
depression of the vortex plane α V ε
vortex plane Figure 5-74: The position of the vortex plane behind the wing
k V∞ Xr
5-2 Equilibrium in steady, straight, symmetric Flight
245
position of the vortex plane α1
α2
∆α xw
∆zh
V
Xr at α1
∆zwake
xh Xr at α2
Figure 5-75: The vertical displacement of the wake and the tailplane relative to the flow field due to an inx x Rh Rh crease in angle of attack ∆α; ∆zh = ∆ α dx = ∆α (xh − xw ), ∆zwake = ∆ ε(x) dx, xw
xw
∆zwake < ∆zh as ∆ε(x) is smaller than ∆α. With increasing angle of attack the horizontal stabilizer moves downwards through the wake
attached (non separated) flow. The dominant influence of the wing aspect ratio is apparent. dε A rough estimate of dα at sufficiently large distances behind the wing is possible for straight wings for not too small aspect ratios. The classical formula to be used is,
dε CLα 2 A 4 =2 ≈ 2π = dα πA πA A+2 A+2
(5-67)
dε This expression results in values for dα for A > 5 which are in very good agreement with the more exact values given in figure 5-77 for an elliptical wing. ℓc Figure 5-78 shows the variation in spanwise direction of the lift distribution c2b for wings of ct different taper ratios. Note that for wings with small values of λ = cr the average downwash angle near the plane of symmetry is larger than for wings with larger values of λ at the same values of CL . This is caused by the higher lift loading near the wing center of former wings. dε at the horizontal tailplane increases with decreasing λ. Sweepback of an As a consequence dα dε untwisted wing causes a lower loading near the wing center. As a consequence, dα decreases with increasing sweep angle Λ, see also figure 5-79.
The downwash angle decreases with increasing distances above and below the vortex plane. Figure 5-80 shows the results of numerical simulations using a source and doublet singularity method (also known as ‘panel methods’). Measurements of the downwash angle in the plane of symmetry behind a tapered wing can be found in reference [146]. From figure 5-80 it is clear that the downwash in a certain distance above the wake is larger than in a point at the same distance below the wake. This is due to the fact that as a result of the depression of the wake and of the deformation of the vortex plane the free and the lifting vortices contribute more to the downwash above the wake than below the wake.
Flight Dynamics
246
Analysis of Steady Symmetric Flight
Figure 5-76: Numerical simulation of the deformation and wake roll-up of the vortex sheet behind a wing
5-2 Equilibrium in steady, straight, symmetric Flight
247
dε dα 1.2
1.0
0.8
x b/2
= 0.5
0.6
1.0
1.5
0.4
2.0 and 2.5 0.2
0 0
2
4
6
8
10
12
14
A V
b 2
X dε Figure 5-77: Computed values of dα behind a wing of elliptical lift distribution, as a function of aspect ratio and distance behind the wing
Flight Dynamics
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
−2
−3
−3
−3
−2 0 X [m]
0
2 2
−2
0 −2
X [m]
Y [m]
0
2 2
0 −2
X [m]
Y [m]
0.1
0.1
0.1
cℓ c/(2b)
0.15
cℓ c/(2b)
0.15
0
0.05
0
0.5
y b/2
1
0
2 2
0.15
0.05
248
Z [m]
3
Z [m]
Z [m]
3
−2
cℓ c/(2b)
Analysis of Steady Symmetric Flight
3
0 −2
Y [m]
0.05
0
0.5
y b/2
1
0
0
0.5
y b/2
1
ℓ ·c Figure 5-78: Numerical simulation of c2b for wings with varying taper ratio λ (CL = 1.0, Λ = 0o , A = 6), computed with a source/doublet singularity method, also see reference [143]
2
2
2
1
1
1
0
−1
−1
−2
−2
−2
−3
−3
−3
−2 0 X [m]
−2 0
2 2
0 −2
X [m]
Y [m]
0
2 2
0 −2
X [m]
Y [m]
0.15
0.1
0.1
0.1
0
cℓ c/(2b)
0.15
0.05
0.05
0
0.5
y b/2
1
0
2 2
0.15
cℓ c/(2b)
cℓ c/(2b)
0
−1
−2
5-2 Equilibrium in steady, straight, symmetric Flight
0
Flight Dynamics
3
Z [m]
3
Z [m]
Z [m]
249
3
0 −2
Y [m]
0.05
0
0.5
y b/2
1
0
0
0.5
y b/2
1
ℓ ·c Figure 5-79: Numerical simulation of c2b for wings with varying sweep angle Λ (CL = 1.0, λ = 1, A = 6), computed with a source/doublet singularity method, also see reference [143]
250
Analysis of Steady Symmetric Flight
Z [m] 3
2
1
0
−1
−2
−3 −2
ε = 2o →
ε = 4o →
ε = 6o →
ε = 8o →
−1 0 1 2
X [m]
3 4 5 6
ε [o ] 11
10
9
8
7
6
5
4
3
2
Figure 5-80: Numerical simulation of downwash angles in the plane of symmetry behind a wing (α = 13o , NACA 23014 airfoil, A = 6, λ = 0.5 and CL = 1.1615, , computed with a source/doublet singularity method, see also reference [146])
5-2 Equilibrium in steady, straight, symmetric Flight
251
The field of flow behind wings with separated flow is significantly different from the case of dε attached flow. Flow separation will cause appreciable changes in dα with increasing angle of attack. A detailed discussion of these changes related to flow separation can be found in reference [117]. For straight wings with just little taper (so a large λ), flow separation starts dε . near the wing root. The reduction in lift over the center wing results in a decrease of dα In addition a broad and deep wake is formed, causing a strong ’inflow’ into the wake. Flow separation near the wing tips, occurring with highly tapered and swept back wings, causes an increase in downwash at the tailplane due to a concentration of lift force to the center of the wing caused by flow separation. Swept wings and delta wings with thin airfoils often exhibit flow separation near the wing leading edge, already at small angles of attack. This leads to characteristic conical vortices above the wing. Such separation vortices may considerably increase the downwash. The camber and nose shape of airfoils, wing twist, boundary layer fences, nose flaps and slats, all serve to influence the flow separation characteristics. It will be clear that these wing geometry parameters will also affect the downwash characteristics at the horizontal tailplane. The fuselage causes a change in the flow behind the wing. This is due to the flow around the fuselage itself and due to the change in the wing lift distribution caused by the fuselage. The resultant influence on the average downwash at the horizontal tailplane is relatively small for slender wings. For wings of small aspect ratio the influence of the fuselage can be considerable and is strongly dependent of the position of the horizontal tailplane relative to the fuselage. If the tailplane is mounted on the fuselage a decrease in the average value of 2 Vh occurs since part of the tailplane is in the boundary layer of the fuselage. In such V 2 of some 5 to 10% is often taken into account. cases a reduction in VVh Example: aircraft with fuselage-mounted nacelles The aerodynamic effects of fuselage-mounted nacelles are shown next on two aircraft models depicted in figures 5-81. These models represent a Cessna Ce550 Citation II with and without nacelles, respectively the top and bottom figure in figures 5-81. dε In figures 5-82 the numerical simulation of dα (for X = 5.0 m, Z = 1.5 m) is given along the wing span (left wing) of the aircraft models presented in figures 5-81. Note that the top figure of figure 5-82 is a 2-dimensional representation (aft view) of the bottom figure. From figures dε 5-82 it follows that in the vicinity of the horizontal stabilizer the parameter dα is increased due to the presence of nacelles. This effect cannot be attributed to the induced velocities of free and bounded vortices but rather to the local airflow conforming to the external shape of the nacelles.
Flap deflection causes a concentration of lift at the center parts of the wing. This leads to dε an increased downwash ε and dα .
Flight Dynamics
252
Analysis of Steady Symmetric Flight
Figure 5-81: Cessna Ce550 Citation II configuration with and without nacelles
5-2 Equilibrium in steady, straight, symmetric Flight
253
4 No nacelles, X = 5.0 m, Z = 1.5 m Including nacelles, X = 5.0 m, Z = 1.5 m 3
Z [m]
1
dε dα ,
2
0
LEFT WING
−1
−2
−3 −10
−9
−8
−7
−6
−5
Y [m]
−4
−3
−2
−1
0
4
3
Z [m]
2
dε dα ,
1
0
−1
−2 0
No nacelles, X = 5.0 m, Z = 1.5 m Including nacelles, X = 5.0 m, Z = 1.5 m
−2 −4 −6
−8
Y [m]
−2
−1
0
1
2
3
4
5
6
7
X [m]
dε Figure 5-82: The effect of nacelles on dα (at X = 5.0 m, Z = 1.5 m) aft of the wing of a Cessna Ce550 Citation II, computed with a source/doublet singularity panel-method, linearized potential flow
Flight Dynamics
254
Analysis of Steady Symmetric Flight
Real aircraft
Ground surface (mirror plane)
Reflected aircraft
Figure 5-83: Calculation of ground effect
In take-off and landing the field of flow around and in particular behind the wing is strongly influenced by the proximity of the ground. Ground effect can be modeled by introducing additional ’mirror’ lifting vortices and trailing vortices, see figure 5-83. The mirror vortices cause a decrease in the induced angles of attack and a corresponding decrease in the effective angle of attack, proportional to the strength of the mirror vortices and thus to CL . The result of ground effect is an increase in CNwα and CNhα . As CNhδ is proportional to CNhα , CNhδ increases as well. The downwash angle behind the wing decreases due to the mirror vortices. This effect can also be understood directly, as the presence of the ground plane inhibits the downward flow behind the wing. Figure 5-84 gives an example of the change in ε at the horizontal tailplane dε caused by the ground effect. This figure also demonstrates that the reduction in dα due to ground effect can be quite important.
An additional consequence of the reduced downwash angles behind the wing is a smaller downward displacement of the wake, as compared with the situation without ground effect, see figure 5-84b . When choosing the vertical position of the horizontal tailplane in the
5-2 Equilibrium in steady, straight, symmetric Flight
255
course of the design process, this effect has to be taken into account. For slender, straight wings without and with deflected flaps references [157, 145] shows how to compute the average downwash angle and the average dynamic pressure at the horizontal tailplane, see also reference [138]. The downwash behind swept wings can be determined with references [73, 52, 48, 50, 158]. An extensive discussion of ground effect is given in reference [59]. The ground effect on the downwash and the wake can be calculated with reference [92]. For wings with a small aspect ratio or large sweep angle or both, the fuselage has a large influence on the flow at the horizontal tailplane and in case of flow separation , the flow field at the horizontal tailplane should be derived from measurements in the wind tunnel or by CFD simulations. An extensive summary and analysis of experimental data is given in reference [117], where also a method is described to estimate the characteristics of the horizontal tailplane.
5-2-5
Effect of airspeed and center of gravity on tail load
As discussed in section 5-2-1, a normal force acts on the horizontal tailplane to produce equilibrium of the longitudinal moment in an equilibrium situation. This normal force is called the tail load. The magnitude of the tail load Nh follows from equation (5-57) multiplied by 12 ρV 2 S¯ c, M = 0 = Cmac
1 2 ρV S¯ c + Nw (xc.g. − xw ) − Nh lh 2
(5-68)
1 2 ρV S¯ c + Nw (xc.g. − xw ) 2
(5-69)
or, 1 Nh = lh
Cmac
At small values of the angle of pitch θ, and if in addition Nh << Nw , it follows with equation (5-54), Nw ≈ W The result becomes, 1 Nh ≈ lh
1 2 Cmac ρV S¯ c + W (xc.g. − xw ) 2
(5-70)
This also follows directly from figure 5-85, where the model of equilibrium of the forces and moment has been given in a strongly simplified form. It can be seen from equation (5-70) that the tail load Nh is needed, 1. to compensate for the moment caused by the coefficient Cmac ; this moment depends on the airspeed but not on center of gravity position; in addition it varies with flap deflection δf Flight Dynamics
256
Analysis of Steady Symmetric Flight ε [o ] 6
δf = 0o 4
without ground effect 2
with ground effect z0.25¯c − zground = 0.19 b/2
0 0
2
4
6
8
10
12
10
12
α [o ]
(A) The downwash angle ε as a function of α
-0.4
0 0
2
4
6
8
with ground effect z0.25¯c − zground = 0.19 b/2
0.4
δf = 0o
without ground effect
0.8
1.2
α [o ]
h b/2
tailplane h wing center line of wake (B) The distance from the wake center line to the horizontal tailplane Figure 5-84: The ground effect on the downwash angle ε and the location of the wake relative to the horizontal tailplane of a Siebel 204 D-1 aircraft
5-2 Equilibrium in steady, straight, symmetric Flight
257
W cosθ + Nh ≈ W
Nh
c Mac = Cmac 12 ρV 2 S¯
acw
c.g. ach
≈ lh
xw − xc.g. W cosθ Figure 5-85: Simplified picture of the equilibrium of moments
2. to balance the moment of Nw (≈ W ) about the center of gravity; this moment is independent of airspeed, but varies with the center of gravity position From equation (5-70) it can be seen that Nh varies quadratically with V and increases in the upward sense with a rearward shift of the center of gravity. Figure 5-86 gives a sketch of the variations of the tail load with airspeed for various c.g. positions and signs of Cmac . The point of intersection of the parabola with the line V = 0, in actual flight the airspeed can, of course, not drop below Vmin , follows from equation (5-70), Nh = W
xc.g. − xw lh
Figure 5-86 illustrates that due to the usually negative sign of Cmac of the wing, fuselage and nacelles, the tail load reaches its largest positive value at the rearmost c.g. position and at minimum airspeed. The largest negative, downward, tail load occurs for a given negative Cmac at the most forward c.g. position and at a maximum airspeed. Figure 5-87 presents tail loads measured in flight on a DeHavilland ‘Mosquito’. The influence of airspeed on Nh as well as of c.g. position agrees in principle with equation (5-70). From Flight Dynamics
258
Analysis of Steady Symmetric Flight Nh Cmac > 0 xc.g. > xw c.g. behind ac
W
Cmac = 0
xc.g. −xw lh
Cmac < 0 Vmin
Cmac > 0
V
c.g. forward of ac xc.g. < xw
Cmac = 0
Cmac < 0 Figure 5-86: The variation of the tail toad with airspeed at different c.g. positions and values of Cmac
this figure it can be derived that for the range of airspeeds 350 < Ve < 500 km/h : xw = 0.08 c¯ and Cmac = −0.021.
5-2-6
Elevator deflection required for moment equilibrium
The way in which the elevator is used to produce moment equilibrium follows from expression (5-57), Cm = Cmac
xc.g. − xw + CNw − CNh c¯
Vh V
2
Sh lh =0 S¯ c
CNw can be written in the linear range as, CNw = CNwα (α − α0 )
(5-71)
In section 5-2-2, CNh has been expressed by, see equation (5-60), CNh = CNhα αh + CNhδ δe while in section 5-2-4 the expression for the angle of attack of the horizontal tailplane αh was found to be, see equation (5-65),
dε αh = (α − α0 ) 1 − dα
+ (α0 + ih )
5-2 Equilibrium in steady, straight, symmetric Flight
259
Nh 700
xc.g.1 = 0.341¯ c 600
500
c xc.g.2 = 0.292¯ 400
300
200
100
0 100
200
300
400
500
600
Ve [km/h]
-100
-200
Figure 5-87: The tail load as a function of airspeed at two c.g. positions for De Havilland ‘Mosquito II F’ (from reference [26]), W = 6800 kg, S = 41.8 m2 , lh = 8.0 m and c¯ = 2.81 m
Flight Dynamics
260
Analysis of Steady Symmetric Flight
Substituting equations (5-71), (5-60) and (5-65) in equation (5-57) results in the condition for moment equilibrium, xc.g. − xw − c¯ 2 dε Vh Sh lh + (α0 + ih ) + CNhδ δe (α − α0 ) 1 − dα V S¯ c Cm = Cmac + CNwα (α − α0 )
CNhα
= 0
(5-72)
or, in an ordered form, Cm = Cm0 + Cmα (α − α0 ) + Cmδe δe = 0
(5-73)
with, • a constant Cm0 , Cm0 = Cmac − CNhα (α0 + ih )
2
Sh lh S¯ c
Vh V
Vh V
(5-74)
• the static longitudinal stability, stick fixed Cmα , Cmα = CNwα
1−
dε dα
Vh V
2
Sh lh S¯ c
xc.g. − xw − CNhα c¯
2
Sh lh S¯ c
(5-75)
• the elevator effectivity Cmδe , Cmδe = −CNhδ
(5-76)
For a given aircraft the pilot can obtain equilibrium of the moment at a certain angle of attack, or (α − α0 ), and c.g. position, or Cmα , see equation (5-75), by deflecting the elevator, i.e. by choosing δe such that the resultant Cm is equal to zero. If the total moment Cm is written as the sum of the moment at δe = 0 and the moment due to the elevator deflection, Cm = (Cm )δe =0 + Cmδe δe
(5-77)
the condition for equilibrium of the total moment, Cm = 0, results in the required elevator deflection, δe = −
1 (Cm )δe =0 Cmδe
(5-78)
5-2 Equilibrium in steady, straight, symmetric Flight
261
Since, according to equation (5-73), (Cm )δe =0 = Cm0 + Cmα (α − α0 )
(5-79)
the elevator angle δe becomes, δe = −
1 Cmδe
{Cm0 + Cmα (α − α0 )}
(5-80)
The above expressions are based on the simple model as derived in section 5-2-1. In addition, use has been made of several linear approximations. As a consequence, the above expressions are valid, strictly speaking, only for those angles of attack and elevator angles where CNw and CNh are proportional to α, or αh and δe respectively. For more accurate calculations it will be better to rely on wind tunnel measurements of Cm for a range of values of interest of α, ih and δe . The above expressions apply to the case of an aircraft with a fixed stabilizer and a movable elevator. In many cases, however, the available power Cmδe δe max is insufficient to obtain moment equilibrium at all desired c.g. positions over the entire range of airspeeds. Often an adjustable stabilizer with a variable ih is employed, and set such that δe = 0. In that case Cm0 in equation (5-74) is of course no longer a constant . Some aircraft types have a movable horizontal tailplane with even no elevator at all, the so-called ‘flying-tail’. Expressions can be derived quite similarly for the value of ih required for equilibrium in such cases. We conclude this section on equilibrium and tail load with the following comments on equations (5-72) through (5-76), • Cmac , CNwα and α0 are fixed by the aerodynamic design of the wing, the fuselage and the nacelles. h lh • The required tailvolume SS¯ c follows primarily from the requirement that the aircraft shall be statically stable (i.e. Cmα < 0)at the rearmost permissable c.g. position, the derivative CNhα following from the the shape and size of the horizontal tailplane, see section 5-2-2.
• If the angle of incidence of the horizontal tailplane, ih , is not adjustable, ih is generally set such that the elevator angle is zero, minimizing the parasite tailplane drag . 2 dε • Both dα and VVh follow from the characteristics of the wing and the location of the horizontal tailplane relative to the wing and the fuselage, see section 5-2-4. • For tail planes with fixed ih , (Cm )δe =0 depends on angle of attack and c.g. position, and has to be compensated for by the moment generated by the elevator Cmδe δe , see equation (5-77). Flight Dynamics
262
Analysis of Steady Symmetric Flight • For most aircraft the maximum, positive value of Cmδe · δe (elevator up) required in any steady flight condition is larger than the minimum, negative (elevator down) value. • The largest positive values of Cmδe ·δe will be needed to compensate for the largest value of (Cm )δe =0 . The latter occurs at the most forward c.g. position and at the maximum angle of attack.
Ground effect usually produces a further negative contribution to (Cm )δe =0 ; as a consequence for many aircraft the required size of the elevator is determined by the requirement that the aircraft can take-off and land at the most forward c.g. position.
5-2-7
Stick forces in steady flight
To obtain equilibrium of the aerodynamic moment moment about the lateral axis through the center of gravity, the elevator must be deflected by applying a force on the control stick or wheel to compensate for the corresponding aerodynamic hinge moment. In many modern aircraft the connection between the pilot’s control manipulator and the elevator is such that this effort provides only part of the total required hinge moment. Sometimes there is even no direct mechanical connection at all between the control stick or wheel and the control surface, such as in case of a Fly by Wire flight control system. these classes of control systems will not be discussed now, however, the concepts developed here and in subsequent chapters for mechanical flight control systems do in fact apply also to these types of flight control systems. In figure 5-88 a classical manual control system is depicted. The positive directions of the control deflections s, control forces F , surface angles δ and hinge moments H are as indicated. In the following the longitudinal control force required for equilibrium about the lateral axis, Fe , will be considered in more detail. We now derive a simple model for the control force on the elevator control stick or wheel. Similar models can be derived for the aileron control force, Fa , and for the control force on the rudder pedals, Fr . If the control stick is displaced over a small distance dse , the work done by the control force Fe is Fe · dse . Neglecting flexibility, the elevator deflects over an angle dδe and the work done by the hinge moment He is He · dδe . If the internal friction in the control system, as well as cable stretch, is neglected, the control system absorbs no work internally and the total amount of external work has to be zero if the stick and the elevator are again at rest after the displacement. This implies that, Fe dse + He dδe = 0
(5-81)
or, see also figure 5-89, Fe = −
dδe He dse
(5-82)
5-2 Equilibrium in steady, straight, symmetric Flight
263
Figure 5-88: The positive direction of control deflections, control forces, control surface deflections and hinge moments
Figure 5-89: The relation between the control force and the hinge moment
Flight Dynamics
264
Analysis of Steady Symmetric Flight
Figure 5-90: Contributions to the hinge moment
where the gear ratio
dδe dse
is positive , see figure 5-88.
The total hinge moment to be balanced by the control force, see figure 5-90, consists of several components, i.e. the aerodynamic hinge moment (Hea ), a hinge moment caused by friction in the control system (Hef ) and, if the control surface is not statically balanced, a hinge moment (Hew ) caused by the static unbalance. In many aircraft a spring is applied in the control system, giving rise to yet another component of the total hinge moment, Hes . The total hinge moment then is, H e = H ea + H ef + H ew + H es For simplicity , the hinge moments due to friction, static unbalance and the spring will be omitted in the following. This results in, H e = H ea Substituting equation (5-61) in (5-82) the longitudinal control force can then be written as, Fe = −
dδe 1 2 ρV Se c¯e Che dse 2 h
and, using the linearized expression as in equation (5-63) for Che , Fe = −
dδe 1 2 ρVh Se c¯e Chα αh + Chδ δe + Chδt δte dse 2
(5-83)
Suppose now, that for a given c.g. position and angle of attack, or airspeed, the elevator angle, δe , required for equilibrium of the moment about the lateral axis has been determined
5-2 Equilibrium in steady, straight, symmetric Flight
265
according to equation (5-80). Now the longitudinal control force required to maintain this elevator angle, for a given tab angle, δte , follows from equation (5-83). Chapter 7 deals in more detail with this control force as a function of airspeed, c.g. position and tab angle. We conclude with some remarks on the required signs and magnitudes of the hinge moment derivatives Chδt , Chδ and Chα . • The required magnitude of Chδt (this is the case of a classical tail plane with fixed incidence of the horizontal tailplane, ih ) follows from the requirement that in steady flight conditions it must be possible to trim the control forces down to zero by adjusting the trim tab angle. A more detailed definition of aircraft configurations and flight conditions for which this requirement holds can be found in the airworthiness requirements, see references [11, 19, 6, 20, 12, 13, 7]. • The required magnitude of Chδt also depends of course on the allowable tab deflecting angles, δte , and on the values of Chα and Chδ . • As has been mentioned before, in order to keep the required control forces within reasonable limits when changing either the aircraft configuration, indicated airspeed the steady flight condition or when executing turns or pull up manoeuvres, Chα and Chδ should be small in an absolute sense. • As shown in chapter 7 and mentioned above, Chδ must always be negative. Usually Chα is negative as well, but small positive values of Chα are permissable, even desirable and occur indeed in practice. When studying the control forces in more detail in chapter 7, this will be discussed in more detail.
Flight Dynamics
266
Analysis of Steady Symmetric Flight
Chapter 6 Longitudinal Stability and Control Derivatives
In the following sections attention is given to the methods used to obtain the various stability and control derivatives for the symmetric motions. The corresponding methods used for the asymmetric motions will be discussed in chapter 7. Evidently, the accuracy to which the derivatives can be determined has a direct influence on the accuracy of the calculated non-steady aircraft motions. The stability derivatives can be found not only by calculation from generalized data, but also from measurements. Flight tests with the actual aircraft can be made, or wind tunnel tests on a model. From comparisons of measured and calculated stability derivatives obtained in different ways, it appears that none of these various ways to obtain the various derivatives gives satisfactory results in all cases of practical interest. However, recent research suggests that Computational Fluid Dynamics’ (CFD) techniques can also be used to estimate the stability and control derivatives with an accuracy comparable to results obtained by flight test techniques. Calculated derivatives may be expected to show an acceptable correspondence with experimentally obtained values, only for conventional aircraft configurations. For less conventional configurations, calculated derivatives may be considered only as a very first approximation. In such situations measured stability derivatives are indispensable for calculations of stability and control characteristics on which some reliance has to be placed. In the following the various stability derivatives are discussed in turn. When considering the forces and moments in the initial equilibrium condition, the contributions from the propulsive system are included; when discussing the stability derivatives these contributions are not considered.
Flight Dynamics
268
Longitudinal Stability and Control Derivatives Zr
L
α
k Xr XS
α0
Tp
α
D
ip
V c.g. k Xr ZS
Xr
Figure 6-1: The attitude of the stability reference frame relative to the Xr - and Zr -axis, after a disturbance from the equilibrium flight condition
Various calculation methods are mentioned, for more detailed discussions and quantitative data reference is made to the literature. All derivatives, with the exception of CZα˙ , Cmα˙ and Cnβ˙ , are calculated under the assumption that the airflow is always stationary.
6-1
Aerodynamic forces in the nominal flight condition
For the following discussions the direction of the XS -axis of the system of stability axes is defined relative to the aircraft, using the fixed and invariable system of the aircraft reference axes, see figure 6-1 and chapter 2. The angle between the negative Xr -axis and the positive XS -axis is α0 . Note: This is not the α0 meant in equation (4-24). This angle remains constant during the disturbed motion and varies only if one other initial, steady flight condition is chosen. The total aerodynamic force acting on the aircraft in symmetric flight is divided in, 1. The thrust of the propulsive system, Tp 2. The remaining aerodynamic force on the aircraft, consisting of the components, (a) lift, L, perpendicular to the direction of the undisturbed flow (b) drag, D, parallel to the direction of the undisturbed flow
6-1 Aerodynamic forces in the nominal flight condition
269
These three forces are depicted in figure 6-1. The angle between Tp and the negative Xr -axis is ip . In the following it is assumed that ip is independent of the flight condition and does not change during the disturbed motion, as is the case with α0 . Furthermore, α0 and ip are assumed to be small, so,
cos (α0 + ip ) ≈ 1 sin (α0 + ip ) ≈ α0 + ip The total aerodynamic moment acting on the aircraft in steady flight is not divided in contributions explicitly assigned to thrust, lift and drag. The components X and Z of the total aerodynamic force along the XS -axis and the ZS -axis of the system of stability axes can now be calculated. From figure 6-1 it follows, X = Z
L sin α
− D cos α +
Tp
= −L cos α − D sin α − Tp (α0 + ip )
In non-dimensional form, after division by 12 ρV 2 S, it follows, CX CZ
=
CL sin α
Tc′
− CD cos α +
= −CL cos α − CD sin α −
(6-1) Tc′
(α0 + ip )
where α is now the change in angle of attack relative to the value in the equilibrium situation. In equation (6-1) is, Tc′ =
Tp 2D2 = Tc 1 2 S 2 ρV S
In the initial, steady flight condition (the equilibrium situation) α = 0, therefore,
CX0
= −CD + Tc′
CZ0
= −CL − Tc′ (α0 + ip )
In addition, the following relations hold because of equations (4-34) and (4-33) and α = 0, W
CX0
=
CZ0
W = − 1 ρV 2 S cos γ0
1 ρV 2 S 2
sin γ0 (6-2)
2
Flight Dynamics
270
Longitudinal Stability and Control Derivatives
or, CX0 = −CZ0 tg γ0 ≈ +CL tg γ0 If the steady, initial condition is one of horizontal steady flight, then according to equation (6-2), CX0 = 0 and, by consequence, CD = Tc′ For the calculation of the moments reference is made to previous sections of this chapter. For the equations of motion it is important that in the initial steady flight condition the moments are balanced and therefore, Cm0 = 0
6-2
Derivatives with respect to airspeed
Experimentally obtained values of the stability derivatives CXu , CZu and Cmu are rare. Nearly all measurements of stability derivatives in flight are restricted to the motions at constant airspeed. Measurements in wind tunnels are also invariably made at constant speed. Due to these circumstances it is usually not possible to compare calculated with measured values of CXu , CZu and Cmu . The expressions for the derivatives with respect to airspeed to be discussed in the following, should be used with care, since they have been calibrated against experimental data to an insufficient degree.
6-2-1
Stability derivative CXu
According to table 4-3, this derivative is equal to, CXu =
1 1 2 ρV
∂X S ∂V
Also, X = CX
1 2 ρV S 2
From this follows, ∂X ∂CX 1 2 = CX ρV S + ρV S ∂V ∂V 2
6-2 Derivatives with respect to airspeed
271
and, by consequence, CXu = 2 CX +
∂CX V ∂V
(6-3)
where, ∂CX ∂CX V = ∂V ∂u ˆ See also equation (4-35). The partial derivative with respect to airspeed is determined for deviations from the steady, initial flight condition in airspeed only, i.e. at α = 0. This means that in equation (6-3), CX = CX0 and, CXu = 2 CX0 +
∂CX V ∂V
The starting point for the determination of the partial derivative CX at α = 0,
(6-4) ∂CX ∂V
is the expression for
CX = −CD + Tc′ Differentiating this expression with respect to V yields, ∂CX ∂Tc′ ∂CD = − ∂V ∂V ∂V
6-2-2
(6-5)
Stability derivative CZu
In analogy with equation (6-4) it follows for the derivative CZu , CZu = 2 CZ0 +
∂CZ V ∂V
(6-6)
In the steady, initial flight condition is, CZ = −CL − Tc′ (α0 + ip ) After differentiating it follows, in analogy with equation (6-5), ∂CZ ∂CL ∂Tc′ =− − (α0 + ip ) ∂V ∂V ∂V
(6-7) Flight Dynamics
272
6-2-3
Longitudinal Stability and Control Derivatives
Stability derivative Cmu
Following equations (6-4) and (6-6), Cmu is written as, Cmu = 2 Cm0 +
∂Cm V ∂V
Since Cm = 0 in the steady, initial condition, Cmu is reduced to, Cmu = The derivatives
∂CD ∂CL ∂V , ∂V
and
∂Cm ∂V
∂Cm V ∂V
(6-8)
may differ from zero due to various causes,
1. The variation of Mach number with airspeed. Due to the effects of compressibility, the aerodynamic coefficients vary with airspeed at Mach numbers higher than 0.6 to 0.7. 2. The variation of the aircraft’s aeroelastic deformation with airspeed, or rather with dynamic pressure. This effect is not further considered here. 3. The variation of the Reynolds number with airspeed. This influence is entirely neglected for the small variations of airspeed considered here. 4. For a propeller-driven aircraft the contribution to the coefficients CL , CD and Cm made by those parts of the wing and the tailplane submerged in the slipstream, varies with ∂CL D airspeed via the thrust coefficient Tc′ . For ∂C ∂V and ∂V the influences of compressibility m and slipstream effects are taken separately. The same applies to ∂C ∂V . The partial derivatives with respect to V then are, ∂CD ∂V
=
∂CD ∂M ∂M ∂V
+
∂CD ∂Tc′ ∂Tc′ ∂V
∂CL ∂V
=
∂CL ∂M ∂M ∂V
+
∂CL ∂Tc′ ∂Tc′ ∂V
∂Cm ∂V
=
∂Cm ∂M ∂M ∂V
+
∂Cm ∂Tc′ ∂Tc′ ∂V
Expressions (6-4), (6-6) and (6-8) can now be written as, CXu = 2 CX0
CZu = 2 CZ0
∂CD + 1− ∂Tc′
dTc′ ∂CD V − M dV ∂M
∂CL dTc′ ∂CL − (α0 + ip ) + V − M ′ ∂Tc dV ∂M
Cmu =
∂Cm dTc′ ∂Cm V + M ′ ∂Tc dV ∂M
(6-9)
(6-10)
(6-11)
6-2 Derivatives with respect to airspeed
273
In equations (6-9), (6-10) and (6-11) the partial derivatives with respect to Mach number are not further considered here. The effects of compressibility, expressed here through the ∂CL ∂Cm D presence of the partial derivatives ∂C ∂M , ∂M and ∂M , are not the subject of this book. The ∂CD ∂CL m slipstream effect, expressed through ∂T ′ , ∂T ′ and the various contributions to ∂C ∂Tc′ , are not c c discussed in this book either. ′
c The derivative ∂T ∂V is now briefly discussed. The basic assumption is always that during the disturbed aircraft motions to be studied, the power setting of the engine(s), i.e. the throttle position and the setting of any other possible control lever, remains unchanged. Another assumption to be made when determining the variation of Tc′ is that changes in airspeed always occur in a quasi-steady manner, i.e. so slowly that Tc′ varies with airspeed just as in ′ c a series of steady flight conditions. If these assumptions are met, ∂T ∂V can be calculated in a ′ c fairly simple manner. In the following ∂T ∂V is derived for various types of propulsive systems.
1. Jet turbines and rocket motors In studies on dynamic stability characteristics the usual assumption is that the thrust of these engines is independent of airspeed at constant throttle setting. This means, Tp = Tc′
1 2 ρV S = constant 2
This results in, dTc′ 2T ′ =− c dV V At supersonic speeds in particular the thrust of a jet turbine does vary with airspeed. ′ c The correct value dT dV then has to be obtained from more detailed calculations or from measurements. 2. Piston or turbine engines driving constant speed propellers In this case, again at constant power setting, the power produced by the engine is approximately independent of airspeed. If the propeller efficiency is also assumed to be constant for relatively small variations in airspeed, then, Tp V = constant or, Tc′ V 3 = constant This results in, dTc′ T′ = −3 c dV V Flight Dynamics
274
Longitudinal Stability and Control Derivatives
3. Gliding flight Finally, it should be noted that for any propulsive system in gliding flight (Tc′ = 0) the following relation holds, dTc′ =0 dV The foregoing expressions for
dTc′ dV
may be summarized as follows, dTc′ T′ = −k c dV V
where k is, 1. in gliding flight, k = 0 2. jet turbines at subsonic speeds, rocket motors, k = 2 3. piston and turbine engines driving constant speed propellers, k = 3 As derived in the previous subsection,
CX0
= −CD + Tc′
CZ0
= −CL − Tc′ (α0 + ip )
Using these relations, equations (6-9), (6-10) and (6-11) can finally be written as, CXu = −2 CD +
CZu = −2 CL +
Tc′
Tc′
∂CD ∂CD 2−k· 1− − M ∂Tc′ ∂M
∂CL (−2 + k) (α0 + ip ) + k · ∂Tc′
Cmu = −k · Tc′ ·
−
∂CL M ∂M
∂Cm ∂Cm + M ∂Tc′ ∂M
(6-12)
(6-13)
(6-14)
If CX0 ≈ CL tg γ0 is used, equation (6-12) becomes, CXu = 2 CL tg γ0 − k A few specific cases are,
· Tc′
∂CD 1− ∂Tc′
−
∂CD M ∂M
(6-15)
6-2 Derivatives with respect to airspeed
275
1. Gliding flight at subsonic speed Or, with k = 0, and, ∂CD ∂CL ∂Cm = = =0 ∂M ∂M ∂M yields, CXu
= 2 CL tg γ0 = −2 CD
CZu
= −2 CL
Cmu
= 0
2. Level flight at subsonic speed for a jet- or rocket-propelled aircraft Or, with γ0 = 0, k = 2, Tc′ = CD and, ∂CL ∂Cm ∂CD = = =0 ∂M ∂M ∂M yields, CXu
= −2 CD
CZu
= −2 CL
Cmu
= −2 CD
∂Cm ∂Tc′
3. Level flight at subsonic speed for a propeller-driven aircraft having constant speed propellers Or, with γ0 = 0, k = 3, Tc′ = CD and, ∂CD ∂CL ∂Cm = = =0 ∂M ∂M ∂M yields, CXu
= −3 CD 1 −
∂CD ∂Tc′
CZu
n = −2 CL + CD − (α0 + ip ) + 3
Cmu
= −3 CD
∂CD ∂Tc′
o
∂Cm ∂Tc′
Flight Dynamics
276
6-3
Longitudinal Stability and Control Derivatives
Derivatives with respect to angle of attack
A change in angle of attack only is obtained by varying the speed w along the ZS -axis. A pure α-motion is thus seen to be a steady motion where the aircraft performs a translation along the ZS -axis (plunging motion). The stability derivatives CXα , CZα and Cmα pertaining to this type of motion may be obtained in a relatively simple way from wind tunnel measurements on a model of the aircraft. CXα and CZα are derived from a measured or calculated polar curve corresponding to the specific aircraft configuration, whereas Cmα follows from a measured or calculated moment curve Cm − α. A discussion of each of the three derivatives is given below.
6-3-1
Stability derivative CXα
According to table 4-3, this derivative is equal to, CXα =
1 1 2 ρV
∂X ∂CX = ∂α S ∂w
From equation (6-1) follows that, CX = CL sin α − CD cos α + Tc′ Taking the derivative with respect to α, leaving Tc′ constant, results in, CXα = CL cos α + CLα sin α + CD sin α − CDα cos α In the steady, initial condition α = 0, leading to, CXα = CL − CDα if CX is expressed using CN and CT , CXα can be written as, CXα = −CTα cos α0 − CNα sin α0 If a parabolic polar curve is assumed, then, CD = CD0 +
CL2 πAe
where CD0 is the drag coefficient at CL = 0. Then, CDα is, CDα = 2 and CXα can be written as,
CLα CL πAe
(6-16)
Flight Dynamics
277 6-3 Derivatives with respect to angle of attack
Figure 6-2: The definition of aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylon’, ‘nacelles’, ‘vertical fin’ and ‘fuselage’ for the Cessna Ce550 ‘Citation II’ model (starting from the left top figure, in clockwise direction)
278
Longitudinal Stability and Control Derivatives 0.16 wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.14
0.12
CX
0.1
0.08
0.06
0.04
0.02
0
−0.02 −10
−8
−6
−4
−2
0
α [Deg.]
2
4
6
8
10
Figure 6-3: Calculated contributions of various aircraft parts to the force curve CX versus α for the Cessna Ce550 ‘Citation II’
CXα = CL
2CLα 1− πAe
Contrary to most stability derivatives, CXα is normally positive. Using linearized potential flow theory (panel methods), absolute values of CX as a function of α have been calculated for the Cessna Ce550 ‘Citation II’, see figure 6-3. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contribution to CX is from the wing (induced drag).
6-3-2
Stability derivative CZα
According to table 4-3, this derivative is equal to, CZα =
1 1 2 ρV
∂Z ∂CZ = ∂α S ∂w
CZ has been written in equation (6-1) as, CZ = −CL cos α − CD sin α − Tc′ (α0 + ip )
6-3 Derivatives with respect to angle of attack
279
This results in CZα , CZα = CL sin α − CLα cos α − CD cos α − CDα sin α which reduces for α = 0 to, CZα = −CLα − CD As normally CD << CLα , a simplified expression for CZα is, CZα ≈ −CLα The dominant contribution to CZα is, of course, provided by the wing. But in some cases the contribution made by the horizontal tailplane is not negligible. Then it may be practical to express CZα , CZα = −CNwα − CNhα
1−
dε dα
Vh V
2
Sh S
(6-17)
Data for a quantitative determination of CZα using equation (6-17) can be found in various places in the literature, see references [72] and [9]. Using linearized potential flow theory (panel methods), absolute values of CZ as a function of α have been calculated for the Cessna Ce550 ‘Citation II’, see figure 6-4. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contribution to CZ is from the wing.
6-3-3
Stability derivative Cmα
According to table 4-3, this derivative is equal to, Cmα =
1 1 2 ρV
∂M ∂Cm = ∂α S¯ c ∂w
In this chapter the simplified case is considered where the contributions of the tangential forces to Cm are neglected and the contributions of the thrust and slipstream effects are also not considered. It was found in chapter 5 for Cmα , Cmα
xc.g. − xw = CNwα − CNhα c¯
dε 1− dα
Vh V
2
Sh lh S¯ c
(6-18)
More general. Cmα is the slope of the moment curve Cm = Cm (α) at the respective angle of attack at which according to section 6-1, Cm0 = 0. A calculation of Cmα is possible using the references of chapter 5. Flight Dynamics
280
Longitudinal Stability and Control Derivatives 1 wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.5
CZ
0
−0.5
−1
−1.5 −10
−8
−6
−4
−2
0
α [Deg.]
2
4
6
8
10
Figure 6-4: Calculated contributions of various aircraft parts to the force curve CZ versus α for the Cessna Ce550 ‘Citation II’
Using linearized potential flow theory (panel methods), absolute values of Cm as a function of α have been calculated for the Cessna Ce550 ‘Citation II’, see figure 6-5. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to Cm are from the fuselage (destabilizing) and horizontal stabilizer (stabilizing).
6-4
Derivatives with respect to pitching velocity
Before discussing the derivatives CXq , CZq and Cmq , the angular motion under consideration will be studied more closely. During the turning, symmetric motion, indicated here as qmotion, the center of rotation lies on the top axis through the aircraft’s center of gravity, see figure 6-6. The distance R to the c.g. is,
R=
V q
At a positive pitching velocity the center of rotation lies on the negative ZS -axis, i.e. above the aircraft. In all points of the aircraft, with the exception of the c.g. and all other points on the ZS -axis, the geometric angle of attack is proportional to the angular velocity and to the distance in
6-4 Derivatives with respect to pitching velocity
281
0.25 wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.2
0.15
Cm
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2 −10
−8
−6
−4
−2
0
α [Deg.]
2
4
6
8
10
Figure 6-5: Calculated contributions of various aircraft parts to the moment curve Cm versus α for the Cessna Ce550 ‘Citation II’
XS -direction to the center of gravity, ∆α =
x − xc.g. x − xc.g. q¯ c = · R c¯ V
(6-19)
See also figure 6-6. The magnitude of the local airspeed varies (in principle) also with the angular velocity. This is because the local velocity in a given point varies proportional with the distance of that point to the center of rotation. Only in points at distance R of the center of rotation, i.e. also in the c.g., the magnitude of the local velocity does not vary with the rate of pitch. The aircraft’s c.g. is the only point where during this q-motion neither the magnitude of the velocity, nor the direction, i.e. the geometric angle of attack, vary. Changes in the magnitude of the local airspeed and the geometric angle of attack at the aircraft’s c.g. are described as the u-motion and the α- or plunging-motion respectively. As far as the airflow at the center of gravity is concerned, the q-motion causes only a curvature of the streamlines. If the angular velocity during a q-motion is not constant, but varies harmonically with time, the aircraft describes due to this q-motion a sinusoidal trajectory, as indicated in figure 6-7. The most characteristic feature of this trajectory is, that the angle of attack at the aircraft’s c.g. remains constant during this motion, i.e. α˙ = 0. The center of gravity moves along an undulating trajectory, such that, Flight Dynamics
282
Longitudinal Stability and Control Derivatives
q
Zr
∆α
R
∆α
xc.g. x
Xr Figure 6-6: The pure q-motion, with ∆α =
x−xc.g. R
=
x−xc.g. q¯ c c¯ V
6-4 Derivatives with respect to pitching velocity
283
XS XS
XS XS ZS
ZS
ZS XS ZS
ZS
Figure 6-7: Harmonic q-motion, α = constant, θ = γ
Flight Dynamics
284
Longitudinal Stability and Control Derivatives
XS
XS
XS XS
ZS
XS
ZS
ZS ZS
ZS
Figure 6-8: Harmonic α-motion, q = 0
6-4 Derivatives with respect to pitching velocity
285
XS XS
XS
XS ZS
ZS
ZS
ZS
Figure 6-9: Combined harmonic q- and α-motion, q = α, ˙ γ = constant
θ˙ = q = α˙ + γ˙ = γ˙ Both the angle of pitch and the flight path angle will vary harmonically. In figure 6-8 the case for harmonic plunge is presented, i.e. α varies harmonically, however, now q = 0. An entirely different motion, to be distinguished very clearly from the q-motion, results if the center of rotation lies in the aircraft’s c.g., see figure 6-9. Then the angle of attack varies also in the center of gravity and, θ˙ = q = α˙
(6-20)
γ˙ = θ˙ − α˙ = 0
(6-21)
and, by consequence,
According to equation (6-21) the center of gravity moves along a straight line, if the aircraft rotates while the c.g. is the center of rotation. This motion may be considered a superposition of a q-motion as defined above and a variable α-motion, such that the condition (6-20) is satisfied. Flight Dynamics
286
Longitudinal Stability and Control Derivatives
center of gravity
XS
ZS Figure 6-10: Aircraft in a curved flow field
Zr ↑ (xc.g. , zc.g. )
xc.g.
(x, z)
zc.g.
z
x
Xr −→ Figure 6-11: Equivalent curved aircraft in a straight flow field with aircraft frame of reference, Fr
6-4 Derivatives with respect to pitching velocity
287
Figure 6-12: Equivalent curved aircraft in a straight flow field
Figure 6-13: Equivalent curved aircraft in a straight flow field, 3-dimensional view
Flight Dynamics
288
Longitudinal Stability and Control Derivatives
This q-motion is here the subject of discussion, the influence on the aerodynamic forces and moments due to an acceleration along the top axis during a non-steady α-motion is discussed in the next section. The variation of the magnitude of the local airspeed with the distance to the center of rotation during the q-motion is usually neglected because of the small dimensions of the aircraft in the ZS -direction if compared with common values of R. The only effect of the rotation is then a variation of the geometric angle of attack in the XS -direction, expressed by equation (6-19), see figure 6-10. A field of flow equivalent to that of the aircraft in curved flow can be obtained by placing a suitable curved aircraft in a field of parallel flow, see figures 6-11, 6-12 and 6-13. The curvature of the aircraft or the wind tunnel model must be such that, ∆α = −
x − xc.g. q¯ dz c =+ dx c¯ V
see equation (6-19), or, z − zc.g. 1 =− c¯ 2
x − xc.g. c¯
2
q¯ c V
c The curvature is parabolic in the XS -direction and proportional to q¯ V . For theoretical calculations it may be advantageous to use the curved aircraft in parallel flow. If, however, the stability derivatives would have to be measured using a curved model in parallel flow, it would c be necessary to use a separate model for each value of q¯ V .
The changes in the forces X and Z and in the moment M caused by the q-motion are discussed in the following. The change in X, expressed by the derivative CXq , CXq =
1 1 2 ρV
∂X ∂CX = q¯c S¯ c ∂q ∂V
is always neglected. The assumption thus is, CXq = 0
Using linearized potential flow theory (panel methods), absolute values of CX as a function of q¯ c V have been calculated for the Cessna Ce550 ‘Citation II’, see figure 6-14. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. All defined aircraft parts have a neglible contribution to CX (as compared to the force curve CX versus α, see figure 6-3).
6-4 Derivatives with respect to pitching velocity
289
0.03 wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.025
0.02
CX
0.015
0.01
0.005
0
−0.005
−0.01 −0.05
−0.04
−0.03
−0.02
−0.01
0.01 q¯ c 0 V [Rad.]
0.02
0.03
0.04
0.05
Figure 6-14: Calculated contributions of various aircraft parts to the force curve CX versus Cessna Ce550 ‘Citation II’
q¯ c V
for the
The two remaining derivatives are CZq and Cmq , CZq =
Cmq =
1 1 2 ρV
∂Z ∂CZ = q¯c ∂q S¯ c ∂V
1 ∂M ∂Cm = q¯c 2 ∂q S¯ c ∂V
1 2 ρV
Of these two derivatives Cmq is the most important. For an aircraft having a horizontal tailplane, the largest contribution to Cmq is provided by this tailplane. The change in wing lift caused by the angular velocity, i.e. by the curvature of the flow field, is small. Therefore, no change in downwash, as a reaction to a change in wing lift, is accounted for either. The contribution from the tailplane is calculated as follows. According to equation (6-19) the change in angle of attack of the horizontal tailplane due to the rotation is, ∆αh =
xh − xc.g. q¯ c lh q¯ c ≈ c¯ V c¯ V
This change in angle of attack causes a normal force at the tailplane, ∆CNh = CNhα
Vh V
2
Sh ∆αh S Flight Dynamics
290
Longitudinal Stability and Control Derivatives
and this generates a moment about the aircraft’s c.g.,
∆Cm = −CNhα ∆αh
Vh V
2
Sh lh = −CNhα S¯ c
Vh V
2
Sh lh2 q¯ c 2 S¯ c V
As a consequence the contribution of the horizontal tailplane to Cmq is,
h
= −CNhα
Vh V
2
Sh lh2 S¯ c2
h
= −CNhα
Vh V
2
Sh lh S¯ c
Cmq
CZq
and the contribution to CZq is,
A rough estimate of CZq for the complete aircraft is sometimes taken as, CZq = 2 CZq
h
= −2 CNhα
Vh V
2
Sh lh S¯ c
For aircraft having a straight, slender wing Cmq of the entire aircraft is usually approximated by multiplying the contribution of the tailplane with a factor 1.1 to 1.2, in order to account for the influence of the wing. For these conventional aircraft Cmq can be expressed by, Cmq = −(1.1) CNhα
Vh V
2
Sh lh2 S¯ c2
Finally, a practical remark should be made. Often, quantitative data for the determination of CZq and Cmq are taken from publications of the American National Aeronautics and Space Administration (NASA). It is of interest to realize that NASA, and some other organizations q¯ c c as well, commonly relate the stability derivatives CZq and Cmq to 2V rather than q¯ V . In contrast to a u- or α-motion, the q-motion changes with a shift in the c.g. position. This can be seen as follows. In the new c.g. the change in angle of attack due to a rotation about the new center of rotation is, of course, again zero. This change in angle of attack at the new c.g. position c.g.2 , ∆αc.g.2 (where ∆αc.g.2 = 0), is the sum of a contribution from a rotation about the old c.g. (c.g.1 ) and a change in angle of attack due to an α-motion along the ZS -axis. From this consideration Cmq can be calculated in a straightforward manner. In table 6-2 the expressions for the variations of CZq and Cmq with the c.g. position are given, based on the above discussion. Using linearized potential flow theory (panel methods), absolute values of CZ and Cm as c a function of q¯ V have been calculated for the Cessna Ce550 ‘Citation II’, see figure 6-15 and figure 6-16. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to CZ
6-5 Derivatives with respect to the acceleration along the top axis
291
0.15
0.1
0.05
0
CZ
−0.05
−0.1
−0.15
−0.2
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
−0.25
−0.3 −0.05
−0.04
−0.03
−0.02
−0.01
0.01 q¯ c 0 V [Rad.]
0.02
0.03
0.04
0.05
Figure 6-15: Calculated contributions of various aircraft parts to the force curve CZ versus Cessna Ce550 ‘Citation II’
q¯ c V
for the
are from the wing and the horizontal stabilizer. The main contribution to Cm is from the c q¯ c horizontal stabilizer. Note that both the slope of CZ − q¯ V and Cm − V are representative for respectively the stability derivatives CZq and Cmq . E.g. from figure 6-15 it is obvious that CZq indeed is approximately, CZq = 2 CZq
h
Furthermore, from figure 6-16 it also holds that Cmq is indeed approximately, Cmq = 1.1 CZq
6-5
lh h c ¯
Derivatives with respect to the acceleration along the top axis
If the components of airspeed u or w or the rate of pitch q of an aircraft experience a sudden change, a certain time interval passes before the pressure distribution over the entire aircraft has adjusted to the new flow condition. Usually, changes in the airspeed, i.e. in the component u, occur so slowly that this delayed adjustment is not noticeable. Changes in u are, therefore, always assumed to occur in a quasi-steady fashion. On the other hand, changes in angle of attack, i.e. in the speed component w along the ZS -axis, may occur much more quickly. Such changes are discussed in the following. Flight Dynamics
292
Longitudinal Stability and Control Derivatives 0.4 wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.3
0.2
Cm
0.1
0
−0.1
−0.2
−0.3
−0.4 −0.05
−0.04
−0.03
−0.02
−0.01
0.01 q¯ c 0 V [Rad.]
0.02
0.03
0.04
0.05
Figure 6-16: Calculated contributions of various aircraft parts to the moment curve Cm versus Cessna Ce550 ‘Citation II’
q¯ c V
for the
In reference [38] Cowley and Glauert were the first to point to the fact that an acceleration w˙ of the aircraft in the direction of the top axis has a non-negligible influence on the longitudinal moment Cm . This effect has an important influence on the damping of the symmetric motions. Following Cowley and Glauert, this effect is usually expressed through a stability derivative Cmα˙ . In order to account for the influence of the c.g. on this derivative Cmα˙ , it is necessary in principle to know as well the derivative CZα˙ which is in itself quite important. As described earlier the relation between the aerodynamic coefficient Cm and the component of the motion α may be expressed in a Taylor series,
∆Cm
∂Cm ∂Cm α¯ ˙c 1 = ∆α + α¯ + ˙ c ∂α 2! ∂V V ∂ 2 Cm ˙c 2 ∂ α¯ V
α¯ ˙c V
(
∂ 2 Cm ∂ 2 Cm α¯ ˙c 2 (∆α) + 2 ∆α + α¯ ˙ c 2 ∂α V ∂α∂ V
2 )
+
2
˙
1 {· · ·} + · · · · ·· 3! 3
(6-22)
¨c ¨c In principle, the derivatives with respect to α¯ , α¯ , etcetera, could also be included in this V2 V3 series expansion. Experimental evidence shows, however, that the changes in angle of attack occur slowly enough to neglect any influences of the higher time derivatives than α. ˙ As usual, in the series (6-22) only the linear terms are maintained under the assumption that the values ˙c of ∆α and α¯ V remain sufficiently small.
6-5 Derivatives with respect to the acceleration along the top axis
293
The change in the moment coefficient, ∆Cm , can then be written as, ∆Cm =
∂Cm ∂Cm α¯ ˙c ∆α + α¯ ˙ c ∂α ∂V V
or, in the common notation used in stability analyses, ∆Cm = Cmα α + Cmα˙
α¯ ˙c V
(6-23)
˙c The term Cmα˙ α¯ V is then supposed to express the influence on Cm of the delayed adjustment of the airflow to changes in α.
For aircraft having a horizontal tailplane, this tailplane usually provides the most important contribution to Cmα˙ . A simplified explanation of this effect is the following. In chapter 5 the angle of attack of the horizontal tailplane in steady, straight flight was expressed by equation (5-64), αh = α − ε + ih During the accelerated translation along the ZS -axis here considered, ih remains constant. In accordance with the conventions made in the present chapter, the explicit indication that a change in a variable is considered, is omitted. As a consequence αh (t) can be written here as, αh (t) = α(t) − ε(t) The horizontal tailplane is always situated in an air mass having passed the wing a brief time interval ∆t, ∆t ≈
lh V
earlier. As a consequence, the downwash angle at the horizontal tailplane is at any time proportional to the wing angle of attack that brief time interval earlier, ε(t) =
dε α(t − ∆t) dα
(6-24)
The angle of attack of the wing at the time t − ∆t can be written as, α(t − ∆t) = α(t) − α˙ ∆t +
1 1 α ¨ (∆t)2 − · · · · · · 2! 3!
In this expression the time derivative of α higher than the first are omitted, as in equation (6-22). The result is, Flight Dynamics
294
Longitudinal Stability and Control Derivatives
α(t − ∆t) = α(t) − α˙ ∆t In equation (6-24) this leads to,
ε(t) =
dε dε lh α(t) − α˙ dα dα V
and,
αh (t) = α(t)
dε 1− dα
dε lh α˙ dα V
+
The term in the change in αh proportional to α˙ is then, ∆αh (t) = +
dε lh α¯ ˙c dα c¯ V
(6-25)
This ∆αh produces a force,
∆CZ = −CNhα ∆αh
Vh V
2
Sh S
and a longitudinal moment,
∆Cm = −CNhα ∆αh
Vh V
2
Sh lh S¯ c
Substituting equation (6-25) in the latter two expressions results in the desired, approximated, expressions for the stability derivatives,
CZα˙ = −CNhα
Vh V
2
dε Sh lh dα S¯ c
Cmα˙ = −CNhα
Vh V
2
dε Sh lh2 dα S¯ c2
The variation of Cmα˙ with the c.g. position is calculated by, ∆Cmα˙ = −CZα˙ ∆
xc.g. c¯
In the calculation of CZα˙ and Cmα˙ attention should be given, as with CZq and Cmq , to the α¯ ˙c ˙c fact that in the literature these derivatives occur referenced to 2V rather than to α¯ V .
6-6 Derivatives with respect to the elevator angle
6-6 6-6-1
295
Derivatives with respect to the elevator angle Control derivative CXδe
According to table 4-3, this derivative is equal to, CXδe =
1 1 2 2 ρV S
∂X ∂CX = ∂δe ∂δe
This derivative expresses the variation of aircraft drag with the elevator angle. Although the total drag caused by the elevator deflection may be non-negligible, especially at supersonic speed (‘trim drag’), the variation in CX with small variations of δe is commonly neglected. Or, CXδe = 0
6-6-2
Control derivative CZδe
According to table 4-3, this derivative is equal to, CZδe =
1 1 2 2 ρV S
∂Z ∂CZ = ∂δe ∂δe
If the aircraft has a horizontal tailplane, CZδe can also be written more explicitly as, CZδe = −CNhδ
e
Vh V
2
Sh S
(6-26)
References [9, 102, 42, 45, 85, 98, 37] provide data to derive CNhδ if the form and the e dimensions of the stabilizer and the elevator are known. For tailless aircraft CZδe should preferably be obtained from measurements on a wind tunnel model, see also references [42, 45, 98, 37].
6-6-3
Control derivative Cmδe
According to table 4-3, this derivative is equal to, Cmδe =
1 1 2 c 2 ρV S¯
∂M ∂Cm = ∂δe ∂δe
If the aircraft has a horizontal tailplane, Cmδe can be simply expressed using CZδe , Cmδe = CZδe
xh − xc.g. c¯
or, Flight Dynamics
296
Longitudinal Stability and Control Derivatives
Cmδe = −CNδe
Vh V
2
Sh xh − xc.g. S c¯
2
An approximation for Cmδe is, Cmδe = −CNδe
Vh V
Sh lh S¯ c
This results in Cmδe , if CZδe has been previously obtained using equation (6-26). Of the three control derivatives Cmδe has the largest effect on the aircraft motions. For aircraft having a horizontal tailplane, the influence of Cmδe dominates to such an extent that very often CZδe for these aircraft is neglected. This approximation is less applicable to tailless aircraft. The derivative Cmδe for tailless aircraft is derived by the same calculation methods or wind tunnel measurements yielding also CZδe . Table 6-1 summarizes the expressions for the stability and control derivatives discussed in the previous sections.
6-7
Symmetric inertial parameters
Reliable quantitative data on moments and products of inertia of aircraft are rare. Quantitative calculations are very time-consuming and may produce rather inaccurate results. The most reliable inertial parameters of aircraft are those obtained experimentally, usually by oscillating the suspended aircraft on the ground. Details on this experimental technique can be found in references [167, 163, 41, 96].
6-7 Symmetric inertial parameters
297
W
CX0 =
−CD + Tc′ cos (α0 + ip ) =
CZ0 =
W −CL − Tc′ sin (α0 + ip ) = − 1 ρV 2 S cosγ0 ≈ −CL
Cm0 =
0
CXu =
2 CL tg γ0
CZu =
−2 CL
Cmu =
0
CXα =
CL − CDα = −CTα cosα0 − CNα sinα0 ; if CD = CD0 +
1 ρV 2 S 2
sinγ0 ≈ −CD + Tc′
2
CXα = CL 1 −
2CLα πAe
−CLα − CD ≈ −CLα ≈ −CNα
Cmα =
CNwα
CXq =
0
CZq =
2 CZq
Cmq =
−(1.1 to 1.2) Cmq
CXα˙ =
0
CZα˙ =
−CNhα
Cmα˙ =
−CNhα
h
− CNhα 1 −
= −2 CNhα
Vh V Vh V
2 2
h
then
CZα =
xc.g. −xw c¯
2 CL πAe
Vh V
dε dα
2
V 2 h
V
Sh lh S¯ c
Sh lh S¯ c
= −(1.1 to 1.2) · CNhα ·
Vh V
2
2 Sh lh 2 S¯ c
dε Sh lh dα S¯ c 2 dε Sh lh dα S¯ c2
Table 6-1: Simplified formulae for the calculation of stability and control derivatives and coefficients in the initial, steady flight condition for the symmetric motions (without the effects of propellers and jets, aeroelasticity and compressibility of the air).
Flight Dynamics
298
Longitudinal Stability and Control Derivatives
CXδe =
0
CZδe =
−CNhδ
Cmδe =
−CNδe
e
Vh V
Vh V
2
2
Sh S Sh lh S¯ c
Table 6-1: (Continued) Simplified formulae for the calculation of stability and control derivatives and coefficients in the initial, steady flight condition for the symmetric motions (without the effects of propellers and jets, aeroelasticity and compressibility of the air).
Cmα2 =
Cmα1 − CZα
CZq2 =
CZq1 − CZα
Cmq2 =
Cmq1 − CZq2
=
Cmα˙ 2 =
xc.g.2 −xc.g.1 c¯
xc.g.2 −xc.g.1 c¯ xc.g.2 −xc.g.1 c¯
Cmq1 − CZq1 + Cmα1
Cmα˙ 1 − CZα˙
− Cmα1
xc.g.2 −xc.g.1 c¯
xc.g.2 −xc.g.1 c¯
+ CZα
= xc.g.2 −xc.g.1 c¯
2
xc.g.2 −xc.g.1 c¯
Table 6-2: Influence of the center of gravity position on some stability derivatives
Chapter 7 Lateral Stability and Control Derivatives
In the previous chapter the equilibrium about the lateral axis and longitudinal control in steady, symmetric flight was studied. In addition, static longitudinal stability was considered since this is usually the most critical condition for dynamic longitudinal stability. Studies in dynamic lateral stability relate to the asymmetric motions relative to a steady, symmetric flight condition. There is, however, not one specific ‘static lateral stability’ characteristic for the asymmetric disturbed motions. These disturbed asymmetric motions appear to be determined by more than just one stability derivative, while the inertial parameters also have an important influence. In this chapter the lateral stability and control derivatives are discussed, whereas in chapter 9 the lateral equilibrium and lateral control in steady, asymmetric flight conditions will be studied, similar to chapter 8.
7-1
Aerodynamic force and moments due to side slipping, rolling and yawing
The asymmetric degrees of freedom of an aircraft are the following, see figure 7-1. • The translation of the aircraft center of gravity along the YB -axis of the aircraft body axes, positive in the direction of this Y -axis. The component of airspeed along the YB -axis is indicated as v. If such a component is present the aircraft is said to be in sideslipping flight. The angle of sideslip β is the angle between the velocity vector V of the center of gravity and the plane of symmetry, see figure 7-2, Flight Dynamics
300
Lateral Stability and Control Derivatives XB
q
p u
YB
v
c.g. w
r
ZB Figure 7-1: The six degrees of freedom of a rigid aircraft
β = arcsin or, at sufficiently small angles of sideslip, β≈
v V
v V
(Radians)
(7-1)
The sideslip angle β is positive if the aircraft moves in the direction of the positive YB -axis, so to the right as seen by the pilot. • The angular velocity about the XB -axis, positive if the right wing moves down. The angular velocity about the XB -axis, the rolling velocity, is indicated as p. Often the pb non-dimensional form 2V is used. • The angular velocity about the ZB -axis, positive if the aircraft’s nose moves to the right as seen by the pilot. The angular velocity about the ZB -axis, the yawing velocity, is pb indicated as r. The non-dimensional yaw velocity about this top axis is 2V . The asymmetric motions in general cause an aerodynamic force Y along the YB -axis, a moment L about the XB -axis and a moment N about the ZB -axis, see figure 7-3. As already noted in section 4-1, asymmetric motions generally do not produce symmetric aerodynamic forces and moments. As a consequence the symmetric aerodynamic forces X and Z and the moment M , with a single exception, do not have to be considered. The rolling moment L is commonly indicated as L′ in situations where confusion with the lift L is possible. The non-dimensional asymmetric forces and moments are defined as,
7-1 Aerodynamic force and moments due to side slipping, rolling and yawing XB
301 YB
u
v
V β α c.g. w
ZB Figure 7-2: Definition of the angle of attack α and sideslip angle β
CY =
Cℓ =
Cn =
Y 1 2 2 ρV S
L
(7-2)
1 2 2 ρV Sb
N 1 2 2 ρV Sb
where Cℓ and Cn are referenced to the wing span and not, like Cm , to the wing m.a.c., c¯. The non-dimensional aerodynamic force and moments resulting from side slipping, rolling and yawing are determined by the partial derivatives with respect to the non-dimensional pb rb asymmetric components of the moment β, 2V and 2V P. These derivatives are, • For sideslipping flight, CYβ =
∂CY ∂β
Cℓβ =
∂Cℓ ∂β
Cnβ =
∂Cn ∂β
• For rolling flight, CYp =
∂CY pb ∂ 2V
Cℓp =
∂Cℓ pb ∂ 2V
Cnp =
∂Cn pb ∂ 2V
• For yawing flight, CYr =
∂CY rb ∂ 2V
Cℓr =
∂Cℓ rb ∂ 2V
Cnr =
∂Cn rb ∂ 2V Flight Dynamics
302
Lateral Stability and Control Derivatives XB
L = Cℓ 12 ρV 2 Sb
YB
V β α pb 2V
v c.g.
Y = CY
1 2 2 ρV S
rb 2V
N = Cn 12 ρV 2 Sb ZB Figure 7-3: Asymmetric force and moments pb rb The derivatives with respect to 2V and 2V are indicated only with the subscript p and r respectively, to simplify the notation. The above mentioned partial derivatives are called the stability derivatives with respect to angle of sideslip, non-dimensional rolling velocity and non-dimensional yawing velocity, as these derivatives are used primarily in relation to the stability of aircraft flight. When using the stability derivatives it is assumed that the forces pb rb and 2V . The additional assumption is made, that and moments vary linearly with β, 2V in symmetric flight CY , Cℓ and Cn are zero. As a result, for instance in steady, straight sideslipping flight (p = r = 0), it can be written,
CY = CYβ β
Cℓ = Cℓβ β
Cn = Cnβ β
For small deviations from the symmetric equilibrium condition to which the study of stability refers, this latter assumption is generally acceptable. Commonly, this assumption also holds pb rb when considering steady, asymmetric flight at values of β, 2V and 2V which are not too large. During the non-steady motion of aircraft caused by a disturbance from the symmetric equilibrium condition the accelerations also generate aerodynamic forces and moments. These are usually small enough to be negligible. An exception is only made for the moment about the top axis caused by an acceleration along the YB -axis. The stability derivative describing this moment, indicated in accordance with the above as Cnβ˙ , can be compared to the stability derivative Cmα˙ for the symmetric motions already discussed in section 6-5. A deviation from the rule that asymmetric motions cause no symmetric forces and moments, is the change in pitching moment caused by a velocity about the lateral axis, v. In the study of stability, where small deviations from the symmetric equilibrium situation are considered, this change in Cm is commonly negligible. In steady asymmetric flight where large angles of sideslip occur, but also during take-off runs and during landing with cross wind, this pitching moment due to sideslip certainly has to be considered. The effect is expressed quantitatively
7-2 Stability derivatives with respect to the sideslip angle
303
by the derivative Cm2 . β
Summarizing the above, the non-dimensional asymmetric aerodynamic forces and moments caused by the lateral motions are written as, CY = CYβ β + CYp
pb rb + CYr 2V 2V
Cℓ = Cℓβ β + Cℓp
pb rb + Cℓr 2V 2V
Cn = Cnβ β + Cnβ˙
(7-3)
˙ βb pb rb + Cnp + Cnr V 2V 2V
In the following the underlying mechanisms, magnitudes and signs of the stability derivatives are discussed. Figures 7-50 give a summary of the forces and moments acting on the aircraft pb rb due to β, 2V and 2V . The usual signs of the corresponding stability derivatives are shown as well. Quantitative values of the stability derivatives have been compiled in appendix B for a number of aircraft in various flight conditions, see also reference [18]. When it comes to the actual calculation of stability derivatives, reference will be made to relevant references. An extensive list of references on the calculation of stability derivatives can be found in reference [154]. The calculation methods indicated in the references often rely on empirical data obtained from similar aircraft configurations. Calculation methods based entirely on theory result in most instances in insufficiently accurate results. Wind tunnel measurements remain necessary in later design stages to arrive at sufficiently accurate data. This applies in particular to the stability derivatives with respect to angle of sideslip, β, which have a predominant influence on lateral stability and lateral control characteristics of aircraft.
7-2 7-2-1
Stability derivatives with respect to the sideslip angle Stability derivative CYβ
Sideslipping motion generates an aerodynamic lateral force, usually negative at positive angles of sideslip, Y = CYβ β
1 2 ρV S 2
(7-4)
where CYβ is negative. The dominant contributions to CYβ are caused by the fuselage and the vertical tailplane, see figure 7-4. The contribution of the wing is generally negligible unless the wing has a large angle of sweep. The propulsion system may also give a contribution to CYβ that is not negligible.
Flight Dynamics
304
Lateral Stability and Control Derivatives
CY 0.3
δf = 0o complete model 0.2
0.1
wing+fuselage fuselage -16
-12
-8
-4
0
4
8
12
β [o ]
wing -0.1
-0.2
-0.3
Figure 7-4: CY as a function of β measured on a model of the Fokker F-27 in gliding flight (from references [148, 150, 24])
7-2 Stability derivatives with respect to the sideslip angle
305
The small contribution of the wing may, if necessary, be obtained for low airspeeds using references [10, 70, 34, 122, 156, 63]. The influence of the compressibility of the air can be accounted for using reference [56]. Calculation methods applicable to supersonic speeds are given in references [10, 70, 34, 3, 139, 2, 76]. The contribution of the fuselage arises in the same way as the normal force on a symmetric fuselage at a non-zero angle of attack. Calculation methods can be found in references [10, 70, 34, 3, 139, 2, 76]. The contribution of the vertical tailplane can, in analogy with the gradient of the normal force on the horizontal tailplane in the symmetric case, be written as, CYβ
v
= CYvα
dαv dβ
Vv V
2
Sv S
(7-5)
Compare this equation to the corresponding expression for the horizontal tailplane, (CNα )h = CNhα
dαh dα
Vh V
2
Sh S
C
In equation (7-5) CYvα = αYvv is the gradient of the normal force on the isolated vertical tailplane referenced to the area Sv of the vertical tailplane and the average local dynamic pressure 12 ρVv2 at the vertical tailplane. The angle of attack αv is the local angle of attack of the vertical tailplane measured in the XB OYB -plane and counted as positive if the airflow hits the tailplane from the left, see figure 7-5. Even when disregarding the sign, αv is generally not equal to β. Due to the presence of the fuselage and the wing-fuselage interaction a change in the component of the local velocity of the air parallel to the YB -axis occurs. The resulting change in the direction of the airflow is described using the sidewash angle σ, comparable to the downwash angle ε in symmetric flow. Consulting figure 7-5 it follows, αv = − (β − σ)
(7-6)
The corresponding expression for the horizontal tailplane was, see equation (5-64), αh = α − ε + ih From equation (7-6) follows, dαv dσ =− 1− dβ dβ
(7-7)
Substituting equation (7-7) in equation (7-5) results in, CYβ
v
= −CYvα
dσ 1− dβ
Vv V
2
Sv S
(7-8) Flight Dynamics
306
Lateral Stability and Control Derivatives
β (> 0)
V
αv (< 0)
αv = − (β − σ)
σ (> 0)
Figure 7-5: Relation between the sideslip angle β, the sidewash angle σ and the angle of attack αv at the vertical tailplane
As a result of the sign convention for αv , CYvα is positive like any normal force gradient. In the quantitative calculation of CYvα the presence of the horizontal tailplane, sometimes acting as an end plate to the vertical tail plane, has to be taken into account. Due to this effect the effective aspect ratio of the vertical tailplane may be considerably larger than its geometric aspect ratio. For subsonic airspeeds the calculation of CYvα can be made using references [10, 70, 103, 91, 116, 135, 131, 31]. For supersonic speeds references [109, 29, 106, 107] provide calculation methods. If the angle of attack is not too large,
Vv 2 V
may usually be taken as 1 if the tailplane is not 2 2 located in a slipstream. In the contrary case, VVv is determined in the same way as VVh ) for the horizontal tailplane. The magnitude of dealing with Cnβ .
dσ dβ
will be discussed in more detail when
The propulsion system provides a negative contribution to CYβ caused by a lateral force in the propeller plane or in the plane of the engine inlet in the case of jet propulsion. This effect is a direct analogy with the normal force generated by a propeller or the engine air inlet if placed under a non-zero angle of attack. Calculation methods to determine this contribution are given in references [133, 134]. Using linearized potential flow theory (panel methods), CY as a function of β has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-6. The definition of the aircraft
7-2 Stability derivatives with respect to the sideslip angle
307
0.1
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.08
0.06
0.04
CY
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1 −10
−8
−6
−4
−2
0
2
β [Deg.]
4
6
8
10
Figure 7-6: Aerodynamic force coefficient CY as a function of β for the Cessna Ce550 ‘Citation II’, α = 0o
parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to CYβ are from the vertical fin and the fuselage.
7-2-2
Stability derivative Cℓβ
The stability derivative Cℓβ is one of the primary lateral stability derivatives, usually indicated as the ‘effective dihedral’ of the aircraft. The explanation of this name lies in the fact that for conventional aircraft without wing sweep Cℓβ can usually be varied primarily changing the dihedral of the wing. To obtain desirable lateral control charactersitics it is required that Cℓβ is negative. This can be illustrated as follows. Suppose a disturbance causes an angle of roll to the right. Under the influence of the component of gravity along the YB -axis, the aircraft starts sideslipping to the right. A negative Cℓβ then causes a rolling moment trying to return the aircraft to an even keel, without interference by the pilot. In a qualitative sense this explains the desirability of a negative Cℓβ . Apart from wing dihedral, wing sweep and the wing-fuselage interaction have a large influence on Cℓβ . The slipstream influence on Cℓβ of propeller driven aircraft can be considerable. The contributions of the fuselage and the tailplane are generally small. The contribution of wing-dihedral to Cℓβ is caused by the difference in geometric angle of attack of the left and right hand halves of the wing in sideslipping flight. From subfigure b of figure 7-7 follows for the right wing without sweep and having not too large a dihedral, Flight Dynamics
308
Lateral Stability and Control Derivatives
Vnr = w cos Γ + v sin Γ ≈ w + v Γ where for small values of α and β, see subfigure a of figure 7-7, w ≈ V sin α ≈ V α and, v ≈ V sin β ≈ V β resulting in, Vnr = V (α + β Γ)
(7-9)
The geometric angle of attack of the right wing then is, see subfigure c of figure 7-7, αwr = arctan
Vnr u
≈α+β Γ
(7-10)
In a similar manner it can be derived for the left wing, Vnl ≈ V (α − β Γ)
(7-11)
αwl ≈ α − β Γ
(7-12)
hence,
The increase in lift on the right wing due to ∆αwr = +β · Γ and the decrease in lift on the left wing due to ∆αwl = −β · Γ are thus proportional to the geometric dihedral Γ. From the above it can be concluded that Cℓβ w will also be approximately proportional to Γ.
Swept wings experience an additional difference in lift between the two halves of the wing, independent of dihedral. This difference in lift is caused by the difference in the components of airspeed perpendicular to the wing leading edge on the 14 -chord line. It can be seen from figure 7-8 that the difference in lift ∆L can be roughly approximated as, ∆L = CL
1 2 S 2 ρV cos (Λ − β) − cos2 (Λ + β) 2 2
(7-13)
Further analysis results for small values of β in, ∆L = CL
1 2 ρV S sin 2Λ · β 2
(7-14)
7-2 Stability derivatives with respect to the sideslip angle w
309
β Γ
α v
u V
YB
XB
ZB (A) Wing with dihedral in sideslipping flight
Vnr
w
w Γ
Vnℓ v
v
Γ YB
− 2b
+ 2b
ZB (B) The normal velocities Vnℓ and Vnr at two sides of the wing Vnℓ
αwℓ
Vnr
αwr
u u (C) The angles of attack at the left and right wing
Figure 7-7: The origin of the difference in angles of attack at the left and right wing for a wing with dihedral in sideslipping flight Flight Dynamics
310
Lateral Stability and Control Derivatives
β
V
Vnl = V cos(Λ + β)
Vnr = V cos(Λ − β)
Λ V Λ−β
Λ+β V
Figure 7-8: The origin of the difference in the velocities over the left and right wing for a swept wing in sideslipping flight
Cℓβ
Λ<0 CL (α)
Cℓβ due to Γ > 0 Λ=0
Λ>0 Figure 7-9: The variation of Cℓβ with CL for swept wings
7-2 Stability derivatives with respect to the sideslip angle
311
As the rolling moment of the wing Cℓβ w · β will be proportional to ∆L, to a first approxi mation, this simple calculation shows that Cℓβ w will be approximately proportional to CL and to sin 2Λ, see figure 7-9. Flap deflection causes, at constant lift on the entire wing, a concentration of the lift at the center parts of the wing. Swept wings will have a less negative Cℓβ due to flap deflection. The same effect is obtained by applying negative wing-twist to swept wings. The influence of dihedral and sweep on Cℓβ w are additive to a first order approximation. In summary it is seen from the foregoing that for a swept wing the contribution Cℓβ w is composed of one part proportional to the wing dihedral Γ (independent of CL and wing sweep Λ) and of one part independent of Γ and proportional to CL and sin 2Λ. A consequence of the variation of Cℓβ with CL is that swept wings often have a zero or even negative geometric dihedral, to avoid the unfavourable effects of too large and effective dihedral at large angles of attack. This will be further discussed when studying the dynamic lateral stability. Calculation methods to determine Cℓβ w are presented in references [10, 70, 122, 44, 130, 127]. The contribution of the fuselage itself to Cℓβ is very small. The wing fuselage interaction in sideslipping flight, however, can cause significant differences in the angles of attack of the two halves of the wing. This again has a strong influence on the rolling moment due to side-slip. Important for the way in which the rolling moment varies with angle of sideslip is the vertical position of the wing relative to the fuselage, see figure 7-13. The forward half wing of a low wing aircraft experiences a decrease in angle of attack due to sideslip. This produces a decrease in lift whereas the receding half wing experiences an increase in angle of attack and thus an increase in lift, due to the presence of the fuselage. The situation is the reverse for a high wing aircraft. A positive angle of sideslip causes for a low wing aircraft an extra positive rolling moment, Cℓβ becomes less negative. On the other hand, Cℓβ of a high wing aircraft becomes more negative. The magnitude of the contribution is strongly influenced by the shape of the fuselage and by its size compared to that of the wing. A detailed description of the influence of the wing-fuselage interaction on Cℓβ is given in reference [79]. A determination of Cℓβ of aircraft without tailplanes is possible using references [122, 156, 25, 44, 130, 127] for subsonic speeds and references [129, 84, 83, 108, 144] for supersonic speeds. For example, see also the following illustrations; for the case of a high-wing-fuselage configuration in side-slipping flight, the configuration and velocity field are plotted in figure 7-10, while for the case of a low-wing-fuselage configuration in side-slipping flight, the configuration and velocity field are plotted in figure 7-11. Figure 7-12 presents the airflow around the Cessna Ce550 ‘Citation II’ in side-slipping flight. Particle traces depict the flow around the bottom side of the fuselage; as is obvious, the right wing induces a downwash in front of it, i.e. the particle traces follow a path below the wing and fuselage, while the left wing induces an upwash in front of it, i.e. the particle lines follow a path over the wing. In this figure a speedbar is given as well, depicting a measure for the local airspeed’s magnitude along particle traces. The results of these simulations were obtained using a panel method (linearized potential flow). Flight Dynamics
312
Lateral Stability and Control Derivatives
The contribution of the vertical tailplane to Cℓβ is usually not large for conventional aircraft, but may not be neglected for aircraft having a relatively large vertical tailplane. The magnitude of Cℓβ v follows from the magnitude and the position of the point of action of the lateral force CYβ v · β. In the calculation of Cℓβ v the choice of the reference frame is important, as will be shown below. In the study of stability, the so-called stability reference frame is commonly used. This is a system of aircraft body axes of which the XS -axis in the plane of symmetry is chosen in the direction of the undisturbed flow. In this situation the rolling moment due to the side force on the vertical tailplane depends on the angle of attack of the aircraft. From figure 7-15 follows for Cℓβ Cℓβ
v
= CYβ
v
v
,
xv − xc.g. zv − zc.g. cos α0 − sin α0 b b
(7-15)
the ordinates xv and zv can be determined using the same references as indicated for the determination of CYvα . The effective dihedral of a propeller driven aircraft having a propeller in front of the wing can experience an important change due to the interaction between the slipstream and the wing. Due to the increased dynamic pressure in the slipstream an extra rolling moment is generated in sideslipping flight, see figure 7-16. At a positive side-slip angle β this rolling moment acts in the right wing down sense, thus giving a positive contribution to Cℓβ , The contribution increases with Tc and CL . Deflection of the landing flaps will increase the effective angle of attack over the center part of the wing. It will further increase the effect of the slipstream on Cℓβ . Using linearized potential flow theory (panel methods), Cℓ as a function of β has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-14. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to Cℓβ are from the vertical fin and the wing.
7-2-3
Stability derivative Cnβ
The stability derivative Cnβ is called the static directional, or Weathercock, stability, as this derivative is geometrically directly comparable to the static longitudinal stability Cmα For good control characteristics it is desirable that the angle of sideslip causes a moment about the top axis trying to reduce the sideslip. The desired sign of Cnβ is thus positive. The magnitude of Cnβ is determined by a small, positive contribution of the wing, an important destabilizing, negative contribution of the fuselage and a usually large stabilizing, positive contribution of the vertical tailplane, see figure 7-17. The contribution of the propulsion
7-2 Stability derivatives with respect to the sideslip angle
313
Right wing
Figure 7-10: High-wing-fuselage configuration with the calculated velocity field in sideslipping flight (β = 10o , α = 0o )
Flight Dynamics
314
Lateral Stability and Control Derivatives
Right wing
Figure 7-11: Low-wing-fuselage configuration with the calculated velocity field in sideslipping flight (β = 10o , α = 0o )
7-2 Stability derivatives with respect to the sideslip angle
315
55.0 m/s 54.5 m/s 54.0 m/s 53.5 m/s 53.0 m/s 52.5 m/s 52.0 m/s 51.5 m/s 51.0 m/s 50.5 m/s 50.0 m/s 55.0 m/s 54.5 m/s 54.0 m/s 53.5 m/s 53.0 m/s 52.5 m/s 52.0 m/s 51.5 m/s 51.0 m/s 50.5 m/s 50.0 m/s Figure 7-12: Calculated particle traces around the bottom half of the fuselage of the Cessna Ce550 ‘Citation II’, sideslipping flight β = 10o , α = 0o (note: the speedbar is related to the magnitude of the airflow’s velocity along the calculated particle traces)
Flight Dynamics
316
Lateral Stability and Control Derivatives
cℓ · c left wing
right wing V sinβ − 2b
+ 2b
cℓ (< 0) cℓ · c
(A) High-wing aircraft
V sinβ
left wing
− 2b
right wing
+ 2b
cℓ (> 0)
(B) Low-wing aircraft Figure 7-13: Origin of a rolling moment caused by wing-fuselage interactions in sideslipping flight
0.02
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.015
0.01
Cℓ
0.005
0
−0.005
−0.01
−0.015
−0.02 −10
−8
−6
−4
−2
0
2
β [Deg.]
4
6
8
10
Figure 7-14: Aerodynamic moment coefficient Cℓ as a function of β for the Cessna Ce550 ‘Citation II’, α = 0o
7-2 Stability derivatives with respect to the sideslip angle
317
Zr
YB,S
(CY )v XB,S
c.g.
α0
zc.g.
zv
xc.g. Xr xv
ZB,S
(zv − zc.g. ) sinα0 (CY )v
a.c.v
(xv − xc.g. ) cosα0 c.g. (zv − zc.g. ) cosα0
α0
(xv − xc.g. ) sinα0 xv − xc.g. (zv − zc.g. ) Figure 7-15: The position of the point of action of (CYv )v relative to the X- and Z-axis in the stability Flight Dynamics reference frame
318
Lateral Stability and Control Derivatives
system may be either positive or negative. The contribution to Cnβ of a wing without sweep is very small. The contribution of a swept back wing may be important especially at high CL values due to the sideforce acting relatively far behind the center of gravity. This wing contribution may be calculated using the same references as for the determination of the contribution of the wing to CYβ . The negative destabilizing contribution of the fuselage arises in a similar manner as the corresponding distribution of the fuselage to Cmα in symmetric flight, see references [114, 2, 76, 3, 139, 22, 128]. The contribution of the vertical tailplane to Cnβ , expressed in the stability reference frame, follows from figure 7-15, Cnβ
v
In this expression CYβ
= − CYβ v
v
xv − xc.g. zv − zc.g. sin α0 + cos α0 b b
(7-16)
is given by equation (7-8).
At small angles of attack with xv − xc.g. = ℓv follows from equations (7-8) and (7-16), Cnβ
v
= CYvα
dσ 1− dβ
Vv V
2
Sv lv Sb
(7-17)
The corresponding expression for the horizontal tailplane is, see equation (8-7), (Cmα )h = −CNhα
1−
dε dα
Vh V
2
Sh lh S¯ c
For the sake of simplicity lv is often taken as the distance between the 14 -chord points on the m.a.c.’s of the wing and the vertical tailplane. Calculation methods to determine CYvα have been mentioned already in section 7-2. The calculation of the sidewash gives rise to consirable difficulties in practice as the flow is strongly determined by the interaction between flow around the fuselage, the wing and the tailplane. The flow around an isolated fuselage can be presented schematically as in figure 7-18. A positive angle of sideslip causes a negative induced sidewash, or dσ dβ < 0. Another numerical example of the flow around a complete aircraft configuration in side-slip is given in figure 7-19. It is obvious from this figure that the determination of σ at the vertical tailplane will be difficult because of strong interactions between the fuselage, vertical tailplane and nacelles. Also in this figure speedbars have been added to give an impression of the magnitude of the local airspeed along the calculated particle traces. Note that the characteristic flow over the top of a fuselage in sideslip is given in figure 7-18, see also the examples given in figures 7-20 where particle traces have been given over the top of a fuselage of a large 4-engined transport aircraft in sideslip.
7-2 Stability derivatives with respect to the sideslip angle
319
V XB
β
YB
(A) The part of the wing submerged in the slipstream cℓ · c
propeller on
propeller off
− 2b
cℓ (> 0)
+ 2b
y
(B) The change in lift distribution Figure 7-16: The contribution of the slipstream to Cℓβ
Flight Dynamics
320
Lateral Stability and Control Derivatives
Cn 0.06
δf = 0o α = 2.5o
0.04
0.02
wing+fuselage fuselage -16
-12
-8
-4
0
4
8
12
β [o ]
wing -0.02
complete model
-0.04
-0.06
Figure 7-17: Cn as a function of β measured on a model of the Fokker F-27 in gliding flight (from references [148, 150, 24])
7-2 Stability derivatives with respect to the sideslip angle
321
V sin (β − σ) V sinβ
(A) Velocity components in a plane perpendicular to the fuselage axes
β
β (> 0) V cosβ V sinβ β −σ σ (< 0) V cosβ V sin (β − σ)
(B) The flow around the fuselage and the vertical tailplane Figure 7-18: The sidewash induced by the fuselage at the vertical tailplane Flight Dynamics
322
Lateral Stability and Control Derivatives
55.0 m/s 54.5 m/s 54.0 m/s 53.5 m/s 53.0 m/s 52.5 m/s 52.0 m/s 51.5 m/s 51.0 m/s 50.5 m/s 50.0 m/s 55.0 m/s 54.5 m/s 54.0 m/s 53.5 m/s 53.0 m/s 52.5 m/s 52.0 m/s 51.5 m/s 51.0 m/s 50.5 m/s 50.0 m/s Figure 7-19: Calculated particle traces around the fuselage of the Cessna Ce550 ‘Citation II’, sideslipping flight β = 10o , α = 0o (note: the speedbar is related to the magnitude of the airflow’s velocity along the calculated particle traces)
7-2 Stability derivatives with respect to the sideslip angle
323
55.0 m/s 54.5 m/s 54.0 m/s 53.5 m/s 53.0 m/s 52.5 m/s 52.0 m/s 51.5 m/s 51.0 m/s 50.5 m/s 50.0 m/s 55.0 m/s 54.5 m/s 54.0 m/s 53.5 m/s 53.0 m/s 52.5 m/s 52.0 m/s 51.5 m/s 51.0 m/s 50.5 m/s 50.0 m/s Flight Dynamics Figure 7-20: Calculated particle traces around the fuselage of a large 4-engined transport aircraft, sideslipping flight β = 10o , α = 0o (note: the speedbar is related to the magnitude of the airflow’s velocity along the calculated particle traces)
324
Lateral Stability and Control Derivatives
Earlier in section 7-2 the change in lift distribution in sideslip due to the presence of the fuselage was discussed. The extra lift distribution caused by the interaction also induces a side-wash at the location of the vertical tailplane. The difference in lift on the two halves of the wing causes a difference between the downwash at either side of the fuselage, see subfigures a and b of figure 7-21. The downwash pattern is displaced laterally along with the main flow, over an angle β with the X-axis, see subfigure b of figure 7-21. The main effect, however, is a circulation about the fuselage. If a low wing aircraft sideslips to the right, the circulation is counter-clockwise, subfigure c of figure 7-21. This circulation generates at the vertical tailplane an extra negative sidewash. This induced cross-flow is stabilizing, as can also be seen from equation (7-17), ∆ dσ dβ < 0. For a high wing aircraft in sideslip to the right, the sidewash is positive, the influence of the wing fuselage interaction on Cnβ v is thus seen to be destabilizing, ∆ dσ dβ > 0. Figure 7-22 shows the results of measurements of the sidewash and the static directional stability illustrating the above. From this figure it can also be seen that the sidewash is greatly influenced by the presence of the horizontal tailplane. A low horizontal tailplane acts as an end plate to the vertical tailplane. This reduces the sidewash considerably. An accurate calculation of the average sidewash σ, or dσ dβ , at the vertical tailplane is not well possible, although references [78, 80, 81, 82, 90, 28] give calculation methods. In many cases results of systematic measurements on similar configurations are used to determine dσ dβ .
The contribution of the propulsion system to Cnβ is caused by the lateral force acting on the propeller or the engine inlet in cross-flow, see figure 7-23. If the propeller or the engine inlet is situated forward of the center of gravity this contribution is destabilizing. For propeller-driven aircraft the increased dynamic pressure in the slipstream causes a higher static directional stability, if the vertical tailplane is placed in the slipstream. Using linearized potential flow theory (panel methods), Cn as a function of β has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-24. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to Cnβ are from the fuselage (destabilizing contribution) and the vertical fin (stabilizing contribution).
7-2-4
Stability derivative Cnβ˙
The stability derivative Cnβ˙ is a measure of the moment about the top axis caused during a change in the angle of sideslip when the aircraft executes an accelerated motion along the Y -axis. The derivative Cnβ˙ corresponds entirely to the stability derivative Cmα˙ for the symmetric motions. After a sudden change in the angle of sideslip a short interval passes before the changed sidewash, caused by the wing-fuselage interaction, has reached the vertical tailplane. As a
7-2 Stability derivatives with respect to the sideslip angle
325
V sinβ YB
ZB β > 0o
cℓ · c
β = 0o
− 2b
+ 2b (A) The lift distribution of a low-wing aircraft sideslipping to the right
− 2b
β > 0o
+ 2b
β = 0o ∆ε
(B) The change in downwash behind the wing and fuselage
Due to Γ
Due to β
V sin (β − σ)
Circulation Γ (C) The change in sidewash at the vertical tailplane Figure 7-21: The effect of wing-fuselage interactions on the sidewash at the vertical tailplane of a low-wing aircraft in sideslipping flight Flight Dynamics
326
Lateral Stability and Control Derivatives
Figure 7-22: The effect of wing-fuselage interactions on the sidewash at the tailplanes and the derivative Cnβ (from reference [142])
7-2 Stability derivatives with respect to the sideslip angle XB
β
327
V
CY (< 0) YB c.g. Cn (< 0)
Figure 7-23: The contribution of the sideforce on the propeller to Cnβ 0.025
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.02
0.015
0.01
Cn
0.005
0
−0.005
−0.01
−0.015
−0.02
−0.025 −10
−8
−6
−4
−2
0
2
β [Deg.]
4
6
8
10
Figure 7-24: Aerodynamic moment coefficient Cn as a function of β for the Cessna Ce550 ‘Citation II’, α = 0o
Flight Dynamics
328
Lateral Stability and Control Derivatives
Figure 7-25: The pitching moment as a function of sideslip angle (from reference [126])
consequence, the change of Cn with β does not occur as sudden as the change in β. This phenomenon is described using the stability derivative Cnβ˙ . For aircraft with a straight wing of sufficiently large aspect ratio (e.g. A > 4 to 5), Cnβ˙ is usually neglected. Also in the discussion of the dynamic lateral stability, Cnβ˙ will not be considered. For aircraft having swept wings of low aspect ratio, Cnβ˙ has to be determined using the results of systematic measurements on oscillating models, see for instance reference [55].
7-2-5
Stability derivative Cmβ 2
A sideslip can cause a pitching moment that may be considerable, especially at large angles of sideslip, see figure 7-25. It can be seen from this figure, as follows also from considerations of symmetry, that the change in pitching moment has the same sign for both positive and
7-3 Stability derivatives with respect to roll rate
329
negative angles of sideslip. The stability derivative expressing the pitching moment due to sideslip is, therefore, written as Cmβ 2 . This moment is caused by the wing-fuselage interaction and the interaction between the fuselage and the tailplanes already discussed in relation to Cnβ . Reference [79] discusses the influence of wing-fuselage interactions on the pitching moment for combinations of a fuselage with a straight wing and a swept wing in a sideslip. Here also the vertical position of the wing relative to the fuselage is important. Especially at large angles of sideslip the change in pitching moment is determined to a large extent by the contribution of the horizontal tailplane, see references [126] and figure 7-25. The tailplane is located in the downwash field behind the wing, modified by the wing-fuselage interaction. If the tailplane is mounted low on the fuselage the field of flow is modified additionally by the fuselage and the vertical tailplane. The influence of the vertical position of the horizontal tailplane on Cm in sideslip is presented in figure 7-26. For a quantitative determination of Cmβ 2 the designer has to rely on wind tunnel measurements and/or CFD simulations.
7-3
Stability derivatives with respect to roll rate
If the aircraft rolls about the XB -axis, the geometric angle of attack of the various wing and tailplane chords varies proportional to the rolling-velocity p and to the distance of the chords to the XB -axis, see figure 7-27. For a wing chord at a distance y from the plane of symmetry, the change in the geometric angle of attack is, ∆α =
pb y py = V 2V b/2
(7-18)
Here y is measured in the system of aircraft body axes. For the sake of simplicity the aircraft pb c.g. is assumed to lie in the plane of symmetry. The non-dimensional rolling velocity 2V is thus equal to the change in geometric angle of attack at the wing tip where y = b/2. This is also the helix angle of the helix described by the wing tip in rolling flight, which is why the rolling velocity is made non-dimensional by multiplying with 2Vb .
7-3-1
Stability derivative CYp
The derivative CYp is always relatively small and is very often neglected. Only if the wing has a large sweep angle, or if a relatively large vertical tailplane is used, this derivative has to be taken into account. The origin of the contribution from a swept wing to CYp will be explained when discussing Cnp w . For swept back wings Cnp w is negative.
The contribution of the vertical tailplane to CYp is caused by the change in angle of attack experienced by the vertical tailplane in rolling flight. A positive rolling velocity gives rise to a negative lateral force, Yv = CYp
v
pb 1 2 ρV S 2V 2
(7-19) Flight Dynamics
330
Lateral Stability and Control Derivatives
Figure 7-26: The influence of the vertical position of the horizontal tailplane on the variation of Cm in sideslipping flight for a fuselage with tailplanes (from reference [126])
Figure 7-27: The variation of the local geometric angle of attack at the wing and the tailplane of a rolling aircraft
7-3 Stability derivatives with respect to roll rate
331
0.03
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.025
0.02
0.015
CY
0.01
0.005
0
−0.005
−0.01
−0.015
−0.02 −0.2
−0.15
−0.1
−0.05
0pb
2V
0.05
[Rad.]
Figure 7-28: Aerodynamic force coefficient CY as a function of α = 0o
pb 2V
0.1
0.15
0.2
for the Cessna Ce550 ‘Citation II’,
Where CYp v is thus negative. In the determination of CYp v the sidewash at the vertical tailplane, induced by the rolling velocity has to be taken into account. This subject will be further discussed when dealing with Cℓp . Calculation methods can be found in references [89, 31, 126, 106, 107]. pb Using linearized potential flow theory (panel methods), CY as a function of 2V has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-28. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to CYp are from the wing, vertical fin and horizontal tailplane. The stability derivative CYp is usually very snall.
7-3-2
Stability derivative Cℓp
The stability derivative Cℓp is a measure of the moment about the XB -axis due to rolling about this axis, L = Cℓp
pb 1 2 ρV Sb 2V 2
(7-20)
In all normal flight conditions this moment opposes the rolling, trying to slow the motion down. Cℓp is normally negative and a measure for the roll damping.
Flight Dynamics
332
Lateral Stability and Control Derivatives
∆α
− 2b
p
pb + 2V
+ 2b y
pb − 2V
(A) The variation in geometric angle of attack ∆αe
− 2b
p
pb + 2V
+ 2b y
pb − 2V
(B) The variation in effective angle of attack Figure 7-29: The variation of the local geometric and effective angle of attack along the span of a rolling wing
7-3 Stability derivatives with respect to roll rate
333
Cℓp
A 0
2
4
6
8
10
Λ 60o -0.4
45o 30o 0o
-0.8
Cℓp
(A) taper ratio λ = 1.0 A 0
2
4
6
8
10
Λ 60o -0.4
45o 30o 0o
-0.8
Cℓp
(B) taper ratio λ = 0.5 A 0
2
4
6
8
10
Λ 60o -0.4
45o 30o 0o
-0.8
(C) taper ratio λ = 0 Figure 7-30: The roll-damping as a function of aspect ratio and sweep for various wing taper ratio’s (from reference [34])
Flight Dynamics
334
Lateral Stability and Control Derivatives
The dominant contribution to Cℓp is generated by the wing. In some cases, the contribution from the tailplanes has to be taken into account as well. The contribution due to the fuselage, the wing-fuselage interaction and the propeller are usually negligible. The contribution of the wing is caused by the change in geometric angle of attack and the resulting change in effective angle of attack along the wing span, see figure 7-29. An additional anti-symmetric lift distribution is generated causing a rolling moment opposite to the sense of the rotation. This has a damping effect on the rotation. Calculated values of Cℓp for various wing planforms have been collected in figure 7-30. It can be seen that the roll damping of the wing increases in the absolute sense with the taper ratio, λ, and aspect ratio, A. The damping decreases with increasing wing sweep. If the airflow over the wing remains attached, the wing contribution to Cℓp is independent of CL but proportional to the lift gradient CLα . If the lift gradient decreases at large angles of attack due to flow separation, Cℓp decreases in the absolute sense as well. At very large angles of attack the down going wing may stall, causing a considerable reduction in wing damping. It is even possible that the influence of the loss in lift caused by the stall of the down going wing dominates the influence of the decrease in lift due to smaller angle of attack of the up going wing. In that case Cℓp may become positive. The lift distribution in rolling flight can be calculated using references [25, 44]. At subsonic speeds the wing contribution to Cℓp can be determined with references [10, 70, 156, 124, 125, 62]. Reference [110] enables the wing contribution to be determined for angles of attack where the relation between cℓ and α of airfoils is non-linear. For supersonic speeds references [10, 70, 84, 83, 67] are available. The contribution of the horizontal and the vertical tailplane to Cℓp , caused by the same mechanism as the wing contribution, see figure 7-27, is generally much smaller than the wing contribution. Due to the smaller span of the tailplanes the maximum change in angle of attack is smaller than that of the wing. A further decrease in the change of angle of attack at the horizontal and vertical tailplanes is caused by the change in downwash behind the rolling wing. Behind the downgoing wing the downwash increases, it decreases behind the upgoing wing, relative to non-rolling flight, see figure 7-31. This extra downwash distribution causes a circulation as depicted in subfigure c of figure 7-31. This downwash decreases the change in angle of attack along the span of the tailplanes. A further study of this phenomenon can be found in reference [89]. For conventional, subsonic aircraft the contributions of the tailplanes to Cℓp can usually be neglected. Modern fighter aircraft and V/STOL-aircraft usually have relatively large tailplanes. Calculation of the tailplane contributions to Cℓp for such aircraft is possible using references [131, 31, 29, 106, 107]. pb Using linearized potential flow theory (panel methods), Cℓ as a function of 2V has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-32. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in
7-3 Stability derivatives with respect to roll rate
335
p
YB
ZB cℓ · c
resultant cℓ · c-distribution
due to α due to
pb 2V
− 2b
+ 2b (A) The change in lift distribution on a rolling wing wi pb due to 2V
− 2b
+ 2b
due to α resultant downwash
(B) The change in downwash behind a rolling wing due to
pb 2V
due to Γ
due to Γ due to
pb 2V
p due to
pb 2V
circulation Γ due to Γ (C) The flow at the horizontal and vertical tailplanes due to p and Γ Figure 7-31: The flow at the tailplanes of a rolling aircraft Flight Dynamics
336
Lateral Stability and Control Derivatives 0.1
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.08
0.06
0.04
Cℓ
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1 −0.2
−0.15
−0.1
−0.05
0pb
2V
0.05
0.1
[Rad.]
Figure 7-32: Aerodynamic moment coefficient Cℓ as a function of α = 0o
pb 2V
0.15
0.2
for the Cessna Ce550 ‘Citation II’,
figure 6-2. The main contributions to Cℓp is from the wing. The wing tries to damp the rolling motion.
7-3-3
Stability derivative Cnp
This derivative is determined mainly by the contribution of the wing. Only a relatively large vertical tailplane produces a non-negligible contribution to Cnp . Under normal conditions Cnp is negative. The generation of the wing’s contribution to Cnp is illustrated in figure 7-33. In rolling flight an extra force in XB -direction acts on each section of the wing. Since the local angle of attack varies along the wing span, it is essential to study the distribution of this extra force in the stability reference frame. The axes of the reference frame have a fixed direction relative to the aircraft. In this reference frame the down going wing experiences an extra forward force due to the increased angle of attack. The up going wing experiences an extra longitudinal force pointing backwards. As a result, a positive rolling velocity causes a negative yawing moment, N = Cnp where Cnp
w
< 0.
w
pb 1 2 ρV Sb 2V 2
(7-21)
7-3 Stability derivatives with respect to roll rate
337
Figure 7-33: The origin of a negative yawing moment about the Z-axis of the stability reference frame for a wing having a positive rate of roll (attached flow)
Flight Dynamics
338
Lateral Stability and Control Derivatives XB
p
∆CXn ∆CXn′ CY
∆CY0
∆CYn Cn
∆CX0 ∆CX0′ Figure 7-34: The side force and yawing moment on a rolling, swept back wing
If flow separation occurs on the down going wing, the drag increases considerably causing the extra force in XB - or XS -direction to become negative. This renders Cnp w less negative or even positive. This applies also to wings with a sharp leading edge. Due to the absence of the suction peak on the airfoil’s nose, the resultant aerodynamic force and the change in the resultant aerodynamic force act here approximately perpendicular to the wing chord. From subfigure b of figure 7-33 it can be seen that for normal wings the extra force in XS direction on the down going wing is positive (pointing forwards) and the extra force on the up going wing acts in the direction of the negative XS direction. For normal wings Cnp w apparently is always negative. For swept wings the change in resultant aerodynamic force acting on a section of the rolling wing has a component in the YB - or YS -direction, see figure 7-34. This lateral force has pb the same direction on both wing halves. This causes a resultant lateral force CYp w · 2V . This force generates a second, less important contribution to Cnp w , depending on the c.g. position. Calculation methods to determine CYp w and Cnp w for subsonic speeds are given in references [10, 70, 122, 156, 64]. For supersonic speeds, see references [84, 83]. The contribution of the vertical tailplane to Cnp follows directly from CYp
v
and the position
7-4 Stability derivatives with respect to yaw rate
339
−3
6
x 10
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
5
4
3
Cn
2
1
0
−1
−2
−3
−4 −0.2
−0.15
−0.1
−0.05
0pb
2V
0.05
0.1
[Rad.]
Figure 7-35: Aerodynamic moment coefficient Cn as a function of α = 0o
pb 2V
0.15
0.2
for the Cessna Ce550 ‘Citation II’,
of the point of action of CYp v using an expression in analogy with equation (7-16). The position of the point of action CYp v can be determined using the same references as given for the determination of the magnitude of CYp w . pb Using linearized potential flow theory (panel methods), Cn as a function of 2V has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-35. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to Cnp are from the vertical fin, wing and horizontal stabilizer.
7-4
Stability derivatives with respect to yaw rate
Before going into a discussion of the derivatives with respect to yawing velocity, CYr , Cℓr and Cnr separately, the motion to be considered needs further description. The notion of a so called ‘r-motion’ is introduced here, in analogy with the ‘q-motion’, see section 6-4. When performing such an r-motion, the aircraft moves along a curved trajectory in the XOY -plane, such that the velocity vector of the center of gravity remains in the plane of symmetry. By definition the angle of sideslip of the aircraft remains zero during aa r-motion, see figure 7-36. Apparently, the r-motion causes merely a curvature of the streamlines in the XOY -plane. At the position of the c.g. the radius of curvature is R, Flight Dynamics
340
Lateral Stability and Control Derivatives
R=
V r
The center of rotation is situated at a distance R from the c.g. on the positive Y -axis if r is positive. The effect of this r-motion is twofold. In the first place, and in analogy with the change in angle of attack caused by the q-motion, a change in the flow direction measured in planes parallel to the XOY -plane occurs at all points of the aircraft, see figure 7-37,
∆α =
rb x − xc.g. 2V b/2
(7-22)
As a consequence the airflow meets the vertical tailplane from the left if r is positive. In the second place, the local speed of the airflow changes at all points of the aircraft by an amount, see figure 7-38, ∆V rb y =− V 2V b/2
(7-23)
As in equation (7-18), y is measured here in the stability reference frame. At positive r, the right wing has a lower airspeed and the left wing a larger airspeed than the aircraft center of gravity. Similar differences in airspeed occur in the plane of symmetry during the q-motion, but they are negligible due to the relatively small dimensions of the aircraft in Z-direction. rb According to equation (7-23), 2V is the non-dimensional change in airspeed occurring at the wing tip. The choice of the factor 2Vb to make the angular velocity non-dimensional thus has a geometric interpretation.
7-4-1
Stability derivative CYr
This derivative is usually of minor importance. The dominant contribution to CYr is due to the vertical tailplane, CYr is commonly positive. The contribution of the wing is very small. Calculation methods are given in references [10, 70, 122, 156] for subsonic speeds and in references [84, 83] for supersonic speeds. The contributions of the fuselage and the wing-fuselage interaction are negligible, as is the contribution of the-horizontal tailplane. The lateral force on the vertical tailplane due to yawing can be written as,
Yv = (CYr )v
rb 1 2 ρV S 2V 2
(7-24)
7-4 Stability derivatives with respect to yaw rate
341
XB
R=
V r
V
r YB
c.g.
Figure 7-36: The pure ‘r- motion’
The lateral force is caused by the change in angle of attack ∆αv of the vertical tailplane, see figure 7-39. Accordingly,
Yv = CYvα ∆αv
1 2 ρV Sv 2 v
(7-25)
The change in angle of attack ∆αv follows from equation (7-22) with xv − xc.g. = lv , ∆αv =
rb lv 2V b/2
(7-26)
From equations (7-24), (7-25) and (7-26) it follows after some reduction,
(CYr )v = 2 CYvα
Vv V
2
Sv lv Sb
(7-27)
From this expression it follows that (CYr )v is positive. The values of (CYr )v calculated using equation (7-27) may not be very accurate due to the unknown influence of the wing-fuselage in the curved flow field on the flow direction at the vertical tailplane. For these reasons it is commonly necessary to use the results of wind tunnel measurements on similar configurations to determine (CYr )v .
Flight Dynamics
342
Lateral Stability and Control Derivatives
XB
Yr
xc.g. r x c.g.
YB ∆α Xr
∆α
∆α =
rb 2V
·
x−xc.g. b/2
Figure 7-37: The change in geometric flow direction at an arbitrary point of the aircraft due to an ‘rmotion’
XB
∆V
∆V r
V c.g.
YB y
R=
∆V V
rb = − 2V ·
V r
y b/2
Figure 7-38: The variation in airspeed in spanwise direction due to an r-motion
7-4 Stability derivatives with respect to yaw rate XB
343
∆αp (< 0)
Yp (< 0) xp − xc.g.
V r c.g.
YB
lv ∆αv (> 0) Yv (> 0)
Figure 7-39: The side forces on the vertical tailplane and the propeller due to an r-motion.
During the r-motion the propeller experiences an oblique inflow also causing a contribution to CYr , see figure 7-39. In analogy with equation (7-27), (CYr )p can be written as,
(CYr )p = 2 CYpα
Vp V
2
The derivative (CYr )p usually is negative. CYpα and [133, 134].
Sv xp − xc.g. S b
Vp V
2
(7-28)
can be determined using references
rb Using linearized potential flow theory (panel methods), CY as a function of 2V has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-40. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to CYr are from the vertical fin, the fuselage and the wing.
7-4-2
Stability derivative Cℓr
The stability derivative Cℓr is determined primarily by the contributions of the wing and the vertical tailplane. The contributions of the fuselage, the wing-fuselage interaction, the horizontal tailplane and the propulsion system may usually be neglected. Cℓr is always positive.
Flight Dynamics
344
Lateral Stability and Control Derivatives 0.06
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.04
CY
0.02
0
−0.02
−0.04
−0.06 −0.2
−0.15
−0.1
−0.05
0
rb 2V
0.05
0.1
[Rad.]
Figure 7-40: Aerodynamic force coefficient CY as a function of α = 0o
rb 2V
0.15
0.2
for the Cessna Ce550 ‘Citation II’,
The contribution of the wing arises due to the differences in airspeed over the two wing halves during the r-motion, as described previously. At a positive yawing velocity, i.e. a rotation to the right, the left wing generates more lift than the right wing, causing a right wing down, positive, rolling moment,
L = (Cℓr )w
rb 1 2 ρV Sb 2V 2
(7-29)
This rolling moment, and thus (Cℓr )w , is proportional to the lift coefficient and at a given CL dependent on the lift distribution over the wing span. Tapered wings show a higher lift concentration near the center of the wing, causing an increase in speed over the outer wing to have less effect. As a consequence, (Cℓr )w decreases with λ at constant CL . The same applies to wings having negative twist and to wings with deflected landing flaps. Swept back wings have a higher lift concentration on the outer wings. This causes (Cℓr )w to increase somewhat with Λ. The derivative (Cℓr )w can be calculated using references [10, 70, 122, 156, 33] for subsonic speeds and references [84, 83] for supersonic speeds. The contribution of the vertical tailplane to Cℓr is found from (CYr )v and the position of the point of action of (CYr )v , (Cℓr )v = (CYr )v
zv − zc.g. xv − xc.g. cos α0 − sin α0 b b
(7-30)
7-4 Stability derivatives with respect to yaw rate
345
0.015
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.01
Cℓ
0.005
0
−0.005
−0.01
−0.015 −0.2
−0.15
−0.1
−0.05
0
rb 2V
0.05
0.1
[Rad.]
Figure 7-41: Aerodynamic moment coefficient Cℓ as a function of α = 0o
rb 2V
0.15
0.2
for the Cessna Ce550 ‘Citation II’,
As zv − zc.g. is usually positive, (Cℓr )v is usually positive as well. rb Using linearized potential flow theory (panel methods), Cℓ as a function of 2V has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-41. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to Cℓr are from the vertical fin and the wing.
7-4-3
Stability derivative Cnr
Similar to the derivatives Cmq and Cℓp , Cnr is a measure of the aerodynamic moment about an axis caused by an angular velocity about the same axis. Under normal circumstances such a moment tries to slow down the motion, which implies that Cnr is normally negative, again similar to Cmq and Cℓp . The dominant contribution to Cnr is provided by the vertical tailplane. Additionally, only the wing, the fuselage and the propulsion system deliver non-negligible contributions. The part provided by the wing is generated by the differences in drag between the two halves of the wing, caused by the differences in local airspeed. Calculation methods for subsonic speeds are found in references [10, 70, 122, 156] and in references [84, 83] for supersonic speeds. The contribution due to the fuselage is small, but it may be non-negligible for modern configurations having a relatively large fuselage in comparison with the wing. A calculation Flight Dynamics
346
Lateral Stability and Control Derivatives
Figure 7-42: The effects of the size of the vertical tailplane and the taillength on Cnr (from reference [88])
is possible using references [139, 88].
The contribution of the vertical tailplane follows directly from, see equation (7-27) and the taillength of the aircraft, see also figure 7-39,
(Cnr )v = − (CYr )v
lv b
(7-31)
For the determination of (Cnr )v in the stability reference frame, at large angles of attack use has to be made of an expression in analogy with equation (7-16). Figure 7-42 gives Cnr of a model for different values of lbv and Sbv . The contribution of the propeller to Cnr follows in a similar manner from (CYr )p . If the propeller is situated in front of the c.g., (CYr )p is negative. The damping moment is then increased by the propeller, see figure 7-39. Additionally, the increased dynamic pressure in the slipstream may cause an increase in the contribution of the vertical tailplane.
rb Using linearized potential flow theory (panel methods), Cn as a function of 2V has been calculated for the Cessna Ce550 ‘Citation II’, see figure 7-43. The definition of the aircraft parts ‘wing’, ‘horizontal stabilizer’, ‘pylons’, ‘fuselage’, ‘vertical fin’ and ‘nacelles’ is given in figure 6-2. The main contributions to Cnr are from the vertical fin and the fuselage.
7-5 The forces and moments due to aileron, rudder and spoiler deflections
347
0.025
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.02
0.015
0.01
Cn
0.005
0
−0.005
−0.01
−0.015
−0.02
−0.025 −0.2
−0.15
−0.1
−0.05
0
rb 2V
0.05
Figure 7-43: Aerodynamic moment coefficient Cn as a function of α = 0o
7-5
0.1
[Rad.] rb 2V
0.15
0.2
for the Cessna Ce550 ‘Citation II’,
The forces and moments due to aileron, rudder and spoiler deflections
For lateral control it is necessary to generate variable moments about the XB -axis and the ZB -axis. The moment about the XB -axis is usually obtained by deflecting ailerons, at high airspeeds the use of spoilers is also quite common. The moment about the ZB -axis is provided by the rudder. The deflection of the ailerons or the spoilers generates in addition to the rolling moment L also a lateral force Y and a yawing moment N . Equally, rudder deflections cause not only a yawing moment, but also a lateral force and a rolling moment. As is the case with the lateral motions of the entire aircraft, in principle no symmetric forces and moments are generated. Measures of the non-dimensional force CY and the nondimensional moments Cℓ and Cn due to control surface deflections are again the partial derivatives with respect to aileron deflection δa and the rudder deflection δr , these control derivatives are, 1. aileron deflection, CYδa =
∂CY ∂δa
Cℓδa =
∂Cℓ ∂δa
Cnδa =
∂Cn ∂δa Flight Dynamics
348
Lateral Stability and Control Derivatives
2. rudder deflection, CYδr =
∂CY ∂δr
Cℓδr =
∂Cℓ ∂δr
Cnδr =
∂Cn ∂δr
When using the control deflections as measures for the forces and moments, in analogy with the stability derivatives, the assumption is made that the forces and moments vary linearly with δa and δr . For not too large control surface deflections, this assumption is usually quite acceptable. Appendix B provides, in addition to stability derivatives, also the control derivatives of some aircraft. Figure 7-44 shows once more the positive direction of the control surface deflections. They are always measured in a plane perpendicular to the hinge axes of the surfaces. The deflection δa is defined as, δa = δaright − δalef t
(7-32)
A positive deflection of the control stick, or a positive deflection of the control wheel (to the left), can be seen to produce a positive δa . Equally, a positive deflection of the rudder pedals (left pedal forward) causes a positive δr . If spoilers are used, derivatives with respect to spoiler deflection δs are defined, in analogy with the above control deflections,
CYδs =
∂CY ∂δs
Cℓδs =
∂Cℓ ∂δs
Cnδs =
∂Cn ∂δs
A spoiler deflection on the left wing is taken as positive. A positive δs corresponds with a positive deflection of the roll control manipulator. When using the derivatives with respect to δs great care is required, as for many spoiler configurations the forces and moments vary strongly non-linear with the spoiler deflection. The control derivatives are discussed in the following. The use of spoilers for roll control is discussed in section 7-7.
7-6 7-6-1
Aileron control derivatives Control derivative CYδa
The lateral force caused by an aileron deflection may be taken as zero for straight wings. On swept wings a lateral force does arise as is the case in rolling flight. The derivative CYδa is very small in magnitude and is usually neglected.
7-6 Aileron control derivatives
349
Figure 7-44: The positive direction of control deflections, control forces, control surface deflections and hinge moments
Flight Dynamics
350
Lateral Stability and Control Derivatives cℓ · c
wing with deflected ailerons
wing only
ailerons only cℓ (< 0)
− 2b
+ 2b y
ailerons only ba
ba
Figure 7-45: The effect of an aileron deflection on the lift distribution in spanwise direction
7-6-2
Control derivative Cℓδa
This control derivative is, of course, the primary one for the ailerons. Due to a positive aileron deflection the lift on the right wing increases, on the left wing the lift decreases. The result is a rolling moment to the left, which is counted as negative, see figure 7-45,
L = Cℓδa δa
1 2 ρV Sb 2
(7-33)
The derivative Cℓδa is negative and is also called the aileron effectivity. If the aeroelastic deformation of the wing is neglected, the influence of the Mach number is not considered and no flow separation on the wing in the area of the ailerons occurs, then Cℓδa may be considered as independent of CL . The magnitude of Cℓδa depends strongly on the dimensions and the location of the ailerons on the wing. Wing sweep and taper ratio also have an important influence on the aileron effectivity. For a given location and dimension of the ailerons, Cℓδa decreases in the absolute sense with increasing wing sweep and taper ratio (lower λ), especially when the wing aspect ratio is large. The calculation of Cℓδa is possible, using references [10, 70, 155, 44, 99] For supersonic speeds Cℓδa can be determined with reference [99]. In many instances measured values of Cℓδa obtained from flight tests turn out to be smaller in magnitude than was expected on the basis of calculations.
7-6 Aileron control derivatives
351
δa < 0
δa = 0
δa > 0
Figure 7-46: The Frise aileron
7-6-3
Control derivative Cnδa
The yawing due to aileron deflection arises because the drag of the wing having the downward deflected aileron increases, while that of the other side decreases, or perhaps increases less. As a consequence, a positive aileron deflection causes a positive yawing moment, N = Cnδa δa
1 2 ρV Sb 2
(7-34)
where Cnδa is positive. To a first approximation Cnδa is proportional to CL . Too large values of Cnδa are undesirable. If, for instance, the pilot wants to initiate a left turn, he deflects the roll control (stick or wheel) to the left to obtain the required negative roll-angle ϕ (left wing down). While the aircraft must obtain a negative yaw-rate (to the left), the positive aileron deflection generates a positive yawing moment, because Cnδa is positive. This yawing moment causes the aircraft to yaw initially to the right. This effect is amplified by the usually positive yawing moment caused by the negative rolling velocity of the aircraft, Cnp was seen to be negative. This initial yaw opposite to the desired direction is usually called the ‘adverse yaw’ of the aircraft. The derivative Cnδa can be kept small in various ways. One of the means is the application of differential deflections of the ailerons when applying roll control. This implies that the Flight Dynamics
352
Lateral Stability and Control Derivatives
downward deflection of the aileron is made smaller than the upward deflection of the other aileron. The difference in drag is reduced in this manner. An other method uses so called ‘Frise’ ailerons. The shape of the nose of these ailerons is such that the drag of the aileron strongly increases if it is deflected upward, see figure 7-46 and reference [155]. A calculation of Cnδa having some accuracy is generally not possible. As a consequence it is always desirable to determine Cnδa on the basis of experimentally obtained data, see reference [70].
7-7
Spoiler control derivatives
At transonic and supersonic airspeeds roll control of the aircraft can often not be effected by ailerons alone. In such cases spoilers, or a combination of spoilers and ailerons are used. The latter solution is the more usual one. A spoiler is nothing but a flap, deflected from the upper surface on one side of the wing to ‘spoil’ the airflow over that part of the wing, see subfigure a of figure 7-47. The local disturbance causes a decrease in lift and thereby generates a rolling moment. A deflection of the spoiler on the left wing is counted as positive. This causes, like an upward deflection of the left aileron, a negative rolling moment, hence Cnδs is negative as Cnδa is. Spoilers are more effective if they are located closer to the wing leading edge. The response of the aircraft to spoiler deflection becomes slightly more sluggish in this way. Therefore, in practice spoilers are located mostly on the rear part of the wing chord. Various causes make the use of spoilers on high speed aircraft more or less mandatory. In the first place, aileron effectivity decreases due to compressibility effects with increasing Mach number. On the other hand, the effectiveness of spoilers slightly increases with increasing Mach number. A second, even more important, phenomenon is the decrease in aileron effectivity due to wing deformation caused by aileron deflection. Figure 7-48 shows how a downward deflection of the aileron causes the wing to twist such that the local angle of attack decreases. This counteracts the increase in lift due to the aileron deflection. Conversely, an upward deflection of the aileron causes an increase of the local angle of attack. Such a decrease in aileron effectivity may be appreciable especially at high airspeeds, both due to the high dynamic pressure and to the rearward shift of the point of action of the local change in lift caused by the aileron deflection with increasing Mach number. In this way the resultant effect of an aileron deflection on highly swept wings with thin wing sections may even be opposite to the desired one. The effect is called ‘control reversal’ and the airspeed from which onward this occurs is the ‘reversal speed’. The point of action of the resultant change in lift for spoilers lies more forward on the chord than for ailerons. Hence, the wing twist is less and as a result the reversal speed is higher.
7-7 Spoiler control derivatives
353
δs
(A) Plate type spoiler
(B) Plugtype spoiler Figure 7-47: Spoiler types
Flight Dynamics
354
Lateral Stability and Control Derivatives
ϕ
δa
Figure 7-48: The local wing twist of a wing cross-section due to elastic deformation caused by an aileron deflection
Finally and quite apart from the foregoing reasons, it is often not very well possible to use large ailerons on wings having a high sweep angle or a low aspect ratio. The trailing edges of such wings are taken up largely by landing flaps to obtain an acceptable low landing speed. A spoiler deflection causes not only a reduction in lift, but also causes a strong increase in drag. A secondary advantage of the use of spoilers is, that due to this drag increase, Cnδs is nearly always negative, contrary to Cnδa . The adverse yaw at the initiation of a turn is thereby decreased or even entirely avoided. The increase in drag created by the deployment of spoilers may cause the designer to use the spoilers not only for roll control but as speed brakes as well. The spoilers are deflected symmetrically for this purpose. Symmetric deflection of spoilers situated in front of the landing flaps, when applied directly after touch down, provides the possibility to drastically decrease the wing lift during the roll out. A larger part of the weight acts on the undercarriage making the wheel brakes more effective, thus yielding an appreciable reduction in the stopping distance. A disadvantage of spoilers is their low effectivity at small deflections, caused by re-attachment of the airflow behind the spoiler. So-called ‘plug type spoilers’, see subfigure b of figure 7-47 have better characteristics in this respect due to the open connection between lower and upper surface of the wing caused by the spoiler deflection. Re-attachment of the airflow is avoided in this manner. At large angles of attack the effectivity of the spoiler may also be low, especially if flow separation occurs at the trailing edge of the wing. A further disadvantage during the approach and landing phases is the decrease in total wing lift caused by the operation of the spoilers. This renders the precise control of the aircraft’s flight path slightly more difficult. The hinge moment of a plain flap type spoiler, see subfigure a of figure 7-47, may be
7-8 Rudder control derivatives
355
appreciable, and it usually changes non-linearly with the deflection angle. On the other hand, the hinge moment of a spoiler in the shape of a half circular plate, see subfigure b of figure 7-47, may be next to negligible. However, the total aerodynamic load on a spoiler of this type may be relatively high. Cℓδs and Cnδs can be determined using references [70]. Reference [57] gives theoretical calculation methods to determine Cℓδs while reference [100] contains an extensive list of references on the use of spoilers.
7-8 7-8-1
Rudder control derivatives Control derivative CYδr
The rudder deflection causes a lateral force, 1 2 ρV S 2
Y = CYδr δr
(7-35)
A positive δr gives rise to a positive lateral force, see figure 7-49. The control derivative CYδr is positive. The lateral force can be written as, Y = CYvδ δr
1 2 ρV Sv 2 v
(7-36)
where CYvδ is the gradient of the normal, here lateral, force on the vertical tailplane. From equations (7-35) and (7-36) it follows, CYδr = CYvδ
Vv V
2
Sv S
(7-37)
The derivative CYvδ is determined in a similar way as the derivative CNhδ of the horizontal tailplane, see references [155, 43, 46, 126]. For supersonic speeds this derivative can be calculated with reference [99].
7-8-2
Control derivative Cℓδr
A positive lateral force acting on the vertical tailplane, caused by a positive rudder deflection, causes a positive rolling moment, L = Cℓδr δr
1 2 ρV Sb 2
(7-38)
where Cℓδr is positive. In analogy with the contribution of the vertical tailplane Cℓβ Cℓp v and (Cℓr )v , the control derivative Cℓδr can be written as, Cℓδr = CYδr
zv − zc.g. xv − xc.g. cos α0 − sin α0 b b
v
,
(7-39) Flight Dynamics
356
Lateral Stability and Control Derivatives
V
XB
Cnδr · δr (< 0) YB c.g. lv
CYδr · δr (> 0)
δr (> 0) Figure 7-49: The side force and yawing moment due to a rudder deflection
7-8 Rudder control derivatives
357
Large positive values of Cℓδr could be detrimental to good lateral control characteristics. In flight, when accurate lateral control is required, small deviations from the desired heading may be corrected by using rudder deflections, without recourse to the ailerons. This may give a result more quickly than changing the aircraft’s heading via a change in angle of roll generated by the ailerons. If in such a situation a positive rudder deflection is used to yaw the aircraft to the left, a large positive Cℓδr causes an angle of roll to the right. Such a roll angle is associated with a turn to the right and opposes the desired yawing motion of the aircraft. The effect just mentioned occurs mainly with aircraft having a highly mounted vertical tailplane, where zv − zc.g. in equation (7-39) is large. In these aircraft the rudder and ailerons may be coupled mechanically such that a positive aileron deflection is generated by a positive rudder deflection. The moment Cℓδa · δa then opposes the moment Cℓδr · δr .
7-8-3
Control derivative Cnδr
The primary effect of a rudder deflection is the yawing moment, N = Cnδr δr
1 2 ρV Sb 2
(7-40)
The control derivative Cnδr follows from figure 7-49 at small α0 and taking xv − xc.g. = lv , Cnδr = −CYδr
lv b
(7-41)
At large angles of attack, Cnδr follows from an expression in analogy with equation (7-16) for Cnβ v . The control derivative Cnδr is negative.
Flight Dynamics
358
Lateral Stability and Control Derivatives
Figure 7-50: Asymmetric stability and control derivatives
Chapter 8 Longitudinal Stability and Control in Steady Flight
8-1
Stick Fixed Static Longitudinal Stability
In chapter 5 we discussed the equilibrium of the aerodynamic and gravitational forces in the plane of symmetry and the aerodynamic moment about the center of gravity. The also showed that the horizontal tailplane and its elevator play a crucial rˆ ole in the equilibrium of the aerodynamic moment about the center of gravity. For the equilibrium, the following equations (5-53), (5-54) and (5-55), were derived, W sin θ 2 2 ρV S
CT = CTw = − 1 CN = CNw + CNh
Cm = Cmac
Vh V
2
Sh W = 1 2 cos θ S 2 ρV S
xcg − xw + CNw + CNh c¯
Vh V
2
Sh xcg − xh =0 S c¯
For the equilibrium of the moments equation (5-57) was introduced,
Cm = Cmac
xcg − xw + CNw − CNh c¯
Vh V
2
Sh lh =0 S¯ c
Equilibrium of forces and moments acting on an aircraft in flight is a necessary, but certainly not a sufficient condition for steady flight. A steady flight condition must also be stable as pointed out for example already in reference [123]. This means that after the occurrence of a small disturbance the aircraft should return to the original state of equilibrium. If this is Flight Dynamics
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Longitudinal Stability and Control in Steady Flight
the case, the equilibrium is called dynamically stable. Although, as remarked in reference [123], stability is strictly a characteristic of an equilibrium, it is common practice to call the aircraft itself dynamically stable if its equilibrium is stable. The motions of aircraft after a disturbance, and along with it the dynamic stability, are determined by the equations of motion of the aircraft. A discussion of aircraft motions, using these equations of motion (see chapters 3 and 4), is postponed at this stage. There is, as in reference [53], a simpler way to study stability. Experience has shown, that most critical requirements to be satisfied for dynamic stability can be expressed for most aircraft by the condition that the aircraft must be statically stable. The discussion of static stability is based on the equations for the equilibrium of the aircraft, see equations (5-53), (5-54) and (5-57). These equations are in fact special cases of the more general equations of motion. When we discuss static longitudinal stability we refer to pitch attitude stability. So we just need to consider the expression for the longitudinal moment, equations (5-57) or (5-55) allowing us to study the change of aerodynamic moment caused by a change in angle of attack in a similar way as we did with the two dimensional wing profile, wings and wing-fuselage composites in section 5-1-3. Here we analize the static stability of complete aircraft including horizontal tail surfaces. New is also that we must consider static stability for the case of stick fixed (fixed elevator deflection) and the case of stick free (floating elevator). It will be shown that a close relationship exists between the way in which the stick position varies with airspeed, or angle of attack, in steady, straight flight on the one side and the static stability, stick fixed on the other. In a similar way it will be shown that static stability with stick free, is closely related to the way in which the stick force varies with airspeed in steady, straight flight. For simplicity , propulsion effects will be neglected and the dimensionless aerodynamic coefficients are assumed to be independent of airspeed. We could readily take these effects into account, at the cost, however, of much more cumbersome derivations. It is interesting to note that steady flight conditions as discussed in the present chapter, are in fact just a special case of dynamic flight conditions described by equations of motion. This fact allows us to present an alternative way to derive static stability and control characteristics directly from the linearized equations of motion. The resulting expressions contain stability and control derivatives which do take account of things as propulsion effects, and so, are in a sense more general than those which were derived for the more simple case of the model above. We present these derivations from the linearized equations of motion in boxes under the heading of ‘Quick derivation of ....’ following the more elaborate analytical derivations based on the simplified model above.
8-1-1
Stick fixed static longitudinal stability in gliding flight
In section 5-1-3 the concept of static stability was presented for two dimensional wing profiles, wings and wing-fuselage composites. This discussion is now extended to a complete aircraft. Suppose the aircraft is in steady flight, so the moment about the center of gravity is zero, Cm = 0. If some external disturbance would cause an increase of the pitch angle and
8-1 Stick Fixed Static Longitudinal Stability
361
so of the angle of attack, we would like to see a negative, nose-down change in Cm , a nose down, restoring moment. The condition for static longitudinal stability then is, dCm = Cmα < 0, at Cm = 0 dα
(8-1)
If Cmα = 0 the static stability is called neutral. If Cmα > 0 the aircraft is statically unstable. To study the various contributions to the derivative Cmα of the complete aircraft, the expression for the moment coefficient is reconsidered as in equation (5-72) Cm = Cmac + CNwα (α − α0 )
xcg − xw − c¯
2 dε Sh lh Vh + (α0 + ih ) + CNhδ δe CNhα (α − α0 ) 1 − e dα V S¯ c
(8-2)
Equation (8-2) can be rearranged to combine the contributions of the wing with fuselage and nacelles (indicated by the index w) and the contribution of the horizontal tailplane (indicated by index h), Cm = Cmw + Cmh
(8-3)
with, Cmw = Cmac + CNwα (α − α0 )
xcg − xw c¯
(8-4)
and,
Cmh
V 2 S l dε h h h + (α0 + ih ) +CNhδ δe = − CNhα (α − α0 ) 1 − e dα V S¯ c | {z } αh | {z }
(8-5)
CNh
Figures 8-1 schematically show the Cm − α-curves of the contributions according to equations (8-4) and (8-5) and of the complete aircraft configuration according to (8-2). Figure 8-2 presents the contributions of the various parts of the aircraft to the Cm − α-curve as measured in a wind tunnel on a model of the Fokker F-27. Figure 8-3 presents calculated moment curves for several parts of the Cessna Ce550 ‘Citation II’. This data was calculated using an inviscid panel method (linearized potential flow). The definition of the aircraft parts is presented in figure 6-2; these parts are: ‘wing’, ‘horizontal stabilizer’, ‘pylon’, ‘nacelles’, ‘vertical fin’ and ‘fuselage’ (in clockwise direction, starting from the left top figure). The results presented Flight Dynamics
362
Longitudinal Stability and Control in Steady Flight
Cmw
α
Cmh (A) Wing-fuselage contribution
α
Cm (B) Horizontal tailplane contribution
α
Equilibrium steady flight
(C) Cm = Cmh + Cmw
Figure 8-1: The contribution of various parts of the aircraft to the moment curve Cm − α
8-1 Stick Fixed Static Longitudinal Stability
363
Cm
model without tailplane
0.2
wing
0 -4
contribution of fuselage and nacelles
0
4
8
12
α [o ]
16
complete model -0.2
contribution of horizontal tailplane -0.4
-0.6
Figure 8-2: Measured contributions of various aircraft parts to the moment curve of the Fokker F- 27, reference point at 0.346 c¯ (from reference [147])
in figure 8-3 clearly show the destabilizing effect of the fuselage and the stabilizing effect of the horizontal tailplane, while the contribution of the wing and nacelles is almost indifferent. The slopes of the moment curves follow from equations (8-4) and (8-5) by differentiating with respect to the angle α. Because the ‘stick fixed’ situation is considered, the elevator angle δe is taken constant, Cmαw = CNwα
Cmαh = −CNhα
xcg − xw c¯
(8-6)
2 dε Vh Sh lh 1− dα V S¯ c
(8-7)
Because the cg usually lies behind the ac of the combination of a wing with fuselage and nacelles, the derivative Cmαw is positive, see equation (8-6). This means, that even if the aircraft without the horizontal tailplane could be at equilibrium at all (Cmw = 0), the equilibrium would be unstable, see also section 5-1-3. The contribution of the tailplane has a stabilizing effect, according to equation (8-7), (Cmα )h < 0. This contribution must be large enough, that Cmα is, see equation (5-75), Cmα
xcg − xw = CNwα − CNhα c¯
dε 1− dα
Vh V
2
Sh lh S¯ c
is negative, Flight Dynamics
364
Longitudinal Stability and Control in Steady Flight 0.25
wing horizontal stabilizer pylons fuselage vertical fin nacelles total
0.2
0.15
Cm
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2 −10
−8
−6
−4
−2
0
α [Deg.]
2
4
6
8
10
Figure 8-3: Calculated contributions of various aircraft parts to the moment curve of the Cessna Ce550 ‘Citation II’
Cmα < 0 The important role of the horizontal tailplane for obtaining static stability is obvious, as already discussed in section 5-2-1. The horizontal tailplane is also essential to obtain equilibrium of the longitudinal moment. As can be seen from a number of measured moment curves in figures 8-4 and 8-5, Cm varies almost linearly with α over a large range of angles of attack. But at large values of α, Cmα is no longer constant. This is due to the contribution of CT to the longitudinal moment and to the non-linearity caused by flow separation. Strictly speaking, static stability is the slope of the moment curve Cmα , only at Cm = 0. By varying ih and δe the moment curve can be shifted up and down. If the angle of attack is not too large, the shifted moment curves are all parallel, see figure 8-4. A change in ih influences Cm according to the expression in equation (5-73), Cm = Cm0 + Cmα (α − α0 ) + Cmδe δe = 0 only by changes in Cm0 , which is independent of α, see equation (5-74) Cm0 = Cmac − CNhα (α0 + ih )
Vh V
2
Sh lh S¯ c
8-1 Stick Fixed Static Longitudinal Stability
365
Cm
0.2
ih = −2o ih = 0o
0 -4
0
4
8
12
16
α [o ]
-0.2
ih = 2o -0.4
-0.6
Figure 8-4: Influence of the tailplane incidence on the moment curve of the Fokker F-27, reference point at 0.346 c¯ (from reference [147])
A change in δe influences Cm via the term Cmδe · δe , in principle again independent of α, see also equation (5-76), Cmδe δe = −CNhδ
e
Vh V
2
Sh lh δe S¯ c
If Cm can be made zero at any angle of attack by a suitable choice of ih or δe or both without influencing Cmα , the slope of the moment curve at an angle of attack where Cm 6= 0 may be considered the static stability at that particular angle of attack α.
8-1-2
Neutral point, stick fixed
It follows from equation (5-75) and figure 8-5 that the cg position has a strong influence on the static stability. We will study this in more detail now. In order to obtain slightly more elegant expressions, we start with the expression for Cmα obtained by differentiating equation (5-55) with respect to α, Cmα = CNwα
xcg − xw + CNhα c¯
dε 1− dα
Vh V
2
Sh xcg − xh S c¯
(8-8)
The only difference between equation (8-8) and the expression in equation (5-75), following from equation (5-57), is, that lh has been replaced by xh − xcg . If the shift in cg positions are small relative to the tail length they are typically about 20 % of the mean aerodynamic chord c¯ and since lh is typically 3 to 4 times c¯, the contribution of the tailplane to Cmα may Flight Dynamics
366
Longitudinal Stability and Control in Steady Flight
be considered as constant to a first approximation. The main effect of a change in cg position is a change in the contribution of the wing, fuselage and nacelles to Cmα . Shifting the cg rearward (xcg increases) will change the contribution of Cmαw , usually already positive, to grow further in the positive sense. The result is, that Cmα of the complete aircraft becomes less negative. A rearward shift of the center gravity thus has a destabilizing effect, see also figure 8-5. At a sufficiently far aft cg position the positive contribution of the wing-fuselage combination just compensates the negative contribution of the horizontal tailplane. Then, Cmα = 0 This center of gravity position, at which the equilibrium of the moment is neutrally stable, stick fixed, is called the neutral point stick fixed (n.p.fix). This is the first interpretation of the neutral point, stick fixed. The abscissa of this point is xnf ix . If the center of gravity coincides with this neutral point, it follows from equation (8-8) that, Cmα = 0 = CNwα
xnf ix − xw + CNhα c¯
dε 1− dα
Vh V
2
Sh xnf ix − xh S c¯
(8-9)
From equation (8-9) the position of the neutral point is derived. To this end CNwα is eliminated from equation (8-9) by multiplying the expression for the normal force gradient, CNα = CNwα + CNhα
1−
dε dα
Vh V
2
Sh S
(8-10)
Sh lh S¯ c
(8-11)
by, xnf ix − xw c¯ The result is, xnf ix − xw CNhα = c¯ CNα
dε 1− dα
Vh V
2
where, as in an equation similar to equation (5-56), lh = xh − xw in an aircraft fixed reference frame. Next, equation (8-9) is substracted from equation (8-8). Using equation (8-10), the resulting expression for Cm becomes, Cmα = CNα
xcg − xnf ix c¯
(8-12)
8-1 Stick Fixed Static Longitudinal Stability
367
Cm
0.2
C 0 -4
0
4
8
12
B
A
-0.2
α [o ]
16
D -0.4
-0.6
Zr D 0.08¯ c A
B
0.135¯ c
C
Xr
0.085¯ c
xcg /¯ c
zcg /¯ c
A
0.211
−0.044
B
0.346
−0.044
C
0.431
−0.044
D
0.346
+0.036
Figure 8-5: Influence of the position of the reference point (center of gravity) on the moment curves of the Fokker F-27, (from reference [147])
Flight Dynamics
368
Longitudinal Stability and Control in Steady Flight Zr
dCm = dCN ·
dCNw
dCN
xw
α
acw
∆α
xcg −xnf ix c¯
dCNh
lh
xcg
cg
n.p.f ix
Vh V
2
Sh S
ach
xnf ix xh
Xr Figure 8-6: Change in the moment dCm due to a change in the angle of attack ∆α
Consider now by means of equation (8-12) a change in the moment dCm due to a change in angle of attack dα such that dCm = Cmα dα. Since the accompanying change in the normal force is dCN = CNα dα, equation (8-12) may be written as, dCm = dCN
xcg − xnf ix c¯
(8-13)
According to equation (8-13), the change in the moment dCm may be interpreted as being xcg −xnf ix caused by a change in the normal force dCN , acting a distance from the center of c¯ gravity, see figure 8-6. From this follows the second interpretation of the neutral point with stick fixed as the point of action on the m.ac of the normal force increment, caused by a step change of the angle of attack with fixed elevator angle deflection. In concurrence with this second interpretation of the neutral point, the non-dimensional xcg −xnf ix distance is often called the ‘stability margin’, stick fixed. c¯ The present interpretation of the neutral point agrees entirely with its first introduction in section 5-1-2as the point of intersection of the line of action of an increment in the aerodynamic force, dC R and the mean aerodynamic chord.
8-1-3
Elevator trim curve and elevator trim stability
A. Elevator trim curve In section 5-2-6 the elevator angle required for equilibrium, i.e. for Cm = 0, was expressed in equation (5-80):
8-1 Stick Fixed Static Longitudinal Stability
δe = −
1 Cmδe
369
{Cm0 + Cmα (α − α0 )}
where Cm is the static stability according to equations (5-75) or (8-8). The constant Cm0 and the elevator efficiency Cmδe were already expressed in equations (5-74) and (5-76) respectively, see section 8-1-1. If Cmδe and Cmα are constant, the elevator angle required for equilibrium varies linearly with α, according to equation (5-80). The graphic representation of the elevator angle required for equilibrium as a function of α or airspeed V is known as the elevator trim curve. The relation between the angle of attack α and the airspeed V is derived from the equilibrium in Z-direction, or in a slightly modified form, CN ≈ CNα (α − α0 ) ≈
W
(8-14)
1 2 2 ρV S
From equation (5-80) follows with equation (8-14) the expression for the elevator trim curve δe as a function of V , δe = −
1 Cmδe
Cm0
Cmα + CNα
W 1 2S ρV 2
!
(8-15)
Figure 8-7 presents schematically some elevator trim curves and the corresponding moment curves for a statically stable (Cmα < 0), a neutrally stable (Cmα = 0) and a statically unstable (Cmα > 0) aircraft. It is common practice to plot an ‘elevator-up’ deflection in the upward direction, implying that the negative δe -axis points upwards. From these figures, and also from the expressions in equations (5-80) and (8-15), it can be seen that there is a close relation between the moment curve and the elevator trim curve. In particular the slope of the elevator-trim curve is directly related to the slope of the moment curve Cmα , i.e. the static longitudinal stability, if Cmδe is exactly constant. B. Elevator stick position stability The following discussion shows how the slope of the elevator trim curve is important for the pilot’s opinion on the handling qualities of aircraft. The slope of the elevator trim curve δe −α follows from equation (5-80) by differentiating with respect to α, dδe Cmα =− dα Cmδe
(8-16)
Considering that Cmδe is always negative, it follows that for a statically stable aircraft (Cmα < 0) the slope of the trim curve δe − α is always negative, i.e., dδe <0 dα
(8-17) Flight Dynamics
370
Longitudinal Stability and Control in Steady Flight
(Cm )δe =0
Cm0
dCm dα
dCm dα
>0
dCm dα
=0
α0 = 0o assumed
<0 α
δe (−)
δe (−) dCm dα dCm dα
<0
<0 α
δe =
C − Cmm0 δ
e
dCm dα
dCm dα
=0
V dCm dα
=0
>0 dCm dα
Figure 8-7: Moment curves and corresponding trim curves
>0
8-1 Stick Fixed Static Longitudinal Stability
371
e The slope dδ dV of the elevator trim curve δe − V follows from equation (8-15) by differentiating with respect to V,
dδe 4W 1 Cmα = 3 dV ρV S Cmδe CNα
(8-18)
For a statically stable aircraft (Cmα < 0) it follows from equation (8-18), dδe >0 dV
(8-19)
If the relation between the elevator angle and the airspeed in steady, straight flight is such that equation (8-19) is satisfied, the aircraft is said to have elevator trim stability, see also reference [161]. This concept of the elevator control position stability can be interpreted in two different ways, i.e., e 1. According to the first interpretation, trim stability (the slope dδ dV of the elevator trim curve δe − V ) provides an indication of the static longitudinal stability of the aircraft. Elevator trim stability is a convenient characteristic to obtain the static stability both qualitatively and quantitatively from measurements in actual flight, see section 8-1-6 for a more detailed discussion.
2. The second interpretation of trim stability lies in the fact that an aircraft possessing such stability is pleasant for the pilot and safe to fly. This will be further illustrated in the following. Suppose that from a given steady flight condition pilot elects to transition to another steady flight condition at a slightly lower airspeed and a corresponding larger angle of attack. He must therefore initiate a nose-up rotation of the aircraft over the small angle ∆θ, to obtain the required new angle of attack. This initial rotation is obtained through an initial moment ∆Cmi about the lateral axis, generated by an initial elevator deflection ∆δei , see figure 8-8. If α and θ are intended to increase and as a consequence V is intended to decrease, the initial elevator displacement must be upward and the corresponding stick displacement directed aft. For any aircraft at any cg position the following holds, ∆δei >0 ∆V and, ∆sei >0 ∆V Now it is known that it is pleasant for the pilot, because it eases the flying of the aircraft, if the initial control displacement ∆sei and the final control displacement ∆seu are in the same direction, ∆seu >0 ∆sei
⇒
∆δeu >0 ∆δei
(8-20) Flight Dynamics
372
Longitudinal Stability and Control in Steady Flight
∆Cmi ∆δei
∆θi
cg
XB Figure 8-8: Initial control surface deflections
with the subscript u as in seu and δeu implying the final, or ultimate, control (surface) displacement. There is no need for ∆δeu and ∆δei to be of equal magnitude. In order to speed up the response of the aircraft, the pilot might well generate an initial control deflection larger than the final control displacement. This has been indicated by the dashed line in figure 8-9a. Evidently, the following holds, ∆δeu = ∆δei
∆δeu ∆V ∆δei ∆V
∆δ
It was argued, that always ∆Vei > 0. As a consequence equation (8-20) corresponds to the eu dδe requirement ∆δ ∆V > 0 or, for infinitesimally small changes : dV > 0. It is thus seen, that equation (8-20) leads to the requirement in equation (8-19) for elevator control position stability. Figure 8-10 may serve as a further illustration of the importance of this desired control characteristic. The figure shows in a highly schematic way how the pilot handles the elevator control, if he or she wants to change the aircraft from one steady flight condition into another steady condition. The case of a trim curve showing control position stability is considered first. Suppose again the pilot wants to reduce airspeed, starting from a steady flight condition (indicated ‘1’ in the figure). Pulling the control wheel aft, will cause α and θ to increase, and soon after V to decrease. For an aircraft with positive elevator control position stability, the new elevator deflection δe2 corresponds to a new steady flight condition at a lower airspeed V2 (see the flight condition indicated as ‘2’ in figure 8-10). The response of the aircraft to the initial control displacement is in the direction of the desired new state equilibrium. Figure 8-10 also shows the case of an aircraft with negative elevator control position stability. It can be seen, that the response of the aircraft to the initial control displacement is in the other direction as needed for the desired new steady flight condition. As a consequence a much more complicated time-history of the control displacement is needed to arrive at the steady flight condition ‘2’. This characteristic where the initial control input and the final control input have different signs is considered to be an objectionable characteristic which pilots are supposed not to appreciate. Worse is, of course, that in this case this control characteristic reflects the fact that the aircraft is statically unstable, and would require a lot
8-1 Stick Fixed Static Longitudinal Stability
373
Figure 8-9: Initial and ultimate control displacement and control surface deflection for the transition to a lower airspeed
Flight Dynamics
374
Longitudinal Stability and Control in Steady Flight
Figure 8-10: Aircraft responses to a control deflection
8-1 Stick Fixed Static Longitudinal Stability
375
(Cm )δe =0
xcg = xnf ix α0 = 0o assumed cg moves rearward α δe (−)
δe (−)
α
V xcg = xnf ix
xcg = xnf ix cg moves rearward cg moves rearward
Figure 8-11: Influence of cg position on the trim curve
of attention and effort by the pilot to keep under control, and it might well be an impossible task for even the most gifted pilot.
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Longitudinal Stability and Control in Steady Flight
A quick derivation of stick position stability We start with the symmetrical equations of motion, in which, for simplicity, the derivatives CZα˙ and CZq were set to zero.
u ˆ CXα CZo 0 CXu − 2µc Dc α CZu CZα − 2µc Dc 0 2µc θ + 0 0 −Dc 1 q¯ c Cmu Cmα + Cmα˙ Dc 0 Cmq − 2µc KY2 Dc V 0 CZ δ 0 .δe = 0 Cmδ For stationary straight flight reduce to: CXu CZu Cmu
c c conditions, where q¯ ˆ = Dc q¯ V = Dc α = Dc u V = 0 these equations CXα CZo u ˆ 0 CZα 0 . α + CZδ .δe = 0 Cmα 0 θ Cmδ
Now , we introduce a small stick deflection ∆δe , in a trimmed flight condition with V0 , α0 , θ0 . This results, after some time, in a new steady state flight condition with V = Vo + ∆V (∆ˆ u= So,
CXu CZu Cmu
or
∆V ), α=αo + ∆α, θ=θo +∆θ Vo
CXα CZo ∆ˆ u 0 CZα 0 . ∆α + CZδ .∆δe = 0 Cmα 0 ∆θ Cmδ Ass .∆x − B.∆δe = 0
where Ass
CXu = CZu Cmu
CXα CZα Cmα
After dividing by ∆δe we get:
CXu CZu Cmu ∆x ∆δe
CZo ∆ˆ u 0 0 , ∆x = ∆α and B = CZδ 0 ∆θ Cmδ
CXα CZo CZα 0 . Cmα 0
∆ˆ u ∆δe ∆α ∆δe ∆θ ∆δe
0 + CZδ = 0 Cmδ
now simply follows from ∆x = −A−1 ss .B ∆δe
8-1 Stick Fixed Static Longitudinal Stability By applying Cramer’s rule we find: 0 CXα −CZ CZα δ −Cm Cm ∆ˆ u α δ = ∆δe |Ass |
377
CZo 0 0
=
−CZδ Cmα + Cmδ CZα CZu Cmα − Cmu CZα
If for simplicity the less usually important derivatives CZδ and Cmu are set to zero, we get: ∆ˆ u Cmδ CZα = ∆δe CZu Cmα Substituting CZα = −CNα , and CZu ≈ −2CL results in ∆V
∆ˆ u Cmδ CNα Cmδ CNα = . or, Vo = . . ∆δe 2CL Cmα ∆δe 2CL Cmα After introducing:
W CL = 1 /2ρVo2 S
we arrive at the well known expression for stick displacement stability: dδe 4W Cmα 1 = . . dV ρVo3 S CNα Cmδ e For a statically stable aeroplane where Cmα < 0 and dδ dV > 0, which implies that for flight at a lower speed, an aft (negative) stick deflection is required.
8-1-4
Influence of various parameters on elevator trim curve
A. Influence of the center of gravity position Using equation (8-12) the expression in equation (8-15) for the elevator trim curve can be rewritten as,
δe = −
1 Cmδe
Cm0
xcg − xnf ix W + 1 2 c¯ 2 ρV S
!
(8-21)
The slope of the trim curve, the trim stability, follows from equations (8-18) and (8-12), dδe 4W 1 xcg − xnf ix = 3 dV ρV S Cmδe c¯
(8-22)
Equations (8-21) and (8-22) reveal the direct influence of the position of the center of gravity relative to the neutral point, stick-fixed, on the elevator angle required for equilibrium and on the elevator control position stability. A rearward cg shift increases the elevator angle required for equilibrium, the elevator trim Flight Dynamics
378
Longitudinal Stability and Control in Steady Flight
Cm
α ih increases δe (−)
δe (−) ih increases ih increases α
V
Figure 8-12: Influence of the tailplane angle of incidence on the trim curve (aircraft has control position stability)
curve δe − V in figure 8-11 shifts downward. In addition when the cg shifts aft the stability margin decreases. The elevator control position stability, i.e. the slope of the trim curve, decreases, see also figure 8-11. B. Influence of the stabilizer setting In the expression for the trim curve the part of the elevator angle independent of airspeed is obtained by letting in equation (8-21) V = ∞ (and by consequence α = α0 ). Using eqution (5-74), the resulting δeV =∞ is,
δeV =∞
Cm0 1 =− =− Cmδe Cmδe
( |
Cmac − CNhα (α0 + ih ) {z
Cm0
Vh V
2
Sh lh S¯ c
) }
(8-23)
For many aircraft the stabilizer setting ih can be varied in flight. From equation (8-23) it follows that such a change has an influence on δeV =∞ . The stabilizer setting is adjusted in
8-1 Stick Fixed Static Longitudinal Stability
379
flight to reduce the elevator control force to zero in a given steady flight condition, see section 8-2. In such an aircraft the stabilizer thus has a function comparable to that of a trim tab. It follows from equation (8-23) that an increase of ih (the stabilizer leading edge moves up) causes an increase in the negative sense of the part of δe independent of airspeed. The trailing edge of the elevator moves up as well. The simple explanation is that at constant airspeeds and cg positions the value of CN required for equilibrium of the moment remains constant. This has been indicated schematically in figure 8-12. C. Influence of the trim tab angle Many aircraft, especially the smaller and slower ones, have a trim tab at the trailing edge of the elevator. If the tab angle δte is changed, the trim curve shifts parallel to itself. Qualitatively, this shift is entirely comparable to the shift caused by a change in the stabilizer angle of incidence ih . Usually, the infuence of a change in δte on the elevator trim curve is neglected, see also equation (5-60). Subfigure a of figure 8-21 shows a number of trim curves measured at constant cg positions and different trim tab angles. D. Shape of the elevator trim curve The shape of the trim curve not only depends on the variables discussed so far: the cg position, the stabilizer angle of incidence ih and the trim tab angle δte . In addition, the influence of engine power setting can be appreciable, in particular for propeller-driven aircraft. Usually, an increase in engine power decreases the elevator control position stability. The Mach number effects and aeroelastic deformation can also have an important influence on the elevator trim curves, because they make the aerodynamic coefficient functions of airspeed. As noted in section 1-2 these phenomena will not be considered here.
8-1-5
Static longitudinal stability of tailless aircraft
The stability of tailless aircraft has already been discussed in section 5-1-3. It was shown, see figure 5-21, that equilibrium (Cm = 0) at positive values of CN is possible only if Cmac > 0. The equilibrium is stable, i.e. Cmα < 0, if the center of gravity lies ahead of the ac, xcg < xac = xw By choosing a sufficiently forward cg position this condition can be satisfied. The requirement for a positive Cmac needs a further discussion. In section 5-1-4 the following expression was derived for the Cmac of a wing, see equation (5-30),
Cmac
b b Z2 Z2 2 2 = cmac c dy − cℓb c (x − x0 ) dy S¯ c 0
0
Flight Dynamics
380
Longitudinal Stability and Control in Steady Flight cℓb c
+ 2b
− 2b
y
cℓroot
ε<0 Λ
cℓtip
Figure 8-13: A positive Cma.c is obtained by combining sweepback and negative wing twist
It is possible to obtain a positive Cmac by choosing wing airfoil sections having a positive cmac . Airfoils showing an S-shaped camber line possess at cℓ = 0 a positive value for the moment coefficient. For wings having such airfoils the first integral in equation (5-30) is positive. A second possibility to obtain a positive Cmac is offered by the second integral in equation (5-30). As was discussed in section 5-1-4, this integral represents the moment due to the basic lift distribution. Figure 8-13 shows that this moment is positive at CL = 0 for a wing having sweep back and negative wing twist. Although it is thus shown that static longitudinal stability can be obtained without resorting to a horizontal tailplane, most aircraft nevertheless have a horizontal tailplane. The elevator of a tailless aircraft is placed at the trailing edge of the wing. The two parts of the elevator are commonly used as ailerons (or named as ‘elevons’) as well. Due to the relatively small distance to the aircraft center of gravity, elevons are less effective for pitch control than an elevator mounted on a tailplane. When the aircraft is in a trimmed condition, in general δe 6= 0. The increased drag due to the elevator deflection, called ‘trimdrag’, of the tailless aircraft may be one of the factors in favour of a design with a tailplane. The horizontal tailplane also contributes to a considerable degree in the damping of the so-called short-period oscillation, occurring after a symmetric disturbance of the equilibrium, see chapter 10. This is an additional advantage of the horizontal tailplane. Summarizing, it can be said that the horizontal tailplane contributes to, • the equilibrium of the moment about the aircraft cg
8-1 Stick Fixed Static Longitudinal Stability
381
• the static longitudinal stability • the aerodynamic damping about the lateral axis The price to be paid is, however, • an increased aerodynamic drag • an increased aircraft weight
8-1-6
Determination Cmδe from measurements in flight
In section 5-2-6 the condition for equilibrium of the moment was written as, see equation (5-77), Cm = (Cm )δe =0 + Cmδe δe Suppose Cmδe were known. From a trim curve, δe − V , measured in flight the moment coefficient (Cm )δe =0 can be obtained as a function of V or α. In many cases Cmδe varies markedly with flight condition, for instance under the influence of the wing wake or the slipstream. It follows that it is necessary to know Cmδe in the flight conditions of interest, since a measured trim curve is not a proper reflection of the moment curve (Cm )δe =0 . Methods to determine the elevator efficiency Cmδe from measurements flight are all based on the principle of applying a known moment about the lateral axis in the flight conditions to be investigated. The extra elevator deflection needed to compensate for this moment is a measure of Cmδe in that flight condition. The simplest manner to generate a known moment employs a shift of the center of gravity in the X-direction, for instance by shifting a known amount of ballast over a known distance. In figure 8-14 cg positions (cg1 and cg2 ) have been indicated as well as the equilibrium at cg1 . At constant elevator angle δe = δe1 no equilibrium exists about cg2 . At constant α and δe , CN does not change and it still passes through cg1 . Equilibrium of the moment at cg2 at constant α is obtained by changing the elevator angle. The change of the moment about cg2 caused by δe2 − δe1 is, ∆Cm = Cmδe (δe2 − δe1 ) = Cmδe ∆δe This moment is balanced by the moment of CN about cg2 , i.e., CN
xcg2 − xcg1 + Cmδe (δe2 − δe1 ) = 0 c¯
(8-24)
The influence of the elevator deflection ∆δe on CN is neglected here, or, ∆CN = CNhδ ∆δe
Vh V
2
Sh << CN S Flight Dynamics
382
Longitudinal Stability and Control in Steady Flight XB
R N
θ
W sinθ
T
cg1 δe1
W cosθ cg2 W
W
δe2 ∆δe
ZB Figure 8-14: The determination of elevator efficiency from flight tests
From equation (8-24) it follows, Cmδe = −
1 ∆xcg CN ∆δe c¯
(8-25)
The above means that if the trim curves have been measured at two cg positions differing in their x-coordinates, Cmδe can be obtained with equation (8-25). In the foregoing the influence of the cg position on Cmδe has been neglected. Because the tail volume equation (5-76) gives,
Sh lh S¯ c
is obtained directly from the dimensions of the aircraft, use of
CNhδ
Vh V
2
= −Cmδe
S¯ c Sh lh
(8-26)
2 The normal force gradient CNhδ can only be obtained from flight tests if in addition VVh has been measured along the span of the horizontal tailplane. Once the elevator efficiency Cmδe has been measured as just described, the stability margin (stick fixed) can be derived from the slope of the trim curve, using equation (8-22). If the aircraft is equipped with an adjustable stabilizer, the derivative Cmih can be obtained as well, once Cmδe has been measured. To this end, the elevator angle required for equilibrium is measured in two flight conditions, differing only in the choice of the stabilizer angle of incidence but identical in all other respects. In section 5-2-4 the angle of attack αh of the horizontal tailplane was derived as, see equation (5-64), αh = α − ε + ih From this expression it follows that if flight conditions do not change (both α and ε remain constant) a change in ih influences only αh , as expressed by,
8-1 Stick Fixed Static Longitudinal Stability
383
∆αh = ∆ih At constant α and ε, CNh has to remain constant as well to maintain the same steady flight conditions, where Cm = 0, CNh = CNhα αh + CNhδ δe = constant or, CNhα ∆αh + CNhδ ∆δe = 0 The result is, CNhα = − As mentioned before, in general CNhδ ·
Vh V
∆δe CN ∆ih hδ
2
will be determined, rather than CNh . If 2 is not known, the procedure just described can only produce CNhα · VVh , CNhα
Vh V
2
=−
∆δe CNhδ ∆ih
Vh V
Vh V
2
2
By way of illustration, the following gives an example of the determination of stability characteristics from measured elevator trim curves. The aircraft in the example is a Fokker F-27, the measured trim curves were taken from reference [51]. Figure 8-15 shows two elevator trim curves δe − Ve . They were measured at an engine power of 420 hp (approximately gliding flight) and a trim tab angle of δte = +4.5o . Apart from cg positions also the aircraft weight was different. For this reason, the trim curve for xcg = 0.227 c¯, pertaining to an aircraft weight of 14,500 kg, was corrected to a weight of 13,000 kg. At a constant flight condition, i.e. at constant CN and δe , the following holds, Ve2 = Ve1
r
W2 W1
The trim curve thus corrected is also shown in figure 8-15. In the first place Cmδe is derived from the two trim curves belonging to the same aircraft weight of 13,000 kg. At Ve = 240 ∆x km/h, figure 8-15 shows that ∆δe = 1.4o . The shift in cg position is c¯cg = 0.058. At an aircraft weight of 13,000 kg the airspeed of Ve = 240 km/h corresponds to a normal force coefficient of CN = 0.681. From equation (8-25) it then follows, Cmδe = −
1 · 0.681 · 0.058 = −0.0282 1.4
with δe in degrees.
Flight Dynamics
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Longitudinal Stability and Control in Steady Flight
Figure 8-15: Trim curves for two cg positions of the Fokker F-27 (from reference [51])
8-2 Stick Free Static Longitudinal Stability The tailplane volume of the present aircraft is follows with equation (8-26),
385 Sh lh S¯ c .
Using this value at Ve = 240 km/h it
CNhδ
Vh V
2
= +0.0303 (δe in degrees)
CNhδ
Vh V
2
= +1.7353 (δe in radians)
Next, the position of the neutral point, stick fixed is derived by means of equation (8-22). Again at Ve = 240 km/h is ρ04W = 0.0204 and at xcg = 0.277 c¯ the slope of the elevator trim V 3S curve is
dδe dVe
e
= +0.0502o /km/h or +0.187o /m/s. This results in, xcg − xnf ix = −0.187 · 0.0282 · 48.9 = −0.258 c¯
With xcg = 0.277 c¯ follows xnf ix = 0.485 c¯. If the same calculation is repeated for the trim curve measured at xcg = 0.285 c¯, the result is xnf ix = 0.490 c¯. Using the data obtained thus far the moment curves Cm versus V and Cm versus CN can be derived. If in addition the relation CN versus α of the aircraft is known, the moment curves Cm versus α can be 2 calculated as well. Figure 8-16a shows the calculated values of CNhα · VVh as a function of α.Figure 8-16b gives the Cm versus α curves and figure 8-16c shows the position of the neutral point, stick fixed, as a function of α and CN .
8-2
Stick Free Static Longitudinal Stability
It was already noted in the previous section that it makes sense to study the static stability in the case of stick free. Two reasons can be given to study the static stability in this condition. The first is the requirement for the aircraft to be remain stable if the pilot would release the stick. The second reason lies in the close relationship between the static stability, with stick free, and a certain characteristic of the way in which the elevator force required to maintain the elevator in the position for steady flight, varies with airspeed. This characteristic will be called the elevator stick (or control) force stability. As in section 8-1, the concept of static stability is based entirely on the variation of the pitching moment with angle of attack. Leaving the elevator free only changes the contribution of the horizontal tailplane, and the elevator, to the pitching moment. First we consider the behavior of the elevator in the control free situation. If the control stick or wheel is free, the control force Fe is zero. Neglecting the friction in the control mechanism this means that in the stick free situation the hinge moment is zero, He = 0, or, Chef ree = 0 Flight Dynamics
386
Longitudinal Stability and Control in Steady Flight
Figure 8-16: Some stability characteristics derived from the measured trim curves shown in figure 8-15 (Fokker F-27)
8-2 Stick Free Static Longitudinal Stability
387 Chef ree = 0
αh V δef ree
Chef ree = Chα αh + Chδ δef ree + Chδt δte = 0 δte Figure 8-17: The equilibrium of the free elevator
In this situation the elevator angle assumes a certain value, depending on the way the elevator is aerodynamically balanced and on the trim tab angle. This elevator angle, stick free, indicated as δef ree see figure 8-17, varies with angle of attack. If the airfoil of the tailplane is symmetric, see equation (5-63), Ch0 = 0 and then, Chef ree = Chα αh + Chδ δef ree + Chδt δte = 0 This means that δef ree becomes, δef ree = −
Chδt Chα αh − δt Chδ Chδ e
(8-27)
Differentiation with respect to αh results in the variation of δef ree , with αh at constant trim tab angle,
dδe dαh
f ree
=−
Chα Chδ
and the variation with α,
dδe dα
f ree
=−
Chα dαh Chδ dα
(8-28)
Using equation (5-66), dαh dε =1− dα dα results in,
dδe dα
f ree
Ch =− α Chδ
dε 1− dα
(8-29)
Flight Dynamics
388
Longitudinal Stability and Control in Steady Flight
8-2-1
Stick free static longitudinal stability in gliding flight
The contribution of the horizontal tailplane to the pitching moment is, see equations (5-60) and (8-5), V 2 S l h h h Cmh = − CNhα αh + CNhδ δe V S¯ c
(8-30)
When studying the static longitudinal stability in section 8-1, the elevator control was assumed to be held fixed, the elevator angle δe was constant and did not change when α was slightly varied. If, however, in equation (8-30) the constant δe is replaced by the variable δef ree , differentiation with respect to α results in the contribution of the horizontal tailplane with free elevator to the static longitudinal stability, stick free, Cmαf ree ,
dCmh dα
f ree
= Cmαh
= − CNhα
f ree
dαh + CNhδ dα
dδe dα
f ree
!
Vh V
2
Sh lh S¯ c
(8-31)
Substituting equation (5-66),
dαh = dα
dε 1− dα
and equation (8-29),
dδe dα
Ch =− α Chδ
f ree
dε 1− dα
results in, Cmαh
f ree
= − CNhα − CNhδ
Chα Chδ
dε 1− dα
Vh V
2
Sh lh S¯ c
(8-32)
If equation (8-32) is compared with the corresponding expression (8-7) for the contribution of the horizontal tailplane in the stick fixed situation, when δe = 0 is constant, Cmαh
f ix
= −CNhα
dε 1− dα
Vh V
2
Sh lh S¯ c
the concept of the normal force gradient of the tailplane in the stick free situation arises, CNhα
f ree
= CNhα − CNhδ
Chα Chδ
(8-33)
Using this gradient in equation (8-32) results in close analogy with equation (8-7), Cmαh
f ree
= −CNhα
f ree
2 dε Vh Sh lh 1− dα V S¯ c
(8-34)
8-2 Stick Free Static Longitudinal Stability
389
The static stability, stick free, Cmαf ree , follows by adding to equation (8-34) the contribution of the wing, fuselage and nacelles, Cmαf ree = CNwα
xcg − xw − CNhα f ree c¯
dε 1− dα
Vh V
2
Sh lh S¯ c
(8-35)
Previously, the static stability, stick fixed, was derived as, Cmα
xcg − xw = CNwα − CNhα c¯
dε 1− dα
The expressions (5-75), (8-35) and (8-33) for CNhα
f ree
Vh V
2
Sh lh S¯ c
indicate that the difference between C
the static stability stick fixed and stick free is entirely due to the term CNhδ Chhα . In this δ term CNhδ is positive. The hinge moment derivatives Chα and Chδ were already discussed in chapter 5. They depend on the way the elevator is aerodynamically balanced. To keep the control forces at reasonably low levels, it is essential that both Chα and Chδ are made small in the absolute sense. The required sign of Chδ follows from the behaviour of the free elevator in steady, trimmed flight, i.e. the situation where Fe = 0. In that situation Che = 0. Suppose the elevator now obtains a small deviation dδe from the equilibrium position, due to some disturbance. The immediate effect is a hinge moment, dChe = Chδ dδe if Chδ > 0, the hinge moment will be positive if dδe is positive. Due to the hinge moment the change in elevator angle will further increase. The general conclusion is that an equilibrium position of a control surface in the controls free condition will be unstable if Chδ is positive; the control surface does not return to the equilibrium position after a disturbance has occurred. For this reason it is strictly necessary that Chδ is negative. The above argument needs a slight extension. Suppose a control surface has a Chδ nearly equal to zero. In such a case actually two hinge moments are present, of nearly equal magnitude but of opposite sign, see the pressure distribution in figure 5-71. The resultant value of Chδ is due to the difference of these two hinge moments. A small variation in one of these moments, as may be caused by a small change in the shape of the control surface, due for instance to the tolerance in the manufacturing process, will have a large effect on Chδ if compared to the intended value. For this reason Chδ has to be not only negative, but should not be allowed to approach zero too closely. Common values are: −0.2 to−0.3 < Chδ < −0.1 (δe in Radians). The sign of Chδ will be discussed in detail in section 8-2-3 in relation with the elevator control force stability. Here it can be said that Chα may be either negative or positive. Too large positive values of Chα , e.g. Chα > approximately 0.1 (with αh in Radians), cannot be used because they may lead to dynamic instability, stick free. In the decision on the required values of Chα and Chδ the influence which ice accretion on the stabilizer and elevator has on the aerodynamic balance of the control surface, has to be taken into account as well. In the Flight Dynamics
390
Longitudinal Stability and Control in Steady Flight Zr
dCmf ree = dCNf ree ·
dCNw
dCNf ree
xw
α
acw
∆α
V
xcg −xnf ree c¯
dCNhf ree
lh
xcg
Vh V
2
Sh S
ach cg
n.p.f ree
xnf ree xh
Xr Figure 8-18: The change in the moment dCmf ree due to a change in angle of attack ∆α
final choice of the elevator balance the non-linear relation between the hinge moment and αh and δe at large angles is an additional important consideration. From equations (8-33) and (8-35) it can be seen that depending on the sign of Chα the aircraft with free elevator control will be less statically stable (Chα < 0), equally stable (Chα = 0) or more stable (Chα > 0) than it is in the control fixed situation.
8-2-2
Neutral point, stick free
As for the aircraft with fixed elevator control a certain center of gravity position exists also in the control free situation where Cmα , here Cmαf ree , is equal to zero. This center of gravity position is called the neutral point, stick free, indicated as n.p.f ree . The abscissa of this point is xnf ree . As in equation (8-11) the position of the neutral point, stick free, is obtained by letting Cmαf ree = 0, see equation (8-35), 2 CNhα xnf ree − xw dε Vh Sh lh f ree = 1− c¯ CNα dα V S¯ c
(8-36)
Just like the neutral point, stick fixed, the neutral point, stick free, can be interpreted in a second way, in addition to the one just given. It is the point of action (on the m.ac) of the total change in normal force coefficient dCN due to a change in angle of attack dα if the elevator control is left free. In analogy with equation (8-12) it follows, Cmαf ree = CNαf ree
xcg − xnf ree c¯
8-2 Stick Free Static Longitudinal Stability
391
with, CNαf ree ≈ CNαf ix The latter expression is obtained by neglecting in equation (8-10) the difference in normal force gradient CNα due to the difference between CNhα and CNhα , as the contribuf ree tion of the tailplane to CNα is relatively small when compared to the contribution of the wing. The change in pitching moment dCmf ree due to a change in angle of attack dα with free elevator control then is, dCmf ree = dCNf ree
xcg − xnf ree c¯
(8-37)
Figure 8-18 gives an illustration of this second interpretation of the neutral point, stick free. The relative position of the two neutral points, stick fixed and stick free, follows from equations (8-11) and (8-36), 2 − CNhα CNhα xnf ree − xnf ix Vh Sh lh dε f ree f ix = 1− c¯ CNα dα V S¯ c
(8-38)
Using equation (8-33), CNhα
f ree
= CNhα − CNhδ
Chα Chδ
the result is, CNhδ Chα xnf ree − xnf ix =− c¯ CNα Chδ
2 Vh Cmδ Chα dε Sh lh dε = 1− 1− dα V S¯ c CNα Chδ dα
(8-39)
According to this latter expression the relative position of the two neutral points is determined primarily by the sign of Chα . A simple explanation of the influence of the sign of Chα on the static stability, stick free, is possible by looking at the behaviour of the free elevator, see figure 8-19. If Chα = 0 the elevator angle δef ree does not vary with angle of attack according to equation (8-29). It is as if the controls were fixed, see subfigure b of figure 8-19. There is no difference between the static stability stick fixed and stick free, the two neutral points coincide. If Chα < 0 the elevator turns with the direction of the flow as the angle of attack varies, see subfigure a of figure 8-19. With an increase in angle of attack the trailing edge of the elevator moves up. The effect of freeing the elevator control is an extra downward force on the tailplane. This extra force promotes a further increase in angle of attack. As a consequence the static stability stick free is less than it is for stick fixed and xnf ree is smaller than xnf ix , if Chα < 0. The influence of a positive Chα can be understood in the same way, see subfigure c of figure 8-19. Flight Dynamics
392
Longitudinal Stability and Control in Steady Flight ∆δef ree δef ree (< 0) α ∆α
V
(A)
Chα < 0 ∆δef ree = 0
α ∆α
V
δef ree (B)
Chα = 0
α ∆α
V
δef ree (C)
Chα > 0 ∆δef ree
Figure 8-19: The behaviour of the free elevator after a change of the angle of attack
8-2-3
Elevator stick force curves and elevator stick force Stability
A. The control force curve The elevator control force needed for equilibrium was derived in chapter 5 as, Fe = −
dδe 1 2 ρV Se c¯e Che dse 2 h
The following discussion is intended primarily for qualitative purposes. It is, therefore, permissible to express Che here as a linear function of αh , δe and δte , see equation (5-63). This leads to the following expression for Fe , see also equation (5-83), Fe = −
dδe 1 2 ρVh Se c¯e Chα αh + Chδ δe + Chδt δte dse 2
For a more accurate calculation of the control force in a given flight condition Che has to be obtained from wind tunnel measurements for the correct values of αh , δe and δte . In chapter 5 αh was derived as, see equation (5-65),
dε αh = (α − α0 ) 1 − dα
+ (α0 + ih )
The elevator angle δe required for equilibrium about the lateral axis is, see equation (5-80),
8-2 Stick Free Static Longitudinal Stability
δe = −
n
1 Cmδe
393
Cm0 + Cmαf ix (α − α0 )
o
Substitution of equations (5-65) and (5-80) in equation (5-83) results after some elaboration in, Fe = − where, Ch′ 0 = − Ch′ α
dδe 1 2 ρVh Se c¯e Ch′ 0 + Ch′ α (α − α0 ) dse 2
Chδ Chδ Cmac − CNhα (α0 + ih ) + Chδt δte f ree Cmδ CNhδ
= Chα
dε 1− dα
=−
−
(8-40)
(8-41)
Chδ Ch Cmαf ix = − δ Cmαf ree Cmδ Cmδ
xcg − xnf ree Chδ CNα Cmδ c¯
(8-42)
Furthermore, (α − α0 ) ≈
W 1 2 2 ρV S
1 1 W ≈ 1 2 CLα CNα 2 ρV S
(8-43)
Substituting equation (8-43) in equation (8-40) results in, dδe Fe = − Se c¯e dse
Vh V
2 1 ′ 1 2 ′ W Ch0 ρV + Chα 2 S CNα
(8-44)
According to equation (8-44) the elevator control force required to maintain steady flight consists of two parts. One is proportional to the dynamic pressure and thus varies with airspeed. The other part is independent of airspeed. The part depending on airspeed is considered first. It can be seen from equations (8-44) and (8-41) that this part varies by changing the trim tab angle or the stabilizer angle of incidence or both. Now suppose that ih is constant. At a certain value of δte , Ch′ 0 will be equal to zero. This particular trim tab angle is called δte0 . It is derived from equation (8-41) by letting, Ch′ 0 = 0 The result is,
δte0 =
Chδ Cmδ
Cmac +
Chδ CNh
δ
CNhα
Chδt
f ree
(α0 + ih ) (8-45)
Flight Dynamics
394
Longitudinal Stability and Control in Steady Flight
Fe (−)
Fe (−) δte < δte0
δte < δte0
δte = δte0
FeV =0
δte > δte0
1 2 2 ρVmin
(A)
1 2 2 ρV
δte = δte0
δte > δte0
FeV =0
Ve
Vemin (B)
Figure 8-20: Schematic form of the elevator control force curve, Fe as a function of (a) dynamic pressure 1 2 2 ρV and (b) Fe as a function of equivalent airspeed Ve
8-2 Stick Free Static Longitudinal Stability
395
Using equations (8-41) and (8-45), Ch′ 0 can now be written as, Ch′ 0 = Chδt
δte − δte0
(8-46)
After some elaboration the final expression for the elevator control force can now be obtained from equation (8-44), using equations (8-46) and (8-42). The result is, dδe Fe = Se c¯e dse
Vh V
2
W Chδ xcg − xnf ree 1 − ρV 2 Chδt S Cmδe c¯ 2
δte − δte0
(8-47)
The variation of the elevator control force with dynamic pressure or airspeed as expressed by equation (8-47) is shown schematically in figure 8-20. Negative elevator control forces, i.e. pull forces exerted by the pilot, are plotted upward in the figure, just like negative elevator angles. If the various aerodynamic characteristics in equation (8-47) may indeed be considered as constants, then Fe varies linearly with the dynamic pressure 12 ρV 2 , see subfigure a of figure 8-20, or quadratically with V , see subfigure b of figure 8-20. The measured trim curves and elevator control force curves shown in figure 8-21 show that in reality the various simplifying assumptions are not always satisfied. The second part of the elevator control force is independent of airspeed, it is indicated as FeV =O . According to equation (8-47) this part is determined by the position of the cg relative to the n.p.f ree or by the sign of the static stability, stick free. If the aircraft is statically stable, stick free (xcg < xnf ree ), FeV =O is negative. The variation of the part of Fe depending on airspeed is determined by δte . If δte = δte0 , Fe does not vary with airspeed. If δte > δte0 the control force increases in the positive sense with increasing airspeed, see figure 8-20, independent of the cg position. It was assumed that Chδ , Cmδe and Chδt have the normal, negative sign. The airspeed at which the control force is zero, is called the trim speed, indicated as Vtr . So, Vtr = VFe =O If the elevator control is let free at this airspeed, the control position does not change.
B. Elevator stick force stability If the elevator control force curve at the trim speed satisfies the condition,
dFe dV
>0
(8-48)
Fe =0
the aircraft is said to have elevator control force stability in that particular flight condition. Elevator control force stability is thus seen to depend only on the slope of the control force curve in the trimmed, i.e. Fe = 0, condition.
Flight Dynamics
396
Longitudinal Stability and Control in Steady Flight
The expression will now be further analyzed. Differentiating equation (8-47) with respect to V yields the slope of the control force curve as, dδe dFe =− Se c¯e dV dse
Vh V
2
ρV Chδt
δte − δte0
(8-49)
This expression holds for both trimmed and untrimmed conditions. The elevator control force stability follows by substituting in equation (8-49) the values of δte pertaining to Vtr . The trim speed Vtr is obtained from equation (8-47) by letting Fe = 0, 1 2 ρV Chδt 2 tr
W Chδ xcg − xnf ree δte − δte0 = S Cmδe c¯
(8-50)
Combining equations (8-49) and (8-50) results in the elevator control force stability,
dFe dV
Fe =0
dδe = −2 Se c¯e dse
Vh V
2
W Chδ xcg − xnf ree 1 S Cmδe c¯ Vtr
(8-51)
The direct relation between the control force stability as defined in equation (8-48) and the position of cg relative to the n.p.f ree position can be seen from equation (8-51). If the aircraft is statically stable, stick free, it is control force stable according to equation (8-51) and vice versa. Just like the control position stability, discussed in section 8-1, control force stability is important in two respects, 1. In the first place, the control force stability provides the possibility to ascertain from measurements in flight if the aircraft is statically stable, stick free, in a certain aircraft configuration and flight condition, see equation (8-51). 2. In the second place, and perhaps most important, an aircraft that is control force stable is more pleasant and safer to fly by the pilot. The latter subject will now be discussed. As a general rule it can be stated that it is highly desirable, if not imperative, that control displacements and the control forces required to generate and maintain these displacements both in manoeuvres and in steady flight conditions, have equal directions. This requirement can be expressed simply as, dFe >0 dse
(8-52)
The control manipulator, whether it is a stick or a wheel, then behaves as if it were pulled back to the neutral position by springs, see figure 8-22. The effect to the pilot is that he feels through the force he has to apply on the control manipulator in which direction and to what extent he has deflected the control. Generally it can be said that the pilot is far more sensitive to changes in the control forces he has to exert than to changes in the control
8-2 Stick Free Static Longitudinal Stability
397
Figure 8-21: Measured trim curves and elevator control force curves for the De Havilland D.H.98 ‘Mosquito’ M II F in gliding flight (from reference [27])
Flight Dynamics
398
Longitudinal Stability and Control in Steady Flight
Figure 8-22: Schematic representation of the concept of ‘positive feel’,
dFe dse
column displacement. Due to the adapted sign convention be written as,
dδe dse
> 0, the requirement as in equation (8-52) can then
dFe >0 dδe It was argued in subsection B of section 8-1-3 that the initial and the ultimate control displacement should be in the same direction. From the above it follows that this applies equally to the changes in the exerted control force. Using the same sign convention for control displacements and control forces and considering the requirement for elevator control position stability, see equation (8-19), dδe >0 dV the corresponding requirement for elevator stick force stability is written as in equation (8-48),
dFe dV
>0 Fe =0
Inevitably, the control mechanism of the aircraft has a certain friction which was neglected so far. The magnitude of this friction, which is primarily static friction, has been expressed in section 5-2-7 in terms of an equivalent hinge moment Hef . A certain control force Fef is required to overcome this friction. If the airspeed in steady differs only slightly from flight Vtr the required control may be small as well. If |Fe | < Fef , no control force is required from the pilot. He can leave the control stick or wheel free, the required control position is maintained by the friction in the control mechanism, see figure 8-23.
8-2 Stick Free Static Longitudinal Stability
399
Fe (−)
Range of possible values of Vtrim due to Fef 6= 0
Fef V
Fef
Vtrim if Fef = 0
Figure 8-23: Uncertainty in trim speed due to friction in the control mechanism
The result is that in actual flight not one single trim speed exists, and any airspeed within the range where Fe > Fef is then a trim speed. In order to keep this uncertainty of the trim speed within acceptable limits the elevator control force stability must have a certain minimum positive value and, in addition, Fef has to be sufficiently small. The U.S. military regulations, see reference [19], require,
dFe dV
> 0.5 (lbs/3 kts) Fe =0
corresponding with,
dFe dV
> 0.041 (kg/km/h) Fe =0
According to the same regulations the control force required to overcome the friction is limited to, • Light aircraft – control stick Fef < 1.3 kg Flight Dynamics
400
Longitudinal Stability and Control in Steady Flight – control wheel Fef < 1.8 kg • Heavy aircraft
– control stick Fef < 2.3 kg – control wheel Fef < 3.2 kg
This control force is to be measured on the ground, the engines not running. In flight the control force needed to overcome the friction may be considerably less due to vibrations of the aircraft. The exact definitions of the terms ‘light’ and ‘heavy’ aircraft can be found in reference [19].
A quick derivation of stick force stability We start with the symmetrical equations of motion, in which, for simplicity, the derivatives CZα˙ and CZq were set to zero.
CXu − 2µc Dc CXα CZo 0 u ˆ α CZu CZα − 2µc Dc 0 2µc θ + 0 0 −Dc 1 q¯ c Cmα + Cmα˙ Dc 0 Cmq − 2µc KY2 Dc Cmu V 0 CZ δ 0 δe = 0 Cmδ
c c In stationary straight flight conditions q¯ ˆ = Dc q¯ V = Dc α = Dc u V = 0, so: CXu CXα CZo u ˆ 0 CZu CZα 0 . α + CZδ .δe = 0 Cmu Cmα 0 θ Cmδ
The stick force can be written as: Fe = −
dδe 1 / ρV 2 .Se c¯e .(Chα .αh + Chδ .δe + Chδt .δte ) dse 2 h
writing it for simplicity as Fe = K.(Chα .αh + Chδ .δe + Chδt .δte ) where K=−
dδe 1 / ρV 2 .Se c¯e dse 2 h
8-2 Stick Free Static Longitudinal Stability
401
We will concern ourselves with small variations ∆Fe of Fe and corresponding small deviations from a trimmed steady state condition at V0 , α0 , θ0 . After some time, a new state of equilibrium will be reached: V = Vo + ∆V (∆ˆ u=
∆V ), α=αo + ∆α, θ=θo +∆θ Vo
As we are interested in stick force, we assume that the trim tab is not changed, so, ∆Fe = K.(Chα .∆αh + Chδ .∆δe ) where, ∆αh = (1 −
dε ) ∆α dα
so ∆Fe = K {Chα ∆α(1 − and ∆δe =
dε ) + Chδ ∆δe } dα
1 Ch dε ∆Fe − α (1 − ) ∆α K Ch δ Chδ dα
If we now substitute ∆δe into the equations of ‘motion’ for the case of steady state we get CXu CXα CZo u ˆ 0 Ch dε CZu CZα 0 α + CZδ {− α (1 − ) ∆α+ Chδ dα Cmu Cmα 0 θ Cmδ 1 ∆Fe = 0 K Ch δ or
CXu CZu Cmu
CZα
CXα Chα − Ch .(1 −
Cmα −
δ Chα Chδ
.(1 −
dε dα ).CZδ dε dα ).Cmδ
0 CZδ Cmδ
CZo ∆ˆ u 0 ∆α + ∆θ 0 1 ∆Fe = 0 K Ch δ
Here we introduce two new stability derivatives for the case of stick free, Chα dε (1 − ). CZδ Chδ dα Ch dε − α .(1 − ).Cmδ Chδ dα
CZα f ree = CZα − Cmα f ree = Cmα so, A ∆xss + B
1 ∆Fe = 0 K.Chδ
Flight Dynamics
402 with
Longitudinal Stability and Control in Steady Flight
CXu A = CZu Cmu
CZo 0 ∆ˆ u 0 , B = CZδ ,∆xss = ∆α Cmδ ∆θ 0
CXα CZα f ree Cmα f ree
Multiplication by the inverse of A results in:
∆xss = −A−1 B or
xss ‘ =
and
Finally, since ∆ˆ u=
∆ˆ u ∆Fe ∆α ∆Fe ∆θ ∆Fe
1 ∆Fe K.Chδ
K Ch δ K Chδ = −A−1 B K Ch δ
∆F K Ch δ = ∆ˆ u xss ‘(1) ∆V V
, we get:
K Ch δ ∆F dδe 1 = = − Ch Se c¯e 1/2ρVh2 ∆V xss ‘(1) V dse δ xss ‘(1)V Applying Cramer’s rule, we find: 0 CXα C C Zαf ree Zδ Cmδ Cmαf ree x’ss (1) = − CXα CXu C Zu CZαf ree Cmu Cmαf ree =−
CZo 0 0 CZo 0 0
C Zo = C Zo
CZαf ree Cmαf ree
CZu Cmu
CZαf ree Cmαf ree
CZδ Cmαf ree − Cmδ CZαf ree
CZu Cmαf ree − Cmu CZαf ree
Now for simplicity we set CZδ = 0, Cmu = 0 resulting in: ′
CZδ Cmδ
xss (1) = −
−Cmδ CZαf ree CZu Cmαf ree
Since CZα ≈ −CNα and CZu ≈ −2CL = −2
W 1/ ρV 2 S 2
=
8-2 Stick Free Static Longitudinal Stability we get: ′
xss (1) = −
403
Cmδ CZαf ree CZu Cmαf ree
With CZαf ree ≈ CZαf ree and CZu ≈ −2 CL = −2 Cmδ CZαf ree ′ xss (1) = − . W 2 1/ ρV Cmαf ree 2S 2 We already derived:
W 1/2ρV 2 S ,
∆F K.Chδ dδe 1 1 = V =− , Ch .Se c¯e 1/2ρVh2 ′ ∆V xss ‘(1) dse δ xss (1) V so:
Cmαf ree 1 ∆F K.Chδ dδe 2 W = V =− Chδ .Se c¯e 1/2ρVh2 . = ∆V xss ‘(1) dse Cmδ 1/2ρV 2 S CNαf ree V =−
xc.g. 2 W dδe Chδ .Se c¯e 1/2ρVh2 . 2 1 dse Cmδ /2ρV S c¯
and since this formula applies to trim speed V = Vtr we find: dF dδe Vh W Chδ xc.g. − xnf ree 1 = −2 Chδ .Se c¯( )2 dV dse V S Cmδ c¯ Vtr
8-2-4
Effect of center of gravity and mass unbalance on stick force stability
A. Influence of the center of gravity position In the foregoing it was shown that the center of gravity has an influence only on the part FeV =0 of the control force which is independent of airspeed, see equation (8-47). If the cg is shifted in X-direction this constant term in equation (8-47) varies and, at a constant trim tab angle, the entire control force curve will shift up or down, parallel to itself. If the cg is moved forward the control force curve will shift upwards (a larger pull force will be required) at constant δte and ih . This agrees with the measured control force curves shown in figures 8-21 and 8-24. The slope of the control force does not change at constant δte , but with the new cg position corresponds to another, higher trim speed. If the original trim speed is re-established by an additional downward deflection of the trim tab, it turns out, see for instance subfigure d of figure 8-24, that the elevator control force stability at constant trim speed has increased. So, strictly speaking, the elevator control force stability cannot be considered independent of the trim speed. B. Trim tab angle and the stabilizer angle of incidence
Flight Dynamics
404
Longitudinal Stability and Control in Steady Flight
Figure 8-24: Measured elevator control force curves for the North American ‘Harvard’ II B in gliding flight (from reference [15])
8-2 Stick Free Static Longitudinal Stability
405
From equation (8-47) it can be concluded that for a stick free, statically stable aircraft, any trim tab angle δte > δte0 results in a certain airspeed where Fe = 0. The trim tab angle for a given Vtr is obtained from equation (8-50), δte = δte0 +
W 1 2 2 ρVtr S
1 Chδ xcg − xnf ree Chδt Cmδe c¯
(8-53)
It follows from equation (8-53) that for a given cg position the pilot can choose a certain trim speed where he can fly ‘hands-off’ by selecting the appropriate δte . If the stability margin stick free is positive, cg in front of the n.p.f ree , the trim speed is reduced by increasing δte , i.e. with increasing downward tab deflection. The trim wheel in the cockpit is turned backward to this end. Once a certain trim speed has been chosen the variation of the control force with airspeed in steady flight is fixed, according to equation (8-51), and so is the elevator control force stability. It can be clearly seen in figure 8-24 from the measured elevator control forces that the slope of the control force curve indeed increases at a given airspeed by a downward deflection of the trim tab. From the foregoing it becomes clear that for a given cg position a fixed relation exists between the trim tab angle and the correspondig trim speed. The aircraft possesses elevator control force stability if according to equation (8-53), dδte <0 dVtr The trim tab angle curve may be used to investigate the elevator control force stability. An example of a measured trim tab angle curve is shown in figure 8-25. As already mentioned in section 5-2-6, in some aircraft the elevator control forces are reduced to zero by adjusting the stabilizer angle of incidence. In such cases the elevator does not need to have a trim tab. It follows from equations (8-41) and (8-44) that both the horizontal stabilizer setting ih and the trim tab angle δte figure in Ch′ 0 and FeV =0 . It also follows from (8-41) and (8-44) that it is irrelevant whether the stabilizer or the trim tab is used to trim the control force to zero. If the stabilizer is used Ch′ 0 can be written in analogy with equation (8-46) as, Ch′ 0 = Chδ (ih − ih0 )
(8-54)
where ih0 is the value of ih for which Ch′ 0 = 0. The influence of ih on the elevator control force stability is entirely comparable to that of δte . C. Influence of a spring or an unbalanced mass in the control mechanism Sometimes it is not possible to obtain satisfactory control forces and elevator control force stability in all required aircraft configurations and flight conditions by aerodynamic balancing of the elevator only. If at a given cg position and trim speed the elevator control force stability Flight Dynamics
406
Longitudinal Stability and Control in Steady Flight
Figure 8-25: Trim curves, hinge moment coefficients and required trim tab angles as functions of CL for the Siebel 204-D-1 aircraft (from reference [16])
8-2 Stick Free Static Longitudinal Stability
407
Figure 8-26: The influence of a bobweight in the control mechanism
Figure 8-27: The influence of a spring in the control mechanism
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408
Longitudinal Stability and Control in Steady Flight
Figure 8-28: The influence of a spring or bobweight on the elevator control force curve
should become too low an unbalanced mass, a ‘bobweight’, see figure 8-26, or a spring, see figure 8-27, may be used in the control mechanism, see also figure 5-90. These devices increase FeV =0 by a constant force ∆FeV =0 , see figure 8-28. A spring to be used for this purpose is chosen in principle such that the applied force remains approximately constant over the entire range of control deflections. The entire control force curve will then be shifted upward, parallel to itself. A more detailed analysis shows, that now Cmf ree has become a function of airspeed. It also turns out that the n.p.f ree position has moved backwards and consequently the stability margin, stick free, has been increased. The use of a spring or an unbalanced spring in the control mechanism is thus seen to have the same effect, as far as the control force curve is concerned, as a forward shift of the center of gravity.
8-2-5
Influence of design variables on control forces
Another look at equation (8-47), dδe Fe = Se c¯e dse
Vh V
2
W Chδ xcg − xnf ree 1 − ρV 2 Chδt S Cmδe c¯ 2
δte − δet0
reveals that the variables of direct interest to the pilot are: 12 ρV 2 , xcg , δte and ih (they have already been discussed in previous sections). Other variables occuring in equation (8-47) are dδe h lh of interest in the design of the aircraft, such as ds , Se , c¯e , the tailvolume SS¯ c and CNhδ (or e xn
Cmδe ), Chδ , Chδt , fc¯ree . The following can be said on these variables. The control forces increase with the wing loading W S . the control forces are in addition proportional to Se and
8-2 Stick Free Static Longitudinal Stability
409
c¯e . Hence, for aircraft of identical shape the control forces vary with the third power of the dimensions. e The control gearing dδ dse is determined by the fact that the extreme deflection of the control surface must correspond with the extreme displacement of the cockpit control column or stick. The range of elevator deflections ∆δemax is restricted aerodynamically, amounting to about 50o or 60o (25o to 30o to either side). The permissable range of control displacements ∆semax is restricted by the dimensions of the human pilot and sometimes by the dimensions dδe of the cockpit. The maximum value is about 40 cm.. The resulting value of ds turns out to e o be approximately 1.25 to 1.5 /cm. (2.2 to 2.6 Radians/m). Some aircraft have a variable gear ratio depending on the aircraft configuration.
The required value of Cmδe is determined by the maximum required elevator power. Usually either the take-off or the landing at the most forward cg position is the most critital condition see section 5-2-6. The derivative Chδt determines the required size of the trim tab, taking into account the (limited) range of the trim tab angles. For a trim tab to be effective at any occurring value of αh and δe the deflection should not be larger than ±15o . The required size of the trim tab then follows from the requirement that, according to the regulations, it must be possible to reduce the control force to zero in certain aircraft configurations, including the cg position, and over certain ranges of airspeed, see section 5-2-7. The only remaining variables which the designer can use to influence the control forces are the stability margin, stick free and Chδ . Earlier was found, see equation (8-39), CNhδ Chα xnf ree − xnf ix =− c¯ CNα Chδ
dε 1− dα
Vh V
2
Cmδe Chα Sh lh = S¯ c CNα Chδ
dε 1− dα
This means that at a given position of the neutral point, stick fixed, Chα and Chδ are the only remaining variables with which the designer can manipulate the control forces. From the foregoing it will have become clear that the general requirement is for the center of gravity to be positioned forward of the neutral points, both stick fixed and stick free. The aircraft then possesses both elevator control position and force stability. If Chα of the elevator as well as Chδ are negative, the n.p.f ree lies forward of the n.p.f ix position. The aft permissable cg position is limited by the n.p.f ree . It is possible to shift this n.p. aft by reducing |Chα |. The n.p.f ree will even lie aft of the n.p.f ix , if Chα > 0. This is one reason why a slight overbalance with respect to angle of attack is sometimes aimed for when balancing the elevator. For many modern aircraft it turns out to be no longer possible to obtain acceptable control forces in all required aircraft configurations and flight conditions, merely by choosing Chα and Chδ . In particular this is true for aircraft having a high wing loading W S , a high maximum airVmax speed Vmax and/or a large ratio of maximum to minimum airspeed Vmin (V/STOL-aircraft). If purely aerodynamic means do not suffice additional use can be made in the first place of a spring or an unbalanced mass or both in the control mechanism. These devices also influence the dynamic stability, stick free, and sometimes in an unfavourable sense. In more advanced cases, such as transonic, supersonic and some V/STOL-aircraft, springs and masses cannot offer a satisfactory solution. In those cases the aerodynamic hinge moments Flight Dynamics
410
Longitudinal Stability and Control in Steady Flight
are balanced in total or in part by, usually hydraulic, control force amplifiers, i.e. control boosters, or by servo controls. When hydraulic servo controls are used it would be sufficient for the pilot to apply only the very small forces needed to operate the servo valve. Experience has shown, however, that in the interest of safety of flight the pilot must have to exert forces in his control stick or wheel having the usual levels of magnitude and varying in the normal way with airspeed, control deflection and trim deflection. This requirement is met by the use of a separate installation in the flight control system, a so called ‘artificial feel unit’. This mechanism applies a force in the control manipulator which has to be balanced by the pilot’s control force. These artificially generated control forces are made to vary with 12 ρV 2 , δe and δte in such a way that the pilot’s control force shows indeed the variations found in an aircraft where the control manipulator has a direct mechanical linkage with the control surface.
8-2-6
Control forces a pilot can exert
The discussion in the previous sections centered around the control forces required to fly the aircraft. The following deals with forces a pilot can exert on his controls. Evidently the designer must ensure that the required control forces remain within the capabilities of even the least strongest amongst the pilot population. Appropriate regulations aim to safeguard this requirement. The maximum control force a pilot can apply depends primarily on, 1. the individual person; strength and endurance differ appreciably from one person to another 2. the type of control manipulator (control wheel, center stick, side stick, rudder pedals) 3. the time during which the force must be exerted 4. the location of the control manipulator relative to the pilot Regarding to these items, the following can be remarked. Figure 8-29 clearly shows the large differences in the maximum possible control forces due to individual differences in strength and endurance of the test subjects. It appears that the largest force a pilot can exert is the rudder pedal force (curve a), the side force for aileron control is the smallest (curves d and e). It is also seen that the forces on a control wheel can be larger than those on a control stick. In the measurements of figure 8-29 the control manipulators were all handled by the right hand. This is in agreement with the situation in most single seat cockpits where the pilot uses his left hand to operate the engine throttle. Figure 8-29 gives an indication of the maximum possible control forces as a function of the uninterrupted time period over which these forces have to be applied. The maximum possible control force decreases approximately logarithmically with increasing duration. The location
8-2 Stick Free Static Longitudinal Stability
411
Figure 8-29: Maximum control forces as a function of the duration (from reference [140])
of the control manipulator is not only important because it influences the maximum possible control forces. It also influences the pilot’s comfort, see references [140, 65, 111, 36, 39]. For a large amount of information on the preferable locations of controls in the cockpit. Occasionally the aircraft flight condition can experience a rather sudden change. In view of such occurrances it is necessary to know how fast a pilot can change the control deflections under various levels of control force gradients. Information on this can be found in references [162, 120, 69]. It is evident that the required control forces may not surpass the possible control forces. The various Airworthiness Regulations contain requirements concerning the maximum permissable control forces under various circumstances. Table 8-1 gives the maximum forces as stipulated in the U.S. and the British Airworthiness Regulations, see references [6] and [7] respectively.
8-2-7
Airworthiness requirements for steady straight symmetric flight
The Airworthiness Regulations, see references [11, 19, 6, 20, 12, 13, 7], require that the aircraft is statically stable in the most important aircraft configurations and flight conditions. This means that the cg must be in front of the neutral point in those conditions. Sometimes Flight Dynamics
412
Longitudinal Stability and Control in Steady Flight
the requirement is expressed in terms of an equivalent requirement regarding the control characteristics. In view of the fact that the pilot is in general more sensitive to control forces and changes in control forces than to control positions and control displacements, the requirement usually stipulates that the aircraft possesses elevator control force stability in the relevant aircraft configurations and flight conditions at all permissable center of gravity positions. As discussed in section 8-2-2 this requirement can easily be reformulated into the condition that the center of gravity must be positioned in front of the neutral point stick free. For the designer the emphasis may be slightly different. In section 8-2-4 it was seen that several possibilities exist to influence the control force through non-aerodynamic means. The arsenal of available tools is certainly not exhausted with the springs and unbalanced masses discussed in section 8-2-4. It is of great practical interest that several of these means may still be applied relatively easy in a late stage of the development of the aircraft, if the need arises, without great expense or drastic changes in the design. To change the elevator control position stability or the static stability, stick fixed, usually turns out to be much more difficult since it requires considerably more drastic modifications in the design, see section 8-1-2. This argument leads to an important design rule. Already in an early stage of the design of an aircraft it is essentual to ensure that, in addition to other requirements to be met, the neutral point, stick fixed, in the prescribed aircraft configurations and flight conditions will lie behind the rearmost envisaged center of gravity position. Experience has shown that during the development of an aircraft the center of gravity often shows the tendency to move a bit due to the successive design modifications. For this reason it may be wise to provide an additional margin when choosing the position of the neutral point, stick fixed, in the various aircraft configurations and flight conditions. In the next section some control characteristics will be discussed concerning turning flight. Requirements to be met by the control characteristics in such flight conditions will prove to influence also the forward and rear limits of the permissable center of gravity positions.
8-3
Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
The previous two sections dealt with the longitudinal control characteristics in steady straight flight. Control in turning and non steady flight, i.e. in manoeuvres, was not considered so far. It is by no means certain that an aircraft showing good control characteristics in steady, straight flight, also has similarly good control characteristics in manoeuvres. It is thus necessary to study these latter control characteristics and to this end express them quantitatively in appropriate terms. This way also allows to formulate quantitative requirements for desired control characteristics. Of the many possible turning manoeuvres only two of the greatest general interest are considered here: the pull-up from a dive and the steady, coordinated, i.e. slip-free, horizontal
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
413
Requirements according to
Rudder pedals
Elevator control
Aileron control
U.S. Civil (reference [7])
British Civil (reference [6])
82 kg (short intervals)
82 kg (short intervals)
9 kg (long intervals)
22.5 kg (long intervals)
35 kg (short intervals)
22.5 kg (short intervals)
4.5 kg (long intervals)
2.6 kg (long intervals)
27 kg (short intervals) 2.3 kg (long intervals)
note that a ‘short interval’ is regarded as a period of time lasting a few seconds while a ‘long interval’ is regarded as a period of time lasting several minutes Table 8-1: Maximum permissable values of the required control forces, according to the U.S. and British Civil Airworthiness Regulations
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Longitudinal Stability and Control in Steady Flight
turn. During these two manoeuvres the aircraft has an angular, pitching, velocity about the lateral axis. In turns the pitching velocity q about the lateral axis is always accompanied by a yawing velocity r about the top axis, but because of the symmetry of the aircraft the influence of the asymmetric component r on the symmetric forces and moments can be neglected, see chapter 4. An important purpose of the following discussion is to make longitudinal control in these manoeuvres accessible to quantitative study. Therefore, the control characteristics have to be expressed in characteristic variables, sufficiently simple to be manageable for the designer. To arrive at such characteristic variables some simplifying assumptions have to be made. Suppose the pilot of a normal stable aircraft wants to pull his aircraft up from a steady, straight, symmetric dive. In this context any descending flight will be called a dive. To achieve his purpose, he moves the elevator control back to a new position, which is assumed to be constant to simplify the discussion. The effect of this schematic control movement can be described as follows. Assuming a stable aircraft, first the pitching velocity q and the angle of attack α quickly increase to new approximately constant values depending on the ultimate control displacement. This motion occurs usually within a few seconds, often via a more or less well damped oscillation during which the airspeed remains very nearly constant. The aircraft motion, usually called the ‘short period oscillation’ will be discussed in more detail in chapter 10. Figure 8-30 gives an example of such a symmetric motion caused by a step elevator deflection. Figure 8-31 presents the various measured components, the elevator deflection and the control force for a pull-up manoeuvre of an Auster J-5B ‘Autocar’. Due to the increase in angle of attack the total lift on the aircraft becomes larger than the aircraft weight. This causes the trajectory of the cg to be curved, the aircraft is pulled up from the dive. If the pilot keeps the elevator angle constant, a second oscillation will occur. In contrast with the first oscillation, the second one is relatively slow and usually has a very low damping. After the second oscillation has also dampened out, the pitching velocity q is zero and the airspeed as well as the angle of attack have assumed the new constant values corresponding to the new elevator angle in steady, straight flight. In actual flight, when pulling up from a dive the pilot will suppress the second, long-period, oscillation by means of suitable, small corrective elevator movements. This second oscillation, usually called ‘the slow oscillation’, or ‘phugoid’ will also be discussed in more detail in chapter 10. In the following, only the first and relatively fast part of the motion is studied. The assumption is made that the airspeed remains constant at the level of the original steady flight condition. In addition to the assumption of a constant airspeed another simplification can be made. The normal load factor, n, is equal to 1 in the original straight flight. Due to the control displacement initiating the pull-up from the dive, the value of n increases. If the short-period oscillation is sufficiently damped, the normal load factor remains very nearly constant during a brief time interval, see figures 8-30 and 8-31. This fact allows the normal load factor to be considered constant in the following, just like the airspeed.
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
415
Figure 8-30: Response curves due to an elevator step deflection, Lockheed 1049 C ‘Super Constellation’
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Longitudinal Stability and Control in Steady Flight
Figure 8-31: Response curves due to an elevator step deflection, Auster J-5B ‘Autocar’ (from reference [23])
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
417
Due to these two assumptions the pull-up trajectory is idealized to a steady flight condition. For steady turns the two simplifying and approximating assumptions concerning the aircraft motion need not to be made. The aircraft has by definition in a steady turn not only a constant airspeed but also a constant normal load factor. In the following two notions will be derived characterizing longitudinal control in ‘steady’ pull-ups and in steady, horizontal, coordinated turns.
8-3-1
Characteristics of longitudinal control in turning flight
A notion characteristic for longitudinal control in turning flight should relate two aspects of the manoeuvre: on the one side the pilot’s action, the control displacement and the control force, and on the other hand a characteristic element of the resultant aircraft motion. For the latter the normal load factor n, mentioned before, is used, n=
N W
(8-55)
In the initial steady flight condition N = W , or n = 1. Using this load factor, the ‘stick displacement per g’ and the ‘stick force per g’ are obtained, where g is the acceleration due to gravity. Just as with the elevator control position stability, the elevator angle is used rather than the stick displacement itself, so, • stick displacement per g, • stick force per g,
dδe dn
dFe dn
The load factor in these two expressions is not only characteristic for the change in the trajectory of the aircraft, it is also closely related to the loads imposed on the aircraft. dFe e The two characteristics dδ dn and dn express the extent to which the control position and control force have changed after the aircraft has changed at constant airspeed from a condition of steady, straight flight to another condition of steady, but turning flight or to steady turn at a load factor n. For the case of the pull-up manoeuvre it can be argued that dδe dFe dn and dn should have a negative sign. This argument is as follows.
In order to achieve a pull-up from a dive at a certain airspeed, an increase in angle of attack is needed. This requires an increase in the angle of pitch of the aircraft. This is the initial aircraft motion, the transition from the steady dive to the ‘steady’ pull-up manoeuvre. For this initial motion a tail-heavy moment (∆Cm > 0) is needed and because Cmδ < 0, an initial backward elevator control movement and an elevator deflection ‘trailing edge up’ (∆δei < 0) are required.
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Longitudinal Stability and Control in Steady Flight
Maximum
Minimum
Trainer and fighter aircraft
− n25.4 L −1
− nL9.5−1
Transport aircraft and bombers
− n54.4 L −1
− n20.4 L −1
Table 8-2: Permissable values of the stick force per g in kg
It is argued in section 8-1-3 that a very general desirable control characteristic requires the final control displacement to have the same sign as the initial displacement. During the ‘steady’ pull-up manoeuvre at a constant, positive ∆n this general characteristic requires that also the final ∆δe is negative. This implies that the required sign is negative, dδe <0 dn Here it is argued that the change in control force required for a control displacement should e have the same direction as the control displacement itself. As a consequence, since dδ ds > 0, dFe <0 dn It is of course desirable that also in a turn both the change in control position and control force are negative. In the pull-up manoeuvre as well as in a turn, a backward control movement and an extra pull force should be needed. The stick force per g is generally assigned both a maximum and a minimum permissable value. If the stick force per g is too large in the absolute sense, the pilot will become tired too soon if he has to perform many manoeuvres. If, on the other hand, the stick force per g is too low, the aircraft is too sensitive to small unintentional variations in the control force. There is the additional risk that the aircraft may become overstrained due to a sudden control force. The dynamic characteristics of the aircraft are also important here. dFe e They have a direct bearing on the desired values of dδ dn and dn , see reference [86]. The following table 8-2 gives the maximum and minimum permissible values of the stick force per g, derived from the U.S. military requirements, see reference [19]. It will be seen that these values are related to the maximum permissable load factor n and thus to the strength of the aircraft.
A general rule is, that the absolute value of the stick force per g should not be lower than 1.4 kg. If the requirements of table 8-2 are applied to fighter aircraft having a nL = 6, the result is that the stick force per g should be between −5.1 kg and −1.9 kg. For transport aircraft,
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
419
dδe dn
ineffective
spongy
heavy and ineffective too light light and effective oversensitive
too heavy dFe dn
too sensitive Figure 8-32: Description of the character of the control feel for various ratio’s of the control force per g to the control displacement per g (from reference [170]).
dse dn
(cm) 2.0
1.5 poor 1.0 acceptable satisfactory
0.5
0
0
2
4
6
8
10
dFe dn
(kg)
Figure 8-33: The influence of the stick displacement per g and the stick force per g on the pilot’s opinion of the control characteristics at constant airspeed (from reference [86]).
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Longitudinal Stability and Control in Steady Flight
usually having a nL = 2.5, the boundaries are −32.2 kg and −13.6 kg. It is not sufficient to give a maximum and a minimum permissable value of the stick force per g. The relation between the stick displacement per g and the stick force per g is also important to achieve pleasant control characteristics. Figure 8-32 shows how various combinations of the stick displacement per g and the stick force per g were judged. Figure 8-33 is another example of the influences of the two criteria on the pilot’s opinion on the control characteristics of a fighter aircraft at constant airspeed. Quantitative requirements for the stick displacement per g do not exist. In the following the stick displacement per g and the stick force per g are expressed as functions of the aerodynamic coefficients and the mass of the aircraft for the two cases, 1. The idealized pull-up manoeuvre previously discussed, where n and V are assumed to be constant. The trajectory of the aircraft then is an arc of a circle in the vertical plane. The extent to which this idealized pull-up manoeuvre agrees with the actual motion of the aircraft may be judged from figures 8-30 and 8-31. It will be seen that n and V indeed remain very nearly constant over a brief time interval. 2. The steady, horizontal, coordinated turn.
8-3-2
The stick displacement per g
The calculation is based on the forces and moments acting on the aircraft both before and after the transition from the steady, straight flight to the idealized pull-up manoeuvre or the steady, horizontal turn. During both manoeuvres the airspeed is assumed to be constant. This implies, that the forces in X-direction are assumed to be in equilibrium and need not be further considered. In the steady initial condition is, N0 = W and, M0 = 0 or in non-dimensional form, CN0 = and,
W 1 2 2 ρV S
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
421
Cm0 = 0 The index 0 here indicates the steady initial condition. After the transition is, using equation (8-55), ∆CN =
W
∆n
1 2 2 ρV S
(8-56)
and, ∆Cm = 0
(8-57)
After the transition the angle of attack has increased an amount ∆α and the aircraft has obtained an angular velocity q about the lateral axis. The elevator angle has changed by ∆δe . The variables describing the aircraft motion now are, α,
q or
q¯ c and δe (V = constant) V
whereas in straight flight, considered so far, the variables are, α, V
and δe (
q¯ c = 0) V
In chapter 3 the assumption was made that V would have no effect on the aerodynamic coefficients. Due to the change in angle of attack the aircraft experiences an extra normal force CNα · ∆α and an extra moment Cmα · ∆α. The moment due to the elevator deflection is Cmδe · ∆δe . The pitching velocity q causes an aerodynamic force along the Z-axis. In section 6-4 a nondimensional stability derivative CZq was introduced, CZq =
∂CZ ∂CN q¯ c = − q¯ ∂V ∂ Vc
c Using this stability derivative, the extra force along the Z-axis can be expressed as CZq · q¯ V . q¯ c q¯ Here V is the non-dimensional pitching velocity, see also section 6-4. A positive force CZq · Vc is directed along the positive Z-axis (downward). Usually CZq is negative. The pitching velocity q gives also rise to an aerodynamic pitching moment about the lateral axis. This moment was expressed using another stability derivative, Cmq , also discussed in 6-4,
Cmq = The pitching moment due to q is Cmq ·
q¯ c V .
∂Cm c ∂ q¯ V
Usually Cmq is negative.
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Longitudinal Stability and Control in Steady Flight
From the foregoing follows for equations (8-56) and (8-57), q¯ c W = 1 2 ∆n V 2 ρV S
(8-58)
q¯ c + Cmδe ∆δe = 0 V
(8-59)
∆CN = CNα ∆α − CZq ∆Cm = Cmα ∆α + Cmq Equation (8-59) results in, ∆δe = −
1 Cmδe
Cmα ∆α + Cmq
q¯ c V
The stick displacement per g results from differentiating this latter expression with respect to n, dδe 1 =− dn Cmδe dα dn
Cmα
d q¯c dα + Cmq V dn dn
!
follows from equation (8-58), dα 1 = dn CNα
c CZq d q¯ W V + 1 2 CNα dn 2 ρV S
(8-60)
This results in, dδe 1 =− dn Cmδe
(
Cmα CNα
W + 1 2 2 ρV S
Cmα CZq + Cmq CNα
c d q¯ V dn
)
(8-61)
c The relation between the non-dimensional pitching velocity q¯ V and the normal load factor n is yet to be determined. A distinction has to be made here between the idealized pull-up manoeuvre and the steady, horizontal turn.
A. Derivation of
c d q¯ V dn
for the pull-up manoeuvre
In the pull-up manoeuvre the bottom of the trajectory is considered, see figure 8-34. In this lowest point of the trajectory, the Z-axis is very nearly vertical and the total force along the Z-axis is N − W . The centripetal acceleration along the Z-axis is V · q. As a consequence, N −W =mV q or, using equation (8-55), q¯ c g¯ c = 2 (n − 1) V V
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
423
q
center of rotation
R N
V
W Figure 8-34: The forces and angular velocity q in the idealized pull-up manoeuvre
This results in, c d q¯ g¯ c V = 2 dn V
(8-62)
In chapter 4 a non-dimensional measure of the aircraft’s mass µc was introduced. This socalled relative density is defined as, µc =
m W = ρ S c¯ g ρ S c¯
Using µc , equation (8-62) can be written as, c d q¯ 1 W V = 1 dn 2µc 2 ρV 2 S
B. Derivation of
c d q¯ V dn
(8-63)
for the steady, horizontal turn
In this steady flight condition the resultant force along the Z-axis, see figure 8-35, is N − W · cos ϕ. The centripetal acceleration along the Z-axis is again V · q, so, N − W cos ϕ = m V q Flight Dynamics
424
Longitudinal Stability and Control in Steady Flight N cosϕ
N
ϕ
q r Ω
W cosϕ
W YB
rear view
ZB
Figure 8-35: The forces and angular velocity in a steady horizontal turn
From figure 8-35 follows also, N cos ϕ = W This relation and equation (8-55) result in, q¯ c g¯ c = 2 V V
1 n− n
and after differentiation, c d q¯ 1 W V = 1 dn 2µc 2 ρV 2 S
1 1+ 2 n
(8-64)
c It follows from (8-64) that in steady, horizontal turns q¯ V is a non-linear function of n. If equation (8-63) is substituted in equation (8-61), the resulting expression for the stick displacement per g in the pull-up manoeuvre can be written as,
dδe 1 =− dn Cmδe
W 1 2 2 ρV S
Cmα CNα
CZq 1+ 2µc
Cmq + 2µc
(8-65)
Using equations (8-64) and (8-61) the resulting expression for the steady turns reads, dδe 1 =− dn Cmδe
W 1 2 2 ρV S
Cmα + CNα
Cmq Cmα CZq + CNα 2µc 2µc
1 1+ 2 n
(8-66)
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
425
A few can now be made. In equation (8-65) 2µc is relatively large if compared simplifications mα to CZq . Also, in equation (8-66) Cmq is large in the absolute sense, if compared to C CNα CZq . This permits both equations (8-65) and (8-66) to be simplified to, Pull-up manoeuvres
dδe 1 =− dn Cmδe
W 1 2 2 ρV S
Cmq Cmα + CNα 2µc
(8-67)
Steady turns
1 dδe =− dn Cmδe
W 1 2 2 ρV S
Cmq Cmα + CNα 2µc
1+
1 n2
(8-68)
It follows from equation (8-67) that in pull-up manoeuvres the control deflection and the elevator angle are linear functions of n. In steady, horizontal turns this is, according to equation (8-68), not the case. dδe dδe e Then dδ dn varies with n. The difference between dn pull−up and dn turns decreases with increasing n.
From equations (8-67) and (8-68) it follows also that both for the pull-up manoeuvres and W 1 e the turns dδ dn is proportional to the wingloading S and to 1 ρV 2 . Figures 8-36 and 8-37 show 2
measurements of the stick displacement per g in pull-up manoeuvres and steady turns of the Auster J-5B ‘Autocar’.
e For both types of manoeuvres dδ dn appears to have the desired negative sign. For this aircraft the relation between δe and n in pull-up manoeuvres is non-linear. From the measurements in figures 8-36 and 8-37 it is clear that the required control displacement for a steady turn is larger than for the pull-up manoeuvre at the same load factor n.
Flight Dynamics
426
Longitudinal Stability and Control in Steady Flight
A quick derivation of the stick displacement per g We start with the approximated equations of motion for constant speed, see chapter 10, α CZδ CZα 2uc + CZq . q¯c + δe = 0 Cmα Cmq Cmδ V We consider a small deviation from steady straight flight with α = ∆α, q = ∆q and ∆n = n − 1 = Vg γ˙ = Vg (θ˙ − α) ˙ = Vg (q − α). ˙ As we consider a steady state pull up manoeuvre we have α˙ = 0, so V q¯ c g¯ c ∆n = q, so = 2 ∆n g V V Substitution results in
CZα Cmα
2uc +C Zq Cmq
or with Axss +B∆δ e = 0, in which CZα A= Cmα This can be solved as: xss =
∆α g¯ c .∆n V2
.
2uc +C Zq Cmq
+
CZδ Cmδ
CZδ Cmδ
, and B =
= −A−1 B∆δe , or
By applying Cramer’s rule we get: CZα CZ δ Cm Cm ∆n α δ = CZα 2uc + CZq ∆δe Cm Cmq α or:
∆α g¯ c .∆n V2
"
∆α ∆δ e g¯ c .∆n V2 ∆δ e
.∆δ e = 0
#
.
= −A−1 B
CZα Cmδ − Cmα CZδ V2 = . CZα Cmq − Cmα (2uc + CZq ) g¯ c
CZα Cmq − Cmα (2uc + CZq ) g¯ ∆δe c = . 2 ∆n CZα Cmδ − Cmα CZδ V
When simplifying this expression by settingCZδ ≈ CZq ≈ 0, and replacing CZα by −CNα and W uc by gρS¯ c we get: Cmq dδe W ∆δe 1 Cmα = =− .( + ) 2 1 dn ∆n Cmδ /2ρV S CNα 2µc
8-3-3
The manoeuvre point, stick fixed
mα In equations (8-67) and (8-68) the first term between parantheses is C CNα . From the discussion of the static longitudinal stability and the neutral point stick fixed, from section 8-1-2 follows,
xcg − xnf ix Cmα = CNα c¯
(8-69)
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
427
Figure 8-36: The incremental elevator deflection ∆δe as a function of the incremental load factor ∆n in pull-up manoeuvres, Auster J-5B ‘Autocar’ (from reference [23]).
In equation (8-69) xnf ix is the x-coordinate of the neutral point, stick fixed, in gliding flight. e The first contribution to dδ dn then appears to be proportional to the distance of the center of gravity to the neutral point. As a consequence the stick displacement per g will decrease in the absolute sense if the cg is shifted backward. This can clearly be seen in figures 8-36 and 8-37. It follows that there is a certain cg position where the stick displacement per g is zero. This is the case for the pull-up manoeuvres, if, see equation (8-67), xcg − xnf ix Cmq Cmq Cmα + = + =0 CNα 2µc c¯ 2µc
(8-70)
The cg position at which the stick displacement per g is zero, is called the manoeuvre point, stick fixed, abbreviated as m.p.f ix . The x-coordinate of this point is xmf ix . From equation (8-70) follows, if xcg = xmf ix , xmf ix − xnf ix Cmq =− c¯ 2µc
(8-71)
Here Cmq has been assumed independent of the cg position, see chapter 6. Combining equations (8-69) and (8-71) results in, xcg − xmf ix Cmq Cmα = + c¯ CNα 2µc
(8-72) Flight Dynamics
428
Longitudinal Stability and Control in Steady Flight
Figure 8-37: The incremental elevator angle ∆δe as a function of the incremental load factor in turns, Auster J-5B ‘Autocar’ (from reference [23])
Next, equation (8-72) is substituted in equation (8-67). This results in an expression for the stick displacement per g as a function of cg position relative to the m.p.f ix , dδe 1 =− dn Cmδe
W 1 2 2 ρV S
xcg − xmf ix c¯
(8-73)
But the manoeuvre point, stick fixed, is not just the cg position at which the stick displacement per g vanishes. The manoeuvre point can be interpreted in yet another way, as explained in the following. The normal force N can generally be written as, see equation (8-55), N =nW Then, dCN W = 1 2 dn 2 ρV S Substitution of equation (8-74) in equation (8-73) results in,
(8-74)
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
429
dδe 1 dCN xcg − xmf ix =− dn Cmδe dn c¯ or, Cmδe dδe = −dCN
xcg − xmf ix c¯
(8-75)
Here Cmδe · dδe is the change in the pitching moment due to the elevator deflection dδe and dCN is the change in the normal force coefficient. According to equation (8-75) Cmδe · dδe xcg −xm
xcg −xm
f ix f ix must be balanced by dCN · . This means that is the arm of the force dCN . c¯ c¯ Evidently, the second interpretation of the manoeuvre point, stick fixed, is the point where the resultant change in normal force acts after the transition from a condition of steady, straight flight to a ‘steady’ pull-up manoeuvre, if the elevator angle were kept constant during this transition.
Since Cmq is negative, it follows from equation (8-71) that the m.p.f ix always lies behind the n.p.f ix . With increasing µc , i.e. with increasing flight altitude, the manoeuvre point moves forward. This can also be seen in figure 8-38 where the calculated position of the m.p.f ix of the Fokker F-27 ‘Friendship’ is shown. Subfigure a of figure 8-38 clearly shows the forward shift of m.p.f ix and also the limiting Cm position: n.p.f ix . The latter position is attained when the term 2µcq becomes vanishingly small. Thus far the manoeuvre point was discussed only in relation to the pull-up manoeuvres. Both interpretations of the manoeuvre point hold equally for the steady turns. The difference, however, is that then the position of the manoeuvre point is a function also of the load factor n. This can be seen as follows. From equations (8-68) and (8-69) follows for the steady turn, xmf ix − xnf ix Cmq =− c¯ 2µc
1 1+ 2 n
(8-76)
and by consequence, xcg − xmf ix Cmq Cmα = + c¯ CNα 2µc
1 1+ 2 n
(8-77)
The expression (8-73) is valid also for the stick displacement per g in steady turns, although the actual position of the manoeuvre point will be different from the position in the pull-up manoeuvre. From equation (8-76) it can be seen that xmf ix now depends not only on the flight altitude, but also on the load factor. If n = 1, xmf ixpull−up − xnf ix xmf ixturns − xnf ix =2 c¯ c¯
(8-78)
With increasing n the manoeuvre point in turns shifts forward. At high values of n the manoeuvre points in the pull-up and in the turn very nearly coincide. This can be seen also Flight Dynamics
430
Longitudinal Stability and Control in Steady Flight
Figure 8-38: Calculated positions of the manoeuvre point, stick fixed, and the stick displacement per g of the Fokker F-27 ‘Friendship’
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
431
in subfigure b of figure 8-38. It should be remembered that n = 2 in a steady, horizontal turn corresponds to an angle of roll ϕ = 60o , (n = cos1 ϕ ). Apart from the stick displacement per g, the stick force per g is of particular interest. It is discussed in the following section.
8-3-4
The stick force per g
The elevator control force was written in section 5-2-7 as, see equation (5-83), Fe = −
dδe 1 2 ρVh Se c¯e Chα αh + Chδ δe + Chδt δte dse 2
The stick force per g is the derivative of the elevator control force with respect to the load factor in a pull-up manoeuvre or a steady turn. It is a measure of the change in elevator control force after the transition from a condition of steady, straight flight to a ‘steady’ pullup manoeuvre or a steady turn. In equation (5-83) both αh and δe are functions of the load factor n. It is assumed that the trim tab angle does not vary in the pull-up manoeuvre or the turn, δte is constant. The airspeed V and thus also Vh in equation (5-83) are assumed to be constant. The stick force per g is then obtained from equation (5-83), dδe 1 2 dFe =− ρV Se c¯e dn dse 2 h
Chα
dαh dδe + Chδ dn dn
(8-79)
e The stick displacement per g, the factor dδ dn on the right hand side of equation (8-79), has dαh e already been determined in section 8-3-2. In order to find dF dn the derivative dn needs to be determined.
The angle of attack αh of the horizontal tailplane varies with n in the steady manoeuvres c because both α and q, q¯ V , vary with n and, q¯ c αh = f α, V
The partial variation of αh with α alone was already discussed in section 5-2-4. For steady, straight flight it was found that, see equation (5-65),
dε αh = (α − α0 ) 1 − dα
+ (α0 + ih )
The partial variation of αh with q alone will now be derived. In the idealized pull-up manoeuvre and in the steady turn the symmetric motion of the aircraft relative to the air can be considered as a pure rotation, with angular velocity q, about a center of rotation situated on the Z-axis above the center of gravity, see figure 8-39. This position of the center of rotation is such that the local angle of attack at the center of gravity does not change due to the rotation. When in chapter 4 the stability derivatives with respect to pitching velocity were discussed, it was shown that the principal effect of such a ‘q-motion’ is a variation of the local, geometric Flight Dynamics
432
Longitudinal Stability and Control in Steady Flight q
center of rotation ∆α =
x−xcg R
=
R=
x−xcg q¯ c c¯ V
∆α
Zr
V
V q
∆α
xcg x Xr
Figure 8-39: The variation of the angle of attack in an arbitrary point of the aircraft caused by a pure q-motion
angle of attack, proportional to the angular velocity q and the distance in x-direction to the center of gravity, see figure 8-39, ∆α =
x − xcg q¯ x − xcg c = R c¯ V
(8-80)
At the horizontal tailplane the change in geometric angle of attack due to a q-motion is, xh − xcg q¯ c lh q¯ c ≈ c¯ V c¯ V
∆αh =
(8-81)
In steady, turning flight the total αh is found by adding equation (5-65) and equation (8-81),
dε αh = (α − α0 ) 1 − dα
+ (α0 + ih ) +
lh q¯ c c¯ V
(8-82)
From equation (8-82) follows the derivative with respect to n, dαh = dn The derivatives (8-64).
dα dn
and
c d q¯ V dn
dε 1− dα
q¯ c dα lh d V + dn c¯ dn
(8-83)
were aready obtained previously, see equations (8-60), (8-63) and
All variables needed to determine and (8-64) results in,
dαh dn
are now known. Substituting equations (8-60), (8-63)
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
433
Pull-up manoeuvres dαh W = 1 2 dn ρV S 2
dε 1− dα
lh 1 1 + CNα c¯ 2µc
(8-84)
Steady turns dαh W = 1 2 dn 2 ρV S
dε 1− dα
lh 1 1 + CNα c¯ 2µc
1 1+ 2 n
(8-85)
All factors in (8-79) are now known. Substituting and using the expressions already derived in sections 5-2-6 and 8-2-1, i.e. equation (5-76),
Cmδe = −CNhδ
Vh V
2
Sh lh S¯ c
equation (8-35),
Cmαf ree = CNwα
xcg − xw − CNhα f ree c¯
dε 1− dα
Vh V
2
Sh lh S¯ c
and equation (8-33), CNhα
= CNhα − CNhδ
f ree
Chα Chδ
the stick force per g is obtained after some reformulation, Pull-up manoeuvres
Vh V
dFe dδe W = dn dse S
dFe dδe W = dn dse S
2
Chδ Se c¯e Cmδe
(
Cmαf ree CNα
1 + 2µc
Cmq − Cmδe
Chα lh Chδ c¯
)
(8-86)
Steady turns Vh V
2
Chδ Se c¯e Cmδe
(
Cmαf ree CNα
1 1+ 2 n
1 + 2µc
Cmq − Cmδe
Chα lh Chδ c¯
· (8-87)
The first term between the brackets of the expressions (8-86) and (8-87) is due to the static stability, stick free. The second term between the brackets in equation (8-86) can be simplified Flight Dynamics
434
Longitudinal Stability and Control in Steady Flight
considerably. From equation (8-81) It can be seen that lc¯h is a measure of the change in angle of attack of the horizontal tailplane caused by the q-motion, i.e., dαh lh q¯ c = c¯ dV
(8-88)
The variation of the elevator angle with the angle of attack of the horizontal tailplane in the stick free situation was expressed in equation (8-27),
dδe dαh
f ree
=−
Chα Chδ
Combining equations (8-88) and (8-27) results in, dδe c d q¯ V
!
=
f ree
dδe dαh
Chα lh dαh q¯ c =− Chδ c¯ dV
f ree
C
Evidently, the term −Cmδe · Chhα · lc¯h in equations (8-86) and (8-87) represents the moment δ due to the q-movement, caused by freeing the elevator for,
∆
dCm c d q¯ V
!
= ∆Cmq = Cmδe
dδe c d q¯ V
!
f ree
= −Cmδe
Chα lh Chδ c¯
(8-89)
If a new derivative, Cmq stick free, is introduced it can be written as, Cmqf ree = Cmqf ix + ∆Cmq or, with equation (8-89), Cmqf ree = Cmq − Cmδe
Chα lh Chδ c¯
(8-90)
With equations (8-90), (8-86) and (8-87) this can be written as, Pull-up manoeuvres dFe dδe W = dn dse S
Vh V
2
Chδ Se c¯e Cmδe
(
Cmαf ree CNα
+
Cmqf ree 2µc
)
(8-91)
Steady turns dFe dδe W = dn dse S
Vh V
2
Chδ Se c¯e Cmδe
(
Cmαf ree CNα
+
Cmqf ree 2µc
1 1+ 2 n
)
(8-92)
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
435
From equations (8-91) and (8-92) the influence of the cg position on the stick force per g can be understood. If the cg moves aft, Cmαf ree decreases in the absolute sense and as a consequence the stick force per g decreases in the absolute sense.
From the above expressions it will be seen that contrary to the stick displacement per g the stick force per g is independent of airspeed if the aerodynamic derivatives in equations (8-91) and (8-92) are invariant with airspeed. The variation of the stick force per g with the cg position is also evident from the measurements shown in figures 8-40 and 8-41. The stick force per g in steady turns is indeed, in the absolute sense, larger than in pull-up manoeuvres at the same load factor n. The relatively large scatter in the data points, visible mainly in figure 8-40, is thought to be caused by friction in the control system.
A quick derivation of stick force per g Starting from the approximated equations of motion for constant speed, see chapter 10, CZα 2uc + CZq α CZδ . q¯c + .δe = 0 Cmα Cmq Cmδ V we substitute α = ∆α, q = ∆q and ∆n = n − 1 = Vg γ˙ = Vg (θ˙ − α) ˙ = in a steady state pull up: V q¯ c g¯ c ∆n = q, so = 2 ∆n g V V
V g
(q − α), ˙ knowing that
which results in:
CZα Cmα
2uc + CZq Cmq
∆α CZδ . + .∆δe = 0 c ∆ q¯ Cmδ V
The corresponding stick force is : ∆Fe = −
dδe 1 . /2ρVh2 Se c¯e {Chα .∆αh + Chδ .∆δe } = K.{Chα .∆αh + Chδ .∆δe } dse
, with K=−
dδe 1 . /2ρVh2 Se c¯e dse
and ∆αh = (1 − So ∆δe =
dε ℓh q¯ c ).∆α + .∆ dα c¯ V
∆Fe 1 Ch dε Ch ℓh q¯ c − α (1 − ).∆α − α . ∆ K Chδ Chδ dα Chδ c¯ V Flight Dynamics
436
Longitudinal Stability and Control in Steady Flight
Substituting this in the equations of motion results in: CZα 2uc + CZq ∆α . + c Cmα Cmq ∆ q¯ V Ch dε ∆Fe 1 Ch ℓh q¯ c CZδ − α (1 − ).∆α − α .∆ } = .{ Cmδ K Chδ Chδ dα Chδ c¯ V C C dε CZα − CZδ Chhα (1 − dα ) 2uc + CZq − CZδ Chhα ℓc¯h ∆α δ δ . + c C C dε ∆ q¯ Cmα − Cmδ Chhα (1 − dα ) Cmq − Cmδ Chhα ℓc¯h V δ δ ∆Fe 1 CZδ . =0 Cmδ K Chδ Now we introduce: dε Chα (1 − )=CZα f ree Chδ dα Ch ℓh CZq − CZδ α = CZq f ree Chδ c¯ Ch dε Cmα − Cmδ α (1 − )=Cmα f ree Chδ dα Ch ℓh Cmq − Cmδ α = Cmq f ree Chδ c¯ CZα − CZδ
resulting in: "
CZα f ree 2uc + CZq f ree Cmα f ree Cmq f ree
# ∆F 1 ∆α CZδ + . =0 . q¯ c Cmδ ∆V K Chδ
or: A.∆xss + B.
∆F 1 =0 K Chδ
Of which the solution is: −1
∆xss = −A
∆F 1 B with ∆xss = K Chδ
∆α c ∆ q¯ V
c We may solve for ∆ q¯ V as:
∆
=
CZα f ree CZδ Cm Cmδ α f ree
q¯ c = V CZα f ree 2uc + CZq f ree Cmq f ree Cmα f ree Cmδ .CZα f ree − CZδ .Cmα f ree
∆F 1 . K Chδ
∆F 1 CZα f ree Cmq f ree − Cmα f ree (2uc + CZq f ree ) K Chδ .
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
437
With good approximation we may set CZq = CZδ = 0, so that CZαf ree = CZα and CZqf ree = CZq = 0 so q¯ c ∆Fe 1 Cmδ .CZα ∆ = . V CZα .Cmqf ree − 2µc Cmαf ree K Chδ c and using ∆ q¯ V =
g¯ c V 2 ∆n
and CZα = −CNα : ( ) Cmαf ree Cmqf ree ∆Fe g¯ c Chδ = K.2µc 2 . + = ∆n V Cmδ CNα 2µc ( ) Cmαf ree Cmqf ree c Chδ dδe 1 2.W g¯ 2 . . /2ρVh Se c¯e . + − dse gρS¯ c V 2 Cmδ CNα 2µc
or: Ch dFe dδe W Vh 2 =− .( ) Se c¯e . δ dn dse S V Cmδ
8-3-5
(
Cmαf ree CNα
+
Cmqf ree 2µc
)
The manoeuvre point, stick free
In section 8-3-3 it was shown that at a certain cg position corresponding with the manoeuvre point, stick fixed, the stick displacement per g becomes zero. For the stick force per g also a e certain cg position exists where dF dn is zero. This point is called the manoeuvre point, stick free. It is also indicated as m.p.f ree , the abscissa of this point is xmf ree . From the discussion in section 8-2-2 follows, Cmαf ree CNα If xcg = xmf ree then
dFe dn
=
xcg − xnf ree c¯
(8-93)
= 0, and, Cmαf ree CNα
+
Cmqf ree 2µc
=0
For the pull-up manoeuvre the above can be written as, see equations (8-91) and (8-93), Cmqf ree xmf ree − xnf ree =− c¯ 2µc
(8-94)
Since Cmqf ree is again negative, the manoeuvre point, stick free, lies behind the neutral point, stick free in gliding flight. According to equations (8-93) and (8-94) is for the pull-up manoeuvres, Cmαf ree Cmqf ree xcg − xmf ree = + c¯ CNα 2µc
(8-95) Flight Dynamics
438
Longitudinal Stability and Control in Steady Flight
Figure 8-40: The incremental control force ∆Fe as a function of the incremental load factor ∆n in pull-up manoeuvres, Auster J-5B ‘Autocar’ (from reference [23])
Figure 8-41: The incremental control force ∆Fe as a function of the incremental load factor ∆n in turns, Auster J-5B ‘Autocar’ (from reference [23])
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
439
For the stick force per g follows then with equations (8-91) and (8-95), dFe dδe W = dn dse S
Vh V
2
Se c¯e
Chδ xcg − xmf ree Cmδe c¯
(8-96)
For the steady turns it can be derived in the same way, Cmqf ree xmf ree − xnf ree =− c¯ 2µc
1 1+ 2 n
(8-97)
and consequently, Cmqf ree Cmαf ree xcg − xmf ree 1 = + 1+ 2 c¯ CNα 2µc n
(8-98)
This means that equation (8-96) is also true for the stick force per g in steady turns, although the position of the manoeuvre point, stick free, differs for the two types of manoeuvres, according to equations (8-94) and (8-97). Just as in the case of the manoeuvre point, stick fixed, a second interpretation can be given of the manoeuvre point, stick free. To this end, the transition from a condition of steady, straight flight to a manoeuvre is separated in two phases. These will be considered separately one after another. In the first phase, the aircraft is supposed to change with free elevator, from steady, straight flight to a turning manoeuvre. As far as the forces along the Z-axis are concerned this is a steady manoeuvre: equilibrium exists along the Z-axis. The change in normal force dCNf ree due to this transition acts in a certain point, the abscissa of which is provisionally indicated as xdCNf ree . The accompanying change in the pitching moment due to this transition then is, dCmf ree = dCN
xcg − xdCNf ree c¯
The second and subsequent phase takes place at constant α, q and V . In this phase equilibrium of the pitching moment is obtained. The moment dCmf ree is balanced by applying an appropriate elevator deflection dδe and the control force dFe . The required elevator angle dδe is, dδe = −
xcg − xdCNf ree dCmf ree 1 =− dCNf ree Cmδe Cmδe c¯
(8-99)
The required elevator control force dFe follows from the change in the hinge moment occuring in the second phase. Note that in the first phase the elevator was free: Fe = 0. Thus, dFe = −
dδe 1 2 ρV Se c¯e dChe dse 2 h
(8-100) Flight Dynamics
440
Longitudinal Stability and Control in Steady Flight
Figure 8-42: Calculated positions of the manoeuvre point, stick free, and the stick force per g of the Fokker F-27 ‘Friendship’
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
altitude
Chα < 0
xmf ree
441
Chα = 0
Ch α > 0
xmf ree
xmf ree = xmf ix
h1
xnf ree
xnf ree = xnf ix
x c¯
xnf ree
dFe (−) dn
Chα < 0
Chα = 0
0 h1
Chα > 0
0 h1 xmf ree = xmf ix
0 h1 xmf ree = xmf ix
x c¯
Figure 8-43: Influence of altitude and magnitude and sign of Chα on the position of the manoeuvre point, stick free, and the stick force per g
Flight Dynamics
442
Longitudinal Stability and Control in Steady Flight
Since αh is constant in the second phase (α and q remain constant), dChe is, dChe = Chδ dδe
(8-101)
Substituting dδe according to equation (8-99) in equation (8-101) and subsequently in equation (8-100) results in,
dFe =
xcg − xdCNf ree Chδ dδe 1 2 ρVh Se c¯e dCNf ree dse 2 Cmδe c¯
(8-102)
Next, the change in elevator control force dFe is considered, following from the expression (8-96) for the stick force per g. With equation (8-74), where dCN = dCNf ree , from equation (8-96) follows for both types of manoeuvres, dFe =
xcg − xmf ree dδe 1 2 Chδ ρVh Se c¯e dCNf ree dse 2 Cmδe c¯
(8-103)
If the expressions (8-102) and (8-103) are now compared, it will be clear that the manoeuvre point, stick free, in equation (8-103) corresponds with the point of action of the change in normal force dCNf ree in equation (8-102), caused by the transition with free elevator from steady, straight flight to a manoeuvre. This explains the second interpretation of the manoeuvre point, stick free. Using equations (8-94) and (8-97) the position of the manoeuvre point was calculated for the Fokker F-27. The results are shown in figure 8-42. The Chα of the elevator of this aircraft is approximately zero. As a result, there is no difference between the situations: control free and control fixed. It can be seen from the figure that the manoeuvre point moves forward with increasing flight altitude, approaching in the limit the neutral point. With increasing load factor n the manoeuvre point in steady turns approaches the manoeuvre point in pull-up manoeuvres. Figure 8-43 indicates schematically how the position of the manoeuvre point, stick free, and dFe dn change with flight altitude and with Chα . Figure 8-44 shows the influence of Chδ on the range of permissible center of gravity positions. The discussion in section 8-3-1 referred to the fact that the stick force per g has an upper and a lower limit of permissable values. Other requirements remain also in force, such as those relating to the elevator control position and force stability. As a consequence it is sometimes not an easy matter to satisfy all requirements on the stability and control characteristics in all aircraft configurations and flight conditions stipulated in the regulations.
8-3-6
Non-aerodynamic means to influence the stick force Per g
Sometimes springs or unbalanced masses are installed in the control mechanism, to influence the elevator control force stability or the stick force per g or both.
8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns
443
Figure 8-44: Increasing the range of permissible cg positions by decreasing |Chδ |
In section 8-2-4 the influence was discussed of a spring or an unbalanced mass on the elevator control force stability and the position of the neutral point, stick free. The spring was assumed to exert a constant hinge moment. Such a spring has no influence on the stick force per g, but an unbalanced mass has. Suppose, a mass has been installed in the control mechanism, see figure 8-45, such that in straight and level flight a static hinge moment Hew is generated. In a manoeuvre at a load factor n an extra hinge moment (n − 1) · Hew occurs. This hinge moment must be balanced by an extra control force ∆Fe , ∆Fe = −
dδe ∆n Hew dse
The influence of the mass, sometimes called a ‘bobweight’, on the stick force per g is,
∆
dFe dδe =− H ew dn dse
(8-104)
In this way it is seen that a modification of the stick force per g can be obtained which is e independent of cg position and flight altitude. In some cases this may bring dF dn within the required limits. By choosing the right combination of springs and bobweights it is possible in principle to vary both the elevator control force stability and the stick force per g independent of one another using non-aerodynamic means. If, however, the hinge moment generated by springs or bobweights or both becomes too large, the dynamic stability, stick free, may be influenced Flight Dynamics
444
Longitudinal Stability and Control in Steady Flight
Figure 8-45: The incremental hinge moment due to a bobweight in the control mechanism in a pull-up manoeuvre at a load factor n
unfavorably. In addition, the possibility of the occurrence of flutter has to be investigated. It was shown in this chapter that the stick displacement per g and the stick force per g must have a negative sign, whereas the magnitude of the stick force per g is additionally limited to lie within a certain range. These requirements may restrict both the forward and the rear limits of permissible cg positions. This remark is supplementary to what was noted in section 8-2-7. There the rear limit of cg positions was said to be determined by the requirement that the aircraft must possess elevator control position and elevator control force stability. The forward limit of permissible cg positions was there shown to be dictated by the available elevator power in take-off or landing.
Chapter 9 Lateral Stability and Control in Steady Flight
9-1
Introduction
Using the equilibrium equations for asymmetric flight, the lateral control characteristics in some important steady, asymmetric flight conditions will be discussed in the following. To simplify the discussion only horizontal flight will be considered. In a condition of steady flight, in which the roll angle is constant by definition, the rolling velocity about the XB -axis is zero. This is the case when the XB -axis, along the velocity vector of the c.g., lies in the horizontal plane, see equation (3-158) on page 116 From the above follows that in steady, asymmetric horizontal flight the rolling velocity is always zero. The ‘steady’ rolling flight to be considered in section 9-6, is an approximated, quasi-steady flight condition giving in a simple manner insight in some important requirements on roll control for aircraft.
9-2
Equations of Equilibrium
In the most general case of a horizontal, steady, asymmetric condition of flight, the aircraft rb sideslips and yaws at constant β and 2V . The angle of roll ϕ of the aircraft differs from zero but the rolling velocity is zero. To maintain this flight condition, generally the ailerons as well as the rudder are deflected. Not only the aerodynamic force Y along the YB -axis and the aerodynamic moments L and N about the XB - and ZB -axis respectively act on the aircraft, but also the component of the weight along the YB -axis. Since the pitch angle θ, measured in the stability reference frame, is zero in horizontal flight, the component of the weight along the YB -axis (or YS -axis) is, Flight Dynamics
446
Lateral Stability and Control in Steady Flight
W sinϕ Y
r
ϕ
W cosϕ W YB
ZB
Figure 9-1: The forces along the YB -axis of an aircraft in steady, horizontal, asymmetric flight
W sin ϕ The combined forces along the YB -axis cause a centripetal acceleration in the YB -direction: V · r, see figure 9-1. The resultant equation for the forces along the YB -axis, both in the aircraft and the stability reference frame, then is, W sin ϕ + Y = m V r
(9-1)
whereas the equilibrium of the moments about the XB - and ZB -axes is expressed by, L=0
(9-2)
N =0
(9-3)
The equations are made non-dimensional by dividing equation (9-1) by 12 ρV 2 S, and both equations (9-2) and (9-3) by 12 ρV 2 Sb. This results for small values of ϕ in, CL ϕ − 4µb
rb 2V
+ CY
= 0
Cℓ = 0 Cn = 0 where,
(9-4)
9-2 Equations of Equilibrium
447
W
CL =
1 2 2 ρV S
and µb =
m ρSb
is the non-dimensional mass parameter for the asymmetric motions, see chapter 4. If the aerodynamic forces and moments in equation (9-4) are now expressed in the contrirb butions arising from β, 2V , δa and δr , according to equations (7-2) and (7-3), the resulting asymmetric equilibrium equations for horizontal steady asymmetric flight become, CL ϕ + CYβ β + (CYr − 4µb ) Cℓβ β
rb 2V rb 2V
+Cℓr
Cnβ β
+Cnr
rb 2V
+ CYδa δa + CYδr δr = 0 + Cℓδa δa + Cℓδr δr = 0
(9-5)
+ Cnδa δa + Cnδr δr = 0
Based on the discussions in chapter 7, the control derivatives CYδa and Cℓδr will be neglected in the following. In a first approximation used to study the various types of steady flight, CYr , Cnδa and CYδr will also be dropped. In the thus simplified form the equilibrium equations read as, CL ϕ + CYβ β − 4µb
rb =0 2V
(9-6)
Cℓβ β + Cℓr
rb + Cℓδa δa = 0 2V
(9-7)
Cnβ β + Cnr
rb + Cnδr δr = 0 2V
(9-8)
Or, using matrix notation,
CL CYβ 0 Cℓ β 0
Cnβ
−4µb
0
0
Cℓr
Cℓδa
0
Cnr
0
Cnδr
ϕ
β rb 2V = 0 δ a δr
Flight Dynamics
448
Lateral Stability and Control in Steady Flight
In section 9-3 to 9-5 the lateral control characteristics of the aircraft in various steady, asymmetric flight conditions are studied, using the above equations. Because of the simplifications introduced, the considerations are primarily qualitative in nature, especially for larger deviations from symmetric flight.
9-3
Steady Horizontal Turns
In the three equations (9-6), (9-7) and (9-8) five variables occur. This implies, that steady rb turns at a given airspeed and yaw-rate 2V can be flown in principle at infinitely many combinations of the remaining four variables δr , δa , β and ϕ. Only if one of these four variables has rb . In the following steady been fixed, the remaining three can be expressed as functions of 2V turns are studied in which each of the four variables mentioned are assumed separately to be equal to zero.
9-3-1
Turns Using the Ailerons Only, δr = 0
Using equations (9-7) and (9-8) and the condition δr = 0, it follows that the variation of the aileron angles with rate of yaw can be expressed as, dδa 1 Cℓβ Cnr − Cℓr Cnβ = rb Cℓδa Cnβ d 2V
(9-9)
Here Cℓδa < 0 and Cnβ > 0. To initiate a turn to the right using the ailerons only, a negative aileron deflection must be given to obtain a positive rolling velocity and a positive angle of roll. Using the same arguments as in chapter 8, it is desirable that in the ultimate steady flight condition the aileron control remains deflected in the direction of the initial control deflection, so ddδrba < 0. According to equation (9-9) the necessary condition for this is, Cℓβ Cnr − Cℓr Cnβ > 0
2V
(9-10)
This latter condition corresponds to the condition for spiral stability which is discussed in chapter 11. If δr = 0 in the expressions (9-6) and (9-8), the variation of angle of roll with rate of yaw can be derived as, 4µb + CYβ dϕ = rb CL d 2V
Cnr Cnβ
>0
(for
CL > 0)
(9-11)
9-3 Steady Horizontal Turns
449 δa [o ] 2
-12
-8
-4
0
4
8
rb 2V
· 103
0
4
8
rb 2V
· 103
0
4
8
rb 2V
· 103
-2
β [o ] 1
-12
-8
-4
-1
ϕ [o ] 60
40
20
0 -12
-8
-4
-20
-40
-60
Figure 9-2: Steady turns using ailerons only, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) Flight Dynamics
450
Lateral Stability and Control in Steady Flight
and also, dβ Cn =− r >0 rb Cnβ d 2V
Cnr < 0,
Cnβ > 0
(9-12)
In a turn to the right, with δr = 0, the aircraft has a roll angle to the right (ϕ > 0) and the rb sideslip is towards the inside of the turn (β > 0). At practical values of ϕ and 2V the angle of sideslip usually remains limited to a few degrees. Figure 9-2 shows the results of some measurements made in steady flight, using the ailerons only.
9-3-2
Turns Using the Rudder Only, δa = 0
Using equations (9-7) and (9-8) and the condition δa = 0, the variation of the rudder angle with rate of yaw is obtained as, dδr 1 Cℓβ Cnr − Cℓr Cnβ =− rb Cnδr Cℓβ d 2V
(9-13)
Here Cnδr < 0 and Cℓβ < 0. It then follows that if the condition equation (9-10) for spiral stability is satisfied, in a turn to the right the rudder is also deflected to the right (δr < 0). In analogy with equations (9-11) and (9-12) the following expressions for the variations of angle of roll and angle of sideslip hold, 4µb + CYβ dϕ = rb CL d 2V
Cℓr Cℓβ
>0
(for
CL > 0)
(9-14)
and, dβ Cℓ =− r >0 rb Cℓβ d 2V
Cℓr > 0,
Cℓβ < 0
(9-15)
In a turn to the right, using the rudder only, the aircraft assumes again a positive angle of roll (to the right) and a positive, usually very small, angle of sideslip. Figure 9-3 shows the results of some measurements made in steady flight using the rudder only. As in figure 9-2, the control surface deflection and the angle of sideslip remain very small, even at angles of roll of ±40o . As a consequence, the influence of the limited accuracy of the measurements and possible non-linearities is large enough to obscure a clear relation between the slopes of some measured curves and the expressions derived in this section, such as equation (9-15).
9-3-3
Coordinated Turns, β = 0
When flying a steady turn, the pilot’s aim is in principle to keep the angle of sideslip zero. Then the drag is at a minmum. In additiion, the coordinated turn is the most comfortable turn for passengers.
9-3 Steady Horizontal Turns
451 δr [o ] 1
-12
-8
-4
0
4
8
rb 2V
· 103
0
4
8
rb 2V
· 103
0
4
8
rb 2V
· 103
-1
β [o ] 1
-12
-8
-4
-1
ϕ [o ] 60
40
20
0 -12
-8
-4
-20
-40
-60
Figure 9-3: Steady turns using rudder only, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14]) Flight Dynamics
452
Lateral Stability and Control in Steady Flight
If β = 0, it follows from equation (9-6) that, CL ϕ = 4µb
rb 2V
(9-16)
This means that in the coordinated turn the component along the YB -axis of the centripetal acceleration is caused only by the lateral component of the aircraft weight. This is true, also for all objects in the aircraft. They do not experience a side force during the coordinated turn. As a consequence the pilot can also feel if he flies a well coordinated turn. The variation of the roll angle with rate of yaw follows from equation (9-16), dϕ 4µb = >0 rb CL d 2V
(for
CL > 0)
(9-17)
The required control surface deflections follow from equations (9-7) and (9-8) with β = 0, dδa Cℓ =− r >0 rb Cℓδa d 2V
Cℓr > 0,
Cℓδa < 0
(9-18)
Cn dδr =− r <0 rb C d 2V nδr
Cnr < 0,
Cnδr < 0
(9-19)
and,
In a coordinated turn to the right the aircraft has an angle of roll to the right. The rudder is deflected in the desirable negative direction (to the right). But the aileron is deflected to the left, in the direction opposite to that of the initial control deflection. As a matter of fact, this is less desirable, however it is inevitable in normal aircraft. Figure 9-4 shows the results of measurements made in steady, coordinated turns. The aileron angle turns out to be very nearly constant. Quantitative calculations show very simply that the control surface deflections in each of the three types of steady turns just described remain usually small, see also figures 9-2, 9-3 and 9-4. It follows then that these flight conditions are not at all critical for the sizing of the control surfaces. In cruising flight and usually also in the approach the differences between the coordinated turn (β = 0) and a turn using the ailerons only (δr = 0) is so small as to make it attractive to the pilot to perform the lateral control using the roll control only. If the initiation of a turn is done not too quickly, the adverse yaw of a conventional aircraft will be small enough to render the use of the rudder unnecessary. Obviously, if the rudder need not be employed, the control of the aircraft is much easier, thereby improving the accuracy of the control of the aircraft.
9-4 Steady, Straight, Sideslipping Flight
9-3-4
453
Flat Turns
From equation (9-6) follows, if ϕ = 0, dβ 1 = 4µb < 0 rb C d 2V Yβ
CYβ < 0
Using equation (9-20) it is simple to derive that, dδa >0 rb d 2V
(9-20)
dδr <0 rb d 2V
In a flat turn to the right, the rudder angle is negative, just as in a turn to the right using the rudder only. In a flat turn the aircraft must perform a sideslip to the outside of the turn (β < 0 if r > 0) in order to obtain the lateral force required for the centripetal acceleration. Due to the combined sideslipping and yawing motions, a positive rolling moment (to the right) is generated. This requires a positive aileron deflection (to the left) for equilibrium about the XB -axis. Figure 9-5 shows the results of some measurements made in steady, flat turns.
9-4
Steady, Straight, Sideslipping Flight
In general the pilot will try to avoid sideslipping in flight. The intentional use of steady, straight, sideslipping flight is limited to landings with cross wind and to the approach in flapless aircraft in order to control the flight path angle. A further discussion of steady, straight sideslipping flight is justified, however, because the aircraft may enter into a sideslip even without the pilot’s intention. Using the lateral control characteristics in steady, straight, sideslips, it is simple to verify if the effective dihedral, Cℓβ , and the static dimensional stability, Cnβ , have the correct sign. These two stability derivatives have a strong influence on the lateral control characteristics in general. They are also two important parameters in obtaining good lateral stability characteristics. The equilibrium equations for steady, straight sideslip follow directly from the expressions rb (9-6), (9-7) and (9-8), by letting 2V = 0. Neglecting small contributions, the equilibrium equations then become,
CL CYβ 0 Cℓ β 0
Cnβ
0
0
Cℓδa
0
0
Cnδr
ϕ
β =0 δa
(9-21)
δr
Flight Dynamics
454
Lateral Stability and Control in Steady Flight δa [o ] 2
-12
-8
-4
0
4
8
rb 2V
· 103
0
4
8
rb 2V
· 103
0
4
8
rb 2V
· 103
-2
δr [o ] 1
-12
-8
-4
-1
ϕ [o ] 60
40
20
0 -12
-8
-4
-20
-40
-60
Figure 9-4: Steady coordinated turns, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14])
9-4 Steady, Straight, Sideslipping Flight
455
The variation of the angle of roll with β follows from the first row of equation (9-21). CY dϕ =− β >0 dβ CL
CYβ < 0,
CL > 0
(9-22)
Apparently, a positive angle of sideslip is associated with a positive angle of roll. A given angle of sideslip creates a large lateral force CYβ · β which has to be balanced in steady flight by a component of the weight along the YB -axis. If the absolute value of CYβ is large, the equilibrium at a given angle of side-slip necessitates a large roll angle, as can be seen from equation (9-22). A suddenly occurring angle of sideslip, caused for instance by a gust, gives also rise to a force along the YB -axis and hence to a sideward acceleration. The pilot senses this acceleration as a sideward force exerted upon him by his seat and seat belts. If, again CYβ is large in the absolute sense, this force will be relatively large, making it easier for the pilot to detect the presence of the sideslipping motion. These arguments lead to the conclusion that a large negative value of CYβ may be considered as desirable. To initiate a straight sideslip with a positive β (velocity vector pointing to the right of Xb axis) the aircraft has to be given a positive rolling moment (to the right) and a negative yawing moment (to the left). This requires an initial aileron deflection to the right (negative) and an initial rudder deflection to the left (positive). It is then desirable that in the ultimate steady flight condition the aileron control is also deflected to the right and the rudder control to the left or, dδa <0 dβ
and
dδr >0 dβ
The required surface deflections follow from equation (9-21), Cℓ dδa =− β dβ Cℓδa
(9-23)
Cn dδr =− β dβ Cnδr
(9-24)
As both Cℓδa and Cnδr are negative, the required directions of the control deflections in steady, straight sideslips are obained, if, Cℓβ < 0
and
Cnβ > 0
It will be shown in chapter 11 that the latter conditions have to be satisfied also to obtain lateral dynamic stability. Measurement of the control deflections in steady, straight sideslips offers a simple method to determine if these conditions are satisfied. Figure 9-6 shows some Flight Dynamics
456
Lateral Stability and Control in Steady Flight δa [o ] 8
4
-3
-2
-1
1
2
rb 2V
· 103
0
1
2
rb 2V
· 103
0
1
2
rb 2V
· 103
0
-4
-8
δr [o ] 8
4
-3
-2
-1
-4
-8
β [o ] 8
4
-3
-2
-1
-4
-8
Figure 9-5: Steady flat turns, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14])
9-4 Steady, Straight, Sideslipping Flight
457 δa [o ] 8
4
-12
-8
-4
4
8
β [o ]
0
4
8
β [o ]
0
4
8
β [o ]
0
-4
-8
δr [o ] 8
4
-12
-8
-4
-4
-8
ϕ [o ] 8
4
-12
-8
-4
-4
-8
Figure 9-6: Steady, straight, sideslipping flight, North American ‘Harvard II B’, gliding flight, CL = 0.31, xc.g. = 0.304 c¯, V = 78 m/sec (from reference [14])
Flight Dynamics
458
Lateral Stability and Control in Steady Flight
results of measurements, made in steady, straight sideslips. As a final remark it can be said that the center of gravity position has an influence on the magnitude Cnβ , in analogy with the influence on Cmα . As a consequence, the c.g. position also has some influence on the lateral stability and control. The importance of this influence is, however, generally much less than is the case for the longitudinal stability and control characteristics.
9-5
Steady Straight Flight with One or More Engines Inoperative
For multi-engined aircraft the requirement exists that the aircraft can continue flight if one engine becomes inoperative. This imposes requirements on the controllability and usually in particular on the available rudder power. The most important consequences of the loss of the engine thrust from an engine not mounted in the plane of symmetry are a yawing moment due to the asymmetric thrust distribution and, in the case of a propeller-driven aircraft, a rolling moment caused by the resultant asymmetric lift distribution. The yawing moment can be written in non-dimensional form as, Cne = k
∆Tp ye 1 2 2 ρV Sb
(9-25)
Here ∆Tp is the difference in thrust from the operative engine and the drag of the inoperative engine; ye is the distance from the line of action of the thrust or drag to the plane of symmetry of the aircraft. ∆Tp may be considerably larger than the thrust of the operative engine if the inoperative engine is windmilling in the airstream. Tn the case of a propeller-driven aircraft, ∆Tp is minimal if the inoperative propeller is feathered, whereby the rotation is stopped. The factor k in equation (9-25), which for propeller-driven aircraft may assume values of 1.5 to 2.0 in some instances, expresses the fact that the yawing moment actually occurring may be larger than would result from the changes in thrust only. Figure 9-7 shows some results of measurements of Cne confirming this fact. The explanation of this effect lies in the increased lift of the part of the wing submerged in the slipstream. This increase in lift creates an extra downwash behind the wing which is now asymmetric. The downwash behind the operative propeller is larger than the downwash behind the inoperative propeller. In analogy with the airflow behind a high or low mounted wing in sideslipping flight, the differences in downwash create a circulation around the rear fuselage, thereby causing a cross-flow at the vertical tailplane. It is not difficult to verify that this cross-flow or sidewash causes an increase in the disturbing yawing moment. The experimental evidence in figure 9-7 shows that the direction of rotation of the propeller and the vertical position of the wing relative to the fuselage have an important influence on the magnitude of Cne . Reference [57] discusses this matter in detail. In general it can
9-5 Steady Straight Flight with One or More Engines Inoperative
459
Figure 9-7: The magnitude of Cne due to an engine failure, as a function of the location and the sense of rotation of the propeller and the vertical position of the wing, measured on a model of a twin-engined propeller-driven aircraft (from reference [105]).
Flight Dynamics
460
Lateral Stability and Control in Steady Flight
be concluded, that if the propellers turn clockwise, the loss of the left engine causes the largest disturbing yawing moment Cne . This engine is then called the critical engine. The rolling moment Cℓe due to an inoperative engine is caused mainly by the difference in lift on both sides of the wing, behind the operative and the in-operative engine. Some results of measurements on Cℓe are shown in figure 9-8. If the right engine cuts out, both a positive Cne and a positive Cℓe arise. Due to Cne the aircraft assumes a positive rate of yaw leading to a negative β. The rolling moment Cℓe causes a positive rate of roll and thus a positive roll angle ϕ. The rolling motion is amplified rb by the contributions Cℓβ · β (> 0) and Cℓr · 2V (> 0) due to the sideslipping and yawing motions. As a consequence, the roll angle may increase very rapidly. The motion is counteracted by the yawing moments generated by p, r and β (Cnp , Cnr , Cnβ , and by the application of both a positive rudder deflection and a positive aileron deflection. A new condition of steady, straight flight may now be established, described by the following equilibrium equations in which a few minor contributions have been neglected,
CL CYβ
0
0
0
CYδr
Cℓβ
Cℓδa
0
Cnβ
0
Cnδr
ϕ
0 β = − Cℓ e δa Cne δr
(9-26)
In flight with asymmetrically distributed engine thrust very large rudder deflections may be required. As a consequence the lateral force caused by δr cannot be omitted in equation (9-26). As the four variables ϕ, β, δa and δr occur in the equilibrium equations (9-26) there is not just one, single steady flight condition possible, but a whole range of conditions. In practice two of the many possible conditions have special significance: steady flight at a roll angle equal to zero and the condition at which the angle of sideslip β is equal to zero. If β = 0 the aircraft drag is at a minimum, resulting in maximum flight performance. If the right engine is inoperative, a positive rudder deflection (to the left) is required to balance Cne . In the flight condition at ϕ = 0, a positive angle of sideslip is necessary according to the top row of equation (9-26), to obtain an aerodynamic lateral force to balance the lateral force CYδr · δr , see figure 9-9. The aircraft performs a sideslipping motion in the direction of the inoperative engine. In the flight condition with β = 0 the pilot must apply a negative roll angle, assuming again that the right engine is stopped, such that the component of the weight CL · ϕ balances the lateral force CYδr · δr . The wing with the operative engine has to be kept low, see figure 9-10.
9-5 Steady Straight Flight with One or More Engines Inoperative
461
Figure 9-8: The magnitude of Cne due to an engine failure, as a function of the location and the sense of rotation of the propeller and the vertical position of the wing, measured on a model of a twin-engined propeller-driven aircraft (from reference [105]).
Flight Dynamics
462
Lateral Stability and Control in Steady Flight
Figure 9-9: Steady, straight, single-engined flight at ϕ = 0
In this situation the rudder deflection required for equilibrium is smaller than in the flight condition with ϕ = 0, where not only Cne but also the moment Cnβ · β acting in the same direction has to be balanced, see the bottom row in equation (9-26). The flight condition at ϕ = 0 is, therefore, more critical for the determination of the required rudder power than the flight condition at β = 0. Contrary to the moments Cnδr · δr and Cnβ · β, the disturbing yawing moment Cne depends strongly on airspeed. Taking the case of the propeller-driven aircraft, the engine power is constant at constant throttle setting. To a first approximation this means, Tp V = constant From equation (9-25) then follows, that Cne in that case is inversely proportional to the cube of airspeed, V 3 . There will be an airspeed below which it is not possible to balance the yawing moment due to an engine failure, Cne , see figure 9-11. This airspeed is called the ‘minimum control speed’, Vm.c. . When determining Vm.c. in flight, the regulations stipulate that |ϕ| may not be larger than 5o . High engine power and low airspeeds, and consequently large values of Cne at engine failure, occur during take-off and in climb immediately after take-off. In the take-off configuration at high CL -values, the extra interference yawing moment, expressed by the factor k in equation (9-25), is also large. An additional factor is the fact that during take-off the aircraft is not allowed to loose height or to obtain large angles of roll.
9-6 Steady Rolling Flight
463
Figure 9-10: Steady, straight, single-engined flight at β = 0
The importance of the minimum control speed as an element in the choice of the take-off procedure to be followed, depends strongly on the type and location of the engine. Jetpropelled aircraft, having the engines mounted on the rear fuselage, often have a minimum control speed lower than the minimum airspeed for sustained flight. For such aircraft the minimum control speed loses its significance. Handling the aircraft on the ground with failed engine may then be an important consideration. For twin-engined, propeller-driven aircraft, however, the mininimum control speed in the take-off configuration is of extreme importance. Figure 9-12 shows some results of measurements on the Fokker F-27 with the right propeller feathered.
9-6
Steady Rolling Flight
One of the measures for the manoeuvrability of an aircraft is the rate of roll obtained by giving a certain aileron deflection. The time required to establish a steady turn, is determined largely by this rate of roll. This is the reason why requirements exist for the attainable rates of roll. In many cases this fixes the required aileron power. If the ailerons are deflected and then maintained in their deflected position, the aircraft will experience at first an angular acceleration about the XB -axis. When the growing damping pb moment Cℓp · 2V has become equal to the rolling moment Cℓδa · δa , the rate of roll will remain Flight Dynamics
464
Lateral Stability and Control in Steady Flight Cn
Cne
Cne + Cnβ · β (if ϕ = 0)
Cnδr · δr
max
V (Vm.c. )β=0
(Vm.c. )ϕ=0
Figure 9-11: Minimum control speed
constant, if the simplifying assumption is made that during this rolling motion the aircraft will neither sideslip nor yaw. The resulting flight condition is that of steady rolling flight. It is possible to demonstrate, that this ‘steady’ condition is usually established relatively quickly. This makes it sensible to assess the manoeuvrability of an aircraft by means of the attainable rate of roll.
Under the influence of for instance the component of the weight CL · ϕ and the yawing moments due to δa and p, the aircraft will not only roll but also sideslip and yaw. This means, that continuous rudder control would be required to make the aircraft perform a rolling flight at constant rate of roll without any sideslippiag and yawing motions.
A real exactly steady rolling flight is thus not possible. The ‘steady’ rolling flight to be discussed below is, therefore, a quasi-steady flight condition in straight flight (r = 0) at constant rudder angle (δr = 0), at small angles of roll ϕ and to the neglect of the angle of sideslip actually occurring (β = 0). For this idealized flight condition the equation for the equilibrium of the aerodynamic moments about the XB -axis is,
Cℓp
pb + Cℓδa δa = 0 2V
(9-27)
9-6 Steady Rolling Flight
465 δa [o ] 40
30
20
10
-12
-8
-4
4
8
β [o ]
0
4
8
β [o ]
0
4
8
β [o ]
0
δr [o ] 40
30
20
10
-12
-8
-4
ϕ [o ] 2
-12
-8
-4
-2
-4
-6
Figure 9-12: Steady, straight, sideslipping flight with the right propeller feathered, Fokker F-27, h = 1850 m, CL = 1.92, xc.g. = 0.28 c¯, V = 46.3 m/sec (from reference [51]) Flight Dynamics
466
Lateral Stability and Control in Steady Flight
or, Cℓ pb = − δa δa 2V Cℓp
(9-28)
where both Cℓδa and Cℓp are negative. A positive aileron deflection is thus seen to cause a negative rate of roll. In the discussion of Cℓδa and Cℓp it was shown that these two derivatives are to a first approximation independent of CL or the airspeed. According to equation (9-28) the rate of roll at a given aileron angle is thus proportional to airspeed, but the non-dimensional rate of roll is independent of V . From early investigations, see reference [60], it appeared that the pilot’s assessmentof the manoeuvrability of the then considered aircraft, showed pb a better correlation with 2V than with pmax . On this basis the requirements on max manoeuvrability were expressed using the non-dimensional rate of roll. As an example, older U.S. military regulations, see reference [19], required for a fighter aircraft,
pb 2V
> 0.09
pb 2V
> 0.07
max
and for the other categories of aircraft,
max
Since the time when the requirements were formulated the range of possible airspeeds has pb increased so much, that the non-dimensional rate of roll, 2V , as the only criterion for the manoeuvrability of an aircraft about the XB -axis, has lost some of its significance. For V/STOL aircraft capable to operate at very low airspeeds, the rate of roll p resulting pb would be too low at low values of V , to ensure good from the above requirements on 2V manoeuvrability. For such aircraft the additional requirement is sometimes used, that in the pb approach 2V , which can be seen as the maximum vertical speed of the wing tip due to max
rolling, must be at least be 10 ft/sec.
On the other hand, the rolling velocity at supersonic airspeed of a fighter aircraft having pb a small wing span according to the requirement on 2V , would be unreasonably large. At those high rates of roll, difficulties with the dynamic stability might occur, as at high rolling velocities the symmetric and the asymmetric aircraft motions at high rates of roll may become quite violently unstable. In later U.S. military Regulations, see reference [13], the requirement is given that fighter aircraft in the combat configuration must be capable to attain an angle of roll of ϕ = 50o within one second. For other aircraft configurations and flight conditions, as well as for other categories of aircraft, requirements on non-dimensional rate of roll are maintained.
9-6 Steady Rolling Flight
467
p
δa3
δa2 δa1
Reversal Speed V
Vmin Figure 9-13: The limitation in the maximum attainable roll-rate due to elastic wing deformation
It was discussed in section 7-7 that due the elastic deformation of the wing, Cℓδa will decrease in absolute value at high airspeed, or rather at high dynamic pressures. Due to this effect, the rate of roll no longer increases linearly with airspeed at a constant aileron deflection, see figure 9-13. At high airspeeds the rate of roll will even decrease. The reversal speed is reached when Cℓδa = 0 and as a consequence also p = 0. The elastic wing twist may be reduced by locating the ailerons not at the wing tips, but more inboard. The sketch In figure 9-14 shows that this may considerably reduce the total twist at the wing tip, thereby reducing the loss in rolling moment generated by aileron deflection. If the ailerons are located inboard, the outer part of the wing, generally possessing less torsional stiffness, will experience no further increase in wing twist. In cases were no satisfactory roll controll can be obtained in this way, it may be necessary to use spoilers for roll control, as discussed in section 7-7. The maximum achievable rate of roll p may be limited at high airspeeds by the maximum possible roll control force, rather than by the maximum aileron deflection. If no hydraulic servos are applied and if no differential aileron deflections are used, the roll control force can be written according to equation (9-40), if in addition δta = 0, as, dδa 1 2 1 Fa = − ρV Sa c¯a Chα ∆αa + Chδ δa dsa 2 2
(9-29)
It will be seen, that the required control force strongly increases with airspeed at a given Flight Dynamics
468
Lateral Stability and Control in Steady Flight
ϕ (due to δa )
1
2
y
2
1
Figure 9-14: Variation of the wing twist angle ϕ along the wing span due to aileron deflection for two locations of the ailerons
9-6 Steady Rolling Flight
469
∆α
∆αgeometric pb 2V
∆αaverage = ∆α0
∆αef f ective ym ba
V b 2
Figure 9-15: The average value of ∆α along the aileron span in steady, rolling flight
aileron deflection. In equation (9-29) ∆αa is the average effective change of angle of attack due to rolling over the span of the aileron. This average effective value can be equated to a geometric change in angle of attack at a distance ym from the plane of symmetry, see figure 9-15. The expression is,
∆αa =
pb 2ym 2V b
(9-30)
For a given wing, ym can be determined using reference [155]. The aileron deflection δa required for a steady, rolling flight at β = 0 follows from equation (9-28),
δa = −
pb Cℓp 2V Cℓδa
(9-31)
Substitution of equation (9-30) and (9-31) in (9-29) results in the following expression for the roll control force in a steady, rolling flight at β = 0, as a function of rate of roll p and airspeed V,
Fa = −
dδa 1 2 2ym 1 Cℓp pb ρV Sa c¯a Chα − Chδ dsa 2 b 2 Cℓδa 2V
(9-32) Flight Dynamics
470
Lateral Stability and Control in Steady Flight p δa = constant |Fa | increasing
δamax δa = constant
|δa | increasing
elastic wing δa = constant Fa = constant Famax Fa = constant Fa = constant V Figure 9-16: Roll-rate p as a function of airspeed V for various values of δa and Fa
or, dδa 1 b 2ym Chδ Cℓp Fa = − ρV p Sa c¯a Chα − dsa 2 2 b 2 Cℓδa
(9-33)
If for a given aircraft the aerodynamic derivatives Chα , Chδ , Cℓp and Cℓδa may be assumed to remain constant, at a given flight altitude (i.e. value of ρ) the roll control force is, Fa = constant · p · V
(9-34)
In a diagram showing rate of roll as a function of V , curves for a constant control force will be hyperbola’s, see figure 9-16. Due to wing twist, Cℓδa will decrease in absolute value with increasing airspeed, as discussed in section 7-7. For a real, elastic wing, the rate of roll achieved at a given control force will be smaller than for a rigid wing, just like the rate of roll resulting from a given aileron deflection, see figure 9-16.
9-7 9-7-1
Control Forces and Hinge Moments for Lateral Control Roll Control
Using the same argument as in chapter 5 for the elevator control, the following expression applies to roll control in analogy with equation (5-77), Fa dsa + Har dδar + Hal dδal = 0
(9-35)
9-7 Control Forces and Hinge Moments for Lateral Control
471
or, Fa = −
dδar dδal Har + Hal dsa dsa
(9-36)
The hinge moments Har and Hal can be written in the usual way as, Har = Char
1 2 ρV Sar c¯ar 2
Hal = Chal
1 2 ρV Sal c¯al 2
If Sar = Sal = Sa (the area of one single aileron) and, c¯ar = c¯al = c¯a the aileron control force in equation (9-36) becomes, Fa = −
dδar dδal Char + Chal dsa dsa
1 2 ρV Sa c¯a 2
(9-37)
If no differential deflection of the ailerons is used, this expression can be futher developed in a straightforward manner. Using equation (7-32), δar = −δal =
δa 2
or, dδar dδa 1 dδa =− l = dsa dsa 2 dsa Substitution in equation (9-37), Fa = −
1 dδa 1 2 ρV Sa c¯a Char − Chal 2 dsa 2
(9-38)
where, Char = Ch0r + Chα αr + Chδ δar + Chδt δtar and, Chal = Ch0l + Chα αl + Chδ δal + Chδt δtal In asymmetric flight is αr 6= αl . Suppose now, Flight Dynamics
472
Lateral Stability and Control in Steady Flight
αr = α + ∆αa then, αl = α − ∆αa The result is, αr − αl = 2 ∆αa In addition, is, δar − δal = δa δtar − δtal = δta Ch0r − Ch0l = 0 and thus, Char − Chal = Chα 2 ∆αa + Chδ δa + Chδt δta
(9-39)
Substitution of equation (9-39) in equation (9-38) finally results in, dδa 1 2 Fa = − ρV Sa c¯a dsa 2
Chα ∆αa + Chδ
δa δt + Chδt a 2 2
(9-40)
The expression between parentheses is Chα of a single aileron.
9-7-2
Rudder Control
The applicable expression for the control force on the rudder pedals is entirely in analogy with equation (5-77) for the elevator control, Fr = −
dδr 1 2 ρVv Sr c¯r Chα αv + Chδ δr + Chδt δtr dsr 2
(9-41)
Calculation methods for the hinge moment derivatives in equation (9-40) and (9-41) have been mentioned in section 5-2-3.
Part III
Dynamic Stability Analysis
Flight Dynamics
Chapter 10 Analysis of Symmetric Equations of Motion
Once the equations of motion of an aircraft have been derived and the required aerodynamic derivatives are known, it is possible to calculate the aircraft motions caused by an arbitrary control deflection or external disturbance. In the most general case, if the equations of motion to be used are non-linear, only a numerical integration of the equations is possible. Many aircraft motions, however, may be considered as deviations from a condition of steady, straight, symmetric flight, small enough to permit linearization of the equations about that reference condition. In chapter 4 the linearization has been given. Thereafter it is possible to use analytical methods, such as the Laplace-transform, to solve the linearized equations. Even then, the computer remains a nearly indispensable tool to carry out the calculations. If the motions of an aircraft about a given equilibrium condition are studied, it is, quite independent of the type of disturbance to be considered, of primary interest to know if the equilibrium is stable. The following discusses how this question is answered. Stability is a characteristic of any equilibrium condition of any arbitrary system. The question of the stability of an equilibrium condition relates to the behaviour of the system after it has been subjected to a small deviation from the equilibrium due to some disturbance, once the disturbance has ceased to act. If, after some time, the system returns to the original equilibrium condition, that condition is stable. If the system does not return to the original condition but deviates increasingly with time, the equilibrium was unstable. Finally, the equilibrium condition is neutrally stable, if the deviation caused by the disturbance neither disappears nor keeps increasing with time. If the motions of the system under study about the initial reference condition can be described by linear equations of motion, the stability of the equilibrium condition is independent of the Flight Dynamics
476
Analysis of Symmetric Equations of Motion
type and magnitude of the disturbance and of the magnitude of the initial deviation from the equilibrium condition. The condition of linearity is, also in the case of aircraft, not always satisfied. In the following, stability is a characteristic of an arbitrary steady flight condition. For the sake of simplicity the ‘stability of an aircraft’ is often referred to, whereas the stability of a certain equilibrium condition of that aircraft is actually meant. This is not quite correct, since any particular aircraft may be in a stable and sometimes also in an unstable equilibrium condition. The behaviour of the aircraft once it has acquired a small deviation from an equilibrium condition is described by the linearized ‘deviation equations’, (4-37) and (4-40). For simplicity’s sake, only the longitudinal stability is discussed in the following sections. The stability criteria to be derived and the methods used to solve the equations of motion are equally applicable to the asymmetric motions, see chapter 11. To begin with, the linearized ‘deviation equations’ for the symmetric motions, where CXq = 0, are repeated here, see equation (4-38),
CXu − 2µc Dc CZu
CXα
CZ0
CZα + (CZα˙ − 2µc ) Dc −CX0
0 CZq + 2µc
0
0
−Dc
1
Cmu
Cmα + Cmα˙ Dc
0
Cmq − 2µc KY2 Dc
−CXδe
−CZδ e = 0
−Cmδe
u ˆ
α = θ q¯ c V
δe
This is a group of four simultaneous, constant coefficient, linear differential equations of first order. The independent variable in the equations is, in the physical sense, the disturbance variable, i.e. the elevator deflection. The dependent variables are the four components of the motion. According to the foregoing, the stability of the equilibrium condition is apparent from the aircraft’s motion, once it has acquired a deviation from the equilibrium situation due to a disturbance and the disturbance has ceased to act. In the equations of motion this means that the time history of the dependent variable is studied, assuming the independent variables to be all zero and assuming furthermore, that the dependent variables have been given non-zero initial conditions. As the equations are linear, the magnitude of the initial
10-1 Solution of the equations of motion
477
conditions, i.e. the magnitude of the assumed disturbances and the resulting deviations, have no influence on the stability of the equilibrium, as noted before. In the more general case, where the motions are described by non-linear equations, the magnitude of the disturbances may influence stability. Since stability is then independent of the input, or the disturbance, the equations to be solved for the disturbed symmetric aircraft motions can now be written in the following homogeneous form,
CXu − 2µc Dc
CXα
CZ0
0
CZu
CZα + (CZα˙ − 2µc ) Dc
−CX0
CZq + 2µc
0
0
−Dc
1
Cmu
Cmα + Cmα˙ Dc
0
Cmq − 2µc KY2 Dc
uˆ
α =0 θ q¯ c V
(10-1) As the elevator angle has been assumed to remain constant, the stability to be determined from these equations is the so-called ‘stability, stick fixed’. The assumption is thus made that the pilot holds the control manipulator and thereby the elevator fixed, such that the elevator angle does indeed remain constant. A highly realistic alternative is the situation where the pilot has trimmed the control force to zero and takes his hands off the control manipulator. In that case the control mechanism is free during the disturbed motion and the elevator may vary. The actual time history of the elevator angle during the disturbed motion then follows from one or more additional equations of motion describing the behaviour of the control mechanism in the ‘stick free’ condition. The stability thus to be determined, is the so-called ‘stability, stick free’. The following general discussions of the stability of an equilibrium condition are valid both for the stability, stick fixed and the stability, stick free.
10-1
Solution of the equations of motion
The analytical solution of the linear differential equations considered here is based on the transformation of these equations into algbraic equations. This can be done in various ways. The solution of the above homogeneous equations has the following general form, see for instance reference [104]. For the symmetric motions, x = Ax eλc sc
(10-2) Flight Dynamics
478
Analysis of Symmetric Equations of Motion
where x represents any of the components of the motion, and the variables sc is the time, made non-dimensional, see table 4-1, where, sc =
V t c¯
The coefficient Ax in equation (10-2) is determined partly by the initial conditions given to the equations of motion. The variable λc in (10-2) to be discussed in detail later, can be either real or complex. From the solution (10-2) it can be seen at once if the equilibrium condition under study is stable. For stability it is both necessary and sufficient that x goes to zero with increasing time. The requirement for stability then is, lim x = 0
sc →∞
For finite values of the coefficient Ax it is the variable λc that determines entirely and only, if the stability requirement is met. It is, therefore, of primary interest to see how this λc is to be found. To this end it is only necessary to substitute the solution (10-2) of the equations of motion back in these equations. The value or values of λc turning these equations into equalities are the ones looked for. Before performing the substitution, it may be realized that in equation (10-1), Dc x =
c¯ d Ax eλc V dt
V c ¯
t
= λc x
(10-3) Dc2 x = λ2c x In the following, the equations for the symmetric motions are further analyzed. Equations (10-2) and (10-3) are substituted in equation (10-1),
CXu − 2µc λc
CXα
CZ0
0
CZu
CZα + (CZα˙ − 2µc ) λc
−CX0
CZq + 2µc
0
0
−λc
Cmu
Cmα + Cmα˙ λc
0
Cmq
Au
Aα Aθ 1 Aq − 2µc KY2 λc
λs e c c =0
(10-4) or, in a more compact notation,
10-1 Solution of the equations of motion
479
[∆] A eλc sc = 0 where [∆] is the matrix and A is the vector with elements Au , Aα , Aθ and Aq . By the substitution the equations of motion change from differential equations into simultaneous, linear, homogeneous algebraic equations. From these equations the non-zero common factor eλc sc can be omitted without influencing the values of λc to be obtained, [∆] A = 0
(10-5)
These homogeneous equations are always satisfied by the so-called trivial solution, A=0 which is a short notation for, Au = Aα = Aθ = Aq = 0 In equation (10-2) this means, u ˆ=α=θ=
q¯ c =0 V
This is the original equilibrium flight condition, which is evidently a particular case of the disturbed motion. This trivial solution is not further considered here. The homogeneous, algebraic equations represented by equation (10-5) are in general inconsistent. A simple example may serve to illustrate this fact. Suppose, the following two homogeneous, algebraic equations have been given, a11 x1 +a12 x2 = 0 a21 x1 +a22 x2 = 0 Or,
a11 a12 a21 a22
x1 x2
=0
From the first equation follows the solution, x2 = −
a11 x1 a12 Flight Dynamics
480
Analysis of Symmetric Equations of Motion
and from the second equation, x2 = −
a21 x1 a22
These two solutions are consistent if, −
a11 a21 =− a12 a22
or, a11 a22 − a12 a21 = 0 Or, expressed in a different way, a11 a12 a21 a22
=0
If the coefficient determinant of the homogeneous, algebraic equations is zero, the equations are no longer inconsistent. But then they have become dependent. In the case of aircraft motions considered here, the equations (10-5) and (10-4) have a solution only for a few particular values of λc . These are the values of λc resolving the inconsistency of the equations by making the equations dependent. These values of λc called the eigenvalues of the differential equations, are found according to the foregoing by equating the determinant of the square matrix [∆] to zero. The determinant of [∆] is called the characteristic determinant. The eigenvalues are then found from, [∆] = 0 This ‘characteristic equation’ can be written more explicitly as, CXu − 2µc λc CXα CZ0 CZu CZα + (CZα˙ − 2µc ) λc −CX0 0 0 −λc Cmu Cmα + Cmα˙ λc 0
CZq + 2µc =0 1 2 Cmq − 2µc KY λc 0
For a further study of stability via the eigenvalues it is necessary to expand the above characteristic equation. The result may be expressed as, A λ4c + B λ3c + C λ2c + D λc + E = 0
(10-6)
10-1 Solution of the equations of motion
481
The four roots of this characteristic equation are the four eigenvalues λc to be found. According to equation (10-2) they determine if the original equilibrium condition is stable. The coefficients A to E in equation (10-6) are given with the more detailed discussion of dynamic longitudinal stability in section 10-3. The resulting eigenvalues may finally be substituted in (10-5). For each λc follow an arbitrary combination of three out of the four equations the three ratio’s, real or complex, of the four components of the motion. They provide further insight into the characteristics of the disturbed motion, but this subject is not further pursued here. In general the roots of the characteristic equation are all different. After substitution in equation (10-2) they give the aircraft motion after a disturbance from the equilibrium situation. Assuming four different eigenvalues, the solution for the symmetric motions is,
u ˆ
α = [A] θ q¯ c V
eλc1 sc
λ s e c2 c eλc3 sc
eλc4 sc
(10-7)
where the matrix [A] can be written more explicitely as,
Au1
Aα1 [A] = Aθ 1 Aq1
Au2 Aα2
Au3 Aα3
Au4 Aα4
Aθ2
Aθ3
Aθ4
Aq2
Aq3
Aq4
(10-8)
Combination of equation (10-7) and (10-8) results in the solution of the homogeneous differential equations, u ˆ
= Au1 eλc1
sc
+Au2 eλc2
sc
+Au3 eλc3
sc
+Au4 eλc4
sc
α
= Aα1 eλc1
sc
+Aα2 eλc2
sc
+Aα3 eλc3
sc
+Aα4 eλc4
sc
θ
= Aθ1 eλc1
sc
+Aθ2 eλc2
sc
+Aθ3 eλc3
sc
+Aθ4 eλc4
sc
q¯ c V
= Aq1 eλc1
sc
+Aq2 eλc2
sc
+Aq3 eλc3
sc
+Aq4 eλc4
sc
Four out of the sixteen constants in [A] are fixed by four initial conditions which must be given if the disturbed motion is to be determined completely. The remaining twelve constants then follow from twelve real or complex ratios of the components of the motion, mentioned Flight Dynamics
482
Analysis of Symmetric Equations of Motion
earlier. From this brief discussion it may be seen, that the constants in [A] have no influence whatsoever on the stability or instability of the equilibrium conditions. The various possible types of eigenvalues λc and the corresponding motions, the so-called ‘eigenmotions’ are now studied, under the assumption that all eigenvalues are different. Real eigenvalues The part of the total aircraft response, i.e. the eigenmotion, corresponding to a real eigenvalue λc is described by an aperiodic exponential function, see figure 10-1. The equilibrium can only be stable, if all real eigenvalues are negative. Only then is, lim Ax eλc sc = 0
sc →∞
If λc = 0, then, Ax eλc sc = Ax The eigenmotion corresponding to this particular value is a constant for any of the components of the motion. For λc > 0, all components of the motion tend to go to infinity with increasing time, because for any value of Ax , lim Ax eλc sc = ±∞
sc →∞
The speed at which for a given negative, real λc the corresponding eigenmotion converges to zero, is commonly expressed by two different characteristic times, • The time to damp to half the amplitude, T 1 . This is the time interval in which the 2 exponential function decreases to half its original value. This means, x(t + T 1 ) = 2
1 x(t) 2
or, λc Vc¯
e
„
t+T 1
2
«
=
1 λc V e c¯ 2
t
From this follows,
T1 = 2
ln 12 c¯ 0.693 c¯ =− λc V λc V
10-1 Solution of the equations of motion
483
1
0.9
0.8
0.7
x
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6 t [sec]
7
5
6 t [sec]
7
5
7
8
9
10
8
9
10
8
9
10
(a) Convergent motion, λ < 0 2
1.8
1.6
1.4
x
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
(b) Constant deviation, λ = 0 150
x
100
50
0
0
1
2
3
4
6 t [sec]
(c) Divergent motion, λ > 0
Figure 10-1: Aperiodic motions corresponding to a real eigenvalue λ
Flight Dynamics
484
Analysis of Symmetric Equations of Motion • The time constant τ . This is the time interval in which the exponent decreases by 1 and the exponential function itself decreases by the factor 1e . Then, x(t + τ ) =
1 x(t) e
or, V
eλc c¯ (t+τ ) =
1 λc V e c¯ e
t
or λc
V τ = −1 c¯
τ =−
1 c¯ λc V
The relations between T 1 and τ are, 2
τ = 1.443 T 1
2
T 1 = 0.693 τ 2
Using the time constant, the eigenmotion corresponding to the real λc can be written as, V
t
Ax eλc c¯ t = Ax e− τ If λc is positive and an aperiodic divergence results, T 1 and τ may still be used to indicate the 2 level of divergence. In that case both T 1 and τ are negative. In those situations use is often 2 made of the ‘time to double amplitude’ T2 . It is the time interval in which the exponential function increases to twice its original value, T2 = −T 1 2
10-1 Solution of the equations of motion
485
Complex Eigenvalues In the case of a complex λc , the eigenvalue is written as, λc = ξc + j ηc
j=
√
−1
Substituting of such an eigenvalue in equation (10-2) or (10-3) would lead to complex expressions for the components of the motion. But complex eigenvalues occur always in complex conjugate pairs, i.e., λc1,2 = ξc ± j ηc The corresponding constants Ax1 and Ax2 in equation (10-2) are also complex conjugate, as the resultant motion can only be real. This means, Ax1,2 = Rx ± j Qx The eigenmotion belonging to the two complex conjugate roots of the characteristic equation, can now be written as, Ax1 eλc1 sc + Ax2 eλc2 sc = (Rx + j Qx ) e(ξc +j ηc )sc + (Rx − j Qx ) e(ξc −j ηc )sc = p Qx ξc sc 2 2 2 Rx + Qx e cos ηc sc + arctan Rx
(10-9)
This part of the total response of the components of the motion is evidently a damped oscillation, i.e. ξc is negative. The period P of the oscillation follows from the imaginary part ηc of the two eigenvalues. If the argument of the harmonic function has increased by 2π, a time P has elapsed, ηc
V P = 2π c¯
or, P =
2π c¯ ηc V
According to equation (10-9), the amplitude of the oscillation is, p 2 Rx2 + Q2x eξc sc
Flight Dynamics
486
Analysis of Symmetric Equations of Motion
1
x
0.5
0
−0.5
0
1
2
3
4
5
6 t [sec]
7
5
6 t [sec]
7
5
7
8
9
10
8
9
10
8
9
10
(a) Convergent motion, ξ < 0 1.5
1
x
0.5
0
−0.5
−1
−1.5
0
1
2
3
4
(b) Constant amplitude, ξ = 0 150
100
x
50
0
−50
−100
−150
0
1
2
3
4
6 t [sec]
(c) Divergent motion, ξ > 0
Figure 10-2: Periodic motion corresponding to a pair of complex, conjugate eigenvalues λ1,2 = ξ ± j η
10-1 Solution of the equations of motion
487
Apparently, only the real part ξc of the eigenvalues determines if the amplitude converges to zero with increasing time, see figure 10-2. For stability it is necessary, that the real parts of all complex eigenvalues are negative. Only then is, p lim 2 Rx2 + Q2x eξc sc = 0
sc →∞
Various measures of the damping of an oscillation are in use. • The time to damp to half amplitude, T 1 . This characteristic has the same meaning as 2 for the aperiodic motions. T 1 is now expressed by, 2
T1 = − 2
0.693 c¯ ξc V
If an oscillation has negative damping, i.e. it diverges, it means that ξc > 0 and T 1 is 2 negative. Usually the time to double amplitude, T2 of the oscillation is then given, T2 = −T 1 2
• The number of periods, C 1 , in which the amplitude decreases to half its original value, 2
C1 = 2
T1 2
P
=
ηc −0.693 ηc = −0.110 2π ξc ξc
In analogy with T2 is, C2 = −C 1 2
• The logarithmic decrement δ. This is the natural logarithm of the ratio of the oscillation’s amplitude in two successive maxima, V
δ = ln
eξc c¯ (t+P ) ξc Vc¯
e
t
V P c¯
= ξc
or, δ = 2π
ξc 0.693 =− ηc C1 2
• The damping ratio ζ. The complex eigenvalues λc1,2 are often written as follows, n
λc1,2 = −ζω0 ± jω0
p
1−
ζ2
o c¯ V
(10-10)
Here ζ is the damping ratio and ω0 is the undamped natural frequency (in rad/sec). If ζ = 0, the eigenvalues are, Flight Dynamics
488
Analysis of Symmetric Equations of Motion
λc1,2 = ±jω0
c¯ = ±ηc V
The real part of λc1,2 is then zero, the oscillation is accordingly undamped and the angular frequency of the oscillation is ω0 . The relation between the period and the eigenfrequency ωn of the damped oscillation is, P = With P =
2π ηc
·
c¯ V
2π ωn
it follows from equation (10-10) that, ωn = ω0
p
1 − ζ2
The relations between ω0 , ζ, ξc and ηc can be derived to be, ω0 =
p
ξc2 + ηc2
ζ=p
V c¯
−ξc
ξc2 + ηc2
if 0 < η < 1 a damped oscillation occurs, because then λc1 and λc2 are complex and the real part −ζ ω0 Vc¯ is negative. With increasing ζ the damping increases as well. The relations between C 1 , δ and ζ are, 2
and,
ζ δ = −2π p 1 − ζ2
C 1 = 0.110 2
p
1 − ζ2 ζ
If ζ = 1, a transition occurs from a periodic to an aperiodic motion. The motion is then called critically damped. The characteristic equation in that particular case has two equal roots. If ζ > 1, the matIoa is aperiodic, as both λc1 and λc2 are then real. Finally, if ζ < 0, the real part of the eigenvalues is positive, see equation (10-10). The considered equilibrium condition is then unstable. The motions diverge, in a periodic manner if −1 < ζ < 0 and aperiodic if ζ < −1. At the end of this review of the stability of an equilibrium condition for the case where all eigenvalues are different, the findings can be summarized as follows.
10-2 Stability criteria
489
The equilibrium is stable if all real eigenvalues and the real parts of the complex eigenvalues are negative. The disturbed motion then converges back to the original equilibrium condition. If at least one real or complex eigenvalue has a positive real part, then the equilibrium condition is unstable. In this case the disturbed motion diverges increasingly from the steady equilibrium flight condition. A transitional situation exists, where no eigenvalue has a positive real opart, but a real eigenvalue or the real part of a pair of complex eigenvalues is just equal to zero. In such a situation, a deviation of constant magnitude or amplitude occurs. The equilibrium condition is then called neutrally stable. Often, the eigenvalues, i. e. the roots of the characteristic equation, are depicted as points in the complex plane. Real eigenvalues are situated on the real axis at a distance of λc of the origin and complex eigenvalues have as coordinates (ξc ± j ηc ). The above stability criterion implies that all eigenvalues must be situated to the left of the imaginary axis. As soon as at least one eigenvalue is placed to the right of the imaginary axes, the equilibrium condition is unstable. If one or more eigenvalues are placed on the imaginary axis, whether or not in the origin of the system axis, and none lies in the right half of the plane, the equilibrium condition is neutrally stable. The solution of the equations of motion for the case where two or more roots of the characteristic equation are equal, is not further discussed here. This topic is further discussed for instance in reference [104]. It appears then that what was said previously about distinct eigenvalues is generally true also for multiple eigenvalues. Differences exist only, if multiple eigenvalues lie on the imaginary axis and none to the right of that axis. The equilibrium is then either neutrally stable or unstable, depending on the multiplicity of the relevant eigenvalues and on the rank of the characteristic determinant. In order to determine the stability of the equilibrium condition according to the foregoing rules, it is always necessary to construct the characteristic determinant and to determine from it the values and types of eigenvalues. In contrast with the foregoing, the following section discusses a method to determine the stability without an explicit calculation of the eigenvalues.
10-2
Stability criteria
Often a need exists to determine the stability of an equilibrium condition without resorting at once to the solution of the characteristic equation. Sometimes it is necessary to study the influence on the stability of systematic changes in the system under consideration. In such cases, fruitful use may be made of various so called ‘stability criteria’. In this chapter only the Routh-Hurwitz criteria will be discussed. A more general stability criterion is that of Lyapunov of which the Routh-Hurwitz criteria are a particular case, see reference [104].
Flight Dynamics
490
Analysis of Symmetric Equations of Motion
The Routh-Hurwitz Stability Criteria
In the foregoing it was shown, that the criterion for stability is that all real eigenvalues and all real parts of the complex eigenvalues are negative. Routh, see reference [137] and Hurwitz, see reference [77], derived criteria which the coefficients of an algebraic equation have to satisfy for all real roots and the real parts of the complex roots to be negative. It will be sufficient here to present only the stability criteria as they apply to a quartic characteristic equation. The quartic is, A λ4c + B λ3c + C λ2c + D λc + E = 0
(10-11)
The coefficients A, B, C, D and E are all real. Without loss in generality of the following discussions it is always assumed that, A>0 The stability criteria then are, A > 0, B > 0, C > 0, D > 0, E > 0 and, BCD − AD2 − B 2 E > 0 The latter expression, R = BCD − AD2 − B 2 E is often called Routh’s discriminant. If A < 0, the signs of B, C, D, E and R must also be negative to obtain negative real parts of the eigenvalues.
10-3
The complete solution of the equations of motion
As indicated already in section 10-1, an idea of the aircraft motions after a disturbance can be obtained by determining the roots of the characteristic equation (10-6). The general character of the symmetric motions is discussed in the following. To this end, a numerical example is discussed, along the lines presented in section 10-1.
10-3 The complete solution of the equations of motion
491
According to equation (10-6) the characteristic equation for the symmetric motions is, A λ4c + B λ3c + C λ2c + D λc + E = 0 The coefficients A to E are obtained by expanding the characteristic determinant, see section 10-1. The coefficients A to E are, A = 4µ2c KY2 (CZα˙ − 2µc )
B = Cmα˙ 2µc CZq + 2µc − Cmq 2µc (CZα˙ − 2µc ) − 2µc KY2 {CXu (CZα˙ − 2µc ) − 2µc CZα }
C = Cmα 2µc CZq + 2µc − Cmα˙ 2µc CX0 + CXu CZq + 2µc + Cmq {CXu (CZα˙ − 2µc ) − 2µc CZα } + 2µc KY2 (CXα CZu − CZα CXu )
(10-12)
D = Cmu CXα CZq + 2µc − CZ0 (CZα˙ − 2µc ) − Cmα 2µc CX0 + CXu CZq + 2µc + Cmα˙ (CX0 CXu − CZ0 CZu ) + Cmq (CXu CZα − CZu CXα )
E = −Cmu (CX0 CXα + CZ0 CZα ) + Cmα (CX0 CXu + CZ0 CZu ) If the various aerodynamic and inertial parameters are known, the quantitative values of the coefficients A to E can be calculated. The eigenvalues λc are next obtained as the roots of the quartic characteristic equation (10-6). To find the roots of the characteristic equation, use is made of the program package MATLAB. In many practical situations a computer is available to obtain the eigenvalues λc . Very often the starting point for the calculations is then the characteristic determinant rather than the characteristic equation (10-6). The required program, e.g. MATLAB, is commonly Flight Dynamics
492
Analysis of Symmetric Equations of Motion
V
=
59.9 m/sec
m
=
4547.8 kg
c¯
=
2.022 m
S
=
24.2 m2
lh
=
5.5 m
µc
=
102.7
KY2
=
0.980
xcg
=
0.30 c¯
CX0
=
0
CZ0
=
−1.1360
CXu
=
−0.2199
CZu
=
−2.2720
Cmu
=
0
CXα
=
0.4653
CZα
=
−5.1600
Cmα
=
−0.4300
CXα˙
=
0
CZα˙
=
−1.4300
Cmα˙
=
−3.7000
CXq
=
0
CZq
=
−3.8600
Cmq
=
−7.0400
CXδ
=
0
CZδ
=
−0.6238
Cmδ
=
−1.5530
Table 10-1: Symmetric stability and control derivatives for the Cessna Ce500 ‘Citation’
available as a standard routine. It is then merely necessary to provide the numerical values of the stability derivatives and the inertial parameters. As an example the symmetric disturbed motions of a Cessna Ce500 ‘Citation’ are considered. The required data are given in table 10-1. From these data follow the coefficients A to E, according to equation (10-12) and from these parameters the characteristic equation. From the coefficients A to E the stability can directly be assessed. The eigenvalues are actually determined by MATLAB using the roots.m routine for root finding of polynomials. The resulting two pairs of complex conjugate eigenvalues are, λc1,2 = −2.9107 · 10−4 ± j 6.6006 · 10−3 λc3,4 = −3.9161 · 10−2 ± j 3.7971 · 10−2 or with λ = λc
V c¯ ,
λ1,2 = −8.6226 · 10−3 ± j 1.9544 · 10−1 λ3,4 = −1.1601 · 100 ± j 1.1240 · 100
10-3 The complete solution of the equations of motion
493
1.5
λ3 1
0.5
jη
λ1 0
λ2 −0.5
−1
λ4 −1.5 −1.5
−1
−0.5
0
ξ
0.5
Figure 10-3: The location of the eigenvalues for the symmetric motions
Apparently, the disturbed motion is the sum of two periodic motions. The one corresponding to the eigenvalues λc1,2 is a relatively slow, lightly damped oscillation, the other mode, corresponding to eigenvalues λc3,4 has a much shorter period and is highly damped. Two such oscillations occur in nearly all situations with most categories of conventional aircraft. They are indicated as the long period oscillation, or ‘phugoid’, and the short period oscillation respectively. The various characteristics of these motions derived in section 10-1 are next calculated from the eigenvalues λc . The results are, • Phugoid mode – λc1,2 = −2.9107 · 10−4 ± j 6.6006 · 10−3
– the period P = 32.1391 seconds
– the time to damp to half the amplitude T 1 ≈ 81 seconds 2
– the undamped natural frequency ω0 = 0.1957 rad/sec. – the damping ratio ζ = 0.0441 • Short period mode – λc3,4 = −3.9161 · 10−2 ± j 3.7971 · 10−2
– the period P = 5.5900 seconds
Flight Dynamics
494
Analysis of Symmetric Equations of Motion – the time to damp to half the amplitude T 1 ≈ 0.6 seconds 2
– the undamped natural frequency ω0 = 1.6153 rad/sec. – the damping ratio ζ = 0.7182
Figure 10-3 shows the position of the eigenvalues in the complex plane. Here the dimensional eigenvalues are used, λ = λc
V c¯
ξ = ξc
V c¯
η = ηc
V c¯
Figures 10-4 and 10-5 show the time responses of the aircraft after some small disturbance. This calculation was made using the program package MATLAB. The actual disturbance used was a negative step elevator deflection (∆δe = −0.005 Rad.). The parameters used in the simulation, as well as the flight condition, are given in table 10-1. In figure 10-4 the simulation of the phugoid mode is clearly seen, the period is approximately 32 seconds. The short period oscillation is best seen in figure 10-5, see the simulation of the pitch-rate q at the larger time scale. The high damping of this motion is also apparent.
10-4
Approximate solutions
. Figures 10-4 and 10-5 show some characteristic differences between the phugoid mode and the short period oscillation. The airspeed varies hardly during the short period oscillation. Due to the phugoid oscillation, airspeed and pitch angle vary in particular. The angular acceleration about the lateral axis, q˙ is almost zero during the long period oscillation. Also during this oscillation the variations of the angle of attack are relatively small if compared with the variations in angle of pitch. Since θ = α + γ, the changes in flight path angle γ will be nearly equal to those of θ. Based on these characteristic differences, some approximating calculation methods for the short period and the long period oscillations can be given. The approximate solutions are useful to quickly and easily get some idea of the various characteristics of the two eigenmotions. Of course, the results are very handy for a quick check on the order of magnitude of results of complete computer calculations. The accuracy of the approximating calculations depends of course on the inacceptability which depends on the type of aircraft. The following gives some calculation methods that generally produce satisfactory results for conventional aircraft. Even among the approximating
10-4 Approximate solutions
495
60
59.5
59
V [m/sec]
58.5
58
57.5
57
56.5
56
55.5
55
0
50
t [sec]
100
150
100
150
0.05
0.04
0.03
α [Rad]
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0
50
t [sec]
Figure 10-4: Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, phugoid response
Flight Dynamics
496
Analysis of Symmetric Equations of Motion
0.1
0.08
0.06
θ [Rad]
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
0
50
t [sec]
100
150
100
150
0.02
0.015
q [Rad/sec]
0.01
0.005
0
−0.005
−0.01
−0.015
−0.02
0
50
t [sec]
Figure 10-4: (Continued) Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, phugoid response
10-4 Approximate solutions
497
60
59.5
59
V [m/sec]
58.5
58
57.5
57
56.5
56
55.5
55
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
t [sec]
0.05
0.04
0.03
α [Rad]
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0
1
2
3
4
t [sec]
Figure 10-5: Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, magnification of figure 10-4, short period response
Flight Dynamics
498
Analysis of Symmetric Equations of Motion
0.1
0.08
0.06
θ [Rad]
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
t [sec]
0.02
0.015
q [Rad/sec]
0.01
0.005
0
−0.005
−0.01
−0.015
−0.02
0
1
2
3
4
t [sec]
Figure 10-5: (Continued) Response curves for a step elevator deflection (∆δe = −0.005 [Rad]) for the Cessna Ce500 ‘Citation’, magnification of figure 10-4, short period response
10-4 Approximate solutions
499
calculations, some are more refined than others. Very often an apparently more refined calculation does not produce more accurate results, if compared with the complete calculation. A more detailed motivation for the various approximations can be found using the so-called eigenvectors of the eigenmotions. They are not discussed here. 1. Short period oscillation (a) V = constant The approximating calculation of the short period oscillation is based on the assumption that during this motion the airspeed remains constant. This implies that in the equations of motion u ˆ = 0 and that the first column in equation (10-1) disappears. This also implies that the forces in the XB -direction must remain in equilibrium during the entire motion. As a consequence, the XB -equation can be entirely omitted. As an additional simplification, the initial steady flight condition is assumed to be level, so γ0 = 0 and CX0 = 0. This latter assumption causes the angle of pitch θ to disappear from the Z- and M equations, allowing the kinematic c relation −Dc θ + q¯ V = 0 to be omitted as well. In this way equations (10-1) are thus reduced to the following set of simpler equations,
CZα + (CZα˙ − 2µc ) Dc Cmα + Cmα˙ Dc
CZq + 2µc Cmq −
2µc KY2 Dc
α q¯ c V
=0
The characteristic equation is obtained, just as in the discussion given in section 10-1, by equating the characteristic determinant to zero, CZα + (CZα˙ − 2µc ) λc CZq + 2µc Cmα + Cmα˙ λc Cmq − 2µc KY2 λc
Expanding the determinant results in,
=0
Aλ2c + Bλc + C = 0 where, A = 2µc KY2 (2µc − CZα˙ ) B = −2µc KY2 CZα − 2µc + CZq Cmα˙ − (2µc − CZα˙ ) Cmq C = CZα Cmq − 2µc + CZq Cmα
The roots of the characteristic equation are the two eigenvalues λc1,2 , λc1,2 = ξc ± j ηc =
−B ± j
√
4AC − B 2 2A Flight Dynamics
500
Analysis of Symmetric Equations of Motion A further simplification is possible by omitting the derivatives CZα˙ and CZq . They occur in the equations of motion in combination with the mass parameter 2µc compared to which they are negligible. The coefficients A, B and C are then reduced to, A = 4µ2c KY2 B = −2µc KY2 CZα + Cmα˙ + Cmq
C = CZα Cmq − 2µc Cmα (b) V = constant, rotations in pitch only (γ is constant) A further simplifying assumption is, that during the short period oscillation the trajectory of the aircraft c.g. is a straight line. If the initial steady flight condition is level flight the c.g. moves along a horizontal straight line. This means that the forces in the ZB -direction remain in balance during the entire motion. As a consequence the Z-equation can be omitted. The only remaining motion is the rotation in pitch, i.e. only the M -equation remains. Because γ = θ − α = 0, α can be replaced by θ and Dc α by Dc θ in the M -equation. c 2 With Dc q¯ V = Dc θ the M -equation is then written as, Cmα + Cmα˙ Dc + Cmq Dc − 2µc KY2 Dc2 θ = 0
The characteristic equation has the coefficients, A = −2µc KY2
B = Cmα˙ + Cmq C = Cmα 2. Phugoid oscillation (a) q˙ = 0, α = 0 In this first, coarsest approximation of the long period oscillation the assumption is made that the angle of attack remains constant during this motion. As a consequence α = 0 and also α˙ = 0. This makes the α-column to disappear in equation (10-1) and one of the equations has become superfluous. Because q˙ = 0, the M equation is omitted. The fact that α = 0, may be seen as a consequence of a relatively high negative value of Cmα . In such a case is also very nearly Cm = 0. If, finally, CZq is neglected relative to 2µc and again CX0 = 0, the equations (10-1) now become,
10-4 Approximate solutions
501
CXu − 2µc Dc
CZ0
CZu
0
0
−Dc
0
u ˆ
θ =0 2µc q¯ c 1 V
The characteristic equation is reduced to a quadratic, the coefficients are, A = −4µ2c B = 2µc CXu C = −CZu CZ0 The result permits a very simple approximation of the period. The undamped natural frequency ω0 and the damping ratio ζ follow from the eigenvalues λc1,2 = ξc ± j ηc expressed in the coefficients A, B and C. V ω0 = c¯
r
C V = A c¯
s
CZu CZ0 4µ2c
(sec−1 )
B −CXu ζ=− √ = p 2 CZu CZ0 2 AC
For CXu , CZ0 and CZu the following approximations are introduced as discussed in sections 6-1, 6-2, CXu = −2 CD CZ0 = −CL CZu = −2 CL For ω0 and ζ then follows, V ω0 = c¯
s
√ 2CL2 g 2 = 4µ2c V
(sec−1 )
2CD 1 √ CD ζ= q = 2 2 CL 2 2CL2
According to this highly simplified calculation the damping of the long period oscillation is determined by the drag to lift ratio CCDL . This ratio usually is in the order of 0.1. The damping of the long period oscillation is, therefore, nearly always very low. In that case the period can be written as, Flight Dynamics
502
Analysis of Symmetric Equations of Motion
P =
2π 2π p ≈ 2 ω0 ω0 1 − ζ
(sec)
With the above result for ω0 , the period of the long period oscillation becomes, 2π P = √ V = 0.453 V g 2
(sec)
with V [m/sec]. According to this aprroximating calculation, the period in seconds of the long period oscillation is roughly 0.5 times the airspeed in [m/sec]. It should be remarked that for powered flight, the influence of Cmu (neglected here) can be of considerable influence on the phugoid motion. (b) q˙ = 0, α˙ = 0 A second, slightly more refined approximation of the long period oscillation goes less far than the previous one. Variations in α are now permitted, however, they are assumed to occur so slowly that α˙ remains negligible. In the equations (10-1) c the contributions due to Dc q¯ V and Dc α disappear, but none of the equations can be omitted. If CZq and CX0 are again neglected, the result is,
CXu − 2µc Dc CXα
CZ0
0
CZu
CZα
0
2µc
0
0
−Dc
1
Cmu
Cmα
0
Cmq
u ˆ
α =0 θ q¯ c V
The coefficients of the characteristic equation, following from an expansion of the characteristic determinant, are, A = 2µc CZα Cmq − 2µc Cmα
B = 2µc (CXu Cmα − Cmu CXα ) + Cmq (CZu CXα − CXu CZα ) C = CZ0 (Cmu CZα − CZu Cmα )
Chapter 11 Analysis of Asymmetric Equations of Motion
The asymmetric equations of motion will be treated along the same line as in chapter 10. Furthermore, the characteristic modes for the asymmetric aircraft motions will be discussed in this chapter, including several approximations for these modes. All simulations will be performed using a model of the Cessna Ce500 ‘Citation’.
11-1
Solution of the equations of motion
The equations for the disturbed asymmetric motions are obtained in the homogeneous form from equation (4-41), by omitting the terms due to the control surface deflections,
CYβ + CYβ˙ − 2µb Db
CL
CYp
0
− 21 Db
1
Cℓβ
0
2 Cℓp − 4µb KX Db
Cnβ + Cnβ˙ Db
0
Cnp + 4µb KXZ Db
Cℓr
β ϕ 0 =0 pb + 4µb KXZ Db 2V
CYr − 4µb
Cnr − 4µb KZ2 Db
rb 2V
(11-1) The character of the asymmetric motions is discussed below, using a quantitative example. Some characteristic parameters were derived for the symmetric eigenmotions in section 10-1. The same expressions also apply to the asymmetric motions, if the mean aerodynamic chord c¯ is replaced by the wing span b. The eigenvalues resulting from the characteristic equation are now indicated as λb .
Flight Dynamics
504
Analysis of Asymmetric Equations of Motion
The expressions for the time to damp to half an amplitude, and for the period, now read as follows, • Aperiodic motion (λb real), ln 12 b 0.693 b =− λb V λb V
T1 = 2
• Periodic motion corresponding to two complex, conjungate eigenvalues λb = ξb ± j ηb P =
T1 = − 2
T1 2
C1 =
P
2
eξb b (t+P ) ξb Vb
e
t
√
−1
2π b ηb V 0.693 b ξb V
−0.693 ηb ηb = −0.110 2π ξb ξb
=
V
δ = ln
j=
= ξb
V ξb 0.693 P = 2π = b ηb C1 2
In chapter 9 it was stated that CYβ˙ and Cnβ˙ are usually neglected, so it will be assumed here that, CYβ˙ = Cnβ˙ = 0
(11-2)
Expanding the characteristic determinant of equation (11-1), C + C − 2µ λ Yβ Yβ˙ b b 0 Cℓβ Cnβ + Cnβ˙ λb
CL
CYp
− 12 λb
1
0
2 Cℓp − 4µb KX λb
0
Cnp + 4µb KXZ λb
0 =0 Cℓr + 4µb KXZ λb Cnr − 4µb KZ2 λb CYr − 4µb
with the assumptions of equation (11-2), results in a quartic characteristic equation, A λ4b + B λ3b + C λ2b + D λb + E = 0
(11-3)
11-1 Solution of the equations of motion
505
The coefficients A to E for the asymmetric motions are derived from the characteristic determinant, see (11-3), 2 2 A = 16µ3b KX KZ2 − KXZ
2 2 2 KZ2 − KXZ + Cnr KX + Cℓp KZ2 + Cℓr + Cnp KXZ B = −4µ2b 2 CYβ KX C = 2µb
2 CYβ Cnr − CYr Cnβ KX + CYβ Cℓp − Cℓβ CYp KZ2 +
CYβ Cnp − Cnβ CYp + CYβ Cℓr − Cℓβ CYr KXZ +
2 4µb Cnβ KX
+ 4µb Cℓβ KXZ
1 + Cℓp Cnr − Cnp Cℓr 2
D = −4µb CL Cℓβ KZ2 + Cnβ KXZ + 2µb Cℓβ Cnp − Cnβ Cℓp + 1 1 CYβ Cℓr Cnp − Cnr Cℓp + CYp Cℓβ Cnr − Cnβ Cℓr + 2 2 1 CYr Cℓp Cnβ − Cnp Cℓβ 2 E = CL Cℓβ Cnr − Cnβ Cℓr
If the stability derivatives and inertial parameters are known for a certain aircraft configuration and flight condition, the coefficients A to E can be calculated. Using one of the common numerical methods, the eigenvalues may be obtained. As an example the asymmetric disturbed motions of a Cessna Ce500 ‘Citation’ are studied. The required data are given in table 11-1. From these data follow the coefficients A to E for the characteristic equation, A λ4b + B λ3b + C λ2b + D λb + E = 0 which can be obtained from equation(11-3). The roots of the characteristic polynomial are obtained by the program package MATLAB using the routine roots.m.
Flight Dynamics
506
Analysis of Asymmetric Equations of Motion
V
=
59.9 m/sec
µb
=
15.5
S
=
24.2 m2
2 KX
=
0.012
b
=
13.36 m
KZ2
=
0.037
CL
=
1.1360
KXZ
=
0.002
CYβ
=
−0.9896
Cℓβ
=
−0.0772
Cnβ
=
0.1638
CYp
=
−0.0870
Cℓp
=
−0.3444
Cnp
=
−0.0108
CYr
=
0.4300
Cℓr
=
0.2800
Cnr
=
−0.1930
CYδa
=
0
Cℓδa
=
−0.2349
Cnδa
=
0.0286
CYδr
=
0.3037
Cℓδr
=
0.0286
Cnδr
=
−0.1261
Table 11-1: Asymmetric stability and control derivatives for the Cessna Ce500 ‘Citation’
For a stable aircraft it follows that all eigenvalues λb must have negative real parts and for the disturbed motion to be positively damped. An explicit determination of the four eigenvalues, see also figure 11-1, for the Cessna Ce500 ‘Citation’ example become, λb1 = −0.3291 λb2 = −0.0108 λb3,4 = −0.0313 ± j 0.3314 The disturbed asymmetric motions consist apparently of two aperiodic eigenmotions and one periodic motion. One of the two aperiodic motions appears to be well damped, the other one is only very lightly damped or may be even slightly undamped (which is the case for most aircraft. The characteristics of the eigenmotions are as follows, • First aperiodic mode – λb1 = −0.3291
– the time to damp to half the amplitude T 1 ≈ 0.223 seconds 2
• Second aperiodic mode
11-2 General character of the asymmetric motions
507
4
λ3 3
2
jη
1
λ2
λ1
0
−1
−2
λ4
−3
−4 −3.5
−3
−2.5
−2
ξ
−1.5
−1
−0.5
0
Figure 11-1: The location of the eigenvalues λ = λb Vb for the asymmetric motions
– λb2 = −0.0108
– the time to damp to half the amplitude T 1 ≈ 6.782 seconds 2
• Periodic mode – λb3,4 = −0.0313 ± j 0.3314
– the period P = 2.004 seconds – the time to damp to half the amplitude T 1 ≈ 2.340 seconds 2
– the undamped natural frequency ω0 = 3.1495 rad/sec. – the damping ratio ζ = 0.0940 Figures 11-2 and 11-3 show the calculated responses for the aircraft, simulated by the program package MATLAB. In this particular case the disturbances are pulse-shaped rudder and aileron deflections.
11-2
General character of the asymmetric motions
From the time responses in figures 11-2 and 11-3 three eigenmotions may be distinguished, as was evident also from the eigenvalues. Flight Dynamics
508
Analysis of Asymmetric Equations of Motion
0.03
0.02
β [Rad]
0.01
0
−0.01
−0.02
−0.03 0
5
t [sec]
10
15
10
15
0.01
0
ϕ [Rad]
−0.01
−0.02
−0.03
−0.04
−0.05
−0.06
−0.07 0
5
t [sec]
Figure 11-2: Response curves for a pulse-shaped rudder deflection (∆δr = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’
11-2 General character of the asymmetric motions
509
0.1 0.08
p [Rad/sec]
0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 0
5
t [sec]
10
15
10
15
0.08
r [Rad/sec]
0.06
0.04
0.02
0
−0.02
−0.04
−0.06 0
5
t [sec]
Figure 11-2: (Continued) Response curves for a pulse-shaped rudder deflection (∆δr = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’
Flight Dynamics
510
Analysis of Asymmetric Equations of Motion 0.015 0.01 0.005
ψ [Rad]
0 −0.005 −0.01 −0.015 −0.02 −0.025 −0.03 −0.035 0
5
t [sec]
10
15
Figure 11-2: (Continued) Response curves for a pulse-shaped rudder deflection (∆δr = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’
• A highly damped aperiodic motion The corresponding eigenvalue is λb11 , real and strongly negative. The motion is apparent mostly in the rate of roll. The damping of this motion is due primarily to the damping rolling moment generated by the wing when rotating about the longitudinal axis, i.e. pb the moment Cℓp · 2V . • Aperiodic mode in which the aircraft sideslips, yaws and rolls The motion is characterized by the eigenvalue λb2 real, lightly positively or negatively damped. Depending on the sign of λb2 this eigenmotion converges or diverges. If the motion diverges (λb2 > 0), the aircraft after a disturbance enters into a sideslipping turn with ever increasing angle of roll and decreasing radius. During this motion the aircraft sideslips towards the inside of the turn, see figure 11-4. The aircraft is than called spirally unstable, as the aircraft describes a descending spiral. The eigenmotion is called the spiral motion. If it is damped, the aircraft has ‘spiral stability’. • Periodic mode in which the aircraft sideslips, yaws and rolls This eigenmotion is determined by the eigenvalues λb3,4 = ξb ± j ηb . Figure 11-5 shows the general character of the motion. Rolling and yawing velocity are presented for the case where the motion is damped ξb < 0. This motion is generally called ‘Dutch roll’. The name seems to be inspired by the resemblance to the motion of a skater on the ice.
11-2 General character of the asymmetric motions
511
−3
4
x 10
2
β [Rad]
0
−2
−4
−6
−8 0
5
t [sec]
10
15
0
−0.05
ϕ [Rad]
−0.1
−0.15
−0.2
−0.25
−0.3 0
5
t [sec]
10
15
Figure 11-3: Response curves for a pulse-shaped aileron deflection (∆δa = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’
Flight Dynamics
512
Analysis of Asymmetric Equations of Motion
0.05
p [Rad/sec]
0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3 0
5
t [sec]
10
15
10
15
0
−0.005
r [Rad/sec]
−0.01
−0.015
−0.02
−0.025
−0.03
−0.035
−0.04 0
5
t [sec]
Figure 11-3: (Continued) Response curves for a pulse-shaped aileron deflection (∆δa = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’
11-3 Routh-Hurwitz stability criteria for the asymmetric motions
513
0 −0.02 −0.04
ψ [Rad]
−0.06 −0.08 −0.1 −0.12 −0.14 −0.16 −0.18 0
5
t [sec]
10
15
Figure 11-3: (Continued) Response curves for a pulse-shaped aileron deflection (∆δa = +0.025 [Rad] during 1 second) for the Cessna Ce500 ‘Citation’
11-3
Routh-Hurwitz stability criteria for the asymmetric motions
The criteria for stability were discussed already in section 11-1. It can be proved, that in the transition from stability to instability the conditions E > 0 and R > 0 are the critical stability criteria. If only E becomes negative, one real eigenvalue changes sign. This means that one of the aperiodic motions changes from convergence to divergence. If R becomes more negative, the real part of the two complex, conjugate eigenvalues changes signs implying periodic divergence. According to the foregoing, the coefficient E determines the character of the aperiodic spiral motion and from the discriminant R follows the character of the (periodic) Dutch roll motion. In the following the stability criteria are used for a further study of the spiral motion and the Dutch roll oscillation of the aircraft. To prepare for this discussion, the factors determining the signs of the coefficients A to E are investigated. As a first approximation, the relatively small derivative CYp is equated to zero and CYr is neglected relative to 4µb . For aircraft having a straight wing without a large dihedral these assumptions are acceptable. Also, the coefficient KXZ , the product of inertia, is omitted. 2 and K 2 . The remaining stability derivatives deterKXZ is relatively small, relative to KX Y mining the magnitude and the signs of the coefficients A to E have, according to chapter 9, the following signs for conventional aircraft in normal flight, Flight Dynamics
514
Analysis of Asymmetric Equations of Motion
Figure 11-4: The spiral motion of an aircraft
11-3 Routh-Hurwitz stability criteria for the asymmetric motions
515
Figure 11-5: Characteristics of the Dutch Roll motion
Flight Dynamics
516
Analysis of Asymmetric Equations of Motion 0.1 roll−rate yaw−rate
0.08
p, r [Rad/sec]
0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 0
5
t [sec]
10
15
Figure 11-6: Roll- and yaw-rate characteristics of the Dutch Roll motion
CYβ < 0
Cℓp < 0
Cnp < 0
Cℓr > 0
Cnr < 0
The stability derivatives Cℓβ and Cnβ are practically the only ones the designer can manipulate to influence the lateral stability. As discussed in chapter 9, pleasant control characteristics require that, Cnβ > 0
Cℓβ < 0
Neglecting CYp , CYr and KXZ and using the known signs of the other derivatives and the inertial parameters the signs of the coefficients A to E can be established as follows, 2 A = 16µ3b KX KZ2 > 0
2 2 B = −4µ2b 2 CYβ KX KZ2 + Cnr KX + Cℓp KZ2 > 0
1 2 2 C = 2µb CYβ Cnr KX + CYβ Cℓp KZ2 + 4µb Cnβ KX + Cℓp Cnr − Cnp Cℓr 2
If Cnβ > 0 (the aircraft is then statically directionally stable), then C > 0.
1 D = −4µb CL Cℓβ KZ2 + 2µb Cℓβ Cnp − Cnβ Cℓp + CYβ Cℓr Cnp − Cnr Cℓp 2
11-4 Spiral and Dutch roll mode, the lateral stability diagram
517
If CL > 0, Cℓβ < 0 and Cnβ > 0, then also D > 0. E = CL Cℓβ Cnr − Cnβ Cℓr
At positive values of CL , the positive sign of E is determined by the condition, Cℓβ Cnr − Cnβ Cℓr > 0 It appears that the above requirement for dynamic stability is identical to the one derived in chapter 9, see section 9-3. The latter was based on the criterion for good control characteristics in steady turns using the ailerons or the rudder only.
11-4
Spiral and Dutch roll mode, the lateral stability diagram
In the previous section it was stated that positive damping of the spiral motion required that E > 0 and for damping of the Dutch roll motion, R > 0. The magnitudes of E and R are influenced to a large extent by the effective dihedral Cℓβ and the static directional stability Cnβ . The derivative Cℓβ can be varied by changing the geometric dihedral of the wing; Cnβ is modified by varying the surface Sv of the vertical tailplane and the taillength lv . But it should be remembered that changing Sv and/or lv influences also some other stability derivatives, such as Cnr and CYβ . The effect of changes in Cnβ and Cℓβ on E and R is graphically depicted in the so-called lateral stability diagram, along the axes, Cℓβ and Cnβ are plotted, see figure 11-7. For a particular aircraft, the boundaries E = 0 and R = 0 are plotted for a given aircraft configuration and flight condition in the lateral stability diagram. The remaining stability derivatives are assumed to remain constant, or to be adjusted to the external shape of the aircraft corresponding to each combination of Cℓβ and Cnβ . The two curves E = 0 and R = 0 indicate the boundaries of areas in the diagram for spiral stability and a damped Dutch roll oscillation. The diagram indicates for any combination of Cℓβ and Cnβ the state of the dynamic lateral stability for the aircraft configuration and flight condition under study. Only the most common case, where Cℓβ < 0 and Cnβ > 0 is further discussed. Using the lateral stability diagram, the spiral motion is considered first. The criterion for spiral stability is, Cℓβ Cnr − Cnβ Cℓr > 0 It follows, that spiral stability can be increased in two ways, e.g. by decreasing Cnβ or by increasing −Cℓβ . Because other requirements on the control characteristic have to be met as well, a decrease in Cnβ may be hardly acceptable. A value of Cℓβ too large negative may not Flight Dynamics
518
Analysis of Asymmetric Equations of Motion
Cn β ↑
E=0 E<0
Spiral instability E > 0, R > 0
Spiral stability and convergent Dutch Roll
R=0
R<0 Divergent Dutch Roll
−Cℓβ Figure 11-7: Lateral stability diagram
→
11-5 Approximate solutions
519
be acceptable either, as will be discussed below. The effect of these conflicting requirements is, that it may be difficult to choose the external form of the aircraft such that it is spirally stable in all required aircraft configurations and flight conditions. The divergence from a spirally unstable equilibrium flight condition is often relatively slow. In order to avoid even less desirable characteristics, a certain slight degree of spiral instability may be unavoidable. The curve for R = 0, determining stability of the m Dutch roll motion, appears to depend largely on the relative density µb = ρSb of the aircraft. The curve for R = 0 moves upward in the lateral stability diagram with increasing µb . This means that for given aerodynamic characteristics (stability derivatives) the damping of the Dutch roll motion decreases with increasing µb . In general, the Dutch roll motion of aircraft having a low wing loading, flying at relatively low altitudes are relatively well damped. A characteristic value of the damping for such aircraft is ζ ≈ 0.15 or C 1 ≈ 0.7. 2
For the Dutch roll motion it does not suffice to determine only the sign of R, because good flying qualities imply a certain minimal damping of the Dutch roll motion. This minimal required damping has been made a function of the period of the motion, see reference [13]. This may be explained as follows. If the period is short, the reactions of the pilot will be too slow to stabilize the motion by suitable control deflections. In such a case a relatively high damping or a low value of T 1 , or C 1 , is required. If the period is longer, the 2 2 damping of the motion may be lower, because in such a situation the pilot has less difficulty to improve the damping of the aircraft’s motions via his control deflections. However, the requirement relating the period and the minimal acceptable time to damp to half amplitude has turned out to provide insufficient guarantee for good flying qualities. More recent requirements, see e.g. reference [13], stipulate that the damping of the Dutch roll has a certain minimal value depending on the roll to yaw ratio |p| |r| occurring during the Dutch roll motion. The interpretation of this ratio
11-5
|p| |r|
is shown in figure 11-6.
Approximate solutions
1. Heavily damped aperiodic rolling motion The aperiodic rolling moment may be approximated by assuming that the aircraft can only roll about the longitudinal axis. This is permissable because this motion often disappears before the other eigenmotions of the aircraft have really started. If only the rolling motion is considered, in the equations (11-1) the angle of sideslip β, and the rb non-dimensional yaw-rate 2V disappear and the equations for the lateral force and the yawing moment can be omitted. In the remaining equation for the rolling moments the pb angle of roll ϕ does not occur, so the kinematic relation − 12 Db ϕ + 2V = 0 is no longer needed. The rolling moment equation then reads, 2 Cℓp − 4 µb KX Db
pb =0 2V Flight Dynamics
520
Analysis of Asymmetric Equations of Motion It is easy to see, that this expression gives the real eigenvalue, λb1 =
Cℓp 2 4 µb KX
2. Dutch roll motion (a) ϕ =
pb 2V
=0
For a large category of conventionial aircraft a certain approximation of the Dutch roll motion results if the rolling component is discarded in the Dutch roll motion. pb pb If ϕ and 2V are set at zero, the ϕ- and 2V -columns in the equations of motion disappear and the rolling moment equation can be omitted since the rolling moments have to remain in balance. Only the aerodynamic force Y - and the moment N -equations remain. As usual, CYβ˙ and Cnβ˙ are neglected, whereas CYr is insignificant relative to 4 µb ,
CYβ − 2µb Db −4µb Cnβ
Cnr −
4µb KZ2 Db
β rb 2V
=0
The coefficients of the quadratic characteristic equation become, A = 8 µ2b KZ2 B = −2µb Cnr + 2KZ2 CYβ
C = 4µb Cnβ + CYβ Cnr From these coefficients ω0 , ζ, P and T 1 can easily be obtained. 2
(b) ϕ =
pb 2V
= 0, yawing rotation only
A more restrictive assumption is that the trajectory of the aircraft’s c.g. is a straight line during the oscillation. This implies that the course angle χ is constant, where χ = β + ψ = 0, or β = −ψ. It implies as well that the Y -equation is superfluous. The remaining yawing moment equation is further simplified by the rb fact that 2V = 12 Db ψ and β = −ψ. The result is,
−Cnβ
1 2 2 + Cnr Db − 2µb KZ Db ψ = 0 2
The coefficients A, B and C of the characteristic equation are, A = −2µb KZ2
11-5 Approximate solutions
521
1 B = Cnr 2 C = −Cnβ The expressions for ω0 and ζ are obtained very simply as, r
V ω0 = b
C V = A b
s
Cnβ 2µb KZ2
B Cnr ζ=− √ =− q 2 AC 4 2µb K 2 Cn Z
3. Aperiodic spiral motion
β
This is usually a very slow eigenmotion, in which the aircraft sideslips, yaws and rolls. It is, therefore, permissible to approximate the motion by assuming all linear and angular accelerations to be negligible. In the equations of motion this means, Db β = Db
pb rb = Db =0 2V 2V
In addition to CYr , also CYp is neglected.
CYβ
0 Cℓ β
Cnβ
CL
0
−4µb
− 12 Db
1
0
0
Cℓp
Cℓr
0
Cnp
Cnr
β
ϕ pb = 0 2V rb 2V
Expanding the characteristic equation results in the eigenvalue,
λb4 =
Cℓp
2CL Cℓβ Cnr − Cnβ Cℓr CYβ Cnr + 4µb Cnβ − Cnp CYβ Cℓr + 4µb Cℓβ
It is easy to verify that the dominator is negative if all stability derivatives have their normal signs. For convergence of this motion it is necessary that λb4 < 0, which means, CL Cℓβ Cnr − Cnβ Cℓr > 0
This corresponds to the requirement E > 0, derived in section 11-3. Flight Dynamics
522
Analysis of Asymmetric Equations of Motion
4. Dutch roll motion and the aperiodic rolling motion If the assumption is made that the c.g. moves along a straight line, but in addition the rolling motion is permitted, only the Y -equation disappears. The following equations pb in ψ and 2V result,
2D −Cℓβ + 12 Cℓr Db + 2µb KXZ Db2 Cℓp − 4µb KX b
−Cnβ +
1 2 Cnr Db
−
2µb KZ2 Db2
Cnp + 4µb KXZ Db
The characteristic equation is a cubic,
ψ pb 2V
=0
A λ3b + B λ2b + C λb + D = 0 The coefficients are then, 2 2 A = 4µ2b KX KZ2 − KXZ
B = −µb
2 + Cℓp KZ2 Cℓr + Cnp KXZ + Cnr KX
1 2 Cℓp Cnr − Cnp Cℓr + C = 2µb Cℓβ KXZ + Cnβ KX 4 D=
1 Cℓβ Cnp − Cnβ Cℓp 2
Evidently no explicit analytical solution to this cubic can be given. By using the approximation for the eigenvalue of the motion in roll, the characteristic equation may be reduced to a quadratic equation.
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Appendix A List of Symbols
Roman Symbols a A
Speed of sound Center of the Earth, origin of FI
A
Wing aspect ratio
b bf B c cf ca,e,r,t cr ct cm
Wing span Local fuselage width Moment of momentum, angular momentum Local wing chord Wing chord flap Chord of aileron, elevator, rudder or tab, behind hinge line Chord length in the plane of symmetry Chord length at wing tip Mean wing chord (Eng. notation: c¯)
c¯
Mean aerodynamic chord (English notation: c¯)
c¯w,h,v
Mean aerodynamic chord of wing, horizontal tailplane, or vertical tailplane Mean aerodynamic chord of control surfaces and tabs, behind hinge line
c¯a,e,r,t cd
Drag coefficient two-dimensional flow
CFD
Computational Fluid Dynamics
CD
Drag coefficient three-dimensional flow
c.g.
Center of gravity
b2 = S
S = b
"
d = 1 2 2 ρV c
#
"
D = 1 2 ρV S 2
#
Flight Dynamics
536
List of Symbols
Che
Hinge moment coefficient of the elevator. The indices a, r or t are used to indicate the hinge moment coefficients of the aileron, the rudder or a tab respectively
Chα Chδ Chδt cℓ Cℓ CL cℓa cℓb
Coefficient of the additional lift distribution Coefficient of the basic lift distribution
cℓα
Derivative of the lift-coefficient with respect to angle of attack, two-dimensional flow
CLα
Derivative of the lift-coefficient with respect to angle of attack, three-dimensional flow Coefficient of the rolling moment due to one inoperative engine
Cℓe Cℓp Cℓr Cℓβ Cℓδa Cℓδr cm
Coefficient of the aerodynamic moment about the Y axis, two-dimensional flow
Cm
Coefficient of the aerodynamic moment about the Y axis, three-dimensional flow Coefficient of the moment about the aerodynamic center, two-dimensional flow Coefficient of the moment about the aerodynamic center, three-dimensional flow
cma.c. Cma.c.
"
He = 1 2 ¯e 2 ρVh Se c
#
∂Ch = ∂αa,h,v ∂Ch = ∂δ a,e,r ∂Ch = " ∂δta,e,r # ℓ = 1 2 ¯ 2 ρV c " # L = 1 2 ρV Sb " 2 # L = 1 2 2 ρV S dcℓ = dα dCL = dα "
∂Cℓ
#
= pb " ∂ 2V # ∂Cℓ = rb ∂ 2V ∂Cℓ = ∂β ∂Cℓ = ∂δa ∂Cℓ = ∂δr # " m = 1 2 2 2 ρV c " # M = 1 2 c 2 ρV S¯ = cm0
537 Cmh
Contribution of the horizontal tailplane to Cm
Cmq "
"
∂Cm = q¯c ∂V
1 ∂M = 1 c ∂u 2 ρV S¯
Cmu Cmw Cm0 Cmαf ix Cmαf ree
contribution of the wing with fuselage and nacelles to Cm Cm at CL = δe = 0 ∂Cm at δe = constant ∂α ∂Cm at Fe = 0 ∂α
Cmα˙ Cmβ 2 Cmδ
Elevator efficiency
cn
Normal force coefficient, two-dimensional flow
Cn
Yawing moment coefficient
CN
Normal force coefficient, three-dimensional flow
Cne
Yawing moment coefficient due to one inoperative engine
Cnp Cnr Cnβ
Static directional stability
Cnδa Cnδr CNh
Normal force coefficient of the horizontal tailplane
CNh0
CNh at αh = δe = δte = 0
CNhα CNhδ CNhδ
t
# #
"
# ∂Cm = α¯ ∂ V˙ c ∂Cm = 2 ∂β ∂Cm = ∂δe # " n = 1 2 2 ρV c # " N = 1 2 ρV Sb " 2 # N = 1 2 2 ρV S "
∂Cn
#
= = =
∂CNh ∂αh ∂CNh ∂δe ∂CNh ∂δte
= pb " ∂ 2V # ∂Cn = rb ∂ 2V ∂Cn = ∂β ∂Cn = ∂δa ∂Cn = ∂δr # " Nh = 1 2 2 ρVh Sh
Flight Dynamics
538
List of Symbols
CNp
Normal force coefficient on the propeller
CNw
Coefficient of the normal force on the wing with fuselage and nacelles
cr
Coefficient of the resultant aerodynamic force, twodimensional flow
CR
Coefficient of the resultant aerodynamic force, threedimensional flow
ct
Coefficient of the tangential force, two-dimensional flow
CT
Coefficient of the tangential force, three-dimensional flow
CTh
Coefficient of the tangential force of the horizontal tailplane
CTw
Coefficient of the tangential force on the wing with fuselage and nacelles
CX CXq CXu CX0 CXα CXδe CY CYp CYr CYβ CYδa CYδr
CX in steady flight
"
# Np = 1 2 ρV Sp " 2 # Nw = 1 2 2 ρV S " # r = 1 2 ¯ 2 ρV c " # R = 1 2 2 ρV S " # t = 1 2 ¯ 2 ρV c " # T = 1 2 2 ρV S " # Th = 1 2 2 ρVh Sh " # Tw = 1 2 2 ρV S # " X = 1 2 ρV S "2 # ∂CX = q¯c ∂V " # 1 ∂X = 1 ∂u 2 ρV S ∂CX = ∂α ∂CX = ∂δe # " Y = 1 2 ρV S "2 # ∂CY = pb " ∂ 2V # ∂CY = rb ∂ 2V ∂CY = ∂β ∂CY = ∂δa ∂CY = ∂δr
539 "
# Z = 1 2 ρV S "2 # ∂CZ = q¯c ∂V " # 1 ∂Z = 1 ∂u 2 ρV S
CZ CZq CZu CZ0
CZ in steady flight
CZα CZα˙ CZδe C1 2
d
Characteristic diameter of a jet engine
d
Drag, two-dimensional flow
D
Diameter of a propeller
D
Drag, three-dimensional flow
D
differential operator
D2 Db
Non-dimensional differential operator, asymmetric motions
Dc
Non-dimensional differential operator, symmetric motions Base of the natural logarithms Distance of the center of pressure of a profile behind the leading edge Aileron, elevator or rudder control force exerted by the pilot Vehicle (aircraft) center of mass Altitude
e e Fa,e,r G h He
Ha,e,ra Ha,e,rf Ha,e,rs Ha,e,rw
Hinge moment about the elevator hinge line, subscripts a, e, r indicate the aileron, elevator or rudder respectively Aerodynamic hinge moment Hinge moment due to friction in the control mechanism Hinge moment due to a spring in the control mechanism Hinge moment due to static unbalance of the control mechanism
∂CZ = ∂α # " ∂CZ = α¯ ∂ V˙ c ∂CZ = " ∂δe # T1 = 2 P 1 2 = cd ρV c¯ 2 1 2 = CD ρV S 2 d = dt d2 = 2 dt d b d = = dsb V dt d c¯ d = = dsc V dt
1 2 = Che ρVh Se c¯e 2
Flight Dynamics
540
List of Symbols
i ih,p
Unit vector along the X-axis Angle of incidence of the horizontal tail or the propeller axis, relative to the Xm -axis
Ix
Moment of inertia about the X-axis
Ix0
Principle moment of inertia
Iy
Moment of inertia about the Y-axis
Iy0
Principle moment of inertia
Iz
Moment of inertia about the Z-axis
Iz0 j
Principle moment of inertia Unit vector along the Y -axis
Jxy
Product of inertia
Jxz
Product of inertia
Jyz
Product of inertia
k k
Factor related to the variation of thrust with speed Unit vector along the Z-axis
kx ky kz
Radius of gyration about the X-axis Radius of gyration about the Y-axis Radius of gyration about the Z-axis
kxz KX
Non-dimensional radius of gyration about the X-axis
KY
Non-dimensional radius of gyration about the Y -axis
KZ
Non-dimensional radius of gyration about the Z-axis
KXZ
Non-dimensional product of inertia
ℓ
Lift, two-dimensional flow
lf lh lv lw LTI
Fuselage length Tail length, horizontal tail Tail length, vertical tail Length of the fuselage ahead of the wing Linear Time Invariant
Z = Z = Z =
y 2 + z 2 dm x2 + z 2 dm 2
x +y
2
dm
Z = xydm Z = xzdm Z = yzdm "
r
= " r
= " r
Ix m Iy m
# #
# Iz = m Jxz = m kx = b ky = b kz = b kxz = 2 b 1 2 = cℓ ρV c 2 [= xh − xw ] [= xv − xw ]
541 1 2 = Cℓ ρV Sb 2 1 2 = CL ρV S 2
L
Rolling moment about the X-axis
L
Lift, three-dimensional flow
Lg Lt
Longitude Latitude
m m mac m.p.
Aerodynamic moment about dimensional flow Mass Mean aerodynamic chord Manoeuvre point
M
Moment about the Y -axis, three-dimensional flow
M
Mach number
M,M
Total moment
n
Normal force, two-dimensional flow
n.p.
Neutral point
N
Normal force, three-dimensional flow
N
Yawing moment about the Z-axis
Nh
Normal force on the horizontal tailplane
Np
Normal force on the propeller
Nw
Normal force on the wing with fuselage and nacelles
O p p P0 P
Origin reference frames FE and FO Static pressure Angular velocity about the X-axis Static pressure in undisturbed flow Period of an oscillation
q
Dynamic pressure
q r r, r
Angular velocity about the Y -axis Angular velocity about the Z-axis Vector indicating a position
r
Resultant aerodynamic force, two-dimensional flow
R
Resultant aerodynamic force, three-dimensional flow
R Re sa,e,r
Routh’s discriminant Reynolds number Pilot’s control deflection; aileron, elevator or rudder control respectively
the
Y -axis,
two-
1 2 2 = cm ρV c 2 [= c¯] 1 2 = Cm ρV Sc 2 h ai = V 1 2 = cn ρV c 2 1 2 = CN ρV S 2 1 2 = Cn ρV Sb 2 1 2 = CNh ρVh Sh 2 1 2 = CNp ρV Sp 2 1 2 = CNw ρV S 2
1 2 = ρV 2
1 2 = cr ρV c 2 1 2 = CR ρV S 2
Flight Dynamics
542
List of Symbols
sb
Non-dimensional parameter of time, asymmetric motions
sc
Non-dimensional parameter of time, symmetric motions
S Sa Se,r,t Sh,v
Wing area Area of one aileron, behind hinge line Area of elevator, rudder or tab respectively, behind hinge line Area of horizontal or vertical tailplane
Sp
Area of the propeller disk
t
Time
t
Tangential force, two-dimensional flow
T
Tangential force, three-dimensional flow
Tc
Thrust coefficient of a propeller
Tc
Thrust coefficient of a jet engine
Tc′
Thrust coefficient
Th Tw T1
Tangential force on the horizontal tailplane Tangential force on the wing with fuselage and nacelles Time to damp to half amplitude
T2
Time to double amplitude
2
u ˆ u ∆u, u v V, V ∆v, v V Va Ve Vm.c. Vtrim Vk w ∆w, w W
Component of V along the X-axis Change in the component of V along the X-axis Component of V along the Y -axis Velocity vector Change in the component of V along the Y -axis Magnitude of the airspeed vector V Aerodynamic velocity vector, velocity relative to a particle in the undisturbed air equivalent airspeed Minimum control speed Trimmed airspeed, V at which Fe = 0 Kinematic velocity vector, velocity relative to the normal Earth-fixed reference frame Component of V along the Z-axis Change in the component of V along the Z-axis Aircraft weight
V = t b V = t c¯
h
π 2i D 4 1 2 = ct ρV c 2 1 2 = CT ρV S 2 Tp = 2 2 " ρV D # Tp = 1 2 2 ρV d " 2 # Tp = 1 2 2 ρV S =
h
i = −T 1 2 du = V0
543 x xa.c. xc.g. xd xh xmf ree xmf ixed xnf ree xnf ixed xp xv xw x0 X y Y z zh zp zv zw z0 Z
x-coordinate, abscissa Abscissa of the a.c. of the wing Abscissa of the c.g. Abscissa of the center of pressure Abscissa of the a.c. of the horizontal tailplane Abscissa of the manoeuver point, stick free Abscissa of the manoeuver point, stick fixed Abscissa of the neutral point, stick free Abscissa of the neutral point, stick fixed Abscissa of the center of the propeller disk Abscissa of the a.c. of the vertical tailplane Abscissa of the a.c. of the wing with fuselage and nacelles Abscissa of the leading edge of the mac Component of the total aerodynamic force along the Xaxis y-coordinate, ordinate Component of the total aerodynamic force along the Y axis, lateral force z-coordinate z-coordinate of the a.c. of the horizontal tailplane z-coordinate of the center of the propeller disk z-coordinate of the a.c. of the vertical tailplane z-coordinate of the a.c. of the wing with fuselage and nacelles z-coordinate of the leading edge of the m.a.c. component of the total aerodynamic force along the Zaxis
1 2 = CX ρV S 2 1 2 = CY ρV S 2
1 2 = CZ ρV S 2
Flight Dynamics
544
List of Symbols
Greek Symbols α α0 α0 αh,v,w
θ0
angle of attack angle of attack of the Xa -axis at CL = 0 angle of attack in steady flight angle of attack of the horizontal or vertical tailplane or the wing angle of sideslip flight path angle, angle between V , relative to the earth, and the horizontal plane flight path angle in steady flight dihedral, angle between the Yr -axis and the projection of the 14 -chord line on the Yr OZr -plane effective dihedral logarithmic decrement Total aileron deflection deflection of the right or left aileron elevator deflection flap deflection rudder deflection spoiler deflection deflection of a trim tab wing twist or washout angle downwash angle, usually at the horizontal tailplane angle between the Xr -axis and the principal inertial Xaxis damping ratio of an oscillation propeller efficiency imaginary part of a complex eigenvalue angle between the principal inertial X-axis and the Xs axis of the stability reference frame angle of pitch, angle between the Xr -axis and the horizontal plane angle of pitch in steady flight
λ
Taper ratio
λ λb λc Λ
eigenvalue non-dimensional eigenvalue, asymmetric motions non-dimensional eigenvalue, symmetric motions angle of sweep, angle between the Yr -axis and the projection of the 14 -chord line on the Xr OYr -plane
β γ γ0 Γ Γe δ δa δar , δaℓ δe δf δr δs δt ε ε ε ζ η η η θ
µb
Relative density, asymmetric motions
µc
Relative density, symmetric motions
[= δar − δaℓ ]
ct λ= cr
m = ρSb m = ρS¯ c
545 ν ξ ρ σ τ τ ϕ Φ χ χ1 χ2 ψ ω ω ω0 Ω Ω
Kinematic viscosity of the air Real part of a complex eigenvalue Air density Sidewash angle, usually at the vertical tailplane Time constant Trailing edge angle of a profile angle of roll, angle of the Y -axis and the intersection of the Y OZ-plane and the horizontal plane Bank angle, angle between the Y -axis and the horizontal plane Track angle Angle between the direction of C R and the Z-axis Angle between the neutral line and the Z-axis Yaw angle Angular velocity Circular frequency Undamped natural frequency Total angular velocity about the center of gravity Angular velocity vector
Flight Dynamics
546
List of Symbols
Subscripts a a a a a.c. b b c.g. e e E f f f f ix f ree h i i I k k n O p r r r s s t t u v w w x y z 0
Aerodynamic reference frame Aerodynamic Aerodynamic (velocity) Aileron Aerodynamic center Balance tab Body-fixed reference frame Center of gravity Elevator Engine Normal Earth-fixed reference frame Flap Fuselage Friction δe = constant Fe = 0, (Che = 0) Horizontal tailplane Initial Interference Inertial reference frame Kinematic reference frame Kinematic (velocity) Nacelle vehicle carried normal reference frame Relating to the propulsive system Vehicle reference frame Rudder Root of the wing Spring in the control mechanism Stability reference frame tip trim tab ultimate, final vertical tailplane static unbalance wing or wing with fuselage and nacelles along the X-axis along the Y -axis along the Z-axis initial, steady flight condition
Appendix B Aircraft Parameters
B-1
Introduction
In this appendix a list of stability and control derivatives are given for several aircraft. Also 2 , K2 , K2 , K the inertial parameters, i.e. µb , µc , KX XZ , etcetera, for these aircraft are Y Z provided.
B-2
Linearized Equations of Motion
In this section the linear equations of motion are given in which the inertia parameters and stability and control derivatives have to be substituted for simulation.
CXu − 2µc Dc CZu
CXα
CZ0
CZα + (CZα˙ − 2µc ) Dc −CX0
CXq CZq + 2µc
0
0
−Dc
1
Cmu
Cmα + Cmα˙ Dc
0
Cmq − 2µc KY2 Dc
−CXδe
−CZδ e = 0
−Cmδe
u ˆ
α = θ q¯ c V
δe
and as in equation (4-41), Flight Dynamics
548
Aircraft Parameters
CYβ + CYβ˙ − 2µb Db
CL
CYp
0
− 12 Db
1
Cℓβ
0
2D Cℓp − 4µb KX b
Cnβ + Cnβ˙ Db
0
Cnp + 4µb KXZ Db
−CYδa
0 = −Cℓ δa
−Cnδa
B-3
CYr − 4µb
β
ϕ 0 = pb Cℓr + 4µb KXZ Db 2V Cnr − 4µb KZ2 Db
rb 2V
−CYδr
δa δr
0 −Cℓδr −Cnδr
LTI-System Representation
The LTI-system, or state-space, representation of the equations of motions are given below. The symmetric and asymmetric equations of motion in state-space form are, see equations (4-46) and (4-50), Symmetric equations of motion
u ˆ˙
xu
xα
xθ
0
α˙ zu zα zθ zq = ˙ V θ 0 0 0 c¯ q¯ ˙c mu mα mθ mq V
u ˆ
xδe
α zδe + θ 0 q¯ c V
mδe
δe
Asymmetric equations of motion
β˙
yβ
yϕ yp
ϕ˙ 0 0 = pb ˙ 2V lβ 0 rb ˙ nβ 0 2V
2 Vb lp np
yr
β
0
0 ϕ 0 pb + lr 2V lδa nr
rb 2V
yδr 0 lδr
nδa nδr
δa δr
B-3 LTI-System Representation
549
We refer to tables 4-9 and 4-10 for the definition of the parameters used in the above-mentioned state-space representations of the equations of motion.
In the following tables inertial parameters and stability and control derivatives for several aircraft are given.
Flight Dynamics
550
Aircraft Parameters
V S KY2
= = =
59.9 m/sec 24.2 m2 0.980
m lh xcg
= = =
4547.8 kg 5.5 m 0.30 c¯
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
b 2 KX
CYβ CYp CYr CYδa CYδr
c¯ µc
= =
2.022 m 102.7
0 −0.2199 0.4653 0 0 0
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−1.1360 −2.2720 −5.1600 −1.4300 −3.8600 −0.6238
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
0 −0.4300 −3.7000 −7.0400 −1.5530
= =
13.36 m 0.012
CL KZ2
= =
1.1360 0.037
µb KXZ
= =
15.5 0.002
= = = = =
−0.9896 −0.0870 0.4300 0 0.3037
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
−0.0772 −0.3444 0.2800 −0.2349 0.0286
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.1638 −0.0108 −0.1930 0.0286 −0.1261
Table B-1: Symmetric and asymmetric stability and control derivatives for the Cessna Ce500 ‘Citation’, Cruise
B-3 LTI-System Representation
V S KY2
= = =
124.5 m/sec 70.0 m2 2.72
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 −0.09 0.15 0 0 0
b 2 KX
= =
29.00 m 0.0127
CYβ CYp CYr CYδa CYδr
= = = = =
−0.90 −0.23 0.48 0 0.29
551
m
=
16200 kg
c¯ µc
= =
2.58 m 137.5
xcg
=
0.32 c¯
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−0.45 −0.90 −5.90 −1.59 −7.36 −0.44
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
0 −0.80 −6.50 −16.50 −1.80
CL KZ2
= =
0.45 0.0342
µb KXZ
= =
12.22 0.0
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
−0.09 −0.60 0.23 −0.086 0.029
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.11 0.02 −0.14 0.0 −0.086
Table B-2: Symmetric and asymmetric stability and control derivatives for the Fokker F-27 ‘Friendship’, Cruise
V S KY2
= = =
66.75 m/sec 16.17 m2 0.6814
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 −0.093 0.18 0 0 0
m
=
1199.8 kg
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−0.310 −0.620 −4.631 −0.850 −1.95 −0.430
c¯ µc
= =
1.494 m 47.05
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
0 −0.890 −2.600 −6.200 −1.28
Table B-3: Symmetric stability and control derivatives for the Cessna Ce-172 ‘Skyhawk’, Cruise
Flight Dynamics
552
Aircraft Parameters
V S KY2
= = =
51.82 m/sec 21.37 m2 0.8979
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 0 0.580 0 0 0
m
=
5897 kg
xcg
=
0.32 c¯
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−1.640 −3.72 −5.5296 −0.80 −2.050 −0.400
c¯ µc
= =
2.134 m 105.56
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
−0.004 −0.660 −2.50 −6.75 −0.980
Table B-4: Symmetric stability and control derivatives for the Learjet I, Approach
V S KY2
= = =
103.63 m/sec 26.01 m2 1.646
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 −0.06 0.06 0 0 0
m
=
3175 kg
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−0.191 −0.402 −5.510 −1.250 −4.05 −0.600
c¯ µc
= =
1.981 m 58.35
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
0 −1.890 −4.550 −17.00 −2.00
Table B-5: Symmetric stability and control derivatives for the Beechcraft M99, Cruise
B-3 LTI-System Representation
V S KY2
= = =
67.36 m/sec 510.97 m2 2.3345
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 0 0.630 0 0 0
553
m lh xcg
= = =
255830 kg 31.09 m 0.25 c¯
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−1.760 −3.3 −5.933 −3.350 −2.825 −0.360
c¯ µc
= =
8.321 m 49.12
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
0.071 −1.450 −1.650 −10.70 −1.400
Table B-6: Symmetric stability and control derivatives for the Boeing 747-100, Approach
V S KY2
= = =
129.1 m/sec 510.97 m2 2.488
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 −0.0478 0.687 0 0 0
m lh xcg
= = =
254240 kg 31.09 m 0.32 c¯
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−0.477 −0.954 −4.487 6.62 −4.27 −0.353
c¯ µc ρ
= = =
8.321 m 56.51 1.058 kg m−3
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
−0.0252 −0.554 −3.39 −19.45 −1.42
Table B-7: Symmetric stability and control derivatives for the Boeing 747-100, Holding, flaps up
Flight Dynamics
554
Aircraft Parameters
V S KY2
= = =
73.0 m/sec 510.97 m2 2.488
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
0 −0.42 1.59 0 0 0
m lh xcg
= = =
254240 kg 31.09 m 0.32 c¯
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−1.49 −2.98 −5.293 6.70 −6.66 −0.353
c¯ µc ρ
= = =
8.321 m 48.81 1.125 kg m−3
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
−0.185 −1.05 −3.45 −21.98 −1.42
Table B-8: Symmetric stability and control derivatives for the Boeing 747-100, Approach, flaps 33o
V S KY2
= = =
66.142 m/sec 510.97 m2 3.77
CX0 CXu CXα CXα˙ CXq CXδe
= = = = = =
−0.0944 −0.90 1.2542 0 0 0
m lh xcg
= = =
249650 kg 31.09 m 0.25 c¯
CZ0 CZu CZα CZα˙ CZq CZδe
= = = = = =
−1.80 −3.60 −5.344 −2.00 −2.84 −0.355
c¯ µc ρ
= = =
8.321 m 48.40 1.125 kg m−3
Cmu Cmα Cmα˙ Cmq Cmδe
= = = = =
0 −1.536 −1.70 −10.75 −1.409
Table B-9: Symmetric stability and control derivatives for the Boeing 747-100, Landing
B-3 LTI-System Representation
V S b m
= = = =
145 m/sec 153.5 m2 37.49 m 59020 kg
CYβ CYp CYr CYδa CYδr
= = = = =
−0.5960 0 0.3690 0 0.2150
555
µb 2 KX KZ2 KXZ
= = = =
17.219 0.0283 0.0471 0
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
−0.1374 −0.5200 0.1440 −0.0975 0
h
=
6900 m
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.1173 −0.0210 −0.1800 0.0052 −0.1030
Table B-10: Asymmetric stability and control derivatives for the Lockheed L1049C ‘Super Constellation’, Cruise
V S b m
= = = =
65.4 m/sec 153.5 m2 37.49 m 51200 kg
CYβ CYp CYr CYδa CYδr
= = = = =
−0.5610 0 0.1945 0 0.2150
µb 2 KX KZ2 KXZ
= = = =
7.26 0.0326 0.0543 0
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
−0.1374 −0.5200 0.2750 −0.0975 0
h
=
0m
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.1173 −0.0626 −0.1800 0.0052 −0.1030
Table B-11: Asymmetric stability and control derivatives for the Lockheed L1049C ‘Super Constellation’, Approach
Flight Dynamics
556
Aircraft Parameters
V S b m
= = = =
77.1 m/sec 470 m2 24.4 m 100000 kg
CYβ CYp CYr CYδa CYδr
= = = = =
−0.3640 0 0 0 0.1290
µb 2 KX KZ2 KXZ
= = = =
7.185 0.052 0.249 −0.066
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
−0.1660 −0.1410 0.2500 −0.1010 0
h
=
0m
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.1360 −0.1430 −0.2100 0 −0.0790
Table B-12: Asymmetric stability and control derivatives for the BAC-Aerospatiale ‘Concorde’, Approach
V S b m
= = = =
660 m/sec 18.6 m2 6.7 m 11800 kg
CYβ CYp CYr CYδa CYδr
= = = = =
−1.4200 0 0 −0.0735 0.3200
µb 2 KX KZ2 KXZ
= = = =
1072 0.0134 0.247 0
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
0.0100 −0.3150 0 −0.0500 0.0100
h
=
20000 m
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.4000 0 −1.4500 0.0490 −0.2000
Table B-13: Asymmetric stability and control derivatives for the North-American X-15 experimental aircraft, Cruise (unspecified)
B-3 LTI-System Representation
V S b m
= = = =
40.2 m/sec 23.23 m2 14.63 m 2315 kg
CYβ CYp CYr CYδa CYδr
= = = = =
−0.6000 0 0.1600 0 0.2420
557
µb 2 KX KZ2 KXZ
= = = =
5.56 0.0845 0.0192 0
Cℓβ Cℓp Cℓr Cℓδa Cℓδr
= = = = =
−0.0560 −0.5500 0.1200 −0.1120 0
h
=
0m
Cnβ Cnp Cnr Cnδa Cnδr
= = = = =
0.0248 −0.0600 −0.0530 0.0015 −0.0970
Table B-14: Asymmetric stability and control derivatives for the De Havilland Canada DHC-2 ‘Beaver’, Approach
Flight Dynamics
558
Aircraft Parameters
Appendix C MATLAB Files
C-1
Introduction
In this appendix a list of MATLAB files, as used in this book, is given. The presented MATLAB files are only to be used regarding the LTI, or state-space, form, i.e., x˙ = A x + B u y =C x+D u All files are current with MATLAB version 5.2.
C-2
Root Finding
MATLAB routine roots.m returns the roots of a polynomial, EIG = roots(POLYNOMIAL) with EIG the eigenvalues, or roots in complex form, EIG = A + j B, and POLYNOMIAL the coefficients of a polynomial in descending order, for instance, POLYNOMIAL = [an an−1 an−2 · · · a2 a1 a0 ] for the polynomial, an xn + an−1 xn−1 + an−2 xn−2 + · · · + a2 x2 + a1 x + a0 or, Flight Dynamics
560 ROOTS
MATLAB Files Find polynomial roots. ROOTS(C) computes the roots of the polynomial whose coefficients are the elements of the vector C. If C has N+1 components, the polynomial is C(1)*X^N + ... + C(N)*X + C(N+1). See also POLY, RESIDUE, FZERO.
C-3
Eigenvalue Computation
Regarding stability analysis of LTI systems, the following program is used, EIG
Eigenvalues and eigenvectors. E = EIG(X) is a vector containing the eigenvalues of a square matrix X. [V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D. [V,D] = EIG(X,’nobalance’) performs the computation with balancing disabled, which sometimes gives more accurate results for certain problems with unusual scaling. E = EIG(A,B) is a vector containing the generalized eigenvalues of square matrices A and B. [V,D] = EIG(A,B) produces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that A*V = B*V*D. See also CONDEIG, EIGS. Overloaded methods help sym/eig.m help lti/eig.m
C-4
Simulation
For the simulation of LTI systems, the following programs are used, Step-response STEP Step response of continuous-time linear systems.
C-4 Simulation
561
Y = STEP(A,B,C,D,iu,T) calculates the response of the system: . x = Ax + Bu y = Cx + Du to a step applied to the iu’th input. Vector T must be a regularly spaced time vector that specifies the time axis for the step response. STEP returns a matrix Y with as many columns as there are outputs y, and with LENGTH(T) rows. [Y,X] = STEP(A,B,C,D,iu,T) also returns the state time history. Y = STEP(NUM,DEN,T) calculates the step response from the transfer function description G(s) = NUM(s)/DEN(s) where NUM and DEN contain the polynomial coefficients in descending powers.
Time-response to arbitrary inputs LSIM Simulation of continuous-time linear systems to arbitrary inputs. LSIM(A,B,C,D,U,T) calculates and plots the time response of the system: . x = Ax + Bu y = Cx + Du to input time history U. Matrix U must have as many columns as there are inputs, U. Each row of U corresponds to a new time point, and U must have LENGTH(T) rows. Y=LSIM(A,B,C,D,U,T) returns, without plotting, a matrix Y with as many columns as there are outputs y, and with LENGTH(T) rows. [Y,X] = LSIM(A,B,C,D,U,T) also returns the state time history. LSIM(A,B,C,D,U,T,X0) can be used if initial conditions exist. LSIM(NUM,DEN,U,T) plots the time response from the transfer function description G(s) = NUM(s)/DEN(s) where NUM and DEN contain the polynomial coefficients in descending powers. LSIM now assumes a straight line approximation between the discrete values of u. This is achieved via a first order hold approximation. Left hand arguments suppress plotting. J.N. Little 4-21-85 Copyright (c) 1985-89 by the MathWorks, Inc. Flight Dynamics
562
MATLAB Files Revised A.C.W.Grace 8-27-89 Synopsis: lsim(a,b,c,d,u,t); lsim(num,den,u,t); lsim(a,b,c,d,u,t,x0); lsim(num,den,u,t,x0); [y,x]=lsim(a,b,c,d,u,t,.) [y,x]=lsim(num,den,u,t,.)
Index
Angle of attack Aerodynamic, 30 Kinematic, 30 Angular velocity vector, xxv Aspect ratio, xxvii Cruising Flight, xxxi Dihedral, xxvii Ecliptic, 22 Equator, 22 Greenwich meridian, 24 Landing, xxxi Latitude Geocentric, 26 Geodesic, 26 Gravitation, 26 Mean aerodynamic chord, xxvi Mean chord, geometric chord, xxvii Polar motion, 22 Powered Approach, xxxi Reference frames, xxiv, 21 Aerodynamic, xxiv, 30 Body-fixed, xxiv, 27 Conventional celestial, 23 Conventional terrestrial, 24 Earth-centered inertial, 22 Earth-centered, Earth-fixed, 24 Inertial, xxiv, 22 International celestial, 22
International terrestrial, 24 Kinematic, xxiv, 30 Normal Earth-fixed, xxiv, 25 Principle axes, xxiv, 28 Stability, xxiv, 28 Vehicle, xxiv, 32 Vehicle carried normal Earth, xxiv, 26 Zero-lift body axes, xxiv, 28 Sideslip angle Aerodynamic, 30 Kinematic, 30 Slipping flight, xxxi Steady flight, xxxi Straight flight, xxxi Symmetric flight, xxxi Taper ratio, xxvii Time derivative, xxiv Transformation matrix, xxiv Velocity Aerodynamic, 30 Kinematic, 30 Vernal equinox, 22 Washout, wing-twist, xxvii Wing airfoil, xxx Wing area, xxvi Wing sweep, xxvii Wingspan, xxvi
Flight Dynamics
