FLUTTER SPEED DETERMINATION USING P-METHOD Consider a typical airfoil section where the linear spring provides the plunge displacement and torsional spring provides the twist (torsional stiffness) as shown in the figure below. The reference point would represent elastic axis. P, C, Q, T represents reference point, Centre of mass, aerodynamic center and three-quarter chord respectively. The dimensionless parameter e and a represents determine the location of C and P.The chord offset of center of mass from the reference point is made dimensionless by airfoil semichord ‘b’ and denoted by .The Langrangean’s equation is used to find the kinetic energy and the potential energy.

In this method the following assumptions are made 1. The flow is steady. 2. A simple aerodynamic theory (thin airfoil theory) is used to express the lift and moment expression.

where where

( ̇ ̇ ̇ ̇) ̇

The langrangean is given by

̇ The generalized coordinates are h and

On solving we obtain a two system of equations

( ̈ ̈ ) ̈ ̈ Lift is given by

The governing equation for the flutter using steady flow theory in matrix form is given by

m mbx q

mbx q h K h + Ip q 0

0 h 0 h 0 2πρU2bq + = K q q 0 -2πρU2b 2(0.5+a) q 0

We assume the solution of the equation in exponential form

̅ ̅ √ √ On substitution we get

mb2ωθ 2 s2 +mb2ωh 2 2 2 2 mb s ωθ x θ

h 0 b = Ips 2ωθ2 +Ip ωθ 2 -2 0.5+a πρb 2U 2 0 θ mb 2s 2ωθ 2x θ +2πρb 2U 2

Now we define non-dimensional parameters

μ

- Mass ratio

σ

- Ratio of uncoupled bending to torsional frequencies

r

- Radius of gyration

V

- Dimensionless free stream speed of air also known as reduced velocity On further simplification

2 2 V2 2 s +σ s x +2 h θ μ b = 0 2 0 V s 2 x θ s 2r 2 +r 2 -2 0.5+a θ μ For non-trivial solution to exist, the determinant of the matrix is made zero The roots of the determinant are complex conjugates

Limitations of p-method The unsteady effects were completely neglected and lift and moment equations were obtained from simple aerodynamic theory.

Matlab code The flutter determinant is solved using a MATLAB code to find the flutter speeds. .The real and imaginary roots of the solution of the determinant are plotted versus reduced speed. The negative of real part of root (Г) corresponds to modal damping and the

imaginary part (Ω) corresponds to modal frequency . The various non-dimensional parameters are changed and the flutter speed variation is studied. Baseline system of parameters considered is as follows[1]

a = -0.2

e = -0.1

μ = 20

σ = 0.4

r2 = 0.24 The mass ratio, ratio of plunge frequency to pitch frequency and radius of gyration is varied independently and the corresponding nature of variation of flutter speed is studied.

1.Effect of change of mass ratio For a constant σ, x θ and r2(square of radius of gyration) as the mass ratio(μ) is varied between 20 to 100 corresponding to density variation from sea level to 10000 ft the flutter speed also increases. Table 1.1 Flutter speeds for different mass ratios μ(mass ratio) 20 40 60 80 100

Flutter speed 1.9 2.7 3.2 3.7 4.2

Fig 1.1 Plot of flutter speed versus mass ratios 5 4 d e e p s r e t t u l f

3 2 1 0 0

20

40

60 μ

80

100

120

Fig.1.2 Plot of imaginary part and real part of root versus Reduced speed for μ=20

Fig1.3 Plot of imaginary and real part of roots versus Reduced speed for μ=40

Fig 1.4 Plot of imaginary and real part versus the reduced speed for μ=60

Fig 1.5 Plot of imaginary and real roots versus Reduced speed for μ=80

Fig 1.6 Plot of imaginary and real roots versus Reduced speed for μ=100

2.Effect of change of σ For a constant mass ratio μ, x θ and r2 (square of radius of gyration) as the ratio of plunge to pitch frequency(σ) is varied from 0.2 to 0.8 the flutter speed decreases. Table 2.1 Flutter speed at different σ σ

0.2 0.4 0.6 0.8

Flutter speed at μ=20 2.2 1.9 1.5 1.1

Flutter speed at μ=40 3.1 2.7 2.1 1.6

Flutter speed at μ=60 3.8 3.2 2.6 1.9

Fig 2.1 Plot of reduced speed versus σ 4 3.5 3

d e e2.5 p s d 2 e c u1.5 d e 1 R

mu=20 mu=40 mu=60

0.5 0 0

0.2

0.4

0.6

0.8

1

σ

Fig. 2.1 plot of imaginary and real part versus reduced speed at σ=0.2

Fig 2.2 Plot of imaginary and real part of roots versus reduced speed at σ=0.4

Fig 2.3 Plot of imaginary and real parts of roots versus Reduced speed at σ=0.6

Fig. 2.4 Plot of imaginary and real part of roots versus reduced speed at σ=0.8

3.Effect of change of r(radius of gyration) For a constant mass ratio μ ,σ and x θ .as the radius of gyration is varied from 0.24 to 0.36 the flutter speed increases .Flutter speed also increases with increase in mass ratios corresponding to constant radius of gyration.

r^2

mu=20 mu=40 mu=60

0.24 1.9

2.6

3.2

0.28 2

2.9

3.5

0.32 2.2

3

3.7

0.36 2.3

3.2

3.9

Fig 3.1 Plot of Flutter speed versus square of radius of gyration (r2) 4.5 4 3.5 d 3 e e2.5 p s r 2 e t t u1.5 l F

mu=20 mu=40 mu=60

1

0.5 0 0

0.1

0.2

0.3

0.4

Radius of gyration(r 2)

Results 1. As the mass ratio was varied from 20 to 100 the flutter speed increased. 2. As the ratio of plunge to pitch frequency is varied 0.2 to 0.8 the flutter speed decreased. 3. As the radius of gyration (square) varied from 0.24 to 0.36 the flutter speed decreased.

APPENDIX Matlab code for determining flutter speed using p-method clear all; clc; clf; % mu is mass ratio mu=20; % R is square of radius of gyration R=0.24; e=-0.1; a=-0.2; %x_theta is offset distance between reference point and center of mass x_theta=e-a; % ratio of pitch frequency to plunge frequency sigma=0.4; %semi chord b=1; % V is reduced speed V_start=0; V_inc=0.1; V_end=3; % Flutter determinant syms s; icount=0; for V=V_start:V_inc:V_end icount=icount+1; f11=s^2 + sigma^2; f12=s^2*x_theta+2*(V^2)/mu; f21=s^2*x_theta; f22=s^2*R+R-(2*V^2/mu)*(0.5+a); FD=[f11 f12;f21 f22]; W=det(FD); roots=solve(W); for jj = 1:4 im(jj,icount) = imag(roots(jj));

re(jj,icount) = real(roots(jj)); end vel(icount)=V; end subplot(2,1,1); plot(vel,abs(im(2:3,1:end)),'k'); title('Imaginary part Vs Non-dimensional velocity(Reduced speed)'); xlabel('V(Reduced speed)'); ylabel('Imaginary part');grid subplot(2,1,2); plot(vel,re,'r'); title('Real part Vs Non-dimensional velocity(Reduced speed)'); xlabel('V(Reduced speed)'); ylabel('Real part');grid

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close