Lecture 5 Harmonic Analysis 14.5 Release
ANSYS Mechanical Linear and Nonlinear Dynamics
Harmonic Analysis Topics Covered A. What What is is Harm Harmoni onicc Anal Analysi ysiss B. Theo Theory ry and and Termi Termino nolo logy gy C. Contac Contactt in in Harmo Harmonic nic Analys Analysis is D. Full Full Harmon Harmonic ic Analys Analysis is E. Dampin Damping g in Full Full Harm Harmoni onicc Analy Analysis sis F. Loads Loads and Bounda Boundary ry Condit Condition ionss G. Anal Analys ysis is set setti ting ngss – Full Harmonic
Harmonic Analysis Topics Covered A. What What is is Harm Harmoni onicc Anal Analysi ysiss B. Theo Theory ry and and Termi Termino nolo logy gy C. Contac Contactt in in Harmo Harmonic nic Analys Analysis is D. Full Full Harmon Harmonic ic Analys Analysis is E. Dampin Damping g in Full Full Harm Harmoni onicc Analy Analysis sis F. Loads Loads and Bounda Boundary ry Condit Condition ionss G. Anal Analys ysis is set setti ting ngss – Full Harmonic
Harmonic Analysis … Topics Covered
H. Mode Mode-s -sup uper erpo posit sitio ion n Harm Harmon onic ic Ana Analy lysi siss I. Dampin Damping g in Mode Mode-su -super perpos positi ition on Harmon Harmonic ic Analy Analysis sis J. Anal Analys ysis is se sett ttin ings gs – Mode Superposition anslysis K. Workshop 5
A. What is Harmonic Analysis •
Input: –
•
Harmonic loads (forces, pressures, and imposed displacements) of known magnitude and frequency.
–
May be multiple loads all at the same frequency. frequency.
–
Forces and displacements can be in-phase or out-of phase.
–
Body loads can only be specified with a phase angle of zero.
Output: –
–
Harmonic displacements at each DOF, DOF, usually out of phase with the applied loads. Other derived quantities, such as stresses and strains. strains.
... What is Harmonic Analysis •
Assumptions and Restrictions: –
The entire structure has constant or frequency-dependent stiffness, damping, and mass effects.
–
No nonlinearities are permitted.
–
Transient effects are not calculated.
–
Acceleration, bearing, and moment loads are assumed to be real (in-phase) only.
... What is Harmonic Analysis •
Assumptions and Restrictions:
–
–
–
All loads and displacements vary sinusoidally at the same known frequency (although not necessarily in phase). All loads and displacements, both input and output, are assumed to occur at the same frequency. Calculated displacements are complex if: –
damping is specified, or
–
applied load is complex.
F i
F i
sin t i
where F
amplitude
freqency
phase angle
---- F1 ---- F2
B. Theory and Terminology •
Governing equation for a mass-springdamper system, subject to a sinusoidal force is
u
uk/f
mu cu ku f sin t u
f k
1 2 2
2
2
n
tan1
/
n
1 2
n
2
n
d
n
1
2 /
... Theory and Terminology •
•
When the imposed frequency approaches a natural frequency in the direction of excitation, resonance occurs. an increase in damping decreases the amplitude of the response for all imposed frequencies,
•
a small change in damping has a large effect on the response near resonance, and
•
the phase angle always passes through ±90° at resonance for any amount of damping.
u
uk/f
/
/
... Theory and Terminology •
The governing equation for a linear structure is:
M u C u K u F •
Assume {F } and {u} are harmonic with frequency
F F u u
e max
e max
Note: The symbols
i
i
e
e
:
i t
i t
an differentiate the input from the output:
= input (imposed) circular frequency = output (natural) circular frequency
... Theory and Terminology •
Take two time derivatives:
u u u •
u iu e u iu e u iu e 1
i
2
1 1
i t
2
i t
2
i t
2
Substitute and simplify:
M iC K u iu F i F 2
1
•
2
1
This can then be solved using one of two methods.
2
... Theory and Terminology •
Solution Techniques: –
Full Harmonic Response Analysis –
solves a system of simultaneous equations directly using a static solver designed for complex arithmetic:
K c uc
F c
M iC K u iu F i F 2
1
2
1
2
K u F c
–
c
c
Mode Superposition Response Analysis –
expresses the displacements as a linear combination of mode shapes.
M iC K u iu F i F 2
1
2
1
2
2
i 2 j j j
y
jc
f jc
2
C. Contact in Harmonic Analysis •
Contact regions are available in harmonic analysis; however, since this is a purely linear analysis, contact behavior will differ for the nonlinear contact types, as shown below: Linear Dynamic Analysis
•
Contact Type
Static Analysis
Bonded
Initially Touching
Inside Pinball Region
Outside Pinball Region
Bonded
Bonded
Bonded
Free
No Separation
No Separation
No Separation
No Separation
Free
Rough
Rough
Bonded
Free
Free
Frictionless
Frictionless
No Separation
Free
Free
Frictional
Frictional
Free
Free
= 0, No Separation > 0, Bonded
Contact behavior will reduce to its linear counterparts.
D. Full Harmonic Analysis •
Exact solution.
•
Generally slower than MSUP.
•
Supports all types of loads and boundary conditions.
•
Solution points must be equally distributed across the frequency domain
•
Solves the full system of simultaneous equations using the Sparse matrix solver for complex arithmetic.
K c uc
F c
M iC K u iu F i F 2
1
2
1
K u F c
c
c
2
E. Damping in Full Harmonic Analysis 1.
Rayleigh Damping:
–
Alpha damping and Beta damping are used to define Rayleigh damping constants α and β. The damping matrix [C] is calculated by using these constants to multiply the mass matrix [M] and stiffness matrix [K]:
C M K
Equivalent damping
2
2
... Damping in Full Harmonic Analysis 1.
Material Damping: •
•
•
Material damping is inherently present in a material (energy is dissipated by internal friction), so it is typically considered in a dynamic analysis.
Energy dissipated by internal friction in a real system does not depend on the cyclic frequency.
The simplest device to represent it is to assume the damping force is proportional to velocity and inversely proportional to frequency
C
2
g K
Equivalent damping
g = constant structural damping ratio
g
... Structural Damping Matrix [C]
•
The complete expression for the structural damping matrix, [C], is Mass damping
N ma
C M im M i i 1 Structural damping
2 2 g K jm g j K j j 1 N mb
Element damping
Gyroscopic damping
Viscoelastic damping
N e
N g
N v
k 1
l 1
l 1
C k Gl •
g is constant damping.
1
C m
…. Structural Damping Matrix [C] The value of g ,
and
can be input using the following:
[ 1 ] M a te r i al -d e p e n d e n t d a m p i n g v a l u e
(Mass-Matrix Damping Multiplier, and k-Matrix Damping Multiplier)
C M i jm N ma i 1
m i
N mb
j 1
2
g j K j
Equivalent damping
i
2 i
i 2
g
…. Structural Damping Matrix [C] [ 2 ] D i r ec t l y a s g l o b a l d a m p i n g v a l u e
(Details section of Analysis Settings)
2 C M g K
Equivalent damping
i
2 i
i 2
g
F. Loads and Boundary Conditions •
Structural loads and supports may also be used in harmonic analyses with the following exceptions: •
•
Loads Not Supported: –
Gravity Loads
–
Thermal Loads
–
Rotational Velocity
–
Pretension Bolt Load
–
Compression Only Support (if present, it behaves similar to a Frictionless Support)
Remember that all structural loads will vary sinusoidally at the same excitation frequency
•
Loads can be out of phase with each other.
•
Transient effects are not calculated.
•
Remote Force, Moment, and Acceleration loads may be defined, although these loads are assumed to act at a phase angle of zero.
... Loads and Boundary Conditions A list of supported loads are shown below:
•
•
Not all available loads support phase input. Accelerations, Bearing Load, and Moment Load will have a phase angle of 0 . °
–
If other loads are present, shift the phase angle of other loads, such that the Acceleration, Bearing, and Moment Loads will remain at a phase angle of 0 . °
... Loads and Boundary Conditions •
Specifying harmonic loads requires: 1. Amplitude F imax 2. phase angle , and 3. Frequency
F i
F max i
where F
i
sin t i
amplitude
freqency
max
phase angle
... Loads and Boundary Conditions •
Amplitude and phase angle •
The load value (magnitude) represents the amplitude ( F 1max and F 2max ).
•
Phase angle is the phase shift between two or more harmonic loads.
•
is not required if only one load is present.
---- F1 ---- F2
Amplitude Phase Angle
F. Analysis Settings – Full Harmonic •
Analysis Settings > Options –
Frequency Range: Specified in cycles per second (Hertz) •
Range Minimum >> Minimum Frequency
•
Range Maximum >> Maximum Frequency
–
Solution Intervals
–
Solution Method
A range of 0-500 Hz with 10 solution intervals gives solutions at frequencies of 50, 100, 150, …, 450, and 500 Hz. Same range with 1 substep gives one solution at 500 Hz.
F. Analysis Settings – Full Harmonic •
Analysis Settings > Options –
Solution Intervals
Evenly-spaced frequency points
... Analysis Settings – Full Harmonic •
Analysis Settings > Output Controls
•
Analysis Settings > Damping Controls
... Full Harmonic Analysis Analysis Setting > Solution Method > Full
... Results- Frequency Response •
Frequency Response: •
display how the response varies with frequency
) m ( e d u t i l p m A
) o ( e l g n A e s a h P
Frequency (Hz)
Frequency (Hz)
... Results- Phase Response •
Phase Response: •
show how much a response lags behind the applied loads.
----- Output
----- Force
Angle (o)
... Results – Contour Plots •
Contour plots include: – stress, – elastic strain, and – deformation.
•
For these results, you must specify a frequency and phase angle.
... Results – Contour Plots •
•
•
A contour result can be created from a Frequency Response. The Phase Angle of the contour result has the same magnitude as the frequency result type but an opposite sign .
The sign of the phase angle is reversed so that the response amplitude of the frequency response plot for that frequency and phase angle matches with the contour results. RMB
... Results – Contour Plots 1. By Frequency
RMB
Note: The sign of the phase angle in the contour result is reversed so that the response amplitude of the frequency response plot for that frequency and phase angle matches with the contour results.
... Results – Contour Plots 2. By: Maximum Over Frequency
RMB
... Results – Contour Plots 3. By: Frequency of Maximum
RMB
... Results – Contour Plots 4. By: Maximum over Phase
RMB
... Results – Contour Plots 5. By: Phase of Maximum
RMB
Note: The sign of the phase angle is reversed.
Mode Sup Harmonic Analysis 14.5 Release
ANSYS Mechanical Linear and Nonlinear Dynamics
G. Mode-superposition Harmonic Analysis •
Approximate solution; accuracy depends on whether an adequate number of modes have been extracted.
•
Generally faster than FULL.
•
Does not support nonzero imposed harmonic displacements.
•
•
Solution points may be either equally distributed across the frequency domain or clustered about the natural frequencies of the structure. Solves an uncoupled system of equations by performing a linear combination of orthogonal vectors (mode shapes).
M iC K u iu F i F 2
1
2
1
2
2
i 2 j j j
y
jc
f jc
2
… Mode Superposition Method •
Example:
y1
+
1
–
–
y2
=
2
Here, the sum of mode shape 1 and mode shape 2 approximates the final response. Since mode shapes are relative, the coefficients y 1 and y 2 are required. Mode shapes (eigenvectors) are also known as generalized coordinates, and in this case, coefficients y 1 and y 2 are the DOF.
H. Damping in Mode-Sup Harmonic Analysis
Constant ratio d
i
Stiff. Coef.
Mass Coef.
m
i
2 i
i 2
I. Analysis Settings – Mode-Sup Harmonic •
Analysis Settings > Options –
Frequency Range •
Range Minimum >> Minimum Frequency
•
Range Maximum >> Maximum Frequency
–
Solution Intervals
–
Solution Method > Mode Superposition
… Analysis Settings – Mode-Sup Harmonic •
Analysis Settings > Options –
Cluster Results > Yes
Without Cluster Option
With Cluster Option
... Analysis Settings – Mode-Sup Harmonic Analysis Settings > Options
•
–
Include Residual Vector
•
In MSUP analysis, the dynamic response will be approximate when the applied loading excites the higher frequency modes of a structure.
•
The residual vector method: –
–
employs additional modal transformation vectors in addition to the eigenvectors in the modal transformation . accounts for high frequency dynamic responses with fewer eigen-modes.
... Mode-superposition Harmonic Analysis •
Setup a mode-sup transient analysis in the schematic by: 1. linking a modal system to a transient structural system at the solution level.
•
Notice in the transient branch, the modal analysis result becomes an initial condition.
... Mode-superposition Harmonic Analysis 2. Or, Analysis Setting > Solution Method > Mode Superposition (Standalone Analysis)
