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David Herrin, Ph.D.University of Kentucky
Modal and Harmonic Analysis using ANSYS
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
g Used to determine the natural frequencies and mode shapes of a continuous structure
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Review of Multi DOF Systems
g Expressed in matrix form as
M1 M2
K2
C2
K1
C1
K3
C3
x1 x2
F1 F2
[ ]{ } [ ]{ } [ ]{ } { }FuKuCuM =++ &&&
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Review of Multi DOF Systems
[ ]{ } [ ]{ } [ ]{ } { }FuKuCuM =++ &&&
g The mass, damping and stiffness matrices are constant with time
g The unknown nodal displacements vary with time
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
g A continuous structure has an infinite number of degrees of freedom
g The finite element method approximates the real structure with a finite number of DOFs
g N mode shapes can be found for a FEM having N DOFs
g Modal Analysis
ü Process for determining the N natural frequencies and mode shapes
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
g Given “suitable” initial conditions, the structure will vibrate
ü at one of its natural frequencies
ü the shape of the vibration will be a scalar multiple of a mode shape
g Given “arbitrary” initial conditions, the resulting vibration will be a
ü Superposition of mode shapes
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
g Determines the vibration characteristics (natural frequencies and mode shapes) of a structural components
g Natural frequencies and mode shapes are a starting point for a transient or harmonic analysis
ü If using the mode superposition method
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mode Shape of a Thin Plate (240 Hz)
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mode Extraction Methods
g Subspace
g Block Lanczos
g PowerDynamics
g Reduced
g Unsymmetric
g Damped and QR damped (Include damping)
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Steps in Modal Analysis
g Build the model
ü Same as for static analysis
ü Use top-down or bottom-up techniques
g Apply loads and obtain solution
ü Only valid loads are zero-value displacement constraints
ü Other loads can be specified but are ignored
g Expand the modes and review results
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
360 inches
IZZ = 450 in4
A = 3 in2
H = 5 in
In-Class Exercise
E = 30E6 psi
2
3
lb s8.031 4
in inE
ρ = −
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
In-Class Exercise
g Set element type to BEAM3
g Set the appropriate real constants and material constants
g Create Keypoints at the start and end of the beam and a Line between them
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mesh the Line and Apply B.C.s
g Set the meshing size control of the line to 5
g Fix the Keypoint at the right end of the beam
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Solution Options
g Change the analysis type to Modal
ü Solution > Analysis Type > New Analysis
ü <Modal>
g Set the analysis options
ü Solution > Analysis Options
ü Extract 10 mode <OK>
ü Enter <1500> for the ending frequency
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Solution Options
g At this point, you have told ANSYS to find a particular quantity of modes and to look within a particular frequency range. If ANSYS finds that quantity before it finishes the frequency range, it will stop the search. If ANSYS does not find that quantity before finishing the frequency range, then it will stop the search.
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Solution Options
g Solve the load set
g ANSYS generates a substep result for each natural frequency and mode shape
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Postprocessing
g List results summary
ü General Postproc > List Results > Results Summary
g Read results for a substep
ü General Postproc > Read Results > First Set
ü Plot deformed geometry
ü General Postproc > Read Results > Next Set
ü Plot deformed geometry
David Herrin, Ph.D.University of Kentucky
Theoretical Modal Analysis
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Structural Vibration at a Single DOF
g At a single degree of freedom
( )tu ωφ= cos
DOF nodal a ofnt displaceme=u
amplitude=φ
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Structural Vibration for the Entire Structure
{ } { } ( )tu ωφ= cos
{ } ntsdisplaceme nodal ofVector =u
{ } DOFeach for amplitudes ofvector =φ
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mode Shapes
g The vector of amplitudes is a mode shape
2φ 3φ 4φ 5φ6φ
7φ8φ
g Use subscript i to differentiate mode shapes and natural frequencies
{ } { } ( )tu ii ωφ= cos
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Appropriate Initial Conditions
g If the I.C.s are a scalar (a) multiple of a specific mode shape then the structure will vibrate in the corresponding mode and natural frequency
{ }ia φ=ICs
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Determining Natural Frequencies
g Consider a multi DOF system
[ ]{ } [ ]{ } { }0=+ uKuM &&
g Note that the system is
ü Undamped
ü Not excited by any external forces
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
{ } { } ( )tu ii ωφ= cos
g If the system vibrates according to a particular mode shape and frequency
{ } ii shape mode=φ
ii frequency natural=ω
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
{ } { } ( )tu iii ωφω sin−=&
g First derivative (Velocity)
g Second derivative (Acceleration)
{ } { } ( )2 cosi iiu t= −ω φ ω&&
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis
[ ] [ ]( ){ } { }2 0i iM K− ω + φ =
g Plug velocity and acceleration into the equation of motion
g Two possible solutions
{ } { }0i
φ =
[ ] [ ]( )2det 0iM K− ω + =
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
The Eigen Problem
g The Eigen Problem
[ ] [ ]( )2det 0iM K− ω + =
λg Eigenvalue
{ }xg Eigenvector
[ ]{ } [ ]{ }xIxA λ=
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis – An Eigen Problem
[ ] [ ]( ){ } [ ]2 0iM K− ω + φ =
g Natural frequencies are eigenvalues
g The mode shapes are eigenvectors
[ ]{ } [ ]{ }2iK Mφ = ω φ
2iω
{ }φ
[ ] [ ]{ } [ ]{ }φω=φ− IKM i21
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Analysis – An Eigen Problem
g Solving the eigen problem
[ ] [ ]( ){ } 0A I x− λ =
[ ] [ ]( )det 0A I− λ =
[ ]{ } [ ]{ }xIxA λ=
[ ] [ ]{ } [ ]{ }φω=φ− IKM i21
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Example Eigen Problem
1 1
2 2
2 33 2 i
x xx x
= λ
2 3 1 0det 0
3 2 0 1i
− λ =
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Find Eigenvalues
2 3det 0
3 2 − λ
= − λ ( )( ) 22 2 9 4 5− λ − λ − = λ − λ −
g Eigenvalues
1 1λ = −
2 5λ =
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
For Eigenvector 1
1 1λ = −
( ) 1
2 1
2 3 1 0 01
3 2 0 1 0xx
− − =
1
2 1
2 1 3 03 2 1 0
xx
+ = +
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
For Eigenvector 1
1
2 1
3 3 03 3 0
xx
=
2 1x x= −
g x1 can be any value
1
2 1
1 .70711 .7071
xx
= = − −
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
For Eigenvector 2
2 5λ =
( ) 1
2 2
2 3 1 0 05
3 2 0 1 0xx
− =
1
2 2
2 5 3 03 2 5 0
xx
− = −
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
For Eigenvector 2
1
2 2
3 3 03 3 0
xx
− = −
2 1x x=
g x2 can be any value
1
2 2
1 .70711 .7071
xx
= =
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Example
g Equations of motions
M1 M2
K1 K2
x1 x2
1 1 1 1 1 2 0M x K x K x+ − =&&
2 2 2 2 1 2 1 1 0M x K x K x K x+ + − =&&
1 5 kgM =
2 10 kgM =
1 800 N/mK =
2 400 N/mK =
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Matrix Form
1 1 1 1 1
2 2 1 1 2 2
0 00 0
M x K K xM x K K K x
− + = − +
&&&&
1 1
2 2
5 0 800 800 00 10 800 1200 0
x xx x
− + = −
&&&&
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Up Eigen Problem
[ ] [ ]{ } { }1 2iM K− φ = ω φ
[ ] [ ]1 160 16080 120
M K− −
= −
160 160 1 0det 0
80 120 0 1 −
− λ = −
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Solve Eigen Problem
160 160det 0
80 120 − λ −
= − − λ 1 1 5.011 rad/sω = λ =
2 2 15.965 rad/sω = λ =
1
2 1
0.76450.6446
φ = φ
1
2 1
0.86010.5101
φ − = φ
1
5.011.7975 Hz
2f = =
π
2
15.9652.5409 Hz
2f = =
π
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Weighted Orthogonality
[ ] 022
1
12
1 =
φφ
φφ
MT
[ ] 022
1
12
1 =
φφ
φφ
KT
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Vector Scaling
[ ] 112
1
12
1 =
φφ
φφ
MT
g Unity Modal Mass
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Modal Vector Scaling
164467645
10005
64467645
11
=
XX
XX
T
.
...
g Unity Modal Mass
376.=X
=
φφ
242287
12
1
.
.
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mode 1
M1 M2
K1 K2
0.7645 0.6446
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mode 2
M1 M2
K1 K2
-0.8601 0.5101
David Herrin, Ph.D.University of Kentucky
Harmonic Response using ANSYS
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Harmonic Response Analysis
g Solves the time-dependent equations of motion for linear structures undergoing steady-state vibration
g All loads and displacements vary sinusoidally at the same frequency
1sin( )iF F t= ω + φ
2sin( )jF F t= ω + φ
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Harmonic Response Analysis
g Analyses can generate plots of displacement amplitudes at given points in the structure as a function of forcing frequency
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
360 inches
IZZ = 450 in4
A = 3 in2
H = 5 in
In-Class Exercise
E = 30E6 psi
2
3
lb s8.031 4
in inE
ρ = −
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
In-Class Exercise
g Set element type to BEAM3
g Set the appropriate real constants and material constants
g Create Keypoints at the start and end of the beam and a Line between them
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Mesh the Line and Apply B.C.s
g Set the meshing size control of the line to 5
g Fix the Keypoint at the right end of the beam
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Solution Options
g Change the analysis type to Modal
ü Solution > Analysis Type > New Analysis
ü <Modal>
g Set the analysis options
ü Solution > Analysis Options
ü Extract 10 mode <OK>
ü Enter <1500> for the ending frequency
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Solution Options
g At this point, you have told ANSYS to find a particular quantity of modes and to look within a particular frequency range. If ANSYS finds that quantity before it finishes the frequency range, it will stop the search. If ANSYS does not find that quantity before finishing the frequency range, then it will stop the search.
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Set Solution Options
g Solve the load set
g ANSYS generates a substep result for each natural frequency and mode shape
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Postprocessing
g List results summary
ü General Postproc > List Results > Results Summary
g Read results for a substep
ü General Postproc > Read Results > First Set
ü Plot deformed geometry
ü General Postproc > Read Results > Next Set
ü Plot deformed geometry
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Forced Response
g Apply a 1.0 N load at the left end of the beam
g Set the Analysis Options
ü Set the solution method to “Full”
ü Set the DOF printout format to “Amplitude and phase” <OK>
ü Set the tolerance to 1e-8 <OK>
g New Analysis > Harmonic
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Forced Response
g Set the frequency substeps
ü Solution > Load Step Opts – Time Frequency > Freq and Substeps
ü Set the Harmonic Frequency Range to between 0 and 50 Hz
ü Set the number of substeps to 100
ü Set to Stepped
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Forced Response
g Set the damping
ü Solution > Load Step Opts – Time Frequency > Damping
ü Set the Constant Damping Ratio to 0.01
g Solve the model
Modal/Harmonic Analysis Using ANSYS
ME 599/699 Vibro-Acoustic Design
Dept. of Mechanical Engineering University of Kentucky
Forced Response
g Enter time history postprocessor
g Define a variable for UY at the leftmost node
g List and graph the variable
g Add a vertical truss member at the center of the beam having an area of 10.