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Module 2
Modal Analysis
A. Define modal analysis and its purpose.
B. Discuss associated concepts, terminology, and mode extraction
methods.
C. Learn how to do a modal analysis in ANSYS.
D. Work on one or two modal analysis exercises.
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Modal Analysis
A. Definition & Purpose
• What is modal analysis?
• A technique used to determine a structure’s vibration
characteristics:
– Natural frequencies
– Mode shapes
– Mode participation factors (how much a given mode participates in a
given direction)
• Most fundamental of all the dynamic analysis types.
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Modal Analysis
… Definition & Purpose
Benefits of modal analysis
• Allows the design to avoid resonant vibrations or to vibrate at a
specified frequency (speakers, for example).
• Gives engineers an idea of how the design will respond to
different types of dynamic loads.
• Helps in calculating solution controls (time steps, etc.) for other
dynamic analyses.
Recommendation: Because a structure’s vibration characteristics
determine how it responds to any type of dynamic load, always perform a
modal analysis first before trying any other dynamic analysis.
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• General equation of motion:
• Assume free vibrations and ignore damping:
• Assume harmonic motion ( i.e. )
• The roots of this equation are i
2, the eigenvalues, where i ranges
from 1 to number of DOF. Corresponding vectors are {u}i, the
eigenvectors.
0
u
M
K 2
0
u
K
u
M
t
F
u
K
u
C
u
M
Modal Analysis
B. Terminology & Concepts
)
t
sin(
U
u
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Modal Analysis
… Terminology & Concepts
• The square roots of the eigenvalues are i , the structure’s natural
circular frequencies (radians/sec). Natural frequencies fi are then
calculated as fi = i /2p cycles/sec. It is the natural frequencies fi
that are input by the user and output by ANSYS.
• The eigenvectors {u}i represent the mode shapes - the shape
assumed by the structure when vibrating at frequency fi.
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Modal Analysis
… Terminology & Concepts (cont.)
• Mode Extraction is the term used to describe the calculation of
eigenvalues and eigenvectors.
• Mode Expansion has a dual meaning. For the reduced method,
mode expansion means calculating the full mode shapes from the
reduced mode shapes. For all other methods, mode expansion
simply means writing mode shapes to the results file.
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Modal Analysis - Terminology & Concepts
Mode Extraction Methods
• Several mode extraction methods are available in ANSYS:
– Block Lanczos (default)
– Subspace
– PowerDynamics
– Reduced
– Unsymmetric
– Damped (full)
– QR Damped
• Which method you choose depends primarily on the model size
(relative to your computer resources) and the particular
application.
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Block Lanczos
• The Block Lanczos method is recommended for most
applications.
– Efficient extraction of large number of modes (40+) in most models
– Typically used in complex models with mixture of
solids/shells/beams etc.
– Efficient extraction of modes in a frequency range
– Handles rigid-body modes well
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Subspace
• When extracting a small number of modes (<40) in similar size
models, the subspace method can be more suitable.
– Requires relatively less memory but large diskspace
– May have convergence problems when rigid body modes are present.
– Not recommended when constraint equations are present.
– Generally superseded by Block Lanczos
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - PowerDynamics
• For large (100K+ DOF) models and a small number of modes
(< 20), use the PowerDynamics method. It can be significantly
faster than Block Lanczos or Subspace, but:
– Requires large amount of memory.
– May not converge with poorly shaped elements or an ill-conditioned
matrix.
– May miss modes (No Sturm sequence check)
– Recommended only as a last resort for large models.
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Reduced
• For models in which lumping mass does not create a local
oscillation, typically beams and spars, use the Reduced method.
– Memory and disk requirements are low.
– In general fastest eigen solver
– Employs matrix reduction, a technique to reduce the size of [K] and
[M] by selecting a subset of DOF called master DOF.
– Reduction of [K] is exact but [M] loses some accuracy
– Accuracy of [M] depends on number and location of master DOF.
– Generally not recommended due to
• Expertise required in picking master DOF
• Efficient alternatives such as Block Lanczos
• reduced cost of hardware
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Unsymmetric
• The unsymmetric method is used for acoustics (with structural
coupling) and other such applications with unsymmetric [K] and [M].
– Calculates complex eigenvalues and eigenvectors:
• Real part is the natural frequency.
• Imaginary part indicates stability - negative means stable, positive
means unstable.
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Damped
• Damping is normally ignored in a modal analysis, but if its effects
are significant, the Damped method is used.
– Typical application is rotor dynamics, where gyroscopic damping
effects are important.
– Two ANSYS elements, BEAM4 and PIPE16, allow gyroscopic effects to
be specified in the form of real constant SPIN (rotational speed,
radians/time).
– Calculates complex eigenvalues and eigenvectors:
• Imaginary part is the natural frequency.
• Real part indicates stability - negative means stable, positive
means unstable.
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Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Q-R damped
• A second mode extraction method that considers damping effects
is the Q-R Damped method.
– Faster and more stable than the existing Damped Solver
– Works with poorly conditioned models
– All forms of damping allowed including damper elements
– Combines the best features of the real eigensolution method (Block
Lanczos) and the Complex Hessenberg method (QR Algorithm)
– Outputs complex eigenvalues ( frequency and stability) and damping
ratio of each mode
– Supports the use of a material dependent damping ratio [MP,DMPR] in
a subsequent mode superposition harmonic analysis
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FEA Model Characteristics:
111,129 active dofs
10 damped modes
Alpha, Beta and Element damping
0
20000
40000
60000
80000
100000
120000
140000
160000
QRDAMP DAMP
CPU (sec)
ELAPSE (sec)
Modal Analysis - Terminology & Concepts
… Mode Extraction Methods - Q-R damped
Comparison Demonstrating the Superior Solution Performance
of the QR Damped Mode Extraction Method
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Extraction
method
Linear Solver
Used
Remarks
Block Lanczos Sparse Matrix Recommended for most applications; Most stable;
Subspace Frontal Solver
Stable but slow; Requires large disk space; Has
difficulty with constraint equations / rigid body
modes
Powerdynamics PCG solver
Same as subspace but with PCG solver; Can
handle very large models; Lumped mass only; May
miss modes; Modes cannot be used in
subsequent spectrum and PSD analyses
Reduced Frontal Solver
In general fastest; Accuracy depends on Master
DOF selection; Limitations similar to Subspace;
Not recommended due to expertise required in
selecting Master DOF.
Modal Analysis - Terminology & Concepts
Summary for symmetric, undamped solvers
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Modal Analysis
C. Procedure
Four main steps in a modal analysis:
• Build the model
• Choose analysis type and options
• Apply boundary conditions and solve
• Review results
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Modal Analysis Procedure
Build the Model
• Remember density!
• Linear elements and materials only. Nonlinearities are ignored.
• See also Modeling Considerations in Module 1.
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Modal Analysis Procedure
Choose Analysis Type & Options
Build the model
Choose analysis type and
options
• Enter Solution and choose
modal analysis.
• Mode extraction options*
• Mode expansion options*
• Other options*
*Discussed next
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Modal Analysis Procedure
… Choose Analysis Type & Options
Mode extraction options
• Method: Block Lanczos
recommended for most applications.
• Number of modes: Must be specified
(except Reduced method).
• Frequency range: Defaults to entire
range, but can be limited to a desired
range (FREQB to FREQE).
Specification of a frequency range
requires additional factorizations and
it is typically faster to simply request
a number of modes which will overlap
the desired range.
• Normalization: Discussed next.
defaults to 1e8
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Modal Analysis Procedure
… Choose Analysis Type & Options
Normalization of mode shapes:
• Only the shape of the DOF solution has real meaning. It is
therefore customary to normalize them for numerical efficiency or
user convenience.
• Modes are normalized either to the mass matrix or to a unit matrix
(unity).
– Normalization to mass matrix is the default, and is required for a
spectrum analysis or if a subsequent mode superposition analysis is
planned.
– Choose normalization to unity when you want to easily compare
relative values of displacements throughout the structure.
• Modes normalized to unity cannot be used in subsequent mode
superposition analyses (transient, harmonic, spectrum or random
vibration)
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Modal Analysis Procedure
… Choose Analysis Type & Options
Mode expansion:
• You need to expand mode shapes if you want to do any of the
following:
– Have element stresses calculated.
– Do a subsequent spectrum or mode superposition analysis.
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Modal Analysis Procedure
… Choose Analysis Type & Options
Mode expansion (continued):
• Recommendation: Always expand as many modes as the number
extracted. The cost of this is minimal.
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Modal Analysis Procedure
… Choose Analysis Type & Options
• Other analysis options:
• Lumped mass matrix
– Mainly used for slender beams and thin shells, or for wave
propagation problems.
– Automatically chosen for PowerDynamics method.
• Pre-stress effects
– For Pre-stressed modal analysis (discussed later).
• Full damping
– Used only if Damped mode extraction method is chosen.
– Damping ratio, alpha damping, and beta damping are allowed.
– BEAM4 and PIPE16 also allow gyroscopic damping.
• QR damping
– All types of damping are allowed.
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Modal Analysis Procedure
Apply BC’s and Solve
Build the model
Choose analysis type and options
Apply boundary conditions and solve
• Displacement constraints: Discussed next.
• External loads: Ignored since free vibrations are assumed.
However, ANSYS creates a load vector which you can use in a
subsequent mode superposition analysis.
• Solve: Discussed next.
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Modal Analysis Procedure
… Apply BC’s and Solve
Displacement constraints:
• Apply as necessary, to simulate actual fixity.
• Rigid body modes will be calculated in directions not constrained.
• Non-zero displacements are not allowed.
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Modal Analysis Procedure
... Apply BC’s and Solve
Displacement constraints (continued):
• Be careful with symmetry
• Symmetry BC’s will only produce
symmetrically shaped modes, so some
modes can be missed.
Full Model
Symmetry BC Anti-Symmetry BC
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Modal Analysis Procedure
… Apply BC’s and Solve
Displacement constraints (continued):
For the plate-with-hole model, the lowest non-zero mode for the full and
the quarter-symmetry case is shown below. The 53-Hz mode was missed
by the anti-symmetry case because ROTX is non-zero along the
symmetry boundaries.
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Modal Analysis Procedure
… Apply BC’s and Solve
Solve:
• Typically one load step.
• Multiple load steps can be used to study the
effect of different displacement constraints
(symmetry BC in one load step and anti-symmetry
BC in another, for example).
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Modal Analysis Procedure
Review Results
Build the model
Choose analysis type and options
Apply boundary conditions and solve
• Review results using POST1, the general postprocessor
• List natural frequencies
• View mode shapes
• Review participation factors
• Review modal stresses
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Modal Analysis Procedure
… Review Results
Listing natural frequencies:
• Choose “Read Results > By Pick” in the General Postproc menu.
• Notice that each mode is stored in a separate substep.
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Modal Analysis Procedure
… Review Results
Viewing mode shapes:
• First read in results for the
desired mode using First Set, Next
Set, or By Load Step.
• Then plot the deformed shape:
General Postproc > Plot Results >
Deformed Shape…
• Notice that the graphics legend
shows mode number (SUB = ) and
the frequency (FREQ = ).
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Modal Analysis Procedure
… Review Results
Viewing mode shapes (continued):
• You can also animate the mode shape: Utility Menu > PlotCtrls >
Animate > Mode Shape...
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Modal Analysis Procedure
… Review Results
Participation Factors:
• Calculated for each mode in global translation and rotation
directions
• High value in a direction indicates that the mode will be excited by
forces in that direction
• Values are relative based on a unit displacement spectrum
• The final participation factor value (ROTZ) can be retrieved into a
parameter using *GET command. A spectrum analysis with a
specified direction (SED,0,1,0) could be used to obtain other
values
• Also printed out (to the output file) is the effective mass. Ideally
the sum of the effective masses in each direction should equal
total mass of structure
• Effective Mass = (participation factor)2
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Modal Analysis Procedure
… Review Results
Modal stresses:
• Available if element stress calculation is activated when choosing analysis
options.
• Stress values have no real meaning, however these can be used to
highlight hot spots
• If mode shapes are normalized to unity, you can compare stresses at
different points for a given mode shape
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Modal Analysis
Procedure
Build the model
Choose analysis type and options
Apply boundary conditions and solve
Review results
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D. Workshop - Modal Analysis
This workshop consists of two problems:
1. Modal analysis of a plate with a hole
– A step-by-step description of how to do the analysis.
– You may choose to run this problem yourself, or your instructor may
show it as a demonstration.
– Follow the instructions in your Dynamics Workshop supplement
(WS2: Modal Analysis - Plate with a Hole, Page WS-17 ).
2. Modal analysis of a model airplane wing
– This is left as an exercise to you.
– Follow the instructions in your Dynamics Workshop supplement
(WS3: Modal Analysis - Model Airplane Wing, Page WS-23 ).
