9/ 9/2015 STRUCTURAL: Chapt er 3: Modal Anal y sis (UP19980818) ht tp: //mostreal.sk /html / guide 55/g- str/GSTR3.htm#S3.4.2. 10 1/42 Chapter 3: Modal Analysis Go to the Ne xt Ch apterGo to the Previous ChapterGo to the Table of Contents for This Manual Go to the Guides Master Index Chapter 1* Chapter 2* Chapter 3* Chapter 4* Chapter 5* Chapter 6* Chapter 7* Chapter 8* Chapter 9 * Chapter 10* Chapter 11* Chapter 12* Chapter 13* Chapter 14 3.1 Definition of Modal Analysis You use modal analysis to determine the vibration characteristics (natural frequencies and mode shapes) of a structure or a machine component while it is being designed. It also can be a starting point for another, more detailed, dynamic analysis, such as a transient dynamic analysis, a harmonic response analysis, or a spectrum analysis. 3.2 Uses for Modal Analysis You use modal analysis to determine the natural frequencies and mode shapes of a structure. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions. They are also required if you want to do a spectrum analysis or a mode superposition harmonic or transient analysis. You can do modal analysis on a prestressed structure, such as a spinning turbine blade. Another useful feature is modal cyclic sy m metry, wh i ch allows yo u to reviewthe modeshapes of a cyclically symmetric structure by m odeling just asector of i t. Modal analysis in the ANSYS family ofprodu cts i s a l i ne ar an al y si s. An y non l i ne ari ti es, su ch as pl asti ci ty an d cont act (gap) elements, areignored even if they are defined. You can choose from several mode extraction meth ods: subspace, Block Lanczos, P owerDynam i cs, reduced, unsy m m etric, and d amped. Th e d amped method allows you to include damping in the structure. Details about mode extraction methods are covered later in this section. 3.3 Commands Used in a Modal Analysis You use the same set of commands to build a model and perform a modal analysis that you use to do any other type of finite element analysis. Likewise, you choose similar options from the graphical user interface (GUI) to bui ld and sol v e m odel s, n o m atter wh at ty pe of an al y si s y ou are doi n g.
The unsymmetric method is used for problems with unsymmetric matrices, such as fluid-structure
interaction problems.
Damped method
The damped method is used for problems where damping cannot be ignored, such as bearing problems.
For most applications, you'll choose the subspace, reduced, Block Lanczos or the PowerDynamics method. The
unsymmetric and damped methods are meant for special applications.
When you specify a mode extraction method, ANSYS automatically chooses the appropriate equation solver.
Note-The damped and unsymmetric methods are not available in the ANSYS/LinearPlus program.
3.4.2.4 Option: Number of Modes to Extract [MODOPT]
This option is required for all mode extraction methods except the reduced method.
For the unsymmetric and damped methods, requesting a larger number of modes than necessary reduces the possibility of missed modes, but results in more solution time.
3.4.2.5 Option: Number of Modes to Expand [MXPAND]
This option is required for the reduced, unsymmetric, and damped methods only. However, if you want element
results, you need to turn on the "Calculate elem results" option, regardless of the mode extraction method. In the
single point response spectrum (SPOPT,SPRS ) and Dynamic Design analysis method (SPOPT, DDAM ), the
modal expansion can be performed after the spectrum analysis, based on the significance factor SIGNIF on the
MXPAND command. If you want to perform modal expansion after the spectrum analysis, choose NO for
mode expansion (MXPAND) on the dialog box for the modal analysis options (MODOPT).
3.4.2.6 Option: Mass Matrix Formulation [LUMPM]
Use this option to specify the default formulation (which is element-dependent) or lumped mass approximation.
We recommend the default formulation for most applications. However, for some problems involving "skinny"
structures such as slender beams or very thin shells, the lumped mass approximation often yields better results.
Also, the lumped mass approximation can result in a shorter run time and lower memory requirements.
modes of interest. We recommend that you define as many MDOF as you can based on your knowledge
of the dynamic characteristics of the structure [M, MGEN], and also let the program choose a few
additional masters based on stiffness-to-mass ratios [TOTAL]. You can list the defined MDOF
[MLIST], and delete extraneous MDOF [MDELE]. For more details about master degrees of freedom
see Section 3.12, "Matrix Reduction."
Command(s):
M
GUI:
Main Menu>Solution>Master DOFs>-User Selected-Define
4. Apply loads on the model. The only "loads" valid in a typical modal analysis are zero-value displacement
constraints. (If you input a non-zero displacement constraint, the program assigns a zero-value constraint to that
DOF instead.) Other loads can be specified, but are ignored (see Note below). For directions in which no
constraints are specified, the program calculates rigid-body (zero-frequency) as well as higher (non-zerofrequency) free body modes. Table 3-2 shows the commands to apply displacement constraints. Notice that you
can apply them either on the solid model (keypoints, lines, and areas) or on the finite element model (nodes and
elements). For a general discussion of solid-model loads versus finite-element loads, see Chapter 2 of the
ANSYS Basic Analysis Procedures Guide.
Note-Other loads-forces, pressures, temperatures, accelerations, etc.-can be specified in a modal analysis, but
they are ignored for the mode extraction. However, the program will calculate a load vector and write it to the
mode shape file ( Jobname.MODE) so that it can be used in a subsequent mode-superposition harmonic or
transient analysis.
Table 3-2 Loads applicable in a modal analysis
Load Type CategoryCmd
FamilyGUI Path
Displacement (UX, UY, UZ, ROTX,
ROTY, ROTZ)Constraints D
Main Menu>Solution>-Loads-Apply> -
Structural-Displacement
In an analysis, loads can be applied, removed, operated on, or listed.
Applying Loads Using Commands
Table 3-3 lists all the commands you can use to apply loads in a modal analysis.
Constant Damping Ratio DMPRATMain Menu>Solution>-Load Step Opts-Time/Frequenc>
Damping
Material-Dependent
Damping RatioMP,DAMP
Main Menu>Solution>-Load Step Opts-Other>Change Mat
Props>-Temp Dependent-Polynomial
3.4.2.10 Damping (Dynamics Options)
Damping is valid only for the damped mode extraction method (and ignored for the other mode extraction
methods; see Note below).
If you include damping, and specify the damped mode extraction method, the eigenvalues calculated are
complex; see "Mode Extraction Methods" for details. See the section "Damping" in Chapter 5 for more
information on damping.
Note-Damping can be specified in a non-damped modal analysis if a single-point response spectrum analysis is
to follow the modal analysis. Although the damping does not affect the eigenvalue solution, it is used to calculatean effective damping ratio for each mode, which is then used to calculate the response to the spectrum. Spectrum
analyses are discussed in Chapter 6.
Alpha (Mass) Damping [ALPHAD]
Beta (Stiffness) Damping [BETAD]
Constant Damping Ratio [DMPRAT]
Material-Dependent Damping Ratio [MP,DAMP]
3.4.2.11 Participation Factor Table Output
The participation factor table lists participation factors, mode coefficients, and mass distribution
percentages for each mode extracted. The participation factors and mode coefficients are calculated
based on an assumed unit displacement spectrum in each of the global Cartesian directions. The reduced
mass distribution is also listed.
Note-You can retrieve a participation factor or mode coefficient by issuing a *GET command. The factor or
coefficient is valid for the excitation (assumed unit displacement spectrum) directed along the last of the
applicable coordinates (z direction for a 3-D analysis). To retrieve a participation factor or mode coefficient for
another direction, perform a spectrum analysis with the excitation set (SED) to the desired direction. Follow with
3. Change the shift point used in eigenvalue extraction.
Command(s):
MODOPT,FREQB
GUI:
Main Menu>Solution>Analysis Options>Subspace
4. Click on OK to display the Subspace Modal Analysis dialog box.
If you use the damped mode extraction method, the eigenvalues and eigenvectors are complex. The
imaginary part of the eigenvalue represents the natural frequency, and the real part is a measure of the
stability of the system.
5. Leave SOLUTION.
Command(s):
FINISH
GUI:
Main Menu>Finish
3.4.3 Expand the Modes
In its strictest sense, the term "expansion" means expanding the reduced solution to the full DOF set . The"reduced solution" is usually in terms of master DOF. In a modal analysis, however, we use the term "expansion"
to mean writing mode shapes to the results file. That is, "expanding the modes" applies not just to reduced
mode shapes from the reduced mode extraction method, but to full mode shapes from the other mode extraction
methods as well. Thus, if you want to review mode shapes in the postprocessor, you must expand them (that is,
write them to the results file). Expanded modes are also required for subsequent spectrum analyses. In the single
point response spectrum (SPOPT,SPRS ) and Dynamic Design analysis method (SPOPT, DDAM ), the modal
expansion can be performed after the spectrum analysis, based on the significance factor SIGNIF on the
MXPAND command. If you want to perform modal expansion after the spectrum analysis, choose NO for
mode expansion (MXPAND) on the dialog box for the modal analysis options (MODOPT). No expansion is
necessary for subsequent mode superposition analyses.
3.4.3.1 Points to Remember
The mode shape file ( Jobname.MODE), Jobname.EMAT, Jobname.ESAV, and Jobname.TRI (if
reduced method) must be available.
The database must contain the same model for which the modal solution was calculated.
In this example, you perform a modal analysis on the wing of a model plane to demonstrate the wing's modal
degrees of freedom.
3.5.1 Problem Description
This is a modal analysis of a wing of a model plane. The wing is of uniform configuration along its length, and its
cross-sectional area is defined to be a straight line and a spline, as shown. It is held fixed to the body on one endand hangs freely at the other. The objective of the problem is to demonstrate the wing's modal degrees of
freedom.
3.5.2 Problem Specifications
The dimensions of the wing are shown in the problem sketch. The wing is made of low density polyethylene with
the following values:
Young's modulus = 38x103 psi
Poisson's ratio = .3
Density = 1.033e-3 slugs/in3
3.5.3 Problem Sketch
Figure 3-1 Diagram of a Model Airplane Wing
3.5.3.1 Specify the Title and Set Preferences
1. Choose menu path Utility Menu>File>Change Title.
2. Enter the text "Modal analysis of a model airplane wing" and click on OK.
VM203 Dynamic Load Effect on Simply-Supported Thick Square Plate
3.8 Prestressed Modal Analysis
Use a prestressed modal analysis to calculate the frequencies and mode shapes of a prestressed structure, such
as a spinning turbine blade. The procedure to do a prestressed modal analysis is essentially the same as a regular
modal analysis, except that you first need to prestress the structure by doing a static analysis:
1. Build the model and obtain a static solution with prestress effects turned on [PSTRES,ON]. The same
lumped mass setting [LUMPM] used here must also be used in the later prestress modal analysis. Chapter 2
describes the procedure to obtain a static solution.
2. Re-enter SOLUTION and obtain the modal solution, also with prestress effects turned on (reissue
PSTRES,ON). Files Jobname.EMAT and Jobname.ESAV from the static analysis must be available.
3. Expand the modes and review them in the postprocessor.
Step 1 above can also be a transient analysis, but you should remember to save the EMAT and ESAV files at
the desired time point.
3.9 Prestressed Modal Analysis of a Large
Deflection Solution
You can also perform a prestressed modal analysis following a large deflection static analysis in order to
calculate the frequencies and mode shapes of a highly deformed structure. Use the prestressed modal analysis procedure, except you use the PSOLVE command to obtain the modal solution instead of the SOLVE
command, as shown in the sample input listing below. Also, you must use the UPCOORD command to update
the coordinates to obtain the correct stresses.
! Initial, large deflection static analysis
!
/PREP7
...
FINISH
/SOLUANTYPE,STATIC ! Static analysis
NLGEOM,ON ! Large deflection analysis
PSTRES,ON ! Flag to calculate the prestress matrix
To understand the procedure for modal cyclic symmetry, you need to understand the concept of nodal
diameters. (The word nodal here is used in the vibrational sense, not in the finite element sense.) The term
"nodal diameter" is derived from the appearance of a simple geometry, like a disk, vibrating in a certain mode.
Most mode shapes contain lines of zero out-of-plane displacement which cross the entire disk as shown below.
These are commonly called nodal diameters.
Figure 3-3 Some examples of nodal diameters, i
For complicated structures with cyclic symmetry (such as a turbine wheel), lines of zero displacement might not
be observable in a mode shape. The mathematical definition of nodal diameter in ANSYS is, therefore, more
general and does not necessarily correspond to the number of lines of zero displacement through the structure.
The number of nodal diameters is an integer that determines the variation in the value of a single DOF at pointsspaced at a circumferential angle equal to the sector angle. For a number of nodal diameters equal to ND, this
variation is described by the function cos(ND*THETA).
The above definition allows a varying number of waves to exist around the circumference for a given nodal
diameter, as long as the DOF at points separated by the sector angle vary by cos(ND*THETA). For example,
nodal diameter = 0 and a 60 degree sector will produce modes with 0, 6, 12, ..., 6n waves around the
circumference. (In some references, the term "mode" is used instead of nodal diameter as defined above, and the
term nodal diameter is used to describe the actual number of observable waves around the structure.)
3.10.3 Standard (Stress-free) Modal Cyclic Symmetry
The procedures for standard (stress-free) modal cyclic symmetry and prestressed modal cyclic symmetry are
described next. Modal cyclic symmetry is available for structures with or without prestress.
Note-The procedure for modal cyclic symmetry uses two predefined ANSYS macros: CYCGEN and
CYCSOL. Both assume a model with structural solid or shell elements.
3.10.3.1 Overview
The procedure for stress-free modal cyclic symmetry is outlined in the flow chart below, followed by a step-by-
step description.
Figure 3-4 Procedure for modal cyclic symmetry (stress-free)
4. Run the CYCGEN macro. This macro creates a second sector that is overlaid on the basic sector. There is a
constant nodal offset (parameter NTOT) between the sectors. The modal analysis is conducted with this two-sector model. The macro copies internal couplings and constraint equations from the basic sector to the second
roughly twice as much as that taken by the "Quick Solution."
7. Run the CYCSOL macro and define your nodal diameter range and the sector angle as follows:
Command(s):
CYCSOL, NDMIN , NDMAX , NSECTOR,`LOW'
GUI:
Main Menu>Solution>Modal Cyclic Sym
NDMIN and NDMAX are the lowest and highest nodal diameters in your range of interest. (The
acceptable range is 0 through for an even value of NSECTOR, and 0 through for
an odd value of NSECTOR.) This command performs a separate eigenvalue extraction for each nodal
diameter, i.
This command (CYCSOL) performs the analysis (no SOLVE required), and computes frequencies and
mode shapes (if requested). The results file ( Jobname.RST) will have multiple load steps. Each load stepcorresponds to a nodal diameter, with the first load step corresponding to NDMIN ; the second, to
NDMIN +1, and so on, up to the last load step corresponding to NDMAX .
Within each load step, the substeps are the modes belonging to that nodal diameter. For example, if
NDMIN = 0, NDMAX = 1, and 2 modes are expanded, then the results file will have:
Load Step Substep Comment
1 1 1st mode of nodal diameter 0
1 2 2nd mode of nodal diameter 0
2 1 1st mode of nodal diameter 1
2 2 2nd mode of nodal diameter 1
8. Enter POSTPROCESSING and expand the model for display. Indicate the number of sectors that you want
Note-The /EXPAND command can also be used to show full model results. See the ANSYS Commands
Reference for more information about the /EXPAND command (Utility Menu>Plot
Cntrls>Style>Symmetry Expansion).
3.10.4 Prestressed Modal Cyclic Symmetry
The procedure for prestressed modal cyclic symmetry is outlined in the flow chart in Figure 3-6.
The steps involved are essentially the same as for the stress-free case, except that a static solution is required to
calculate the prestress in the basic sector. Thus, steps 1-4 and 7 and 8 are the same as for the stress-free case.
Steps 5 and 6 are described below.
5. Enter SOLUTION and define static loads and boundary conditions that induce the prestress. Use the
PSTRES command to achieve prestress calculation, and obtain the static solution [SOLVE].
6. Re-enter PREP7 and define modal analysis and its options, as explained for the stress-free case. Be sure toinclude prestress effects with the PSTRES command.
Note-Symmetry boundary conditions must be removed after the static solution is obtained.
Figure 3-6 Procedure for modal cyclic symmetry (prestressed)
The subspace method uses the subspace iteration technique, which internally uses the generalized Jacobi iteration
algorithm. It is highly accurate because it uses the full [K] and [M] matrices. For the same reason, however, the
subspace method is slower than the reduced method. This method is typically used in cases where high accuracy
is required or where selecting master DOF is not practical.
When doing a modal analysis with a large number of constraint equations, use the subspace iterations method
with the frontal solver instead of the JCG solver, or use the block Lanczos mode extraction method. Using the
JCG solver when your analysis has many constraint equations could result in an internal element stiffnessassembly that requires large amounts of memory.
3.11.2 Block Lanczos Method
The Block Lanczos eigenvalue solver uses the Lanczos algorithm where the Lanczos recursion is performed with
a block of vectors. This method is as accurate as the subspace method, but faster. The Block Lanczos method
uses the sparse matrix solver, overriding any solver specified via the EQSLV command.
The Block Lanczos method is especially powerful when searching for eigenfrequencies in a given part of the
eigenvalue spectrum of a given system. The convergence rate of the eigenfrequencies will be about the same
when extracting modes in the midrange and higher end of the spectrum as when extracting the lowest modes.
Therefore, when you use a shift frequency (FREQB) to extract n modes beyond the starting value of FREQB,
the algorithm extracts the n modes beyond FREQB at about the same speed as it extracts the lowest n modes.
3.11.3 PowerDynamics Method
The PowerDynamics method internally uses the subspace iterations, but uses the PCG iterative solver. This
method may be significantly faster than either the subspace or the Block Lanczos methods, but may not converge
if the model contains poorly-shaped elements, or if the matrix is ill-conditioned. This method is especially useful invery large models (100,000+ DOFs) to obtain a solution for the first few modes.
The PowerDynamics method does not perform a Sturm sequence check (that is, does not check for missing
modes), which might affect problems with multiple repeated frequencies. This method always uses lumped mass
approximation.
Note-If you use PowerDynamics to solve a model that includes rigid body modes, be sure to issue the RIGID
command or choose one of its GUI equivalents (Main Menu>Solution>Analysis Options or Main
Menu>Preprocessor> -Loads->Analysis Options).
3.11.4 Reduced Method
The reduced method uses the HBI algorithm (Householder-Bisection-Inverse iteration) to calculate the
eigenvalues and eigenvectors. It is relatively fast because it works with a small subset of degrees of freedom
called master DOF. Using master DOF leads to an exact [K] matrix but an approximate [M] matrix (usually with
some loss in mass). The accuracy of the results, therefore, depends on how well [M] is approximated, which in
turn depends on the number and location of masters. Section 3.12,"Matrix Reduction," presents guidelines to
The unsymmetric method, which also uses the full [K] and [M] matrices, is meant for problems where the
stiffness and mass matrices are unsymmetric (for example, acoustic fluid-structure interaction problems). It uses
the Lanczos algorithm which calculates complex eigenvalues and eigenvectors if the system is non-conservative
(for example, a shaft mounted on bearings). The real part of the eigenvalue represents the natural frequency and
the imaginary part is a measure of the stability of the system-a negative value means the system is stable, whereasa positive value means the system is unstable. Sturm sequence checking is not available for this method.
Therefore, missed modes are a possibility at the higher end of the frequencies extracted.
3.11.6 Damped Method
The damped method is meant for problems where damping cannot be ignored, such as rotor dynamics
applications. It uses full matrices ([K], [M], and the damping matrix [C]). It uses the Lanczos algorithm and
calculates complex eigenvalues and eigenvectors (as described below). Sturm sequence checking is not availablefor this method. Therefore, missed modes are a possibility at the higher end of the frequencies extracted.
3.11.6.1 Damped Method-Real and Imaginary Parts of the Eigenvalue
The imaginary part of the eigenvalue, , represents the steady-state circular frequency of the system. The real
part of the eigenvalue, , represents the stability of the system. If is less than zero, then the displacement
amplitude will decay exponentially, in accordance with EXP( ). If is greater than zero, then the amplitude will
increase exponentially. (Or, in other words, negative gives an exponentially decreasing, or stable, response;
and positive gives an exponentially increasing, or unstable, response.) If there is no damping, the realcomponent of the eigenvalue will be zero.
Note-The eigenvalue results reported by ANSYS are actually divided by 2* . This gives the frequency in Hz
(cycles/second). In other words:
Imaginary part of eigenvalue, as reported = /(2* )
Real part of eigenvalue, as reported = /(2* )
3.11.6.2 Damped Method-Real and Imaginary Parts of the Eigenvector
In a damped system, the response at different nodes can be out of phase. At any given node, the amplitude will
be the vector sum of the real and imaginary components of the eigenvector.
3.12 Matrix Reduction
Matrix reduction is a way to reduce the size of the matrices of a model and perform a quicker and cheaper
analysis. It is mainly used in dynamic analyses such as modal, harmonic, and transient analyses. Matrix reduction
