ANSYS Structural Analysis Guide MODAL ANALYSIS

Chapter 3: Modal Analysis

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 fre-

quencies 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 symmetry, which allows you to review the mode shapes of a cyclically symmetric structure by

modeling just a sector of it.

Modal analysis in the ANSYS family of products is a linear analysis. Any nonlinearities, such as plasticity and

contact (gap) elements, are ignored even if they are defined. You can choose from several mode-extraction

methods: Block Lanczos (default), subspace, PowerDynamics, reduced, unsymmetric, damped, and QR damped.

The damped and QR damped methods allow you to include damping in the structure. The QR Damped method

also allows for unsymmetrical damping and stiffness matrices. Details about mode-extraction methods are

covered later in this section.

3.3. Overview of Steps in a Modal Analysis

The procedure for a modal analysis consists of four main steps:

1. Build the model.

2. Apply loads and obtain the solution.

3. Expand the modes.

4. Review the results.

3.4. Build the Model

See Section 1.2: Building a Model in the ANSYS Basic Analysis Guide. For further details, see the ANSYS Modeling

and Meshing Guide.

When building your model, remember these points:

• Only linear behavior is valid in a modal analysis. If you specify nonlinear elements, they are treated as linear.

For example, if you include contact elements, their stiffnesses are calculated based on their initial status

and never change.

• Material properties can be linear, isotropic or orthotropic, and constant or temperature-dependent. You

must define both Young's modulus (EX) (or stiffness in some form) and density (DENS) (or mass in some

form) for a modal analysis. Nonlinear properties are ignored. If applying element damping, you must

ANSYS Structural Analysis Guide . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

define the required real constants for the specific element type (COMBIN7, COMBIN14, COMBIN37, and

so on).

3.5. Apply Loads and Obtain the Solution

In this step you define the analysis type and options, apply loads, specify load step options, and begin the finite

element solution for the natural frequencies.

3.5.1. Enter the Solution Processor

1. Enter the ANSYS solution processor.

Command(s): /SOLU

GUI: Main Menu> Solution

3.5.2. Define Analysis Type and Options

After you have entered the solution processor, you define the analysis type and analysis options. ANSYS offers

the options listed in Table 3.1: “Analysis Types and Options” for a modal analysis. Each of the options is explained

in detail below.

Table 3.1 Analysis Types and Options

GUI PathCommandOption

Main Menu> Solution> Analysis Type> New AnalysisANTYPENew Analysis

Main Menu> Solution> Analysis Type> New Analysis>

Modal

ANTYPEAnalysis Type: Modal (see Note below)

Main Menu> Solution> Analysis Type> Analysis OptionsMODOPTmode-extraction Method

Main Menu> Solution> Analysis Type> Analysis OptionsMODOPTNumber of Modes to Extract

Main Menu> Solution> Analysis Type> Analysis OptionsMXPANDNo. of Modes to Expand (see Note below)

Main Menu> Solution> Analysis Type> Analysis OptionsLUMPMMass Matrix Formulation

Main Menu> Solution> Analysis Type> Analysis OptionsPSTRESPrestress Effects Calculation

Note — When you specify a modal analysis, a Solution menu that is appropriate for modal analyses ap-

pears. The Solution menu will be either “abridged” or “unabridged,” depending on the actions you took

prior to this step in your ANSYS session. The abridged menu contains only those solution options that

are valid and/or recommended for modal analyses. If you are on the abridged Solution menu and you

want to access other solution options (that is, solution options that are valid for you to use, but their use

may not be encouraged for this type of analysis), select the Unabridged Menu option. For details, see

Section 3.5.1: Using Abridged Solution Menus in the ANSYS Basic Analysis Guide.

Note — In the single point response spectrum (SPOPT,SPRS) and Dynamic Design analysis method (SP-

OPT,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).

3.5.2.1. Option: New Analysis [ANTYPE]

Choose New Analysis.

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Note — Restarts are not valid in a modal analysis. If you need to apply different sets of boundary conditions,

do a new analysis each time (or use the "partial solution" procedure described in Chapter 3, “Solution”

in the ANSYS Basic Analysis Guide).

3.5.2.2. Option: Analysis Type: Modal [ANTYPE]

Use this option to specify a modal analysis.

3.5.2.3. Option: Mode-Extraction Method [MODOPT]

Choose one of the extraction methods listed below. (For more detailed information, see Section 3.13: Comparing

Mode-Extraction Methods.)

• Block Lanczos method (default)

The Block Lanczos method is used for large symmetric eigenvalue problems. You can use this method for

the same types of problems for which you use the subspace method, but you achieve a faster convergence

rate. The Block Lanczos method uses the sparse matrix solver, overriding any solver specified via the

EQSLV command.

• Subspace method

The subspace method is used for large symmetric eigenvalue problems. Several solution controls are

available to control the subspace iteration process.

When doing a modal analysis with a large number of constraint equations, use the subspace method with

the frontal solver instead of the JCG solver, or use the Block Lanczos mode-extraction method.

• PowerDynamics method

The PowerDynamics method is used for very large models (100,000+ DOFs), and is especially useful to

obtain a solution for the first several modes to learn how the model will behave. You can then choose the

most appropriate extraction method (subspace or Block Lanczos) for running the final solution. This

method automatically uses the lumped mass approximation (LUMPM,ON)

To use the PowerDynamics method when running in batch or command mode, you first issue MOD-

OPT,SUBSPACE, followed by EQSLV,PCG.

• Reduced (Householder) method

The reduced method is faster than the subspace method because it uses reduced (condensed) system

matrices to calculate the solution. However, it is less accurate because the reduced mass matrix is approx-

imate. (See Section 3.13: Comparing Mode-Extraction Methods.)

• Unsymmetric method

The unsymmetric method is used for problems with unsymmetric matrices, such as fluid-structure inter-

action problems.

• Damped method

The damped method is used for problems where damping cannot be ignored, such as bearing problems.

• QR Damped method

The QR damped method is faster and achieves better calculation efficiency than the damped method. It

uses the reduced modal damped matrix to calculate complex damped frequencies in modal coordinates.

Section 3.5: Apply Loads and Obtain the Solution

3–3ANSYS Structural Analysis Guide . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

For most applications, you will use the Block Lanczos, subspace, reduced, or PowerDynamics method. The un-

symmetric, damped, and QR damped methods are meant for special applications.

When you specify a mode-extraction method, ANSYS automatically chooses the appropriate equation solver.

Note — The damped, unsymmetric, and QR damped methods are not available in the ANSYS Professional

program.

3.5.2.4. 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.

3.5.2.5. 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). If you want the mode shapes

normalized to unity for the Block Lanczos or subspace methods, you will need to expand the modes as well.

3.5.2.6. 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.

3.5.2.7. Option: Prestress Effects Calculation [PSTRES]

Use this option to calculate the modes of a prestressed structure. By default, no prestress effects are included;

that is, the structure is assumed to be stress-free. To include prestress effects, element files from a previous

static (or transient) analysis must be available; see Section 3.11: Prestressed Modal Analysis. If prestress effects

are turned on, the lumped mass setting [LUMPM] in this and subsequent solutions must be the same as it was

in the prestress static analysis.

Note — You can use only axisymmetric loads for prestressing harmonic elements such as PLANE25 and

SHELL61.

3.5.2.8. Additional Modal Analysis Options

After you complete the fields on the Modal Analysis Options dialog box, click OK. A dialog box specific to the

selected extraction method appears. You see some combination of the following fields: FREQB, FREQE, PRMODE,

Nrmkey, RIGID, SUBOPT. Refer to the MODOPT and RIGID command descriptions for the meaning of these

fields.



3.5.3. Define Master Degrees of Freedom

In a modal analysis, you also need to define master degrees of freedom. These are required only for the reduced

mode-extraction method.

Master degrees of freedom (MDOF) are significant degrees of freedom that characterize the dynamic behavior

of the structure. You should choose at least twice as many MDOF as the number of modes of interest. We recom-

mend 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 3.14: Matrix Reduction.

Command(s): M

GUI: Main Menu> Solution> Master DOFs> User Selected> Define

3.5.4. Apply Loads

After defining master degrees of freedom, apply loads on the model. The only "loads" valid in a typical modal

analysis are zero-value displacement constraints. (If you input a nonzero 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 (nonzero frequency) free body modes. Table 3.2: “Loads Applicable in a Modal Analysis” 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, “Loading” in the ANSYS Basic Analysis Guide.

Note — Other loads - forces, pressures, temperatures, accelerations, and so on - 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

GUI PathCmd FamilyCategoryLoad Type

Main Menu> Solution> Define Loads> Apply>

Structural> Displacement

DConstraintsDisplacement (UX, UY, UZ, ROTX,

ROTY, ROTZ)

In an analysis, loads can be applied, removed, operated on, or listed.

3.5.4.1. Applying Loads Using Commands

Table 3.3: “Load Commands for a Modal Analysis” lists all the commands you can use to apply loads in a modal

analysis.

Table 3.3 Load Commands for a Modal Analysis

Apply SettingsOperateListDeleteApplyEntitySolid Model

or FE

Load Type

-DTRANDKLISTDKDELEDKKeypointsSolid ModelDisplacement

-DTRANDLLISTDLDELEDLLinesSolid Model

-DTRANDALISTDADELEDAAreasSolid Model

DSYM, DCUMDSCALEDLISTDDELEDNodesFinite Elem



3.5.4.2. Applying Loads Using the GUI

All loading operations (except List; see Section 3.5.4.3: Listing Loads) are accessed through a series of cascading

menus. From the Solution menu, you select the operation (apply, delete, and so on), then the load type (displace-

ment, force, and so on), and then the object to which you are applying the load (keypoint, line, node, and so on).

For example, to apply a displacement load to a line, follow this GUI path:

GUI:

Main Menu> Solution> Define Loads> Apply> Structural> Displacement> On lines

3.5.4.3. Listing Loads

To list existing loads, follow this GUI path:

GUI:

Utility Menu> List>Loads> load type

3.5.5. Specify Load Step Options

The only load step options available for a modal analysis are damping options.

Table 3.4 Load Step Options

CommandOption

Damping (Dynamics) Options

ALPHADAlpha (mass) Damping

BETADBeta (stiffness) Damping

MP,DAMPMaterial-Dependent Damping Ratio

RElement Damping (applied via element real constant)

MP,DMPRConstant Material Damping Coefficient

Damping is valid only for the damped and QR damped mode-extraction methods. Damping is ignored for the

other mode-extraction methods; see the Note below.

If you include damping and specify the damped mode-extraction method, the calculated eigenvalues and eigen-

vectors are complex. If you include damping and specify the QR damped mode-extraction method, the eigenvalues

are complex. However, the real eigenvectors are used for the mode superposition analysis. See Section 3.13:

Comparing Mode-Extraction Methods for details. Also see the section Section 5.10.3: Damping in Chapter 5,

“Transient Dynamic Analysis” for more information on damping.

Only the QR damped method supports the constant material damping coefficient application in a downstream

mode superposition harmonic analysis. The QR damped eigen analysis itself, however, does not include the effect

of the constant material damping coefficient. The corresponding modal damping matrix is formulated during

modal harmonic analysis.

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 calculate an effective damping ratio for each mode, which is then used to calculate the re-

sponse to the spectrum. Spectrum analyses are discussed in Chapter 6, “Spectrum Analysis”.



3.5.6. 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 and rotation about each of these axes.

The reduced mass distribution is also listed. Rotational participation factors will be calculated when a real eigen-

solver mode-extraction method (such as Block Lanczos, subspace, or QRDAMP) is used.

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 (rotation about the Z axis 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, then issue another *GET command.

3.5.7. Solve

Before you solve, you should save (SAVE) a back-up copy of the database to a named file. You can then retrieve

your model by reentering the ANSYS program and issuing RESUME.

Now start the solution calculations.

Command(s): SOLVE

GUI: Main Menu> Solution> Solve> Current LS

3.5.7.1. Output

The output from the solution consists mainly of the natural frequencies, which are printed as part of the printed

output (Jobname.OUT) and also written to the mode shape file (Jobname.MODE). The printed output may include

reduced mode shapes and the participation factor table, depending on your analysis options and output controls.

No mode shapes are written to the database or to the results file, so you cannot postprocess the results yet. To

do this, you need to expand the modes (explained next).

3.5.7.1.1. Output From Subspace Mode-Extraction Method

If you use the subspace mode-extraction method, you might see the following warning in the solution printout:

"STURM number = n should be m," where n and m are integer numbers. This indicates that a mode has been

missed, or that the mth

and n

th mode gave the same frequencies and only m modes were requested.

There are two methods that you can use to investigate the missed mode: use more iteration vectors, or change

the shift point used in the eigenvalue extraction. Both methods are briefly described below. (See the ANSYS, Inc.

Theory Reference for more information.)

To use more iteration vectors, you can issue the SUBOPT,,NPAD command. If you prefer to use the GUI to adjust

the number of iteration vectors, follow these steps:

1. Choose menu path Main Menu> Solution> Analysis Type> Analysis Options. The Modal Analysis

dialog box appears.

2. Choose “Subspace” as the mode-extraction method and specify the number of modes to extract; then

click OK. The Subspace Modal Analysis dialog box appears.

3. Change the value of the NPAD field and click OK.

To change the shift point that was used in the eigenvalue extraction, you can issue the MODOPT,,,FREQB com-

mand. If you prefer to use the GUI to change the shift point, follow these steps:



1. Choose menu path Main Menu> Solution> Analysis Type> Analysis Options. The Modal Analysis

dialog box appears.

2. Choose “Subspace” as the mode-extraction method and specify the number of modes to extract; then

click OK. The Subspace Modal Analysis dialog box appears.

3. Change the value of the FREQB field and click OK.

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. If you use the QR damped mode-extraction method, the eigenvalues are complex. However, the real ei-

genvectors are used for the mode superposition analysis.

3.5.8. Exit the Solution Processor

You must now exit the solution processor.

Command(s): FINISH

GUI: Main Menu> Finish

3.6. 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 re-

sponse 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.

3.6.1. 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.

3.6.2. Expanding the Modes

1. Reenter the ANSYS solution processor.

Command(s): /SOLU

GUI: Main Menu> Solution

Note — You must explicitly leave SOLUTION (using the FINISH command) and reenter (/SOLU)

before performing the expansion pass.

2. Activate the expansion pass and its options. ANSYS offers these options for the expansion pass:



Table 3.5 Expansion Pass Options

GUI PathCommandOption

Main Menu> Solution> Analysis Type> ExpansionPassEXPASSExpansion Pass On/Off

Main Menu> Solution> Load Step Opts> Expansion-

Pass> Single Expand> Expand Modes

MXPANDNo. of Modes to Expand



MXPANDFreq. Range for Expansion



MXPANDStress Calc. On/Off

Each of these options is explained in detail below.

Expansion Pass On/Off [EXPASS]

Choose ON.

Number of Modes to Expand [MXPAND, NMODE]

Specify the number. Remember that only expanded modes can be reviewed in the postprocessor.

Default is no modes expanded.

Frequency Range for Expansion [MXPAND,, FREQB, FREQE]

This is another way to control the number of modes expanded. If you specify a frequency range, only

modes within that range are expanded.

Stress Calculations On/Off [MXPAND,,,, Elcalc]

Choose ON only if you plan to do a subsequent spectrum analysis and are interested in stresses or

forces to do the spectrum. "Stresses" from a modal analysis do not represent actual stresses in the

structure, but give you an idea of the relative stress distributions for each mode. Default is no stresses

calculated.

3. Specify load step options. The only options valid in a modal expansion pass are output controls:

• Printed output

Use this option to include any results data (expanded mode shapes, stresses, and forces) on the

printed output file (Jobname.OUT).

Command(s): OUTPR

GUI: Main Menu> Solution> Load Step Opts> Output Ctrls> Solu Printout

• Database and results file output

Use this option to control the data on the results file (Jobname.RST). The FREQ field on OUTRES can

be only ALL or NONE; that is, the data are written for all modes or no modes. For example, you cannot

write information for every other mode.

Command(s): OUTRES

GUI: Main Menu> Solution> Load Step Opts> Output Ctrls> DB/Results File

4. Start expansion pass calculations.

The output consists of expanded mode shapes and, if requested, relative stress distributions for each

mode.

Command(s): SOLVE

GUI: Main Menu> Solution> Solve> Current LS

Section 3.6: Expand the Modes


5. Repeat steps 2, 3, and 4 for additional modes to be expanded (in different frequency ranges, for example).

Each expansion pass is stored as a separate load step on the results file.

Caution: Spectrum analyses expect all expanded modes to be in one load step. 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).

6. Leave SOLUTION. You can now review results in the postprocessor.

Command(s): FINISH

GUI: Close the Solution menu.

Note — The expansion pass has been presented here as a separate step. However, if you include the

MXPAND command in the modal solution step, the program not only extracts the eigenvalues and ei-

genvectors, but also expands the specified mode shapes.

3.7. Review the Results

Results from a modal analysis (that is, the modal expansion pass) are written to the structural results file, Job-

name.RST. Results consist of:

• Natural frequencies

• Expanded mode shapes

• Relative stress and force distributions (if requested).

You can review these results in POST1 [/POST1], the general postprocessor. Some typical postprocessing oper-

ations for a modal analysis are described below. For a complete description of all postprocessing functions, see

Chapter 4, “An Overview of Postprocessing” in the ANSYS Basic Analysis Guide.

3.7.1. Points to Remember

• If you want to review results in POST1, the database must contain the same model for which the solution

was calculated.

• The results file (Jobname.RST) must be available.

3.7.2. Reviewing Results Data

1. Read in results data from the appropriate substep. Each mode is stored on the results file as a separate

substep. If you expand six modes, for instance, your results file will have one load step consisting of six

substeps.

Command(s): SET,SBSTEP

GUI: Main Menu> General Postproc> Read Results> substep

2. Perform any desired POST1 operations. Typical modal analysis POST1 operations are explained below:

3.7.3. Option: Listing All Frequencies

You may want to list the frequencies of all modes expanded. A sample output from this command is shown below.



***** INDEX OF DATA SETS ON RESULTS FILE *****

SET TIME/FREQ LOAD STEP SUBSTEP CUMULATIVE

1 22.973 1 1 1

2 40.476 1 2 2

3 78.082 1 3 3

4 188.34 1 4 4

Command(s): SET,LIST

GUI: Main Menu> General Postproc> List Results

3.7.4. Option: Display Deformed ShapeCommand(s): PLDISP

GUI: Main Menu> General Postproc> Plot Results> Deformed Shape

Use the KUND field on PLDISP to overlay the nondeformed shape on the display.

3.7.5. Option: List Master DOFCommand(s): MLIST,ALL

GUI: Main Menu> Solution> Master DOFs> User Selected> List All

Note — To display the master DOFs graphically, plot the nodes (Utility Menu> Plot> Nodes or command

NLIST).

3.7.6. Option: Line Element ResultsCommand(s): ETABLE

GUI: Main Menu> General Postproc> Element Table> Define Table

For line elements, such as beams, spars, and pipes, use the ETABLE command to access derived data (stresses,

strains, and so on). Results data are identified by a combination of a label and a sequence number or component

name on the ETABLE command. See the ETABLE discussion in The General Postprocessor (POST1) in the ANSYS

Basic Analysis Guide for details.

3.7.7. Option: Contour DisplaysCommand(s): PLNSOL or PLESOL

GUI: Main Menu> General Postproc> Plot Results> Contour Plot> Nodal Solu or Element Solu

Use these options to contour almost any result item, such as stresses (SX, SY, SZ...), strains (EPELX, EPELY, EPELZ...),

and displacements (UX, UY, UZ...).

The KUND field on PLNSOL and PLESOL gives you the option of overlaying the nondeformed shape on the display.

You can also contour element table data and line element data:

Command(s): PLETAB, PLLS

GUI: Main Menu> General Postproc> Element Table> Plot Element Table

Main Menu> General Postproc> Plot Results> Contour Plot> Line Elem Res

Caution: Derived data, such as stresses and strains, are averaged at the nodes by the PLNSOL command.

This averaging results in "smeared" values at nodes where elements of different materials, different shell

thicknesses, or other discontinuities meet. To avoid the smearing effect, use selecting (described in

Chapter 7, “Selecting and Components” in the ANSYS Basic Analysis Guide) to select elements of the same

material, same shell thickness, and so on before issuing PLNSOL.

Section 3.7: Review the Results


3.7.8. Option: Tabular ListingsCommand(s): PRNSOL (nodal results) PRESOL (element-by-element results) PRRSOL (reaction data),

and so on NSORT, ESORT

GUI: Main Menu> General Postproc> List Results> solution option

Main Menu> General Postproc> List Results> Sorted Listing> Sort Nodes

Main Menu> General Postproc> List Results> Sorted Listing> Sort Elems

Use the NSORT and ESORT commands to sort the data before listing them.

3.7.9. Other Capabilities

Many other postprocessing functions - mapping results onto a path, load case combinations, and so on - are

available in POST1. See The General Postprocessor (POST1) in the ANSYS Basic Analysis Guide for details.

See the ANSYS Commands Reference for a discussion of the ANTYPE, MODOPT, M, TOTAL, EXPASS, MXPAND,

SET, and PLDISP commands.

3.8. A Sample Modal Analysis (GUI Method)

In this example, you perform a modal analysis on the wing of a model plane to demonstrate the wing's modal

degrees of freedom.

3.8.1. 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 end

and hangs freely at the other. The objective of the problem is to demonstrate the wing's modal degrees of freedom.

3.8.2. 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 = 8.3e-5 lb-sec2/in

4



3.8.3. Problem Sketch

Figure 3.1 Diagram of a Model Airplane Wing

The detailed step-by-step procedure for this example, Modal Analysis of a Model Airplane Wing, is included in

the Modal Tutorial.

3.9. A Sample Modal Analysis (Command or Batch Method)

You can perform the example modal analysis of a model airplane wing using the ANSYS commands shown below

instead of GUI choices. Items prefaced with an exclamation point (!) are comments.

You may receive warning messages when you run this problem. The version of the problem that appears in the

Modal Tutorial contains an explanation of the warnings.

/FILNAM,MODAL

/TITLE,Modal Analysis of a Model Airplane Wing

/PREP7

ET,1,PLANE42 ! Define PLANE42 as element type 1

ET,2,SOLID45 ! Define SOLID45 as element type 2

MP,EX,1,38000

MP,DENS,1,8.3E-5

MP,NUXY,1,.3

K,1 ! Define keypoint 1 at 0,0,0

K,2,2 ! Define keypoint 2 at 2,0,0

K,3,2.3,.2 ! Define keypoint 3 at 2.3,.2,0

K,4,1.9,.45 ! Define keypoint 4 at 1.9,.45,0

K,5,1,.25 ! Define keypoint 5 at 1,.25,0

LSTR,1,2 ! Create a straight line between keypoints 1 and 2

LSTR,5,1 ! Create a straight line between keypoints 5 and 1

BSPLIN,2,3,4,5,,,-1,,,-1,-.25 ! Create a B-spline

AL,1,3,2

ESIZE,.25

AMESH,1

ESIZE,,10

TYPE,2

VEXT,ALL,,,,,10

/VIEW,,1,1,1

/ANG,1

/REP

EPLOT

FINISH

/SOLU

ANTYPE,MODAL ! Choose modal analysis type

MODOPT,SUBSP,5 ! Choose the subspace mode-extraction method,

Section 3.9: A Sample Modal Analysis (Command or Batch Method)


! extracting 5 modes

ESEL,U,TYPE,,1 ! Unselect element type 1

NSEL,S,LOC,Z,0

D,ALL,ALL

NSEL,ALL

MXPAND,5

SOLVE

FINISH

/POST1

SET,LIST,2

SET,FIRST

PLDISP,0

ANMODE,10,.5E-1

SET,NEXT

PLDISP,0

ANMODE,10,.5E-1

SET,NEXT

PLDISP,0

ANMODE,10,.5E-1

SET,NEXT

PLDISP,0

ANMODE,10,.5E-1

SET,NEXT

PLDISP,0

ANMODE,10,.5E-1

FINISH

/EXIT

3.10. Where to Find Other Examples

Several ANSYS publications, particularly the ANSYS Verification Manual, describe additional modal analyses.

The ANSYS Verification Manual consists of test case analyses demonstrating the analysis capabilities of the ANSYS

family of products. While these test cases demonstrate solutions to realistic analysis problems, the ANSYS Verific-

ation Manual does not present them as step-by-step examples with lengthy data input instructions and printouts.

However, most ANSYS users who have at least limited finite element experience should be able to fill in the

missing details by reviewing each test case's finite element model and input data with accompanying comments.

The ANSYS Verification Manual includes variety of modal analysis test cases:

VM45 - Natural Frequency of a Spring-mass System

VM47 - Torsional Frequency of a Suspended Disk

VM48 - Natural Frequency of a Motor-generator

VM50 - Fundamental Frequency of a Simply Supported Beam

VM52 - Automobile Suspension System Vibrations

VM53 - Vibration of a String Under Tension

VM54 - Vibration of a Rotating Cantilever Blade

VM55 - Vibration of a Stretched Circular Membrane

VM57 - Torsional Frequencies of a Drill Pipe

VM59 - Lateral Vibration of an Axially-loaded Bar

VM60 - Natural Frequency of a Cross-ply Laminated Spherical Shell

VM61 - Longitudinal Vibration of a Free-free Rod

VM62 - Vibration of a Wedge

VM66 - Vibration of a Flat Plate

VM67 - Radial Vibrations of a Circular Ring from an Axisymmetric Model

VM68 - PSD Response of a Two DOF Spring-mass System

VM69 - Seismic Response

VM70 - Seismic Response of a Beam Structure

VM76 - Harmonic Response of a Guitar String



VM89 - Natural Frequencies of a Two-mass-spring System

VM151 - Nonaxisymmetric Vibration of a Circular Plate

VM152 - Nonaxisymmetric Vibration of a Stretched Circular Membrane (Harmonic Els)

VM153 - Nonaxisymmetric Vibration of a Stretched Circular Membrane (Modal)

VM154 - Vibration of a Fluid Coupling

VM175 - Natural Frequency of a Piezoelectric Transducer

VM181 - Natural Frequency of a Flat Circular Plate with a Clamped Edge

VM182 - Transient Response of a Spring-mass System

VM183 - Harmonic Response of a Spring-mass System

VM202 - Transverse Vibrations of a Shear Beam

VM203 - Dynamic Load Effect on Simply-supported Thick Square Plate

VM212 - Modal Analysis of a Rectangular Cavity

3.11. 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, “Structural Static Analysis” describes the procedure to obtain a static solution.

2. Reenter SOLUTION and obtain the modal solution, also with prestress effects turned on (reissue

PSTRES,ON). Files Jobname.EMAT (if ANSYS creates it) and Jobname.ESAV from the static analysis

must be available. If the model is spinning, include spin-softening effects (via the OMEGA command's

KSPIN option) if necessary.

3. Expand the modes and review them in the postprocessor.

Step 1 above can also be a transient analysis. In such a case, save the EMAT and ESAV files at the desired time

point.

If the deformed shape from the static solution differs significantly from its nondeformed shape, you can perform

a prestressed modal analysis of a large-deflection solution instead.

3.12. Prestressed Modal Analysis of a Large-Deflection Solution

You can also perform a prestressed modal analysis following a large deflection (NLGEOM,ON) static analysis in

order to calculate the frequencies and mode shapes of a highly deformed structure. Use the prestressed modal

analysis procedure, but use the PSOLVE command (rather than the SOLVE command) to obtain the modal

solution, as shown in the sample input listing below.

Along with the PSOLVE command, you must issue the UPCOORD command to update the coordinates necessary

for obtaining the correct stresses.

This procedure uses the element matrices and element load vectors (for example, from pressures, temperature

or acceleration loads) from a previous static analysis. These loads will be passed through to a subsequent mode

superposition analysis if specified (LVSCALE command).

Prestress must be applied (PSTRES,ON) during the static portion of the analysis. However, in cases where stress-

stiffening helps convergence:

Section 3.12: Prestressed Modal Analysis of a Large-Deflection Solution


• Stress-stiffening (SSTIF,ON) must be applied instead. (This requirement applies to elements outside of

the 18x family of elements only.)

• The EMATWRITE,YES command is also necessary to write the element matrices to File.EMAT.

Issuing either a PSTRES,OFF or SSTIF,OFF command prevents all previously specified prestressing from being

applied.

If the model is spinning, include spin-softening effects (via the OMEGA command's KSPIN option) in the modal

solution if necessary.

! Initial, large deflection static analysis

!

/PREP7

...

FINISH

/SOLU

ANTYPE,STATIC ! Static analysis

NLGEOM,ON ! Large deflection analysis

PSTRES,ON ! Flag to calculate the prestress matrix

...

SOLVE

FINISH

!

! Prestressed modal analysis

!

/SOLU

ANTYPE,MODAL ! Modal analysis

UPCOORD,1.0,ON ! Display mode shapes relative to deformed geometry

! in the postprocessor.

PSTRES,ON ! Prestress effects ON

MODOPT,... ! Select eigensolver

MXPAND,... ! Specify the number of modes to expand, if desired.

PSOLVE,EIGxxxx ! Calculate the eigenvalues and eigenvectors.

! Use EIGLANB or EIGFULL to match MODOPT command.

FINISH

/SOLU !Additional solution step for expansion.

EXPASS,ON

PSOLVE,EIGEXP ! Expand the eigenvector solution. (Required if you

! want to review mode shapes in the postprocessor.)

FINISH

Note — You may also use one of the other eigensolvers (PSOLVE,EIGSYM, PSOLVE,EIGUNSY, PSOLVE,EIG-

DAMP or PSOLVE,EIGREDUC). In such a case, a PSOLVE,TRIANG command must precede the

PSOLVE,EIGxxx command.

3.13. Comparing Mode-Extraction Methods

The basic equation solved in a typical undamped modal analysis is the classical eigenvalue problem:

[ ] [ ] K Mi i iφ ω φ=2

where:

[K] = stiffness matrix

Φi = mode shape vector (eigenvector) of mode i

Ωi = natural circular frequency of mode i (ωi2

is the eigenvalue)

[M] = mass matrix

Many numerical methods are available to solve the above equation. ANSYS offers these methods:



• Block Lanczos method (default)

• Subspace method

• PowerDynamics method

• Reduced (Householder) method

• Unsymmetric method

• Damped method (The damped method solves a different equation; see the ANSYS, Inc. Theory Reference

for more information.)

• QR damped method (The QR damped method solves a different equation; see the ANSYS, Inc. Theory Ref-

erence for more information.)

Note — The damped, unsymmetric, and QR damped methods are not available in the ANSYS Professional

program.

The first four methods (Block Lanczos, subspace, PowerDynamics, and reduced) are the most commonly used.

Table 3.6: “Symmetric System Eigensolver Choices” compares these four mode-extraction methods. Following

the table is a brief description of each mode-extraction method.

Table 3.6 Symmetric System Eigensolver Choices

Disk Re-

quired

Memory

Required

ApplicationEigensolver

LowMediumDefault. To find many modes (about 40+) of large models. Recommen-

ded when the model consists of poorly shaped solid and shell ele-

ments. This solver performs well when the model consists of shells or

a combination of shells and solids. Works faster but requires about

50% more memory than subspace.

Block Lanczos

HighLowTo find few modes (up to about 40) of large models. Recommended

when the model consists of well-shaped solid and shell elements.

Works well if memory availability is limited.

Subspace

LowHighTo find few modes (up to about 20) of large models. Recommended

for fast computation of eigenvalues of over 100K DOF models. On

coarse mesh models, the frequencies are approximate. Missed modes

are possible when repeated frequencies are present.

Power Dynamics

LowLowTo find all modes of small to medium models (less than 10K DOF). Can

be used to find few modes (up to about 40) of large models with

proper selection of master DOF, but accuracy of frequencies depends

on the master DOF selected.

Reduced

3.13.1. Block Lanczos Method

The Block Lanczos eigenvalue solver is the default. It 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 ei-

genvalue 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.

Section 3.13: Comparing Mode-Extraction Methods


3.13.2. Subspace Method

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 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 stiffness assembly that

requires large amounts of memory.

3.13.3. 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

in very large models (100,000+ DOFs) to obtain a solution for the first few modes. Do not use this method if you

will be running a subsequent spectrum analysis.

The PowerDynamics method does not perform a Sturm sequence check (that is, it 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).

3.13.4. 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 3.14: Matrix Reduction presents guidelines to select master DOFs.

3.13.5. Unsymmetric Method

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,

whereas a 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.

3.13.6. Damped Method

The damped method is meant for problems where damping cannot be ignored, such as rotor dynamics applica-

tions. 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 available for this

method. Therefore, missed modes are a possibility at the higher end of the frequencies extracted.



3.13.6.1. Damped Method-Real and Imaginary Parts of the Eigenvalue

The imaginary part of the eigenvalue, Ω, represents the steady-state circular frequency of the system. The real

part of the eigenvalue, σ, represents the stability of the system. If σ is less than zero, then the displacement

amplitude will decay exponentially, in accordance with EXP(σ). If σ is greater than zero, then the amplitude will

increase exponentially. (Or, in other words, negative σ gives an exponentially decreasing, or stable, response;

and positive σ gives an exponentially increasing, or unstable, response.) If there is no damping, the real component

of the eigenvalue will be zero.

Note — The eigenvalue results reported by ANSYS are actually divided by 2* π. This gives the frequency

in Hz (cycles/second). In other words:

Imaginary part of eigenvalue, as reported = Ω/(2* π)

Real part of eigenvalue, as reported = σ/(2* π)

3.13.6.2. 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.

3.13.7. QR Damped Method

The QR damped method combines the advantages of the Block Lanczos method with the complex Hessenberg

method. The key concept is to approximately represent the first few complex damped eigenvalues by modal

transformation using a small number of eigenvectors of the undamped system. After the undamped mode

shapes are evaluated by using the real eigensolution (Block Lanczos method), the equations of motion are

transformed to these modal coordinates.

Using the QR algorithm, a smaller eigenvalue problem is then solved in the modal subspace. This approach gives

good results for lightly damped systems and can also be applicable to any arbitrary damping type (proportional

or non-proportional symmetric damping or nonsymmetrical gyroscopic damping matrix). This approach also

supports nonsymmetrical stiffness if present in the model.

The QR Damp eigensolver is applicable for models with an unsymmetrical global stiffness matrix where only a

few elements contribute nonsymmetrical element stiffness matrices. Example: Brake friction problem wherein

the local part of a model with friction contacts is producing nonsymmetrical stiffness matrix in contact elements.

Because the accuracy of this method is dependent on the number of modes used in the calculations, a sufficient

number of fundamental modes are recommended, especially for highly damped systems to provide good results.

This method is not recommended for critically damped or overdamped systems. This method outputs both the

real and imaginary eigenvalues (frequencies), but outputs only the real eigenvectors (mode shapes). When re-

quested however, complex mode shapes of damped systems are computed.

3.14. Matrix Reduction

Matrix reduction is a way to reduce the size of the matrices of a model and perform a quicker and cheaper ana-

lysis. It is mainly used in dynamic analyses such as modal, harmonic, and transient analyses. Matrix reduction is

also used in substructure analyses to generate a superelement.

Matrix reduction allows you to build a detailed model, as you would for a static stress analysis, and use only a

"dynamic" portion of it for a dynamic analysis. You choose the "dynamic" portion by identifying key degrees of

freedom, called master degrees of freedom, that characterize the dynamic behavior of the model. The ANSYS

program then calculates reduced matrices and the reduced DOF solution in terms of the master DOF. You can

Section 3.14: Matrix Reduction


then expand the solution to the full DOF set by performing an expansion pass. The main advantage of this pro-

cedure is the savings in CPU time to obtain the reduced solution, especially for dynamic analyses of large problems.

3.14.1. Theoretical Basis of Matrix Reduction

The ANSYS program uses the Guyan Reduction procedure to calculate the reduced matrices. The key assumption

in this procedure is that for the lower frequencies, inertia forces on the slave DOF (those DOF being reduced out)

are negligible compared to elastic forces transmitted by the master DOF. Therefore, the total mass of the structure

is apportioned among only the master DOF. The net result is that the reduced stiffness matrix is exact, whereas

the reduced mass and damping matrices are approximate. For details about how the reduced matrices are cal-

culated, refer to the ANSYS, Inc. Theory Reference.

3.14.1.1. Guidelines for Selecting Master DOF

Choosing master DOF is an important step in a reduced analysis. The accuracy of the reduced mass matrix (and

hence the accuracy of the solution) depends on the number and location of masters. For a given problem, you

can choose many different sets of master DOF and will probably obtain acceptable results in all cases.

You can choose masters using M and MGEN commands, or you can have the program choose masters during

solution using the TOTAL command. We recommend that you do both: choose a few masters yourself, and also

have the ANSYS program choose masters. This way, the program can pick up any modes that you may have

missed. The following list summarizes the guidelines for selecting master DOF:

• The total number of master DOF should be at least twice the number of modes of interest.

• Choose master DOF in directions in which you expect the structure or component to vibrate. For a flat

plate, for example, you should choose at least a few masters in the out-of-plane direction (see Fig-

ure 3.2: “Choose Master DOF” (a)). In cases where motion in one direction induces a significant motion in

another direction, choose master DOF in both directions (see Figure 3.2: “Choose Master DOF” (b)).

Figure 3.2 Choose Master DOF

(a) Possible out-of-plane masters for a flat plate(b) Motion in X induces motion in Y

• Choose masters at locations having relatively large mass or rotary inertia and relatively low stiffness (see

Figure 3.3: “Choosing Master DOFs”). Examples of such locations are overhangs and "loosely" connected

structures. Conversely, do not choose masters at locations with relatively small mass, or at locations with

high stiffness (such as DOF close to constraints).



Figure 3.3 Choosing Master DOFs

Choose masters at locations with (a) large rotary inertia, (b) large mass

• If your primary interest is in bending modes, you can neglect rotational and "stretching" DOF.

• If the degree of freedom to be chosen belongs to a coupled set, choose only the first (primary) DOF of

the coupled set.

• Choose master DOF at locations where forces or nonzero displacements are to be applied.

• For axisymmetric shell models (SHELL51 or SHELL61), choose as masters the global UX degree of freedom

at all nodes on those sections of the model that are parallel to or nearly parallel to the center line, so os-

cillatory motions between master DOF can be avoided (see Figure 3.4: “Choosing Masters in an Axisym-

metric Shell Model”). This recommendation can be relaxed if the motion is primarily parallel to the

centerline. For axisymmetric harmonic elements with MODE = 2 or greater, choose as masters both UX

and UZ degrees of freedom.

Figure 3.4 Choosing Masters in an Axisymmetric Shell Model

The best way to check the validity of the master DOF set is to rerun the analysis with twice (or half) the number

of masters and to compare the results. Another way is to review the reduced mass distribution printed during a

modal solution. The reduced mass should be, at least in the dominant direction of motion, within 10-15 percent

of the total mass of the structure.

Section 3.14: Matrix Reduction


3.14.1.2. A Note About Program-Selected Masters

If you let the ANSYS program select masters [TOTAL], the distribution of masters selected will depend on the

order in which elements are processed during the solution. For example, different master DOF sets may be se-

lected depending on whether the elements are processed from left to right or from right to left. However, this

difference usually yields insignificant differences in the results.

For meshes with uniform element sizes and properties (for example, a flat plate), the distribution of masters will,

in general, not be uniform. In such cases, you should specify some master DOF of your own [M, MGEN]. The same

recommendation applies to structures with an irregular mass distribution, where the program-selected master

DOF may be concentrated in the higher-mass regions.



ANSYS Structural Analysis Guide MODAL ANALYSIS

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