ABAQUS关于固有频率的提取方法_图文

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Abaqus固有频率提取

6.3.5 Natural frequency extraction

Products: Abaqus/Standard Abaqus/CAE Abaqus/AMS

References

•

•

•

•

•

“Procedures: overview,” Section 6.1.1

“General and linear perturbation procedures,” Section 6.1.2

“Dynamic analysis procedures: overview,” Section 6.3.1

*FREQUENCY

“Configuring a frequency procedure” in “Configuring linear

perturbation analysis procedures,” Section 14.11.2 of the

Abaqus/CAE User's Manual

Overview

The frequency extraction procedure:

•

performs eigenvalue extraction to calculate the natural

frequencies and the corresponding mode shapes of a system;

•

will include initial stress and load stiffness effects due to

preloads and initial conditions if geometric nonlinearity is

accounted for in the base state, so that small vibrations of a

preloaded structure can be modeled;

•

will compute residual modes if requested;

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•

•

is a linear perturbation procedure;

can be performed using the traditional Abaqus software architecture

or, if appropriate, the high-performance SIM architecture

(see “Using the SIM architecture for modal superposition dynamic

analyses” in “Dynamic analysis procedures: overview,” Section

6.3.1); and

•

solves the eigenfrequency problem only for symmetric mass and

stiffness matrices; the complex eigenfrequency solver must be

used if unsymmetric contributions, such as the load stiffness,

are needed.

Eigenvalue extraction

The eigenvalue problem for the natural frequencies of an undamped

finite element model is

where

is the mass matrix (which is symmetric and positive definite);

is the stiffness matrix (which includes initial stiffness effects if the

base state included the effects of nonlinear geometry);

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is the eigenvector (the mode of vibration); and

M

and

N

are degrees of freedom.

When is positive definite, all eigenvalues are positive. Rigid body

to be indefinite. Rigid body modes modes and instabilities cause

produce zero eigenvalues. Instabilities produce negative eigenvalues and

occur when you include initial stress effects. Abaqus/Standard solves the

eigenfrequency problem only for symmetric matrices.

Selecting the eigenvalue extraction method

Abaqus/Standard provides three eigenvalue extraction methods:

•

•

Lanczos

Automatic multi-level substructuring (AMS), an add-on analysis

capability for Abaqus/Standard

•

Subspace iteration

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In addition, you must consider the software architecture that will be used

for the subsequent modal superposition procedures. The choice of

architecture has minimal impact on the frequency extraction procedure,

but the SIM architecture can offer significant performance improvements

over the traditional architecture for subsequent mode-based steady-state

or transient dynamic procedures (see “Using the SIM architecture for

modal superposition dynamic analyses” in “Dynamic analysis procedures:

overview,” Section 6.3.1). The architecture that you use for the

frequency extraction procedure is used for all subsequent mode-based

linear dynamic procedures; you cannot switch architectures during an

analysis. The software architectures used by the different eigensolvers

are outlined in Table 6.3.5–1.

Table 6.3.5–1 Software architectures available with different

eigensolvers.

Eigensolver

Software Architecture

Lanczos AMS Subspace Iteration

Traditional

SIM

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The Lanczos solver with the traditional architecture is the default

eigenvalue extraction method because it has the most general capabilities.

However, the Lanczos method is generally slower than the AMS method. The

increased speed of the AMS eigensolver is particularly evident when you

require a large number of eigenmodes for a system with many degrees of

freedom. However, the AMS method has the following limitations:

•

All restrictions imposed on SIM-based linear dynamic procedures

also apply to mode-based linear dynamic analyses based on mode

shapes computed by the AMS eigensolver. See “Using the SIM

architecture for modal superposition dynamic analyses” in

“Dynamic analysis procedures: overview,” Section 6.3.1, for

details.

•

The AMS eigensolver does not compute composite modal damping

factors, participation factors, or modal effective masses.

However, if participation factors are needed for primary base

motions, they will be computed but are not written to the printed

data (.dat) file.

•

You cannot use the AMS eigensolver in an analysis that contains

piezoelectric elements.

•

You cannot request output to the results (.fil) file in an AMS

frequency extraction step.

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If your model has many degrees of freedom and these limitations are

acceptable, you should use the AMS eigensolver. Otherwise, you should use

the Lanczos eigensolver. The Lanczos eigensolver and the subspace

iteration method are described in“Eigenvalue extraction,” Section

2.5.1 of the Abaqus Theory Manual.

Lanczos eigensolver

For the Lanczos method you need to provide the maximum frequency of

interest or the number of eigenvalues required; Abaqus/Standard will

determine a suitable block size (although you can override this choice,

if needed). If you specify both the maximum frequency of interest and the

number of eigenvalues required and the actual number of eigenvalues is

underestimated, Abaqus/Standard will issue a corresponding warning

message; the remaining eigenmodes can be found by restarting the frequency

extraction.

You can also specify the minimum frequencies of interest; Abaqus/Standard

will extract eigenvalues until either the requested number of eigenvalues

has been extracted in the given range or all the frequencies in the given

range have been extracted.

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See “Using the SIM architecture for modal superposition dynamic

analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1,

for information on using the SIM architecture with the Lanczos

eigensolver.

Input File Usage:

*FREQUENCY, EIGENSOLVER=LANCZOS

Abaqus/CAE UsageStep

:

module: Step

r: Lanczos

Choosing a block size for the Lanczos method

Create: Frequency: Basic: Eigensolve

In general, the block size for the Lanczos method should be as large as

the largest expected multiplicity of eigenvalues (that is, the largest

number of modes with the same frequency). A block size larger than 10 is

not recommended. If the number of eigenvalues requested is

n

, the default

block size is the minimum of (7,

n

). The choice of 7 for block size proves

to be efficient for problems with rigid body modes. The number of block

Lanczos steps within each Lanczos run is usually determined by

Abaqus/Standard but can be changed by you. In general, if a particular

type of eigenproblem converges slowly, providing more block Lanczos steps

will reduce the analysis cost. On the other hand, if you know that a

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particular type of problem converges quickly, providing fewer block

Lanczos steps will reduce the amount of in-core memory used. The default

values are

Block size Maximum number of block Lanczos steps

1

2

3

≥ 4

80

50

45

35

Automatic multi-level substructuring (AMS) eigensolver

For the AMS method you need only specify the maximum frequency of interest

(the global frequency), and Abaqus/Standard will extract all the modes

up to this frequency. You can also specify the minimum frequencies of

interest and/or the number of requested modes. However, specifying these

values will not affect the number of modes extracted by the eigensolver;

it will affect only the number of modes that are stored for output or for

a subsequent modal analysis.

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The execution of the AMS eigensolver can be controlled by specifying three

parameters: , , and . These three

parameters multiplied by the maximum frequency of interest define three

cut-off frequencies. (default value of 5) controls the cutoff

frequency for substructure eigenproblems in the reduction phase,

while and (default values of 1.7 and 1.1,

respectively) control the cutoff frequencies used to define a starting

subspace in the reduced eigensolution phase. Generally, increasing the

value of and improves the accuracy of the results

but may affect the performance of the analysis.

Requesting eigenvectors at all nodes

By default, the AMS eigensolver computes eigenvectors at every node of

the model.

Input File Usage:

*FREQUENCY, EIGENSOLVER=AMS

Abaqus/CAE Usage:Step

module: Step

er: AMS

Requesting eigenvectors only at specified nodes

Create: Frequency: Basic: Eigensolv

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Alternatively, you can specify a node set, and eigenvectors will be

computed and stored only at the nodes that belong to that node set. The

node set that you specify must include all nodes at which loads are applied

or output is requested in any subsequent modal analysis (this includes

any restarted analysis). If element output is requested or element-based

loading is applied, the nodes attached to the associated elements must

also be included in this node set. Computing eigenvectors at only selected

nodes improves performance and reduces the amount of stored data.

Therefore, it is recommended that you use this option for large problems.

Input File Usage:

*FREQUENCY, EIGENSOLVER=AMS, NSET=

name

Abaqus/CAE UsageStep

:

module: StepCreate: Frequency: Basic: Eigensolve

r: AMS: Limit region of saved eigenvectors

Controlling the AMS eigensolver

The AMS method consists of the following three phases:

Reduction phase: In this phase Abaqus/Standard uses a multi-level

substructuring technique to reduce the full system in a way that allows

a very efficient eigensolution of the reduced system. The approach

combines a sparse factorization based on a multi-level supernode

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elimination tree and a local eigensolution at each supernode. Starting

from the lowest level supernodes, we use a Craig-Bampton substructure

reduction technique to successively reduce the size of the system as we

progress upward in the elimination tree. At each supernode a local

eigensolution is obtained based on fixing the degrees of freedom connected

to the next higher level supernode (these are the local retained or

“fixed-interface” degrees of freedom). At the end of the reduction phase

the full system has been reduced such that the reduced stiffness matrix

is diagonal and the reduced mass matrix has unit diagonal values but

contains off-diagonal blocks of nonzero values representing the coupling

between the cost of the reduction phase depends on the

system size and the number of eigenvalues extracted (the number of

eigenvalues extracted is controlled indirectly by specifying the highest

eigenfrequency desired). You can make trade-offs between cost and

accuracy during the reduction phase through the parameter.

This parameter multiplied by the highest eigenfrequency specified for the

full model yields the highest eigenfrequency that is extracted in the

local supernode eigensolutions. Increasing the value

of increases the accuracy of the reduction since more local

eigenmodes are retained. However, increasing the number of retained modes

also increases the cost of the reduced eigensolution phase, which is

discussed next.

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Reduced eigensolution phase: In this phase Abaqus/Standard computes the

eigensolution of the reduced system that comes from the previous phase.

Although the reduced system typically is two orders of magnitude smaller

in size than the original system, generally it still is too large to solve

directly. Thus, the system is further reduced mainly by truncating the

retained eigenmodes and then solved using a single subspace iteration step.

The two AMS parameters, and , define a starting

subspace of the subspace iteration step. The default values of these

parameters are carefully chosen and provide accurate results in most cases.

When a more accurate solution is needed, the recommended procedure is to

increase both parameters proportionally from their respective default

values.

Recovery phase: In this phase the eigenvectors of the original system are

recovered using eigenvectors of the reduced problem and local

substructure modes. If you request recovery at specified nodes, the

eigenvectors are computed only at those nodes.

Subspace iteration method

For the subspace iteration procedure you need only specify the number of

eigenvalues required; Abaqus/Standard chooses a suitable number of

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vectors for the iteration. If the subspace iteration technique is

requested, you can also specify the maximum frequency of interest;

Abaqus/Standard extracts eigenvalues until either the requested number

of eigenvalues has been extracted or the last frequency extracted exceeds

the maximum frequency of interest.

Input File Usage:

*FREQUENCY, EIGENSOLVER=SUBSPACE

Abaqus/CAE UsageStep

:

module: Step

r: Subspace

Create: Frequency: Basic: Eigensolve

Structural-acoustic coupling

Structural-acoustic coupling affects the natural frequency response of

systems. In Abaqus only the Lanczos eigensolver fully includes this effect.

In Abaqus/AMS and the subspace eigensolver the effect of coupling is

neglected for the purpose of computing the modes and frequencies; these

are computed using natural boundary conditions at the structural-acoustic

coupling surface. An intermediate degree of consideration of the

structural-acoustic coupling operator is the default in Abaqus/AMS and

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the Lanczos eigensolver, which is based on the SIM architecture: the

coupling is projected onto the modal space and stored for later use.

Structural-acoustic coupling using the Lanczos eigensolver without the

SIM architecture

If structural-acoustic coupling is present in the model and the Lanczos

method not based on the SIM architecture is used, Abaqus/Standard extracts

the coupled modes by default. Because these modes fully account for

coupling, they represent the mathematically optimal basis for subsequent

modal procedures. The effect is most noticeable in strongly coupled

systems such as steel shells and water. However, coupled

structural-acoustic modes cannot be used in subsequent random response

or response spectrum analyses. You can define the coupling using either

acoustic-structural interaction elements (see “Acoustic interface

elements,” Section 29.14.1) or the surface-based tie constraint

(see “Acoustic, shock, and coupled acoustic-structural

analysis,” Section 6.10.1). It is possible to ignore coupling when

extracting acoustic and structural modes; in this case the coupling

boundary is treated as traction-free on the structural side and rigid on

the acoustic side.

Input File Usage: Use the following option to account for

structural-acoustic coupling during the

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frequency extraction:

*FREQUENCY, EIGENSOLVER=LANCZOS, ACOUSTIC

COUPLING=ON (default if the SIM architecture

is not used)

Use the following option to ignore

structural-acoustic coupling during the

frequency extraction:

*FREQUENCY, EIGENSOLVER=LANCZOS, ACOUSTIC

COUPLING=OFF

Abaqus/CAE Usage: Step

module: StepCreate: Frequency: Basic: Eigensol

ver: Lanczos, toggle Include acoustic-structural

coupling where applicable

Structural-acoustic coupling using the AMS and Lanczos eigensolver based

on the SIM architecture

For frequency extractions that use the AMS eigensolver or the Lanczos

eigensolver based on the SIM architecture, the modes are computed using

traction-free boundary conditions on the structural side of the coupling

boundary and rigid boundary conditions on the acoustic side.

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Structural-acoustic coupling operators (see “Acoustic, shock, and

coupled acoustic-structural analysis,” Section 6.10.1) are projected by

default onto the subspace of eigenvectors. Contributions to these global

operators, which come from surface-based tie constraints defined between

structural and acoustic surfaces, are assembled into global matrices that

are projected onto the mode shapes and used in subsequent SIM-based modal

dynamic procedures.

User-defined acoustic-structural interaction elements (see “Acoustic

interface elements,” Section 29.14.1) cannot be used in an AMS

eigenvalue extraction analysis.

Input File Usage: Use either of the following options to

project structural-acoustic coupling

operators onto the subspace of eigenvectors:

*FREQUENCY, EIGENSOLVER=AMS, ACOUSTIC

COUPLING=PROJECTION (default for the AMS

eigensolver)

or

*FREQUENCY, EIGENSOLVER=LANCZOS, SIM,

ACOUSTIC COUPLING=PROJECTION (default in

SIM-based analysis)

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Abaqus/CAE Usage:

. .

Use the following option to disable the

projection of structural-acoustic coupling

operators:

*FREQUENCY, ACOUSTIC COUPLING=OFF

Use the following option to project

structural-acoustic coupling operators onto the

subspace of eigenvectors:

Step

module: StepCreate: Frequency: Basic: Eigensol

ver: AMS, toggle on Project acoustic-structural

coupling where applicable

Use the following option to disable the projection of

structural-acoustic coupling operators:

Step

module: StepCreate: Frequency: Basic: Eigensol

ver: AMS, toggle off Project acoustic-structural

coupling where applicable

Projection of structural-acoustic coupling operators

using the Lanczos eigensolver based on the SIM

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architecture is not supported in Abaqus/CAE.

Specifying a frequency range for the acoustic modes

Because structural-acoustic coupling is ignored during the AMS and

SIM-based Lanczos eigenanalysis, the computed resonances will, in

principle, be higher than those of the fully coupled system. This may be

understood as a consequence of neglecting the mass of the fluid in the

structural phase and vice versa. For the common metal and air case, the

structural resonances may be relatively unaffected; however, some

acoustic modes that are significant in the coupled response may be omitted

due to the air's upward frequency shift during eigenanalysis. Therefore,

Abaqus allows you to specify a multiplier, so that the maximum acoustic

frequency in the analysis is taken to be higher than the structural

maximum.

Input File Usage: Use either of the following options:

*FREQUENCY, EIGENSOLVER=AMS , , , , , ,

acoustic range factor

or

*FREQUENCY, EIGENSOLVER=LANCZOS,

SIM , , , , , ,

acoustic range factor

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Abaqus/CAE Usage: Step

module: StepCreate: Frequency: Basic: Eigensol

ver: AMS, Acoustic range factor:

acoustic range

factor

Specifying a frequency range for the acoustic modes

when using the SIM-based Lanczos eigenanalysis is not

supported in Abaqus/CAE.

Effects of fluid motion on natural frequency analysis of acoustic systems

To extract natural frequencies from an acoustic-only or coupled

structural-acoustic system in which fluid motion is prescribed using an

acoustic flow velocity, either the Lanczos method or the complex

eigenvalue extraction procedure can be used. In the former case Abaqus

extracts real-only eigenvalues and considers the fluid motion's effects

only on the acoustic stiffness matrix. Thus, these results are of primary

interest as a basis for subsequent linear perturbation procedures. When

the complex eigenvalue extraction procedure is used, the fluid motion

effects are included in their entirety; that is, the acoustic stiffness

and damping matrices are included in the analysis.

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Frequency shift

For the Lanczos and subspace iteration eigensolvers you can specify a

positive or negative shifted squared frequency,

S

. This feature is useful

when a particular frequency is of concern or when the natural frequencies

of an unrestrained structure or a structure that uses secondary base

motions (large mass approach) are needed. In the latter case a shift from

zero (the frequency of the rigid body modes) will avoid singularity

problems or round-off errors for the large mass approach; a negative

frequency shift is normally used. The default is no shift.

If the Lanczos eigensolver is in use and the user-specified shift is

outside the requested frequency range, the shift will be adjusted

automatically to a value close to the requested range.

Normalization

For the Lanczos and subspace iteration eigensolvers both displacement and

mass eigenvector normalization are available. Displacement normalization

is the default. Mass normalization is the only option available for

SIM-based natural frequency extraction.

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The choice of eigenvector normalization type has no influence on the

results of subsequent modal dynamic steps (see “Linear analysis of a rod

under dynamic loading,” Section 1.4.9 of the Abaqus Benchmarks Manual).

The normalization type determines only the manner in which the

eigenvectors are represented.

In addition to extracting the natural frequencies and mode shapes, the

Lanczos and subspace iteration eigensolvers automatically calculate the

generalized mass, the participation factor, the effective mass, and the

composite modal damping for each mode; therefore, these variables are

available for use in subsequent linear dynamic analyses. The AMS

eigensolver computes only the generalized mass.

Displacement normalization

If displacement normalization is selected, the eigenvectors are

normalized so that the largest displacement entry in each vector is unity.

If the displacements are negligible, as in a torsional mode, the

eigenvectors are normalized so that the largest rotation entry in each

vector is unity. In a coupled acoustic-structural extraction, if the

displacements and rotations in a particular eigenvector are small when

compared to the acoustic pressures, the eigenvector is normalized so that

the largest acoustic pressure in the eigenvector is unity. The

normalization is done before the recovery of dependent degrees of freedom

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that have been previously eliminated with multi-point constraints or

equation constraints. Therefore, it is possible that such degrees of

freedom may have values greater than unity.

Input File Usage:

*FREQUENCY, NORMALIZATION=DISPLACEMENT

Abaqus/CAE Usage: Step

module: StepCreate: Frequency: Other: Normali

ze eigenvectors by: Displacement

Mass normalization

Alternatively, the eigenvectors can be normalized so that the generalized

mass for each vector is unity.

The “generalized mass” associated with mode is

where is the structure's mass matrix and is the eigenvector

for mode . The superscripts

N

and

M

refer to degrees of freedom of the

finite element model.

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If the eigenvectors are normalized with respect to mass, all the

eigenvectors are scaled so that =1. For coupled acoustic-structural

analyses, an acoustic contribution fraction to the generalized mass is

computed as well.

Input File Usage:

*FREQUENCY, NORMALIZATION=MASS

Abaqus/CAE Usage: Step

module: StepCreate: Frequency: Other: Normali

ze eigenvectors by: Mass

Modal participation factors

The participation factor for mode in direction

i

, , is a variable

that indicates how strongly motion in the global

x

-,

y

-, or

z

-direction

or rigid body rotation about one of these axes is represented in the

eigenvector of that mode. The six possible rigid body motions are

indicated by ,

2

, ,

6

. The participation factor is defined as

where defines the magnitude of the rigid body response of degree of

freedom

N

in the model to imposed rigid body motion (displacement or

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infinitesimal rotation) of type

i

. For example, at a node with three

displacement and three rotation components, is

where is unity and all other are zero;

x

,

y

, and

z

are the

coordinates of the node; and , , and represent the coordinates of

the center of rotation. The participation factors are, thus, defined for

the translational degrees of freedom and for rotation around the center

of rotation. For coupled acoustic-structural eigenfrequency analysis, an

additional acoustic participation factor is computed as outlined

in “Coupled acoustic-structural medium analysis,” Section 2.9.1 of the

Abaqus Theory Manual.

Modal effective mass

The effective mass for mode associated with kinematic

direction

i

(,

2

, ,

6

) is defined as

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If the effective masses of all modes are added in any global translational

direction, the sum should give the total mass of the model (except for

mass at kinematically restrained degrees of freedom). Thus, if the

effective masses of the modes used in the analysis add up to a value that

is significantly less than the model's total mass, this result suggests

that modes that have significant participation in a certain excitation

direction have not been extracted.

For coupled acoustic-structural eigenfrequency analysis, an additional

acoustic effective mass is computed as outlined in “Coupled

acoustic-structural medium analysis,” Section 2.9.1 of the Abaqus

Theory Manual.

Composite modal damping

You can define composite damping factors for each material (“Material

damping,” Section 23.1.1), which are assembled into fractions of

critical damping values for each mode, , according to

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where is the critical damping fraction given for

material

a

and

material

a

.

is the part of the structure's mass matrix made of

A composite damping value will be calculated for each mode. These values

are weighted damping values based on each material's participation in each

mode.

Input File Usage:

*DAMPING, COMPOSITE

Abaqus/CAE Usage:Property

module: Material

Composite

Create: MechanicalDamping:

Obtaining residual modes for use in mode-based procedures

Several analysis types in Abaqus/Standard are based on the eigenmodes and

eigenvalues of the system. For example, in a mode-based steady-state

dynamic analysis the mass and stiffness matrices and load vector of the

physical system are projected onto a set of eigenmodes resulting in a

diagonal system in terms of modal amplitudes (or generalized degrees of

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freedom). The solution to the physical system is obtained by scaling each

eigenmode by its corresponding modal amplitude and superimposing the

results (for more information, see “Linear dynamic analysis using modal

superposition,” Section 2.5.3 of the Abaqus Theory Manual).

Due to cost, usually only a small subset of the total possible eigenmodes

of the system are extracted, with the subset consisting of eigenmodes

corresponding to eigenfrequencies that are close to the excitation

frequency. Since excitation frequencies typically fall in the range of

the lower modes, it is usually the higher frequency modes that are left

out. Depending on the nature of the loading, the accuracy of the modal

solution may suffer if too few higher frequency modes are used. Thus, a

trade-off exists between accuracy and cost. To minimize the number of

modes required for a sufficient degree of accuracy, the set of eigenmodes

used in the projection and superposition can be augmented with additional

modes known as

residual modes

. The residual modes help correct for errors

introduced by mode truncation. In Abaqus/Standard a residual mode,

R

,

represents the static response of the structure subjected to a nominal

(or unit) load,

P

, corresponding to the actual load that will be used in

the mode-based analysis orthogonalized against the extracted eigenmodes,

followed by an orthogonalization of the residual modes against each other.

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This orthogonalization is required to retain the orthogonality properties

of the modes (residual and eigen) with respect to mass and stiffness. As

a consequence of the mass and stiffness matrices being available, the

orthogonalization can be done efficiently during the frequency extraction.

Hence, if you wish to include residual modes in subsequent mode-based

procedures, you must activate the residual mode calculations in the

frequency extraction step. If the static responses are linearly dependent

on each other or on the extracted eigenmodes, Abaqus/Standard

automatically eliminates the redundant responses for the purpose of

computing the residual modes.

For the Lanczos eigensolver you must ensure that the static perturbation

response of the load that will be applied in the subsequent mode-based

analysis (i.e., ) is available by specifying that load in a static

perturbation step immediately preceding the frequency extraction step.

If multiple load cases are specified in this static perturbation analysis,

one residual mode is calculated for each load case; otherwise, it is

assumed that all loads are part of a single load case, and only one residual

mode will be calculated. When residual modes are requested, the boundary

conditions applied in the frequency extraction step must match those

applied in the preceding static perturbation step. In addition, in the

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immediately preceding static perturbation step Abaqus/Standard requires

that (1) if multiple load cases are used, the boundary conditions applied

in each load case must be identical, and (2) the boundary condition

magnitudes are zero. When generating dynamic substructures

(see “Generating a reduced structural damping matrix for a substructure”

in “Defining substructures,” Section 10.1.2), residual modes usually

will provide the most benefit if the loading patterns defined in each of

the load cases in the preceding static perturbation step match the loading

patterns defined under the corresponding substructure load cases in the

substructure generation step.

If you use the AMS eigensolver, you do not need to specify the loads in

a preceding static perturbation step. Residual modes are computed at all

degrees of freedom at which a concentrated load is applied in the following

mode-based procedure. You can request additional residual modes by

specifying degrees of freedom. One residual mode is computed for every

requested degree of freedom.

As an outcome of the orthogonalization process, a pseudo-eigenvalue

corresponding to each residual mode, , is computed and given by

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Henceforth, and in other Abaqus/Standard documentation, the term

eigenvalue is used generally to refer to actual eigenvalues and

pseudo-eigenvalues. All data (e.g., participation factors, etc.;

see “Output”) associated with the modes (eigenmodes and residual modes)

are ordered by increasing eigenvalue. Therefore, both eigenmodes and

residual modes are assigned mode numbers. In the printed output file

Abaqus/Standard clearly identifies which modes are eigenmodes and which

modes are residual modes so that you can easily distinguish between them.

By default, if you activate residual modes, all the calculated eigenmodes

and residual modes will be used in subsequent mode-based procedures,

unless:

•

You choose to obtain a new set of eigenmodes and residual modes in

a new frequency extraction step.

•

You choose to select a subset of the available eigenmodes and

residual modes in the mode-based procedure (selection of modes

is described in each of the mode-based analysis type sections).

Residual modes cannot be calculated if the cyclic symmetric modeling

capability is used. In addition, the Lanczos or AMS eigensolver must be

used if you wish to activate residual mode calculations.

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Input File Usage:

*FREQUENCY, RESIDUAL MODES

Abaqus/CAE Usage: Step

module: Step

residual modes

Create: Frequency: Basic: Include

Evaluating frequency-dependent material properties

When frequency-dependent material properties are specified,

Abaqus/Standard offers the option of choosing the frequency at which these

properties are evaluated for use in the frequency extraction procedure.

This evaluation is necessary because the stiffness cannot be modified

during the eigenvalue extraction procedure. If you do not choose the

frequency, Abaqus/Standard evaluates the stiffness associated with

frequency-dependent springs and dashpots at zero frequency and does not

consider the stiffness contributions from frequency domain

viscoelasticity. If you do specify a frequency, only the real part of the

stiffness contributions from frequency domain viscoelasticity is

considered.

Evaluating the properties at a specified frequency is particularly useful

in analyses in which the eigenfrequency extraction step is followed by

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a subspace projection steady-state dynamic step (see “Subspace-based

steady-state dynamic analysis,”Section 6.3.9). In these analyses the

eigenmodes extracted in the frequency extraction step are used as global

basis functions to compute the steady-state dynamic response of a system

subjected to harmonic excitation at a number of output frequencies. The

accuracy of the results in the subspace projection steady-state dynamic

step is improved if you choose to evaluate the material properties at a

frequency in the vicinity of the center of the range spanned by the

frequencies specified for the steady-state dynamic step.

Input File Usage:

*FREQUENCY, PROPERTY EVALUATION=

frequency

Abaqus/CAE Usage: Step

module: StepCreate: Frequency: Other: Evaluat

e dependent properties at frequency

Initial conditions

If the frequency extraction procedure is the first step in an analysis,

the initial conditions form the

base state

for the procedure (except for

initial stresses, which cannot be included in the frequency extraction

if it is the first step). Otherwise, the base state is the current state

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of the model at the end of the last general analysis step (“General and

linear perturbation procedures,” Section 6.1.2). Initial stress

stiffness effects (specified either through defining initial stresses or

through loading in a general analysis step) will be included in the

eigenvalue extraction only if geometric nonlinearity is considered in a

general analysis procedure prior to the frequency extraction procedure.

If initial stresses must be included in the frequency extraction and there

is not a general nonlinear step prior to the frequency extraction step,

a “dummy” static step—which includes geometric nonlinearity and which

maintains the initial stresses with appropriate boundary conditions and

loads—must be included before the frequency extraction step.

“Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section

30.2.1, describes all of the available initial conditions.

Boundary conditions

Nonzero magnitudes of boundary conditions in a frequency extraction step

will be ignored; the degrees of freedom specified will be fixed

(“Boundary conditions in Abaqus/Standard and

Abaqus/Explicit,” Section 30.3.1).

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Boundary conditions defined in a frequency extraction step will not be

used in subsequent general analysis steps (unless they are respecified).

In a frequency extraction step involving piezoelectric elements, the

electric potential degree of freedom must be constrained at least at one

node to remove numerical singularities arising from the dielectric part

of the element operator.

Defining primary and secondary bases for modal superposition procedures

If displacements or rotations are to be prescribed in subsequent dynamic

modal superposition procedures, boundary conditions must be applied in

the frequency extraction step; these degrees of freedom are grouped into

“bases.” The bases are then used for prescribing motion in the modal

superposition procedure—see “Transient modal dynamic

analysis,” Section 6.3.7.

Boundary conditions defined in the frequency extraction step supersede

boundary conditions defined in previous steps. Hence, degrees of freedom

that were fixed prior to the frequency extraction step will be associated

with a specific base if they are redefined with reference to such a base

in the frequency extraction step.

The primary base

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By default, all degrees of freedom listed for a boundary condition will

be assigned to an unnamed “primary” base. If the same motion will be

prescribed at all fixed points, the boundary condition is defined only

once; and all prescribed degrees of freedom belong to the primary base.

Unless removed in the frequency extraction step, boundary conditions from

the last general analysis step become fixed boundary conditions for the

frequency step and belong to the primary base.

If all rigid body motions are not suppressed by the boundary conditions

that make up the primary base, you must apply a suitable frequency shift

to avoid numerical problems.

Input File Usage:

*BOUNDARY

The *BOUNDARY option without the BASE

NAME parameter can appear only once in a

frequency extraction step.

Abaqus/CAE Usage: Load module: Create Boundary Condition

Secondary bases

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If the modal superposition procedure will have more than one independent

base motion, the driven nodes must be grouped together into “secondary”

bases in addition to the primary base. The secondary bases must be named.

(See “Base motions in modal-based procedures,” Section 2.5.9 of the

Abaqus Theory Manual.) Secondary bases are used only in modal dynamic and

steady-state dynamic (not direct) procedures.

The degrees of freedom associated with secondary bases are not suppressed;

instead, a “big” mass is added to each of them. To provide six digits

of numerical accuracy, Abaqus/Standard sets each “big” mass equal to

10

6

times the total mass of the structure and each “big” rotary inertia

equal to 10

6

times the total moment of inertia of the structure. Hence,

an artificial low frequency mode is introduced for every degree of freedom

in a secondary base. To keep the requested range of frequencies unchanged,

Abaqus/Standard automatically increases the number of eigenvalues

extracted. Consequently, the cost of the eigenvalue extraction step will

increase as more degrees of freedom are included in the secondary bases.

To reduce the analysis cost, keep the number of degrees of freedom

associated with secondary bases to a minimum. This can sometimes be done

by reducing several secondary bases that all have the same prescribed

motion to a single node by using BEAM type MPCs (“General multi-point

constraints,” Section 31.2.2).

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For the Lanczos and subspace iteration methods a negative shift must be

used with either the rigid body modes or secondary bases.

The “big” masses are not included in the model statistics, and the total

mass of the structure and the printed messages about masses and inertia

for the entire model are not affected. However, the presence of the masses

will be noticeable in the output tables printed for the eigenvalue

extraction step, as well as in the information for the generalized masses

and effective masses. See “Double cantilever subjected to multiple base

motions,” Section 1.4.12 of the Abaqus Benchmarks Manual, for an example

of the use of the base motion feature.

More than one secondary base can be defined by repeating the boundary

condition definition and assigning different base names.

Input File Usage:

*BOUNDARY, BASE NAME=

name

Abaqus/CAE Usage: Secondary bases are not supported in Abaqus/CAE.

Loads

Applied loads (“Applying loads: overview,” Section 30.4.1) are ignored

during a frequency extraction analysis. If loads were applied in a

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previous general analysis step and geometric nonlinearity was considered

for that prior step, the load stiffness determined at the end of the

previous general analysis step is included in the eigenvalue extraction

(“General and linear perturbation procedures,” Section 6.1.2).

Predefined fields

Predefined fields cannot be prescribed during natural frequency

extraction.

Material options

The density of the material must be defined (“Density,” Section 18.2.1).

The following material properties are not active during a frequency

extraction: plasticity and other inelastic effects, rate-dependent

material properties, thermal properties, mass diffusion properties,

electrical properties (although piezoelectric materials are active), and

pore fluid flow properties—see “General and linear perturbation

procedures,” Section 6.1.2.

Elements

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Because they contribute nonsymmetric damping terms, acoustic flow

velocity and acoustic infinite elements cannot be used with the Abaqus/AMS

eigensolver. Other than generalized axisymmetric elements with twist, any

of the stress/displacement or acoustic elements in Abaqus/Standard

(including those with temperature, pressure, or electrical degrees of

freedom) can be used in a frequency extraction procedure.

Output

The eigenvalues (EIGVAL), eigenfrequencies in cycles/time (EIGFREQ),

generalized masses (GM), composite modal damping factors (CD),

participation factors for displacement degrees of freedom 1–6 (PF1–PF6)

and acoustic pressure (PF7), and modal effective masses for displacement

degrees of freedom 1–6 (EM1–EM6) and acoustic pressure (EM7) are written

automatically to the output database as history data. Output variables

such as stress, strain, and displacement (which represent mode shapes)

are also available for each eigenvalue; these quantities are perturbation

values and represent mode shapes, not absolute values.

The eigenvalues and corresponding frequencies (in both radians/time and

cycles/time) will also be automatically listed in the printed output file,

along with the generalized masses, composite modal damping factors,

participation factors, and modal effective masses.

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The only energy density available in eigenvalue extraction procedures is

the elastic strain energy density, SENER. All of the output variable

identifiers are outlined in “Abaqus/Standard output variable

identifiers,” Section 4.2.1.

The AMS eigensolver does not compute composite modal damping factors,

participation factors, or modal effective masses. In addition, you cannot

request output to the results (.fil) file.

You can restrict output to the results, data, and output database files

by selecting the modes for which output is desired (see “Output to the

data and results files,” Section 4.1.2, and “Output to the output

database,” Section 4.1.3).

Input File Usage: Use one of the following options:

*EL FILE, MODE, LAST MODE *EL PRINT, MODE,

LAST MODE *OUTPUT, MODE LIST

Abaqus/CAE Usage:

Step module: Output

Requests

Field Output

Create: Frequency: Specify modes

Input file template

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*HEADING

…

*BOUNDARY

Data lines to specify zero-valued boundary conditions

*INITIAL CONDITIONS

Data lines to specify initial conditions

**

*STEP (,NLGEOM)

If NLGEOM is used, initial stress and preload stiffness effects

will be included in the frequency extraction step

*STATIC

…

*CLOAD and/or *DLOAD

Data lines to specify loads

*TEMPERATURE and/or *FIELD

Data lines to specify values of predefined fields

*BOUNDARY

Data lines to specify zero-valued or nonzero boundary conditions

*END STEP

**

*STEP, PERTURBATION

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*STATIC

…

*LOAD CASE, NAME=

load case name

Keywords and data lines to define loading for this load case

*END LOAD CASE

…

*END STEP**

*STEP

*FREQUENCY, EIGENSOLVER=LANCZOS, RESIDUAL MODES

Data line to control eigenvalue extraction

*BOUNDARY

*BOUNDARY, BASE NAME=

name

Data lines to assign degrees of freedom to a secondary base

*END STEP

来源：abaqus 帮助文件

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