Products: Abaqus/Standard  Abaqus/CAE  Abaqus/AMS  

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;

• 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.

The eigenvalue problem for the natural frequencies of an undamped finite element model is

Abaqus/Standard provides three eigenvalue extraction methods:

• Lanczos

• Automatic multi-level substructuring (AMS), an add-on analysis capability for Abaqus/Standard

• Subspace iteration

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.

Software Architecture	Eigensolver
	Lanczos	AMS	Subspace Iteration
Traditional
SIM

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.

*FREQUENCY, EIGENSOLVER=LANCZOS

*FREQUENCY, EIGENSOLVER=AMS

*FREQUENCY, EIGENSOLVER=AMS, NSET=name

*FREQUENCY, EIGENSOLVER=SUBSPACE

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 the Lanczos eigensolver, which is based on the SIM architecture: the coupling is projected onto the modal space and stored for later use.

*FREQUENCY, EIGENSOLVER=LANCZOS, 
ACOUSTIC COUPLING=ON (default if the SIM architecture is not used)

*FREQUENCY, EIGENSOLVER=LANCZOS, 
ACOUSTIC COUPLING=OFF

*FREQUENCY, EIGENSOLVER=AMS, 
ACOUSTIC COUPLING=PROJECTION (default for the AMS eigensolver)

*FREQUENCY, EIGENSOLVER=LANCZOS, SIM, 
ACOUSTIC COUPLING=PROJECTION (default in SIM-based analysis)

*FREQUENCY, ACOUSTIC COUPLING=OFF

*FREQUENCY, EIGENSOLVER=AMS
, , , , , , acoustic range factor

*FREQUENCY, EIGENSOLVER=LANCZOS, SIM
, , , , , , acoustic range factor

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.

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.

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.

*FREQUENCY, NORMALIZATION=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.

 

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: Normalize 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 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.

 

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

where  is the critical damping fraction given for material a and  is the part of the structure's mass matrix made of material a.

 

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: MaterialCreate: MechanicalDamping: Composite

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 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,

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 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

• 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).

*FREQUENCY, RESIDUAL MODES

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 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.

*FREQUENCY, PROPERTY EVALUATION=frequency

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 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.

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).

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.

*BOUNDARY

*BOUNDARY, BASE NAME=name

Applied loads (“Applying loads: overview,” Section 30.4.1) are ignored during a frequency extraction analysis. If loads were applied in a 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 cannot be prescribed during natural frequency extraction.

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.

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.

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.

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).

*EL FILE, MODE, LAST MODE
*EL PRINT, MODE, LAST MODE
*OUTPUT, MODE LIST

*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



*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



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