Subspace-Based Steady-State Dynamic Analysis

Steady-state dynamic analysis provides the steady-state amplitude and phase of the response of a system subjected to harmonic excitation at a given frequency. Usually such analysis is done as a frequency sweep, by applying the loading at a series of different frequencies and recording the response. In Abaqus/Standard the subspace-based steady-state dynamic analysis procedure is used to conduct the frequency sweep.

In a subspace-based steady-state dynamic analysis the response is based on direct solution of the steady-state dynamic equations projected onto a subspace of modes. The modes of the undamped, symmetric system must first be extracted using the eigenfrequency extraction procedure. The modes will include eigenmodes and, if activated in the eigenfrequency extraction step, residual modes. The procedure is based on the assumption that the forced steady-state vibration can be represented accurately by a number of modes of the undamped system that are in the range of the excitation frequencies of interest. The number of modes extracted must be sufficient to model the dynamic response of the system adequately, which is a matter of judgment on your part. The projection of the dynamic equilibrium equations onto a subspace of selected modes leads to a small system of complex equations that is solved for modal amplitudes, which are then used to compute nodal displacements, stresses, etc.

When defining a subspace-based steady-state dynamic step, you specify the frequency ranges of interest and the number of frequencies at which results are required in each range (including the bounding frequencies of the range). In addition, you can specify the type of frequency spacing (linear or logarithmic) to be used, as described below (Selecting the Frequency Spacing). Logarithmic frequency spacing is the default if the frequency ranges are specified directly or by eigenfrequencies. If the frequency ranges are specified by the frequency spread, only linear spacing can be used. Frequencies should be given in cycles/time.

The frequency points for which results are required can be spaced equally along the frequency axis (on a linear or a logarithmic scale), or they can be biased toward the ends of the user-defined frequency range by introducing a bias parameter (see The Bias Parameter below).

The subspace-based steady-state dynamic analysis procedure can be used:

• for nonsymmetric stiffness;

• when any form of damping (except modal damping) is included; and

• when viscoelastic material properties must be taken into account.

While the response in this procedure is for linear vibrations, the prior response can be nonlinear. Initial stress effects (stress stiffening) will be included in the steady-state dynamic response if nonlinear geometric effects (General and Perturbation Procedures) were included in any general analysis step prior to the eigenfrequency extraction step preceding the subspace-based steady-state dynamic procedure.

STEADY STATE DYNAMICS, SUBSPACE PROJECTION

Step module: Create Step: Linear perturbation: Steady-state 
dynamics, Subspace

The bias parameter can be used to provide closer spacing of the results points either toward the middle or toward the ends of each frequency interval. Figure 3 shows a few examples of the effect of the bias parameter on the frequency spacing.

Figure 3. Effect of the bias parameter on the frequency spacing for a number of points $n=7$.

The bias formula used in subspace-based steady-state dynamics is

where

y

$=-1+2\left(k-1\right)/\left(n-1\right)$;

n

is the number of frequency points at which results are to be given within a frequency interval (discussed above);

k

is one such frequency point ($k=1,2,\mathrm{\dots },n$);

${\widehat{f}}_1$

is the lower limit of the frequency interval;

${\widehat{f}}_2$

is the upper limit of the frequency interval;

${\widehat{f}}_k$

is the frequency at which the kth results are given;

p

is the bias parameter value; and

$\widehat{f}$

is the frequency or the logarithm of the frequency, depending on the value chosen for the frequency scale.

A bias parameter, p, that is greater than 1.0 provides closer spacing of the results points toward the ends of the frequency interval, while values of p that are less than 1.0 provide closer spacing toward the middle of the frequency interval. The default bias parameter is 3.0 for an eigenfrequency interval and 1.0 for a range frequency interval.

The frequency scale factor can be used to scale frequency points. All the frequency points, except the lower and upper limit of the frequency range, are multiplied by this factor. This scale factor can be used only when the frequency interval is specified by using the system's eigenfrequencies (see Specifying the Frequency Ranges by Using the System's Eigenfrequencies above).

If damping is absent, the response of a structure will be unbounded if the forcing frequency is equal to an eigenfrequency of the structure. To get quantitatively accurate results, especially near natural frequencies, accurate specification of damping properties is essential. The various damping options available are discussed in Material Damping.

In subspace-based steady-state dynamic analysis damping can be created by the following:

• dashpots (see Dashpots),

• “Rayleigh” damping associated with materials and elements (see Material Damping),

• structural damping (see Damping in Dynamic Analysis),

• viscoelasticity included in the material definitions (see Frequency Domain Viscoelasticity),

• contributions from infinite elements (see Infinite Elements) or defined impedance conditions (see Acoustic and Shock Loads) on acoustic elements, and

• “volumetric drag” (viscous Rayleigh damping) in acoustic elements (see Acoustic Medium).

If you specify that a real-only system matrix be generated and projected (see Ignoring Damping above), all forms of damping are ignored, including quiet boundaries on infinite elements and nonreflecting boundaries on acoustic elements.

Abaqus/Standard automatically detects the contact nodes that are slipping due to velocity differences imposed by the motion of the reference frame or the transport velocity in prior steps. At those nodes the tangential degrees of freedom are not constrained and the effect of friction results in an unsymmetric contribution to the stiffness matrix. At other contact nodes the tangential degrees of freedom are constrained.

Friction at contact nodes at which a velocity differential is imposed can give rise to damping terms. There are two kinds of friction-induced damping effects. The first effect is caused by the friction forces stabilizing the vibrations in the direction perpendicular to the slip direction. This effect exists only in three-dimensional analysis. The second effect is caused by a velocity-dependent friction coefficient. If the friction coefficient decreases with velocity (which is usually the case), the effect is destabilizing and is also known as “negative damping.” For more details, see Coulomb friction. Subspace-based steady-state dynamics analysis allows you to include these friction-induced contributions to the damping matrix.

STEADY STATE DYNAMICS, SUBSPACE PROJECTION, FRICTION DAMPING=YES

Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace: Include friction-induced damping effects

You can select modes by specifying the mode numbers individually, by requesting that Abaqus/Standard generate the mode numbers automatically, or by requesting the modes that belong to specified frequency ranges. If you do not select the modes, all modes extracted in the prior eigenfrequency extraction step, including residual modes if they were activated, are used in the modal superposition.

SELECT EIGENMODES, DEFINITION=MODE NUMBERS

SELECT EIGENMODES, GENERATE, DEFINITION=MODE NUMBERS

SELECT EIGENMODES, DEFINITION=FREQUENCY RANGE

You can control the frequency of the subspace projections. By default, the dynamic equations are projected onto the subspace at each frequency you request. However, considerable computational savings can be obtained if the projection onto the subspace is performed only at selected frequency points.