The basic concept of modal superposition is that the response of the
structure is expressed in terms of a relatively small number of eigenmodes of
the system. The orthogonality of the eigenmodes uncouples this system.
Furthermore, only eigenmodes that are close to the frequencies of interest are
usually needed; for example, only the lowest few frequencies are usually
required to obtain an accurate estimate of a structure's linear dynamic
response to relatively long-term loading (for example, its steady-state
response to low frequency excitation). The technique can be extended in a
limited way into the nonlinear régime, but the superposition and orthogonality
principles apply only to purely linear systems: for this reason the methods
described in this section are implemented only for linear analysis.
Abaqus/Standard
has two “subspace” procedures—one for nonlinear dynamic and the other one for
steady-state dynamic analysis—that use some of the eigenmodes of the system on
which the equilibrium equations are projected. In both cases the system's
eigenmodes are used as a set of global basis vectors for computing the dynamic
response, even though the system exhibits nonlinear or frequency-dependent
effects during the dynamic response. These methods are cost-effective compared
to fully nonlinear dynamic response analysis developed in terms of all the
system's degrees of freedom. The subspace projection method for steady-state
response is described in
Subspace-based steady-state dynamic analysis.
The time-domain subspace projection method is described in
Subspace dynamics.
The procedures provided for modal dynamic analysis of linear systems are
summarized below:
Modal dynamic time history analysis (see
Modal dynamic analysis).
This procedure can be used to obtain the time history response of a
system to loading conditions that are given as functions of time. The response
is integrated through time: the integration method used is exact for loadings
that vary piecewise linearly with time. Thus, the only approximations in this
analysis procedure are the linearization of the problem, the spatial modeling
(that is, the choice of the finite element model), the loading definitions, and
the choice of the number of eigenmodes used to represent the system.
Response spectrum analysis (see
Response spectrum analysis).
Response spectrum analysis is often used to obtain an approximate upper
bound to the peak significant response of a system to an input spectrum as a
function of frequency: it gives the maximum response of a one degree of freedom
system as a function of its fundamental frequency of vibration and of its
damping ratio. The method has very low computational cost and gives useful
information about the spectral behavior of a system with respect to frequency.
Steady-state harmonic response analysis (see
Steady-state linear dynamic analysis
and
Subspace-based steady-state dynamic analysis).
This procedure is used when the steady-state response of a system to
harmonic excitation is required. The solution is given as the peak amplitudes
and phase relationships of the solution variables (stress, displacement, etc.)
as functions of frequency: postprocessing options are provided to display such
results conveniently.
A similar option is provided for direct harmonic response analysis
without using the eigenmodes as a basis. The direct method is significantly
more expensive computationally than the modal method: it is needed if the
system is nonsymmetric (because
Abaqus
presently does not have a nonsymmetric eigenvalue extraction capability) or if
the system's behavior includes frequency-dependent parameters.
The “subspace” method is typically less expensive than the direct
method. It is generally used for nonsymmetric systems or when the system's
behavior includes frequency-dependent parameters or discrete damping.
Random response analysis (see
Random response analysis).
This procedure is used when the structure is excited continuously and
the loading can be expressed statistically in terms of a “Power Spectral
Density Function.” The response is calculated in terms of statistical
quantities, such as the mean value and the standard deviation of nodal and
element variables.