Linear dynamic analysis using modal superposition

Linear dynamic analysis using modal superposition is computationally inexpensive and can provide useful insight into the dynamic behavior of a system.

With modern eigenvalue/eigenvector extraction techniques—such as the subspace method available in Abaqus/Standard—the cost of obtaining a sufficient basis of eigensolutions is not excessive; and the subsequent computational effort involved in obtaining the dynamic response by modal superposition methods is relatively small, especially when compared to the cost of the direct integration methods used for general nonlinear analysis (Implicit dynamic analysis).

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About dynamic analysis procedures

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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:

  1. 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.

  2. 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.

  3. 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.

  4. 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.