Random response analysis

Random response linear dynamic analysis is used to predict the response of a structure subjected to a nondeterministic continuous excitation that is expressed in a statistical sense by a cross-spectral density (CSD) matrix.

The random response procedure uses the set of eigenmodes extracted in a previous eigenfrequency step to calculate the corresponding power spectral densities (PSD) of response variables (stresses, strains, displacements, etc.) and, hence—if required—the variance and root mean square values of these same variables. This section provides brief definitions and explanations of the terms used in this type of analysis. Detailed discussion of the theory of random response analysis is provided in the books by Clough and Penzien (1975), Hurty and Rubinstein (1964), and Thompson (1988).

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Random response analysis

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Examples of random response analysis are the study of the response of an airplane to turbulence; the response of a car to road surface imperfections; the response of a structure to noise, such as the “jet noise” emitted by a jet engine; and the response of a building to an earthquake.

Since the loading is nondeterministic, it can be characterized only in a statistical sense. We need some assumptions to make this characterization possible. Although the excitation varies in time, in some sense it must be stationary—its statistical properties must not vary with time. Thus, if x(t) is the variable being considered (such as the height of the road surface in the case of a car driving down a rough road), then any statistical function of x, f(x), must have the same value regardless of what time origin we use to compute f:

f(x(t))=f(x(t+τ))    for any τ.

We also need the excitation to be ergodic. This term means that, if we take several samples of the excitation, the time average of each sample is the same.

These restrictions ensure that the excitation is, statistically, constant. In the following discussion we also assume that the random variables are real, which is the case for the variables that we need to consider.