Non-stationary Response of the Linear System to Random Excitation

Abstract

In relation to the statistical design method of anti-earthquake structures for moderately intense excitations, the basic studies on the statistical quantities such as the covariance and spectral density in the non-stationary stochastic process are described and the input and output relations of such quantities in the case of a multi-input and -output, linear discrete system having time-variant coefficients are presented. As the basic statistical quantities in the time and frequency domain, the local covariance matrix and the local spectral density matrices are considered in this paper. At first, the local co-variance matrix is defined as the product of a two-dimensional cutoff operator and the co-variance matrix in a non-stationary stochastic process. And then, the two-dimensional local spectral density matrix and the several kinds of one-dimensional local spectral density matrices are introduced by defining them as the double and single Fourier transform of the local co-variance matrix, respectively. It is found that the appropriately defined one-dimensional spectral density matrices containing a time variable have the meaning of the power spectral density matrix in the non-stationary stochastic process in the sense that the integral of these quantities over the finite time domain results in the local energy spectral density matrix defined in the square time domain. And also, it is shown that as a limiting case, the one-dimensional local Hermitian spectral density matrix presented in this paper is reduced to the spectral density matrix introduced by D. G. Lampard. Moreover it is shown that the local spectral density matrices are expressed as the weighted averages of the corresponding total spectral density matrices associated with the full time domain. For the general case of a multi-input and -output linear discrete system having timevariant, complex-valued coefficients, the input and output relations of the local co-variance matrix and the one- or two-dimensional local spectral density matrices are presented. And it is shown that as a special case of a linear discrete system having time-invariant, real-valued coefficients, the input and output relation of the two-dimensional total spectral density matrix is reduced to the relation presented by J. S. Bendat. As an example of the non-stationary input process most applicable to earthquake engineering, the quasistationary random process introduced by V. V. Bolotin as well as the locally stationary random process presented by R. A. Silverman are considered, and the basic statistical quantities of the output of a linear system subjected to these random inputs are estimated. And also, it is shown that as a special case of a time-invariant, linear discrete system subjected to a stationary input, the input and output relations of the co-variance matrix and the total spectral density matrices are reduced to the well-known results in the stationary stochastic process. Finally, in the appendix, it is shown that the ensemble averages of the short-time correlation and power spectral density matrix which are introduced by R. M. Fano, are expressed as the weighted time averages of the local co-variance matrix and the local spectral density matrices, respectively.