Nonstationary pandom vibpation analysis of linear elastic structures with finite element method

Abstract

At present, the finite element method is an efficient method for analyzing structural dynamic problems. When the physical quantities such as displacements and stresses are resolved in the spectra and the dynamic matrices are obtained in spectral resolving form, the relative equations cannot be solved by the vibration mode resolving method as usual. For solving such problems, a general method is put forward in this paper. The excitations considered with respect to nonstationary processes are as follows:

$$P(t) = \left\{ {P_1 (t)} \right\},{\text{ }}P_1 (t) = a_1 (t)P_1^0 (t),$$

a₁(t) is a time function already known. We make Fourier transformation for the discretized equations obtained by finite element method, and by utilizing the behaviour of orthogonal increment of spectral quantities in random process^[1], some formula of relations about the spectra of excitation and response are derived. The cross power spectral denisty matrices of responses can be found by these formulas, then the structrual safety analysis can be made. When a(1)=1 (i=1.2, ...,n), the method stated in this paper will be reduced to that which is used in the special case of stationary process.