A Study On Non-stationary Random Vibration Analysis For Large-scale Nonlinear Systems

The random vibration theory has experienced great development. The research on random vibration analysis of linear systems has matured and various methods developed are now applied in engineering practice. Although much important progress has been made for the research on random vibration analysis of nonlinear systems, the traditional methods, including the FPK equation method, the stochastic average method, the moment equation method and so on, are limited to nonlinear systems with a limited number of degrees of freedom, and are mainly suitable for problems with stationary random excitations. Therefore, the more practical and effective random vibration analysis method for nonlinear systems needs to be investigated urgently. The major work in this dissertation is to develop the efficient time-domain explicit Monte Carlo simulation method and the time-domain explicit equivalent linearization method, which provide an effetive solution for the non-stationary random vibration analysis of large-scale nonlinear systems. Besides, the random vibration problems of nonlinear systems containing random parameters and the dynamic reliability of nonlinear systems are another two important research work in this dissertation.The major work in this dissertation is described as follows:(1) The research on the random vibration problems for linear and nonlinear systems is presented first, followed by an introduction of random vibration analysis methods for the linear and nonlinear systems. The dynamical reliability theory is summarized, and the research history of the dynamic reliability for nonlinear systems is presented.(2) The highly efficient Monte Carlo simulation method is investigated for the non-stationary random vibration problem of large-scale nonlinear systems. Motion equations of nonlinear systems are transformed into the quasi-linear motion equations by introducing the concept of equivalent excitations. And the quasi-linear motion equations are solved by either the precise time integration or the Newmark-β integration method. An explicit iteration method is then developed for dynamic response analysis of nonlinear systems. Based on the explicit iteration method which improves the efficiency of the sample analysis for Monte Carlo simulation method greatly, the time-domain explicit Monte Carlo simulation method is proposed for the non-stationary random vibration analysis of nonlinear systems.(3) The fast equivalent linearization method is investigated for the non-stationary random vibration analysis of large-scale nonlinear systems. The equivalent linearization method is applied to the non-stationary random vibration problems of nonlinear systems. And the method is used to obtain the statistical moments of the time-invariant equivalent linear systems corresponding to different time instants in a iterative manner, which makes the random vibration problem of the original nonlinear systems be transformed to the random vibration problem of a series of linear systems. Based on the time-domain explicit formulation method which is highly efficient for the random vibration analysis of time-invariant equivalent linear systems, the time-domain explicit equivalent linearization method is proposed for the non-stationary random vibration analysis of nonlinear systems.(4) The non-stationary random vibration analysis method is investigated for large-scale nonlinear stochastic systems. In consideration of the mutual independence of randomness of structural parameters and excitation parameters, a two-step strategy is adopted to reflect the influences of excitation parameters and structural parameters, respectively. Based on the time-domain explicit equivalent linearization method and the conditional mathematical expectation in probability theory, a total mathematical expectation method for non-stationary random response analysis of nonlinear stochastic systems is proposed.(5) The highly efficient analysis method is investigated for the dynamic reliability of large-scale nonlinear systems. By introduing the subset simulation technique which can reduce the number of sample in the time-domain explicit Monte Carlo simulation method, the time-domain explicit subset simulation method is proposed for the dynamic reliability of nonlinear systems. As for the dynamic reliability of nonlinear stochastic systems, based on the time-domain explicit subset simulation method and the conditional probability in probability theory, a total probability method for the dynamic reliability of nonlinear stochastic systems is proposed.It has been shown that, based on the time-domain explicit approach, the proposed series of methods can solve the non-stationary random vibration problem and the dynamic reliability problem of large-scale nonlinear systems, effectively breaking the bottleneck of traditional nonlinear random vibration anaylsis methods which are restricted to scale problems of systems and the stationarity of random excitations. Numerical examples show that the proposed methods have good accuracy and high efficiency.