Study On Evolutionary Random Response Problems Of Stochastic Linear Systems And Elementary Study On Bifurcation And Chaos Of Stochastic Duffing System

This thesis is devoted to the evolutionary random response problems of linear random systems, and to the response problems of the random Duffing system due to harmonic excitations.In the first part of the thesis, we generalize the unified approach to the evolutionary random response problems of a deterministic system (the unified approach in brief), to a random linear system in the following ways.First, the unified approach is combined with the Monte Carlo method. As an illustrative example, the evolutionary random response of a random shear column under the Niigata earthquake excitation is analysed. The unified approach is first applied to the sample systems. Then the Monte Carlo method is applied to simulate the random parameter, the length of the column, to acquire ensemble mean square response. Comparing with the method using Monte Carlo method alone to simulate both the random parameter and the random excitation, the suggested method can reduce the computational effort greatly.Secondly, the response problem of a random structure under evolutionary random excitation is solved by the stochastic perturbation method combined with the unified approach. Our derivation is based on the following three indispensable assumptions: 1, the random perturbations must be small fluctuations; 2, the random parameters of the system are mutually independent; 3, the random parameters and the random excitation are mutually independent too. It is said that the stochastic perturbation method basically could not fulfil the requirement of dynamical random response problems. However, so far in the dissipative systems, this is not the case.Thirdly, the unified approach is combined with the orthogonal polynomial approximation. By orthogonal polynomial approximation method, we first reduce the random system into its deterministic equivalent one, so the response problem of a random system can be transformed into that of a deterministic system. Then the unified approach can be applied to it to acquire the ensemble random evolutionary response. Since the normal probability density function (PDF) may lead to instability of some sample systems when the random parameters taking sufficiently small negative values, an arch-like PDF and a more adaptable -PDF, together with the matching Chebyshev polynomial approximation and Gegenbauer polynomial approximation, are suggested. Numerical examples show that the suggested methods are effective.In the second part, we try to apply orthogonal polynomial approximations to the dynamical response problem of the Duffing equation with random parameters under harmonic excitations. We first reduce the random Duffing system into its non-linear deterministic equivalent one. Then, using numerical method, we study the elementary non-linear phenomena in the system, such as saddle-node bifurcation, symmetry break bifurcation,phenomena in the system, such as saddle-node bifurcation, symmetry break bifurcation, period-doubling bifurcation and chaos. The method we present here seems to be a new approach to dynamical response problems of non-linear random systems.