Response And Stability Of Multi-Degree-of-Freedom Strongly Nonlinear Stochastic Systems

The resporse prediction and stability analysis are always hot topics of research in stochastic dynamics. However, these topics are much more complicated for multi-degree-of-freedom (MDOF) strongly nonlinear system under stochastic excitations. Some of the topics are investigated deeply and comprehensively in the present dissertation.In the first part, the response prediction of nonlinear stochastic system is studied. By using exterior differentiation and inverse method, exact stationary solutions of strongly nonlinear MDOF systems subject to Gaussian white noises are obtained, which are generally independent of energy. The obtained exact stationary solutions are the most general class of the exact stationary solutions so far, and some classes of the known ones dependent on energy belong to the special cases of them. Besides stationary solutions, the nonstationary responses of some systems are also studied. The stochastic averaging method based on generalized harmonic functions is applied and the nonstationary solution is approximately expressed as a multiple series expansion in terms of Laguerre orthogonal basis functions with time-dependent coefficients. Then the approximate nonstationary probability density for amplitude response can be obtained by using the Galerkin method, from which the approximate nonstationary probability densities for displacement and velocity can be derived. The proposed procedure is applied to study the nonstationary probability densities of MDOF nonlinear systems and strongly nonlinear systems with time-delayed feedback control subject to Gaussian white noise excitations.In the second part, the stability of quasi Hamiltonian system is analyzed. Base on the existing researches, the asymptotic Lyapunov stability with probability one of MDOF quasi Hamiltonian systems is studied by using Lyapunov function. For quasi-integrable and resonant Hamilton systems and quasi-partially-integrable Hamiltonian systems, the optima linear combination of the independent first integrals in involution is taken as the Lyapunov function. Then, the Ito stochastic differential equation for the Lyapunov function can be obtained by using stochastic averaging method. And the sufficient condition for the asymptotic Lyapunov stability with probability one of MDOF quasi Hamiltonian systems can be determined. It should be pointed out that determining the asymptotic Lyapunov stability with probability one of the system is finally transformed into the problems of evaluating the eigenvalues and eigenvectors associated with the linearized drift coefficients of the averaged Ito equation.In the last part, the response and stability of stochastic system containing fractional derivative which is of great significance in engineering field are investigated. For linear system, the response of stochastic system including two terms of fractional derivative with real and arbitrary orders can be described as a Duhamel integral-type close-form expression by using Green's functions obtained based on a Laplace transform approach and the weighted generalized Mittag-Leffler function. The statistical behavior of the system is subsequently obtained. For nonlinear system, a stochastic averaging procedure for strongly nonlinear system with light damping modeled by fractional derivative under stochastic excitations is developed by using the so-called generalized harmonic functions. With the application of the developed averaging method, the stationary probability density for system response and the largest Lyapunov exponent are approximately obtained.