Stochastic Dynamics and Bifurcation Behavior of Nonlinear Nonconservative Systemsin the Presence of Noise

摘要

The main objectives of the completed work are to develop mathematical techniquesto reduce the dimensionality of multidegree-of-freedom nonlinear systems near bifurcation points and to solve for the response statistics of the reduced system. The asymptotic behavior of nonlinear dynamical systems in the presence of noise is studied using the method of stochastic normal forms. The crucial point in the normal form computations is to find the resonant terms that cannot be eliminated through a nonlinear change of variables. Subsequent to reduction of the dimensionality, a Markovian approximation is used to obtain the associated stochastic normal forms. The key result is that the second order stochastic terms have to be retained in the normal form computations in order to capture the contributions of the stable modes stochastic components to the critical modes drift terms. It is also shown that the method of extended stochastic averaging is in fact 'equivalent' to stochastic normal forms for a specified class of nonlinear systems. In addition, mean square stability of the response is obtained and the bifurcation behavior and associated stationary and transient probability density functions for the reduced stochastic system are determined. Finally, the general results are applied to the study of the dynamics of aircraft at high angles of attack, plates under gas flow, structures under follower forces, and propellant lines conveying pulsating fluid. (JHD)