Stochastic Averaging Technique Based On Jacobi Elliptic Functions And Its Application

In the present thesis, the stochastic averaging technique based on Jacobi elliptic functions is generalized to evaluate the random responses of Duffing system subject to Gaussian white noises or bounded noise. Based on the Jacobi elliptic functions transformation including Jacobi elliptic sine, cosine and delta functions, the stochastic differential equations with respect to the system amplitude and phase are derived.For system under Gaussian white noise, with the application of the stochastic averaging principle, the amplitude response is approximated as a Markov diffusion process and the associated averaged Ito stochastic differential equation is established. Solving the corresponding reduced Fokker-Planck-Kolmogorov equation yields the stationary probability density of the amplitude response, from which the stationary probability densities of the displacement and velocity are derived. Numerical results for a Duffing-Van der Pol oscillator with hardening, softening stiffness and bistable potential are given to verify the applicability and accuracy of the proposed procedure by comparing with the results from Monte-Carlo simulations.For system under bounded noises and considering the resonance case, the system can be governed by the stochastic differential equations with respect to the system amplitude and the phase difference between the imposed excitation and the system response. Through the stochastic averaging principle, a two-dimensional diffusion process with respect to the system amplitude and the phase difference between the imposed excitation and the system response is derived. The associated Ito stochastic differential equations can be established. The stationary joint probability density of the amplitude and the phase difference is obtained by solving the corresponding Fokker-Planck-Kolmogorov equation. Numerical results for a Duffing-Van der Pol oscillator with hardening and softening stiffness are given to verify the feasibility and accuracy of the proposed procedure. Compared to the stochastic averaging based on generalized harmonic functions, the present procedure is of higher accuracy as it is based on the exact solution of the associated conservative nonlinear system.