Stochastic Response Of Nonlinear Systems Research Articles

A Reduced-Order Wiener Path Integral Formalism for Determining the Stochastic Response of Nonlinear Systems With Fractional Derivative Elements

Abstract A technique based on the Wiener path integral (WPI) is developed for determining the stochastic response of diverse nonlinear systems with fractional derivative elements. Specifically, a reduced-order WPI formulation is proposed, which can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the herein developed technique can determine, directly, any lower-dimensional joint response probability density function corresponding to a subset only of the response vector components. This is done by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in few only specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system. An indicative numerical example is considered pertaining to a stochastically excited tuned mass-damper-inerter nonlinear system with a fractional derivative element. Comparisons with relevant Monte Carlo simulation data demonstrate the accuracy and computational efficiency of the technique.

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.)