Nonlinear Stochastic Dynamics, Chaos, and Reliability Analysis for a Single Degree of Freedom Model of a Rotor Blade

In turbomachinery, the analysis of systems subjected to stochastic or periodic excitation becomes highly complex in the presence of nonlinearities. Nonlinear rotor systems exhibit a variety of dynamic behaviors that include periodic, quasiperiodic, chaotic motion, limit cycle, jump phenomena, etc. The transitional probability density function (PDF) for the random response of nonlinear systems under white or colored noise excitation (delta-correlated) is governed by both the forward Fokker–Planck (FP) and backward Kolmogorov equations. This paper presents efficient numerical solution of the stationary and transient form of the forward FP equation corresponding to two state nonlinear systems by standard sequential finite element (FE) method using $C0$ shape functions and Crank–Nicholson time integration scheme. For computing the reliability of system, the transient FP equation is solved on the safe domain defined by D barriers using the FE method. A new approach for numerical implementation of path integral (PI) method based on non-Gaussian transition PDF and Gauss–Legendre scheme is developed. In this study, PI solution procedure is employed to solve the FP equation numerically to examine some features of chaotic and stochastic responses of nonlinear rotor systems.