PHASE-SPACE LINEARIZATION FOR NON-LINEAR OSCILLATORS: DETERMINISTIC AND STOCHASTIC SYSTEMS

A new and efficient semi-analytical phase-space linearization (PSL) scheme for a class of non-linear oscillators is developed in this paper. The method is based on replacement of the non-linear vector field by a set of linear ones, each valid over a short segment of the evolving trajectory or over sufficiently small interval of time. Based on this concept, a few explicit and implicit integration schemes are first proposed and applied to a class of low-dimensional non-linear dynamical systems to accurately determine their response trajectories. This approach of local linearization is further extended to non-linear oscillators excited by formal derivatives of one or a combination of Gauss–Markov processes. Since the present methodology reduces the non-linear operator by a set of linear operators, it is also demonstrated that the principles of linear random vibration may be suitable exploited to arrive at a faster and semi-analytical Monte-Carlo scheme for computing the response statistics, both in stationary and non-stationary regimes. Limited examples are presented and compared with exact solutions whenever available, to illustrate the efficiency and versatility of the proposed schemes.