Modeling nonlinear random vibration: Implication of the energy conservation law

TL;DR

This paper develops a stochastic differential equation model for nonlinear random vibrations driven by Gaussian and Poisson noises, ensuring energy conservation by choosing appropriate stochastic integrals, and demonstrates the model's validity through numerical examples.

Contribution

It establishes the correct stochastic integrals (Stratonovich and Di Paola-Falsone) for modeling energy-conserving nonlinear vibrations under different noise types.

Findings

The model satisfies energy conservation law.

Stratonovich integral is suitable for Gaussian noise.

Di Paola-Falsone integral is suitable for Poisson noise.

Abstract

Nonlinear random vibration under excitations of both Gaussian and Poisson white noises is considered. The model is based on stochastic differential equations, and the corresponding stochastic integrals are defined in such a way that the energy conservation law is satisfied. It is shown that Stratonovich integral and Di Paola-Falsone integral should be used for excitations of Gaussian and Poisson white noises, respectively, in order for the model to satisfy the underlining physical laws (e.g., energy conservation). Numerical examples are presented to illustrate the theoretical results.