Nonlinear Model Reduction to Random Spectral Submanifolds in Random Vibrations

TL;DR

This paper introduces a method to reduce high-dimensional nonlinear systems to low-dimensional random spectral submanifolds, enabling faster Monte Carlo simulations for systems under irregular, random excitations.

Contribution

It generalizes spectral submanifolds to stochastic systems and demonstrates their effectiveness in reducing computational cost for complex mechanical models.

Findings

Significant reduction in simulation time using random SSMs

High accuracy in approximating power spectral densities

Applicable to a range of mechanical systems from simple to complex

Abstract

Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.