Nonautonomous Spectral Submanifolds for Model Reduction of Nonlinear Mechanical Systems under Parametric Resonance

TL;DR

This paper advances the theory of Spectral Submanifolds (SSM) for nonlinear mechanical systems under parametric excitation, providing new computational methods, theoretical simplifications, and practical tools for analyzing resonance and forced response.

Contribution

It introduces higher-order nonautonomous SSM parameterizations, simplifies SSM computation for second-order systems, and offers explicit formulas for steady-state response analysis.

Findings

Developed expressions for nonautonomous SSM terms under periodic and quasiperiodic excitation.

Provided explicit formulas for steady-state response in two-dimensional SSMs.

Demonstrated methods on mechanical system examples, including finite-element models.

Abstract

We use the recent theory of Spectral Submanifolds (SSM) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results both for general first-order as well as second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.