Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems

TL;DR

This paper extends spectral submanifold theory to forced nonlinear mechanical systems, providing explicit formulas for forced responses and backbone curves, and demonstrating their effectiveness through numerical examples.

Contribution

It introduces analytical expressions for forced responses on spectral submanifolds in forced-damped nonlinear systems, expanding the applicability of SSM-based reduction methods.

Findings

Analytical formulas for forced responses derived

Comparison with normal form methods shows accuracy

Effective reduction of complex dynamics demonstrated

Abstract

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time dependent SSMs. A judicious choice of the parameterization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal form methods.