Time-varying Spectral Submanifolds: Analytic Calculation of Backbone Curves and Forced Response

Abstract

To understand and analyze the behavior of realistic nonlinear structures it is desirable to reduce the dimensionality of the system, as well as simplify the equation of motion. Reduction to Spectral submanifolds (SSMs) has recently been shown to provide such a dimension reduction, yielding 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. We demonstrate our analytical formulae on numerical examples and compare them to results obtained from alternative methods.