Fast computation and characterization of forced response surfaces via spectral submanifolds and parameter continuation

TL;DR

This paper introduces a fast method for computing forced response surfaces in nonlinear mechanical systems using spectral submanifolds and parameter continuation, enabling efficient analysis of complex vibrational behaviors.

Contribution

The authors develop a spectral submanifold-based reduced-order modeling approach combined with manifold continuation to efficiently compute and analyze forced response surfaces in high-dimensional systems.

Findings

Successfully computed FRSs for complex resonant systems

Identified ridges and trenches without full FRS computation

Demonstrated efficiency on mechanical structures with internal resonances

Abstract

For mechanical systems subject to periodic excitation, forced response curves (FRCs) depict the relationship between the amplitude of the periodic response and the forcing frequency. For nonlinear systems, this functional relationship is different for different forcing amplitudes. Forced response surfaces (FRSs), which relate the response amplitude to both forcing frequency and forcing amplitude, are then required in such settings. Yet, FRSs have been rarely computed in the literature due to the higher numerical effort they require. Here, we use spectral submanifolds (SSMs) to construct reduced-order models (ROMs) for high-dimensional mechanical systems and then use multidimensional manifold continuation of fixed points of the SSM-based ROMs to efficiently extract the FRSs. Ridges and trenches in an FRS characterize the main features of the forced response. We show how to extract these ridges and trenches directly without computing the FRS via reduced optimization problems on the ROMs. We demonstrate the effectiveness and efficiency of the proposed approach by calculating the FRSs and their ridges and trenches for a plate with a 1:1 internal resonance and for a shallow shell with a 1:2 internal resonance.