Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve

TL;DR

This paper introduces a spectral submanifold-based reduction method for analyzing forced nonlinear mechanical systems with internal resonance, enabling efficient computation of periodic responses and response curves.

Contribution

It develops a reduced-order modeling approach that captures internal resonance effects, simplifying the analysis of high-dimensional systems without losing essential dynamics.

Findings

Effective reduction of high-dimensional models

Accurate computation of periodic responses

Demonstrated on finite-element mechanical systems

Abstract

We show how spectral submanifold theory can be used to construct reduced-order models for harmonically excited mechanical systems with internal resonances. Efficient calculations of periodic and quasi-periodic responses with the reduced-order models are discussed in this paper and its companion, Part II, respectively. The dimension of a reduced-order model is determined by the number of modes involved in the internal resonance, independently of the dimension of the full system. The periodic responses of the full system are obtained as equilibria of the reduced-order model on spectral submanifolds. The forced response curve of periodic orbits then becomes a manifold of equilibria, which can be easily extracted using parameter continuation. To demonstrate the effectiveness and efficiency of the reduction, we compute the forced response curves of several high-dimensional nonlinear mechanical systems, including the finite-element models of a von K\'arm\'an beam and a plate.