Analysis and design of nonlinear resonances via singularity theory

TL;DR

This paper demonstrates how singularity theory complements bifurcation analysis in understanding the organization of nonlinear resonances and structural changes in parameter space, exemplified through a two-degree-of-freedom system.

Contribution

It introduces the application of singularity theory with a distinguished parameter to analyze nonlinear resonances, providing new insights beyond traditional bifurcation methods.

Findings

Singularity theory reveals structural organization of nonlinear resonances.

Complementary insights to bifurcation analysis in nonlinear systems.

Analysis of detached resonance curves in a two-degree-of-freedom system.

Abstract

Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behaviour, tracking singularities reveals how structural changes are themselves organised in parameter space. The complementarity of that information is demonstrated in the analysis of detached resonance curves in a two-degree-of-freedom system.