The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems

TL;DR

This paper extends the harmonic balance method to bifurcation analysis in large-scale nonlinear mechanical systems, enabling detection and tracking of bifurcations through novel algorithms and numerical experiments.

Contribution

It introduces an algorithm combining Floquet exponents with bordering techniques and a new method for tracking Neimark-Sacker bifurcations using eigenvalue derivatives.

Findings

Successfully applied to a spacecraft structure with nonlinear vibration isolation.

Demonstrated effective detection of bifurcations in complex mechanical systems.

Provided a robust numerical framework for bifurcation analysis.

Abstract

The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed. A new procedure for the tracking of Neimark-Sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed. The HB method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device.