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

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.

Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical systems

“…The first one relies on so-called standard extended systems and consists in introducing one or more additional equations characterizing the bifurcation [19,26]. The second approach relies on bordering techniques and minimally extended systems in which only one scalar function is added [32,40,47]. The approach based on standard extended systems is used in the present paper.…”

“…In MATCONT, the continuation of codimension-1 bifurcations of dynamical systems is performed by means of minimally extended systems. In [47], Detroux et al combined this approach with the HBM for the tracking of limit point, branch point and Neimark-Sacker bifurcations of large-scale mechanical systems. In this paper, we combine HBM and standard extended systems.…”

Uncovering inner detached resonance curves in coupled oscillators with nonlinearity

“…Moreover, magenta dashed lines indicate the existence of only complex conjugate eigenvalues with positive real parts, and the point at which stability is lost, i.e. where the eigenvalues cross the imaginary axis, is associated with a Neimark-Sacker bifurcations [6]. In this latter case, a quasi-periodic motion manifests in the system response.…”

Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation

“…LP bifurcation curves contain valuable information about the system's dynamics. For instance, they were used to predict the existence of isolated periodic solutions in the NLFR of various systems [Kuether et al, 2015;Detroux et al, 2015a;Detroux et al, 2015b;Gatti, 2016]. LP bifurcations also represent stability boundaries and mark out the region where hysteretic behavior can be observed when sweeping back and forth the resonance of nonlinear mechanical systems.…”