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

Abstract

The aim of this paper is to provide an efficient frequency-domain method for bifurcation analysis of nonlinear dynamical systems. The proposed method consists in directly tracking the bifurcation points when a system parameter such as the excitation or nonlinearity level is varied. To this end, a so-called extended system comprising the equation of motion and an additional equation characterizing the bifurcation of interest is solved by means of the Harmonic Balance Method coupled with an arc-length continuation technique. In particular, an original extended system for the detection and tracking of Neimark-Sacker (secondary Hopf) bifurcations is introduced. By applying the methodology to a nonlinear energy sink and to a rotor-stator rubbing system, it is shown that the bifurcation tracking can be used to efficiently compute the boundaries of stability and/or dynamical regimes, i.e., safe operating zones.

Numerical antiresonance continuation of structural systems

“…The governing equation can be written: shows the evolution of the hardening response as the amplitude of the external force increases. One can see that the fold bifurcation appears beyond a given amplitude of motion and its tracking (see [25,26] and Section 3.2), depicted by the dashed line, is useful to predict for instance the critical value of the force amplitude from which the jump phenomenon occurs. Fig.…”

“…However, the procedure can require to add a phase equation to (17), as explained in [18], since the driving force is zero. More specific methods, like bifurcation tracking, consist in following the branch of solution at a given particular point [26] (saddle node, branch point bifurcation . .…”

“…It is also able to track resonance points, defined as a maximum of a frequency response. Our contribution provides an additional tool for the development of dynamic absorbers, in addition to particular procedures such as bifurcation tracking methods [25,26]. In this article, we have implemented our strategy of continuation of extrema in the framework of the ANM/HBM and the software Manlab2.0 [27], but since it is very close to fold bifurcation continuation, it could also be implemented in generic continuation solvers like AUTO [19] and MATCONT [20].…”

Nonlinear Harmonic Analysis of a Blade Model Subjected to Large Geometrical Deflection and Internal Resonance

“…For a full rank n the bifurcation is a turning point, while for a rank at most equal to n − 1 the bifurcation is a branching point and it is necessary to go on with step 3. An alternative to the system (19), is described in [22]. It prevents the singularity of the Jacobian of the system (19) near BPs.…”

Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances

“…Xie et al [5] characterized and tracked bifurcations of cycles from frequency-response curves by using pseudo arclength continuation on fully extended systems in the frequency domain. Minimally extended systems for the same purpose were proposed by Detroux et al [6].…”