Keywords: nonlinear vibrations, bifurcation tracking, period doubling, isolas

Intermittent contacts are commonplace in numerous mechanical engineering applications. Since the resulting localized forces are strongly non-linear, multiple dynamical regimes may co-exist for a given set of conditions, and small perturbations then suffice to induce a change from a desirable to a potentially unfavourable state. For this reason, prediction of global dynamics is critical, especially in regard to the systems' parameters. Numerical bifurcation tracking offers a means to efficiently compute stability boundaries, as done by Xie et al. [1] for systems under forced vibration. In this approach, constraints are appended to the equations of motion, and the solution of the extended systems thus constructed gives the loci of bifurcation points for a varying parameter. Even isolated resonance curves, which are usually difficult to detect, have been found through this technique, see e.g. Kuether et al. [2]. However, this has only been done for the case when both the regimes on the isolated and main resonance branches share a common fundamental frequency, whereas it is known that sub-harmonic isolated resonances are also possible for vibrating systems undergoing asymmetric contacts [3].
In the present contribution, we first propose an extended system characterizing period doubling bifurcations through the harmonic balance method (HBM), then proceed to track these bifurcations. We show that, given a judicious choice of tracking parameter, extremal points on the ensuing stability boundaries correspond to the birth of isolated sub-harmonic resonances. Moreover, we further characterize and track these points with respect to a second parameter. This is interesting from a design viewpoint, since it provides a way to avoid these regimes altogether. Afterwards, we present an example application on an academic system, where an "asymmetry parameter" controls the existence of sub-harmonic isolas. Finally, numerical results are confronted with experimental measurements, which validate the methods proposed herein.

References:

[1] L. Xie, S. Baguet, B. Prabel, R. Dufour, Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical systems, J. of Mechanical Systems and Signal Processing, 88 (2017) 445-461
[2] R. J. Kuether, L. Renson, T. Detroux, C. Grappasonni, G. Kerschen, M.S. Allen, Nonlinear normal modes, modal interactions and isolated resonance curves, J. of Sound and Vibration, 351 (2015) 299-310
[3] C. Duan and R. Singh, Isolated sub-harmonic resonance branch in the frequency response of an oscillator with slight asymmetry in the clearance, J. of Sound and Vibration, 314 (2008) 12-18