Double Grazing Periodic Motions and Bifurcations in a Vibroimpact System with Bilateral Stops

Abstract

The double grazing periodic motions and bifurcations are investigated for a two-degree-of-freedom vibroimpact system with symmetrical rigid stops in this paper. From the initial condition and periodicity, existence of the double grazing periodic motion of the system is discussed. Using the existence condition derived, a set of parameter values is found that generates a double grazing periodic motion in the considered system. By extending the discontinuity mapping of one constraint surface to that of two constraint surfaces, the Poincaré map of the vibroimpact system is constructed in the proximity of the grazing point of a double grazing periodic orbit, which has a more complex form than that of the single grazing periodic orbit. The grazing bifurcation of the system is analyzed through the Poincaré map with clearance as a bifurcation parameter. Numerical simulations show that there is a continuous transition from the chaotic band to a period-1 periodic motion, which is confirmed by the numerical simulation of the original system.

Existence and stability of the periodic orbits induced by grazing bifurcation in a cantilever beam system with single rigid impacting constraint

“…Bernard [7] analyzed the slip bifurcations of n-dimensional piecewise smooth dynamical systems, gave the corresponding discontinuous mapping, and derived the complete expression of the grazing-induced bifurcation normal form mapping. Li [8] considered a two-degree-of-freedom impact oscillator with symmetrical rigid constraints, found the dynamic behavior induced by grazing bifurcation numerically. Zhang [9] derived the discontinuous maps of a rub-impacting rotor system, used the sequence limit theory to analyze the bifurcation situation induced by grazing of the rotor system.…”

Double grazing bifurcations of the non-smooth railway wheelset systems

“…di Bernardo et al [23] extended the results presented in [17] to n-dimensional piecewise-smooth systems and derived normal form maps for grazing bifurcation. For additional literature about grazing bifurcation, see, e.g., [24,25].…”

Anti-phase synchronization and symmetry-breaking bifurcation of impulsively coupled oscillators

“…Blazejczyk-Okolewska et al [25] studied the dynamics of two mechanical oscillators coupled by impacts and observed the anti-phase synchronization phenomenon when the distance between these oscillators exceeded some critical thresholds. In [26][27][28][29], the dynamics of multi-degree-of-freedom vibro-impact systems has been studied by using numerical simulations and qualitative analysis. This paper will use the physical model in [25] to study the dynamics and the synchronization of impulsively coupled oscillators under variations of amplitude and frequency of external excitation.…”