Arbitrary Periodic Motions and Grazing Switching of a Forced Piecewise Linear, Impacting Oscillator

Abstract

The grazing bifurcation and periodic motion switching of the harmonically forced, piecewise linear system with impacting are investigated. The generic mappings relative to the discontinuous boundaries of this piecewise system are introduced. Based on such mappings, the corresponding grazing conditions are obtained. The mapping structures are developed for the analytical prediction of periodic motions in such a system. The local stability and bifurcation conditions for specified periodic motions are obtained. The regular and grazing, periodic motions are illustrated. The grazing is the origin of the periodic motion switching for this system. Such a grazing bifurcation cannot be estimated through the local stability analysis. This model is applicable to prediction of periodic motions in nonlinear dynamics of gear transmission systems.

Flow switchability and periodic motions in a periodically forced, discontinuous dynamical system

“…From the aforementioned causes, the onset, existence and disappearance of the stick motion should be determined through the normal vector field criteria in Eqs. (21) and (22). The grazing bifurcation will be determined by Eq.…”

“…Another simple system possessing the displacement boundary is the impacting oscillator, which extensively exists in mechanical systems such as rattling gears [13][14][15][16], vibration absorber [17], and impact hammers [18]. In 2005, Luo and Chen [19] developed a piecewise linear system with impacts to investigate the gear transmission system, and the chaotic and periodic motions and grazing mechanism for such a oscillator were discussed in Luo and Chen [20,21].…”

“…Chaotic motion of P(21) k (k → ∞): (a) phase plane, (b) Poincare mapping section, (c) switching normal vector fields, and (d) switching normal vector field product. Initial condition is Ωt 0 ≈ 2.8192, x 0 ≈ −0.9657 and y 0 ≈ −7.8283.…”

Periodic Motions and Chaos With Impacting Chatter and Stick in a Gear Transmission System

“…The switching between the two periodic motions causes motion complexity, and the asymmetric periodic motions exhibit the period-doubling bifurcation in such an oscillator. Luo and Chen [2007] used the mapping technique to analytically predict arbitrary periodic motions for such an oscillator. As in [Luo & O'Connor, 2007], it was stated that the impacting location in the model of Luo and Chen [2007] was fixed and the perfect plastic impact was considered in this piecewise linear model.…”

“…Luo and Chen [2007] used the mapping technique to analytically predict arbitrary periodic motions for such an oscillator. As in [Luo & O'Connor, 2007], it was stated that the impacting location in the model of Luo and Chen [2007] was fixed and the perfect plastic impact was considered in this piecewise linear model. The separation of two gears occurs at the same location of gear impacts.…”