The influence of localized randomness on regular grazing bifurcations with applications to impacting dynamics

Abstract

This paper concerns stochastic perturbations of piecewise-smooth ODE systems relevant for vibro-impacting dynamics, where impact events constitute the primary source of randomness. Such systems are characterized by the existence of switching manifolds that divide the phase space into regions where the system is smooth. The initiation of impacts is captured by a grazing bifurcation, at which a periodic orbit describing motion without impacts develops a tangential intersection with a switching manifold. Oscillatory dynamics near regular grazing bifurcations are described by piecewisesmooth maps involving a square-root singularity, known as Nordmark maps. We consider three scenarios where colored noise only affects impacting dynamics, and derive three two-dimensional stochastic Nordmark maps with the noise appearing in different nonlinear or multiplicative ways, depending on the source of the noise. Consequently the stochastic dynamics differs between the three noise sources, and is fundamentally different to that of a Nordmark map with additive noise. This critical dependence on the nature of the noise is illustrated with a prototypical one-degree-of-freedom impact oscillator.

Post-grazing dynamics of a vibro-impacting energy generator

“…For β = π/6 we numerically detect a different type of critical point for the 2:1 period-2 solutions, namely, grazing bifurcations as indicated by the vertical lines at d = G 1 and d = G 2 in Figure 5 (j)-(l), at which Żj = 0 and Z j = d/2 [42,44,47]. Figure 8 zooms in on the bifurcation branches near these values.…”

“…Vibro-impact systems have rich phenomenological behaviors, manifesting various nonlinear phenomena like bifurcations, grazing and chaos. These effects have been studied in deterministic and stochastic vibroimpact systems, as in [38][39][40][41][42][43][44][45][46][47] among others. The models of vibro-impact systems include piecewise linear stiffness [48,49] as well as rigid barriers and instantaneous impacts leading to a velocity jump for inelastic impacts.…”

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

“…Shen [12][13] considered an impact oscillator, suppressed the jumping phenomenon at the grazing bifurcation point based on the special dynamic characteristics of the continuous transition at the degenerated grazing bifurcation point. Simpson [14] considered a kind of impact oscillators with random excitation, and gave the dynamic behavior induced by grazing under different random excitation. Simpson [15] considered a class of single-degree-of-degree impact oscillators, which revealed that some truncated terms neglected in the Nordmark mapping had important effects on the dynamics characteristics of the real primary system.…”

Existence and Stability of the Periodic Orbits Induced by Grazing Bifurcation in a Cantilever Beam System with Single Rigid Impacting Constraint

“…Shen [12][13] considered an impact oscillator, suppressed the jumping phenomenon at the grazing bifurcation point based on the special dynamic characteristics of the continuous transition at the degenerated grazing bifurcation point. Simpson [14] considered a kind of impact oscillators with random excitation, and gave the dynamic behavior induced by grazing under different random excitation. Simpson [15] considered a class of single-degree-of-degree impact oscillators, which revealed that some truncated terms neglected in the Nordmark mapping had important effects on the dynamics characteristics of the real primary system.…”