Control of near‐grazing dynamics and discontinuity‐induced bifurcations in piecewise‐smooth dynamical systems

Abstract

SUMMARYThis paper develops a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. In particular, it is shown that a discrete-in-time linear feedback correction to a parameter governing a state-space discontinuity surface can suppress discontinuity-induced fold bifurcations of limit cycles that achieve near-tangential intersections with the discontinuity surface. The methodology ensures a persistent branch of limit cycles over an interval of parameter values near the critical condition of tangential contact that is an order of magnitude larger than that in the absence of control. The theoretical treatment is illustrated with a harmonically excited damped harmonic oscillator with a piecewise-linear spring stiffness as well as with a piecewise-nonlinear model of a capacitively excited mechanical oscillator.

Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

“…Dankowicz and Svahn [21] presented for the existence of event-driven control strategies that guarantee the local persistence of system attractors with at most lowvelocity contact in vibro impacting oscillators. Misra and Dankowicz [22] developed a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. Yin et al [23] analyzed the stability for near-grazing periodone impact motion to suppress grazing-induced instabilities.…”

“…Based on the concept of controlling the persistence of local attractors near the grazing trajectory in impact oscillator with unilateral constraints mentioned in [20][21][22][23][24], this paper aims to control the stability of grazing bifurcation or control the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Compared with impact oscillator with unilateral constraint, the instability problems near grazing trajectory become more complex for impact oscillator with symmetrical constraints as mentioned in [17].…”