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

Zihan Wang,Jieqiong Xu,Shuai Wu,Quan Yuan

doi:10.1155/2020/7893451

Abstract

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

Highlights

Grazing bifurcation, one type of discontinuity-induced bifurcations, has been extensively studied in vibroimpact system as it has complex dynamics and is widely encountered in many engineering examples
E analysis is usually carried out by finding an appropriate local map describing the system dynamics in neighborhood of the grazing event. e local map can be combined with an analytic Poincaremap to give the so-called grazing normal form whose dynamics can be shown to be topologically equivalent to those of the underlying flow. e grazing normal form derived by the discontinuity-mapping approach is used to analyze the local dynamics in the vicinity of a grazing trajectory
As shown in [5,6,7,8], the normal form map of the rigid impact oscillator contains a square-root term causing a singularity in the first derivative, which results in an abrupt loss of the stability

Summary

Introduction

One type of discontinuity-induced bifurcations, has been extensively studied in vibroimpact system as it has complex dynamics and is widely encountered in many engineering examples. Some conditions for the persistence of a local attractor in the immediate vicinity of quasiperiodic grazing trajectories in an impacting dynamical system were formulated by ota and Dankowicz [10]. Ota et al [13] studied the distribution of codimension-two grazing bifurcation point according to the discontinuous mapping in the single-degree-offreedom collision oscillator and discussed the possible dynamic characteristics of the system response near this bifurcation point. E stability of double grazing motion bifurcation in the system is lost in three ways, and the existence conditions of the codimension-two grazing bifurcation occur in four different cases For this complex unstable problem, analytic expressions of stability criterion are obtained in this paper.

Mechanical Model and Double Grazing Periodic Motion

Near-Grazing Dynamics

Controlling the Persistence of NearGrazing Attractors

Numerical Experiments