Codimension-Two Grazing Bifurcations in Three-Degree-of-Freedom Impact Oscillator with Symmetrical Constraints

Qunhong Li,Pu Chen,Jieqiong Xu

doi:10.1155/2015/353581

Abstract

This paper investigates the codimension-two grazing bifurcations of a three-degree-of-freedom vibroimpact system with symmetrical rigid stops since little research can be found on this important issue. The criterion for existence of double grazing periodic motion is presented. Using the classical discontinuity mapping method, the Poincaré mapping of double grazing periodic motion is obtained. Based on it, the sufficient condition of codimension-two bifurcation of double grazing periodic motion is formulated, which is simplified further using the Jacobian matrix of smooth Poincaré mapping. At the end, the existence regions of different types of periodic-impact motions in the vicinity of the codimension-two grazing bifurcation point are displayed numerically by unfolding diagram and phase diagrams.

Highlights

Impacting phenomena exist in a large number of mechanical systems
The early work mainly focuses on the singledegree-of-freedom impact oscillators, for example, [1,2,3]
Luo [7] developed a method to investigate the symmetry of solutions in nonsmooth dynamical systems and obtained all possible stable and unstable motions

Summary

Introduction

Impacting phenomena exist in a large number of mechanical systems. Because the collision introduces essential nonlinearity and discontinuity, the vibroimpact systems can exhibit rich and complicated dynamical behavior. For multidegree-of-freedom vibroimpact systems, detailed studies of dynamics (including stability and bifurcations) using numerical simulations and qualitative analyses were carried out in decades, for example, [4,5,6,7,8]. Li et al [14, 15] investigated the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibroimpact system with unilateral constraint and symmetric constraints, respectively. Mason et al [20] analyzed a model of a periodically forced impact oscillator with two discontinuity surfaces and provided new insights into the extremely rich dynamical behavior including codimension-one, codimension-two, and codimension-three bifurcations by discontinuity-geometry methodology. We investigated codimension-two grazing bifurcations in three-degree-of-freedom impact oscillator with symmetrical constraints.

Double Grazing Periodic Motion in Three-Degree-of-Freedom Impact Oscillator

Stability at Grazing and Codimension-Two Grazing Bifurcation

Numerical Simulations

Conclusions