Bifurcation Analysis of a Rigid Impact Oscillator with Bilinear Damping

Liping Zhang,Yang Liu,Haibo Jiang

doi:10.1155/2018/1617650

Liping Zhang, Yang Liu + Show 1 more

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https://doi.org/10.1155/2018/1617650

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Abstract

This paper studies a rigid impact oscillator with bilinear damping developed as the mechanical model of an impulsive switched system. The stability and the bifurcation of periodic orbits in the impact oscillator are determined by using the mapping methods. One-parameter bifurcation analyses under variation of forcing frequency and amplitude of external excitation are carried out. Coexisting attractors and various types of bifurcations, such as grazing, period-doubling, and saddle-node, are observed, which show the complex phenomena inhered in this impact oscillator.

Highlights

The rigid impact, known as the impulsive reactions whenever rigid bodies collide, widely exists in many engineering applications, such as rotating machinery, car suspension systems, and cutting processes
Foale [5, 6] classified the types of grazing bifurcation in a class of rigid impact oscillators and presented analytical results to show how the type of grazing bifurcation changed with control parameter
The rigid impact oscillator with bilinear damping constructed as the mechanical model of an impulsive switched system was investigated through one-parameter bifurcation analysis in this paper

Summary

Introduction

The rigid impact, known as the impulsive reactions whenever rigid bodies collide, widely exists in many engineering applications, such as rotating machinery, car suspension systems, and cutting processes. Nordmark [3] studied the singularities of grazing impacts in a single-degree-of-freedom periodically forced oscillator subjected to a rigid amplitude constraint using analytical methods. In [4], Ivanov developed the linear theory for analyzing the stability and the bifurcation of a class of rigid impact oscillators. We will study a rigid impact oscillator with bilinear damping through one-parameter bifurcation analysis. In [20], Natsiavas presented an appropriate stability analysis for periodic solutions of harmonically excited piecewise linear systems with dry friction and damping coefficients depending on the velocity direction. In [22], Verros et al studied the dynamic response of a controlled single-degree-of-freedom quarter-car model subjected to road excitation, and the control strategy applied to the system was based on the selection of two values of the damping ratio. Numerical simulations show that the system exhibits complex phenomena, including periodic and chaotic orbits

Physical Model

Local Maps and Stability Analysis

Bifurcation Analysis of the Impact Oscillator with Bilinear Damping

P21543

Conclusions