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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.

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. In general, such impact can be studied using the rigid impact oscillator, which is modelled using the coefficient of restitution rule assuming the instantaneous reversal of velocity for the collision body. The rigid impact oscillator is a nonsmooth dynamical system which can exhibit complex dynamical behavior, so the stability and the bifurcation of the rigid impact oscillator have received great attention; see, for example, [

In this paper, we will study a rigid impact oscillator with bilinear damping through one-parameter bifurcation analysis. In general, linear damping is always considered in the physical models of mechanical systems. However, in some engineering practices, piecewise linear or viscous dampers are widely designed and used owing to their adaptability in the changing environment. For example, shock absorber with bilinear characteristics is often used for improving the driving safety and traveling comfort of vehicles [

Recently, there has been an increasing interest in the analysis and synthesis of impulsive switched systems with the presence of two nonsmooth effects, impulsive disturbances and switching, because of their significance in theory and applications in physics, biology, engineering, and information science [

The physical model of the rigid impact oscillator with one-sided constraint is shown in Figure

(a) Physical model and (b) the dual-rate damper model.

The governing equations of the impact oscillator can be written as

The impact oscillator (

Given an initial value

Representative trajectories and segments of (a) the nonimpacting period-1 orbit, (b) the period-1 orbit with one impact per period, and (c) the local maps.

In the following, the mapping methods proposed in [

In this section, one-parameter bifurcation analysis of the impact oscillator with bilinear damping is carried out to investigate the influence of forcing frequency and amplitude on the dynamics of the system.

To investigate the influence of forcing frequency on the dynamics, we chose

(a) Bifurcation diagrams constructed for the mass displacement under varying forcing frequency

Figure

Figure

Evolution of the basins of attraction for the impact oscillator with bilinear damping obtained for

To investigate the influence of forcing amplitude on the dynamics, we fixed

Bifurcation diagrams constructed for the mass displacement under varying forcing amplitude (a)

Figure

Evolution of the basins of attraction for the impact oscillator with bilinear damping obtained for

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. The stability and the bifurcation of the periodic orbits in the impact oscillator were studied by using the mapping methods. In order to describe the periodic orbit, we divided the phase space of the system into four subspaces and defined four discontinuous boundaries. Local maps were built to describe the motion of the system, and the eigenvalues of the Jacobian matrix of these global maps were used to determine the stability and bifurcation of the system.

Our one-parameter bifurcation analyses show that, as the forcing frequency varies for

Our one-parameter bifurcation analyses using the forcing amplitude as the branching parameter reveal that the impact oscillator encounters two grazing, one period-doubling, and two saddle-node bifurcations when the forcing amplitude varies for

Compared to the research on the rigid impact oscillators with linear damping in [

Consider the equation

This paper was presented at the International Conference on Engineering Vibration (Sofia, Bulgaria, September 4–7, 2017).

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is partially supported by the National Natural Science Foundation of China (Grants nos. 11402224, 11672257, and 11202180), the Natural Science Foundation of Jiangsu Province of China (Grants nos. BK20161314 and BK20151295), the Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-Aged Teachers and Presidents, the Excellent Scientific and Technological Innovation Team of Jiangsu University, and Jiangsu Key Laboratory for Big Data of Psychology and Cognitive Science. The authors also thank Professor Marian Wiercigroch for helpful comments.