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Impulsive switched systems have been extensively investigated. However, there are few papers on the mechanical model. In this paper, a rigid impact oscillator with bilinear damping is developed as a mechanical model for impulsive switched systems. Firstly, the dual-rate damper model is used to describe the bilinear damping and the physical model of a rigid impact oscillator with bilinear damping is constructed. In the dual-rate damper model, the damping coefficient takes a different value in compression than in extension, depending on the sign of the velocity, which is typical in automotive engineering practice. Secondly, to describe the segments of periodic orbits of the impact oscillators, we divide the phase space into four subspaces and obtain four discontinuity boundaries. Then five local maps are defined between these discontinuity boundaries and periodic orbits can be described by the composite of these local maps. So the stability and bifurcations can be determined by the eigenvalues of the Jacobian matrix of global maps. Thirdly, bifurcation analysis of the impact oscillator with bilinear damping are carried out to show the influence of forcing frequency, forcing amplitude and damping on the dynamics of the system. Numerical simulations show that the system could exhibit complex phenomena, including periodic orbits and chaotic orbits.