Dynamics of SDOF Oscillators with Hysteretic Motion-Limiting Stop

Interest in impact vibrations is two-fold: (i) a wide range ofpractical problems involve bodies colliding with one another or/and withobstacles, and (ii) the complex dynamics of such problems is a goodtesting bench for nonlinear theories. The assumption of rigid stop isquite popular, although unfortunately it does not allow us to simulatethe actual dissipative character of the impact response, but via apriori fixed coefficient of restitution. However, the correctdescription of the energy dissipated during impact is very important.In this paper, the dynamic response of a single-degree-of-freedomsystem is studied, where hysteretic stop, which allows the simulation ofthe real behaviour of a wide range of material pairings, is assumed; adistinction between hard and soft contacts is made according to theimpulsive or nonimpulsive nature of the contact reaction. The evolutionthrough stable closed orbits and period-doubling routes to chaos arestudied in terms of the clearance between the mass in the initial placeand the obstacle. For different clearances, strange attractors arerevealed and their evolution illustrated. Furthermore, in the case ofhard contact, an equivalent coefficient of restitution is proposed whichdepends, in a simple way, on some characteristic parameters of thehysteretic contact law. Such a coefficient, not given a priori butobtained via simulation of physical behaviour, provides the definitionof an equivalent impact oscillator (i.e. with rigid stop).