Impact oscillators are a special class of continuous time dynamical systems which undergo intermittent impact collisions. This class of oscillator systems have dynamical trajectories in state space which are piecewise continuous, with discontinuities in the velocities resulting from the collisions. Even if a system without impact is linear, and therefore well behaved, the overall dynamics of the system exhibits a rich variety of behavior because of the nonlinearity introduced by the impacts.

Impact oscillators can be used to model a variety of systems arising naturally in many applications. Some of the first impacting oscillators to be studied were models of various atomic and sub-atomic particle interactions. In 1949 Fermi [1] proposed a mechanism for the acceleration of cosmic rays that involved collisions with magnetic field structures. This mathematically reduces to the study of a particle moving between two walls, one which is oscillating and the other stationary, bouncing elastically off each wall. More recently many engineering systems like rattling gears [2,3], vibration absorbers [4], car suspensions [5], impact print hammers [6], articulated mooring tower [7] and slider-crank joints [8] have been studied using impact oscillator models. These systems have motion limiting constraints occurring naturally in their operation. Repeated impacts, referred to as vibro-impact response, possibly allied with sliding, is a potent damage producing mechanism for many types of mechanical devices. They lead to excessive noise, wear and fatigue.

Impact phenomena are also relevant to the study of biological and electrical systems, such as studies of walking [9] and in DC-DC buck converters [10], respectively. In buck converters, the action of the pulse width modulation of the supply is very similar to an impacting process.

Investigation of such systems is particularly useful because these chaotic oscillations can be related to distinct types of dynamic phenomena and serve as reliable indicators of specific problems [11]. Furthermore, understanding the dynamics of the system could help improve the overall system performance by being able to control the system in some desired regime or by preventing the system from going into some undesirable regime which would eventually result in system failure.

In the study of vibro-impact problems, two different approaches are generally used to model the impact phenomenon. In the first model, it is assumed that the impacting bodies are elastic with linear or bilinear or nonlinear stiffness and damping. In the second model, it is assumed that the impacting bodies are rigid and that the velocity changes instantly, with the outgoing velocity being a function of the incoming velocity. In the second approach to modelling the impact phenomena is essentially like assuming an infinite stiffness coefficient with dissipation and the impacting process is instantaneous. In our investigations we follow the latter approach to model impact process. Though, we use the bilinear model of the impact process to understand the phenomena of hysteresis.

REFERENCES

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E. Fermi, Phys. Rev. 75, 1169-1174, (1949).

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K. Karagiannis and F. Pfeiffer, Nonlinear Dynamics 2, 367-387 (1991).

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C. Knudsen, R. Feldberg and H. True, Philos. Trans. R. Soc. London. A 338, 445-469 (1992).