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A shooting method is a very powerful numerical method to obtain periodic solutions of nonlinear systems. However, as a variational equation of motion is needed in the shooting method and it is very difficult to obtain it in the impact systems, the shooting method for impact systems has not been developed. In this report, a shooting method for impact systems is presented by solving this problem of variational equation. Namely, the variational equation with the delta function and its differentiation is derived. It is shown that the calculation speed of this method is very fast and complicated periodic solutions are easily obtainable in high accuracy. The stabilities of periodic solutions obtained in the shooting method are in good accordance with those obtained by the analytical method. The discontinuities in the stability of the periodic solutions are shown using characteristic multiplier. Lyapunov exponents are also calculated by applying the integral technique of variational equation.

This paper deals with a two-degree-of-freedom forced system composed of a main system and an impact damper which has been analyzed by many researchers. Periodic solutions, bifurcations, chaos, and vibration quenching are discussed. A shooting method for impact systems presented by authors is used in numerical calculations. The following results were obtained : (1) Unsynmetric periodic solution with four collisions per period and subharmonic vibrations with many collisions were found. (2) Discontinuities in the stability of the periodic solutions caused by impact were shown using characteristic multipliers. (3) Two routes to chaos were found, namely, the period doubling route and the torus doubling route. (4) Hyper chaos was found for the first time in the impact damper system. (5) The vibration quenching problems for the narrow frequency region near the resonance point and for the wide frequency region were discussed.

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