Double impact orbits of periodically forced impact oscillators

We consider a periodically forced oscillator which impacts against a rigid constraint. The forcing is both sinusoidal and non-sinusoidal and is made up of a combination of the first and second harmonics of a Fourier series. A simple restitution law is used to describe the impact. This paper centres around the analysis of double-impact periodic solutions, that is motions which repeat after every second impact. Solutions for these double-impact orbits are found, including non-physical orbits and unstable orbits. Behaviour typical of nonlinear dynamical systems, such as period-doubling bifurcations are detected and grazing bifurcations, which are unique to discontinuous dynamical systems are also observed. A combination of analytical and numerical techniques are used to uncover the behaviour of the system at odd multiples of the natural frequency, where numerical experiments tell us that stable double-impact periodic solutions exist. The persistence of these stable double-impact orbits are also examined as parameters are varied. We conclude by presenting some numerical calculations of two more general cases of impacting systems.