Lyapunov exponents of impact oscillators with Hertz׳s and Newton׳s contact models

In this paper, investigations of a harmonically excited one-degree-of-freedom mechanical system having an amplitude constraint are presented. The contact between the oscillated mass and the barrier is modeled by Hertz׳s law with a non-linear damping as well as by Newton׳s law. The influence of the frequency of excitation force on the system׳s behavior is studied in a wide range of the control parameter by determining and analyzing the corresponding spectra of Lyapunov exponents. The dynamical behaviors of two systems with impacts: a system with Hertz׳s undamped impacts and a system with perfectly elastic hard impacts, which are equivalent in the sense of the same rate of impact energy dissipation, are compared and strong qualitative and quantitative similarities are observed. As an application example, a simple cantilever beam system with impacts is considered and the combined effects of the nonlinearities due to beam deflection and impacts of Hertz׳s as well as Newton׳s types are investigated.