Mirror hysteresis and Lyapunov exponents of impact oscillator with symmetrical soft stops

• The dynamics of 1DOF impact oscillator with two Hertz type stops is explored.
• The spectra of Lyapunov exponents are calculated.
• The mirror hysteresis phenomenon has been detected.
• The effect of the nonlinearities due to beam deflection and impacts is investigated.
• A simple cantilever beam system is considered as an application example and the compound effect of the nonlinearities due to beam deflection and two-sided impacts is investigated.

The paper is concerned with the analysis of motion of a harmonically excited one-degree-of-freedom mechanical system having two symmetrical amplitude constraints. The contact between the oscillated mass and the barriers is modeled by Hertz׳s law with non-linear damping. We study the influence of the frequency of excitation force on the system׳s behavior in a wide range of the control parameter by determining the spectra of Lyapunov exponents. Detailed analysis of the system responses allowed to detect a mirror hysteresis phenomenon. Its characteristic feature is the appearance of two branches in a bifurcation diagram such that the dynamic behaviors corresponding to increasing and decreasing values of the control parameter are symmetric in the sense that their phase portraits are mirror images. As an application example, we consider a cantilever beam system with two-sided impacts and investigate the combined effects of the nonlinearities due to beam deflection and two-sided impacts of Hertz׳s type as well as linear elastic type.