Stability and dynamics of a harmonically excited elastic-perfectly plastic oscillator

This paper deals with the stability and the dynamics of a harmonically excited elastic-perfectly plastic oscillator. The hysteretical system is written as a non-smooth forced autonomous system. It is shown that the dimension of the phase space can be reduced using adapted variables. Free vibrations of such a system are then considered for the damped system. The extended direct method of Liapounov is applied to this non-smooth mechanical system and the asymptotic stability of the origin is proven in the new phase space. The forced vibration of such an oscillator is treated by numerical approach, by using the time locating techniques. The stability of the limit cycles is analytically investigated with a perturbation approach. The boundary between elastoplastic shakedown and alternating plasticity is given in closed form. It is shown that this boundary corresponds to a bifurcation boundary for the undamped system (period-doubling bifurcation). Finally, the equivalent damping of this hysteretic system is characterized from dynamical properties.