Vázquez-González B., Silva-Navarro G.

Some practical systems exhibit slight nonlinear behavior under radical operating conditions. The autoparametric vibration absorber based on a pendulum has been widely studied when employed to quench the resonant excitation response of a primary linear mass, spring and viscous damped system. In this work we report the results obtained when the primary system includes nonlinearities as those typical in the Duffing equation. Using the multiple scales method it is found that, for external and internal resonances, the external excitation signal and the cubic nonlinearity are absorbed by the pendulum. In both cases, the steady-state resonant-amplitude response for the primary system is the same, either with linear and nonlinear springs. It is observed that in the case of the Duffing primary system, three fixed points are added to the frequency response of the pendulum, compared to the linear primary system. The nonlinearity of the primary Duffing system, which is transferred to the pendulum, modifies its steady state-response, bending the jumps. On the other hand, the new fixed points are defined by the nonlinearity, particularly, on the amplitude and phase responses and, as a consequence, the stability is regulated by complex expressions attributed to the nonlinearity. To guarantee the effectiveness of the absorber’s dynamic performance, additional mathematical relations between the nonlinearity and the external excitation must be established. Some numerical simulations of the frequency response of the system were performed to analyze its steady-state and dynamic behavior.