In this work we study the frequency and dynamic response of a damped Duffing system attached to a parametrically excited pendulum vibration absorber. The multiple scales method is applied to get the autoparametric resonance conditions and the results are compared with a similar application of a pendulum absorber for a linear primary system. The approximate frequency analysis reveals that the nonlinear dynamics of the externally excited system are suppressed by the pendulum absorber and, under this condition, the primary Duffing system yields a time response almost equivalent to that obtained for a linear primary system, although the absorber frequency response is drastically modified and affected by the cubic stiffness, thus modifying the jumps defined by the fixed points. In the absorber frequency response can be appreciated a good absorption capability for certain ranges of nonlinear stiffness and the internal coupling is maintained by the existing damping between the pendulum and the primary system. Moreover, the stability of the coupled system is also affected by some extra fixed points introduced by the cubic stiffness, which is illustrated with several amplitude-force responses. Some numerical simulations of the approximate frequency responses and dynamic behavior are performed to show the steady-state and transient responses.
Vazquez-Gonzalez, B., & Silva-Navarro, G. (2014). Evaluation of the Autoparametric Pendulum Vibration Absorber for a Duffing System. Shock and Vibration, 15(3–4), 355–368. https://doi.org/10.1155/2008/827129