The multi-time scale responses of a two degrees of freedom pendulum coupled with a nonlinear absorber is studied. The absorber is positioned in an arbitrary direction with respect to those of the pendulum oscillations. Phase-dependent slow invariant manifold of the system and its stable zones are traced at fast time scale while studying system responses at slow time scale around the slow invariant manifold leads to detection of equilibrium and singular points. Moreover, the amplitude-frequency curves of the system are detected showing the possibility of existence of isolated branches, which could correspond to high energy levels for pendulum oscillations. All analytic developments are confronted with numerical results collected from direct numerical integration of system equations. Depending on characteristics of external excitations, the system can face periodic or modulated regimes.
A pendulum, which can oscillate in two directions, is subjected to a generalized external force. A non-smooth absorber is coupled to the pendulum with an arbitrary location and orientation. The equations of the system are derived and are treated with a multiple scale method. At fast time scale, the topology of the slow invariant manifold is described with its stable and unstable zones. The equilibrium and singular points of the system are detected at the first slow time scale. The responses of the main system, given as a function of the frequency of the external force, show reductions of the vibration levels. The analytic predictions are compared by direct numerical time integration of the equations of the system. They illustrate the operationality of the non-smooth absorber in several cases.
We seek to understand the behavior of a pendulum under parametric excitation coupled with a nonlinear absorber. First, the reference system without any coupled absorber, i.e., a simple pendulum, is analyzed with a multiple scale method thanks to supposed assumptions about the excitation. The equilibrium points of the system are calculated, and their stability is determined. The phase portraits are introduced in order to better predict the behavior of the system. Then the same analysis is performed on the pendulum coupled with the nonlinear absorber leading to detection of the slow invariant manifold and its dynamic characteristic points. Both systems are compared to estimate the effects of the absorber on the vibratory behaviors of the pendulum.
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