Dynamics of a semi-active impact damper: Regular and chaotic motions

A study is made of the general behavior of a semi-active impact damper. The system consists of an undamped forced torsional oscillator, and a flywheel which can be locked to the oscillator through a clutch. Clutch engagement takes place instantaneously when the two rotors move in opposite directions. The resulting impact is effective in reducing the vibration amplitude level of the oscillator when it is subjected to bounded excitation.; All solutions of the system are shown to be bounded when the input is bounded. Emphasis is placed on 2 impacts/cycle periodic solutions. Exact symmetric and nonsymmetric solutions are derived analytically and the region of asymptotic stability is determined. The stability analysis leads to the definition of a transition matrix which determines the state of the system immediately after impact from its state after the previous impact. It also leads to the definition of a nonlinear map associated with the impact damper so that periodic solutions of the system correspond to fixed points of the map. The region of stability is defined as the region in parameter space where the eigenvalues of the transition matrix have modulus less than unity.; In the region of instability solutions are quasiperiodic or chaotic. As the parameter of the problem varies, the fixed points of the associated map undergo Hopf bifurcation which results in an invariant circle. Breakdown of the invariant circle leads to chaotic motions by the impact damper. Time histories, phase plane portraits, power spectra and Poincare maps are used as descriptors to observe the evolution of chaotic motions. Computation of the largest Lyapunov exponent verifies the fact that when the structure of the invariant circle in resonance breaks down motions of the system are indeed chaotic.; It is found that under practically realizable conditions the mechanism is an effective damper. Applications include rotating shafts, and numerical simulations of a two-degree-of-freedom torsion bar were also carried out to observe the effects and effectiveness of impact in a multidegree of freedom primary system.