On stability and stabilization of mechanical systems with nonlinear energy sinks

Direct energy pumping phenomenon, i. e. passive irreversible transfer of mechanical energy from a linear oscillator to a nonlinear one has been studied intensively during the past decade. On the base of this phenomenon, numerous seismic mitigation devices were developed. Therefore, the important problems are those of stability analysis and stabilizing control synthesis for complex mechanical systems composed from a linear part, a nonlinear energy sink and essentially nonlinear interconnections. In the present paper, by the use of the Lyapunov direct method, the sufficient conditions of asymptotic stability of equilibrium positions for such systems are obtained. The theorems proved make it possible to study stability of an equilibrium position on the basis of decomposition of the original complex mechanical system into several isolated subsystems. For systems with incomplete measurement of a generalized coordinates vector the problems of stabilization of an equilibrium position by means of nonlinear feedback using only measured coordinates and auxiliary variables are investigated. The results obtained are applied in the problem of stabilisation of a three-mass system with a single measurible coordinate.