Transient energy exchange between a primary structure and a set of oscillators: return time and apparent damping

In this paper we examine the conditions that influence the return time, the time it takes before energy returns from a set of satellite oscillators attached to a primary structure. Two methods are presented to estimate the return time. One estimate is based on an analysis of the reaction force on a rigid base by a finite number of oscillators as compared with an infinite number of continuously distributed oscillators. The result gives a lower-bound estimate for the return time. A more accurate estimation results from considering the dynamic behavior of a set of oscillators as waves in a waveguide. Such an analogy explains energy flow between a primary structure and the oscillators in terms of pseudowaves and shows that a nonlinear frequency distribution of the oscillators leads to pseudodispersive waves. The resulting approximate expressions show the influence of the natural frequency distribution within the set of oscillators, and of their number, on the return time as compared with the asymptotic case of a continuous set with infinite oscillators. In the paper we also introduce a new method based on a Hilbert envelope to estimate the apparent damping loss factor of the primary structure during the return time considering transient energy flow from the primary structure before any energy reflects back from the attached oscillators. The expressions developed for return time and damping factor show close agreement with direct numerical simulations. The paper concludes with a discussion of the return time and its relation to apparent damping and optimum frequency distribution within a set of oscillators that maximize these quantities.