INDUCED DAMPING BY A NEARLY CONTINUOUS DISTRIBUTION OF NEARLY UNDAMPED OSCILLATORS: LINEAR ANALYSIS

The induced damping in a master oscillator contributed by a set of satellite oscillators is obtained in terms of a summation over a discrete distribution of the set. (The distribution is with respect to the resonance frequencies of the satellite oscillators in the set. The distribution is cast in an ascending order and is assumed to be centered about the resonance frequency of the master oscillator in isolation). If the modal overlap parameters are less than unity, significant undulations are present in the induced damping; the less the modal overlap parameters are compared with unity, the more prominent are the undulations. The undulations are largely suppressed when the local modal overlap parameters exceed unity. Moreover, appropriately averaging the undulations yields values for the induced damping that coincide with those obtained when the modal overlap parameters exceed unity. Further, it transpires that these common values are independent of the individual modal overlap parameters. When the summation is replaced by an integration, the first order results are undulations-free and the values, so obtained, again, coincide with those pertaining to modal overlap parameters that exceed unity. Without defining the excursions in the undulations, the transition from a discrete-to-a continuous distribution, that is implied by a summation-to-an integration, must assume that the modal overlap parameters exceed unity. In particular, without careful and meaningful qualifications, it may be misleading to assume, a priori, that the modal overlap parameters are equal to zero.