Modal overlap and dissipation effects in a fuzzy structure containing a continuous master element

Abstract

This work was prompted by a study by Strasberg [J. Acoust. Soc. Am. 102, 3130(A) (1997)] in which a large suspended mass had numerous small spring–mass–damper systems attached to it. The system parameters in his model were selected such that the isolated natural frequency of each attached system was close to the natural frequency of the isolated master structure. Strasberg found when an impulse excitation is applied to the master structure, the critical issue is the bandwidth of the isolated attached systems in comparison to the spacing between the natural frequencies of those systems. Modal overlap, which corresponds to bandwidths that exceed the spacing of those frequencies, was shown to lead to an impulse response that decays exponentially. In contrast, light damping leads to an impulse response that consists of a sequence of exponentially decaying pulses, due to delayed energy return from the subsystems. The present work explores these issues for continuous systems by replacing the one-degree-of-freedom master structure with a cantilevered beam. The system parameters are selected to match Strasberg’s model, with the suspended subsystems placed randomly along the beam. The beam displacement is represented as a Ritz series whose basis functions are cantilever beam modes. The coupled equations are solved by a state–space eigenmode analysis that yields a simple closed-form representation of the response in terms of the complex eigenmode properties. The continuous fuzzy structure is shown not to display the transfer of energy between the master structure and the substructures exhibited by the discrete fuzzy structure, apparently because the attachment points move asynchronously due to the spatial dependence of the beam’s displacement.