Modal Overlap and Dissipation Effects of a Cantilever Beam with Multiple Attached Oscillators

Abstract

This work was prompted by a study performed by Strasberg [7] in which numerous small spring-mass-damper systems are attached to a large suspended mass representing the master structure. The isolated natural frequency of each attached system was selected to match in average the natural frequency of the isolated master structure. Strasberg found that the critical issue when an impulse excitation is applied to the master structure is the bandwidth of the isolated attached systems in comparison to the spacing between the natural frequencies of the system. Modal overlap, which corresponds to bandwidths that exceed the spacing of those frequencies, was shown to greatly influence the response of the master structure. Light damping, for which there is little or no modal overlap, corresponds to an impulse response that consists of a sequence of nearly periodic exponentially decaying pulses, and the transfer function for harmonic excitation of the master structure indicates that the substructure acts as a vibration absorber for the master structure. Increased damping leads to modal overlap, with the result that the impulse response consists of a single decaying pulse. The frequency domain transfer function indicates that the vibration absorber effect is enhanced. The present work explores these issues for continuous systems by replacing the one degree of freedom master structure with a cantilever beam. The system parameters are selected to match Strasberg’s model, with the suspended oscillators placed randomly along the beam. The beam displacement is represented as a Ritz series whose basis functions are the cantilever beam modes. The coupled equations are solved by a state-space eigenmode analysis that yields a 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 substructure that was exhibited by the discrete fuzzy structure, apparently because of the asynchronous motion of the attachment points resulting from the spatial variability of the beam’s motion. The vibration absorber effect for harmonic excitation is only obtained for the heavy damping in the case of a beam.