Modal overlap and dissipation effects of a cantilever beam with multiple attached oscillators

This work was prompted by a study performed by Strasberg [7] in which numer ous 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 march in average the natural frequency of t he isolated master structure. Strasberg found that the critical issue when an impulse excitation is applied to the master structure is the bandwidth o f the isolated attached systems in comparison to the spacing between the na tural frequencies of the system. Modal overlap, which corresponds to bandwi dths that exceed the sparing of those frequencies, was shown to greatly inf luence the response of the master structure. Light damping, for which there is little or no modal overlap, corresponds to an impulse response that con sists of a sequence of nearly periodic exponentially decaying pulses, and t he transfer function for harmonic excitation of the master structure indica tes that the substructure acts as a vibration absorber for the master struc ture. Increased damping leads to modal overlap, with the result that the im pulse response consists of a single decaying pulse. The frequency domain tr ansfer function indicates that the vibration absorber effect is enhanced. T he present work explores these issues for continuous systems by replacing t he one degree of freedom master structure with a cantilever beam. The syste m parameters are selected to march Strasberg's model, with the suspended os cillators placed randomly along the beam. The beam displacement is represen ted as a Ritz series Ir-hose basis functions are the cantilever beam nodes. 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 nor to displ ay the transfer of energy between the master structure and the substructure that was exhibited by the discrete fuzzy structure. apparently be cause of the asynchronous motion of the attachment points resulting from the spatia l variability of the beam's motion. The vibration absorber effect for harmo nic excitation is only obtained for the heavy damping in the case of a beam .