Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments

We consider the response of a linear structural system when coupled to an attachment containing strong or even essential nonlinearities. For this system, the attachment, designated as a nonlinear energy sink, is designed as a nonlinear vibration absorber, serving to dissipate energy from the structural system. Moreover, the attachment not only leads to a reduction in the total energy of the system, but also nonlinearly couples together the vibration modes of the linear structural system. When the structure is impulsively loaded, the nonlinear energy sink serves to both dissipate and redistribute energy, thus enhancing the observed structural dissipation. The effect of the nonlinear attachment on the linear primary system can be quantified in terms of equivalent measures for the damping and frequency of each mode, derived through consideration of the instantaneous energy balance in each mode. The influence of the nonlinear energy sink on the structural response is illustrated with an impulsively forced two-degree-of-freedom primary system, representing a two-story structure, with different types of nonlinear energy sink attached to the top floor. We perform optimization studies in order to design the nonlinear energy sink for optimal shock mitigation of the primary system. The proposed methodology is based only on measured time series without resorting to frequency analysis. As such, it is valid for strongly nonlinear systems as well as for systems with nonsmooth nonlinearities, and is suited to both simulated and experimental results. Finally, an experimental validation of the enhanced dissipation introduced by the nonlinear energy sink is provided, and the experimental response is compared against numerical simulation of the corresponding analytical model to illustrate the effectiveness of the nonlinear energy sink design. Thus, we analytically predict and experimentally verify the efficacy of the nonlinear energy sink to significantly reduce the response of the two degree-of-freedom system subject to shock excitation.