Energy equipartitioning and frequency distribution in complex attachments

As reported in several recent publications, an undamped simple oscillator with a complex attachment that consists of a set of undamped parallel resonators can exhibit unusual energy sharing properties. The conservative set of oscillators of the attachment can absorb nearly all the impulsive energy applied to the primary oscillator to which it is connected. The key factor in the ability of the attachment to absorb energy with near irreversibility correlates with the natural frequency distribution of the resonators within it. The reported results also show that a family of optimal frequency distributions can be determined on the basis of a variational approach, minimizing a certain functional related to the system response. The present paper establishes a link between these optimal frequency distributions and the energy equipartition principle: optimal frequency distributions are those that spread the injected energy as uniformly as possible over the degrees of freedom or over the modes of the system. Theoretical as well as numerical results presented support this point of view.