  A mechanical system consisting of an elastic beam under harmonic excitation
and an attached sliding body is investigated. Recent experimental observations
suggest that the system passively (self-)adapts the axial location of the
slider to achieve and maintain a condition of self-resonance, which could be
useful in applications such as energy harvesting. The purpose of this work is
to provide a theoretical explanation of this phenomenon based on an appropriate
model. A key feature of the proposed model is a small clearance between the
slider and the beam. This clearance gives rise to backlash and frictional
contact interactions, both of which are found to be essential for the
self-adaptive behavior. Contact is modeled in terms of the Coulomb and
Signorini laws, together with the Newton impact law. The set-valued character
of the contact laws is accounted for in a measure differential inclusion
formulation. Numerical integration is carried out using Moreau's time-stepping
scheme. The proposed model reproduces qualitatively most experimental
observations. However, although the system showed a distinct self-adaptive
character, the behavior was found to be non-resonant for the considered set of
parameters. Beside estimating the relationship between resonance frequency and
slider location, the model permits predicting the operating limits with regard
to excitation level and frequency. Finally, some specific dynamical phenomena
such as hysteresis effects and transient resonance captures underline the rich
dynamical behavior of the system.
