Predictive design of impact absorbers for mitigating resonances of flexible structures using a semi-analytical approach

Analytical conditions are available for the optimum design of impact absorbers (or Vibro-Impact Nonlinear Energy Sinks) for the special case where the host structure is well described as a rigid body. Accordingly, the analysis relies on the assumption that the impacts cause immediate dissipation in the contact region, which is modeled in terms of a coefficient of restitution (Newton's impact law) that is assumed to be known and fixed. When a flexible host structure is considered instead, the impact absorber not only dissipates energy at the time instances of impact, but, and perhaps more importantly, it inflicts nonlinear energy scattering between structural modes. Hence, it is crucial to account for such nonlinear energy transfers yielding energy redistribution within the modal space of the structure. In the present work, we develop a design approach for flexible host structures. We consider the case of a forced excitation near primary resonance with a well-separated mode. We demonstrate that the time scales of the impact and the resonant vibration can be decoupled. On the long time scale, the dynamics of the host structure can be properly reduced to the fundamental harmonic of the resonant mode. A light impact absorber responds to this enforced motion, and we show that the conventional Slow Invariant Manifold of the dynamics is recovered for the commonly considered regime of two symmetric impacts per period. On the short time scale of the impact dynamics, the contact mechanics and elasto-dynamics must be finely resolved. We show that it is sufficient to run a numerical simulation of a single impact event with representative pre-impact velocity. From this short-time simulation, we derive an effective modal coefficient of restitution and the properties of the contact force pulse, needed to approximate the behavior on the long time scale. We derive a closed-form expression of the periodic resonant response and establish that the design problem can be reduced to four dimensionless parameters. We demonstrate the approach for the numerical example of a cantilevered beam with a spherical impact absorber. We conclude that the proposed semi-analytical procedure enables deep qualitative understanding of the problem and, at the same time, yields a quantitatively accurate prediction of the optimum design.