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 for the case where the host structure is well described as 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 known coefficient of restitution. When a flexible host structure is considered instead, the impact absorber not only dissipates energy at the time instances of impact, but 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 reonantly-driven, flexible host structures. We demonstrate decoupling of the time scales of the impact and the resonant vibration. On the long time scale, the dynamics can be properly reduced to the fundamental harmonic of the resonant mode. A light impact absorber responds to this enforced motion, and we recover the Slow Invariant Manifold of the dynamics for the regime of two impacts per period. On the short time scale, 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 adequate pre-impact velocity. From this simulation, we derive a modal coefficient of restitution and the properties of the contact force pulse, needed to approximate the behavior on the long time scale. We establish that the design problem can be reduced to four dimensionless parameters and demonstrate the approach for the numerical example of a cantilevered beam with an 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.