  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.
