Design and analysis of spatially symmetric multi-dimensional delayed resonators

Abstract

The contribution of the paper is three-fold. First, a novel, spatially symmetric two-dimensional vibration absorber is proposed. Structural conditions are given for unified frequencies and damping ratios, which lead to a one-parameter family of absorbers. Local proportional delayed feedback control laws are then derived that turn the absorber into an ideal absorber with assignable resonance frequency, which generalizes the delayed resonator. Second, it is proven that the time-averaged power requirement of the actuators in operation, deployed to compensate damping and adjust the resonance frequency to the one of a harmonic excitation force, as well as the stability of the controlled absorber, do not depend on the geometry of the unified absorber. However, the power distribution over the links strongly depend on the geometry, as illustrated with a comparison between the new symmetric configuration and a previously proposed asymmetric one. Third, the generalization to spatially symmetric three-dimensional absorbers is addressed. Among other results, it is shown that all admissible spatial configurations for unified frequencies and damping ratios are identical up to reflection and rotation, in sharp contrast to the two-dimensional setting.