Band gap synthesis in elastic monatomic lattices via input shaping

This work describes control-theoretic methods for inducing a band gap-like behavior in elastic monatomic lattices. The dynamics of the system under consideration are derived in detail. Open-loop pre-filtering techniques, in the form of Posicast and user-selected time delay filters, are then utilized to eliminate specific frequency contents from the response of the multi-degree of freedom spring-mass chain. The input shaping approach is used to synthesize one or more band gaps in a homogenous lattice that can be designed to resemble Bragg-scattering in phononic crystals, as well as local resonance effects in acoustic metamaterials. The presence of the synthesized band gaps in the lattice’s dispersion behavior is validated using a spatiotemporal Fourier transform of the system’s impulse response as well as the frequency response of the end-to-end transfer function. Finally, an analysis of the control effort required to cancel the targeted system poles is carried out.