A Green’s function approach for nonreciprocal vibrations of finite elastic structures with spatiotemporally modulated material properties

The spatiotemporal modulation of material properties in acoustic and elastic metamaterials is one way to realize dynamic nonreciprocity for mechanical wave propagation and vibration of finite systems. A recent study on nonreciprocal vibrations of finite elastic beams has shown that strong nonreciprocal input-output responses resulting from the spatiotemporal modulation only occur for a small subset of modulation parameters, which may complicate the design of nonreciprocal systems [Goldsberry et al., Phys. Rev. B 102 (2020)]. In this work, we derive a Green’s function approach for finite elastic systems with spatiotemporally modulated material properties, which mathematically describes the modulated system response for a given set of boundary conditions. As a case example, we investigate two-dimensional flexural vibrations of a plate whose stiffness is modulated in a traveling wave form. We then investigate the first order corrections to the unmodulated system by utilizing an asymptotic expansion in the small parameter defining the modulation strength. Finally, we find conditions on the material and modulation parameters that yield a large degree of nonreciprocity. The present analysis leads to potential applications in acoustic communications, vibration suppression, and energy harvesting. [Work supported by NSF EFRI.]