Nonreciprocal vibrations of discretized finite elastic structures with spatiotemporally modulated material properties

Elastic waveguides with time- and space-dependent material properties have received great attention as a means to realize nonreciprocal propagation of small-amplitude mechanical waves in unbounded elastic media. Previous works have shown that propagating waves in a modulated medium violate reciprocity by means of asymmetric frequency and wave number conversion between two counterpropagating modes. We previously investigated nonreciprocal longitudinal and transverse vibrations in a finite elastic waveguide with time- and space-dependent material properties using a semi-analytical coupled-mode theory The model employs the mode shapes of the non-modulated beam as the orthogonal basis functions to describe the nonreciprocal vibrations of beams with spatiotemporally modulated properties. The present study extends previous work to consider vibrations of Euler beams with spatially discretized stiffness that are modulated in time and phased relative to each other to mimic spatiotemporal modulation. This is achieved by approximating the discretized stiffness and then employing the mode shapes of the beam with unmodulated discretized stiffness to predict the nonreciprocal vibrational response of the beam when subjected to spatiotemporal modulation. We perform a parametric study of the system parameters and discuss how this model can be used to define requirements of future systems that can be used for experimental demonstration.