Computation of response envelopes in a lattice material with spatiotemporal modulations

Abstract

Materials with spatiotemporal modulations possess effective properties that vary in space and time periodically. Because of the wave-like properties of a modulated material, propagation of incident waves through the material depends on their directions of travel. This direction-dependent transmission occurs due to a scattering effect that is caused because of the modulations. If the modulated material is long enough, it can act as a unidirectional wave isolator, preventing waves from propagating in one direction. However, such unidirectional transmission does not occur in very short modulated materials, i.e. two degrees of freedom (2 DOF) systems. In this work, we study nonreciprocal vibration transmission in a discrete onedimensional (1-D) modulated material, with a focus on computing the change in the amplitude and phase of transmitted vibrations along opposite directions. Because the response of a modulated system is not periodic in time, this process requires either brute force computations or asymptotic analysis with a limited range of validity. To overcome this shortcoming, we develop and utilize the envelopes of the steady-state output displacements to investigate nonreciprocity. Furthermore, we highlight the application of envelope equations in identifying nonreciprocal response regimes characterized by a nonreciprocal phase shift in transmitted vibrations. The role of the length of 1-D modulated materials on determining the amplitude difference is highlighted. The analysis method based on envelopes of the steady-state response facilitates future parametric studies on nonreciprocity in discrete modulated materials.