Abstract

Reciprocity, a fundamental property of linear, time-invariant systems, implies that wave transmission characteristics between two points in a material or structure remain unchanged upon interchanging the locations of the source and receiver. Thus, reciprocal systems cannot have direction-dependent transmission properties. One way to overcome this limitation is to utilize nonlinearity.

There has been a surge of interest in the past two decades in nonlinear nonreciprocity in the context of phononic crystals, metamaterials and lattice materials. This thesis contributes to this body of knowledge by providing a detailed account of three nonreciprocal transmission regimes: energy-preserving, phase-preserving and unilateral transmission. This computational investigation is focused exclusively on the steady-state response of spatially periodic systems to external harmonic excitation.

The most salient indicator of nonreciprocity is the ability of a system to support unidirectional transmission. This occurs when there is a large difference between the energies transmitted in opposite directions. This energy bias is accompanied by a difference in the phase of the transmitted vibrations. The role of the phase bias in nonreciprocity has primarily been overlooked in the literature. To highlight the role of phase, we consider two limiting cases. We demonstrate the existence of response regimes in which the energy bias is zero and nonreciprocity is solely caused by the phase bias. Moreover, we show that energy bias alone, without any contribution from phase, can still lead to nonreciprocity, but only at very finely tuned system parameters. We provide methodologies for systematically realizing response regimes of energy-preserving and phase-preserving nonreciprocity.

Furthermore, we investigate unilateral transmission, a phenomenon in which transmitted vibrations remain purely in tension or compression. We investigate unilateral transmission in a system with different effective elasticity in compression and tension. We show that breaking the mirror symmetry of the system in either the elastic or inertial properties enables unilateral transmission to occur near the primary resonances.

This dissertation advances the understanding of nonlinear nonreciprocity in vibration systems, with a focus on three response regimes that are characterized by distinct dynamic features. These findings provide new insights into the design of nonlinear waveguides and mechanical systems with tunable nonreciprocal properties.