Unilateral and nonreciprocal transmission through bilinear spring systems

Abstract

Longitudinal wave propagation is considered in a pair of waveguides connected by bilinear spring systems. The nature of the nonlinearity causes the compressive and tensile force-displacement relations of the bilinear spring to behave in a piece-wise linear manner, and all transmitted and reflected waves scale linearly with the incident wave amplitude. We first concentrate on a single bilinear spring connecting two waveguides. By controlling the bilinear stiffness parameters it is possible to convert a time harmonic incident wave into a transmitted wave of the same period but with particle displacement of a single sign, positive or negative, an effect we call unilateral transmission. Nonreciprocal wave phenomena are obtained by introducing spatial asymmetry. A simple combination of a single bilinear spring with a mass and a linear spring shows significant nonreciprocity with transmission relatively high in one direction and low in the opposite direction.

Two Masses in Relative Hooke’s Potential and Elliptic Integrals

“…Equivalent form of the formula for energy is the following Now, let us determine the remaining roots r 3 , r 4 of the quartic (17). The Vieta formulas and (18) lead to following formulas:…”

“…In theoretical mechanics vibrating systems are of particular importance (cf. [2,9,15,17]). There is a big variety of such systems, which constitute models for various sometimes very complicated processes.…”

Computation of phase nonreciprocity in a nonlinear periodic system

“…Here are some comprehensive reviews on nonreciprocity in electromagnetics [10], electronics [11], and fluid dynamics [12]. The breakdown of reciprocity might take place in systems with time-dependent (active) [13] or nonlinear (passive) [14][15][16] material properties. Our focus in this research is on the passive approach, particularly on nonreciprocity caused by nonlinear elasticity: when the local relation between the restoring force and deformation is not linear.…”

“…Boechler et al [19] used a granular crystal composed of a one-dimension array of spherical beads to achieve nonreciprocal transmission thresholds. Lu and Norris [20] used bilinear stiffness to design an amplitude-independent diode-like waveguide. Their design caused nonreciprocity which urged the system to transmit waves with different patterns from different directions without dependence on the wave amplitude.…”

Nonreciprocal and directional wave propagation in a two-dimensional lattice with bilinear properties

“…The energetic approaches fall into two types of methods: introducing moving parts or circulating flows in the propagation medium [2,3], and performing spatial-temporal modulations of the system properties [4][5][6][7][8][9][10]. The passive methods typically make use of various nonlinear mechanical properties and the necessary ingredient of spatial asymmetry [11][12][13][14][15][16][17]. Many researchers have successfully realized breaking reciprocity in 1D domain via active and passive approaches, demonstrating effects such as one-way acoustic and elastic wave propagation [1, 4-8, 15, 16, 18], asymmetric energy transfer [1,10,13,17] and nonreciprocal phase shift [1,3].…”