Non-Reciprocal Wave Transmission in a Bilinear Spring-Mass System

Abstract

Significant amplitude-independent and passive non-reciprocal wave motion can be achieved in a one dimensional (1D) discrete chain of masses and springs with bilinear elastic stiffness. Some fundamental asymmetric spatial modulations of the bilinear spring stiffness are first examined for their non-reciprocal properties. These are combined as building blocks into more complex configurations with the objective of maximizing non-reciprocal wave behavior. The non-reciprocal property is demonstrated by the significant difference between the transmitted pulse displacement amplitudes and energies for incidence in opposite directions. Extreme non-reciprocity is realized when almost-zero transmission is achieved for the propagation from one direction with a noticeable transmitted pulse for incidence from the other. These models provide the basis for a class of simple 1D non-reciprocal designs and can serve as the building blocks for more complex and higher dimensional non-reciprocal wave systems.Bilinear springs present a unique case of nonlinearity, consisting of two different linear load-deformation relations. Unlike other nonlinearities, such as cubic [12,13], the bilinear relation is amplitude-independent; the nonlinearity enters only through the sign of the displacement. The analogous phenomenon in continuum mechanics occurs in materials with bilinear (also known as heteromodular or bimodular) constitutive elastic behavior, which have been proposed as nonlinear models for studying contact forces [14], elastic solids containing cracks [15] and for the dynamics of geophysical systems, including granular media [16]. The discontinuity of the piecewise linear relation gives rise to a strong nonlinearity, for which it is difficult to find analytical solutions for simple wave problems. Wave motion in bimodular media has been studied extensively [17,18,19,20,21,22,23,24,25,26]. Even a small difference between the moduli in tension and compression immediately causes the appearance of shock waves [22]; however linear viscosity eliminates the shocks. A good review of the literature of wave motion in continuous bimodular media, particularly the considerable work done by Russian researchers, can be found in [22], while [27, p. 32] provides an earlier review. There are far fewer studies of wave motion in discrete spring-mass chains with bilinear spring forces. Of particular interest is the study [16] which analyzed impulse harmonic wave propagation in a 1D system of bilinear oscillators. Although they did not emphasize non-reciprocal effects, the authors noticed that sign inversion of a signal can be obtained, from tension to compression that can lead to pulse spreading or shortening and possible shock formation. However, none of these prior studies of either continuous or discrete systems considered spatial inhomogeneity and asymmetry, which are necessary for producing non-reciprocal wave motion in the presence of material nonlinearity.Here we leverage the bilinear property to break wave reciprocity in a simple mechanical struct...

Unilateral and nonreciprocal transmission through bilinear spring systems

“…With the same incident wave of (12), we have for the interval τ ∈ (τ j , τ j+1 ), j ∈ N + , that T = f − R and R = sin(τ + θ) − e −(τ −τj) tan θ sin(τ j + θ) cos θ. (18) Note that θ of eq. (15) depends on the sign of R through eq.…”

“…To realize the geometric asymmetry, we simply use a bilinear and a linear spring in series and connected by a mass. This idea comes from our previous bilinear spring-mass chain system with spatial stiffness modulation in [18], showing nonreciprocal pulse propagation. Additionally, we present an interesting reciprocal phenomenon of an oscillatory incident wave converted into a transmitted wave with particle displacement of a single sign.…”

Spacetime modulation in floating thin elastic plates

“…This is the analog of time-modulation of the permittivity in photonic crystals [45] by allowing a periodic evolution of the flexural rigidity, i.e., D(t + T ) = D(t), with a temporal period T . Regarding a possible experiment, in order to change the elastic parameters directly, one can add shunted external circuits, with tunable electrical properties (resistance, capacitance, and inductance) (see SM [43] for more details on the piezo-electric potential modelling [46][47][48][49][50][51][52]). Equation ( 1) shall be re-written in time-domain and in the one-dimensional (1D) scenario…”

Упругие Волны В Средах С Разномодульной Нелинейностью С Учетом Эффектов Отражения От Ударных Фронтов

“…Изучению нелинейного распространения продольных упругих волн в разномодульных твердых телах без дисперсии (и без учета отражения от разрывов) посвящено довольно много работ [8][9][10][11][12][13][14][15][16][17]. В таких средах нелинейный режим распространения имеет место только для разнополярных волн, а однополярные одиночные импульсы распространяются линейно, с постоянными, но различными скоростями, зависящими от их полярности.…”