Unilateral vibration transmission in mechanical systems with bilinear coupling

TL;DR

This study addresses unilateral vibration transmission in bilinearly coupled mechanical systems under harmonic forcing, showing that symmetry breaking via mass or stiffness imbalance enables pure-tension or pure-compression transmission near resonances. The authors model a 2DoF system with bilinear coupling, regularize the stiffness with $K_c = \frac{k(\beta-1)}{\pi}\arctan(B(x_2-x_1)) + \frac{k(\beta+1)}{2}$ (with $B=1000$) and use numerical continuation to compute steady-state periodic responses, assessing stability through Floquet multipliers. Transmission and nonreciprocity are quantified by the unilateral ratio $R_u = |C|/|A|$ and the reciprocity bias $R$, revealing that symmetry-breaking parameters $r$ and $\mu$ place unilateral transmission near primary resonances and yield direction-dependent harmonic content. The analysis extends to a periodic lattice of bilinearly coupled units, showing that the number of units and energy dissipation shape the persistence of unilateral transmission, with low damping in long chains capable of restoring unilateral behavior; these findings offer design guidance for directional vibration control in engineering systems.

Abstract

Unilateral transmission refers to the scenario in which the waves transmitted through a system remain in pure tension or pure compression. This transmission phenomenon may occur in systems that exhibit different effective elasticity in compression and tension; i.e. bilinear elasticity. We present a computational investigation of unilateral transmission in the steady-state response of harmonically driven mechanical systems with bilinear coupling. Starting with two bilinearly coupled oscillators, we find that breaking the mirror symmetry of the system, in either elastic or inertial properties, facilitates unilateral transmission by allowing it to occur near a primary resonance. This asymmetry also enables nonreciprocal transmission to occur. We then investigate the nonreciprocal dynamics of the system, including linear stability analysis, with a focus on unilateral transmission. We also extend our discussion to a bilinear periodic structure, for which we investigate the influence of the number of units and energy dissipation on unilateral transmission. We report on the existence of stable nonreciprocal unilateral transmission near primary and internal resonances of the system, as well as other nonreciprocal features such as period-doubled and quasiperiodic response characteristics.

Figures (14)

• Figure 1: Schematic of the 2DoF system consists of two linear damped oscillators coupled with a bilinear spring.
• Figure 2: Periodic orbits represented by $A$ and $C$ (a) non-unilateral (b) unilateral
• Figure 3: (a) Frequency response of the symmetric 2DoF ($r=\mu=1$) system for the forward configuration. Thick lines indicate response that exhibits unilateral transmission and dashed lines indicate unstable response. Diamond markers indicate period-doubling bifurcation points. The upper right inset shows the unilateral time response at the frequency highlighted by the black square. (b) Unilateral transmission $R_u$ as a function of forcing frequency.
• Figure 4: Locus of the onset of unilateral transmission $R_u=1$ as a function of the stiffness ratio $r$ for (a) Output amplitude (b) Forcing frequency.
• Figure 5: (a) Frequency response of the asymmetric system $r=0.28$ for the onset of unilateral transmission at the peak frequency (b) Frequency response of the asymmetric system $r=0.25$ for the onset of unilateral transmission before the peak frequency