Extreme Localization via a Singular Edge Mode in an Asymmetric Linkage System

Abstract

Mechanical localized modes generally retain exponentially decaying tails, whereas confinement to a single structural unit is difficult to achieve without resonators, flat-band lattices or topological design. To address this need, an asymmetric linkage system is introduced in which geometry tunes a boundary mode from finite localization to its extreme limit. Each cell comprises two unequal rigid bars connected by a torsional spring and is described by one generalized coordinate. Linearization and coordinate transformation yield a tridiagonal model and closed-form expressions for the edge mode frequency and spatial decay ratio. The analysis identifies the configurations supporting a singular edge mode at which its decay ratio vanishes, leaving only the boundary cell active in the linear regime. Simulations show near-complete localization under finite-amplitude excitation, and experiments reproduce the configuration-dependent transition from bulk transmission to boundary confinement. These results establish a minimal and analytically tunable platform for mechanical localization