Elastic hyperbolic strip lattices

We investigate the dynamic properties of elastic lattices defined by tessellations of a hyperbolic strip domain. These strip lattices are generated using a conformal map of tessellations of a hyperbolic disk. The resulting hyperbolically distributed lattice sites are coupled by Euclidean edges, leading to a natural grading of coupling strengths along the strip width. Their vibrational modes are organized into three distinct classes: boundary localized, interior localized, and global. This mode classification is governed by a computed localization index quantifying the spatial localization of each mode along the strip's width. We show that, like hyperbolic lattices in the disk, hyperbolic lattices in the strip exhibit a dynamic spectrum populated by a majority of localized modes. This finding is supported by numerical studies of the dynamics of a strip lattice of structural beams. Computation of the integrated density of states for the boundary, interior, and global modes reveals the predominance of localized modes, and the local density of states allows for the identification of spectral bands dominated by particular mode classes. This analysis informs time-domain simulations of the lattice response to dynamic forcing by bandlimited inputs dominated by each mode class. The results illustrate distinctive wave-propagation behavior when the excited frequency band is dominated by boundary-localized, interior-localized, or global modes. We confirm these observations via vibrometry experiments in the frequency and time domains. In the frequency domain, the measured response confirms that the spectral neighborhoods of each excitation are indeed populated by the mode class predicted by numerical investigations. We further show that the time-averaged responses are consistent with simulations. Through this work, elastic hyperbolic strips emerge as a previously unexplored class of lattices with characteristic beam-like, truss-core architectures and as-yet unseen site arrangements. The considered configuration shows promising capabilities to confine and guide elastic waves along varying spatial regions depending on the frequency content of the excitation.