The study of phononic materials and structures is an emerging discipline that lies at the crossroads of vibration and acoustics engineering and condensed matter physics. Broadly speaking, a phononic medium is a material or structural system that usually exhibits some form of periodicity, which can be in the constituent material phases, or the internal geometry, or even the boundary conditions. As such, its overall dynamical characteristics are compactly described by a frequency band structure, in analogy to an electronic band diagram. With roots extended to early studies of periodic systems by Newton and Rayleigh, the field has grown to encompass engineering configurations ranging from trusses and ribbed shells to phononic crystals and metamaterials. While applied research in this area has been abundant in recent years, treatment from a fundamental mechanics perspective, and particularly from the standpoint of dynamical systems, is needed to advance the field in new directions. For example, techniques already developed for the incorporation of damping and nonlinearities have recently been applied to wave propagation in phononic materials and structures. Similarly, numerical and experimental approaches originally developed for the characterization of conventional materials and structures are now being employed toward better understanding and exploitation of phononic systems. This article starts with an overview of historical developments and follows with an in-depth literature and technical review of recent progress in the field with special consideration given to aspects pertaining to the fundamentals of dynamics, vibrations, and acoustics. Finally, an outlook is projected onto the future on the basis of the current trajectories of the field.

Wave propagation in one-dimensional nonlinear periodic structures is investigated through a novel perturbation analysis and accompanying numerical simulations. Several chain unit cells are considered featuring a sequence of masses connected by linear and cubic springs. Approximate closed-form, first-order dispersion relations capture the effect of nonlinearities on harmonic wave propagation. These relationships document amplitude-dependent behavior to include tunable dispersion curves and cutoff frequencies, which shift with wave amplitude. Numerical simulations verify the dispersion relations obtained from the perturbation analysis. The simulation of an infinite domain is accomplished by employing viscous-based perfectly matched layers appended to the chain ends. Numerically estimated wavenumbers show good agreement with the perturbation predictions. Several example chain unit cells demonstrate the manner in which nonlinearities in periodic systems may be exploited to achieve amplitude-dependent dispersion properties for the design of tunable acoustic devices.

Enhancement of structure-borne wave energy harvesting is investigated by exploiting metamaterial-based and metamaterial-inspired electroelastic systems. The concepts of wave focusing, localization, and funneling are leveraged to establish novel metamaterial energy harvester (MEH) configurations. The MEH systems transform the incoming structure-borne wave energy into electrical energy by coupling the metamaterial and electroelastic domains. The energy harvesting component of the work employs piezoelectric transduction due to the high power density and ease of application offered by piezoelectric materials. Therefore, in all MEH configurations studied in this work, the metamaterial system is combined with piezoelectric energy harvesting for enhanced electricity generation from waves propagating in elastic structures. Experiments are conducted to validate the dramatic performance enhancement in MEH systems as compared to using the same volume of piezoelectric patch in the absence of the metamaterial component. It is shown that MEH systems can be used for both broadband and tuned wave energy harvesting. The MEH concepts covered in this paper are (1) wave focusing using a metamaterial-inspired parabolic acoustic mirror (for broadband energy harvesting), (2) energy localization using an imperfection in a 2D lattice structure (for tuned energy harvesting), and (3) wave guiding using an acoustic funnel (for narrow-to-broadband energy harvesting). It is shown that MEH systems can boost the harvested power by more than an order of magnitude.

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In this study, a dynamic finite element model is developed for pulley belt-drive systems and is employed to determine the transient and steady-state response of a prototypical belt-drive. The belt is modeled using standard truss elements, while the pulleys are modeled using rotating circular constraints, for which the driver pulley’s angular velocity is prescribed. Frictional contact between the pulleys and the belt is modeled using a penalty formulation with frictional contact governed by a Coulomb-like tri-linear friction law. One-way clutch elements are modeled using a proportional torque law supporting torque transmission in a single direction. The dynamic response of the drive is then studied by incorporating the model into an explicit finite element code, which can maintain time-accuracy for large rotations and for long simulation times. The finite element solution is validated through comparison to an exact analytical solution of a steadily-rotating, two-pulley drive. Several response quantities are compared, including the normal and tangential (friction) force distributions between the pulleys and the belt, the driven pulley angular velocity, and the belt span tensions. Excellent agreement is found. Transient response results for a second belt-drive example involving a one-way clutch are used to demonstrate the utility and flexibility of the finite element solution approach.

This work investigates the effect of nonlinearities on topologically protected edge states in one- and two-dimensional phononic lattices. We first show that localized modes arise at the interface between two spring-mass chains that are inverted copies of each other. Explicit expressions derived for the frequencies of the localized modes guide the study of the effect of cubic nonlinearities on the resonant characteristics of the interface, which are shown to be described by a Duffing-like equation. Nonlinearities produce amplitude-dependent frequency shifts, which in the case of a softening nonlinearity cause the localized mode to migrate to the bulk spectrum. The case of a hexagonal lattice implementing a phononic analog of a crystal exhibiting the quantum spin Hall effect is also investigated in the presence of weakly nonlinear cubic springs. An asymptotic analysis provides estimates of the amplitude dependence of the localized modes, while numerical simulations illustrate how the lattice response transitions from bulk-to-edge mode-dominated by varying the excitation amplitude. In contrast with the interface mode of the first example studies, this occurs both for hardening and softening springs. The results of this study provide a theoretical framework for the investigation of nonlinear effects that induce and control topologically protected wave modes through nonlinear interactions and amplitude tuning.

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The paper investigates wave dispersion in two-dimensional, weakly nonlinear periodic lattices. A perturbation approach, originally developed for one-dimensional systems and extended herein, allows for closed-form determination of the effects nonlinearities have on dispersion and group velocity. These expressions are used to identify amplitude-dependent bandgaps, and wave directivity in the anisotropic setting. The predictions from the perturbation technique are verified by numerically integrating the lattice equations of motion. For small amplitude waves, excellent agreement is documented for dispersion relationships and directivity patterns. Further, numerical simulations demonstrate that the response in anisotropic nonlinear lattices is characterized by amplitude-dependent “dead zones.”