Elastic/acoustic metamaterials made from locally resonant arrays can exhibit bandgaps at wavelengths much longer than the lattice size for various applications spanning from low-frequency vibration/sound attenuation to wave guiding and filtering in mechanical and electromechanical devices. For an effective use of such locally resonant metamaterial concepts in finite structures, it is required to bridge the gap between the lattice dispersion characteristics and modal behavior of the host structure with its resonators. To this end, we develop a novel argument for bandgap formation in finite-length elastic metamaterial beams, relying on the modal analysis and the assumption of infinitely many resonators. We show that the dual problem to wave propagation through an infinite periodic beam is the modal analysis of a finite beam with an infinite number of resonators. A simple formula that depends only on the resonator natural frequency and total mass ratio is derived for placing the bandgap in a desired frequency range, yielding an analytical insight and a rule of thumb for design purposes. A method for understanding the importance of a resonator location and mass is discussed in the context of a Riemann sum approximation of an integral, and a method for determining the optimal number of resonators for a given set of boundary conditions and target frequency is introduced. The simulations of the theoretical framework are validated by experiments for bending vibrations of a locally resonant cantilever beam. Published by AIP Publishing.

Locally resonant metamaterials are characterized by bandgaps at wavelengths that are much larger than the lattice size, enabling low-frequency vibration attenuation. Typically, bandgap analyses and predictions rely on the assumption of traveling waves in an infinite medium, and do not take advantage of modal representations typically used for the analysis of the dynamic behavior of finite structures. Recently, we developed a method for understanding the locally resonant bandgap in uniform finite metamaterial beams using modal analysis. Here we extend that framework to general (potentially nonuniform, 1D or 2D) locally resonant metastructures with specified boundary conditions using a general operator formulation. Using this approach, along with the assumption of an infinite number of resonators tuned to the same frequency, the frequency range of the locally resonant bandgap is easily derived in closed form. Furthermore, the bandgap expression is shown to be the same regardless of the type of vibration problem under consideration, depending only on the added mass ratio and target frequency. For practical designs with a finite number of resonators, it is shown that the number of resonators required for the bandgap to appear increases with the target frequency range, i.e. respective modal neighborhood. Furthermore, it is observed that there is an optimal, finite number of resonators which gives a bandgap that is wider than the infinite-absorber bandgap, and that the optimal number of resonators increases with target frequency and added mass ratio. As the number of resonators becomes sufficiently large, the bandgap converges to the derived infinite-absorber bandgap. Additionally, the derived bandgap edge frequencies are shown to agree with results from dispersion analysis using the plane wave expansion method. The model is validated experimentally for a locally resonant cantilever beam under base excitation. Numerical and experimental investigations are performed regarding the effects of mass ratio, non-uniform spacing of resonators, and parameter variations among the resonators.

Locally resonant electromechanical metastructures made from flexible substrates with piezoelectric layers connected to resonant shunt circuits exhibit vibration attenuation properties similar to those of purely mechanical metastructures. Thus, in analogy, these locally resonant electromechanical metastructures can exhibit electroelastic bandgaps at wavelengths much larger than the lattice size. In order to effectively design such metastructures, the modal behavior of the finite structure with given boundary conditions must be reconciled with the electromechanical behavior of the piezoelectric layers and shunt circuits. To this end, we develop the theory for a piezoelectric bimorph beam with segmented electrodes under transverse vibrations, and extract analytical results for bandgap estimation using modal analysis. Under the assumption of an infinite number of segmented electrodes, the locally resonant bandgap is estimated in closed form and shown to depend only on the target frequency and the system-level electromechanical coupling. It is shown that bandgap formation in piezoelectric metastructures is associated with a frequency-dependent modal stiffness, unlike the frequency-dependent modal mass in mechanical metastructures. The relevant electromechanical coupling term and the normalized bandgap size are calculated for a representative structure and a selection of piezoelectric ceramics and single crystals, revealing that single crystals (e.g. PMN-PT) result in significantly wider bandgap than ceramics (e.g. PZT-5A). Numerical studies are performed to demonstrate that the closed-form bandgap expression derived in this work holds for a finite number of electrode segments. It is shown that the number of electrodes required to create the bandgap increases as the target frequency is increased.

Vibration-based energy harvesting is a growing field for generating low-power electricity to use in wireless electronic devices, such as the sensor networks used in structural health monitoring applications. Locally resonant metastructures, which are structures that comprise locally resonant metamaterial components, enable bandgap formation at wavelengths much longer than the lattice size, for critical applications such as low-frequency vibration attenuation in flexible structures. This work aims to bridge the domains of energy harvesting and locally resonant metamaterials to form multifunctional structures that exhibit both low-power electricity generation and vibration attenuation capabilities. A fully coupled electromechanical modeling framework is developed for two characteristic systems and their modal analysis is presented. Simulations are performed to explore the vibration and electrical power frequency response maps for varying electrical load resistance, and optimal loading conditions are presented. Case studies are presented to understand the interaction of bandgap formation and energy harvesting capabilities of this new class of multifunctional energyharvesting locally resonant metastructures. It is shown that useful energy can be harvested from locally resonant metastructures without significantly diminishing their dramatic vibration attenuation in the locally resonant bandgap. Thus, integrating energy harvesters into a locally resonant metastructure enables a new potential for multifunctional locally resonant metastructures that can host self-powered sensors.

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