Christopher Sugino

On the mechanism of bandgap formation in locally resonant finite elastic metamaterials

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.

A general theory for bandgap estimation in locally resonant metastructures

Abstract 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 locally resonant 1D and 2D metastructures (i.e. locally resonant metamaterial-based finite structures) 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 increased target frequency, i.e. more resonators are required for higher vibration modes. Additionally, it is observed that there is an optimal, finite number of resonators which gives a bandgap that is wider than the infinite-resonator 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-resonator bandgap. Furthermore, 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.

Adaptive locally resonant metamaterials leveraging shape memory alloys

Locally resonant metamaterials leveraging shape memory alloy (SMA) springs are explored in this work in an effort to develop adaptive metamaterial configurations that can exhibit tunable bandgap properties as well as enhanced damping capabilities. An analytical model for a locally resonant metamaterial beam in transverse vibrations is combined with an SMA model for the resonator springs to investigate and leverage the potential of temperature-induced phase transformations and stress-induced hysteretic behavior of the springs. Two case studies are presented for this new class of smart metamaterials and the resulting finite metastructures. In one case, SMA resonators operate in the linear elastic regime, first at low temperature (martensitic behavior) and then at high temperature (austenitic behavior), demonstrating how the bandgap can be tuned to a different frequency range by altering the SMA elastic modulus with temperature. In the second case, the SMA springs are kept at high temperature at all times to operate in the nonlinear regime, so that the hysteresis associated with the SMA pseudoelastic effect is manifested, yielding additional dissipation over a range of frequencies, especially for the modes right outside the bandgap.Locally resonant metamaterials leveraging shape memory alloy (SMA) springs are explored in this work in an effort to develop adaptive metamaterial configurations that can exhibit tunable bandgap properties as well as enhanced damping capabilities. An analytical model for a locally resonant metamaterial beam in transverse vibrations is combined with an SMA model for the resonator springs to investigate and leverage the potential of temperature-induced phase transformations and stress-induced hysteretic behavior of the springs. Two case studies are presented for this new class of smart metamaterials and the resulting finite metastructures. In one case, SMA resonators operate in the linear elastic regime, first at low temperature (martensitic behavior) and then at high temperature (austenitic behavior), demonstrating how the bandgap can be tuned to a different frequency range by altering the SMA elastic modulus with temperature. In the second case, the SMA springs are kept at high temperature at all times to...

Locally resonant metamaterials with shape-memory alloy springs

Locally resonant metamaterials offer bandgap formation for wavelengths much longer than the lattice size, en- abling low-frequency and wideband vibration attenuation. Acoustic/elastic metamaterials made from resonating components usually do not exhibit reconfigurable and adaptive characteristics since the bandgap frequency range (i.e. target frequency and bandwidth combination) is fixed for a given mass ratio and stiffness of the resonators. In this work, we explore locally resonant metamaterials that exploit shape-memory alloy springs in an effort to develop adaptive metamaterials that can exhibit tunable bandgap properties. An analytical model for locally res- onant metastructures (i.e. metamaterials with specific boundary conditions) is combined with a shape-memory spring model of the resonator springs to investigate and exploit the potential of temperature-induced phase transformations and stress-induced hysteretic behavior of the springs. Various case studies are presented for this new class of smart metamaterials and metastructures.

Dispersion tailoring in varying-inductance piezoelectric metamaterials

Inductive shunt circuits have thus far been explored mainly for low-frequency bandgap formation in locally reso- nant piezoelectric metamaterials. Other than the well-studied bandgap phenomenon, the substantial sensitivity of the dispersion curves to variations in the target frequency (i.e. to variations in the inductance value) right below or above the bandgap offers a very rich potential for applications ranging from on-demand spatial tailoring of the refractive index profile to dynamic stiffness modification for leveraging in novel problems of wave propa- gation with spatial and temporal property modulation. As a specific instance, if the unit cells of a metamaterial are shunted to resonate at gradually varying frequencies above the design frequency, one can achieve a smooth variation of both group velocity and phase velocity in space for wavelengths much longer than the lattice size, as a low-frequency electromechanical gradient-index metamaterial. In this work, we explore flexural waves in a one-dimensional piezoelectric metamaterial with unit cells that are shunted to inductive circuits of varying inductance values. Unit cell dispersion characteristics for an infinite metamaterial are studied to demonstrate various phenomena, such as the modification of the phase velocity and the refractive index. Specifically, case studies are given for the formation of a hyperbolic secant refractive index profile to enable plane wave focusing. The effects of dissipation and frequency variation are also studied, revealing that the proposed concepts can enable significant refractive index variation even in the presence of damping (e.g. sufficient for lens design). The advantages of this approach span from low-frequency gradient-index metamaterial design to stiffness modulation beyond the limits of short- and open-circuit values even in the absence of a negative capacitance circuit.

Metamaterial piezoelectric beam with synthetic impedance shunts

We present a metamaterial beam based on a piezoelectric bimorph with segmented electrodes. Previously, we found the theoretical response of the beam using the assumed-modes method, and derived the effect of the shunt circuit impedance applied to each pair of electrodes. The structural response is governed by a frequency- dependent stiffness term, which depends on a material/geometry-based electromechanical coupling parameter and the impedance of the shunt circuits. A simple way to interpret the response of the system with frequency- dependent stiffness is the root locus method, which immediately yields the poles of each mode of the system using simple geometric rules. Case studies are shown for creating locally resonant bandgap with or without negative capacitance. To justify the use of these admittances that often require power input to the system, the concept of synthetic impedance is extended to symmetric voltages, as are encountered in series-connected piezoelectric bimorphs. Synthetic impedance or admittance is a method for obtaining an arbitrary impedance across a load by measuring the voltage and applying the corresponding current using digital signal processing and an analog circuit. Time domain simulations using these synthetic impedance circuits are compared to the ideal frequency domain results with good agreement. Surprisingly, the necessary digital sampling rate for stability is significantly higher than the Nyquist frequency.

Toward structurally integrated locally resonant metamaterials for vibration attenuation

In this contribution, we explore the use of locally resonant metamaterials for multi-functional structural load- bearing concepts using analytical, numerical, and experimental techniques. Locally resonant metamaterials exhibit bandgaps at wavelengths much larger than the lattice dimension. This is a promising feature for low- frequency vibration attenuation. The presented work aims to investigate highly integrated structural concepts and experimentally validated prototypes for vibration reduction in load-bearing applications. The goal is to explore and extend the design space of lightweight structural systems, by designing multi-functional periodic structural elements, preserving structural stiffness while concurrently enabling sufficiently wideband damping performance over a target frequency range of interest. Following a generalized theoretical modeling framework for bandgap design and analysis in finite structures, the focus is placed on the design, fabrication, and analysis of a load-carrying frame development with internally resonant components. Finite-element modeling is employed to design and analyze the frequency response of the frame and simplified analytical solution is compared with this numerical solution. Experimental validations are presented for a 3D-printed prototype. The effects of various parameters are reported both based on numerical and experimental findings.

Dramatic effect of fluid damping on the performance of a nonlinear M-shaped broadband energy harvester

This work aims to demonstrate the detrimental effect of fluid damping on the bandwidth of a flexible nonlinear energy harvester and thereby further enhance the performance by minimizing nonlinear damping. A vacuum setup has been introduced to conduct nonlinear base excitation experiments at different air pressure levels in an effort to control the quadratic (velocity-squared) damping coefficient. It is shown that reduced air pressure substantially enhances the frequency bandwidth for primary resonance excitation. The empirical electromechanical model is modified to express the fluid damping in terms of fluid pressure and validated experimentally for different excitation levels.