Localized Resonance Research Articles

A novel broadband underwater sound absorption metastructure with multi-oscillators

Underwater absorptive metamaterials are rapidly developed duo local resonance mechanisms, but they still suffer from the disadvantages of poor hydrostatic resistance. The poor acoustic absorption performance at high hydrostatic pressure mainly attributes to the decreased damping coefficient and increased modulus of substrate at high hydrostatic pressure. Acoustic metastructures with helical oscillators are proposed and optimized with simulating annealing algorithm, where the helical oscillators are beneficial for easing the stress in the polymer substrate and resulting in good acoustic absorption abilities. Four acoustic absorptive metastructures without any oscillators (pure polymer substrate), with cylindrical oscillators, with helical oscillators, with both cylindrical and helical oscillators are proposed and experimentally verified in this study. All three metastructures with oscillators demonstrate excellent acoustic absorption coefficient with the absence of the hydrostatic pressure, while the metastructures with helical oscillators exhibit better hydrostatic pressure resistance compared with the metastructures with cylindrical oscillators. Besides, the coupling of helical oscillators and helical oscillators enhanced the acoustic performance furtherly. It is revealed that the acoustic metamaterials with a combination of cylindrical oscillator and helical oscillator exhibit superior performance under hydrostatic pressure up to 3 MPa in the frequency range of 0.5kHz∼10 kHz compared with the others, demonstrating a new avenue for underwater acoustic metamaterials application.

Flexural wave bandgaps of corrugated-core sandwich metabeam with inertial amplified resonators using the spectral element method

In this paper, we present a theoretical investigation of flexural wave propagation and vibration bandgap properties in corrugated-core sandwich metabeam with inertial amplified resonators. The complete analytical solutions for the flexural wave propagation with free boundary conditions are obtained through the spectral element method. Finite element calculation of transmission characteristics of sandwich metabeam with inertial amplified resonators are carried out to validate the accuracy of the proposed complete analytical solutions. In addition, the vibration reduction effect of sandwich metabeam with inertial amplified resonators is compared with that of the corrugated-core metabeam with local resonance. The influences of system parameters on the flexural wave attenuation are analyzed. Results show that the complete analytical solution for the proposed sandwich metabeam structure can be effectively used to predict the vibrational properties. In addition, significant enlargement of low-frequency bandgap can be obtained by using an inertial amplification sandwich metabeam structure. The proposed sandwich metabeam structure has good bandgap adjustment ability.

A simple image correlation technique for imaging subsurface damage from low-velocity impacts in composite structures

A robust computer vision system is proposed to visualize subsurface barely visible impact damage (BVID) in composite structures through a simple image correlation technique together with a damage imaging condition. This system uses a digital camera to record a video of the surface motion, capturing micron-scale dynamic movement from guided waves propagating on the surface of the structure generated via a sweeping frequency excitation (chirp) up to the ultrasonic frequency range. As the excitation frequency changes during the chirp, waves become trapped within this damaged region, forming standing waves to generate local resonance at specific frequencies. This localized resonance accumulates high wave energy in the region, generating a higher transverse displacement at the damage site compared to the remaining part of the structure. In this work, a simple image correlation technique is proposed to correlate each filtered video frame with the temporal mean of the filtered wavefield video to highlight standing waves at localized damage. The proposed image correlation technique is distinct from digital image correlation (DIC) since it does not use correlation for subset matching to track the movement of surface patterns between two images (video frames). Instead, it uses correlation to directly quantify similarities between corresponding image pixels (windows). Utilizing a single camera greatly simplifies system complexity and hence enhances the practicality and potential for real-time performance over a recently developed technique that utilized 3D DIC with a stereo camera for vision-based BVID detection. To realize the aim, work was conducted in two sequential steps: (1) off-axis 2D DIC was employed rather than 3D DIC to examine the potential of employing a single camera to capture images and extract not only in-plane displacement but also a fraction of transverse displacement in which local resonance is dominant and (2) image correlation was then employed, supplanting the off-axis 2D DIC image processing involved in the first step, to highlight the damage with significantly less processing complexity. This proposed technique using a zero-mean normalized cross-correlation imaging condition, rather than the total wave energy used for DIC-based approaches, is efficient and effective for identifying regions of minute surface movement from local resonance within the damage region without the use of the computationally intensive interpolation to get the sub-pixel gray level information followed by subset matching employed in DIC-based image processing. Two geometrically identical CFRP composite honeycomb panels that had been subjected to low-velocity impacts were used for verification and validation, and two excitation location configurations were tested for each panel. Damaged images produced with correlation for a 100 mm × 100 mm field of view using a single 3-s video, or a total of 19,200 video frames, show accurate damage imaging capabilities regardless of excitation location that exceed that of DIC techniques and is comparable to benchmark damage images obtained from laser Doppler vibrometry, ultrasonic C-scans, and X-ray CT scans. This success of correlation-based imaging of subsurface BVID demonstrates substantial improvements in efficiency and practicality and shows high potential for in situ and real-time computer vision-based nondestructive inspection or structural health monitoring of subsurface BVID in composite aircraft and other critical structures.

Studies on Dual Helmholtz Resonators and Asymmetric Waveguides for Ventilated Soundproofing.

Achieving the simultaneity of ventilation and soundproofing is a significant challenge in applied acoustics. Ventilated soundproofing relies on the interplay between local resonance and nonlocal coupling of acoustic waves within a sub-wavelength structure. However, previously studied structures possess limited types of fundamental resonators and lack modifications from the basic arrangement. These constraints often force the specified position of each attenuation peak and low absorption performance. Here, we suggest the in-duct-type sound barrier with dual Helmholtz resonators, which are positioned around the symmetry-breaking waveguides. The numerical simulations for curated dimensions and scattered fields show the aperiodic migrations and effective amplifications of the two absorptive domains. Collaborating with the subsequent reflective domains, the designed structure holds two effective attenuation bands under the first Fabry-Pérot resonance frequency. This study would serve as a valuable example for understanding the local and non-local behaviors of sub-wavelength resonating structures. Additionally, it could be applied in selective noise absorption and reflection more flexibly.

Electrical and Optical Switching in Vanadium Dioxide Nanostructures Decorated with Gold Nanoparticles

The electrical parameters of the semiconductor-metal phase transition in vanadium dioxide nanostructures synthesized by chemical vapor deposition on a silicon substrate (100) and decorated with gold nanoparticles with a surface concentration from 3∙109 to 3∙1010 cm–2 are studied. X-ray phase analysis revealed that the synthesized nanostructures of vanadium dioxide contain a monoclinic M1 phase undergoing a phase transition at a temperature of about 68 °C. The morphology of the surface of vanadium dioxide nanostructures coated with gold nanoparticles was studied using a scanning electron microscope and an atomic force microscope. The characteristics of the temperature phase transition of the initial nanostructures and nanostructures decorated with gold nanoparticles are determined. The temperature dependence of the resistance near the phase transition point of the initial nanostructures showed that the resistance jump is about four orders of magnitude, which confirms their high quality. It is shown that an increase in the surface concentration of gold particles to a value of 3∙1010 cm–2 increases the conductivity of vanadium dioxide at room temperature by about two times, and shifts the phase transition temperature by 5 °C: from 68 °C to 63 °C. Optical switching in vanadium dioxide with an array of gold particles with a size of 9 nm is considered by numerical modeling methods. It is established that the response of the electromagnetic wave from the VO2 material during the phase transition is enhanced due to the excitation of localized plasmon resonance in gold nanoparticles and reaches a local maximum in the region of 600 nm. Additionally, this effect is enhanced at angles of incidence near the pseudo-Brewster angle for vanadium dioxide. The considered hybrid VO2–Au nanostructures are promising as basic nanoelements for next-generation computers, as well as for ultrafast and highly sensitive sensors.

Auto-adaptive metastructure for active tunable ultra-low frequency vibration suppression

Metastructures provide a new way to solve the problem of low-frequency vibration suppression because of their unique low-frequency bandgap characteristics. Traditional metastructures usually suffer from the problem of narrow bandgap at low frequencies, which cannot tolerant to uncertainties in manufacturing process of metastructures and cannot adapt to changing working conditions. One way to overcome the shortcomings is to design metastructures with tunable bandgaps. However, it is still a challenge to realize an auto-adaptive metastructure with a highly efficient tunability. In this work, we propose an active metaplate consisting of a host plate and a periodic array of local resonators that can be collectively and actively tuned using stepping motor. We demonstrate numerically and experimentally that the proposed active metaplate can realize continuous tunable ultra-low frequency bandgap. By introducing an auto-adaptive control strategy, it can realize auto-adaptive vibration attenuation at ultra-low frequencies. The proposed active metaplate has promising application prospects since it can achieve real-time active vibration suppression by using only a few active components.

Optical Properties of AgAu Alloy Clusters: Effect of Chemical Configuration along a Rearrangement Pathway

Gold and silver are, for all their chemical similarities, optically very different. Small Ag clusters show a localized surface-plasmon resonance (LSPR), whereas in Au clusters smaller than about 300 atoms, the resonance is absent due to the coupling with the interband transitions from the d electrons. This opens the possibility of tuning the cluster properties depending on their composition and chemical configuration. Earlier work on AgAu alloy clusters has shown that the outermost shell of atoms is crucial to their overall optical properties. In the present contribution, we consider the optical spectroscopic properties associated with the structural rearrangement in 55-atom AgAu alloy clusters in which the core transforms from pure silver to pure gold. Calculations using time-dependent density-functional theory are complemented by an in-depth study of the subtle effects that the chemical configuration has on the details of the materials’ d bands. Although the cluster surface remains alloyed, the geometrical changes translate into strong variations in the optical properties.

Realizing Minimally Perturbed, Nonlocal Chiral Metasurfaces for Direct Stokes Parameter Detection.

Recent development in nonlocal resonance based chiral metasurfaces draws great attention due to their abilities to strongly interact with circularly polarized light at a relatively narrow spectral bandwidth. However, there still remain challenges in realizing effective nonlocal chiral metasurfaces in optical frequency due to demanding fabrications such as 3D-multilayered or nanoscaled chiral geometry, which, in particular, limit their applications to polarimetric detection with high-Q spectra. Here, we study the underlying working principles and reveal the important role of the interaction between high-Q nonlocal resonance and low-Q localized Mie resonance in realizing effective nonlocal chiral metasurfaces. Based on the working principles, we demonstrate one of the simplest types of nonlocal chiral metasurfaces which directly detects a set of Stokes parameters without the numerical combination of transmitted values presented from typical Stokes metasurfaces. This is achieved by minimally altering the geometry and filling ratio of every constituent nanostructure in a unit cell, facilitating consistent-sized nanolithography for all samples experimentally at a targeted wavelength with relatively high-Q spectra. This work provides an alternative design rule to realizing effective polarimetric metasurfaces and the potential applications of nonlocal Stokes parameters detection.

A novel multi-resonator honeycomb metamaterial with enhanced impact mitigation

Local resonant metamaterials show significant promise in applications involving the attenuation of impact waves. This paper introduces a honeycomb metamaterial featuring a circular distribution of multiple mass-beam resonators designed to enhance impact mitigation by broadening low frequency bandgaps. The bandgaps of unit cells with different number of mass-beam resonators (MBR) are calculated. It is found that the unit cell with circularly distributed resonators (C-MBR) has strong low frequency local resonance in all directions, thus opening a complete wide bandgap in about 500–1200Hz. Then, a method of equivalent stiffness reduction related to the corresponding mode shapes is used to research the negative effective mass density characteristic, explaining the generation of the low frequency bandgaps. Subsequently, the impact protection capacity of local resonant metamaterial is compared with that of ordinary honeycomb metamaterial through experiment and simulation. Interestingly, the maximum acceleration values under the impact load of local resonant metamaterial are about 45% lower than that of honeycomb metamaterial, and the reaction force under the blast load is about 80% reduction. Besides, by analyzing the acceleration curve waveform and transmission spectrum, the local resonant metamaterial is found to exhibit continuous short period vibration, which blocks the wave in low frequency band. Finally, the energy dissipation performance of the local resonant metamaterial explains that it converts more impact energy into the kinetic energy of the core with resonators, thereby reducing the energy transferred to the bottom panel. These results underscore the potential application of the proposed metamaterial with multi-resonators in impact mitigation and energy dissipation.

Low-frequency bandgap characteristics and vibration attenuation performance of metamaterial-tailored concrete-filled steel tube columns

Low-frequency vibrations, such as earthquakes and wind loads, may negatively affect the proper operation of buildings and bridges and even lead to structural failure. Even though, in the past two decades, metamaterial-based structures have been utilized in structure vibration suppression, they are mainly used in the mitigation of high-frequency vibrations, and it has been an open challenge to suppress low-frequency vibrations below 100Hz and ensure adequate bearing capacity in engineering structures. In this work, we present metamaterial composite column structures with built-in local resonators to achieve the low-frequency vibration mitigation effect. Compared to ordinary concrete-filled steel tube (CFST) columns, the metamaterial columns (meta-columns) can utilize the rubber shear modulus to create a low-frequency flexural bandgap in 30–50 Hz. Dynamic stiffness is introduced in the Timoshenko beam and Bloch's theorem, the metamaterial analytical model is established, and the bandgap generation mechanism and vibration reduction effect are analyzed in conjunction with numerical simulations. Then, the bandgap effect and vibration attenuation law of the meta-columns are investigated by shaker tests at different acceleration amplitudes. Guided by theory and simulations, we have successfully fabricated several meta-composite columns with low-frequency bandgaps, which attenuate up to half of the vibrational energy near the bandgap center. The bandgap ranges derived from the theoretical and numerical models show a high degree of consistency with the experimental results, with deviations generally within 10 %, all indicating that a bandgap exists near the resonator self-oscillation frequency. Additionally, the bandgap is significantly affected by the number of unit cells and resonators, especially the meta-column with fewer resonators fails to form a bandgap where the top response is smaller than the bottom response. As the number of resonators and metamaterial units increases, the meta-columns exhibit wider bandgap and stronger vibration reduction performance.

Noble Metal Nanoparticles with Nanogel Coatings: Coinage Metal Thiolate-Stabilized Glutathione Hydrogel Shells.

Developing biocompatible nanocoatings is crucial for biomedical applications. Noble metal colloidal nanoparticles with biomolecular shells are thought to combine diverse chemical and optothermal functionalities with biocompatibility. Herein, we present nanoparticles with peptide hydrogel shells that feature an unusual combination of properties: the metal core possesses localized plasmon resonance, whereas a few-nanometer-thick shells open opportunities to employ their soft framework for loading and scaffolding. We demonstrate this concept with gold and silver nanoparticles capped by glutathione peptides stacked into parallel β-sheets as they aggregate on the surface. A key role in the formation of the ordered structure is played by coinage metal(I) thiolates, i.e., Ag(I), Cu(I), and Au(I). The shell thickness can be controlled via the concentration of either metal ions or peptides. Theoretical modeling of the shell's molecular structure suggests that the thiolates have a similar conformation for all the metals and that the parallel β-sheet-like structure is a kinetic product of the peptide aggregation. Using third-order nonlinear two-dimensional infrared spectroscopy, we revealed that the ordered secondary structure is similar to the bulk hydrogels of the coinage metal thiolates of glutathione, which also consist of aggregated stacked parallel β-sheets. We expect that nanoparticles with hydrogel shells will be useful additions to the nanomaterial toolbox. The present method of nanogel coating can be applied to arbitrary surfaces where the initial deposition of the seed glutathione monolayer is possible.

Magnetic anisotropy and ferromagnetic resonance in inhomogeneous demagnetizing fields near edges of thin magnetic films

Using local ferromagnetic resonance spectroscopy, we have studied the magnetic properties near edges of thin tangentially magnetized permalloy films, in which a well-defined uniaxial magnetic anisotropy was induced perpendicular to one of the edges. In the experiment, two samples with thicknesses of 90 and 300 nm and with slightly different compositions were examined. To explain the magnetization dynamics near edges, we propose a simple yet effective model of a film in the form of a rectangular prism, which yields the modified Kittel formula for the resonance frequency. In this formula, the locally averaged distance-dependent demagnetizing field that emerges near the edges is included as an additional uniaxial anisotropy term. The measurements reveal that at a certain distance from the edge, the resulting (apparent) anisotropy, determined from the angular dependencies of the resonance field, almost vanishes. Moreover, its easy axis reorients to become parallel to the edge. The model predictions agree well with these results, proving that the main resonance mode behavior near the film edges can be accurately described by introducing additional effective uniaxial anisotropy, provided the measuring area is relatively large. However, for the thick (300 nm) sample, additional precession modes are also observed. These modes distort the angular dependence of the main mode, thus demonstrating the limitations of the model.

Magnetic Domain Wall Networked Films – Configurational Switching for Tunable Magnetization Dynamics by Perforated Thin Films

AbstractSetting the magnetization and its properties in magnetic materials and films is of the utmost fundamental and technological importance for magnetic applications and devices. Magnetic permeabilities in a material are generally defined by the acting magnetic anisotropies together with an applied magnetic field. Here, deterministic switching of the effective dynamic magnetic properties is demonstrated by tunable networks of magnetic domain walls. By perforating magnetic films, a reversible switching between different magnetization states is achieved. Due to the resulting local effective dynamic fields for the changed domain configurations, different zero‐field permeability spectra are attained for the same material structure. Using local configurational magnetic resonances confined by magnetic domain walls, deterministic switchable bimodal dynamic behavior is achieved. Dynamic magnetooptical Kerr microscopy with picosecond temporal resolution proves that the varying periodic magnetic domain configurations are responsible for the dynamic magnetic bistability. In particular, the field‐free precessional magnetic response can be reproducibly switched between 1.3 and 2.3 GHz. The magnetization dynamics can be modified by small interventions in a repeatable manner beyond fixed material and structural properties. The described fundamental mechanism enables to design new energy efficient configurational microwave devices by switching between zero‐field magnetic arrangements.

Tunable Ultralow-Frequency Bandgaps Based on Locally Resonant Plate with Quasi-Zero-Stiffness Resonators

In order to suppress the transverse vibration of a plate, a quasi-zero-stiffness (QZS) resonator with tunable ultralow frequency bandgaps was introduced and analyzed. The resonator was designed by introducing the quasi-zero-stiffness systems into mass-in-mass resonators. The plane wave expansion method was employed to derive the bandgap characteristics of the locally resonant (LR) plate with QZS resonators, and corresponding simulations were carried out by finite element method (FEM). The results show that an LR plate with a QZS resonator can provide two bandgaps, and the ranges of the bandgaps agree well with the vibration attenuation bands calculated by FEM. Owing to the introduction of the QZS system, the bandgaps can be easily transferred to a lower frequency or even an ultralow frequency. The damping of the QZS resonators can effectively broaden the vibration attenuation bands. In addition, the differentiated design of the bandgap frequencies can be realized to obtain broadband low-frequency transverse wave suppression performance. Finally, a mechanical structure design scheme was proposed in order to achieve flexible adjustment of the bandgap frequency, which significantly increases the engineering applicability of QZS resonators.

Linear optical properties of organic microcavity polaritons with non-Markovian quantum state diffusion

Abstract Hybridisation of the cavity modes and the excitons to polariton states together with the coupling to the vibrational modes determine the linear optical properties of organic semiconductors in microcavities. In this article we compute the refractive index for such system using the Holstein–Tavis–Cummings model and determine then the linear optical properties using the transfer matrix method. We first extract the parameters for the exciton in our model from fitting to experimentally measured absorption of a 2,7-bis[9,9-di(4-methylphenyl)-fluoren-2-yl]-9,9-di(4-methylphenyl) fluorene (TDAF) molecular thin film. Then we compute the reflectivity of such a thin film in a metal clad microcavity system by including the dispersive microcavity mode to the model. We compute susceptibility of the model systems evolving just a single state vector by using the non-Markovian quantum state diffusion. The computed location and height of the lower and upper polaritons agree with the experiment within the estimated errorbars for small angles ( ≤ 30 ° ) $(\le 30{}^{\circ})$ . For larger angles the location of the polariton resonances are within the estimated error.

Characterization of wireless power transfer based on Fano resonant-like surface

This paper presents a new way to enhance the transmission efficiency of a dual-coils wireless power transfer system. That method is to introduce a Fano resonant-like surface in the dual-coils wireless power transfer system. This surface, positioned opposite the transmission direction of the transmitter coil in the system, adopts a four-armed helical structure. The results demonstrate that introducing the Fano resonant-like surface significantly enhances system transmission efficiency, attributed to two primary factors. First, the Fano local resonance effect in the dual-coils wireless power transfer system with Fano resonant-like surface leads to the enhance the transmission efficiency. Second, the Fano resonant-like surface shields the energy propagation of the nearby magnetic field in the direction opposite to transmission. Further, compared to a wireless power transfer system comprising only two coils, the overall improvement in transmission efficiency is 30%–40%. Owing to its simplicity, more compact size, cost-effectiveness, and ease of integration without having to be placed between the transmitting and receiving coils, the Fano resonant-like surface can lay the groundwork for practical applications in wireless charging.

Experimental investigation of Rayleigh wave propagation in a locally resonant metamaterial layer resting on an elastic half-space

In this experimental investigation, we explore the propagation characteristics of surface Rayleigh waves in a Locally Resonant Metamaterial (LRM) layer positioned on an elastic half-space. The study focuses on characterizing the dispersion and attenuation properties of these waves and validating analytical and numerical models of the LRM. For practical purposes, we utilize a thin-plate sample and construct the LRM layer, featuring multiple rows of sub-wavelength resonators, by machining the resonators at one edge of the plate. Employing a piezoelectric transducer coupled to the plate and a laser vibrometer, we actuate and receive the surface-like waves propagating at the plate edge. Two resonant layer configurations, comprising 3 and 5 rows of resonators, corresponding to heights of ∼0.6λh and λh, where λh represents the reference wavelength of Rayleigh waves, are examined. The experimental observations reveal the hybridization of the fundamental surface mode at the resonant frequency of the embedded resonators, leading to the creation of a low-frequency bandgap. This bandgap, attributed to the local resonance mechanism, exhibits a remarkable attenuation of surface wave amplitudes. To support our experimental findings, we conduct both analytical and numerical studies. These analyses demonstrate the confinement of the lowest-order surface mode within the frequency ranges proximate to the resonators’ resonance. The insights gained from this experimental study contribute to the advancement of strategies for mitigating surface waves through the application of resonant metamaterials and metastructures.

Resonant-type Luneburg lens for broadband low-frequency focusing

In this paper, a novel structural Luneburg lens with local resonators is proposed. This lens allows the realization of subwavelength focusing in low-frequency range. The lens is achieved by graded refractive index from the lens's centre to the outer surface. Numerical simulations are conducted to obtain data on wave propagation waveform, maximum displacement amplitude, and full width at half-maximum (FWHM) of the lens's focal region. The results show that a broadband frequency range can be achieved for subwavelength focusing. This provides a straightforward and adaptable method for designing the structural Luneburg lens for numerous applications.

A Novel Approach for Harmonic Assessment of Power Systems With Large Penetration of IBRs – A U.K. Case Study

This work provides a high-level assessment of the harmonic performance of the future UK power system with high penetration of Renewable Energy Sources (RESs). A few variations to the standard harmonic analysis are proposed to allow for a more accurate representation of Inverter-Based Resources (IBRs) in frequency-domain studies. In the first part of the paper, two reduced network models (for 2020 and 2050, respectively) are developed, including both transmission and distribution system representation. A frequency-dependent impedance model for individual and aggregated IBRs is adopted, and a novel distribution load model with harmonic injection is proposed. Additionally, the variability of harmonic phase angles is modelled taking into account the physical behavior of the IBRs. Probabilistic Harmonic Load Flows (PHLFs) are carried out on the reduced networks by varying the harmonic injection magnitudes and phase angles of each harmonic source. This work shows that, while overall harmonic levels are dependent on the changes in demand, the inverter impedance has significant impact on local resonances, and accurate models for IBRs are required.

Characteristics of truncation resonances in periodic bilayer rods and beams with symmetric and asymmetric unit cellsa).

Truncation resonances are resonant frequencies that occur within bandgaps and are a prominent feature of finite phononic crystals. While recent studies have shed light on the existence conditions and modal characteristics of truncation resonances in discrete systems, much remains to be understood about their behavior in continuous structures. To address this knowledge gap, this paper investigates the existence and modal characteristics of truncation resonances in periodic bilayer beams, both numerically and experimentally. Specifically, the effect of symmetry of the unit cells, boundary conditions, material/geometric properties, and the number of unit cells are studied. To this end, we introduce impedance and phase velocity ratios based on the material and geometric properties and show how they affect the existence of truncation resonances, relative location of the truncation resonances within the bandgap, and spatial attenuation or degree of localization of the truncation resonance mode shapes. Finally, the existence and mode shapes of truncation resonances are experimentally validated for both longitudinal and flexural cases using three-dimensional (3D) printed periodic beams. This paper highlights the potential impact of these results on the design of finite phononic crystals for various applications, including energy harvesting and passive flow control.