Phononic Crystal Beam Research Articles

Defect coupling behavior and flexural wave energy harvesting of phononic crystal beams with double defects in thermal environments

As an effective means to manipulate elastic wave energy, the defect mode of phononic crystals (PCs) has recently been employed to implement vibration energy harvesting by localizing waves within certain areas at defect frequencies. Prior research has indicated that the thermal environment affects the defect behavior and energy harvesting of single defects. However, the dynamic characteristics of multi-defect phononic crystals have not yet been discussed. This paper presents an investigation of the coupling behavior and energy harvesting of double defects in heated PC beams. The frequencies and mode shapes of defect modes are studied for different defect pairs within the first bandgap. Results indicate that the difference in the single-defect frequency between the two defects plays a dominant role in determining the intensity of the defect interaction which splits the defect modes. Meanwhile, the influence of the energy distribution around the defects on the coupling behavior becomes prominent when the two defects are quite close to each other. These two factors could cause degeneration of the defect modes. As an external factor, thermal load could enhance the defect coupling intensity, and may reactivate the degenerated defect modes. This effect would increase as the initial defect interaction gets stronger. For vibration energy harvesting, temperature rise may lead to a significant reduction in defect-based harvested energy, even though the number of defect-induced resonance peaks could be increased in the thermal environment. The thermosensitivity of harvesting performance varies greatly for different combinations of defects. Thermal effects should be considered in the design of defect-based harvesters.

Elastic Foundation Induced Wide Bandgaps for Actively-tuned Topologically Protected Wave Propagation in Phononic Crystal Beams

This work presents the active control effects on a piezoelectric phononic crystal beam resting on elastic foundation. The bending of passive periodic beams that rest on elastic foundation can exhibit distinguished applications for geotechnical and railroad structures. Generally, elastic waves propagate in these structures in a certain frequency range and these structures are able to block the other input waves outside this range. These two frequency ranges are usually called the pass band and the bandgap of structures. With periodic attachment of piezoelectric patches having negative capacities, the structural band properties can be easily tuned to a wider range. However, the beam system with negative capacities is always restricted by the presence of electrical boundary conditions in the structural stability response. It leads to tuning inconvenience and tuning disabilities of the structures. In view of these tuning shortcomings, this study proposes to capitalize the effects of elastic foundation or elastic supports to improve the active control of piezoelectric beams for bandgap generation in wave propagation. By artificially positioning periodic negative capacities and piezoelectric patches, the system sub-structure is designed as an A-B-A phononic crystal beam. Further, the plane wave expansion method is applied to the governing equation to derive analytical solutions and to compare with numerical solutions. The topologically protected phase transition is observed by analyzing mode shapes of the specific cases and the Zak phase across the Brillouin zone. The topologically protected interface modes are also captured in this system and the topologically protected manner provides a new paradigm for manipulating wave propagations in the bending of beam system resting on the elastic foundation. Finally, the efficiency of electrical parameters and foundation stiffness on system tuning is examined. The controllable system is further justified to allow operation in a wider frequency range.

A double-layer metastructured beam with contact-separation switchability

In this paper, we use shape memory alloy springs, electromagnets, ordinary springs and two metal beams to construct a double-layer phononic crystal beam. The electromagnets can be controlled to be in contact or separated by means of temperature and electric current. Firstly, we use transfer matrix method to derive the theorical solutions of the bandgap properties, and compare them with finite element results. Then, we focus on the influences of different parameters on bandgap properties to see how they can be effectively regulated. Finally, we take account of the symmetry of the structure and excitations for simplification and get two basic solutions, based on which the bandgap characteristics for an arbitrary excitation are obtained via superposition. It is shown that temperature and electric current can change the bandgap characteristics effectively, and while both the lattice constant and SMA spring stiffness have a global and huge influence, the ordinary spring stiffness and magnetic force all exhibit a local and detailed modification to the bandgaps. When the metastructured beam is perfectly symmetric but subjected to an antisymmetric excitation, we can obtain an extremely low and wide bandgap. The structure designed here is expected to play a role in actively controlled low-frequency noise reduction.

Study on the vibration attenuation property of one finite and hybrid piezoelectric phononic crystal beam

In this paper, a theoretical method is proposed to effectively improve the vibration attenuation characteristics of a finite hybrid piezoelectric phononic crystal (PC) beam utilizing the non-uniform distribution of shunting circuits. Considering the damping of resonators, the vibration transmissibility method (TM) for the finite electromechanical system is derived based on the Timoshenko beam theory. Numerical results are validated by the finite element method (FEM). Subsequently, a comprehensive study is conducted to investigate the influences of the structural and electrical parameters on the vibration attenuation property of the uniform PC. Furthermore, the organized disorder of the shunting circuits is introduced into the system. Combining with various types of non-uniform circuit configurations, the variation of the vibration attenuation is also examined. It is demonstrated that the hybrid PCs offer more tunable mechanisms to design target band-gaps, compared with the purely mechanical and piezoelectric PCs. The low-frequency vibration reduction over a broadband range could be achieved through the selective coupling ways of the mechanical local resonant band-gap (LRG), electromechanical LRG and Bragg band-gap (BG). It is expected to enhance the vibration attenuation and availability of hybrid PCs using the suitable distribution of shunting circuits in practice.

A method for determining the minimum period number in finite locally resonant phononic crystal beams

To achieve the target band-gap in finite locally resonant phononic crystal beams, a method for determining the minimum period number is proposed. The vibration transmissibility method is extended to deal with the finite locally resonant phononic crystal beam. Comparing the vibration attenuation region obtained from the transmissibility method with the band-gap from the conventional transfer matrix method, the minimum period number can be calculated. Based on two forming patterns of locally resonant phononic crystal beams, the effects of the lattice constant and structural parameters of resonators on the band-gap as well as the influence of the period number on the vibration transmission characteristic are investigated. The minimum period number method can improve the applicability of the transmissibility method in the design of band-gaps and overcome the drawback that the transfer matrix method lacks the actual vibration attenuation. Finally, a comprehensive index is introduced to evaluate the effect of vibration reduction.

Active control of flexural waves in a phononic crystal beam with staggered periodic properties

The band gaps of a phononic crystal beam with staggered periodic structure are investigated. The periodic system consists of a pure elastic (i.e. PMMA) matrix beam and some piezoelectric (i.e. PZT) patches with coupling between the mechanical–electrical components. The PZT patches connected by negative capacitance circuits are applied to function as the active control system. Based on the condition at the interface between adjacent unit cells, the transfer matrix and localization factor are derived. The influence of the degree of interlacing and negative capacitance circuits are discussed. The numerical results show that another band gap can be generated by the staggered periodic structure of PZT patches. The widths and locations of the band gaps can be changed by the degree of interlacing.

A shape memory alloy-based tunable phononic crystal beam attached with concentrated masses

Shape memory alloys (SMAs) in phononic crystals (PCs) or metamaterials mostly serve as smart inserts or local resonators to generate tunable bandgaps for other structures. In this letter, we address the method for creating tunable bandgaps in structures made of SMA. Specifically, we consider concentrated masses, realized by steel balls, as attachments on a SMA. To provide a design guideline, we derived the closed-form bandgap formula of the proposed SMA PC beam. As the bandgap order increases, the coefficient of the nth Bragg bandgap width decreases. In addition, the practical final beam enlarges bandgap edges than ideally infinite PCs or metamaterials. Our experimental results on the PC beam with different sizes of the steel ball arrays agree with the analytical predictions of the bandgap behaviors. With the bandgaps formed by the steel balls and the shape memory effect, wide tunability of the bandgaps are achieved on the SMA beam.

Extracting torsional band gaps and transient waves in phononic crystal beams: Method and validation

Due to an absence of proper measuring techniques, torsional waves/vibrations in one-dimensional phononic crystals (PCs) or elastic metamaterials are usually theoretically or numerically studied without experimental validations. This work proposes a measurement method capable of extracting torsional band gaps and transient torsional waves from the total responses of PC and metamaterial beams. Specifically, a pair of point-wise self-demodulated fiber Bragg grating (SFBG) displacement sensors, which can be arranged at points very close to each other on a thin PC beam, is utilized along with the proposed extraction method. The SFBGs are symmetrically set up at two locations about the central line on the surface of the PC beam. The torsional band gaps and the transient torsional waves are extracted from the differential responses of the two SFBGs, in which the common bending waves are cancelled out. In addition to the torsional behaviors, the bending waves can also be extracted from the summation responses of the two SFBGs. We first examine the dynamic linearity of the SFBGs. Then, steady-state detection of the torsional and bending band gaps of a PC beam is performed. Finally, a cantilever PC beam subjected to an eccentric steel ball impact, whose loading history is recorded by a pair of polyvinylidene fluoride (PVDF) films, is investigated to demonstrate the extraction of the torsional/bending bandgaps and waves from the early short time transient responses. The feasibility of the proposed torsional (or bending) extraction method is justified by the method of reverberation-ray matrix (MRRM), which is capable of providing “pure” torsional (or bending) behaviors of the PC beam. With the proposed extraction method, we also observe surface localized modes in the transmission of the PC beam, which are strongly related to the arrangement of the constituent components with different impedances in the unit cells. We believe that the proposed extraction method for torsional wave, along with the SFBG sensing technique and the MRRM, can provide future researchers with measurement and analysis method to validate their designs of torsional phononic crystals or metamaterials.

Nonlocal coupled thermoelastic wave propagation band structures of nano-scale phononic crystal beams based on GN theory with energy dissipation: An analytical solution

This paper deals with the band structures of thermoelastic waves in nano-scale phononic crystal or metamaterial beams considering nano-size effects. The nonlocal coupled thermoelastic governing equations are derived using the Green–Naghdi theory of the generalized coupled thermoelasticity with energy dissipation and Eringen’s nonlocal theory. The derived governing equations are analytically solved and the field quantities including the temperature and the deflection are obtained in the closed forms. Using the proposed analytical solution, the transfer matrix between two unit-cells are obtained using the thermal and mechanical continuity conditions on the interfaces between the unit-cells and between the two sections of each unit-cell. The band structures of the phononic crystals are obtained using the Bloch–Floquet theorem. The detailed discussions are presented for the band structures of nonlocal thermoelastic waves in nano-scale aluminum/epoxy phononic crystal or metamaterial beams. Also, the effects of the small-scale parameter and the thickness of the epoxy layers on the band structures are studied and discussed by using the derived analytical solution.

Highly Localized and Efficient Energy Harvesting in a Phononic Crystal Beam: Defect Placement and Experimental Validation

We study energy harvesting in a binary phononic crystal (PC) beam at the defect mode. Specifically, we consider the placement of a mismatched unit cell related to the excitation point. The mismatched unit cell contains a perfect segment and a geometrically mismatched one with a lower flexural rigidity which serves as a point defect. We show that the strain in the defect PC beam is much larger than those in homogeneous beams with a defect segment. We suggest that the defect segment should be arranged in the first unit cell, but not directly connected to the excitation source, to achieve efficient less-attenuated localized energy harvesting. To harvest the energy, a polyvinylidene fluoride (PVDF) film is attached on top of the mismatched segment. Our numerical and experimental results indicate that the placement of the mismatched segment, which has not been addressed for PC beams under mechanical excitation, plays an important role in efficient energy harvesting based on the defect mode.

Vibration and energy harvesting performance of a piezoelectric phononic crystal beam

Phononic crystal could be used in the vibration energy harvesting devices as its capacity to modulate elastic wave propagation. This paper presents a piezoelectric phononic crystal cantilever beam with periodic variable thicknesses for vibration energy harvesting. The transfer matrix method (TMM) is introduced to analyze the natural frequencies and the band structure of the phononic crystal beam. The influences of structural parameters, such as thicknesses-ratio and lengths-ratio, on the natural frequencies and first band gap of the beam are investigated in detail. The results of first two band gap ranges obtained via the TMM are validated by means of the finite element method (FEM) simulation. Moreover, the electromechanical coupling responses of the phononic crystal beam are explored by using the FEM simulation. The results demonstrate that the piezoelectric phononic crystal beam has more efficient voltage output in the first band gap comparing with the uniform piezoelectric cantilever beam. In conclusion, the proposed piezoelectric phononic crystal beam can extend the bandwidth of the energy harvester in low-frequency vibrations.

A Numerical Method for Flexural Vibration Band Gaps in A Phononic Crystal Beam with Locally Resonant Oscillators

The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.

A bandgap switchable elastic metamaterial using shape memory alloys

Adaptive elastic metamaterials are generally tunable but not switchable. Here, the word “switchable” means switching between different bandgap mechanisms, such as from the local resonance bandgaps to the Bragg scattering bandgaps and vice versa. In this work, to achieve switchable bandgaps, we report a new class of elastic metamaterials whose transmission properties can be significantly tuned by curved two-way shape memory alloy (SMA) resonators. The proposed switchable metamaterial possesses bandgaps capable of being switched back and forth between the Bragg scattering and localized resonance ones. Without thermally activating the curved SMA resonators, the metamaterial beam behaves as a phononic crystal beam whose bandgaps are formed by a periodic array of concentrated masses. By heating the SMAs, the SMAs recover their original curved shape and lift the concentrated masses to form the local resonance bandgaps. The reversible dramatic variation in shape and the stiffness of the SMA resonators allows the bandgaps to be switchable and of course tunable. In addition, the equivalent spring stiffness of a curved beam at two possible directions for the first two modes is derived based on Castigliano's second theorem and is experimentally validated. Compared to SMA springs, the curved shape SMAs allow the generation of high-order local resonance bandgaps. If thermally activated at different temperatures, the negative effective mass density can further be tunable. To the author's knowledge, this work is the first theoretical and careful experimental investigation realizing switchable metamaterials using the two-way shape memory alloys.

Flexural wave manipulation and energy harvesting characteristics of a defect phononic crystal beam with thermal effects

This paper investigates the dynamic and vibration energy harvesting behaviors of defect phononic crystal beams in thermal environments. Variations in the first bandgap and the defect band for flexural waves are discussed with different geometric parameters of the defect cell and structural temperatures by using the supercell technique for infinite defect phononic crystal beams. Transmission characters of defect models with a finite length are studied, considering the influence of thermal loads, and the performance of the defect based energy harvesting is discussed as well. Predicted results indicate that temperature increments reduce the frequency ranges of bandgap and defect band due to the softening effect of thermal load on the phononic crystal beam, and the performance in defect based energy harvesting is obviously depressed with temperature rise. Defect cells with a higher content of the soft component are beneficial for generating defect modes which have relatively stable frequencies in thermal environments. However, higher levels of open-circuit voltage and output power of harvested energy are more likely to be generated with defect cells which have a higher content of the hard component.

Defect mode-induced unidirectional flexural wave transmission using prismatic beams with concentrated gradient masses

We study and realize unidirectional flexural wave transmission in finite phononic crystal beams based on the boundary defect modes. First, we show that by carrying a periodic array of concentrated masses, conventional prismatic beams become phononic crystal beams having multiple transmittance peaks in odd-order bandgaps. We point out that these bandgap transmittance peaks are induced by pass-band splitting and are essentially defect modes due to the existence of the imperfect boundary in finite beam structures. Significant asymmetric flexural wave propagation can be observed in these defect modes by gradually changing each concentrated mass. Using the spectral element method (SEM), the relationship between the concentrated gradient masses and the directivity at the defect modes is discussed. To realize concentrated gradient masses, we periodically attach near-point-contact steel balls with gradient diameters on a prismatic beam. The formation of the bandgaps and unidirectional displacement transmission are experimentally validated with a high-sensitive point-wise fiber Bragg grating displacement sensing system. Asymmetric one-way flexural wave propagation is further demonstrated in the time domain with a Hanning-windowed tone burst signal excited at the two ends of the phononic crystal beam. Agreements between the SEM and experimental results clearly indicate that the asymmetric one-way flexural wave propagation can be achieved in prismatic beams carrying a periodic array of concentrated gradient masses.

A self-demodulated fiber Bragg grating for investigating impact-induced transient responses of phononic crystal beams

This work presents a fiber Bragg grating (FBG) displacement sensing system to experimentally investigate the dynamic behaviors of phononic crystal (PC) beams through impact-induced early short time or long time transient responses. Based on the couple-mode theory and the optical transfer matrix (T-matrix) formulation, we first show that it is feasible to achieve linear displacement sensing using a single FBG without extra demodulators such as matching gratings. To validate its effectiveness, the proposed self-demodulated FBG system is applied to measure the point-wise transient displacement responses of a cantilever PC beam subjected to steel ball impacts. To demonstrate the transient sensing performance, the measured early short time transient responses are compared with the corresponding ones predicted by the method of reverberation-ray matrix (MRRM) theoretically and the finite element method (FEM) numerically. The excellent agreements verify together the experimental, theoretical and numerical results. Finally, conducting Fast-Fourier transform (FFT) to the early short time or long time transient responses gives the frequency responses that clearly show the existence of the band gaps. In addition to providing a new displacement sensing method using the self-demodulated FBGs, this work also offers another route to investigate the band structures of PC beams through the frequency responses transformed from the early short time transient responses, long before the structural damping becomes dominant.

Flexural Wave Band Gaps in Phononic Crystal Euler-Bernoulli Beams Using Wave Finite Element and Plane Wave Expansion Methods

We investigate theoretically and experimentally the forced response of flexural waves propagating in a 1D phononic crystal (PC) Euler-Bernoulli beam, composed by steel and polyethylene, and its band structure. The finite element, spectral element, wave finite element, wave spectral element, conventional and improved plane wave expansion methods are applied. We demonstrate that the vibration attenuation of the unit cell can be improved choosing correctly the polyethylene and steel quantities and we suggest the best percentages of these materials, considering different unit cell lengths. An experiment with a 1D PC beam is proposed and the numerical results can localize the band gap position and width close to the experimental results. A small Bragg-type band gap with low attenuation is observed between 405 Hz - 720 Hz. The 1D PC beam with unit cells of steel and polyethylene presents potential application for vibration control.

Analysis of Bending Waves in Phononic Crystal Beams with Defects

Existing investigations on imperfect phononic crystal beams mainly concern periodic multi-span beams carrying either one or two channel waves with random or deterministic disorder in span-length. This paper studies the two channel bending waves in phononic crystal beams consisting of many phases of materials with defects introduced as one structural segment having different cross-sectional dimensions or material parameters. The method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory, which can conduct high-frequency analysis, is extended for the theoretical analysis of dispersion and transmission of bending waves. The supercell technique and the Floquet–Bloch theorem are adopted for modeling the dispersion characteristics, and the whole finite structural model is used to calculate the transmission spectra. Experimental measurements and numerical calculations are provided to validate the displacement transmission obtained by the proposed MRRM, with the effect of damping on transmission spectra being concerned. The high-frequency calculation applicability of the proposed MRRM is also confirmed by comparing the present results with the corresponding ones either using the transfer matrix method (TMM) or MRRM based on Euler—Bernoulli beam theory. The influences of defect size, defect form, and unit-cell number on the transmission spectra and the band structures are discussed. The drawn conclusions may be useful for designing or evaluating the defected phononic crystal beams in bending wave control. In addition, our conclusions are especially potential for identifying the defect location through bending wave signals.

Experimental and Theoretical Investigation of Lowering the Band Gaps of Phononic Crystal Beams through Fluid-Solid Coupling

We experimentally and theoretically investigate the band-gap and transmission properties of phononic crystal (PC) beams immersed in water. Spectral element method (SEM) is developed for theoretical analysis in which the hydrodynamic loading is taken into consideration. Influence of the hydrodynamic loading on band-gap and transmission properties of the PC beams are studied. To directly detect the displacement transmission of a fully or partially submerged PC beam, a fiber Bragg grating (FBG) displacement sensing system is set up. Agreement between the experimental results and theoretical/numerical calculations also indicates the excellent dynamic sensing performance of the FBG sensing system in the research of the fluid-structure interaction (FSI) problem. Obvious lowering of the band gaps due to fluid-solid coupling is clearly demonstrated. The results in this work might be useful in research such as active tuning of the band gap and transmission properties of the PCs through fluid-solid coupling.

Surface effect on band structure of flexural wave propagating in magneto-elastic phononic crystal nanobeam

A theoretical model is established to study the size-dependent performance of flexural wave propagation in magneto-elastic phononic crystal (PC) nanobeam with surface effect based on Euler–Bernoulli beam theory and Gurtin–Murdoch theory. Considering the magneto-mechanical coupling constitutive relation of magnetostrictive material, the influence of surface effect on band structure is calculated by the plane wave expansion method for PC nanobeam subjected to pre-stress and magnetic field loadings. Through the example of an epoxy/Terfenol-D PC nanobeam, it can be observed that the characteristics of flexural wave band structures are size-dependent, and remarkably affected by surface effect when the dimension of the PC beam reduces to the nanoscale. The edges and width of the band gap with surface effect are higher than those without surface effect, especially for high frequency region. And surface effect gradually reduces with the increasing of bulk layer-to-surface layer thickness ratio until the band gap descends to a constant for the conventional one in the absence of surface effect. The effects of surface elasticity and piezomagneticity on band gap are more prominent than the residual surface stress. In addition, a distinctly nonlinear variation of band gap appears under the combined effects of pre-stress and magnetic field. Moreover, with the varying of filling fraction, multi-peaks of the width of the band gap are obtained and discussed. These results could be helpful for the intelligent regulation of magneto-elastic PC nanobeam and the design of nanobeam-based devices.