Locally Resonant Phononic Crystal Research Articles

Band gap characteristics of new composite multiple locally resonant phononic crystal metamaterial

Locally resonant phononic crystal (LRPC) exhibit elastic wave band gap characteristics within a specific low-frequency range, but their band gap width is relatively narrow, which has certain limitations in practical engineering applications. In order to open a lower frequency band gap and broaden the band gap range, this paper proposes a new composite multiple locally resonant phononic crystal (CMLRPC). Firstly, the band structure of the CMLRPC is calculated by using the finite element method, and then the formation mechanism of the band gap of the CMLRPC is studied by analyzing its vibration mode, and the band gap width is expanded by adjusting the size of the single primitive cell in the supercell model of the CMLRPC. Secondly, an equivalent mass-spring system model for CMLRPC is established to calculate the starting frequency and cut-off frequency of the band gap, and the calculated results are in good agreement with the finite element calculation. Finally, the frequency response function of the CMLRPC is calculated and its attenuation characteristics are analyzed. Within the band gap frequency range, the attenuation values of the CMLRPC are mostly above 20 dB, indicating a good attenuation effect. Compared with traditional LRPC, this new CMLRPC opens multiple band gaps in the frequency range of 200 Hz, with a wider band gap width and better attenuation effect. In addition, considering both the contact between single primitive cell and the adjustment of their spacing in the supercell model of the CMLRPC, lower and wider band gap can be obtained. The research results of this paper provide a new design idea and method for obtaining low-frequency band gap in LRPC, and can provide reference for the design of vibration reduction and isolation structures in the field of low-frequency vibration control.

Wave Propagation in a LRPC Composite Double Panel Structure with Periodically Attached Pillars and Etched Holes

The locally resonant phononic crystal (LRPC) composite double panel structure (DPS) made of a two-dimensional periodic array of a two-component cylindrical LR pillar connected between the upper and lower composite plates is proposed. The plates are composed of two kinds of materials and periodically etched holes. In order to reveal the bandgap properties of structure theoretically, the band structures, displacement fields of eigenmodes and transmission power spectrums of corresponding 8 × 8 finite structure are calculated and displayed by using finite element method (FEM). Numerical results and further analysis demonstrate that if the excitation and response points are picked on different sides of the structure, a wide band gap with low starting frequency is opened, which can be treated as the coupling between dominant vibrations of pillars and plate modes. In addition, the influences of filled-in rubber, etched hole and viscidity of soft material on band gap are studied and understood with the help of “base-spring-mass” simplified model.

Defect-mode-induced energy localization/harvesting of a locally resonant phononic crystal plate: Analysis of line defects

Phononic crystals that are artificially engineered structures have recently been introduced for vibration energy harvesting and sensing applications due to their unique features of band gaps and wave propagation control. Conventional energy harvesters made of phononic crystals are mainly designed for acoustic energy harvesting at a high-frequency vibration source (i.e., kHz levels). In this work, a defect-mode-induced energy harvester is designed for low-frequency excitations in the range of 0–300 ​Hz. The entire system that is a locally resonant phononic crystal (LRPC) plate with line defect patterns is consisted of elastic-wrapped core scatterers periodically embedded in epoxy resin. A two-dimensional (2D) three-component unit cell structure is arranged on the plate and the band gap property is analyzed to optimize the geometric parameters. Defects are then introduced to the LRPC plate with a 7 ​× ​7 point array for analysis. In addition, numerical and experimental studies are conducted to investigate the performance of energy harvesting when attaching a piezoelectric patch on the defect points. The results demonstrate that the proposed LRPC vibration energy harvester having a line defect mode (with continuous or alternate points) shows good performance in energy harvesting, in which a peak power output of 42.72 ​mV can be achieved under 10 ​m/s2 and 252 ​Hz. The performance is almost 6 times more than that of the single-point defect model under the same excitation conditions. The present LRPC-type energy harvester with a line defect mode is more suitable for energy harvesting for low-frequency and broadband conditions.

Influence Mechanism of a New-Style Resonator on Band Gap of Locally Resonant Phononic Crystal Double Panel Structure

Based on the previous studies on the stubbed-on locally resonant phononic crystal (LRPC) double panel structure (DPS) made of a two-dimensional periodic array of a two-component cylindrical LR pillar connected between the upper and lower plates, the stubbed-on LRPC DPS with soft shell surrounded and simplified model with additional springs surrounded are proposed. According to the changes in structural form, the wider band gap is opened, and the novel formation mechanism of the band gap is revealed. Finite element method (FEM) is applied to calculate the band structures. Numerical results and further analysis demonstrate that the soft shell only affects the bands corresponding to symmetric vibration mode and makes the bands shift up. In addition, the influences of density and Poisson’s ratio of soft shell on the band gap can be ignored, but the starting frequency keeps still, and band gap width increases first and then keeps constant with the increase of elasticity modulus. All the results provide a theoretical basis for the study of vibration and noise resistance in engineering.

Vibration Attenuation Investigations on a Distributed Phononic Crystals Beam for Rubber Concrete Structures

Locally resonant phononic crystals (LRPCs) beam is characterized by the band gaps; some frequency ranges within which flexural waves cannot propagate freely. So, the LRPCs beam can be used for noise or vibration isolation. In this paper, a LRPCs beam with distributed oscillators is proposed, and the general formula of band gaps and transmission spectrum are derived by the transfer matrix method (TMM) and spectrum element method (SEM). Subsequently, the parameter effects on band gaps are investigated in detail. Finally, a rubber concrete beam is designed to demonstrate the application of distributed LRPCs beam in civil engineering. Results reveal that the distributed LRPCs beam has multifrequency band gaps and the number of the band gaps is equal to that of the oscillators. Compared with others, the distributed LRPCs beam can reduce the stress concentration when subjected to vibration. The oscillator interval has no effect on the band gaps, which makes it more convenient to design structures. Individual changes of oscillator mass or stiffness affect the band gap location and width. When the resonance frequency of oscillator is fixed, the starting frequency of the band gap remains constant, and increasing oscillator mass of high-frequency band gap widens the high-frequency band gap, while increasing oscillator mass of low-frequency gap widens both high-frequency and low-frequency band gaps. External loads, such as the common uniform spring force provided by foundation in civil engineering, are conducive to the band gap, and when the spring force increases, all the band gaps are widened. Taken together, a configuration of LRPCs rubber concrete beam is designed, and it shows good isolation on the vibration induced by the railway. By the presented design flow chart, the research can serve as a reference for vibration isolation of LRPCs beams in civil engineering.

The effects of composite primitive cells on band gap property of locally resonant phononic crystal

Locally resonant phononic crystal (LRPC) has the extraordinary property to prohibit the wave propagation in specific low-frequency ranges, however it exists limitation in engineering application due to narrow band gap width. Extensive achievements have been obtained on the locally resonant band gap (LRBG) tunability, whereas existing investigations mainly concern the independent primitive cells structure, which have the limitation in obtaining low-frequency and broadband simultaneously. In this paper, the composited locally resonant phononic crystals (CLRPC) are proposed and the effects of primitive cells contact state on the LRBG properties are investigated. The dispersion curves are applied to obtain the LRBG, and the corresponding modal features are analyzed to explain the band gap formation mechanism. The band structure indicates the design of composite primitive cells is able to increase the band gap number and obtain lower band gap, which is verified by the frequency response function (FRF). For the band gap formation mechanism, the asymmetric vibration due to primitive cells contact leads to diverse and strong coupling response, which generates more band gaps and reduces the band gap starting frequency, therefore the band gaps can be tuned by designing carefully the geometry structure of CLRPC. Further researches on band gap optimization demonstrate that the smaller cell spacing, smaller lattice constant and larger damping of coating layer should be satisfied to obtain broader LRBG and considerable attenuation synchronously. This investigation can provide references for the locally resonant isolation structure design in the low-frequency vibration control field.

Multiple topological interface states in broadband locally resonant phononic crystals

We design a one-dimensional locally resonant phononic crystal (LRPC) comprised of a substrate beam periodically attached with twin resonators. By alternating the position of the resonators, the bandgap forming mechanisms of the LRPC, namely, the band folding-induced bandgaps (BFBGs) and the locally resonant bandgap (LRBG), are analyzed. A broadband “pseudo-bandgap” formed by the LRBG and BFBG can be achieved. The topological properties of the LRPC are then studied, and it is found that the topological interface states can occur only in the BFBGs but not in the LRBG. By constructing a finite LRPC formed by two PCs with distinct topological properties connecting with each other, we numerically and experimentally demonstrate the existence of multiple topological interface states in the BFBGs. The interface state within the subwavelength regime can be achieved with strong energy localization and is little affected by material damping, while for the interface state at high frequencies, it is shown that damping could considerably weaken the energy localization. This work provides guidelines for the design of low-frequency elastic topological systems.

Research on multi-band structural noise reduction of vehicle body based on two-degree-of-freedom locally resonant phononic crystal

First, the dispersion curve of a two-degree-of-freedom locally resonant phononic crystal (LRPC) is calculated by the plane wave expansion method. The result shows that the structure has the characteristics of double bandgaps in the mid- and low-frequency range, which is verified by the finite element method. Then, the frequency response analysis of the interior noise and the acoustic contribution analysis of car panels are carried out, and the front left floor is found out to be the panel with a larger contribution of the peak value of noise. Therefore, the LRPC is arranged on the front left floor based on the modal strain energy, and then the total vibration level and the radiated sound power level of the panel are calculated, which are both decreased by 11 dB to 13 dB after the arrangement of LRPC. The acoustic simulation analysis of the whole vehicle after the arrangement is also carried out, which shows that the original sound pressure level peaks at 270 Hz, 346 Hz and 378 Hz are reduced by 6.2 dB(A), 5.56 dB(A) and 9.35 dB(A), respectively. The results provide guidance for the application of LRPC in the field of structural noise reduction of engineering plates.

Isolating low-frequency vibration from power systems on a ship using spiral phononic crystals

There is increasing concern that the locally resonant phononic crystals (LRPCs) are used to reduce low-frequency noise and vibration in ship and ocean engineering. In this work, the vertical velocity amplitude spectrums in “ChangJing 9″ are measured following the ISO international standards. According to the low-frequency mechanical vibration generated by the power systems on the ship, a single-phase spiral-shaped phononic crystal is designed and analyzed based on the locally resonant modes. The vibration characteristics of the phononic crystal plates with different boundary conditions and loads are numerically simulated, and groups of experimental tests are conducted to verify the abilities of vibration isolation for three kinds of phononic crystals. Furthermore, a vibration isolation platform installed on a steel plate is also studied by numerical simulations and experiments. The results show that the spiral phononic crystal has good adaptability for a wide range of low-frequency band, around 15–45 Hz, by altering the cylinders conveniently. The LRPC has a potential application to isolate vibration for protecting electronic devices and precision instruments despite the complex vibration sources and boundaries on the ship. In short, this work is an active exploration of putting the LRPCs into the field of ship vibration and noise control.

Wave-resistance sleeper with locally resonant phononic crystals: Bandgap property and vibration reduction mechanism

As an artificially engineered metamaterial, locally resonant phononic crystals (LRPCs) exhibit great potential for the vibration reduction of urban rail transit due to their capability to prohibit the propagation of elastic waves with long wavelengths. This study proposes a wave-resistance sleeper with negative dynamic mass density by periodically embedding LRPCs into the concrete sleeper and provides new approaches to improve its bandgap property. Furthermore, the vibration reduction effect of the wave-resistance sleeper applied in a slab track was evaluated with a three-dimensional half-track model. The results show that the wave-resistance sleeper has two bandgaps below the frequency of 2000 Hz and can effectively prohibit the vibration within the bandwidth from transmitting to the track bed. Setting intervals in the rubber coating and increasing the void fraction can significantly reduce boundary frequencies of the first bandgap, whereas the second bandgap attenuates completely as the vibration can no longer be localized in the discontinuous rubber coating. A larger metal core is beneficial for obtaining a wider bandgap in the low-frequency range, whereas a thicker rubber coating can be used to obtain a wider bandgap in the medium-frequency range. To obtain wider bandgaps, the steel tube used should be as thin as possible and the use of an oval metal core should be avoided. It is hoped that the research results can provide a useful reference for the vibration reduction of track in rail transit.

Optimal geometric parameters for a double panel structure with low frequency forbidden characteristics

In this paper, a locally resonant phononic crystal (LRPC) double panel structure with optimal geometric parameters is investigated. The present LRPC structure can open band gaps (BGs) with low frequency forbidden characteristics. The effects of several parameters such as the thickness of steel and aluminum plates, the width of elastic beams, the side length and height of cuboid scatterer onto BGs are analyzed. The thickness of the steel and aluminum plates as well as the width of elastic beams is identified to be the most influential factors on BGs. Seventy-seven experimental runs based on the three-factor and seven-level experimental design are performed to complete response surface analysis, three functional relationships are obtained between the three-factor and the terminating frequency of the second BG, the bandwidth of the first BG, and the bandwidth of the second BG. Additionally, the interior point method is also applied to optimize the closed-form expressions of functional relationships to obtain the optimal LRPC double panel structure with the low-frequency forbidden properties from 90 Hz to 250 Hz.

Bandgap Properties of Two-Layered Locally Resonant Phononic Crystals

Multi-layered locally resonant phononic crystals (LRPCs) with wider and multiple bandgaps (BGs) in low frequency range and small size of the unit cell have promising applications in noise and vibration controls. In this paper, a 2D two-layered ternary LRPC consisting of a periodical array of cylindrical inclusions embedded in an epoxy matrix is investigated by the finite element method (FEM), where the inclusion is comprised of two coaxial cylindrical steel cores with rubber coating. It is found that the size of the inclusion of the 2D two-layered ternary LRPC has significant effects on the BG properties. With the increase of the core radius and coating thickness, the first BG would shift to lower frequency range with its width decreasing, and the second BG width would become wider until the third BG appears. Especially, with the increase of the coating thickness, more bands and BGs would appear in the lower frequency range. Based on the formation mechanisms of the BGs, several mass-spring models to predict the frequencies of the first two BG edges are developed. The results calculated by these mass-spring models are in good agreement with those by the FEM except for the upper edge frequency of the second BG when the rubber coating thickness exceeds a certain value and the third BG is opened up. These proposed mass-spring models would allow for quick pre-estimation of the resonance frequencies, and facilitate the selection of possible parameters for the wider and lower frequency BGs to obtain the desired attenuation bands. The studies would also benefit the design of multiple BGs for some device applications.

Low-frequency band gaps in Y-type phononic crystals

In this paper, an Y-type of locally resonant phononic crystal (LRPC) material model is analyzed in detail. Some geometric parameters of the material model are investigated using FEM to assess the trends of band gaps (BGs). The BGs results show that the material model can open a low-frequency BG less than 200 Hz. When the lattice constant of the material model cell is determined, the geometrical parameters of scatterers are found to be the key to open BGs. The results show that the effect of the scatterer on BGs is studied by adjusting the two ratios (the ratio of the square bottom edge length to height of the scatterer composed of lead and the ratio of the square bottom edge length to height of the scatterer composed of aluminium), respectively. The changes of the two ratios have little effect on the upper edge of BGs. Similarly, the change of the ratio of the scatterer composed of aluminum has little effect on the lower edge of BGs. However, the variation of the ratio of the scatterer composed of lead can make the lower edge of BGs lower and increase BGs width. The vibration and noise reduction devices with different working frequencies can be designed by adjusting the geometric parameters of the scatterer.

Extending bandgap method of concentric ring locally resonant phononic crystals

Locally resonant phononic crystals (LRPCs) have the capacity to adjust elastic waves with the structure sizes much smaller than the incident wavelengths, the unique property is called low-frequency bandgap, but it is not easily applied in practical engineering because of narrow bandgap width. Multilayered LRPCs are helpful in generating several bandgaps, in the meanwhile the designs of multilayered LRPCs proposed in previous study result in the larger filling fraction, whereas the bandwidth of LRPCs increases monotonically with filling fraction, thus the pure contribution of concentric ring configuration to the bandwidth extending is less involved. Keeping the filling fraction constant, this paper carefully designs the microstructure of concentric ring locally resonant phononic crystals, and investigates the effects of structure configuration on the bandgap property. To this end, an updated improved plane wave expansion (UIPWE) method is developed to calculate the band structure, and finite element method (FEM) is used to obtain transmission spectra and vibration mode. The results demonstrate that UIPWE method is valid and is able to give precise outcomes, which is verified by FEM. In addition, the concentric ring configuration equivalently produces dual-oscillator system, relative movements between the oscillators generate coupling effect, thus, the bandgaps can be extended by configurating rightly the microstructure of single cell. Further studies about different models indicate that the combination of smaller inner scatterers and larger inner coating layers are beneficial to wider bandgap. These conclusions presented herein provide insights in the design of three-component PCs in multi-frequency vibration control field.

Structural parameters optimization of a comb-like structure using locally resonant phononic crystals

In this paper, a comb-like locally resonant phononic crystal (LRPC) with optimal structural parameters, which has good low frequency and broadband band gaps (BGs) between 20–250 Hz, is investigated numerically. With the intention of obtaining the optimal structural parameters, based on the structures with different number of the short elastic beams, 2-factor (the two key structural parameters, i.e. the width of the scatterer and the thickness of elastic beams) and 7-level numerical experiments are designed to obtain simulations data using finite element method (FEM). The functional relationships are further constructed using the response surface method (RSM) analysis, i.e. the relationship of the starting frequency of the first BG and the two factors, the terminating frequency of the second BG and the two factors, and the relationship of the total bandwidth of the first two BG and the two factors. After calculation using interior point method, the BGs of LRPC with optimal structural parameters are determined with lower and wider BGs below 250 Hz.

Optimal bandgaps of a spiral structure based on locally resonant phononic crystals

A spiral locally resonant phononic crystal (LRPC) with the optimal bandgaps (BGs) between 20 and 250 Hz is proposed. The single factor analysis of three key geometric parameters (the thickness of spiral elastic beams, the side length of square scatterers and the spirals’ turns) are performed to obtain corresponding influences on BGs, two-factor (the thickness of spiral elastic beams and the side length of the square scatterer) and seven-level experiment under the four fixed spirals’ turns are designed to obtain optimal BGs with better low-frequency broadband properties. According to numerical experiments, BG data of present spiral LRPCs can be calculated by finite element method (FEM). Relationships (i.e., the relationship between the starting frequency of the first BG and the two factors, the relationship between the bandwidth of the first BG and the two factors, and the relationship between the bandwidth of the second BG and the two factors) are further obtained using the response surface methodology (RSM). The optimal BGs with better properties are finally calculated using the interior point method.

Locally resonant phononic crystals band-gap analysis on a two dimensional phononic crystal with a square and a triangular lattice

Locally resonant phononic crystals (LRPC) are a new type of sound insulating material. Using the plane wave expansion method based on the Bloch theorem, we compute the band structure of two dimensional (2D) phononic crystals (PC) with square and triangular lattices. Such PC typically consists of infinitely long carbon rods coated with silicon rubber and embedded in an elastic background. Computational results show that gaps appear at the lower frequency range, which are lower than those expected from the Bragg mechanism. Those gaps are generated due to local resonances; the optimum gap is obtained by tuning the thickness ratio of the coating layer. The gap created by the LRPC depends on the filling fraction of the coating cylinders.

Vibration reduction of car body based on 2D dual-base locally resonant phononic crystal

Due to the coating feature and bandgap adjustability of dual-base locally resonant phononic crystal (LRPC), the purpose behind this paper is to explore the application of dual-base LRPC in vibration reduction of car body. In previous research, the band structure and its influencing factors of dual-base LRPC have been investigated theoretically. As an artificial periodic structure, the key factors of application, including the number of unit cells, the curvature of base panel and the boundary constraint, have been taken account into analyzing bandgap characteristics of dual-base LRPC by finite element method (FEM). The results indicate that the non-infinite dual-base LRPC structure with small curvature and boundary constraint can still produce the expected bandgap. Then, according to the acoustic contribution analysis, the floor has been determined as the target panel for vibration reduction, on which the dual-base LRPC has been conducted. In order to obtain the wider bandgap with less added mass, the lattice constant and stiffness have been matched and optimized. With the final arrangement, the total body vibration level sharply reduces by 15 dB, and the interior noise reduces by 9 dB as well. It provides a new method for the vibration reduction of car body.

Optimization for a Locally Resonant Phononic Crystal of Square Spiral With Circle Inside

This paper presents an optimization scheme to better design phononic crystals. A locally resonant phononic crystal (LRPC) structure called square spiral with circle inside is employed to verify the performance of the present scheme. Four geometric parameters, i.e., the side length of square scatterer, the length of each elastic beam, the thickness of elastic beams, and the radii of inner circles, are considered to obtain the corresponding influences on band gaps (BGs) using finite element method (FEM). According to the significant influences of the late two key parameters, a 2-factor (the radii of inner circles, and the thickness of elastic beams) and 7-level experiment is designed to obtain optimal BGs with better low-frequency broadband properties. By 29 times calculations using FEM for the different combination of levels, three relationships between the 2-factor and the first BG’s starting frequency, the first BG’s bandwidth, and the second BG’s bandwidth, are obtained and severed as inputs to the software of response surface methodology (RSM). The closed-form expressions of the three relationships are finally obtained to construct optimization models and result in the optimal band gaps (BGs) between 190–300 Hz and 500–600 Hz . It is expected that the present optimization scheme can be extended to material design of phononic/photonic structures in a reasonable way.

The band gap properties of the three-component semi-infinite plate-like LRPC by using PWE/FE method

This paper applies coupled plane wave expansion and finite element (PWE/FE) method to calculate the band structure of the proposed three-component semi-infinite plate-like locally resonant phononic crystal (LRPC). In order to verify the accuracy of the result, the band structure calculated by PWE/FE method is compared to that calculated by the traditional finite element (FE) method, and the frequency range of the band gap in the band structure is compared to that of the attenuation in the transmission power spectrum. Numerical results and further analysis demonstrate that a band gap is opened by the coupling between the dominant vibrations of the rubber layer and the matrix modes. In addition, the influences of the geometry parameters on the band gap are studied and understood with the help of the simple “base-spring-mass” model, the influence of the viscidity of rubber layer on the band gap is also investigated.