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Enhanced acoustic localization in the two-dimensional phononic crystals with slit tube defect
Physics Letters A 383 (2019) 125918
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Physics Letters A www.elsevier.com/locate/pla
Enhanced acoustic localization in the two-dimensional phononic crystals with slit tube defect Xiaopeng Wang ∗ , Hui Sun, Tianning Chen, Xingguo Wang School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 15 June 2019 Received in revised form 21 August 2019 Accepted 23 August 2019 Available online 29 August 2019 Communicated by B. Malomed Keywords: Phononic crystal Enhanced acoustic localization Band gap Structural design
a b s t r a c t In this paper, the acoustic localization characteristics of the two-dimensional phononic crystals with slit tube defect are investigated theoretically and experimentally. In contrast to the typical formation pattern of defect states, the proposed defect states are created by replacing a slotted tube in the center of perfect phononic crystal. Compared to perfect phononic crystal, the proposed structure can effectively localize waves of specific frequencies in the point defect and improve the acoustic pressure amplification. Then the effects of the geometric parameters of the slotted tube on the acoustic localization characteristics are studied. Numerical results show that the resonant frequency and acoustic pressure amplification amplitude could be effectively modulated by the geometric parameters of the slotted tube. Experimental results are in good agreement with the simulation results. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Phononic crystal (PC) is a new type of artificial periodic structure, which shows unique bandgap characteristics. When acoustic waves propagate into the phononic crystals, incident waves at certain frequency range are forbidden to propagate in the structure, thereby forming an acoustically forbidden band [1–3]. At present, the methods to calculate the dispersion relation of the PCs mainly are plane wave expansion method (PWE) [4], transfer matrix method (TMM) [5], multiple scattering method (MSM) [6], time domain finite difference method (TDFD) [7] and finite element method (FEM) [8]. The PCs could be divided into two types according to the formation mechanism of band gaps, one is the Bragg mechanism [9–11] and the other is the local resonance mechanism [12–14]. With the presence of a defect in the perfect PCs, the elastic wave at the particular frequency will be concentrated at the point defect or propagate along the defect, and the acoustic pressure in the defect center will be amplified, thus show the characteristics of acoustic resonant cavity or waveguide [15–18]. Therefore, the phononic crystal structure with point defect is usually called as phononic crystal resonator (PCR). Phononic crystals have potential applications in vibration reduction and noise reduction due to their unique band gap properties. Cui et al. [19] introduced the cavity into the phononic
*
Corresponding author. E-mail address: [email protected] (X.P. Wang).
https://doi.org/10.1016/j.physleta.2019.125918 0375-9601/© 2019 Elsevier B.V. All rights reserved.
crystal by means of slotting to achieve low-frequency acoustic absorption, and constructed a generalized phononic crystal using a bandgap method to achieve the purpose of broadband and low frequency sound insulation. Romero-García et al. [20] introduced multi-resonant elements in phononic crystals for low-frequency sound insulation. On the basis of the phononic crystal structure of the slotted tube, Bao et al. [21] introduced the arc and flat bar structure inside the circular tube, calculated the band structure and transmission characteristics of the structure, and further reduced the frequency band gap below 500 Hz. Zhou et al. [22] investigated band structures and transmission spectra of two dimensional multilayered locally resonant phononic crystals (LRPCs) by the finite element method, and found that the band gaps of the LRPCs can be extended to several frequency ranges by periodically embedding multilayered coaxial inclusions into a matrix. In addition to the band gap properties, the defect state characteristics of phononic crystals also show excellent applications in acoustic waveguide and energy harvesting. Therefore, it is of great significance to explore them in depth. Khelif et al. [23–25] studied theoretically and experimentally the characteristics of elastic wave propagation in two-dimensional phononic crystals with point defects and line defects. It was found that localized defect band exist in the band gap of phononic crystals. He et al. [26] studied the effect of defects formed by removing scatterers and changing the position of scatterers on the defect state of phononic crystals in supercells. Peng et al. [27] proposed an effective method to induce valley polarization in SL MoSSe by magnetic doping, and studied the defect state of two-dimensional valleytronic materials. Analo-
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Fig. 1. (a) Schematics of the perfect supercell. (b) Schematics of the normal PCR. (c) Schematics of the PCR with slotted tube defect. (d) Finite PCR simulation model.
gous to electrons in semiconductors, introducing defect states into phononic crystals could change the bandgap and control the propagation of sound waves. Li et al. [28] introduced point defects by changing the scatterer material parameters, studied the effect of changes in defect material parameters on the defect state characteristics, and provided new ideas for the formation of defects. Rostami-Dogolsara et al. [29] designed a new type of differential filter based on phononic crystals. This structure consists of two line-defects and a ring-like phononic crystal resonant cavity. It has a very high selective transmission for specific wavelengths of acoustic waves. Increase transmission properties by increasing the number of scatterers. Ouyang et al. [30] used a combination of the phononic crystal waveguide and the symmetric and asymmetric modes of the resonant cavity to design a one-way acoustic waveguide. The structure achieves the conduction and non-conduction of the waveguide through the modal matching and mismatch between the resonant cavity and the line defect. Zhou et al. [31] investigated the flexural elastic wave propagation properties in phononic crystals, and found that point and line defects can be created by properly applying an electric field to some of the cells. These defects bring defect bands inside the original band gaps and lead to elastic wave confinement in the point defect or along the line defect. At present, the research about acoustic energy harvesting using phononic crystal attracted the attention of scholars. To improve the efficiency of acoustic energy harvesting, most of harvesters are consisted of the Helmholtz resonator and piezoelectric materials. Jiang et al. [32] produced a sound energy acquisition device that can be powered by a micro-electromechanical system based on the local modal design of the acoustic defect of a line defect phononic crystal. Sun et al. [33] designed a new sound energy collection system, which mainly includes an acoustic resonator and a piezoelectric bimorph. Oudich [34] et al. designed a phononic crystal plate with point defects. Resonant beads were arranged on a thin elastic plate through a spring connection cycle. Park [35] et al. proposed a systematic design of two-dimensional octagonal phononic crystals through geometric and band gap optimization process. Using the defect states of phononic crystals, the piezoelectric energy acquisition was greatly improved. The acoustic localization of the phononic crystal structure is a key factor affecting the working efficiency and transmission quality of the acoustic device. If the phononic crystal structure has enhanced localization of the acoustic wave at the defect state frequency, the transmission quality of the acoustic control device will be effectively improved. The study of the acoustic localization ability of different types of phononic crystal resonator and the design of defect structures with strong acoustic confinement are beneficial to improve the transmission quality of acoustic energy harvester and waveguide based on phononic crystal defect states. In this paper, the point defect structure of phononic crystal is formed by the slotted tube to replace the seamless tube. The acoustic localization characteristics of the PCR with slotted tube defect are studied. The dispersion relationships and acoustic pres-
sure amplification spectra of structures are calculated by using the finite element method. The reasons for the enhanced acoustic pressure amplification in the resonator are analyzed. Then the defect state adjustment characteristics of the resonator are studied. The resonant frequency and acoustic pressure amplification value can be tuned by changing the geometric parameters of the slotted tube defect. Finally, the acoustic pressure amplification ability of the resonator is verified experimentally. The experimental results are in good agreement with the simulation calculation. 2. Model and method of calculation The PCR presented in this paper consists of periodical circular tubes and a slotted tube defect. The defect tube is designed to be slit in the +x, −x, + y, and − y directions. The seamless tube is replaced by a four-slot circular tube at the center of the supercell, forming a slotted tube defect resonator. As shown in Fig. 1(a), the lattice constant of the perfect supercell is a = 0.08 m, the radius of the seamless tube is r = 0.036 m. The phononic crystal resonator is normally formed by removing scatterers as shown in Fig. 1(b). Fig. 1(c) shows the PCR with slotted tube defect, the radius of the slotted tube is R = 0.02 m, slot width w = 0.005 m, tube thickness t = 0.005 m; the phononic crystal resonator involved in this paper are all made up of polymethyl methacrylate (PMMA) tubes arranged periodically in the air. The velocity of longitudinal wave in PMMA is υ = 3000 m/s, and the density is ρ = 1390 kg/m3 ; the wave velocity of air is υ = 343 m/s, and the density is ρ = 1.25 kg/m3 . Dispersion relationships, power transmission spectra and intrinsic modal analysis are the most effective means on PCs research. This paper will use the FEM combined with Bloch theorem to calculate the PCs’ dispersion relationships and power transmission spectra. All the FEM calculations in this paper are implemented by the MULTIPHYSICS COMSOL 5.0. The dispersion relationship shows the relationship between the angular frequency ω and the Bloch wave vector, from which the band gaps of the PC can be observed directly. According to Bloch theorem [18]:
U (r + a) = e ika U (r )
(1)
Because of the symmetry and periodicity of the PCs in both x and y directions, only the supercell shown in Fig. 1(b) and (c) need to be considered. Periodic boundary condition is applied on the boundary of the adjacent cell, and the dispersion relationships will be obtained by simply scanning the Bloch wave vector along the border line of irreducible Brillouin zone. In order to investigate the localization characteristics of the nested PCRs, the transmission spectra of the structure are also calculated by FEM. Fig. 1(d) shows the finite PCR simulation model, the incident plane acoustic pressure of 1 Pa is applied to the left boundary of the PCR, then the acoustic pressure P at the center of the PCR was extracted. The acoustic pressure amplification is defined as the ratio of the acoustic pressure in the cavity center to
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Fig. 2. The dispersion relations of the perfect supercell. (b) The dispersion relations of the normal PCR. (c) The dispersion relations of the PCR with slotted tube defect.
the incident acoustic pressure. In order to reduce the influence of acoustic reflection at the boundary, the other boundaries are applied plane wave radiation. 3. Numerical results and discussions 3.1. Band structures and acoustic pressure amplification spectra of multiple nested PCRs There are two conditions for the localization of phonon crystal defect states. First, the perfect phononic crystal has a complete band. Secondly, the frequency of defect state should be located in the range of band gap. Fig. 2 shows the band characteristics of the perfect phononic crystal supercell and two phononic crystal resonators shown in Fig. 1(b) and Fig. 1(c). The perfect phononic crystal supercell has a complete band gap between 1740-2920 Hz. In this frequency range, acoustic wave cannot propagate to the other side of the structure. While the PCR formed by removing the scatterer, its complete band gap in the band structure is basically consistent with the perfect supercell. Due to the formation of the point defect by removing scatterer, the defect state flat band appears in the band structure, of which the corresponding frequency is around 2560 Hz. With the introduction of the slotted tube defect in the perfect PC supercell, the complete band gap in the band structure remains unchanged, but the defect state flat band appears in the band gap. The corresponding frequency is around 2400 Hz, which is lower than normal PCR. By introducing point defects into perfect phononic crystal supercells, waves of specific frequencies are localized in the point defects. The acoustic pressure transmission characteristics of a single finite PCR structure at the resonator center are further studied. The calculation result is shown in Fig. 3. Fig. 3(a) shows the acoustic pressure amplification of two PCR. The resonant acoustic pressure amplification of the normal PCR (NPCR) is 8.5, and the resonant frequency is 2560 Hz, while the acoustic pressure amplification of the PCR with slotted tube defect (SPCR) is 24.3, which is much better than that of the normal PCR and it has two peaks of acoustic pressure amplification in the range of 1000-4000 Hz. The peak frequency of SPCR’s first sound pressure amplification is close to that of conventional PCR. It is due to the introduction of slotted tube defect in the center of phononic crystal and a flat band is generated in the band gap. The acoustic wave at the corresponding frequency is focused on the point defect. In the vicinity of 3000 Hz, a new flat band is generated due to the introduction of point defect, but the forbidden band width is narrow and the defect state is not obvious. However, due to the Helmholtz resonance effect of the slit tube, the sound pressure amplification capability at the center
of the cavity is enhanced, so a smaller sound pressure amplification peak appears in the SPCR. Then the resonant acoustic pressure distribution at the centerline of the resonator (x = 0) are extracted, as shown in Fig. 3(b). The acoustic pressure at the center of the PCR with slotted tube defect is higher than the normal PCR. The acoustic pressure distributions of the two resonators at the resonance frequency are shown in Fig. 3(c) and (d) respectively. The acoustic pressure at the center of the resonators is all significantly higher than that of the surrounding space. Compared to the normal PCR, the PCR with slotted tube defect shows enhanced acoustic localization, which has larger acoustic pressure at the center of the cavity, and smaller and more concentrated acoustic energy convergence area. As shown in Fig. 4(a), the single-slot tube can be equivalent to a Helmholtz cavity. According to the Helmholtz resonance theory, there is an abrupt change in the acoustic pressure at the slot position at the resonant frequency. As R = 0.015 m, w = 0.01 m, t = 0.005 m, simulation results show that the resonance frequency is 2242 Hz. The acoustic pressure distribution on the slotted tube and the sound pressure distribution along the transverse and longitudinal center lines of the slotted tube are extracted respectively, as shown in Fig. 4(b) and (c). It clearly shows that the acoustic energy is mainly concentrated in the slotted tube cavity. The acoustic pressure amplification in the cavity is 5.3. There is an acoustic pressure mutation at the slot position. Due to the existence of the acoustic pressure mutation at the slit position, the acoustic pressure presents an asymmetrical distribution at horizontal axes. The slotted tube shows Helmholtz resonance effect, and the normal PCR shows acoustic localization characteristics at the defect state frequency as well. As the resonant frequency of the slotted tube is close to resonant frequency of the normal PCR, the acoustic pressure amplification capability of the PCR with slotted tube defect would be enhanced due to the coupling of local resonant characteristics. 3.2. Parametric study on the acoustic localization characteristics The enhanced acoustic localization of the PCR with slotted tube is mainly caused by the coupling enhancement of two resonant cavities. While the lattice constant and scatterer radius of the outer tubes keep constant, the acoustic localization characteristics of the combined structure are mainly related to the internal slotted tube. First, we studied the regulation of the acoustic localization characteristics by the slotted tube radius R. The PC lattice constant a = 0.08 m, the scatterer radius r = 0.036 m, the slit width w = 0.005 m, and tube thickness t = 0.005 m remained unchanged. The localization characteristics of the structure were calculated when the
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Fig. 3. (a) Acoustic pressure amplification spectra of the PCRs. (b) Acoustic pressure distribution in the center line (the coordinate x = 0) of the PCRs. (c) Acoustic field distribution of normal PCR at resonant frequency. (d) Acoustic field distribution of PCR with slotted tube defect at resonant frequency. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 4. (a) Schematics of the slotted tube. (b) Acoustic pressure distribution of the slotted tube at resonant frequency. (c) Acoustic pressure distribution in the center line (the coordinate x = 0 and y = 0) of the slotted tube.
slotted tube radius are R = 0.015 m, 0.02 m, 0.025 m, 0.03 m, and 0.035 m. Fig. 5(a) shows the acoustic pressure amplification of the PCR with slotted tube defect. It can be seen that the resonant acoustic pressure amplification has the best corresponding slotted diameter. When R = 0.02 m and 0.025 m, the resonant pressure amplification is more than 24. Fig. 5(b) is consistent with the results in Fig. 5(a), further confirming that the slotted tube radius has an optimal value for the adjustment of the acoustic localization characteristics. Keeping other geometric parameters unchanged, only changing the slit width of the PCR, the adjustment rule for acoustic localization characteristics is shown in Fig. 6. As the slit width is increased from 0.006 m to 0.014 m, the resonant acoustic pressure amplification of the PCR with slotted tube gradually decreases, and the resonant frequency gradually increases. According to the Helmholtz
equivalent model of the slotted tube, the slit width increases, this is equivalent to the increase of the cross-sectional area of the short tube in the Helmholtz resonator, and therefore the resonant frequency increases. The resonant acoustic pressure distribution and acoustic pressure amplification spectrum results are consistent. Then we calculate the acoustic pressure amplification spectrum and acoustic pressure distribution of the PCR. The PC lattice constant a = 0.08 m, the scatterer radius r = 0.036 m, the slotted tube radius R = 0.02 m, the slit width w = 0.003 m, while the tube thickness t = 0.001 m, 0.002 m, 0.003 m, 0.004 m, 0.005 m, and 0.006 m, respectively. Fig. 7 shows the effect of tube thickness on acoustic localization characteristics. When t = 0.001 m increases to 0.003 m, the resonant frequency gradually shifts to low frequency; as the tube thickness increases further, the resonant frequency starts to shift to high frequency again, but the overall frequency
X.P. Wang et al. / Physics Letters A 383 (2019) 125918
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Fig. 5. (a) Acoustic pressure amplification spectrum of the slotted defect PCRs with different slotted tube radius. (b) Resonant acoustic pressure distribution in the center line (the coordinate x = 0) of the slotted defect PCRs. (c) Acoustic pressure distribution of the slotted defect PCRs at resonant frequency.
Fig. 6. (a) Acoustic pressure amplification spectrum of the slotted defect PCRs with different slit width. (b) Resonant acoustic pressure distribution in the center line (the coordinate x = 0) of the slotted defect PCRs. (c) Acoustic pressure distribution of the slotted defect PCRs at resonant frequency.
shift is not large. With the increase of the tube thickness, the resonant acoustic pressure amplification of the PCR increases gradually. When t = 0.005 m, the acoustic pressure amplification is up to 38. The resonant acoustic pressure distribution of the PCR shown in Fig. 7(b) and (c) further proves this point. 3.3. Experimental result and discussion In order to verify the acoustic localization characteristics of phononic crystal resonators, an experimental platform for the acoustic pressure amplification of the PCR with slotted tube was
established in this paper, and the test sample was prepared. As shown in Fig. 8(a), the experimental sample consists of 25 tubes; the tubes are periodically fixed on the organic glass plate. The lattice constant of the PCR sample is 0.08 m. The outer radius of the seamless tube is 0.035 m, and the tube thickness is 0.002 m. The outer radius of the slotted-defect is 0.02 m, and its tube thickness is 0.004 m, the slit width is 0.008 m. All the constituent tubes in the test sample have a length of 0.5 m. In order to verify the simulation results, the acoustic localization characteristics are reflected by testing the acoustic pressure amplification at the defect center of the PCR. The environment and
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Fig. 7. (a) Acoustic pressure amplification spectrum of the slotted defect PCRs with different tube thickness. (b) Resonant acoustic pressure distribution in the center line (the coordinate x = 0) of the slotted defect PCRs. (c) Acoustic pressure distribution of the slotted defect PCRs at resonant frequency.
Fig. 8. (a) The experiment sample of the PCR with slotted tube. (b) The environment and apparatus of the experiment.
apparatus of the experiment are shown in Fig. 8(b). The experimental apparatus consists of a signal generator, a power amplifier, a speaker, a microphone, an M+P data acquisition card, and a tripod. The experimental test is completed in a half anechoic chamber. The experimental and simulation results are shown in Fig. 9. Experimental results show that the resonant frequency of the normal PCR is 2506 Hz, and the resonant acoustic pressure amplification is 1.83, while the resonant frequency of the PCR with slotted tube defect is 2470 Hz, and the resonant acoustic pressure amplification is 5.47. Comparing the test results, it is found that the acoustic localization capacity of the PCR with slotted tube defect is stronger than that of the normal PCR, and there are two resonant pressure amplification peaks in the 1500-4000 Hz range. Comparing the test results and the simulation results, the test results of the resonant frequency corresponding to the two PCRs are basically consistent with the simulated values. However, there is a big difference between the experimental test results and the simulation results of the resonant acoustic pressure amplification. The experimental results of the acoustic pressure amplification of the two PCRs are obviously smaller than the simulation results. The reason for this result is analyzed as follows. Firstly, the main reason for the defect state of the PC is the periodic arrangement of the cells, which has a higher requirement for periodicity. Due to the errors in the processing and assembly of experimental samples, the acoustic localization characteristics of the PCRs are weakened. Secondly, while the acoustic pressure at the center of PCRs is cal-
culated by simulation, the acoustic pressure pickup point is at the geometric center of the resonant cavity. But in the actual test process, the microphone is fixed by a tripod and then projected into the center of the cavity to measure the acoustic pressure. Due to the inability to fully guarantee that the measurement position of the microphone completely coincides with the geometric center of the resonator, an error occurs in the measurement value. Thirdly, while the simulation model is a two-dimensional structure, the experimental sample is a three-dimensional structure. The reflection of acoustic wave in the Z direction during the measurement affects the accuracy of the experimental results. Through experimental tests, the PCR with slotted tube shows better acoustic localization capability than the normal PCR. The experimental results are basically consistent with the simulation results. 4. Conclusion In this paper, the slotted circular tube is introduced into the phononic crystal as a novel point defect type. The acoustic localization capability of the PCR with slotted tube defect is studied by the finite element method. Through the calculation, the band structure and acoustic pressure distribution in the cavity of the phononic crystal resonator with slotted tube defect has been studied. Compared with the Helmholtz resonator, this structure can realize the acoustic concentration of multi-directional incident acoustic wave. Compared with the normal PCR, this structure has stronger acous-
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Fig. 9. (a) The experimental and simulation acoustic pressure amplification spectra of the PCR with slotted tube. (b) The experimental and simulation acoustic pressure amplification spectra of the normal PCR.
tic localization ability. The mechanism of the enhanced acoustic confinement in the cavity of the structure is explained as the coupling of Helmholtz resonator and normal PCR. The regulation of the resonant frequency and acoustic pressure amplification by the slotted tube are studied; the geometric parameters of slotted tube can effectively adjust the acoustic localization capacity of the resonator. The resonant acoustic pressure amplification of the PCR with slotted tube defect is investigated experimentally; the experiment result is basically consistent with the calculation result in the error tolerance range. The PCR with slotted tube defect shows better acoustic localization ability than normal PCR, and could be applied to the design of acoustic manipulation devices such as acoustic waveguides and acoustic sensors. Acknowledgements The authors gratefully acknowledge financial support from the Project of National Natural Science Foundation of China (Grant No. 51675402). References [1] Y. Yao, Z. Hou, F. Wu, X. Zhang, Physica B 406 (2011) 11. [2] J.F. Robillard, O.B. Matar, J.O. Vasseur, P.A. Deymier, Appl. Phys. Lett. 95 (2009) 12. [3] C. Lagarrigue, J.P. Groby, V. Tournat, J. Acoust. Soc. Am. 133 (2013) 1. [4] J.O. Vasseur, P.A. Deymier, B. Chenni, B. Djafari Rouhani, L. Dobrzynski, D. Prevost, Phys. Rev. Lett. 86 (2001) 3012. [5] Z.Y. Li, K.M. Ho, Phys. Rev. B 68 (2003) 155101. [6] L.M. Li, Z.Q. Zhang, Phys. Rev. B 58 (1998) 9587. [7] Y. Tanaka, Y. Tomoyasu, S.I. Tamura, Phys. Rev. B 62 (2000) 7387. [8] C. Goffaux, J. Sanchez Dehesa, Phys. Rev. B 67 (2003) 144301. [9] C.S. Kee, J.E. Kim, H.Y. Park, K.J. Chang, J. Appl. Phys. 87 (2000) 4. [10] Z.L. Hou, X.J. Fu, Y.Y. Liu, Phys. Lett. A 317 (2003) 1.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Enhanced acoustic localization in the two-dimensional phononic crystals with slit tube defect Xiaopeng Wang ∗ , Hui Sun, Tianning Chen, Xingguo Wang School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 15 June 2019 Received in revised form 21 August 2019 Accepted 23 August 2019 Available online 29 August 2019 Communicated by B. Malomed Keywords: Phononic crystal Enhanced acoustic localization Band gap Structural design
a b s t r a c t In this paper, the acoustic localization characteristics of the two-dimensional phononic crystals with slit tube defect are investigated theoretically and experimentally. In contrast to the typical formation pattern of defect states, the proposed defect states are created by replacing a slotted tube in the center of perfect phononic crystal. Compared to perfect phononic crystal, the proposed structure can effectively localize waves of specific frequencies in the point defect and improve the acoustic pressure amplification. Then the effects of the geometric parameters of the slotted tube on the acoustic localization characteristics are studied. Numerical results show that the resonant frequency and acoustic pressure amplification amplitude could be effectively modulated by the geometric parameters of the slotted tube. Experimental results are in good agreement with the simulation results. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Phononic crystal (PC) is a new type of artificial periodic structure, which shows unique bandgap characteristics. When acoustic waves propagate into the phononic crystals, incident waves at certain frequency range are forbidden to propagate in the structure, thereby forming an acoustically forbidden band [1–3]. At present, the methods to calculate the dispersion relation of the PCs mainly are plane wave expansion method (PWE) [4], transfer matrix method (TMM) [5], multiple scattering method (MSM) [6], time domain finite difference method (TDFD) [7] and finite element method (FEM) [8]. The PCs could be divided into two types according to the formation mechanism of band gaps, one is the Bragg mechanism [9–11] and the other is the local resonance mechanism [12–14]. With the presence of a defect in the perfect PCs, the elastic wave at the particular frequency will be concentrated at the point defect or propagate along the defect, and the acoustic pressure in the defect center will be amplified, thus show the characteristics of acoustic resonant cavity or waveguide [15–18]. Therefore, the phononic crystal structure with point defect is usually called as phononic crystal resonator (PCR). Phononic crystals have potential applications in vibration reduction and noise reduction due to their unique band gap properties. Cui et al. [19] introduced the cavity into the phononic
*
Corresponding author. E-mail address: [email protected] (X.P. Wang).
https://doi.org/10.1016/j.physleta.2019.125918 0375-9601/© 2019 Elsevier B.V. All rights reserved.
crystal by means of slotting to achieve low-frequency acoustic absorption, and constructed a generalized phononic crystal using a bandgap method to achieve the purpose of broadband and low frequency sound insulation. Romero-García et al. [20] introduced multi-resonant elements in phononic crystals for low-frequency sound insulation. On the basis of the phononic crystal structure of the slotted tube, Bao et al. [21] introduced the arc and flat bar structure inside the circular tube, calculated the band structure and transmission characteristics of the structure, and further reduced the frequency band gap below 500 Hz. Zhou et al. [22] investigated band structures and transmission spectra of two dimensional multilayered locally resonant phononic crystals (LRPCs) by the finite element method, and found that the band gaps of the LRPCs can be extended to several frequency ranges by periodically embedding multilayered coaxial inclusions into a matrix. In addition to the band gap properties, the defect state characteristics of phononic crystals also show excellent applications in acoustic waveguide and energy harvesting. Therefore, it is of great significance to explore them in depth. Khelif et al. [23–25] studied theoretically and experimentally the characteristics of elastic wave propagation in two-dimensional phononic crystals with point defects and line defects. It was found that localized defect band exist in the band gap of phononic crystals. He et al. [26] studied the effect of defects formed by removing scatterers and changing the position of scatterers on the defect state of phononic crystals in supercells. Peng et al. [27] proposed an effective method to induce valley polarization in SL MoSSe by magnetic doping, and studied the defect state of two-dimensional valleytronic materials. Analo-
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Fig. 1. (a) Schematics of the perfect supercell. (b) Schematics of the normal PCR. (c) Schematics of the PCR with slotted tube defect. (d) Finite PCR simulation model.
gous to electrons in semiconductors, introducing defect states into phononic crystals could change the bandgap and control the propagation of sound waves. Li et al. [28] introduced point defects by changing the scatterer material parameters, studied the effect of changes in defect material parameters on the defect state characteristics, and provided new ideas for the formation of defects. Rostami-Dogolsara et al. [29] designed a new type of differential filter based on phononic crystals. This structure consists of two line-defects and a ring-like phononic crystal resonant cavity. It has a very high selective transmission for specific wavelengths of acoustic waves. Increase transmission properties by increasing the number of scatterers. Ouyang et al. [30] used a combination of the phononic crystal waveguide and the symmetric and asymmetric modes of the resonant cavity to design a one-way acoustic waveguide. The structure achieves the conduction and non-conduction of the waveguide through the modal matching and mismatch between the resonant cavity and the line defect. Zhou et al. [31] investigated the flexural elastic wave propagation properties in phononic crystals, and found that point and line defects can be created by properly applying an electric field to some of the cells. These defects bring defect bands inside the original band gaps and lead to elastic wave confinement in the point defect or along the line defect. At present, the research about acoustic energy harvesting using phononic crystal attracted the attention of scholars. To improve the efficiency of acoustic energy harvesting, most of harvesters are consisted of the Helmholtz resonator and piezoelectric materials. Jiang et al. [32] produced a sound energy acquisition device that can be powered by a micro-electromechanical system based on the local modal design of the acoustic defect of a line defect phononic crystal. Sun et al. [33] designed a new sound energy collection system, which mainly includes an acoustic resonator and a piezoelectric bimorph. Oudich [34] et al. designed a phononic crystal plate with point defects. Resonant beads were arranged on a thin elastic plate through a spring connection cycle. Park [35] et al. proposed a systematic design of two-dimensional octagonal phononic crystals through geometric and band gap optimization process. Using the defect states of phononic crystals, the piezoelectric energy acquisition was greatly improved. The acoustic localization of the phononic crystal structure is a key factor affecting the working efficiency and transmission quality of the acoustic device. If the phononic crystal structure has enhanced localization of the acoustic wave at the defect state frequency, the transmission quality of the acoustic control device will be effectively improved. The study of the acoustic localization ability of different types of phononic crystal resonator and the design of defect structures with strong acoustic confinement are beneficial to improve the transmission quality of acoustic energy harvester and waveguide based on phononic crystal defect states. In this paper, the point defect structure of phononic crystal is formed by the slotted tube to replace the seamless tube. The acoustic localization characteristics of the PCR with slotted tube defect are studied. The dispersion relationships and acoustic pres-
sure amplification spectra of structures are calculated by using the finite element method. The reasons for the enhanced acoustic pressure amplification in the resonator are analyzed. Then the defect state adjustment characteristics of the resonator are studied. The resonant frequency and acoustic pressure amplification value can be tuned by changing the geometric parameters of the slotted tube defect. Finally, the acoustic pressure amplification ability of the resonator is verified experimentally. The experimental results are in good agreement with the simulation calculation. 2. Model and method of calculation The PCR presented in this paper consists of periodical circular tubes and a slotted tube defect. The defect tube is designed to be slit in the +x, −x, + y, and − y directions. The seamless tube is replaced by a four-slot circular tube at the center of the supercell, forming a slotted tube defect resonator. As shown in Fig. 1(a), the lattice constant of the perfect supercell is a = 0.08 m, the radius of the seamless tube is r = 0.036 m. The phononic crystal resonator is normally formed by removing scatterers as shown in Fig. 1(b). Fig. 1(c) shows the PCR with slotted tube defect, the radius of the slotted tube is R = 0.02 m, slot width w = 0.005 m, tube thickness t = 0.005 m; the phononic crystal resonator involved in this paper are all made up of polymethyl methacrylate (PMMA) tubes arranged periodically in the air. The velocity of longitudinal wave in PMMA is υ = 3000 m/s, and the density is ρ = 1390 kg/m3 ; the wave velocity of air is υ = 343 m/s, and the density is ρ = 1.25 kg/m3 . Dispersion relationships, power transmission spectra and intrinsic modal analysis are the most effective means on PCs research. This paper will use the FEM combined with Bloch theorem to calculate the PCs’ dispersion relationships and power transmission spectra. All the FEM calculations in this paper are implemented by the MULTIPHYSICS COMSOL 5.0. The dispersion relationship shows the relationship between the angular frequency ω and the Bloch wave vector, from which the band gaps of the PC can be observed directly. According to Bloch theorem [18]:
U (r + a) = e ika U (r )
(1)
Because of the symmetry and periodicity of the PCs in both x and y directions, only the supercell shown in Fig. 1(b) and (c) need to be considered. Periodic boundary condition is applied on the boundary of the adjacent cell, and the dispersion relationships will be obtained by simply scanning the Bloch wave vector along the border line of irreducible Brillouin zone. In order to investigate the localization characteristics of the nested PCRs, the transmission spectra of the structure are also calculated by FEM. Fig. 1(d) shows the finite PCR simulation model, the incident plane acoustic pressure of 1 Pa is applied to the left boundary of the PCR, then the acoustic pressure P at the center of the PCR was extracted. The acoustic pressure amplification is defined as the ratio of the acoustic pressure in the cavity center to
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Fig. 2. The dispersion relations of the perfect supercell. (b) The dispersion relations of the normal PCR. (c) The dispersion relations of the PCR with slotted tube defect.
the incident acoustic pressure. In order to reduce the influence of acoustic reflection at the boundary, the other boundaries are applied plane wave radiation. 3. Numerical results and discussions 3.1. Band structures and acoustic pressure amplification spectra of multiple nested PCRs There are two conditions for the localization of phonon crystal defect states. First, the perfect phononic crystal has a complete band. Secondly, the frequency of defect state should be located in the range of band gap. Fig. 2 shows the band characteristics of the perfect phononic crystal supercell and two phononic crystal resonators shown in Fig. 1(b) and Fig. 1(c). The perfect phononic crystal supercell has a complete band gap between 1740-2920 Hz. In this frequency range, acoustic wave cannot propagate to the other side of the structure. While the PCR formed by removing the scatterer, its complete band gap in the band structure is basically consistent with the perfect supercell. Due to the formation of the point defect by removing scatterer, the defect state flat band appears in the band structure, of which the corresponding frequency is around 2560 Hz. With the introduction of the slotted tube defect in the perfect PC supercell, the complete band gap in the band structure remains unchanged, but the defect state flat band appears in the band gap. The corresponding frequency is around 2400 Hz, which is lower than normal PCR. By introducing point defects into perfect phononic crystal supercells, waves of specific frequencies are localized in the point defects. The acoustic pressure transmission characteristics of a single finite PCR structure at the resonator center are further studied. The calculation result is shown in Fig. 3. Fig. 3(a) shows the acoustic pressure amplification of two PCR. The resonant acoustic pressure amplification of the normal PCR (NPCR) is 8.5, and the resonant frequency is 2560 Hz, while the acoustic pressure amplification of the PCR with slotted tube defect (SPCR) is 24.3, which is much better than that of the normal PCR and it has two peaks of acoustic pressure amplification in the range of 1000-4000 Hz. The peak frequency of SPCR’s first sound pressure amplification is close to that of conventional PCR. It is due to the introduction of slotted tube defect in the center of phononic crystal and a flat band is generated in the band gap. The acoustic wave at the corresponding frequency is focused on the point defect. In the vicinity of 3000 Hz, a new flat band is generated due to the introduction of point defect, but the forbidden band width is narrow and the defect state is not obvious. However, due to the Helmholtz resonance effect of the slit tube, the sound pressure amplification capability at the center
of the cavity is enhanced, so a smaller sound pressure amplification peak appears in the SPCR. Then the resonant acoustic pressure distribution at the centerline of the resonator (x = 0) are extracted, as shown in Fig. 3(b). The acoustic pressure at the center of the PCR with slotted tube defect is higher than the normal PCR. The acoustic pressure distributions of the two resonators at the resonance frequency are shown in Fig. 3(c) and (d) respectively. The acoustic pressure at the center of the resonators is all significantly higher than that of the surrounding space. Compared to the normal PCR, the PCR with slotted tube defect shows enhanced acoustic localization, which has larger acoustic pressure at the center of the cavity, and smaller and more concentrated acoustic energy convergence area. As shown in Fig. 4(a), the single-slot tube can be equivalent to a Helmholtz cavity. According to the Helmholtz resonance theory, there is an abrupt change in the acoustic pressure at the slot position at the resonant frequency. As R = 0.015 m, w = 0.01 m, t = 0.005 m, simulation results show that the resonance frequency is 2242 Hz. The acoustic pressure distribution on the slotted tube and the sound pressure distribution along the transverse and longitudinal center lines of the slotted tube are extracted respectively, as shown in Fig. 4(b) and (c). It clearly shows that the acoustic energy is mainly concentrated in the slotted tube cavity. The acoustic pressure amplification in the cavity is 5.3. There is an acoustic pressure mutation at the slot position. Due to the existence of the acoustic pressure mutation at the slit position, the acoustic pressure presents an asymmetrical distribution at horizontal axes. The slotted tube shows Helmholtz resonance effect, and the normal PCR shows acoustic localization characteristics at the defect state frequency as well. As the resonant frequency of the slotted tube is close to resonant frequency of the normal PCR, the acoustic pressure amplification capability of the PCR with slotted tube defect would be enhanced due to the coupling of local resonant characteristics. 3.2. Parametric study on the acoustic localization characteristics The enhanced acoustic localization of the PCR with slotted tube is mainly caused by the coupling enhancement of two resonant cavities. While the lattice constant and scatterer radius of the outer tubes keep constant, the acoustic localization characteristics of the combined structure are mainly related to the internal slotted tube. First, we studied the regulation of the acoustic localization characteristics by the slotted tube radius R. The PC lattice constant a = 0.08 m, the scatterer radius r = 0.036 m, the slit width w = 0.005 m, and tube thickness t = 0.005 m remained unchanged. The localization characteristics of the structure were calculated when the
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Fig. 3. (a) Acoustic pressure amplification spectra of the PCRs. (b) Acoustic pressure distribution in the center line (the coordinate x = 0) of the PCRs. (c) Acoustic field distribution of normal PCR at resonant frequency. (d) Acoustic field distribution of PCR with slotted tube defect at resonant frequency. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 4. (a) Schematics of the slotted tube. (b) Acoustic pressure distribution of the slotted tube at resonant frequency. (c) Acoustic pressure distribution in the center line (the coordinate x = 0 and y = 0) of the slotted tube.
slotted tube radius are R = 0.015 m, 0.02 m, 0.025 m, 0.03 m, and 0.035 m. Fig. 5(a) shows the acoustic pressure amplification of the PCR with slotted tube defect. It can be seen that the resonant acoustic pressure amplification has the best corresponding slotted diameter. When R = 0.02 m and 0.025 m, the resonant pressure amplification is more than 24. Fig. 5(b) is consistent with the results in Fig. 5(a), further confirming that the slotted tube radius has an optimal value for the adjustment of the acoustic localization characteristics. Keeping other geometric parameters unchanged, only changing the slit width of the PCR, the adjustment rule for acoustic localization characteristics is shown in Fig. 6. As the slit width is increased from 0.006 m to 0.014 m, the resonant acoustic pressure amplification of the PCR with slotted tube gradually decreases, and the resonant frequency gradually increases. According to the Helmholtz
equivalent model of the slotted tube, the slit width increases, this is equivalent to the increase of the cross-sectional area of the short tube in the Helmholtz resonator, and therefore the resonant frequency increases. The resonant acoustic pressure distribution and acoustic pressure amplification spectrum results are consistent. Then we calculate the acoustic pressure amplification spectrum and acoustic pressure distribution of the PCR. The PC lattice constant a = 0.08 m, the scatterer radius r = 0.036 m, the slotted tube radius R = 0.02 m, the slit width w = 0.003 m, while the tube thickness t = 0.001 m, 0.002 m, 0.003 m, 0.004 m, 0.005 m, and 0.006 m, respectively. Fig. 7 shows the effect of tube thickness on acoustic localization characteristics. When t = 0.001 m increases to 0.003 m, the resonant frequency gradually shifts to low frequency; as the tube thickness increases further, the resonant frequency starts to shift to high frequency again, but the overall frequency
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Fig. 5. (a) Acoustic pressure amplification spectrum of the slotted defect PCRs with different slotted tube radius. (b) Resonant acoustic pressure distribution in the center line (the coordinate x = 0) of the slotted defect PCRs. (c) Acoustic pressure distribution of the slotted defect PCRs at resonant frequency.
Fig. 6. (a) Acoustic pressure amplification spectrum of the slotted defect PCRs with different slit width. (b) Resonant acoustic pressure distribution in the center line (the coordinate x = 0) of the slotted defect PCRs. (c) Acoustic pressure distribution of the slotted defect PCRs at resonant frequency.
shift is not large. With the increase of the tube thickness, the resonant acoustic pressure amplification of the PCR increases gradually. When t = 0.005 m, the acoustic pressure amplification is up to 38. The resonant acoustic pressure distribution of the PCR shown in Fig. 7(b) and (c) further proves this point. 3.3. Experimental result and discussion In order to verify the acoustic localization characteristics of phononic crystal resonators, an experimental platform for the acoustic pressure amplification of the PCR with slotted tube was
established in this paper, and the test sample was prepared. As shown in Fig. 8(a), the experimental sample consists of 25 tubes; the tubes are periodically fixed on the organic glass plate. The lattice constant of the PCR sample is 0.08 m. The outer radius of the seamless tube is 0.035 m, and the tube thickness is 0.002 m. The outer radius of the slotted-defect is 0.02 m, and its tube thickness is 0.004 m, the slit width is 0.008 m. All the constituent tubes in the test sample have a length of 0.5 m. In order to verify the simulation results, the acoustic localization characteristics are reflected by testing the acoustic pressure amplification at the defect center of the PCR. The environment and
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Fig. 7. (a) Acoustic pressure amplification spectrum of the slotted defect PCRs with different tube thickness. (b) Resonant acoustic pressure distribution in the center line (the coordinate x = 0) of the slotted defect PCRs. (c) Acoustic pressure distribution of the slotted defect PCRs at resonant frequency.
Fig. 8. (a) The experiment sample of the PCR with slotted tube. (b) The environment and apparatus of the experiment.
apparatus of the experiment are shown in Fig. 8(b). The experimental apparatus consists of a signal generator, a power amplifier, a speaker, a microphone, an M+P data acquisition card, and a tripod. The experimental test is completed in a half anechoic chamber. The experimental and simulation results are shown in Fig. 9. Experimental results show that the resonant frequency of the normal PCR is 2506 Hz, and the resonant acoustic pressure amplification is 1.83, while the resonant frequency of the PCR with slotted tube defect is 2470 Hz, and the resonant acoustic pressure amplification is 5.47. Comparing the test results, it is found that the acoustic localization capacity of the PCR with slotted tube defect is stronger than that of the normal PCR, and there are two resonant pressure amplification peaks in the 1500-4000 Hz range. Comparing the test results and the simulation results, the test results of the resonant frequency corresponding to the two PCRs are basically consistent with the simulated values. However, there is a big difference between the experimental test results and the simulation results of the resonant acoustic pressure amplification. The experimental results of the acoustic pressure amplification of the two PCRs are obviously smaller than the simulation results. The reason for this result is analyzed as follows. Firstly, the main reason for the defect state of the PC is the periodic arrangement of the cells, which has a higher requirement for periodicity. Due to the errors in the processing and assembly of experimental samples, the acoustic localization characteristics of the PCRs are weakened. Secondly, while the acoustic pressure at the center of PCRs is cal-
culated by simulation, the acoustic pressure pickup point is at the geometric center of the resonant cavity. But in the actual test process, the microphone is fixed by a tripod and then projected into the center of the cavity to measure the acoustic pressure. Due to the inability to fully guarantee that the measurement position of the microphone completely coincides with the geometric center of the resonator, an error occurs in the measurement value. Thirdly, while the simulation model is a two-dimensional structure, the experimental sample is a three-dimensional structure. The reflection of acoustic wave in the Z direction during the measurement affects the accuracy of the experimental results. Through experimental tests, the PCR with slotted tube shows better acoustic localization capability than the normal PCR. The experimental results are basically consistent with the simulation results. 4. Conclusion In this paper, the slotted circular tube is introduced into the phononic crystal as a novel point defect type. The acoustic localization capability of the PCR with slotted tube defect is studied by the finite element method. Through the calculation, the band structure and acoustic pressure distribution in the cavity of the phononic crystal resonator with slotted tube defect has been studied. Compared with the Helmholtz resonator, this structure can realize the acoustic concentration of multi-directional incident acoustic wave. Compared with the normal PCR, this structure has stronger acous-
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Fig. 9. (a) The experimental and simulation acoustic pressure amplification spectra of the PCR with slotted tube. (b) The experimental and simulation acoustic pressure amplification spectra of the normal PCR.
tic localization ability. The mechanism of the enhanced acoustic confinement in the cavity of the structure is explained as the coupling of Helmholtz resonator and normal PCR. The regulation of the resonant frequency and acoustic pressure amplification by the slotted tube are studied; the geometric parameters of slotted tube can effectively adjust the acoustic localization capacity of the resonator. The resonant acoustic pressure amplification of the PCR with slotted tube defect is investigated experimentally; the experiment result is basically consistent with the calculation result in the error tolerance range. The PCR with slotted tube defect shows better acoustic localization ability than normal PCR, and could be applied to the design of acoustic manipulation devices such as acoustic waveguides and acoustic sensors. Acknowledgements The authors gratefully acknowledge financial support from the Project of National Natural Science Foundation of China (Grant No. 51675402). References [1] Y. Yao, Z. Hou, F. Wu, X. Zhang, Physica B 406 (2011) 11. [2] J.F. Robillard, O.B. Matar, J.O. Vasseur, P.A. Deymier, Appl. Phys. Lett. 95 (2009) 12. [3] C. Lagarrigue, J.P. Groby, V. Tournat, J. Acoust. Soc. Am. 133 (2013) 1. [4] J.O. Vasseur, P.A. Deymier, B. Chenni, B. Djafari Rouhani, L. Dobrzynski, D. Prevost, Phys. Rev. Lett. 86 (2001) 3012. [5] Z.Y. Li, K.M. Ho, Phys. Rev. B 68 (2003) 155101. [6] L.M. Li, Z.Q. Zhang, Phys. Rev. B 58 (1998) 9587. [7] Y. Tanaka, Y. Tomoyasu, S.I. Tamura, Phys. Rev. B 62 (2000) 7387. [8] C. Goffaux, J. Sanchez Dehesa, Phys. Rev. B 67 (2003) 144301. [9] C.S. Kee, J.E. Kim, H.Y. Park, K.J. Chang, J. Appl. Phys. 87 (2000) 4. [10] Z.L. Hou, X.J. Fu, Y.Y. Liu, Phys. Lett. A 317 (2003) 1.
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