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Extending and lowering band gaps by multilayered locally resonant phononic crystals
Applied Acoustics 133 (2018) 97–106
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Extending and lowering band gaps by multilayered locally resonant phononic crystals ⁎
Xiaoling Zhoua, , Yanlong Xub, Yu Liua, Liangliang Lva, Fujun Penga, Longqi Wangc,
T
⁎
a
Shanghai Institute of Aerospace System Engineering, Shanghai 201109, China School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China c School of Civil & Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore b
A R T I C L E I N F O
A B S T R A C T
Keywords: LRPCs Multilayered inclusions Band gaps Transmission spectra
Locally resonant phononic crystals (LRPCs) have band gaps with low frequencies and the size of the cell is much smaller than the wave lengths of the band gaps. This unique property makes them good candidates for vibration control. Considering that the band gaps of the LRPCs are determined by the single cell rather than the periodic arrangement, it is important to design the microstructure of the cell. In this paper, band structures and transmission spectra of two dimensional multilayered LRPCs are investigated by the finite element method (FEM). It is found that the band gaps of the LRPCs can be extended to several frequency ranges by periodically embedding multilayered coaxial inclusions (i.e., alternate shells of soft rubber and hard metal) into a matrix. Moreover, the frequency responses of the multilayered periodic structures with different number of cells are simulated. There are several sharp dips which refer to high attenuation efficiency in the transmission spectra of the periodic structures with multilayered inclusions. The increase in the number of cells is also found to enhance the attenuation efficiency. Meanwhile, the geometrical effects of the layers on the band structures are investigated. By carefully designing the size and number of the shells in the multilayered inclusions, the band gaps can be extended and lowered. These results are helpful for the applications of LRPCs in noise and vibration control.
1. Introduction Phononic crystals (PCs) have attracted a growing attention for the band gap properties in recent years [1–6]. The mechanisms [7–9], calculation methods [10] and tunability [11–16] of the band structures of PCs are the main topics for scientific researchers. Since noise and vibration transmit in the form of elastic waves in structures, PCs have great potentials in noise and vibration control. However, for the Bragg scattering PCs, the wave lengths of the band gaps almost equal the size of the single periodic cell [1]. This makes the Bragg scattering PCs hardly meet the demand of vibration control in low frequency range since it is difficult and costly to create periodic structures with huge size in real applications. In 2000, Liu et al. [2] proposed the locally resonant phononic crystals (LRPCs), which refer to structures consisting of hard cores coated with soft layers and then embedding into a matrix periodically. The frequency of the first band gap is as low as ∼400 Hz while the size of the single periodic cell is as small as 1.55 cm. The LRPCs bring a brand-new sight for the vibration and noise control in engineering applications. Due to the potential applications of LRPCs, many researchers
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investigated the mechanisms of the low frequency band gaps [2,8]. As stated by Liu et al. [2], the local resonances of the cell result in the band gaps. Yao et al. [17] embedded rubber into a matrix to form a periodic beam and opened the locally resonant band gaps. Hirsekorn [8] and Wang et al. [18] proposed simplified theoretical models to calculate the frequency boundaries of the first band gap of two dimensional LRPCs. In the theoretical models, it is found that the frequency boundaries of the first band gap can be obtained by analyzing the natural frequencies of the mass-spring systems. For the LRPCs, the core and the matrix act as the mass while the soft layers act as the spring [8,18]. From the above studies, researchers found that it is the single cell rather than the periodic arrangement of the LRPCs that determines the band gaps. And hence, the band gaps of LRPCs can be tuned by changing the microstructure of the cell. Numerous methods are proposed to extend and lower the band gaps of PCs [19–26]. For example, Wang et al. [19] investigated the tuning characteristics of band gaps in a multi-stub LRPC plate and they discussed the effects of the stub structure and defect state on the band gap frequencies. Assouar and Oudich [20] used double stubs to enlarge the band gaps of LRPC, which composed of lead-rubber stubs deposited on
Corresponding authors. E-mail addresses: [email protected] (X. Zhou), [email protected] (L. Wang).
https://doi.org/10.1016/j.apacoust.2017.12.012 Received 3 April 2017; Received in revised form 29 November 2017; Accepted 11 December 2017 0003-682X/ © 2017 Elsevier Ltd. All rights reserved.
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the aluminum plate. Zhang et al. [21] obtained 42–150 Hz band gaps in the plate with periodic spiral resonators and they pointed out that the low frequency locally resonant band gaps can be tuned by changing the size of the spiral resonators. Ma et al. [22] investigated the Lamb wave propagation in phononic crystal consisting of periodic arrangement of bilayered materials in the radial direction. New lower band gaps appeared in the radial phononic crystal by applying crystal gliding. Some researchers designed self-similar hierarchy [23] or learned from bioinspired structures [24,25] to form PCs and got ultrawide low frequency band gaps. Zhou and Chen [26] studied the initial stress effects on the band gaps of LRPCs and they also used electric field to tune the band gaps. They investigated two dimensional LRPCs with single or double layers of elastomer and found that LRPCs with double layers of soft elastomer have two wide band gaps compared with the one wide band gap in single layer models. Larabi et al. [27] used a finite difference domain method to study the transmission spectra of LRPCs with multicoaxial cylindrical inclusions embedded in a fluid matrix. They indicated that the transmission spectra of the multilayered models can exhibit several sharp dips instead of one in the same frequency range. Moreover, the dips can be overlapped to create a larger acoustic band gap by appropriately combining the cells with different layers. For LRPCs, it is important to design the microstructure of the cell, and the LRPC with multilayered inclusions may show some novel characteristics. In this paper, the band structures of two dimensional multilayered LRPCs with multicoaxial cylindrical inclusions are investigated. The band gap mechanisms of the multilayered LRPCs with different inclusions are discussed. Moreover, the effects of the cell number on the transmission spectra are studied. It is proved that the band gaps can be extended and lowered by carefully designing the inclusions.
Table 1 Material constants of the components.
Tungsten Rubber Resin
Density (kg/m3)
Lame constants (GPa)
ρ
λ
G
19,100 1300 1180
306 6.051e−4 4.52
131.1 4e−5 1.59
rubber layers nr = 1, 2, 3 and 4 are studied. The thickness of the rubber and coating tungsten layers are trub = ttun = 0.002 m. The size of the LRPC models in the out of plane direction is infinite. So only the inplane wave motions are considered in this paper. The material constants of the models are given in Table 1. For the multilayered LRPC models, the equation of elastic wave propagation can be written as,
∇ [[λ (r) + 2G (r)](∇·u)]−∇ × [G (r) ∇ × u] = ρ u ¨
(1)
where u is the displacement. r the coordinate vector. λ and G are the Lame constants. u ¨ refers to second order derivative of the time. ρ is the density. For the multilayered LRPCs, the Bloch boundary condition must be satisfied, which gives
u (r) = u (r)·e i (k·r − ωt)
(2)
where k is the wave vector. ω is the angular frequency and t is time. For the periodic models, the Bloch boundary condition can be written in the form of,
2. Models and method
u (r) = u (r + n1 a1 + n2 a2)
To elucidate the band structures and mechanisms of the multilayered LRPCs, two dimensional models with inclusions composed by alternate shells of soft silicone rubber and hard tungsten are investigated by the finite element method (FEM). As shown in Fig. 1, the multilayered inclusions with radius of 0.018 m are periodically embedded into a resin matrix. The periodic models have square lattice structure and the irreducible Brillouin zone is shown in Fig. 1. The lattice constant keeps at 0.04 m. Here, models with the number of the
(3)
where a1 and a2 are the lattice vectors; n1 and n2 represent integers. Band structures of the multilayered LRPCs and the transmission spectra of the periodic structures with N × N cells are studied by using the commercial finite element software COMSOL [28]. Since the size of the models in the out of plane direction is infinite, the plane strain condition is adopted in all the simulations. The linear triangular elements are utilized for the models. Fig. 1. Schematic illustration of the multilayered LRPC cells and the irreducible Brillouin zone. The component materials are tungsten, silicone rubber and resin matrix.
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Fig. 2. Band structures of the multilayered LRPCs with different inclusions as shown in Fig. 1. Fig. 3. Displacement fields of (a) the lower band gap boundary and (b) the upper band gap boundary of the LRPC with nr = 1.
Fig. 4. Displacement fields of the band gap boundaries of the multilayered LRPC with nr = 2. (a) The lower boundary and (b) the upper boundary of the first band gap; (c) The lower boundary and (d) the upper boundary of the second band gap.
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Fig. 5. Displacement fields of the band gap boundaries of the multilayered LRPC with nr = 3. The snapshots of (a), (c) and (e) are the upper boundaries of the first, second and third band gaps while the snapshots of (b), (d) and (f) are the lower boundaries of the first, second and third band gaps, respectively.
Fig. 6. Displacement fields of the band gap boundaries of the multilayered LRPC with nr = 4. The snapshots of (a), (c), (e) and (g) are the upper boundaries of the first, second, third and fourth band gaps while the snapshots of (b), (d), (f) and (h) are the lower boundaries of the first, second, third and fourth band gaps, respectively.
3. Numerical results
multilayered LRPC with three rubber layers are 147.3–260.5 Hz, 373.1–533.8 Hz and 588.5–860.1 Hz as shown in Fig. 2(c). Fig. 2(d) shows that the band gaps of the LRPC with four rubber layers are 160.2–258.5 Hz, 337.7–441.2 Hz, 496.6–595.2 Hz and 620.5–861.8 Hz respectively. For the multilayered LRPC models with nr = 3 and 4, there are also straight lines in the first band gaps and that is because rotation vibrations occur in the coating rubber layers. Moreover, there are narrow band gaps at higher frequencies in Fig. 2(c) and (d). These narrow band gaps are not taken into account in this paper. It is found from Fig. 2 that the number of the band gaps in the multilayered LRPCs increases as the rubber layers number nr grows. The reason is that the core and the coating tungsten layers act as localized resonators and the localized vibrations open the band gaps. To further understand the mechanisms of the band gaps in the multilayered LRPCs, the eigen modes of the band gap boundaries are discussed in this section. The corresponding displacement fields of the band gap boundaries are shown in Figs. 3–6. The arrows and colors refer to the displacements of the cells. As shown in Fig. 3(a), the core moves as a whole while the matrix almost keeps stationary at the beginning of the band gap in the LRPC
3.1. Band structures and eigen modes Fig. 2 gives the band structures of the multilayered LRPCs which shown in Fig. 1. For the LRPC with a single rubber layer (see Fig. 2(a)), there is a wide band gap with the frequency range of 176.5–780.3 Hz in the band structure. There are two wide band gaps in the band structure for the model with double rubber layers as shown in Fig. 2(b). The frequency boundaries of the first band gap are 147.6 Hz and 340.8 Hz. The frequency boundaries of the second band gap are 493.8 Hz and 856.3 Hz. One can notice that a flat straight line which refers to a rotation vibration mode of the coating rubber layer appears in the first band gap in Fig. 2(b). Besides, a narrow band gap at higher frequencies appears in the band structures in Fig. 2(a) and (b). Because the frequency ranges are much narrower and there are some straight lines in the frequency ranges, these band gaps at higher frequencies can hardly be used in engineering applications. For the models with nr = 3 and 4, it is interesting to see that the numbers of the wide band gaps are 3 and 4 respectively. The frequency ranges of the band gaps for the
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Fig. 7. Schematic illustration of the transmission spectra simulation of the finite periodic plate.
Fig. 8. Transmission spectra of the periodic plates with different number of multilayered inclusions shown in Fig. 1.
whole, which causes the coating layers to move but hardly the matrix. At the end of the first band gap as shown in Fig. 4(b), the cores and the matrix move toward opposite directions. These mechanisms are similar to those of the LRPC with one rubber layer. However, the eigen modes of the second band gap boundaries are very different. The coating tungsten layer and the tungsten core move to opposite directions at the lower boundary of the second band gap, which is shown in Fig. 4(c). For the upper boundary of the second band gap, the coating tungsten layer
with single rubber layer. At the end of the band gap, the core and the matrix move in opposite directions as Fig. 3(b) has shown. These mechanisms agree well with the results mentioned earlier [2,8,18]. The rubber layer acts as the spring. The core and matrix act as the lumped mass. The localized resonances of the spring-mass resonator result in the band gap. For the multilayered LRPC with two rubber layers, the displacement fields of the band gap boundaries are shown in Fig. 4. At the beginning of the first band gap (see Fig. 4(a)), the core moves as a 101
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Fig. 9. Displacement amplitude attenuation fields of the 5 × 5 models of (a) f = 180 Hz, nr = 1; (b) f = 150 Hz, nr = 2 and (c) f = 500 Hz, nr = 2.
multilayered LRPC cell can be changed to fit in different situations.
and the matrix move oppositely. It is found that the coating tungsten, the core and the matrix can all act as lumped mass. To further elucidate the effect of the coating layers, the eigen modes at the band gap boundaries of the multilayered LRPC with nr = 3 are shown in Fig. 5. For the lower boundaries of the band gaps (Fig. 5(a), (c), (e)), the cores and the coating tungsten layers move as lumped mass while the matrix almost keeps stable. In Fig. 5(a), the core moves dominantly as a whole. In Fig. 5(c), the coating tungsten layers and the cores move to opposite directions. However, at the beginning of the third band gap, as shown in Fig. 5(e), the two coating tungsten layers move to the opposite directions. For the upper boundaries of the band gaps (Fig. 5(b), (d), (f)), the epoxy matrix and the tungsten move to opposite directions. The core, the inner coating tungsten layer and the outer coating tungsten layer move oppositely to the matrix at the end of the first, second and third band gaps respectively. As the number of the coating layers increases, more local resonance modes appear and result in more band gaps. The eigen modes at the band gap boundaries of the multilayered LRPC with nr = 4 are shown in Fig. 6. The mechanisms shown in Fig. 6 are similar to those in Fig. 5. The core, the coating tungsten layers and the matrix act as lumped mass while the coating rubber layers act as spring. At the lower boundaries (see Fig. 6(a), (c), (e), (g)), the matrix almost keeps unmoved. The core and the coating tungsten layers move in different modes at different band gaps. On the other hand, the core and the coating tungsten layers move oppositely to the matrix at the upper boundaries. According to the above analysis, one can see that the number of the band gaps increases with the number of the coating layers due to the presence of more local resonance modes. The increase in the number of band gaps in multilayered LRPCs benefits the applications of noise and vibration control. Moreover, the material constants and the size of the
3.2. Transmission spectra To investigate the attenuation effects of the multilayered LRPCs, the transmission spectra of the corresponding periodic plates with different number of cells are calculated via the frequency response analysis in COMSOL [28]. As shown in Fig. 7, to avoid reflections, two perfect matched layers (PMLs) with the length of 8 m are applied in the x direction while periodic boundaries are used in the y direction. The parameters of the PMLs are identified by the software automatically. A prescribed displacement with the average value of Si is imposed on the excitation port of the periodic plate in the x direction. The integration of the displacement amplitude in the x direction on the receive port is obtained and the average value on the boundary (denoted by So) is calculated. So the transmission can be obtained by 20 × log(So/Si). The periodic plates consist of 5 × 5, 8 × 8 and 10 × 10 periodic cells, respectively. Then it is easy to find the band gap boundaries from the intersections of the transmission spectra of the periodic plates with different number of cells. For efficiency’s sake, the frequency step is set to 10 Hz in the simulations. Fig. 8 gives the transmission spectra of the periodic plates with different number of multilayered inclusions. The shadow regions in Fig. 8(a)–(d) refer to the band gap frequency ranges which shown in Fig. 2(a)–(d). It can be seen that there are wave attenuation in the band gap frequencies. For the multilayered LRPCs, there are sharp dips at the beginning frequencies of the band gaps. This is consistent with the results in Ref. [2]. One can notice that there are some straight lines in the first band gap in the multilayered LRPCs which don’t appear in the transmission spectra. This is because that the frequency step is 10 Hz 102
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Fig. 10. Displacement amplitude attenuation fields of the 5 × 5 models of nr = 3 at (a) f = 150 Hz; (b) f = 380 Hz and (c) f = 590 Hz.
transmission dips in the band gaps are given in Figs. 9–11. To show the attenuation effects clearly, only the displacement amplitude attenuation fields of the periodic cells are shown here. The fields of the whole models with PMLs are given in the Supplementary Materials. For the LRPCs, the equivalent mass of the resonances, the vibration amplitude and the frequency all have influences on the transmission efficiency. Large mass and high vibration amplitude are beneficial for consuming energy. For a given size, higher frequency wave with smaller wave length will be easier to reduce. As shown in Figs. 9–11, the localized vibration of the resonances is the key factor contributing to vibration attenuation. For the model of nr = 2, the rubber and tungsten layers vibrate at a much higher frequency of 500 Hz. It leads to a larger attenuation (see Fig. 9(c)). For the model of nr = 3, it is found that the largest displacement amplitude of the localized resonance mode is at the beginning of the third band gap as shown in Fig. 10. The localized vibration with the largest displacement amplitude at a high frequency of 590 Hz benefits energy consumption and results in a minimum transmission. The displacement amplitudes of the nr = 4 model at different frequencies are shown in Fig. 11. One can see from Fig. 11 that the attenuation effects in Fig. 11(b) and (c) are better than those in Fig. 11(a) and (d). The displacement amplitude of the resonance in Fig. 11(b) is the largest and hence the minimum transmission is in the second band gap as shown in Fig. 8. Although the increasing number of layers may not reduce the minimum transmission further, it can still improve the number of the transmission dips. For the LRPCs, there are transmission dips at the beginning of the band gap, and then the curves increase sharply. Hence the transmission dips are of great importance for low frequency vibration control in engineering. The multilayered LRPCs provide a way to enhance the vibration attenuation.
and it may omit some frequencies. Moreover, there are some narrow band gaps appearing in the transmission spectra of the periodic plates as shown in Fig. 8(a)–(d) and the frequency ranges of the narrow band gaps are consistent with those in Fig. 2(a)–(d). Vibration can be reduced in the band gaps depicted as shadow regions in Fig. 8. However, the attenuation effects are more obvious at the beginning of the band gaps as we can see from the sharp dips in the transmission spectra. As a result, the plates with multilayered inclusion of nr = 4 exhibit an advantage in vibration control due to the four sharp dips in the transmission spectrum shown in Fig. 8(d). Besides, it is found that the attenuation effects can be enhanced by increasing the number of the periodic cells. For the plates with periodic inclusion of nr = 1 (see Fig. 8(a)), the largest attenuation of the vibration is about 28 dB in the 5 × 5 model. When the number of the cells increases to 8 × 8, the largest attenuation can be improved to about 47 dB. For the 10 × 10 model, the largest attenuation is 60 dB. The largest attenuation of the nr = 2 plate is located in the second band gap as shown in Fig. 8(b). As the number of cell increases, the largest attenuation increases from 27 dB to 44 dB and then to 56 dB. In Fig. 8(c), the largest attenuation of the periodic plate with periodic cell of nr = 3 are 59 dB, 95 dB and 118 dB for the 5 × 5, 8 × 8 and 10 × 10 model, respectively. Fig. 8(d) displays the transmission spectra of the nr = 4 model, the attenuation in the sharp dips rises with the number of cells as well. The largest attenuation in the second dip for the 10 × 10 model is 72 dB. Consequently, the structures containing periodic arrangement of the multilayered inclusions can reduce the vibration efficiently. The attenuation effects can be enhanced through the careful design of the inclusions as well as the appropriate increase of the cell number. To further investigate the attenuation mechanisms of the periodic structures, the displacement amplitude fields of the 5 × 5 models at the 103
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Fig. 11. Displacement amplitude attenuation fields of the 5 × 5 models of nr = 4 at (a) f = 170 Hz; (b) f = 340 Hz; (c) f = 500 Hz and (d) f = 630 Hz.
Fig. 12. Dependence of the band gap boundaries on the thickness of the rubber layers. (a) Multilayered LRPC with nr = 3 and (b) multilayered LRPC with nr = 4.
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gaps in the multilayered LRPCs increases as the rubber layer thickness decreasing, the layer thicknesses of the rubbers in Models G and H are reduced to 0.0005 m. It can be seen that the band gaps of Models G and H are much wider than those of Models A–F. Model H has only one coating rubber layer, hence there is only one band gap. For Model G, the number of the coating rubber layers is six so that there are six band gaps. Comparing Model G with Model H, it is obvious that the boundary of the first band gap can be lowered. In this section, it can be seen that the band gaps can be extended and lowered in multilayered LRPCs by changing the size and numbers of the coating layers. As is known, LRPCs have the advantage of low frequency band gaps. But the band gaps can hardly be tuned by the arrangement of the periodic cells. The multilayered LRPCs provide chances for more different characteristics in low frequency band gaps and this is helpful in real-life applications in various situations. 5. Conclusions
Fig. 13. Band gaps of the hierarchical LRPCs with different microstructures of the single cell. Models A–D refer to the corresponding LRPCs shown in Fig. 1. Model E and F represent the LRPCs of three coating rubber layers of which the thicknesses change in arithmetic progresses. From the inner to outer layers, the thicknesses of rubber layers for Model E are 3 mm, 2 mm and 1 mm while those of Model F are 1 mm, 2 mm and 3 mm. Model G and H refer to LRPCs with the thickness of the rubber layers of 0.5 mm. Model G has six coating rubber layers while Model H has only one coating rubber layers. The matrix keeps constant in size for all the Models.
In summary, band structures of multilayered LRPCs are investigated by FEM. By embedding multicoaxial cylindrical inclusions into a matrix, the band gaps can be extended to several ranges because of the presence of more localized resonance modes. The periodic structures consisting of multilayered inclusions are proved to reduce the vibration in the band gaps by calculating the transmission spectra. The attenuation effects can be enhanced by increasing the number of coating layers and the number of multilayered cells. Lastly, the effects of the geometrical sizes are discussed. It is found that the band gap frequencies can be extended and lowered by carefully designing the multilayered inclusions of the LRPCs. The wide low frequency band gaps in the multilayered LRPCs are useful in real-life applications. The experimental verification and engineering applications of the multilayered LRPCs will be discussed in the future work.
4. Discussions 4.1. Band gaps with different geometrical sizes From the simulation results, it is known that the multilayered LRPCs show more band gaps than the LRPC with one rubber layer. The multilayered inclusions can also bring more changes in the size of the microstructures which may influence the band gaps. In this section, the geometrical size effects on the band gaps of the multilayered LRPCs of nr = 3 and 4 are discussed. Here, the size of the matrix keeps unchanged. When changing the size of the coating layers, the thickness of all the coating rubber layers keep equal to each other. If the thickness of the coating rubber layers reduces, the coating tungsten layers will increase the same size in thickness. Fig. 12 shows the dependence of the band gap boundaries on the thickness of the rubber layers. Since the narrow band gaps are not considered in this paper, only the wide band gaps are shown in Figs. 12 and 13. It can be seen that the width of the band gaps increases as the thickness of the rubber layers decreases. As reducing the thickness of the rubber layers, the upper boundaries of the band gaps rise quickly while the lower boundaries of the band gaps change much less. It means one can widen the band gaps by changing the size of the coating layers.
Acknowledgement This work is sponsored by the Shanghai Sailing Program (Grant No 17YF1419400). Dr. X.L. Zhou also thanks the support of the Tsien Hsueshen foundation for the Youth Innovation by China Aerospace Science and Technology Corporation. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apacoust.2017.12.012. References [1] Kushwaha MS, Halevi P, Dobrzynski L, Djafari-Rouhani B. Acoustic band structure of periodic elastic composites. Phys Rev Lett 1993;71:2022. [2] Liu ZY, Zhang XX, Mao YW, Zhu YY, Yang ZY, Chan CT, et al. Locally resonant sonic materials. Science 2000;289:1734. [3] Xiao W, Zeng GW, Cheng YS. Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs. Appl Acous 2008;69:255–61. [4] Cicek A, Kaya OA, Ulug B. Impacts of uniaxial elongation on the bandstructures of two-dimensional sonic crystals and associated applications. Appl Acous 2012;73:28–36. [5] Nouri MB, Moradi M. Presentation and investigation of a new two dimensional heterostructure phononic crystal to obtain extended band gap. Physica B 2016;489:28–32. [6] Wu LY, Chen LW. Propagation of acoustic waves in the woodpile sonic crystal with a defect. Appl Acous 2012;73:312–22. [7] Liu M, Xiang J, Zhong Y. The band gap and transmission characteristics investigation of local resonant quaternary phononic crystals with periodic coating. Appl Acous 2015;100:10–7. [8] Hirsekorn M. Small-size sonic crystals with strong attenuation bands in the audible frequency range. Appl Phys Lett 2004;84:3364. [9] Xia B, Chen N, Xie L, Qin Y, Yu D. Temperature-controlled tunable acoustic metamaterial with active band gap and negative bulk modulus. Appl Acous 2016;112:1–9. [10] Åberg M, Gudmundson P. The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure. J Acous Soc
4.2. Extending and lowering the band gaps In the previous section, the rubber layers are designed with the same size in thickness and one can also make a different size for each coating layers. Here, multilayered LRPC models with different microstructures in the inclusions are studied. For all the models, the matrix keeps constant size. The band gaps of different multilayered LRPC models are displayed in Fig. 13. For the sake of comparison, the band gaps of the multilayered LRPCs shown in Fig. 1 are shown in Fig. 13 as Models A–D. Models E–H possess a constant thickness of the coating tungsten layers of 2 mm. Models E and F refer to the LRPCs with three coating rubber layers whose thicknesses change in arithmetic progresses. The thicknesses of the rubber layers in Model E are 3 mm, 2 mm and 1 mm from inner to outer region. On the other hand, the corresponding rubber layer thicknesses of Model F are 1 mm, 2 mm and 3 mm. From Fig. 13, it can be seen that the band gaps of Models E–F are extended and lowered compared with Models A–D. Considering that the width of the band 105
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Extending and lowering band gaps by multilayered locally resonant phononic crystals ⁎
Xiaoling Zhoua, , Yanlong Xub, Yu Liua, Liangliang Lva, Fujun Penga, Longqi Wangc,
T
⁎
a
Shanghai Institute of Aerospace System Engineering, Shanghai 201109, China School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China c School of Civil & Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore b
A R T I C L E I N F O
A B S T R A C T
Keywords: LRPCs Multilayered inclusions Band gaps Transmission spectra
Locally resonant phononic crystals (LRPCs) have band gaps with low frequencies and the size of the cell is much smaller than the wave lengths of the band gaps. This unique property makes them good candidates for vibration control. Considering that the band gaps of the LRPCs are determined by the single cell rather than the periodic arrangement, it is important to design the microstructure of the cell. In this paper, band structures and transmission spectra of two dimensional multilayered LRPCs are investigated by the finite element method (FEM). It is found that the band gaps of the LRPCs can be extended to several frequency ranges by periodically embedding multilayered coaxial inclusions (i.e., alternate shells of soft rubber and hard metal) into a matrix. Moreover, the frequency responses of the multilayered periodic structures with different number of cells are simulated. There are several sharp dips which refer to high attenuation efficiency in the transmission spectra of the periodic structures with multilayered inclusions. The increase in the number of cells is also found to enhance the attenuation efficiency. Meanwhile, the geometrical effects of the layers on the band structures are investigated. By carefully designing the size and number of the shells in the multilayered inclusions, the band gaps can be extended and lowered. These results are helpful for the applications of LRPCs in noise and vibration control.
1. Introduction Phononic crystals (PCs) have attracted a growing attention for the band gap properties in recent years [1–6]. The mechanisms [7–9], calculation methods [10] and tunability [11–16] of the band structures of PCs are the main topics for scientific researchers. Since noise and vibration transmit in the form of elastic waves in structures, PCs have great potentials in noise and vibration control. However, for the Bragg scattering PCs, the wave lengths of the band gaps almost equal the size of the single periodic cell [1]. This makes the Bragg scattering PCs hardly meet the demand of vibration control in low frequency range since it is difficult and costly to create periodic structures with huge size in real applications. In 2000, Liu et al. [2] proposed the locally resonant phononic crystals (LRPCs), which refer to structures consisting of hard cores coated with soft layers and then embedding into a matrix periodically. The frequency of the first band gap is as low as ∼400 Hz while the size of the single periodic cell is as small as 1.55 cm. The LRPCs bring a brand-new sight for the vibration and noise control in engineering applications. Due to the potential applications of LRPCs, many researchers
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investigated the mechanisms of the low frequency band gaps [2,8]. As stated by Liu et al. [2], the local resonances of the cell result in the band gaps. Yao et al. [17] embedded rubber into a matrix to form a periodic beam and opened the locally resonant band gaps. Hirsekorn [8] and Wang et al. [18] proposed simplified theoretical models to calculate the frequency boundaries of the first band gap of two dimensional LRPCs. In the theoretical models, it is found that the frequency boundaries of the first band gap can be obtained by analyzing the natural frequencies of the mass-spring systems. For the LRPCs, the core and the matrix act as the mass while the soft layers act as the spring [8,18]. From the above studies, researchers found that it is the single cell rather than the periodic arrangement of the LRPCs that determines the band gaps. And hence, the band gaps of LRPCs can be tuned by changing the microstructure of the cell. Numerous methods are proposed to extend and lower the band gaps of PCs [19–26]. For example, Wang et al. [19] investigated the tuning characteristics of band gaps in a multi-stub LRPC plate and they discussed the effects of the stub structure and defect state on the band gap frequencies. Assouar and Oudich [20] used double stubs to enlarge the band gaps of LRPC, which composed of lead-rubber stubs deposited on
Corresponding authors. E-mail addresses: [email protected] (X. Zhou), [email protected] (L. Wang).
https://doi.org/10.1016/j.apacoust.2017.12.012 Received 3 April 2017; Received in revised form 29 November 2017; Accepted 11 December 2017 0003-682X/ © 2017 Elsevier Ltd. All rights reserved.
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the aluminum plate. Zhang et al. [21] obtained 42–150 Hz band gaps in the plate with periodic spiral resonators and they pointed out that the low frequency locally resonant band gaps can be tuned by changing the size of the spiral resonators. Ma et al. [22] investigated the Lamb wave propagation in phononic crystal consisting of periodic arrangement of bilayered materials in the radial direction. New lower band gaps appeared in the radial phononic crystal by applying crystal gliding. Some researchers designed self-similar hierarchy [23] or learned from bioinspired structures [24,25] to form PCs and got ultrawide low frequency band gaps. Zhou and Chen [26] studied the initial stress effects on the band gaps of LRPCs and they also used electric field to tune the band gaps. They investigated two dimensional LRPCs with single or double layers of elastomer and found that LRPCs with double layers of soft elastomer have two wide band gaps compared with the one wide band gap in single layer models. Larabi et al. [27] used a finite difference domain method to study the transmission spectra of LRPCs with multicoaxial cylindrical inclusions embedded in a fluid matrix. They indicated that the transmission spectra of the multilayered models can exhibit several sharp dips instead of one in the same frequency range. Moreover, the dips can be overlapped to create a larger acoustic band gap by appropriately combining the cells with different layers. For LRPCs, it is important to design the microstructure of the cell, and the LRPC with multilayered inclusions may show some novel characteristics. In this paper, the band structures of two dimensional multilayered LRPCs with multicoaxial cylindrical inclusions are investigated. The band gap mechanisms of the multilayered LRPCs with different inclusions are discussed. Moreover, the effects of the cell number on the transmission spectra are studied. It is proved that the band gaps can be extended and lowered by carefully designing the inclusions.
Table 1 Material constants of the components.
Tungsten Rubber Resin
Density (kg/m3)
Lame constants (GPa)
ρ
λ
G
19,100 1300 1180
306 6.051e−4 4.52
131.1 4e−5 1.59
rubber layers nr = 1, 2, 3 and 4 are studied. The thickness of the rubber and coating tungsten layers are trub = ttun = 0.002 m. The size of the LRPC models in the out of plane direction is infinite. So only the inplane wave motions are considered in this paper. The material constants of the models are given in Table 1. For the multilayered LRPC models, the equation of elastic wave propagation can be written as,
∇ [[λ (r) + 2G (r)](∇·u)]−∇ × [G (r) ∇ × u] = ρ u ¨
(1)
where u is the displacement. r the coordinate vector. λ and G are the Lame constants. u ¨ refers to second order derivative of the time. ρ is the density. For the multilayered LRPCs, the Bloch boundary condition must be satisfied, which gives
u (r) = u (r)·e i (k·r − ωt)
(2)
where k is the wave vector. ω is the angular frequency and t is time. For the periodic models, the Bloch boundary condition can be written in the form of,
2. Models and method
u (r) = u (r + n1 a1 + n2 a2)
To elucidate the band structures and mechanisms of the multilayered LRPCs, two dimensional models with inclusions composed by alternate shells of soft silicone rubber and hard tungsten are investigated by the finite element method (FEM). As shown in Fig. 1, the multilayered inclusions with radius of 0.018 m are periodically embedded into a resin matrix. The periodic models have square lattice structure and the irreducible Brillouin zone is shown in Fig. 1. The lattice constant keeps at 0.04 m. Here, models with the number of the
(3)
where a1 and a2 are the lattice vectors; n1 and n2 represent integers. Band structures of the multilayered LRPCs and the transmission spectra of the periodic structures with N × N cells are studied by using the commercial finite element software COMSOL [28]. Since the size of the models in the out of plane direction is infinite, the plane strain condition is adopted in all the simulations. The linear triangular elements are utilized for the models. Fig. 1. Schematic illustration of the multilayered LRPC cells and the irreducible Brillouin zone. The component materials are tungsten, silicone rubber and resin matrix.
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Fig. 2. Band structures of the multilayered LRPCs with different inclusions as shown in Fig. 1. Fig. 3. Displacement fields of (a) the lower band gap boundary and (b) the upper band gap boundary of the LRPC with nr = 1.
Fig. 4. Displacement fields of the band gap boundaries of the multilayered LRPC with nr = 2. (a) The lower boundary and (b) the upper boundary of the first band gap; (c) The lower boundary and (d) the upper boundary of the second band gap.
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Fig. 5. Displacement fields of the band gap boundaries of the multilayered LRPC with nr = 3. The snapshots of (a), (c) and (e) are the upper boundaries of the first, second and third band gaps while the snapshots of (b), (d) and (f) are the lower boundaries of the first, second and third band gaps, respectively.
Fig. 6. Displacement fields of the band gap boundaries of the multilayered LRPC with nr = 4. The snapshots of (a), (c), (e) and (g) are the upper boundaries of the first, second, third and fourth band gaps while the snapshots of (b), (d), (f) and (h) are the lower boundaries of the first, second, third and fourth band gaps, respectively.
3. Numerical results
multilayered LRPC with three rubber layers are 147.3–260.5 Hz, 373.1–533.8 Hz and 588.5–860.1 Hz as shown in Fig. 2(c). Fig. 2(d) shows that the band gaps of the LRPC with four rubber layers are 160.2–258.5 Hz, 337.7–441.2 Hz, 496.6–595.2 Hz and 620.5–861.8 Hz respectively. For the multilayered LRPC models with nr = 3 and 4, there are also straight lines in the first band gaps and that is because rotation vibrations occur in the coating rubber layers. Moreover, there are narrow band gaps at higher frequencies in Fig. 2(c) and (d). These narrow band gaps are not taken into account in this paper. It is found from Fig. 2 that the number of the band gaps in the multilayered LRPCs increases as the rubber layers number nr grows. The reason is that the core and the coating tungsten layers act as localized resonators and the localized vibrations open the band gaps. To further understand the mechanisms of the band gaps in the multilayered LRPCs, the eigen modes of the band gap boundaries are discussed in this section. The corresponding displacement fields of the band gap boundaries are shown in Figs. 3–6. The arrows and colors refer to the displacements of the cells. As shown in Fig. 3(a), the core moves as a whole while the matrix almost keeps stationary at the beginning of the band gap in the LRPC
3.1. Band structures and eigen modes Fig. 2 gives the band structures of the multilayered LRPCs which shown in Fig. 1. For the LRPC with a single rubber layer (see Fig. 2(a)), there is a wide band gap with the frequency range of 176.5–780.3 Hz in the band structure. There are two wide band gaps in the band structure for the model with double rubber layers as shown in Fig. 2(b). The frequency boundaries of the first band gap are 147.6 Hz and 340.8 Hz. The frequency boundaries of the second band gap are 493.8 Hz and 856.3 Hz. One can notice that a flat straight line which refers to a rotation vibration mode of the coating rubber layer appears in the first band gap in Fig. 2(b). Besides, a narrow band gap at higher frequencies appears in the band structures in Fig. 2(a) and (b). Because the frequency ranges are much narrower and there are some straight lines in the frequency ranges, these band gaps at higher frequencies can hardly be used in engineering applications. For the models with nr = 3 and 4, it is interesting to see that the numbers of the wide band gaps are 3 and 4 respectively. The frequency ranges of the band gaps for the
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Fig. 7. Schematic illustration of the transmission spectra simulation of the finite periodic plate.
Fig. 8. Transmission spectra of the periodic plates with different number of multilayered inclusions shown in Fig. 1.
whole, which causes the coating layers to move but hardly the matrix. At the end of the first band gap as shown in Fig. 4(b), the cores and the matrix move toward opposite directions. These mechanisms are similar to those of the LRPC with one rubber layer. However, the eigen modes of the second band gap boundaries are very different. The coating tungsten layer and the tungsten core move to opposite directions at the lower boundary of the second band gap, which is shown in Fig. 4(c). For the upper boundary of the second band gap, the coating tungsten layer
with single rubber layer. At the end of the band gap, the core and the matrix move in opposite directions as Fig. 3(b) has shown. These mechanisms agree well with the results mentioned earlier [2,8,18]. The rubber layer acts as the spring. The core and matrix act as the lumped mass. The localized resonances of the spring-mass resonator result in the band gap. For the multilayered LRPC with two rubber layers, the displacement fields of the band gap boundaries are shown in Fig. 4. At the beginning of the first band gap (see Fig. 4(a)), the core moves as a 101
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Fig. 9. Displacement amplitude attenuation fields of the 5 × 5 models of (a) f = 180 Hz, nr = 1; (b) f = 150 Hz, nr = 2 and (c) f = 500 Hz, nr = 2.
multilayered LRPC cell can be changed to fit in different situations.
and the matrix move oppositely. It is found that the coating tungsten, the core and the matrix can all act as lumped mass. To further elucidate the effect of the coating layers, the eigen modes at the band gap boundaries of the multilayered LRPC with nr = 3 are shown in Fig. 5. For the lower boundaries of the band gaps (Fig. 5(a), (c), (e)), the cores and the coating tungsten layers move as lumped mass while the matrix almost keeps stable. In Fig. 5(a), the core moves dominantly as a whole. In Fig. 5(c), the coating tungsten layers and the cores move to opposite directions. However, at the beginning of the third band gap, as shown in Fig. 5(e), the two coating tungsten layers move to the opposite directions. For the upper boundaries of the band gaps (Fig. 5(b), (d), (f)), the epoxy matrix and the tungsten move to opposite directions. The core, the inner coating tungsten layer and the outer coating tungsten layer move oppositely to the matrix at the end of the first, second and third band gaps respectively. As the number of the coating layers increases, more local resonance modes appear and result in more band gaps. The eigen modes at the band gap boundaries of the multilayered LRPC with nr = 4 are shown in Fig. 6. The mechanisms shown in Fig. 6 are similar to those in Fig. 5. The core, the coating tungsten layers and the matrix act as lumped mass while the coating rubber layers act as spring. At the lower boundaries (see Fig. 6(a), (c), (e), (g)), the matrix almost keeps unmoved. The core and the coating tungsten layers move in different modes at different band gaps. On the other hand, the core and the coating tungsten layers move oppositely to the matrix at the upper boundaries. According to the above analysis, one can see that the number of the band gaps increases with the number of the coating layers due to the presence of more local resonance modes. The increase in the number of band gaps in multilayered LRPCs benefits the applications of noise and vibration control. Moreover, the material constants and the size of the
3.2. Transmission spectra To investigate the attenuation effects of the multilayered LRPCs, the transmission spectra of the corresponding periodic plates with different number of cells are calculated via the frequency response analysis in COMSOL [28]. As shown in Fig. 7, to avoid reflections, two perfect matched layers (PMLs) with the length of 8 m are applied in the x direction while periodic boundaries are used in the y direction. The parameters of the PMLs are identified by the software automatically. A prescribed displacement with the average value of Si is imposed on the excitation port of the periodic plate in the x direction. The integration of the displacement amplitude in the x direction on the receive port is obtained and the average value on the boundary (denoted by So) is calculated. So the transmission can be obtained by 20 × log(So/Si). The periodic plates consist of 5 × 5, 8 × 8 and 10 × 10 periodic cells, respectively. Then it is easy to find the band gap boundaries from the intersections of the transmission spectra of the periodic plates with different number of cells. For efficiency’s sake, the frequency step is set to 10 Hz in the simulations. Fig. 8 gives the transmission spectra of the periodic plates with different number of multilayered inclusions. The shadow regions in Fig. 8(a)–(d) refer to the band gap frequency ranges which shown in Fig. 2(a)–(d). It can be seen that there are wave attenuation in the band gap frequencies. For the multilayered LRPCs, there are sharp dips at the beginning frequencies of the band gaps. This is consistent with the results in Ref. [2]. One can notice that there are some straight lines in the first band gap in the multilayered LRPCs which don’t appear in the transmission spectra. This is because that the frequency step is 10 Hz 102
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Fig. 10. Displacement amplitude attenuation fields of the 5 × 5 models of nr = 3 at (a) f = 150 Hz; (b) f = 380 Hz and (c) f = 590 Hz.
transmission dips in the band gaps are given in Figs. 9–11. To show the attenuation effects clearly, only the displacement amplitude attenuation fields of the periodic cells are shown here. The fields of the whole models with PMLs are given in the Supplementary Materials. For the LRPCs, the equivalent mass of the resonances, the vibration amplitude and the frequency all have influences on the transmission efficiency. Large mass and high vibration amplitude are beneficial for consuming energy. For a given size, higher frequency wave with smaller wave length will be easier to reduce. As shown in Figs. 9–11, the localized vibration of the resonances is the key factor contributing to vibration attenuation. For the model of nr = 2, the rubber and tungsten layers vibrate at a much higher frequency of 500 Hz. It leads to a larger attenuation (see Fig. 9(c)). For the model of nr = 3, it is found that the largest displacement amplitude of the localized resonance mode is at the beginning of the third band gap as shown in Fig. 10. The localized vibration with the largest displacement amplitude at a high frequency of 590 Hz benefits energy consumption and results in a minimum transmission. The displacement amplitudes of the nr = 4 model at different frequencies are shown in Fig. 11. One can see from Fig. 11 that the attenuation effects in Fig. 11(b) and (c) are better than those in Fig. 11(a) and (d). The displacement amplitude of the resonance in Fig. 11(b) is the largest and hence the minimum transmission is in the second band gap as shown in Fig. 8. Although the increasing number of layers may not reduce the minimum transmission further, it can still improve the number of the transmission dips. For the LRPCs, there are transmission dips at the beginning of the band gap, and then the curves increase sharply. Hence the transmission dips are of great importance for low frequency vibration control in engineering. The multilayered LRPCs provide a way to enhance the vibration attenuation.
and it may omit some frequencies. Moreover, there are some narrow band gaps appearing in the transmission spectra of the periodic plates as shown in Fig. 8(a)–(d) and the frequency ranges of the narrow band gaps are consistent with those in Fig. 2(a)–(d). Vibration can be reduced in the band gaps depicted as shadow regions in Fig. 8. However, the attenuation effects are more obvious at the beginning of the band gaps as we can see from the sharp dips in the transmission spectra. As a result, the plates with multilayered inclusion of nr = 4 exhibit an advantage in vibration control due to the four sharp dips in the transmission spectrum shown in Fig. 8(d). Besides, it is found that the attenuation effects can be enhanced by increasing the number of the periodic cells. For the plates with periodic inclusion of nr = 1 (see Fig. 8(a)), the largest attenuation of the vibration is about 28 dB in the 5 × 5 model. When the number of the cells increases to 8 × 8, the largest attenuation can be improved to about 47 dB. For the 10 × 10 model, the largest attenuation is 60 dB. The largest attenuation of the nr = 2 plate is located in the second band gap as shown in Fig. 8(b). As the number of cell increases, the largest attenuation increases from 27 dB to 44 dB and then to 56 dB. In Fig. 8(c), the largest attenuation of the periodic plate with periodic cell of nr = 3 are 59 dB, 95 dB and 118 dB for the 5 × 5, 8 × 8 and 10 × 10 model, respectively. Fig. 8(d) displays the transmission spectra of the nr = 4 model, the attenuation in the sharp dips rises with the number of cells as well. The largest attenuation in the second dip for the 10 × 10 model is 72 dB. Consequently, the structures containing periodic arrangement of the multilayered inclusions can reduce the vibration efficiently. The attenuation effects can be enhanced through the careful design of the inclusions as well as the appropriate increase of the cell number. To further investigate the attenuation mechanisms of the periodic structures, the displacement amplitude fields of the 5 × 5 models at the 103
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Fig. 11. Displacement amplitude attenuation fields of the 5 × 5 models of nr = 4 at (a) f = 170 Hz; (b) f = 340 Hz; (c) f = 500 Hz and (d) f = 630 Hz.
Fig. 12. Dependence of the band gap boundaries on the thickness of the rubber layers. (a) Multilayered LRPC with nr = 3 and (b) multilayered LRPC with nr = 4.
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gaps in the multilayered LRPCs increases as the rubber layer thickness decreasing, the layer thicknesses of the rubbers in Models G and H are reduced to 0.0005 m. It can be seen that the band gaps of Models G and H are much wider than those of Models A–F. Model H has only one coating rubber layer, hence there is only one band gap. For Model G, the number of the coating rubber layers is six so that there are six band gaps. Comparing Model G with Model H, it is obvious that the boundary of the first band gap can be lowered. In this section, it can be seen that the band gaps can be extended and lowered in multilayered LRPCs by changing the size and numbers of the coating layers. As is known, LRPCs have the advantage of low frequency band gaps. But the band gaps can hardly be tuned by the arrangement of the periodic cells. The multilayered LRPCs provide chances for more different characteristics in low frequency band gaps and this is helpful in real-life applications in various situations. 5. Conclusions
Fig. 13. Band gaps of the hierarchical LRPCs with different microstructures of the single cell. Models A–D refer to the corresponding LRPCs shown in Fig. 1. Model E and F represent the LRPCs of three coating rubber layers of which the thicknesses change in arithmetic progresses. From the inner to outer layers, the thicknesses of rubber layers for Model E are 3 mm, 2 mm and 1 mm while those of Model F are 1 mm, 2 mm and 3 mm. Model G and H refer to LRPCs with the thickness of the rubber layers of 0.5 mm. Model G has six coating rubber layers while Model H has only one coating rubber layers. The matrix keeps constant in size for all the Models.
In summary, band structures of multilayered LRPCs are investigated by FEM. By embedding multicoaxial cylindrical inclusions into a matrix, the band gaps can be extended to several ranges because of the presence of more localized resonance modes. The periodic structures consisting of multilayered inclusions are proved to reduce the vibration in the band gaps by calculating the transmission spectra. The attenuation effects can be enhanced by increasing the number of coating layers and the number of multilayered cells. Lastly, the effects of the geometrical sizes are discussed. It is found that the band gap frequencies can be extended and lowered by carefully designing the multilayered inclusions of the LRPCs. The wide low frequency band gaps in the multilayered LRPCs are useful in real-life applications. The experimental verification and engineering applications of the multilayered LRPCs will be discussed in the future work.
4. Discussions 4.1. Band gaps with different geometrical sizes From the simulation results, it is known that the multilayered LRPCs show more band gaps than the LRPC with one rubber layer. The multilayered inclusions can also bring more changes in the size of the microstructures which may influence the band gaps. In this section, the geometrical size effects on the band gaps of the multilayered LRPCs of nr = 3 and 4 are discussed. Here, the size of the matrix keeps unchanged. When changing the size of the coating layers, the thickness of all the coating rubber layers keep equal to each other. If the thickness of the coating rubber layers reduces, the coating tungsten layers will increase the same size in thickness. Fig. 12 shows the dependence of the band gap boundaries on the thickness of the rubber layers. Since the narrow band gaps are not considered in this paper, only the wide band gaps are shown in Figs. 12 and 13. It can be seen that the width of the band gaps increases as the thickness of the rubber layers decreases. As reducing the thickness of the rubber layers, the upper boundaries of the band gaps rise quickly while the lower boundaries of the band gaps change much less. It means one can widen the band gaps by changing the size of the coating layers.
Acknowledgement This work is sponsored by the Shanghai Sailing Program (Grant No 17YF1419400). Dr. X.L. Zhou also thanks the support of the Tsien Hsueshen foundation for the Youth Innovation by China Aerospace Science and Technology Corporation. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apacoust.2017.12.012. References [1] Kushwaha MS, Halevi P, Dobrzynski L, Djafari-Rouhani B. Acoustic band structure of periodic elastic composites. Phys Rev Lett 1993;71:2022. [2] Liu ZY, Zhang XX, Mao YW, Zhu YY, Yang ZY, Chan CT, et al. Locally resonant sonic materials. Science 2000;289:1734. [3] Xiao W, Zeng GW, Cheng YS. Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs. Appl Acous 2008;69:255–61. [4] Cicek A, Kaya OA, Ulug B. Impacts of uniaxial elongation on the bandstructures of two-dimensional sonic crystals and associated applications. Appl Acous 2012;73:28–36. [5] Nouri MB, Moradi M. Presentation and investigation of a new two dimensional heterostructure phononic crystal to obtain extended band gap. Physica B 2016;489:28–32. [6] Wu LY, Chen LW. Propagation of acoustic waves in the woodpile sonic crystal with a defect. Appl Acous 2012;73:312–22. [7] Liu M, Xiang J, Zhong Y. The band gap and transmission characteristics investigation of local resonant quaternary phononic crystals with periodic coating. Appl Acous 2015;100:10–7. [8] Hirsekorn M. Small-size sonic crystals with strong attenuation bands in the audible frequency range. Appl Phys Lett 2004;84:3364. [9] Xia B, Chen N, Xie L, Qin Y, Yu D. Temperature-controlled tunable acoustic metamaterial with active band gap and negative bulk modulus. Appl Acous 2016;112:1–9. [10] Åberg M, Gudmundson P. The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure. J Acous Soc
4.2. Extending and lowering the band gaps In the previous section, the rubber layers are designed with the same size in thickness and one can also make a different size for each coating layers. Here, multilayered LRPC models with different microstructures in the inclusions are studied. For all the models, the matrix keeps constant size. The band gaps of different multilayered LRPC models are displayed in Fig. 13. For the sake of comparison, the band gaps of the multilayered LRPCs shown in Fig. 1 are shown in Fig. 13 as Models A–D. Models E–H possess a constant thickness of the coating tungsten layers of 2 mm. Models E and F refer to the LRPCs with three coating rubber layers whose thicknesses change in arithmetic progresses. The thicknesses of the rubber layers in Model E are 3 mm, 2 mm and 1 mm from inner to outer region. On the other hand, the corresponding rubber layer thicknesses of Model F are 1 mm, 2 mm and 3 mm. From Fig. 13, it can be seen that the band gaps of Models E–F are extended and lowered compared with Models A–D. Considering that the width of the band 105
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