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Research on bandgap property of a novel small size multi-band phononic crystal
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Research on bandgap property of a novel small size multi-band phononic crystal
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a
Yake Dong , Hong Yao
a, b
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, Jun Du , Jingbo Zhao
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, Chao Ding
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Air Force Engineering University, School of Aeronautics and Astronautics Engineering, Xi’an 710038, China b Air Force Engineering University, Science College, Xi’an 710051, China
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i n f o
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Article history: Received 30 August 2018 Received in revised form 9 October 2018 Accepted 27 October 2018 Available online xxxx Communicated by M. Wu
A novel small size multi-band gaps phononic crystal (hPC) composed of the periodic plumbum and aluminum mass has been investigated in this paper, we calculated the dispersion relations and the displacement fields with finite element method (FEM). The influences of the geometrical parameters and materials parameters on the band structure are further investigated. We studied the influence factors of band gap by the model of origin anti-resonance. There are many flat bands in the characteristic frequency curve. The band gap covers a large range in low frequency. © 2018 Elsevier B.V. All rights reserved.
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Keywords: Local resonance Bragg scattering Multi-band gaps Origin anti-resonance
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The propagation of elastic waves in periodic structures has been researched for many years, the propagation of elastic waves in the periodic composite materials have received much attention for their renewed physical properties and potential applications in various fields [1–8]. There are three different formation mechanisms of the band gaps, namely Bragg scattering, local resonance and hybridization. For the first mechanism, which is caused by multiple scattering of the periodic inclusions, the wavelength of the band gap frequencies is as the same order of the structural period [9, 10]. The local resonance employs the resonant vibration of the local resonance mass working against the excitation of the incident elastic wave to attenuate the vibration [9,10]. The frequency range of the band gap based on local resonance is almost two orders of magnitude lower than that of the Bragg scattering. The third kind of the band gap formation mechanism is attributed to the coupling effects of local resonances and Bragg scattering. Such gaps are particularly interesting, since it can be shown that they allow the definition of negative effective material parameters in some cases [11,12]. To broaden the band gap, the researchers have designed many kind of hybrid phonon crystal. Zhang et al. designed a novel hybrid phononic crystal (PC) to obtain wider band gaps in low frequency range. The multi-band phononic crystal consists
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1. Introduction
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E-mail address: [email protected] (Y. Dong). https://doi.org/10.1016/j.physleta.2018.10.042 0375-9601/© 2018 Elsevier B.V. All rights reserved.
of rubber slab with periodic holes and plumbum stubs. In comparison with the phononic crystal without periodic holes, the new designed phononic crystal can obtain wider band gaps and better vibration damping characteristics [13]. Wang et al. designed a hybrid structure composed of a local resonance mass and an external oscillator. The hybrid structure can produce multi-band gaps wider than the multi-resonator acoustic metamaterials [14]. M.B. Assouar et al. proposed a hybrid phononic crystal plates which are composed of periodic stepped pillars and periodic holes. The acoustic waves scattered simultaneously by the pillars and holes in a relevant frequency range can generate low and wide acoustic forbidden bands [15]. Yoon et al. designed a hybrid phononic crystal consisting of a bi-prism. A stop band is formed in a central frequency range while positive–positive and positive–negative refractions occur in lower and higher frequency ranges to concentrate acoustic energy in a central localized zone [16]. L. D’Alessandro et al. presented a strategy to design three-dimensional elastic periodic structures endowed with complete bandgaps, the first of which is ultra-wide, where the top limits of the first two bandgaps are overstepped in terms of wave transmission in the finite structure. Thus, subsequent bandgaps are merged, approaching the behavior of a three-dimensional low-pass mechanical filter [17]. In this paper, we investigated a two-dimensional novel multiband gaps phononic crystal (PC). The sections of this paper are arranged as follows: In Section 2, we show the structural model and band gap calculation method. In section 3, the dispersion relations, the displacement fields of the eigenmodes, and the power
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Table 1 Materials parameters. Material
Mass density (kg/m3 )
Elasticity modulus (1010 Pa)
Poisson ratio (1010 Pa)
Plumbum Rubber Plastomer
11600(ρb ) 1300(ρc ) 1190(ρa )
4.08 1.17e−5 0.22
0.37 0.47 0.34
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Table 2 Structure parameters. a (mm)
b (mm)
c (mm)
d (mm)
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2. Model and method
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∂ −ρ (r )ω u i = ∇ · μ(r )∇ u i + ∇ · μ(r ) u ∂ xi ∂ + λ(r )∇ · u (i = x, y ) ∂ xi 2
(2)
Brillouin zone, and uk is a periodic vector function having spatial periodicity the same as the periodic elastic system. The spatial part of the time-harmonic displacement vector is:
(r + p ) = u (r ) exp(ik · p ) u
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is the spatial period vector of the PC structures. where p
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H = 10 log
po
(3)
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pi
102 103 104 105
(4)
where p o and p i are the number value of the transmitted acceleration and incident acceleration, respectively. The wave propagates along direction X . The right side is of response output in Fig. 1(b). Meanwhile, the perfect matching layers are added at both sides of the structure to guarantee the accuracy of the result. 3. Numerical result and discussion
where k = (kx , k y ) is the Bloch wavevector in the irreducible first
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(1)
In Eqs. (1), r = (x, y ) denotes the position vector, ω is the angu = (u x , u y ) is the displacement lar frequency, ρ is mass density, u vector in the transverse plane, λ and μ are the Lame constant and shear modulus, and ∇ = ( ∂∂x , ∂∂y ) is the 2D vector differential operator. Periodic elastic system has the following form:
(r ) = u k (r ) exp(ik · r ) u
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COMSOL is used for the implementation of the above mentioned model for the characteristic frequency of the proposed multi-band PC. The transmission function is defined as:
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A multi-band PC unit is shown in Fig. 1. The materials parameters and structure parameters used in the numerical calculations are shown in Table 1 and Table 2, respectively. The governing field equations in the elastic structures for the in-plane mode can be expressed as [18]:
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Fig. 2. Characteristic frequency for first 50 order and transmission loss curve. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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transmission spectra were calculated. The generation reasons of band gap and material parameters were discussed. The conclusion was described in Section 4. This paper used the finite element software, COMSOL Mutiphysics 4.3a, to calculate eigenfrequency, and used Lagrange quadratic element for mesh dissection. The periodic boundary condition was applied to the unit cell in the simulation.
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Fig. 1. Schematics of elastic phononic crystal plate (a) the unite of the elastic metamaterial plate, (b) elastic metamaterial plate, (c) the first Brillouin zone of a unit cell.
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3.1. Band structures of the new multi-band phononic crystal
In this section, we present the numerical results of the first 80th characteristic frequency and the transmission loss of the PC. The materials parameters and structure parameters used in the numerical calculations are shown in Table 1 and Table 2, respectively. The band structure of the elastic wave is calculated for the PC structures in Fig. 2. The numerical transmission spectrum considering losses is reported as the blue dashed line in Fig. 2. The first 12th characteristic frequency is shown in Fig. 3(a). The first BG (green band) is located between third and fourth band, the first BG is from 399 Hz to 550 Hz as shown in Fig. 3(a). The second BG (yellow band) is from 659 Hz to 1266 Hz. The third BG (blue band) is from 1321 Hz to 2753 Hz. In order to prove the effectiveness of the band gap, we calculate the transmission loss of 10 ∗ 10
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Fig. 3. (a) Characteristic frequency for first 12 order. (b) Characteristic frequency from 13th to 80th order.
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3.2. Formation mechanisms of the band gap In this section, we presented the numerical results and analyzed the propagation behaviors of elastic waves associated with different bands. The transmission loss is shown as Fig. 2. The greater loss of transmission (gray part) is the same as the BG frequency. In order to investigate the formation mechanism of the BG, we analyzed the vibrational modes of the first few BGs. There are several modes from mode A to mode G in Fig. 3(a). We calculated the displacement vectors of the seven modes, respectively. The vibration of mode A is the translational movement of Plumbum vibrators as shown in Fig. 4(A), the vibration of mode B and mode C is the translational movement of plastomer vibrators as shown in Fig. 4(B) and Fig. 4(C). The vibration mode B is the vibration of four plastic vibrators. The vibration of the starting frequency of first BG is the translational movement of Plumbum vibrators as mode A. The vibration of the cutoff frequency of first BG is the translational movement of four plastomer vibrators, the thither plastomer vibrators move in opposite directions with the same frequency in mode B. The second BG (yellow band) is from 659 Hz to 1266 Hz, the vibration mode of starting frequency of second BG is translational movement of two plastomer vibrators in mode E. The vibration mode of cutoff frequency of second BG is translational movement of two plastomer vibrators in mode E, two plastomer vibrators moved in the same direction. The vibration of mode D is torsional movement of plastomer vibrator as shown in Fig. 4(D). The vibration of mode E is translational movement of plastomer. The third BG (blue band) is from 1321 Hz to 2753 Hz. The starting frequency of BG is translational movement of two plastomer vibrators in mode F. The vibration of mode G is translational movement of rubber. The vibrations of mode H, mode M, mode K, mode S are movement of rubber in Fig. 4. Elastic waves are reflected back and forth between the hard boundaries. Elastic waves are localized in a unit and cannot propagate, this is Bragg scattering. According to the above analysis, the first two band gaps are local resonance band gaps, and the third band gap is caused by the interaction of local resonance and Bragg scattering. The local resonance band gap is caused by the original anti-resonance of different oscillators. The first (green part) band gap is initiated by resonance of the larger mass lead oscillator, while the smaller mass oscillator remains stationary in the same direction of motion. The cut-off frequency of the band gap is due to the anti-resonance of the smaller mass oscillator while the larger mass oscillator remains stationary. We theoretically derive the conditions for anti-
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periodic structures in Fig. 1(b). The frequency of the larger attenuation (grey fraction) are equal to the band gaps, which verify the accuracy of band gaps. In order to investigate the physical mechanism of band gaps, we calculated the displacement vector of high symmetry points of the Brillouin zone.
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Fig. 4. Eigenmodes shapes and displacement vector fields of the modes at the edges of the BG.
resonance [19,20]. The second band gap (yellow part) is caused by the local resonance of the small mass oscillator, compared with the mode at the cut-off frequency of the first band gap. Both the starting and cut-off frequencies of the second band gap are caused by the reverse displacement of two different small mass oscillators. The starting frequency of the third band gap is generated by the local resonance of the small mass oscillator, and the cut-off frequency is caused by the elastic wave scattering, i.e. Bragg scat-
K tering. Because of the local resonance frequency f = 21π M , local resonance is sensitive to the oscillator mass. We calculated the characteristic frequencies of different oscillator densities as shown in Fig. 5 and Fig. 6. The band gap change along with the different oscillator density. The red characteristic frequency curve decreases with the increase of mass density. The different oscillator mass has no effect on the higher order eigen-frequency. It can be seen that
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Fig. 5. The dispersion curve of different density
ρb .
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Fig. 6. The dispersion curve of different density
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the large mass oscillator only affects the starting frequency of band gap, while the small mass oscillator only affects the corresponding cut-off frequency of band gap. It is also proved that the first and the second band gaps are locally resonant band gaps and the higher order band gaps are Bragg scattering band gaps. We change the density of rubber as shown in Fig. 7. With increasing of density of rubber, the band gap frequency increases, but the shape of the dispersion curve does not change, the magnitude of the characteristic frequency changes. Therefore, this structure can produce a wide low frequency band gap, both the local resonance band gap and Bragg scattering band gap are in the dispersion curve. Dispersion curve is determined by the structure.
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3.3. Influence of geometry and material parameters on the low-frequency BG
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Fig. 7. The dispersion curve of different density
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In order to further study the effect of parameters on the first band gap, we explain the physical mechanism of the effect of parameters on the band gap by the equivalent model [21]. The
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ρr of rubber.
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Fig. 8. Mode of origin anti-resonance.
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equivalent model of the original resonance structure is shown in Fig. 8. M and m are mass of adjacent oscillators ( M > m), and K is equivalent spring stiffness. The vibration system can be equivalent to two single degree of freedom systems with different mass. It has been proved that the system has the effect of origin antiresonance when the frequency of incident elastic wave is equal to the natural frequency of the adjacent subsystem. One oscillator is vibrating, and the other oscillator of the adjacent subsystem remains stationary. The natural frequencies of adjacent subsystems are F =
1 2π
K M
and f =
1 2π
K . m
In the local resonance band gap,
the starting frequency of the band gap is F = 21π
K , M
and the cut-
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Fig. 9. Spring-mass model.
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M = ρb · V b +
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Kt =
C1 S
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K 2 = 2K s
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fc =
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1
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(5) (6) (7)
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ρb in different methods.
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Fig. 10. The first band gap varies with
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K off frequency of the band gap is f = 21π m . We can see that the boundary frequency of the band gap is only affected by the natural frequency of the corresponding vibration system. The law of band gap changing with parameters conforms to the band gap changing law of the original anti-resonance. To prove the validity of the proposed equivalent model, the frequency of the local resonant band gap with different densities is calculated by the equivalent model method (EMM), and the results are compared with those obtained by the finite element method (FEM). According to the vibration mode of the first band gap boundary frequency and cell model, we calculate eigen-frequency with the equivalent mass-spring model. The vibration mode of starting frequency of the band gap is corresponding to the model in Fig. 9(a), and the vibration mode of cut-off frequency is corresponding to the model in Fig. 9(b).
(11)
(12)
m is mass of equivalent small mass oscillator, M is mass of equivalent large mass oscillator, K t is the stiffness of the effective tension/compression spring. K s is a part of stiffness of the shear spring contributing to the tension/compression spring. α = 0.1 is a weighting coefficient, and elastic constant C 1 = λ + 2μ and C 2 = μ, K 1 is the stiffness of model for starting frequency of BG, K 2 is the stiffness of model for cut-off frequency of band gap, f s is starting frequency of band gap, f c is cutoff frequency of band gap. It can be seen that the calculation results of the two different methods are the same roughly as show in Fig. 10. The effectiveness of the proposed equivalent model is illustrated. The influences of the materials parameters and structure parameters on the BG are investigated to further study the multiband PC characteristics. We change the density ρa , ρb , the length of rubber b and the length of plastomer c in Fig. 1. The starting frequency of band gap decreases with the increase of density ρb , but the cutoff frequency of band gap is almost unchanged. It indicates that the starting frequency is more sensitive to the ρb than the cutoff frequency as shown in Fig. 11(a). The
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Fig. 11. (a) Variation of the frequency of the BG of the multi-band PCs when ρb is changed. (b) Variation of the frequency of the BG of the multi-band PCs when ρa is changed. (c) Variation of the frequency of the BG of the multi-band PCs when b is changed. (d) Variation of the frequency of the BG of the multi-band PCs when c is changed.
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cutoff frequency of band gap remain unchanged with the increase of density ρa , but the starting frequency of band gap decreases gradually as show in Fig. 11(b). With the increase of b, the starting frequency of band gap increases gradually, but the cutoff frequency of band gap varies greatly in Fig. 11(c). The cutoff frequency is more sensitive to the b. With the increase of c, the starting frequency of band gap remains the same, but the cutoff frequency of band gap varies greatly at c = 8 in Fig. 11(d). We can conclude that, to increase the width of the band gap, the ρb , ρa and b will be increased, the c will be decrease.
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4. Conclusions
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In this paper, we present a novel multi-band gaps phononic crystal with broad band gap in low frequency, and calculated the frequency of band gap, transmission loss and displacement vector by the finite element method. Subsequently, the formation mechanism of the band gaps is analyzed by the displacement field of the eigenmodes at the band gap edges. Finally, the effects of the geometrical parameters and materials parameters on the band gap are studied and discussed in detail. The conclusions are that: The new multi-band gaps phononic crystal has a broad band gap in low frequency. There are many flat bands in low frequency,
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[1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, et al., Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (13) (1993) 2022. [2] R. Martínez-Sala, Sound attenuation by sculpture, Nature 378 (1995) 241. [3] Z. Liu, X. Zhang, Y. Mao, et al., Locally resonant sonic materials, Science 289 (5485) (2000) 1734–1736. [4] Z. Liu, C.T. Chan, P. Sheng, Three-component elastic wave band-gap material, Phys. Rev. B 65 (16) (2002) 165116. [5] G. Wang, D. Yu, J. Wen, et al., One-dimensional phononic crystals with locally resonant structures, Phys. Lett. A 327 (5–6) (2004) 512–521. [6] G. Wang, X. Wen, J. Wen, et al., Two-dimensional locally resonant phononic crystals with binary structures, Phys. Rev. Lett. 93 (15) (2004) 154302. [7] C.Q. Chen, J.Z. Cui, H.L. Duan, et al., Perspectives in mechanics of heterogeneous solids, Acta Mech. Solida Sin. 24 (1) (2011) 1–26. [8] M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure, J. Sound Vib. 158 (1992) 377–382.
[9] M.M. Sigalas, E.N. Economou, Attenuation of multiple-scattered sound, Europhys. Lett. 36 (4) (1996) 241. [10] Z. Liu, C.T. Chan, P. Sheng, Three-component elastic wave band-gap material, Phys. Rev. B 65 (16) (2002) 165116. [11] C. Croënne, E.J.S. Lee, H. Hu, et al., Band gaps in phononic crystals: generation mechanisms and interaction effects, AIP Adv. 1 (4) (2011) 041401. [12] Y. Chen, L. Wang, Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps, Appl. Phys. Lett. 105 (19) (2014) 191907. [13] Z. Zhang, X.K. Han, A new hybrid phononic crystal in low frequencies, Phys. Lett. A 380 (45) (2016) 3766–3772. [14] T. Wang, M.P. Sheng, H.B. Guo, Multi-large low-frequency band gaps in a periodic hybrid structure, Mod. Phys. Lett. B 30 (08) (2016) 1650116. [15] M.B. Assouar, J.H. Sun, F.S. Lin, et al., Hybrid phononic crystal plates for lowering and widening acoustic band gaps, Ultrasonics 54 (8) (2014) 2159–2164. [16] S. Yoo, Y.J. Kim, Y.Y. Kim, Hybrid phononic crystals for broad-band frequency noise control by sound blocking and localization, J. Acoust. Soc. Am. 132 (5) (2012) EL411–EL416. [17] L. D’Alessandro, E. Belloni, R. Ardito, et al., Mechanical low-frequency filter via modes separation in 3D periodic structures, Appl. Phys. Lett. 111 (23) (2017) 231902. [18] Y.F. Wang, Y.S. Wang, X.X. Su, Large bandgaps of two-dimensional phononic crystals with cross-like holes, J. Appl. Phys. 110 (2011) 113520. [19] Y. Dong, H. Yao, J. Du, et al., Research on local resonance and Bragg scattering coexistence in phononic crystal, Mod. Phys. Lett. B 31 (11) (2017) 1750127. [20] Y. Dong, H. Yao, J. Du, et al., Research on low-frequency band gap property of a hybrid phononic crystal, Mod. Phys. Lett. B 32 (15) (2018) 1850165. [21] L. Brillouin, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Courier Corporation, 2003.
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which increases the number of band gaps. There is no waveform conversion in the X direction and Y direction, the characteristic frequency curve is straight line, which produces a lot of flat band gap, the band gaps cover more than 95% below 3000 Hz. The boundary frequency of local resonance band gap is only influenced by the parameters of the corresponding vibration system, the coupling factor is less, and it is easy to be adjusted. In order to increase the width of the local resonance band gap, the mass of the large mass oscillator should be increased and the mass of the light mass oscillator will be reduced.
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Research on bandgap property of a novel small size multi-band phononic crystal
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Yake Dong , Hong Yao
a, b
a
, Jun Du , Jingbo Zhao
a, b
, Chao Ding
a
a
Air Force Engineering University, School of Aeronautics and Astronautics Engineering, Xi’an 710038, China b Air Force Engineering University, Science College, Xi’an 710051, China
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Article history: Received 30 August 2018 Received in revised form 9 October 2018 Accepted 27 October 2018 Available online xxxx Communicated by M. Wu
A novel small size multi-band gaps phononic crystal (hPC) composed of the periodic plumbum and aluminum mass has been investigated in this paper, we calculated the dispersion relations and the displacement fields with finite element method (FEM). The influences of the geometrical parameters and materials parameters on the band structure are further investigated. We studied the influence factors of band gap by the model of origin anti-resonance. There are many flat bands in the characteristic frequency curve. The band gap covers a large range in low frequency. © 2018 Elsevier B.V. All rights reserved.
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Keywords: Local resonance Bragg scattering Multi-band gaps Origin anti-resonance
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The propagation of elastic waves in periodic structures has been researched for many years, the propagation of elastic waves in the periodic composite materials have received much attention for their renewed physical properties and potential applications in various fields [1–8]. There are three different formation mechanisms of the band gaps, namely Bragg scattering, local resonance and hybridization. For the first mechanism, which is caused by multiple scattering of the periodic inclusions, the wavelength of the band gap frequencies is as the same order of the structural period [9, 10]. The local resonance employs the resonant vibration of the local resonance mass working against the excitation of the incident elastic wave to attenuate the vibration [9,10]. The frequency range of the band gap based on local resonance is almost two orders of magnitude lower than that of the Bragg scattering. The third kind of the band gap formation mechanism is attributed to the coupling effects of local resonances and Bragg scattering. Such gaps are particularly interesting, since it can be shown that they allow the definition of negative effective material parameters in some cases [11,12]. To broaden the band gap, the researchers have designed many kind of hybrid phonon crystal. Zhang et al. designed a novel hybrid phononic crystal (PC) to obtain wider band gaps in low frequency range. The multi-band phononic crystal consists
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1. Introduction
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E-mail address: [email protected] (Y. Dong). https://doi.org/10.1016/j.physleta.2018.10.042 0375-9601/© 2018 Elsevier B.V. All rights reserved.
of rubber slab with periodic holes and plumbum stubs. In comparison with the phononic crystal without periodic holes, the new designed phononic crystal can obtain wider band gaps and better vibration damping characteristics [13]. Wang et al. designed a hybrid structure composed of a local resonance mass and an external oscillator. The hybrid structure can produce multi-band gaps wider than the multi-resonator acoustic metamaterials [14]. M.B. Assouar et al. proposed a hybrid phononic crystal plates which are composed of periodic stepped pillars and periodic holes. The acoustic waves scattered simultaneously by the pillars and holes in a relevant frequency range can generate low and wide acoustic forbidden bands [15]. Yoon et al. designed a hybrid phononic crystal consisting of a bi-prism. A stop band is formed in a central frequency range while positive–positive and positive–negative refractions occur in lower and higher frequency ranges to concentrate acoustic energy in a central localized zone [16]. L. D’Alessandro et al. presented a strategy to design three-dimensional elastic periodic structures endowed with complete bandgaps, the first of which is ultra-wide, where the top limits of the first two bandgaps are overstepped in terms of wave transmission in the finite structure. Thus, subsequent bandgaps are merged, approaching the behavior of a three-dimensional low-pass mechanical filter [17]. In this paper, we investigated a two-dimensional novel multiband gaps phononic crystal (PC). The sections of this paper are arranged as follows: In Section 2, we show the structural model and band gap calculation method. In section 3, the dispersion relations, the displacement fields of the eigenmodes, and the power
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Table 1 Materials parameters. Material
Mass density (kg/m3 )
Elasticity modulus (1010 Pa)
Poisson ratio (1010 Pa)
Plumbum Rubber Plastomer
11600(ρb ) 1300(ρc ) 1190(ρa )
4.08 1.17e−5 0.22
0.37 0.47 0.34
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Table 2 Structure parameters. a (mm)
b (mm)
c (mm)
d (mm)
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2. Model and method
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∂ −ρ (r )ω u i = ∇ · μ(r )∇ u i + ∇ · μ(r ) u ∂ xi ∂ + λ(r )∇ · u (i = x, y ) ∂ xi 2
(2)
Brillouin zone, and uk is a periodic vector function having spatial periodicity the same as the periodic elastic system. The spatial part of the time-harmonic displacement vector is:
(r + p ) = u (r ) exp(ik · p ) u
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is the spatial period vector of the PC structures. where p
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H = 10 log
po
(3)
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pi
102 103 104 105
(4)
where p o and p i are the number value of the transmitted acceleration and incident acceleration, respectively. The wave propagates along direction X . The right side is of response output in Fig. 1(b). Meanwhile, the perfect matching layers are added at both sides of the structure to guarantee the accuracy of the result. 3. Numerical result and discussion
where k = (kx , k y ) is the Bloch wavevector in the irreducible first
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(1)
In Eqs. (1), r = (x, y ) denotes the position vector, ω is the angu = (u x , u y ) is the displacement lar frequency, ρ is mass density, u vector in the transverse plane, λ and μ are the Lame constant and shear modulus, and ∇ = ( ∂∂x , ∂∂y ) is the 2D vector differential operator. Periodic elastic system has the following form:
(r ) = u k (r ) exp(ik · r ) u
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COMSOL is used for the implementation of the above mentioned model for the characteristic frequency of the proposed multi-band PC. The transmission function is defined as:
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A multi-band PC unit is shown in Fig. 1. The materials parameters and structure parameters used in the numerical calculations are shown in Table 1 and Table 2, respectively. The governing field equations in the elastic structures for the in-plane mode can be expressed as [18]:
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Fig. 2. Characteristic frequency for first 50 order and transmission loss curve. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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transmission spectra were calculated. The generation reasons of band gap and material parameters were discussed. The conclusion was described in Section 4. This paper used the finite element software, COMSOL Mutiphysics 4.3a, to calculate eigenfrequency, and used Lagrange quadratic element for mesh dissection. The periodic boundary condition was applied to the unit cell in the simulation.
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Fig. 1. Schematics of elastic phononic crystal plate (a) the unite of the elastic metamaterial plate, (b) elastic metamaterial plate, (c) the first Brillouin zone of a unit cell.
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3.1. Band structures of the new multi-band phononic crystal
In this section, we present the numerical results of the first 80th characteristic frequency and the transmission loss of the PC. The materials parameters and structure parameters used in the numerical calculations are shown in Table 1 and Table 2, respectively. The band structure of the elastic wave is calculated for the PC structures in Fig. 2. The numerical transmission spectrum considering losses is reported as the blue dashed line in Fig. 2. The first 12th characteristic frequency is shown in Fig. 3(a). The first BG (green band) is located between third and fourth band, the first BG is from 399 Hz to 550 Hz as shown in Fig. 3(a). The second BG (yellow band) is from 659 Hz to 1266 Hz. The third BG (blue band) is from 1321 Hz to 2753 Hz. In order to prove the effectiveness of the band gap, we calculate the transmission loss of 10 ∗ 10
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Fig. 3. (a) Characteristic frequency for first 12 order. (b) Characteristic frequency from 13th to 80th order.
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3.2. Formation mechanisms of the band gap In this section, we presented the numerical results and analyzed the propagation behaviors of elastic waves associated with different bands. The transmission loss is shown as Fig. 2. The greater loss of transmission (gray part) is the same as the BG frequency. In order to investigate the formation mechanism of the BG, we analyzed the vibrational modes of the first few BGs. There are several modes from mode A to mode G in Fig. 3(a). We calculated the displacement vectors of the seven modes, respectively. The vibration of mode A is the translational movement of Plumbum vibrators as shown in Fig. 4(A), the vibration of mode B and mode C is the translational movement of plastomer vibrators as shown in Fig. 4(B) and Fig. 4(C). The vibration mode B is the vibration of four plastic vibrators. The vibration of the starting frequency of first BG is the translational movement of Plumbum vibrators as mode A. The vibration of the cutoff frequency of first BG is the translational movement of four plastomer vibrators, the thither plastomer vibrators move in opposite directions with the same frequency in mode B. The second BG (yellow band) is from 659 Hz to 1266 Hz, the vibration mode of starting frequency of second BG is translational movement of two plastomer vibrators in mode E. The vibration mode of cutoff frequency of second BG is translational movement of two plastomer vibrators in mode E, two plastomer vibrators moved in the same direction. The vibration of mode D is torsional movement of plastomer vibrator as shown in Fig. 4(D). The vibration of mode E is translational movement of plastomer. The third BG (blue band) is from 1321 Hz to 2753 Hz. The starting frequency of BG is translational movement of two plastomer vibrators in mode F. The vibration of mode G is translational movement of rubber. The vibrations of mode H, mode M, mode K, mode S are movement of rubber in Fig. 4. Elastic waves are reflected back and forth between the hard boundaries. Elastic waves are localized in a unit and cannot propagate, this is Bragg scattering. According to the above analysis, the first two band gaps are local resonance band gaps, and the third band gap is caused by the interaction of local resonance and Bragg scattering. The local resonance band gap is caused by the original anti-resonance of different oscillators. The first (green part) band gap is initiated by resonance of the larger mass lead oscillator, while the smaller mass oscillator remains stationary in the same direction of motion. The cut-off frequency of the band gap is due to the anti-resonance of the smaller mass oscillator while the larger mass oscillator remains stationary. We theoretically derive the conditions for anti-
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periodic structures in Fig. 1(b). The frequency of the larger attenuation (grey fraction) are equal to the band gaps, which verify the accuracy of band gaps. In order to investigate the physical mechanism of band gaps, we calculated the displacement vector of high symmetry points of the Brillouin zone.
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Fig. 4. Eigenmodes shapes and displacement vector fields of the modes at the edges of the BG.
resonance [19,20]. The second band gap (yellow part) is caused by the local resonance of the small mass oscillator, compared with the mode at the cut-off frequency of the first band gap. Both the starting and cut-off frequencies of the second band gap are caused by the reverse displacement of two different small mass oscillators. The starting frequency of the third band gap is generated by the local resonance of the small mass oscillator, and the cut-off frequency is caused by the elastic wave scattering, i.e. Bragg scat-
K tering. Because of the local resonance frequency f = 21π M , local resonance is sensitive to the oscillator mass. We calculated the characteristic frequencies of different oscillator densities as shown in Fig. 5 and Fig. 6. The band gap change along with the different oscillator density. The red characteristic frequency curve decreases with the increase of mass density. The different oscillator mass has no effect on the higher order eigen-frequency. It can be seen that
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Fig. 5. The dispersion curve of different density
ρb .
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Fig. 6. The dispersion curve of different density
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the large mass oscillator only affects the starting frequency of band gap, while the small mass oscillator only affects the corresponding cut-off frequency of band gap. It is also proved that the first and the second band gaps are locally resonant band gaps and the higher order band gaps are Bragg scattering band gaps. We change the density of rubber as shown in Fig. 7. With increasing of density of rubber, the band gap frequency increases, but the shape of the dispersion curve does not change, the magnitude of the characteristic frequency changes. Therefore, this structure can produce a wide low frequency band gap, both the local resonance band gap and Bragg scattering band gap are in the dispersion curve. Dispersion curve is determined by the structure.
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3.3. Influence of geometry and material parameters on the low-frequency BG
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Fig. 7. The dispersion curve of different density
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In order to further study the effect of parameters on the first band gap, we explain the physical mechanism of the effect of parameters on the band gap by the equivalent model [21]. The
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ρr of rubber.
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Fig. 8. Mode of origin anti-resonance.
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equivalent model of the original resonance structure is shown in Fig. 8. M and m are mass of adjacent oscillators ( M > m), and K is equivalent spring stiffness. The vibration system can be equivalent to two single degree of freedom systems with different mass. It has been proved that the system has the effect of origin antiresonance when the frequency of incident elastic wave is equal to the natural frequency of the adjacent subsystem. One oscillator is vibrating, and the other oscillator of the adjacent subsystem remains stationary. The natural frequencies of adjacent subsystems are F =
1 2π
K M
and f =
1 2π
K . m
In the local resonance band gap,
the starting frequency of the band gap is F = 21π
K , M
and the cut-
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Fig. 9. Spring-mass model.
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M = ρb · V b +
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Kt =
C1 S
Ks = α
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K 2 = 2K s
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fs =
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fc =
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1
(ρc · V c ) · 2 = 2 · ρa · a · c + ρc · b · a
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(5) (6) (7)
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ρb in different methods.
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Fig. 10. The first band gap varies with
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K off frequency of the band gap is f = 21π m . We can see that the boundary frequency of the band gap is only affected by the natural frequency of the corresponding vibration system. The law of band gap changing with parameters conforms to the band gap changing law of the original anti-resonance. To prove the validity of the proposed equivalent model, the frequency of the local resonant band gap with different densities is calculated by the equivalent model method (EMM), and the results are compared with those obtained by the finite element method (FEM). According to the vibration mode of the first band gap boundary frequency and cell model, we calculate eigen-frequency with the equivalent mass-spring model. The vibration mode of starting frequency of the band gap is corresponding to the model in Fig. 9(a), and the vibration mode of cut-off frequency is corresponding to the model in Fig. 9(b).
(11)
(12)
m is mass of equivalent small mass oscillator, M is mass of equivalent large mass oscillator, K t is the stiffness of the effective tension/compression spring. K s is a part of stiffness of the shear spring contributing to the tension/compression spring. α = 0.1 is a weighting coefficient, and elastic constant C 1 = λ + 2μ and C 2 = μ, K 1 is the stiffness of model for starting frequency of BG, K 2 is the stiffness of model for cut-off frequency of band gap, f s is starting frequency of band gap, f c is cutoff frequency of band gap. It can be seen that the calculation results of the two different methods are the same roughly as show in Fig. 10. The effectiveness of the proposed equivalent model is illustrated. The influences of the materials parameters and structure parameters on the BG are investigated to further study the multiband PC characteristics. We change the density ρa , ρb , the length of rubber b and the length of plastomer c in Fig. 1. The starting frequency of band gap decreases with the increase of density ρb , but the cutoff frequency of band gap is almost unchanged. It indicates that the starting frequency is more sensitive to the ρb than the cutoff frequency as shown in Fig. 11(a). The
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Fig. 11. (a) Variation of the frequency of the BG of the multi-band PCs when ρb is changed. (b) Variation of the frequency of the BG of the multi-band PCs when ρa is changed. (c) Variation of the frequency of the BG of the multi-band PCs when b is changed. (d) Variation of the frequency of the BG of the multi-band PCs when c is changed.
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cutoff frequency of band gap remain unchanged with the increase of density ρa , but the starting frequency of band gap decreases gradually as show in Fig. 11(b). With the increase of b, the starting frequency of band gap increases gradually, but the cutoff frequency of band gap varies greatly in Fig. 11(c). The cutoff frequency is more sensitive to the b. With the increase of c, the starting frequency of band gap remains the same, but the cutoff frequency of band gap varies greatly at c = 8 in Fig. 11(d). We can conclude that, to increase the width of the band gap, the ρb , ρa and b will be increased, the c will be decrease.
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4. Conclusions
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In this paper, we present a novel multi-band gaps phononic crystal with broad band gap in low frequency, and calculated the frequency of band gap, transmission loss and displacement vector by the finite element method. Subsequently, the formation mechanism of the band gaps is analyzed by the displacement field of the eigenmodes at the band gap edges. Finally, the effects of the geometrical parameters and materials parameters on the band gap are studied and discussed in detail. The conclusions are that: The new multi-band gaps phononic crystal has a broad band gap in low frequency. There are many flat bands in low frequency,
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[9] M.M. Sigalas, E.N. Economou, Attenuation of multiple-scattered sound, Europhys. Lett. 36 (4) (1996) 241. [10] Z. Liu, C.T. Chan, P. Sheng, Three-component elastic wave band-gap material, Phys. Rev. B 65 (16) (2002) 165116. [11] C. Croënne, E.J.S. Lee, H. Hu, et al., Band gaps in phononic crystals: generation mechanisms and interaction effects, AIP Adv. 1 (4) (2011) 041401. [12] Y. Chen, L. Wang, Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps, Appl. Phys. Lett. 105 (19) (2014) 191907. [13] Z. Zhang, X.K. Han, A new hybrid phononic crystal in low frequencies, Phys. Lett. A 380 (45) (2016) 3766–3772. [14] T. Wang, M.P. Sheng, H.B. Guo, Multi-large low-frequency band gaps in a periodic hybrid structure, Mod. Phys. Lett. B 30 (08) (2016) 1650116. [15] M.B. Assouar, J.H. Sun, F.S. Lin, et al., Hybrid phononic crystal plates for lowering and widening acoustic band gaps, Ultrasonics 54 (8) (2014) 2159–2164. [16] S. Yoo, Y.J. Kim, Y.Y. Kim, Hybrid phononic crystals for broad-band frequency noise control by sound blocking and localization, J. Acoust. Soc. Am. 132 (5) (2012) EL411–EL416. [17] L. D’Alessandro, E. Belloni, R. Ardito, et al., Mechanical low-frequency filter via modes separation in 3D periodic structures, Appl. Phys. Lett. 111 (23) (2017) 231902. [18] Y.F. Wang, Y.S. Wang, X.X. Su, Large bandgaps of two-dimensional phononic crystals with cross-like holes, J. Appl. Phys. 110 (2011) 113520. [19] Y. Dong, H. Yao, J. Du, et al., Research on local resonance and Bragg scattering coexistence in phononic crystal, Mod. Phys. Lett. B 31 (11) (2017) 1750127. [20] Y. Dong, H. Yao, J. Du, et al., Research on low-frequency band gap property of a hybrid phononic crystal, Mod. Phys. Lett. B 32 (15) (2018) 1850165. [21] L. Brillouin, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Courier Corporation, 2003.
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which increases the number of band gaps. There is no waveform conversion in the X direction and Y direction, the characteristic frequency curve is straight line, which produces a lot of flat band gap, the band gaps cover more than 95% below 3000 Hz. The boundary frequency of local resonance band gap is only influenced by the parameters of the corresponding vibration system, the coupling factor is less, and it is easy to be adjusted. In order to increase the width of the local resonance band gap, the mass of the large mass oscillator should be increased and the mass of the light mass oscillator will be reduced.
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