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Research on bandgaps in two-dimensional phononic crystal with two resonators Nansha Gao a,1, Jiu Hui Wu a,⇑, Lie Yu b a b

School of Mechanical Engineering and the State Laboratory for Strength and Vibranon of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 14 March 2014 Received in revised form 24 July 2014 Accepted 9 August 2014 Available online 27 August 2014 Keywords: Phononic crystal Resonators Low-frequency range

a b s t r a c t In this paper, the bandgap properties of a two-dimensional phononic crystal with the two resonators is studied and embedded in a homogenous matrix. The resonators are not connected with the matrix but linked with connectors directly. The dispersion relationship, transmission spectra, and displacement ﬁelds of the eigenmodes of this phononic crystal are studied with ﬁnite-element method. In contrast to the phononic crystals with one resonators and hollow structure, the proposed structures with two resonators can open bandgaps at lower frequencies. This is a very interesting and useful phenomenon. Results show that, the opening of the bandgaps is because of the local resonance and the scattering interaction between two resonators and matrix. An equivalent spring-pendulum model can be developed in order to evaluate the frequencies of the bandgap edge. The study in this paper is beneﬁcial to the design of opening and tuning bandgaps in phononic crystals and isolators in low-frequency range. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Over the last two decades, much effort has been devoted to the study of the propagation of elastic waves in the periodic composite structures, the phononic crystals (PCs) [1–6]. PCs are artiﬁcial media composed of a periodical repetition of inclusions in a matrix with various topologies [7–10]. The great attention on these composites is mainly due to their rich physical properties such as negative refraction, localized defect modes, and phononic bandgaps (PBGs) [11–14]. Especially, the existence of the PBGs, in which sound and vibration are fully prohibited in any direction [15–17], makes the PCs being found in abundant potential applications in the design of acoustic devices such as acoustic ﬁlters, noise isolators, and acoustic mirrors [18–22]. To promote the application of PCs in noise control and mechanical engineering, the acquisition of tunable bandgaps with large bandwidth in low-frequency range is of great importance. For the two-dimensional PBG materials, a lot of researches have been carried out to explore PC structures with excellent bandgaps. Min et al. studied the effect of symmetry on the PBGs in two-dimensional binary phononic crystals including ﬁve types of straight ⇑ Corresponding author. Tel.: +86 13572873296. E-mail addresses: [email protected] (N. Gao), [email protected] (J.H. Wu). 1 Tel.: +86 13720538741. http://dx.doi.org/10.1016/j.ultras.2014.08.006 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.

rod arranged in hexagonal lattices [23]. Hu et al. presented a type of phononic crystal composed of an array of two-dimensional Helmholtz resonators and analyzed its properties of the relative acoustic refractive index and acoustic impedance mismatch of the structure [24]. Liu et al. designed a novel PC structure that exhibits PBGs two orders of magnitude smaller than the lattice constant and accordingly proposed the localized resonance (LR) mechanism [25]. Laude et al. identiﬁed the full bandgap for surface acoustic waves in a piezoelectric phononic crystal composed of a square-lattice Y-cut lithium niobate matrix with circular void inclusions [26]. Vasseur et al. demonstrated the existence of the guided modes inside a linear defect created by removing one row of holes in both a freestanding PZT5A slab and a slab deposited on a silicon substrate [27]. Hsu et al. reported the locally resonant bandgap of Lamb waves in binary PC slabs [28]. Pennec et al. studied the band structure of a PC slab constituted of a periodical array of cylindrical dots deposited on a thin slab of a homogeneous material [29]. Brunet et al. explored a silicon slab made of centered rectangular and square arrays of holes. No complete bandgap was found [30]. Mohammadi et al. fabricated a PC slab by etching a hexagonal array of air holes through a free standing slab of silicon. The measured high frequency bandgap matched very well with the theoretical simulations [22]. Sun et al. presented the theoretical and experimental investigation of the elasticwave propagation in PC slabs with membranes [31]. Straight and bent guiding are theoretically reported by Oudich et al. in a locally resonant PC

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structures composed of 2D silicon rubber stubs squarely deposited onto an epoxy ﬁnite plate [32]. Negative refraction experiments were conducted by Lee et al. with guided shear-horizontal waves in a thin PC plate with circular holes [33]. Wu et al. demonstrated the focusing of Lamb waves in a gradient-index PC slab [34]. The beam width of the lowest antisymmetric Lamb mode can be efﬁciently compressed. Zhu et al. investigated a thin metamaterial plate with different cantilever-mass systems. No complete bandgap is found in any system [35]. Kuo and Piazza examined the porous fractal PC slabs. Ultra high frequency bandgaps were found in both theory and experiment [36]. Cheng et al. studied the propagation of Lamb waves in PC slabs composed of periodic layers bilaterally deposited on both sides of the homogeneous core layer [37]. A theoretical study of Lamb wave propagation in a 2D PC slab composed of an array of solid Helmholtz resonators on a thin plate is reported by Hsu [38]. Wang et al. investigated the optimized design of alternate-hole-defect on a silicon PC slab in a square lattice. They found out that the Q factors are generally in an inverse relationship with the standard deviation of the band [39]. Leamy studied the dispersion relations of the honeycomb structures with square, diamond and hexagonal lattices and found out that only the hexagonal honeycomb exhibits a low frequency bandgap [40]. In a recent paper, Wang and Wang proposed a kind of PC slabs with cross-like holes [41]. Multiple wide complete bandgaps are found in a large range of thickness. Up to now, research on the system with the resonators inside the plate has seldom been reported. Yu and Chen studied lamb waves in two-dimensional phononic crystal slabs with neck structure [42]. With the resonators depositing inside, the whole structure will be lighter as some material is removed, and meanwhile the volume is smaller. Therefore, the study on the propagation of Lamb waves in a PC plate with the resonators inside is more interesting. But, there is no report on the phononic crystal with two resonators. In this paper, the two rectangular inclusions with neck structure are introduced into PCs as local resonators. The main effect is the possibility of opening a low-frequency bandgap for the slab waves, which may be applicable to the isolation of the low-frequency vibrations and manipulation of the low-frequency elastic wave with a smaller and lighter structure than a typical PC. Numerical simulation is implemented by use of ﬁnite element method. The effect of the connector layout and the inﬂuence of the geometry parameters on the band structure are investigated. The eigen modes at the bandgap edges are calculated to analyze the mechanism of the bandgap generation, and an optimal design of PC slab with two resonators which producing a large complete bandgap is presented as well. 2. Model and formulation A kind of structure of PCs of an isotropic solid with two resonators in a square lattice is proposed. As shown in Fig. 1(a) and (b),

the PC structure considered here is a slab with two rectangular inclusions embedded periodically along the X–Y plane. The inclusions are not connected with the slab directly but linked through some rectangular connectors. The structure is inﬁnite in the Zdirection and the axes of the resonators are parallel to the surface of the PCs. The sidelength of the Model 1 is lattice constant a, and the four sides are equal in length. Inside the Model 1, the sidelength of hollow part is c. In the same cavity, there are two separate resonators, their lengths and widths are k1, w1, k2, and w2 respectively. The connector geomotry parameters are h1, b1, h2, and b2. The corresponding irreducible Brillouin zone of unit cell is shown in Fig. 1(c). In the present work, to investigate the gap characteristics of these new kinds of PC structure, a series of calculations on the dispersion relationship and transmission spectra are conducted with FEA method based on the Bloch theorem [43–46]. For the calculation of the dispersion relations of the proposed structure referring to an inﬁnite system, the governing ﬁeld equations are given by 3 3 X 3 X @ X @ 2 ui cijkl 2 @xj l¼1 k¼1 @t j¼1

! ¼q

@ 2 ui @t 2

ði ¼ 1; 2; 3Þ;

ð1Þ

where q is the mass density, ui is the displacement, t is the time, cijkl are the elastic constants, and xj (j = 1, 2, 3) represents the coordinate variables x, y, and z respectively. Since the inﬁnite system is periodic along the x and y directions simultaneously, according to the Bloch theorem, the displacement ﬁeld can be expressed as

uðrÞ ¼ eiðkrÞ uk ðrÞ

ð2Þ

where k = (kx, ky) is the wave vector limited to the ﬁrst Brillouin zone of the repeated lattice and uk(r) is a periodical vector function with same periodicity as the crystal lattice. In the present work, the ﬁnite element method (FEA) is used to calculate the structures of the PCs. a series of calculation on the dispersion relations and transmission spectra are conducted with the FEM. Due to the periodicity of PCs, the calculation can be implemented in a representative unit cell. The eigenvalue equations in the unit cell can be written as

ðK x2 MÞU ¼ 0

ð3Þ

where U is the displacement at the nodes and K and M are the stiffness and mass matrices of the unit cell, respectively. The Bloch theorem of Eq. (2) should be applied to the boundaries of the unit cell, yielding

Uðr þ aÞ ¼ eiðkaÞ UðrÞ

ð4Þ

where r is located at the boundary nodes and a is the vector that generates the point lattice associated with the phononic crystals. COMSOL Multiphysics 3.5a is utilized to directly solve the eigenvalue Eq. (3) under complex boundary condition of Eq. (4).

Fig. 1. Schematic view of the cross-sections of the PC structure Model 1 and the corresponding irreducible Brillouin zone.

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In the present work, the structural mechanics module operating under the 2D plane strain application mode (smpn) is applied. The free boundary condition is imposed on the surface of the hole, and the Bloch boundary conditions on the two opposite boundaries of the unit cell. The unit cell is meshed by using the default triangular mesh with Lagrange quadratic elements provided by COMSOL. Eigenfrequency analysis is chosen as the solver mode, and the direct SPOOLES is selected as the linear system solver. For the calculation of the transmission spectra, a ﬁnite system must be deﬁned. In the process of using COMSOL simulation, incentive conditions should be put on the left side of the unit throughout the research. We consider here the structure being ﬁnite in the x direction that contains N units. In the y direction, the periodic boundary conditions are still used to represent the inﬁnite units. In this case, a ﬁnite array structure composed of N 1 units is modeled for the calculation. The plane waves with single-frequency, supplied by the acceleration excitation source, are incident from the left side of the ﬁnite array and propagate along the x direction, and the corresponding transmitted acceleration value is recorded on the right side. The transmission is deﬁned as

TL ¼ 10 log

a0 ; ai

ð5Þ

where a0 and ai are the values of the transmitted and incident acceleration, respectively. By varying the excitation frequency of the incident acceleration, the transmission spectra can be obtained. 3. Numerical results and discussions Fig. 2(a) shows the calculated band structure for an inﬁnite PC structure with two resonators, where a = 50 mm, c = 40 mm, w1 = w2 = 15 mm, k1 = k2 = 36 mm, h1 = h2 = 2 mm and b1 = b2 = 3 mm respectively. The material of matrix is rubber, while resonators and connectors are made of both plumbum. The material parameters are chosen as follows: the density qr = 1300 kg/m3, the Young‘s modulus E = 0.2 GPa, and the Poisson’s ratio c = 0.4 for rubber, the density qp = 11,600 kg/m3, the Young‘s modulus

289

E = 40.8 GPa, and the Poisson’s ratio c = 0.369 for plumbum. The free space among A, B and C is assumed to be vacuum during the calculation. It can be observed from Fig. 2(a) that, in the frequency range from 0 to 50 Hz, seven bands are contained and two bandgaps are shown in Fig. 2(a) (shaded region). The ﬁrst gap locates between the ﬁfth and sixth band and ranges from 34.2 Hz to 36.1 Hz, with a gap width of 1.9 Hz. The relative gap width deﬁned as gap width/midgap is 5.4%. The second gap is located between the sixth and seventh band, ranging from 42.5 Hz to 45 Hz. The bandgap width and the relative gap width are 2.5 Hz and 5.71%, respectively. In order to conﬁrm the analysis of the band structure, the transmission spectra through the structure along the x-direction in Fig. 1(b). Clearly existing are two frequency ranges where the attenuation is so large that it can be treated as bandgaps. The location and gap width of both gaps are corresponding well with the result in Fig. 2(a), validating the numerical results of the band structure calculations. From the Fig. 2, in panel (e), there is a dip between 50 Hz and 60 Hz that has a bigger attenuation than that of the second peak indicated in panel (d). This is a very interesting phenomenon. Because of the hollow structure internal space is larger relative to the other structure which has two or one resonator(s). In the process of plane wave propagation, it is more likely to produce scattering and reﬂection, so for the local resonance structure, the hollow structure will have a lot of attenuation in a particular area. As a comparison, the bandgap behavior of the PC slab with hollow structure and one resonator embedded in the rubber matrix is also calculated. During the calculation, the lattice constant a and materials are chosen to be the same as those used in Fig. 2(a). From Fig. 2(b) and (c), it can be found out that there is no bandgap existing from 0 to 100 Hz. The transmission spectra of the ﬁnite slab composed of 12 units are also calculated. To conﬁrm the analysis of the band structure, the results are displayed in Fig. 2(e) and (f). Comparing Fig. 2(a)–(c), it can be found that, with a tworesonators structure, bandgaps appear while there is no bandgap with one resonator or hollow structure. So, this is a very interesting and important conclusion. Compared with one resonator, some

Fig. 2. The band structure (a) and the transmission spectra (d) is of Model 1, (b) and (e) are of PC with hollow structure, (c) and (f) are of PC with one resonator. The insets are the schematic view of their corresponding irreducible Brillouin zone and the unit cell of the PC.

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Fig. 3. Eugenmode shapes and displacement vector ﬁelds of the modes marked in Fig. 2(a): (a) mode A, (b) mode B, (c) mode C, and (d) mode D.

material is removed from the slab structure, the whole weight of the PC is lighter. With these excellent features, the PC slab with two resonators will promote the applications in the low frequency noise control in engineering ﬁeld and the design of low frequency isolators. In order to identify the physical mechanism for the occurrence of the low frequency bandgaps in the proposed PC structure, we calculate the vibration eigenmodes at the edges of the two complete bandgaps. The results are shown in Fig. 3, and their behavior is discussed in the following section. The color map of Fig. 3 denotes the magnitude of the total displacement vector ﬁeld. The modes on the lower and upper edges of the ﬁrst gap are discussed ﬁrstly. From Fig. 3(a), it can be observed that mode A, corresponding to the lower edge of the ﬁrst bandgap, is mainly the torsion of the resonators and ﬂexural vibration of the slab matrix. The rod resonators can compress and extrude the slab matrix through connector structure. The vibration process can be seen as a massspring system, in which the two rods work as resonators and the connector structure plays the role of the spring. As the whole unit cell extends inﬁnitely along the z-direction, the slab matrix can be treated as four thin plates located around the rod resonators. The plate at the right side is linked with the resonators through the connector structure. On the one hand, in-plane Lamb wave modes will extrude the left and right plates, and accordingly promote the torsion of the rod resonators through the connector structure. One the other hand, it can be found out that, the local resonance of the rod resonators will distort the plate at the right side, and then, inﬂuence the Lamb wave modes of the matrix. Due to the interaction between the local resonance of the rod and Lamb wave modes of the slab matrix, the bandgap with low frequency appears. Unlike mode A, mode B, corresponding to the upper edge of the ﬁrst band, is the ﬁrst order in-plane Lamb wave mode of the matrix. The resonators, the connector structure and the plate on the right and left side remain stationary during the process. It should be noted that the sixth band mode B belongs to is also the lower edge of the second bandgap (Fig. 2(a)). It can be deduced that the introduction of the connector structure has little impact on the upper edge of the ﬁrst bandgap and the lower edge of the second bandgap. Similar phenomenon occurs on mode C and mode D which corresponds to the lower edge and upper edge of the second bandgap. Relative to the constructure which has one resonator or none resonator, the phononic crystal which has two resonators can produce bandgap. This is a crucial conclusion. Combined with band diagram and transmission diagram in Fig. 2(c), we can see that, although it is not produce completely forbidden band in the entire brillouin zone boundary, but on the MC and XM direction, the direction bandgaps exist, these results declare the phononic crystal which contains one resonator can inhibit the plane wave propagation in the particular direction. But for the structure which has two resonators, the situation is not exactly the same. Two resonators, in the same collective, act as the role of the two masses in the system respectively.

Because a resonator is divided into two resonators, the internal free surface increases, the scattering and reﬂection function of plane wave strengthen between the matrix and the scattering. In brief, the opening of the forbidden band, is the result of the combined action, include local resonance and bragg scattering mechanism. From the analysis above, we can conclude that the occurrence of the low frequency bandgap is mainly attributed to the existence of the localized resonance modes. In the following, we will investigate the effects of the resonators geometrical parameters, the connector structure distribution and cover thickness on the low frequency bandgaps of the proposed PC structure. 3.1. Effect of the layout of the connector structure on the low frequency bandgaps First, the inﬂuence of the connector layout on the bandgap is investigated. As shown in Fig. 4, there are four kinds of connectors layouts. Fig. 4 displays the different kinds of PC structures and their band structures. From the above band structure, it is easy to draw a conclusion, that within the limits of 0–100 Hz, Model 3 can produce the best ﬁrst bandgap (from 30 to 42.5 Hz), its bandgap width is 12.5 Hz. For Model 2, it can be seen that, its ﬁrst bandgap width is only 5 Hz (from 40 to 45 Hz). But Model 4 cannot produce bandgap at low frequencies. Compared with Fig. 2(a), it is found out that, the movement of distribution of one connector will lead to the number of the bandgaps decrease and the ﬁrst bandgap width decrease (from 1.9 to 12.5 Hz). Therefore, Model 3 is the optimal structure. In order to identify the physical mechanism for the low frequency bandgap in the Model 3, speciﬁc vibration eigenmodes related to the bandgap are investigated. The results are shown in Fig. 5, where mode E corresponds to the upper edge of the ﬁrst bandgap, and mode F to the lower edge of the ﬁrst bandgap. From Fig. 5, it can be observed that, the occurrence of the ﬁrst bandgap is due to the torsion of the resonators and ﬂexural vibration of the slab matrix. Unlike Model 1, mode F has only one center of local resonance, but mode A has two centers. In mode B, ﬂexural vibration happens on the up and down sides of the plate, but in mode E, ﬂexural vibration appears at all the sides of the plate. These causes of different results are connector layouts modiﬁcation. In the following section, we will focus on the effect of the resonators size of Model 3 on the bandgap. 3.2. Effect of the resonators size of Model 3 on the bandgaps Keep the lattice constant a = 50 mm, c = 40 mm, h1 = h2 = 2 mm, b1 = b2 = 3 mm, the effect of the resonator size in Fig. 4(c) on the bandgap as a function is investigated. Fig. 6 shows the band structure as a function of the factor of the resonator size. From Fig. 6(a) and (c), it can be observed that, when the parameter k1 or w1 increases, the lower edge of the ﬁrst bandgap will decrease, while the upper edge is almost invariant, so the width of the bandgap

N. Gao et al. / Ultrasonics 56 (2015) 287–293

291

Fig. 4. Schematic view of four kinds of connectors layouts of PC structure. (a) is of Model 1, (b) is of Model 2, (c) is of Model 3, and (d) is of Model 4. (e)–(g) are the band structures of Model 2, Model 3, and Model 4.

Fig. 5. Eugenmode shapes and displacement vector ﬁelds of the modes marked in Fig. 4(f): (a) mode E, and (b) mode F.

will increase. We can deduce that the parameter k1 or w1 mainly affects the lower edge of the bandgap and has limited effect on the upper edge. For the lower edge, with the increase of the parameter k1 or w1, the mass of the resonator in the mass-spring system increases rapidly, is quicker than that of the spring stiffness, and will result in the resonance frequency declining accordingly. For the upper edge, whose corresponding mode is the shearing vibration of the matrix, the increase of the resonator yields little inﬂuence on as the multiplied ampliﬁcation of the resonator has minor effect on the relevant mode. From Fig. 6(b) and (d), show that, when the k2 or w2 changes, the width of the ﬁrst bandgap will not change. From these results, it can be inferred that, the effects of various resonators size on the bandgap are different. The ﬁrst bandgap is sensitive to the upper resonator size (as shown in Fig. 4(c)), and insensitive to the change of the lower resonator size.

3.3. Effects of the cover thickness of Model 3 on the bandgaps The variation of the bandgap edges with the cover thickness of Model 3 is presented in Fig. 7. When parameter c = 40 mm is constant, and parameter a will be changed from 42 mm to 60 mm. From the FEA analysis, it is obviously that, the upper edge is stabilized around 42 Hz, and the lower edge achieves a growth from 22 Hz to 31 Hz, and the growth trend is slow down. From Fig. 7, we can also analyze the trend of the width of the ﬁrst bandgap, as the parameter a declines from 20 Hz to 12 Hz. So the biggest width of the bandgap occurs when a = 42 mm (cover thickness is 2 mm). All the above analysis indicates that the low-frequency bandgap of the proposed PC structure can be modulated in a rather wide range by changing the geometrical parameters and layout of the connector structure. These ﬁndings would be beneﬁcial to the

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Fig. 6. The evolution of the bandgap as a function of the resonators size (a = 50 mm, c = 40 mm, h1 = h2 = 2 mm, b1 = b2 = 3 mm).

equivalent spring-pendulum model can be suitable for this model in this paper. Spring pendulum model in this paper is shown in the Fig. 8. Assume the oscillatory motion is harmonic and expressed by the equation

u ¼ U sinðxtÞ

ð6Þ

where u is the swing angle; and U is the amplitude of the oscillation. The maximum kinetic energy is

T max ¼

1 JðUxÞ2 2

ð7Þ

where J = J0 + mD2 is the inertia moment of the lump with respect to the rotation center; J0 is the inertia moment of the lump with respect to its center; m = qc2h is the mass of the lump; and D is the distance between the mass center and the rotation center of the lump. The maximum potential energy is Fig. 7. Effects of the cover thickness (parameter a is from 42 mm to 60 mm, parameter c is invariable) of Model 3 on the bandgaps.

application of the PCs in the low-frequency domain, some PCs with two resonators can be designed to obtain the noise control on the mechanical engineering and the design of the isolators.

3.4. Equivalent spring pendulum models for lower bandgap edge modes In previous studies, there are two main models for the estimation of bandgap edge: the spring mass system model and spring pendulum model [47,48]. For the rotational vibration, an

V max ¼

1 1 2 K i D2i ¼ K i ðli UÞ 2 2

ð8Þ

K i is the total effective stiffness of the connectors which play the same role in the vibration; and li is the distance between the associated springs and the rotation center. From the principle of the energy conservation, the maximum kinetic energy is equal to the maximum potential energy. Equating the above two equations, we arrive at the expression for the eigen frequency

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 K 1 l1 þ K 2 l2 x¼ J1 þ J2

ð9Þ

From three model of energy band diagram and displacement diagram, it can be seen that different two different connection bodies

N. Gao et al. / Ultrasonics 56 (2015) 287–293

l1

J1

l2 K1

nectors structure on the low frequency bandgap are evaluated numerically. It is indicated that the location and width of the bandgap can be regulated signiﬁcantly by the geometrical parameters and connectors layouts.

J2

K2

Acknowledgements

Frequency (Hz)

Fig. 8. Spring-pendulum model schematic diagram.

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51075325, the Program for New Century Excellent Talents in University (NCET-09-0644) of Ministry of Education of China and the Fundamental Research Funds for the Central Universities.

45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

Lower edge(FEA) Upper edge(FEA) Lower edge(analytical model) Upper edge(analytical model)

References [1] [2] [3] [4] [5] [6] [7] [8]

18

20

22

24

26

28

30

32

34

36

38

The length of K1(mm) Fig. 9. Contrast ﬁgure (FEA and proposed analytical model).

have different functions, one of them to play the role of a stretched spring, the other to play the role of a shear spring, so the corresponding stiffness calculation is not the same.

K 1 ¼ K t ðcos bÞ2 þ K 0t ðsin bÞ

2

C 11 S L

and K 0t ¼ g

ð10Þ

[9] [10] [11] [12] [13] [14] [15] [16] [17]

K 2 ¼ K 0t ¼ gK s Kt ¼

293

ð11Þ C 44 S 0 l

ð12Þ

b is the angle between the connector and the force. Kt is the tensile stiffness. K 0t ¼ gK s is the contribution of the shear stiffness. The eigen frequency can also evaluated by Eq. (9) with above the parameters. (Fig. 9) shows the use of ﬁnite element calculation results and the calculation results of spring pendulum model, it can be seen that the results of the ﬁnite element calculation on the high side, but the results of the two methods are within 10%, so the proposed analytical model can forecast well below result of forbidden band. 4. Conclusions In this paper, the band properties of a two-dimensional PC with two resonators are investigated with FEM. Numerical calculations show that large bandgaps at low frequency are observed in this novel structure. But there are no bang gaps in the PC structure with hollow structure and one resonator. The transmission spectra of the ﬁnite structures are calculated to verify the numerical results of the dispersion relationships. The displacement ﬁelds of the eigenmodes are studied to reveal the physical mechanism for the existence of the two resonators on the low-frequency bandgaps. Results show that, with the introduction of the two resonators, the PC unit works as a mass-spring system, in which the resonators and connectors are equivalent to the mass and the spring respectively. The strong resonance of the resonators and the interaction between the resonators and matrix result in the formation of the large bandgaps at low frequencies. Furthermore, the effects of the geometrical parameters of the structure and layouts of the con-

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