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A new hybrid phononic crystal in low frequencies
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Physics Letters A ••• (••••) •••–•••
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A new hybrid phononic crystal in low frequencies
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Z. Zhang ∗ , X.K. Han
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State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China
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Article history: Received 18 June 2016 Received in revised form 8 September 2016 Accepted 16 September 2016 Available online xxxx Communicated by R. Wu
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Keywords: Phononic crystal Band gap Bragg scattering Local resonance
A novel hybrid phononic crystal is designed to obtain wider band gaps in low frequency range. The hybrid phononic crystal consists 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. The wider band gap can be attributed to the interaction of local resonance and Bragg scattering. The controlling of the BG is explained by the strain energy of the hybrid PC and the introduced effective mass. The effects of the geometrical parameters and the shapes of the stubs and holes on the controlling of waves are further studied. © 2016 Published by Elsevier B.V.
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1. Introduction
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Over the past two decades, phononic crystal (PC) has attracted considerable interests and attentions due to its excellent performance in controlling elastic waves [1–6]. Both theoretical [7–13] and experimental [14,15] works have demonstrated the existence of the full band gaps (BGs) in PC, which can be used for sound insulation and environmental noise control [16–20]. The formation of BGs is attributed to the Bragg scattering or the local resonance. Different to the Bragg scattering PC, the local resonance PC lattice constant is two orders of magnitude smaller than the relevant wavelength of BG according to Liu et al. [16]. Li et al. [21] presented an original structure composed of a square array of composite stubs on both sides of a composite plate to expand the band gap. Yu et al. [22] analyzed the novel two-dimensional local resonance (LR) PC plates which consist of a thin rubber plate with periodic steel stubs. Li et al. [23] showed a kind of composite plate-type acoustic metamaterial which consists of one-sided cylindrical stubs deposited on two-dimensional binary plate. The BG enlargement of this structure is attributed to the local resonant modes of the stubs and the plate. Oudich et al. [24] studied the acoustic properties of PC structure to obtain extremely low frequency BG. The PC structure includes single-layer or two-layer concentric stubs periodically deposited on the surface of a thin ho-
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Corresponding author. Fax: +86 411 84708432. E-mail address: [email protected] (Z. Zhang).
http://dx.doi.org/10.1016/j.physleta.2016.09.019 0375-9601/© 2016 Published by Elsevier B.V.
mogeneous plate. Hsu [25] introduced PC structure composed of an array of stepped resonators on a thin slab. Both the local resonance BGs at low frequencies and Bragg scattering BGs can be found in such structures. Khelif [26] considered a structure constituted by a periodic array of cylindrical silicon pillars on the surface of a semi-infinite silicon substrate. This structure supports surface propagating modes in the nonradiative region of the substrate, outside the sound cone. In this paper, a novel two-dimensional hybrid PC plate, which consists of rubber slab with periodic holes and periodic plumbum stubs, is proposed. The dispersion relations and the displacement fields of the eigenmodes are calculated by the finite element method (FEM). In comparison with the classical PC slab with single or double stubs, the novel hybrid PC slab shows wider BG and better vibration damping characteristics in low frequency range. The influences of the geometrical shapes and the parameters on the band structure are further investigated. 2. Model and method
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A 2D hybrid PC slab is proposed. As shown in Fig. 1(a), the smallest unit cell is a slab with two circular stubs arranged along one diagonal and two holes arranged along the other. Fig. 1(b) shows the cell cross-section along the diagonal. In this structure, the holes and the stubs lie on the center of each quarter of the slab. The lattice constant is a = 0.01 m, and the radii of the stubs are r1 and the radii of the holes are r2 . The thickness of the slab is t and the height of the stubs is h. The material of the slab
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JID:PLA
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Fig. 1. Unit cell of phononic crystal. (a) The novel PC slab; (b) cross-section of the novel PC slab; (c) slab with single stub; (d) slab with double stubs.
and the stubs are rubber and plumbum, respectively. The material parameters are chosen as follows: the density ρr = 1300 kg/m3 , the Young’s modulus E r = 1.175 × 105 Pa and the Poisson’s ratio γr = 0.46875 for rubber; the density ρp = 11600 kg/m3 , the Young’s modulus E p = 4.08 × 1010 Pa and the Poisson’s ratio γp = 0.3691 for plumbum. The holes in the slab are assumed to be vacuum in the calculation. Two kinds of traditional PC slabs with single or double stubs as shown in Fig. 1(c) and Fig. 1(d) are also considered for comparisons. In the calculation, the lattice constant, the materials and the filling rate are the same in the selected PCs. The eigenvalue equations in the unit cell can be expressed as:
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K−ω M U=0
(1)
where U is the displacement at the nodes, and K and M are the stiffness and mass matrices of the unit cell, respectively. The governing field equations in the elastic structures can be expressed as:
3 3 3 ∂ ∂ 2ui ∂ 2ui c i jkl 2 = ρ 2 ∂xj ∂t ∂t j =1
( i = 1, 2, 3)
(2)
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where ρ is the mass density, u i is the displacement, t is the time, c ijkl are the elastic constants, and x j ( j = 1, 2, 3) represents the coordinate variables x, y and z respectively. According to the Bloch theorem, the elastic displacement field u(r) are given by:
u(r) = uk (r)e (ik·r)
(3)
where k = (kx , k y ) is the Bloch wavevector in the irreducible Brillouin zone and uk (r) is a periodical vector function with same periodicity as the unit cell. Based on the Bloch theorem, the periodic boundary conditions can be obtained according to Eq. (3):
u(r + a) = u(r)e
(ik·a)
(4)
where a is the spatial vector of the PC structure. COMSOL is used for the implementation of the above mentioned model for the eigenfrequency analysis and the calculations of the dispersion relations of the proposed hybrid PC. The incident direction is shown in Fig. 2. The transmission function is defined as:
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Fig. 3. Validation of numerical model.
H = 10 log
po
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pi
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(5)
where p o and p i are the number value of the transmitted acceleration and incident acceleration, respectively.
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3. Results and discussions
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3.1. Band gaps of new designed hybrid PC
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To validate the current numerical model, the band structures of PCs with epoxy slab and single rubber stub proposed in Ref. [23] is recalculated. The physical parameters of the PCs are identically the same with Ref. [23]. The results computed in this numerical model can agree well with the results from Ref. [23], as shown in Fig. 3. The band structure of the elastic wave is calculated for the PC structures shown in Fig. 1. As shown in Fig. 4(a), there is only one complete BG for the hybrid PC when r1 = r2 = 0.00237 m, t = 0.001 m and h = 0.002 m. The BG is located between the twelfth and the thirteenth band, with the frequency range from 247.996 Hz to 979.1 Hz. From Fig. 4(b) and (c), it can be found that there are several BGs from 0 to 1000 Hz for the slabs with single or double stubs and every BG is narrow. The BG of the hybrid PC is much wider than the traditional PCs. The BG of the hybrid PC is located between the twelfth and the thirteenth band from the Fig. 4(a). From Fig. 5(a), it can be found that the mode of the lower edge is mainly composed of the torsion of the stubs and the bend vibration of the boundary of the slab. The mode of the upper edge is only the bend vibration of the free rubber as shown in Fig. 5(d). The modes of the edges of the first band gaps of the hybrid PC are similar with the PCs with single or double stubs in Fig. 5. The difference of the twelfth and the thirteenth modes leads to the formation of the band gap. When the height of the pillars h increases, the relative starting frequency of the BG decreases to below 0.1 as shown in Fig. 6. The local resonance BG appears. The BG corresponding to different order of magnitude wavelength can be obtained by adjusting the h. At the same time, the attenuation of transmission spectra of the hybrid
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JID:PLA AID:24071 /SCO Doctopic: General physics
[m5G; v1.188; Prn:23/09/2016; 10:34] P.3 (1-7)
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Fig. 4. Band structures. (a) Hybrid PC slab; (b) slab with single stub; (c) slab with double stubs.
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Fig. 5. Eigenmodes shapes and displacement vector fields of the modes at the edges of the first BGs. Lower edges: (a) hybrid PC slab, (b) slab with single stub, (c) slab with double stubs; Upper edges: (d) hybrid PC slab, (e) slab with single stub, (f) slab with double stubs.
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band when the radii of the stubs r1 and holes r2 are changed, respectively. The propagation of the elastic waves can be seen the propagation of the mechanical energy in elastic medium. When elastic waves propagate in the medium, the energy can be reserved in the structure. The potential energy which is reserved in the structure is model strain energy W ,
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Fig. 6. Effect of h on the starting frequency.
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PC is larger than the other two PC structures in band gap range, as shown in Fig. 7. It means that the hybrid PC has better vibration damping characteristics.
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The influences of the radii of the stubs and holes on the BG are investigated to further study the hybrid PC slab characteristics. Fig. 8 displays the changes of the twelfth band and thirteenth
f =
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kx2max
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where k is the stiffness coefficient, xmax is the maximum displacement. Due to the regularization and normalization, U , W , and xmax are considered to be dimensionless. The eigenfrequency of the system can be calculated by the following equation:
3.2. Effects of the radii of the stubs and holes
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where σi j is the stress, εi j is the strain. When the energy is all converted into the model stain energy, the potential energy U reaches the maximum and U = W at the moment. The equation of the U is:
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AID:24071 /SCO Doctopic: General physics
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Fig. 7. The transmission spectra of the PC slabs. (a) Hybrid PC slab; (b) slab with single stub; (c) slab with double stubs.
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Fig. 8. Variation of the BG of the novel PC slab when the geometrical parameters are changed. (a) The radii of the stubs r1 ; (b) the radii of the holes r2 .
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Table 1 W and xmax of the modes of the PCs with different stubs.
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xmax of each eigenmode can be calculated by Comsol and then ¯ is decided by the vibration k can be calculated. In Eq. (8) m state. When r1 is smaller than 0.0012 m, the eigenmodes are shown in Figs. 9(a) and (b). The vibration is all concentrated in the slab so there is no BG. With the increase of the radii of the stubs r1 from 0.0014 m to 0.0024 m, the twelfth frequency c declines gradually from 368 Hz to 245 Hz as shown in Fig. 8(a). For the 12th band mode, the vibration is composed of the torsion of the stubs and the bend vibration of the boundary of the slab as shown in Figs. 9(a) and (c). The mass of the stubs m1 is much bigger than the mass of the slab m2 . The vibration is concentrated in the stub and δ = 0.5. W and xmax of each 12th mode of PCs with different stubs are showed in Table 1. The 12th frequency calculated by the formula is shown in Fig. 8(a). The comparison shows that the 12th frequency calculated by Comsol agrees well with the calculated value. It shows that the calculation method for the frequency ¯ is reasonable and effective. by W and xmax , and the selection of m When r1 is increased, k changes very little according to Table 1
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Fig. 9. Eigenmodes shapes and displacement vector fields of the modes at the edges of the BG of different PC structures. r1 = 0.0012 m: (a) lower edge, (b) upper edge; r1 = 0.0014 m: (c) lower edge, (d) upper edge; r1 = 0.0012 m: (e) lower edge, (f) upper edge.
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¯ becomes larger so that the 12th frequency is decreased. and m The thirteenth frequency calculated by Comsol moves to higher
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JID:PLA AID:24071 /SCO Doctopic: General physics
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Fig. 10. Eigenmodes shapes and displacement vector fields of the modes at the upper edges of the BG of different PC structures. (a) r2 = 0.0012 m; (b) r2 = 0.0018 m; (c) r2 = 0.0024 m.
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frequency rapidly from 343 Hz to 1006 Hz with the increase of the r1 from 0.0014 m to 0.0024 m. For the 13th band mode, the vibration is only the bending vibration of the slab as shown in Figs. 9(b) and (d) and δ = 0.2. W and xmax of each 13th mode of PCs with different stubs are also shown in the Table 1. The 13th frequency calculated by formula agrees well with the value by the Comsol. When the r1 is increased, the 13th frequency increases because the k increases from the Table 1 and then m decreases. From Fig. 8(b), the 12th frequency moves from 302 Hz to 244 Hz and the 13th frequency changes from 498 Hz to 97 8Hz when the radii of the holes r2 are increased from 0.001 m to 0.0024 m (Table 2). It shows that the cutoff frequency is more sensitive to the radii of the holes than the starting frequency. Accord¯ = 0.5m1 , ing to eigenmodes the effective mass can be selected, m ¯ = 0.2m2 for the 12th frequency and 13th frequency, respectively. m The results are shown in Fig. 8(b). It can be found that the 12th frequency by formula agrees well with the value by Comsol. But for 13th frequency, the calculated value is much smaller than the value by Comsol. According to Fig. 10, it can be found that the vibration of the slab is focused near the holes when r2 is increased ¯ should be smaller than the 0.2m2 . m ¯ is assigned to be so the m 0.11m2 . The new calculated value is shown in Fig. 10. It can be found that it agrees better with the value by Comsol. The hybrid PC slab can obtain wider BG when the radii of the stubs and the holes are increased (Fig. 11). The attenuation of transmission spectra becomes larger in band gap range when the radii of the stubs or the holes are increased, as shown in Fig. 12.
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3.3. Effects of the shapes of the stubs and holes
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The influences of the shapes of the stubs and holes on the band structure are further considered. The cross sectional areas (CSA) of the stubs or the holes remain invariable relative to the initial structure during the calculation. From the Figs. 13(a)–(d), it can be found that the BG frequency range of the PC slab with circular stubs and square holes is from 234.4–1078.8 Hz, with square stubs and circular holes is from 304.4–1248.9 Hz, and with square stubs and square holes is from 276.2–1515.2 Hz while the BG frequency range of the initial structure is from 247.996–979.1 Hz. It shows that the width of the BG becomes larger when the shape of the cross section of
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Fig. 11. Variation of the 13th frequency of the BG of the hybrid PCs when r2 is changed.
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the stubs and holes are changed to square singly or simultaneously. The wider BG are obtained when the shapes of the cross section of both the stubs and holes are changed from circular to square as shown in Fig. 13(d). By comparison of Figs. 13(a) and (b), or Figs. 13(c) and (d), the starting frequency of the PC slab with square holes is lower than that of the slab with circular holes, and the starting or the cutoff frequency increases when the stubs are changed from circular to square. The frequency is calculated using W and xmax (Table 3). As shown in Figs. 14(a), (c), (e), (f), the vibration of the square ¯ = 0.4m1 is chosen for stubs is weaker than the circular stubs. m the 12th frequency when the stubs are square. For the thirteenth frequency, the vibration part of the structure with square stubs and circular holes and the structure with square stubs and square ¯ = 0.03m2 and m ¯ = 0.1m2 are chosen, reholes is much smaller. m spectively. The calculated value is shown in the Table 4. As we can see, the calculated value is close to the value by Comsol. To study the influence of the shapes of the stubs and holes on damping characteristics, the transmission spectra of the two PC slabs are calculated. From the Fig. 15, the damping characteristic
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Fig. 12. The transmission spectra of the PC slabs. (a) Slabs with different r1 stubs; (b) slabs with different r2 holes.
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of the PC slab with square stubs and holes is better than the PC slab with circular stubs and holes.
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4. Conclusions
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A novel PC holey slab with stubs arranged diagonally is proposed in this paper. The band structure is calculated by FEM. The transmission spectra of the structures are calculated to verify the numerical results of the dispersion relationships. In comparison with the traditional PC slab with single or double pillar, the novel hybrid PC slab appears wider BG in low frequency and better vibration damping characteristics. The BG can be changed
by adjusting the size and the shape of the stubs and holes. The width is broadened when the radiuses of the stubs and holes are increased. When the shapes of the cross section of the stubs and holes are changed to square singly or simultaneously, the width of the BG also becomes larger. The strain energy W is used to calculate stiffness k. The frequency is calculated by proposed formula and agrees well with the value by Comsol. The controlling of the BG is explained by the strain energy of the hybrid PC and the introduced effective mass. By the changes of the size or shape of the stubs and holes, the starting frequency and cutoff frequency are changed to obtain suitable BG.
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Fig. 15. The transmission spectra of the PC slabs with different shape stubs and holes.
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References
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[8]
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[9] [10]
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Fig. 14. Eigenmodes shapes and displacement vector fields of the modes at the edges of the BG of different PC structures. The structure with circular stubs and circular holes: (a) lower edge, (b) upper edge; The structure with circular stubs and square holes: (c) lower edge, (d) upper edge; The structure with square stubs and circular holes: (e) lower edge, (f) upper edge; The structure with square stubs and square holes: (g) lower edge, (h) upper edge.
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[13] [14] [15] [16] [17] [18] [19] [20]
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[1] [2] [3] [4] [5] [6] [7]
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Acknowledgements This work was supported by Program for New Century Excellent Talents in University, the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China (Nos. 11172057 and 11572074) and the National Key Basic Research Special Foundation of China (2011CB013401).
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A new hybrid phononic crystal in low frequencies
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Z. Zhang ∗ , X.K. Han
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State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China
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Article history: Received 18 June 2016 Received in revised form 8 September 2016 Accepted 16 September 2016 Available online xxxx Communicated by R. Wu
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Keywords: Phononic crystal Band gap Bragg scattering Local resonance
A novel hybrid phononic crystal is designed to obtain wider band gaps in low frequency range. The hybrid phononic crystal consists 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. The wider band gap can be attributed to the interaction of local resonance and Bragg scattering. The controlling of the BG is explained by the strain energy of the hybrid PC and the introduced effective mass. The effects of the geometrical parameters and the shapes of the stubs and holes on the controlling of waves are further studied. © 2016 Published by Elsevier B.V.
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1. Introduction
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Over the past two decades, phononic crystal (PC) has attracted considerable interests and attentions due to its excellent performance in controlling elastic waves [1–6]. Both theoretical [7–13] and experimental [14,15] works have demonstrated the existence of the full band gaps (BGs) in PC, which can be used for sound insulation and environmental noise control [16–20]. The formation of BGs is attributed to the Bragg scattering or the local resonance. Different to the Bragg scattering PC, the local resonance PC lattice constant is two orders of magnitude smaller than the relevant wavelength of BG according to Liu et al. [16]. Li et al. [21] presented an original structure composed of a square array of composite stubs on both sides of a composite plate to expand the band gap. Yu et al. [22] analyzed the novel two-dimensional local resonance (LR) PC plates which consist of a thin rubber plate with periodic steel stubs. Li et al. [23] showed a kind of composite plate-type acoustic metamaterial which consists of one-sided cylindrical stubs deposited on two-dimensional binary plate. The BG enlargement of this structure is attributed to the local resonant modes of the stubs and the plate. Oudich et al. [24] studied the acoustic properties of PC structure to obtain extremely low frequency BG. The PC structure includes single-layer or two-layer concentric stubs periodically deposited on the surface of a thin ho-
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*
Corresponding author. Fax: +86 411 84708432. E-mail address: [email protected] (Z. Zhang).
http://dx.doi.org/10.1016/j.physleta.2016.09.019 0375-9601/© 2016 Published by Elsevier B.V.
mogeneous plate. Hsu [25] introduced PC structure composed of an array of stepped resonators on a thin slab. Both the local resonance BGs at low frequencies and Bragg scattering BGs can be found in such structures. Khelif [26] considered a structure constituted by a periodic array of cylindrical silicon pillars on the surface of a semi-infinite silicon substrate. This structure supports surface propagating modes in the nonradiative region of the substrate, outside the sound cone. In this paper, a novel two-dimensional hybrid PC plate, which consists of rubber slab with periodic holes and periodic plumbum stubs, is proposed. The dispersion relations and the displacement fields of the eigenmodes are calculated by the finite element method (FEM). In comparison with the classical PC slab with single or double stubs, the novel hybrid PC slab shows wider BG and better vibration damping characteristics in low frequency range. The influences of the geometrical shapes and the parameters on the band structure are further investigated. 2. Model and method
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A 2D hybrid PC slab is proposed. As shown in Fig. 1(a), the smallest unit cell is a slab with two circular stubs arranged along one diagonal and two holes arranged along the other. Fig. 1(b) shows the cell cross-section along the diagonal. In this structure, the holes and the stubs lie on the center of each quarter of the slab. The lattice constant is a = 0.01 m, and the radii of the stubs are r1 and the radii of the holes are r2 . The thickness of the slab is t and the height of the stubs is h. The material of the slab
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Fig. 1. Unit cell of phononic crystal. (a) The novel PC slab; (b) cross-section of the novel PC slab; (c) slab with single stub; (d) slab with double stubs.
and the stubs are rubber and plumbum, respectively. The material parameters are chosen as follows: the density ρr = 1300 kg/m3 , the Young’s modulus E r = 1.175 × 105 Pa and the Poisson’s ratio γr = 0.46875 for rubber; the density ρp = 11600 kg/m3 , the Young’s modulus E p = 4.08 × 1010 Pa and the Poisson’s ratio γp = 0.3691 for plumbum. The holes in the slab are assumed to be vacuum in the calculation. Two kinds of traditional PC slabs with single or double stubs as shown in Fig. 1(c) and Fig. 1(d) are also considered for comparisons. In the calculation, the lattice constant, the materials and the filling rate are the same in the selected PCs. The eigenvalue equations in the unit cell can be expressed as:
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K−ω M U=0
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where U is the displacement at the nodes, and K and M are the stiffness and mass matrices of the unit cell, respectively. The governing field equations in the elastic structures can be expressed as:
3 3 3 ∂ ∂ 2ui ∂ 2ui c i jkl 2 = ρ 2 ∂xj ∂t ∂t j =1
( i = 1, 2, 3)
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where ρ is the mass density, u i is the displacement, t is the time, c ijkl are the elastic constants, and x j ( j = 1, 2, 3) represents the coordinate variables x, y and z respectively. According to the Bloch theorem, the elastic displacement field u(r) are given by:
u(r) = uk (r)e (ik·r)
(3)
where k = (kx , k y ) is the Bloch wavevector in the irreducible Brillouin zone and uk (r) is a periodical vector function with same periodicity as the unit cell. Based on the Bloch theorem, the periodic boundary conditions can be obtained according to Eq. (3):
u(r + a) = u(r)e
(ik·a)
(4)
where a is the spatial vector of the PC structure. COMSOL is used for the implementation of the above mentioned model for the eigenfrequency analysis and the calculations of the dispersion relations of the proposed hybrid PC. The incident direction is shown in Fig. 2. The transmission function is defined as:
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Fig. 3. Validation of numerical model.
H = 10 log
po
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(5)
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3. Results and discussions
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3.1. Band gaps of new designed hybrid PC
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To validate the current numerical model, the band structures of PCs with epoxy slab and single rubber stub proposed in Ref. [23] is recalculated. The physical parameters of the PCs are identically the same with Ref. [23]. The results computed in this numerical model can agree well with the results from Ref. [23], as shown in Fig. 3. The band structure of the elastic wave is calculated for the PC structures shown in Fig. 1. As shown in Fig. 4(a), there is only one complete BG for the hybrid PC when r1 = r2 = 0.00237 m, t = 0.001 m and h = 0.002 m. The BG is located between the twelfth and the thirteenth band, with the frequency range from 247.996 Hz to 979.1 Hz. From Fig. 4(b) and (c), it can be found that there are several BGs from 0 to 1000 Hz for the slabs with single or double stubs and every BG is narrow. The BG of the hybrid PC is much wider than the traditional PCs. The BG of the hybrid PC is located between the twelfth and the thirteenth band from the Fig. 4(a). From Fig. 5(a), it can be found that the mode of the lower edge is mainly composed of the torsion of the stubs and the bend vibration of the boundary of the slab. The mode of the upper edge is only the bend vibration of the free rubber as shown in Fig. 5(d). The modes of the edges of the first band gaps of the hybrid PC are similar with the PCs with single or double stubs in Fig. 5. The difference of the twelfth and the thirteenth modes leads to the formation of the band gap. When the height of the pillars h increases, the relative starting frequency of the BG decreases to below 0.1 as shown in Fig. 6. The local resonance BG appears. The BG corresponding to different order of magnitude wavelength can be obtained by adjusting the h. At the same time, the attenuation of transmission spectra of the hybrid
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Fig. 4. Band structures. (a) Hybrid PC slab; (b) slab with single stub; (c) slab with double stubs.
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Fig. 5. Eigenmodes shapes and displacement vector fields of the modes at the edges of the first BGs. Lower edges: (a) hybrid PC slab, (b) slab with single stub, (c) slab with double stubs; Upper edges: (d) hybrid PC slab, (e) slab with single stub, (f) slab with double stubs.
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band when the radii of the stubs r1 and holes r2 are changed, respectively. The propagation of the elastic waves can be seen the propagation of the mechanical energy in elastic medium. When elastic waves propagate in the medium, the energy can be reserved in the structure. The potential energy which is reserved in the structure is model strain energy W ,
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PC is larger than the other two PC structures in band gap range, as shown in Fig. 7. It means that the hybrid PC has better vibration damping characteristics.
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The influences of the radii of the stubs and holes on the BG are investigated to further study the hybrid PC slab characteristics. Fig. 8 displays the changes of the twelfth band and thirteenth
f =
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3.2. Effects of the radii of the stubs and holes
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Table 1 W and xmax of the modes of the PCs with different stubs.
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xmax of each eigenmode can be calculated by Comsol and then ¯ is decided by the vibration k can be calculated. In Eq. (8) m state. When r1 is smaller than 0.0012 m, the eigenmodes are shown in Figs. 9(a) and (b). The vibration is all concentrated in the slab so there is no BG. With the increase of the radii of the stubs r1 from 0.0014 m to 0.0024 m, the twelfth frequency c declines gradually from 368 Hz to 245 Hz as shown in Fig. 8(a). For the 12th band mode, the vibration is composed of the torsion of the stubs and the bend vibration of the boundary of the slab as shown in Figs. 9(a) and (c). The mass of the stubs m1 is much bigger than the mass of the slab m2 . The vibration is concentrated in the stub and δ = 0.5. W and xmax of each 12th mode of PCs with different stubs are showed in Table 1. The 12th frequency calculated by the formula is shown in Fig. 8(a). The comparison shows that the 12th frequency calculated by Comsol agrees well with the calculated value. It shows that the calculation method for the frequency ¯ is reasonable and effective. by W and xmax , and the selection of m When r1 is increased, k changes very little according to Table 1
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Fig. 9. Eigenmodes shapes and displacement vector fields of the modes at the edges of the BG of different PC structures. r1 = 0.0012 m: (a) lower edge, (b) upper edge; r1 = 0.0014 m: (c) lower edge, (d) upper edge; r1 = 0.0012 m: (e) lower edge, (f) upper edge.
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¯ becomes larger so that the 12th frequency is decreased. and m The thirteenth frequency calculated by Comsol moves to higher
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frequency rapidly from 343 Hz to 1006 Hz with the increase of the r1 from 0.0014 m to 0.0024 m. For the 13th band mode, the vibration is only the bending vibration of the slab as shown in Figs. 9(b) and (d) and δ = 0.2. W and xmax of each 13th mode of PCs with different stubs are also shown in the Table 1. The 13th frequency calculated by formula agrees well with the value by the Comsol. When the r1 is increased, the 13th frequency increases because the k increases from the Table 1 and then m decreases. From Fig. 8(b), the 12th frequency moves from 302 Hz to 244 Hz and the 13th frequency changes from 498 Hz to 97 8Hz when the radii of the holes r2 are increased from 0.001 m to 0.0024 m (Table 2). It shows that the cutoff frequency is more sensitive to the radii of the holes than the starting frequency. Accord¯ = 0.5m1 , ing to eigenmodes the effective mass can be selected, m ¯ = 0.2m2 for the 12th frequency and 13th frequency, respectively. m The results are shown in Fig. 8(b). It can be found that the 12th frequency by formula agrees well with the value by Comsol. But for 13th frequency, the calculated value is much smaller than the value by Comsol. According to Fig. 10, it can be found that the vibration of the slab is focused near the holes when r2 is increased ¯ should be smaller than the 0.2m2 . m ¯ is assigned to be so the m 0.11m2 . The new calculated value is shown in Fig. 10. It can be found that it agrees better with the value by Comsol. The hybrid PC slab can obtain wider BG when the radii of the stubs and the holes are increased (Fig. 11). The attenuation of transmission spectra becomes larger in band gap range when the radii of the stubs or the holes are increased, as shown in Fig. 12.
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The influences of the shapes of the stubs and holes on the band structure are further considered. The cross sectional areas (CSA) of the stubs or the holes remain invariable relative to the initial structure during the calculation. From the Figs. 13(a)–(d), it can be found that the BG frequency range of the PC slab with circular stubs and square holes is from 234.4–1078.8 Hz, with square stubs and circular holes is from 304.4–1248.9 Hz, and with square stubs and square holes is from 276.2–1515.2 Hz while the BG frequency range of the initial structure is from 247.996–979.1 Hz. It shows that the width of the BG becomes larger when the shape of the cross section of
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Fig. 11. Variation of the 13th frequency of the BG of the hybrid PCs when r2 is changed.
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the stubs and holes are changed to square singly or simultaneously. The wider BG are obtained when the shapes of the cross section of both the stubs and holes are changed from circular to square as shown in Fig. 13(d). By comparison of Figs. 13(a) and (b), or Figs. 13(c) and (d), the starting frequency of the PC slab with square holes is lower than that of the slab with circular holes, and the starting or the cutoff frequency increases when the stubs are changed from circular to square. The frequency is calculated using W and xmax (Table 3). As shown in Figs. 14(a), (c), (e), (f), the vibration of the square ¯ = 0.4m1 is chosen for stubs is weaker than the circular stubs. m the 12th frequency when the stubs are square. For the thirteenth frequency, the vibration part of the structure with square stubs and circular holes and the structure with square stubs and square ¯ = 0.03m2 and m ¯ = 0.1m2 are chosen, reholes is much smaller. m spectively. The calculated value is shown in the Table 4. As we can see, the calculated value is close to the value by Comsol. To study the influence of the shapes of the stubs and holes on damping characteristics, the transmission spectra of the two PC slabs are calculated. From the Fig. 15, the damping characteristic
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4. Conclusions
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A novel PC holey slab with stubs arranged diagonally is proposed in this paper. The band structure is calculated by FEM. The transmission spectra of the structures are calculated to verify the numerical results of the dispersion relationships. In comparison with the traditional PC slab with single or double pillar, the novel hybrid PC slab appears wider BG in low frequency and better vibration damping characteristics. The BG can be changed
by adjusting the size and the shape of the stubs and holes. The width is broadened when the radiuses of the stubs and holes are increased. When the shapes of the cross section of the stubs and holes are changed to square singly or simultaneously, the width of the BG also becomes larger. The strain energy W is used to calculate stiffness k. The frequency is calculated by proposed formula and agrees well with the value by Comsol. The controlling of the BG is explained by the strain energy of the hybrid PC and the introduced effective mass. By the changes of the size or shape of the stubs and holes, the starting frequency and cutoff frequency are changed to obtain suitable BG.
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Fig. 15. The transmission spectra of the PC slabs with different shape stubs and holes.
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Fig. 14. Eigenmodes shapes and displacement vector fields of the modes at the edges of the BG of different PC structures. The structure with circular stubs and circular holes: (a) lower edge, (b) upper edge; The structure with circular stubs and square holes: (c) lower edge, (d) upper edge; The structure with square stubs and circular holes: (e) lower edge, (f) upper edge; The structure with square stubs and square holes: (g) lower edge, (h) upper edge.
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Acknowledgements This work was supported by Program for New Century Excellent Talents in University, the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China (Nos. 11172057 and 11572074) and the National Key Basic Research Special Foundation of China (2011CB013401).
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