- Email: [email protected]
Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring–mass resonators
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.1 (1-7)
Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring–mass resonators Denghui Qian, Zhiyu Shi ∗ State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street No. 29, Nanjing, Jiangsu, 210016, China
a r t i c l e
i n f o
Article history: Received 15 June 2016 Received in revised form 16 July 2016 Accepted 30 July 2016 Available online xxxx Communicated by M. Wu Keywords: Bandgap property Phononic crystal Double panel Symmetric and antisymmetric modes Quality factor
a b s t r a c t Bandgap properties of the locally resonant phononic crystal double panel structure made of a twodimensional periodic array of a spring–mass resonator surrounded by n springs (n equals to zero at the beginning of the study) connected between the upper and lower plates are investigated in this paper. The finite element method is applied to calculate the band structure, of which the accuracy is confirmed in comparison with the one calculated by the extended plane wave expansion (PWE) method and the transmission spectrum. Numerical results and further analysis demonstrate that two bands corresponding to the antisymmetric vibration mode open a wide band gap but is cut narrower by a band corresponding to the symmetric mode. One of the regulation rules shows that the lowest frequency on the symmetric mode band is proportional to the spring stiffness. Then, a new design idea of adding springs around the resonator in a unit cell (n is not equal to zero now) is proposed in the need of widening the bandwidth and lowering the starting frequency. Results show that the bandwidth of the band gap increases from 50 Hz to nearly 200 Hz. By introducing the quality factor, the regulation rules with the comprehensive consideration of the whole structure quality limitation, the wide band gap and the low starting frequency are also discussed. © 2016 Elsevier B.V. All rights reserved.
1. Introduction As is well-known, how to decrease the vibration and reduce the noise is the hot issue that researchers have long been concerning and making efforts to solve. Existing studies show that vibrations mostly propagate along the surrounding structures from the vibration sources of the existing industrial products such as the aircraft, submarine, automobile and so on, by which the noise radiated exactly constitutes the important part of the cabin noise. In addition, the complex sandwich plate structures with the multifunction properties such as light-quality, large-stiffness, excellent shock resistance, good thermal diffusivity and so on are widely applied to the containment structures of the aircraft, submarine and automobile [1–4]. Thus, the researches on the vibration transmission control of the complex sandwich plate structures will play an important role in restraining the structure vibration and reducing the noise in the cabin. The bringing forward of the phononic crystal concept provides a new idea for the study of the theory of the vibration damping and noise reduction propagated along the structure. Phononic crys-
*
Corresponding author. E-mail addresses: [email protected] (D. Qian), [email protected] (Z. Shi).
http://dx.doi.org/10.1016/j.physleta.2016.07.068 0375-9601/© 2016 Elsevier B.V. All rights reserved.
tal is a kind of periodic composite material with the existence of acoustic/elastic wave band gap which promises a broad application prospect in the field of vibration and noise control. Over the past two decades, the propagation of elastic wave in phononic crystal has attracted a lot of attention which is mainly focused on calculation methods and properties of the band gap such as formation mechanisms and regulation rules, but the application researches particularly on the field of vibration and noise control are still immature. Presently, Brag scattering mechanism [5–8] and locally resonant mechanism [9–12] are developed, and the frequency range of the band gap based on the first mechanism is almost two orders of magnitude higher than that based on the second mechanism [9]. Hence, the studies on the complex sandwich plate structures with the design idea of the traditional locally resonant phononic crystal introduced will provide a new idea for restraining the structure vibration and reducing the noise in the unmanageable low frequency region [13–15] of the cabin of some industrial products. Presently, bandgap properties of the complex sandwich plate structures with an array of resonant elements periodically mounted have rarely been studied. However, such an idea has been widely implemented in some basic elastic structures such as rods, beams and single plates in recent years. Wang et al. [16] studied the propagation of longitudinal elastic waves in quasi-one-dimensional
JID:PLA
AID:23995 /SCO Doctopic: General physics
2
[m5G; v1.185; Prn:18/08/2016; 10:42] P.2 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
Fig. 1. (a) An infinitely double panel structure with periodically attached spring–mass resonators. (b) The unit cell of the structure (the upper plate is ignored) with n springs surrounded in the case of n = 0, 1, 2, 4. (c) The first Brillouin zone.
structure consisting of harmonic oscillators periodically jointed on a slender beam and Yu et al. [17] investigated the flexural vibration in Timoshenko beams with periodically attached local resonators theoretically and experimentally. As for the locally resonant phononic crystal single plate, the so-called filled-in system and stubbed-on system, which are formed by etching holes periodically in a solid matrix plate and then filling them with scatterers as well as stubbing resonant units periodically onto the free surfaces of the plate respectively, were investigated. In Ref. [18] and Ref. [19], the plates with filled-in rubber resonant units and filled-in rubber-coated heavy mass resonant units were researched separately. Similarly to the filled-in structure, the locally resonant phononic crystal single plates constructed by periodically depositing rubber stubs with and without Pb capped on the surface of the plate were studied by Oudich et al. [20]. Besides, Xiao et al. [15] researched the propagation of flexural waves in a locally resonant thin plate made of a two-dimensional periodic array of spring– mass resonators attached on a thin homogeneous plate, which can be regarded as the simplified model of the stubbed-on system. All the papers show that a band gap in the low frequency region can be opened up by the resonant responses of the resonant units and the bands corresponding to the in-plane and the out-of-plane modes are staggered, based on which, a new structure with the three-layered spherical resonant units was proposed and a large sub-wavelength full band gap was opened in Ref. [21]. Based on the researches on the filled-in system and stubbed-on system, Li et al. [22] investigated the propagation characteristics of Lamb waves in a locally resonant phononic crystal single plate whose resonant unit was combined by the filled-in and stubbed-on units. In addition, the formation mechanisms and regulation rules of the band gap in a sandwich plate with a periodic composite core constituted by a square array of elastic cylinders embedded in a solid matrix was researched by Liu et al. [23]. In this paper, we investigate the propagation characteristics of flexural waves in a locally resonant double panel system consisting of a two-layer uniform thin plate with periodically attached spring–mass resonators in the cavity, each of which is surrounded by n springs. At first, for the proposed locally resonant phononic crystal double panel structure without spring surrounded (i.e. n = 0), the accurate band structure is studied. Then, the formation mechanisms and regulation rules of the band gap of the n = 0 system are researched in detail, based on which the improved system with n springs surrounded is proposed in the need of widening the band gap and lowering the starting frequency. The new bandgap properties are discussed and presented. Besides, the relations between the minimum value of the total stiffness and the number, the location of the additional springs are researched. All the results are expected to be of theoretic significances and engineering application prospects in the field of vibration and noise reduction. 2. Model and method As the model shown in Fig. 1(a), an infinitely double panel structure with periodically attached resonators is taken into consideration. What should be noted in this paper is that the effect of
the air between the two plates is ignored. The parameters of the isotropic plates are separately defined as follows: the mass density, Young’s modulus, Poisson’s ratio and thickness of the lower plate are ρ1 , E 1 , μ1 and h1 while the ones of the upper plate are ρ2 , E 2 , μ2 and h2 . In addition, each resonator of the structure consists of two springs with the same stiffness kR and a mass mR surrounded by n springs with the uniform stiffness kS (Fig. 1(b) shows the unit cell of the structure (the upper plate is ignored) in the case of n = 0, 1, 2, 4). In this paper, the rectangular lattice, whose basis vectors are a1 = (a1 , 0) and a2 = (0, a2 ), is adopted for numerical calculation. So, as shown in Fig. 1(a), we define the neutral surface of the lower thin plate as the X Y plane which is used as the basis of building the coordinate system, and array each resonator between two plates at the point denoted by the basis vectors a 1 and a2 :
R = ma1 + na2 ,
(1)
where m and n are integers. For the structure proposed in this paper, the resulting finite element model with the damping term not considered can be represented as follows:
K − ω 2 M w = F,
(2)
where K and M are the structural stiffness and mass matrices, ω represents the circular frequency of structural vibration, w and F denote the displacement and external load vectors. It is well known that the periodic boundary conditions according to the Bloch–Floquet theorem [20] should be used for the interfaces between the nearest unit cells:
w(x + a1 , y + a2 ) = exp i (kx a1 + k y a2 ) w(x, y ),
(3)
where (x, y ) denotes the position vector, kx and k y are the components of the Bloch wave vector limited in the irreducible first Brillouin zone (1BZ), as shown in Fig. 1(c). In the present work, based on all the formulas presented above, the band structure of the infinite system and the vibration transmittance of the corresponding finite system with countable periodic array of the unit cell are calculated, respectively. Here, the four node quadrilateral plate element is adopted as the discussed finite element. Besides, the spring element and mass element are used to describe the resonator. Besides, the size of the plate meshing must be adapted to the variation behavior of the solution. As is well-known, the smaller the mesh size is used, the better the convergence of the computation can be obtained, but the longer the calculation time has to be used. So it is important to choose an adaptive mesh that allows us to have sufficiently good convergence in an acceptable calculation time. Fig. 2 shows the finite element mesh division of a unit cell. For the divided elements, according to equation (3), the nodes on the right and up boundaries of the unit cell can be replaced by the ones on the left and down boundaries. As a consequence, the node number and the matrix dimension are reduced in equation (2).
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.3 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
3
3. Numerical results and analyses In this section, the band gap of the proposed locally resonant phononic crystal double panel structure with no spring surrounded (i.e. n = 0) is studied firstly. What should be indicated is that, the upper plate and the lower plate are considered as completely uniform with the material aluminum. In addition, all the parameters are as follows: E 1 = E 2 = E = 77.6 GPa, μ1 = μ2 = μ = 0.35, ρ1 = ρ2 = ρ = 2730 kg m−3 , h1 = h2 = h = 0.002 m, a1 = a2 = a = 0.1 m, mR = 0.1 kg and kR = 4 × 105 N m−1 unless special illustrations are added. Based on the researches on the bandgap properties of the n = 0 system, the original structure is improved and the n = 0 system is proposed and investigated to gain the band gap with the lower starting frequency and wider band gap.
Fig. 2. The finite element mesh division of a unit cell.
3.1. The accuracy of the band structure calculated by the FE method
Fig. 3. The finite element model of the finite structure with countable periodic array of the unit cell.
The band structure of the proposed system can be calculated by using the condition that the equation (2) added periodic boundary condition has non-zero solution when the external load vector F equals zero, that is:
K − ω2 M w = 0.
(4)
Particularly noted, in equation (4), the elements in the matrices K and M are coupled with the items containing the Bloch wave vector such as e ikx a1 , e ik y a2 and so on, which are not the original structural stiffness matrix and mass matrix anymore. Equation (4) is the typical generalized eigenvalue problem and the corresponding characteristic frequencies ω can be get by solving it for the given Bloch wave vector k, based on which the band structure will be obtained. Fig. 3 shows the finite element model of the finite structure with countable periodic array of the unit cell. In this work, the finite double panel structure is made of 12 × 12 unit cells, the model is vibrated without any boundary constraints, the excitation point is picked at the midpoint of the lower plate to avoid the free boundary effects mentioned in Ref. [15] and the response point is picked at the corner of the upper plate. In addition, the finite element software MSC/PATRAN is applied to draw the transmission spectrum and the steady-state wave profile graphs to help study the bandgap properties.
In this subsection, the band structure of the infinite system with no springs surrounded and the transmission spectrum of the corresponding finite system are calculated respectively. Moreover, the band structure is also calculated by using the extended PWE method [15]. Fig. 4(a) shows the band structures of the infinite n = 0 system calculated by the FE method and the PWE method as a comparison, and Fig. 4(b) displays the transmission spectrum of the corresponding finite system. From Fig. 4(a), within the frequency 800 Hz, the band structures calculated by the FE method and the PWE method are coincident very well, which can be attributed to the thin thickness of the plates. In addition, by comparing Fig. 4(a) with (b), what can be found is that there indeed exists a big attenuation in the transmission spectrum, of which the frequency range matches very well with the one of the band gap. In general, the band structure calculated by the FE method is accurate. 3.2. The bandgap properties of the infinite n = 0 system At first, the band structure of the locally resonant double panel is compared to the one of the locally resonant single plate proposed in Ref. [15]. Here, the single plate is formed by taking away the upper plate and the upper spring and the parameters of the two structures are identical. Fig. 5 shows the band structures of the locally resonant double panel and the locally resonant single plate with the same parameters as a comparison. For the band structure of the single plate, a complete band gap is formed between ‘g1 ’ and ‘g2 , g3 , g4 ’. But for the double panel structure, some changes can be seen obviously. For example, ‘g1 ’ is offset upward to ‘G1 ’; ‘g2 ’ and ‘g4 ’ are
Fig. 4. (a) The band structures of the infinite n = 0 system calculated by the FE method and the PWE method. (b) The transmission spectrum of the corresponding finite system.
JID:PLA
4
AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.4 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
Fig. 5. Band structure of the locally resonant double panel. As a comparison, the band structure of the locally resonant single plate with the same parameters is also shown in the figure.
Fig. 6. The steady-state wave profiles of the finite structure under six specific frequencies in the band structure.
divided into ‘G2,1 ’, ‘G2,2 ’ and ‘G4,1 ’, ‘G4,2 ’; but the band ‘g3 ’ is kept still to ‘G3 ’, which is called the Bragg frequency band and just decided by the characteristics of the plate [15]. Eventually, a narrow band gap between ‘G1 ’ and ‘G2,1 , G3 , G4,1 ’ comes into being. Then, some steady-state wave profiles of the finite double panel structure are considered to help study the formation mechanisms
of the band gap. Fig. 6 shows the steady-state wave profiles of the finite structure under six specific frequencies. In the figure, (a), (d) and (e) are corresponding to the wave profiles of the frequencies ‘280 Hz’, ‘400 Hz’ and ‘535 Hz’ located inside the pass bands ‘G1 ’, ‘G2,1 ’ and ‘G2,2 ’ respectively, and (b), (c) and (f) display the wave profiles of the frequencies ‘310 Hz’, ‘385 Hz’ and ‘550 Hz’ lain around the boundary frequencies ‘F1 ’, ‘F2,1 ’ and ‘F2,2 ’ separately. Observing carefully and estimating roughly, Figs. 6(a) and (b) show that in the pass band ‘G1 ’, the upper plate and the lower plate vibrate in phase with the same vibration mode and the frequency ‘F1 ’ is the ending frequency to vibrate in this type, which can be called as the antisymmetric mode of the two plates. Besides, Figs. 6(c) and (d) show that in the pass band ‘G1, 2 ’, the two plates vibrate with the same vibration mode but in the opposite phase and the starting frequency of this vibration form is ‘F2,1 ’, which can be called as the symmetric mode of the two plates. What can be seen from Fig. 6(e) is that when the frequency is between ‘F3 ’ and ‘F2,2 ’, the vibration form is the coupling of the above two vibration modes. Fig. 6(f) shows that when the frequency is close to ‘F2,2 ’, the vibration form appears the antisymmetric vibration mode, which is on account of that the antisymmetric mode plays the leading role. What’s more, by comparing Fig. 6(a) with (f), we can find that though in two wave profiles, the two plates vibrate in phase with the same vibration mode, but the vibrating phases of the masses in the resonators are quite different (labeled by the circles in Fig. 6(a) and (f)), which Fig. 6(a) exhibits the masses vibrate in phase with the two plates while Fig. 6(f) displays the completely opposite phenomenon. To further confirm the vibration forms of the two plates, the band structures corresponding to the symmetric and antisymmetric vibration modes divided from Fig. 6 are calculated, separately. In detail, we firstly divide the displacement vector w to wl , wu and wr , which represent the displacement vectors of the lower plate, the upper plate and the mass in a resonator separately, and then consider wl = −wu and wl = wu in equation (4) respectively. What should be noted is that when the two plates vibrate in symmetric mode, the masses in the resonators keep still without any vibration apparently, that is wr = 0. Substituting the equations above into it, equation (4) can be reduced to calculate the band structures of the symmetric and antisymmetric vibration modes of the system, respectively. Fig. 7 shows the band structures of the locally resonant double panel structure vibrating in the forms of symmetric vibration (Fig. 7(a)) and antisymmetric vibration (Fig. 7(b)). By comparing Fig. 7 with Fig. 5, we can find that ‘G1 ’, ‘G2,2 ’ and ‘G4,2 ’ represent the bands of the antisymmetric vibration mode while ‘G2,1 ’ and ‘G4,1 ’ are opposite. Besides, what can be seen is that both the vi-
Fig. 7. The band structures of the locally resonant doubling plates vibrating in the forms of symmetric vibration (a) and antisymmetric vibration (b).
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.5 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
5
Fig. 8. The relation curves between the frequencies ‘F1 ’, ‘F2,1 ’, ‘F2,2 ’, ‘F3 ’ and (a) the mass mR as well as (b) the spring stiffness kR , respectively.
bration forms share the Bragg frequency band ‘G3 ’, which can be used to explain the coupling of the two vibration modes displayed in Fig. 6(e). In general, all the phenomena are fairly consistent with which described by Fig. 6. In summary, the formation mechanisms of the first band gap can be described as: the bands ‘G1 ’, ‘G2,2 ’, ‘G3 ’ and ‘G4,2 ’ vibrating in antisymmetric mode open a wide band gap, but the common band ‘G3 ’ and the bands ‘G2,1 ’, ‘G4,1 ’ in symmetric vibration mode cut the formative gap to narrower. In addition, the frequency ‘F1 ’ refers to the starting frequency of the band gap and the minimum of the frequencies ‘F2,1 ’, ‘F2,2 ’ and ‘F3 ’ represents the ending frequency. Therefore, in order to satisfy the need of widening the band gap and lowering the starting frequency, the regulation rules of the specific frequencies should be discussed. Fig. 8(a) shows the relation curves between the frequencies ‘F1 ’, ‘F2,1 ’, ‘F2,2 ’, ‘F3 ’ and the mass mR , and Fig. 8(b) displays the relation curves between the frequencies mentioned above and the spring stiffness kR . During the study, the rest parameters are same as the ones of the example shown in Fig. 4(a) while considering the influences of one parameter on the specific frequencies. From the figure, some phenomena displayed agree very well with the formation mechanisms analysed before. For example, the frequency ‘F3 ’ has nothing to do with the mass and the spring stiffness because it lies in the Bragg frequency band ‘G3 ’ just decided by the characteristics of the plate; the frequency ‘F2,1 ’ is not relevant to the mass on account of that it represents the symmetric vibration mode, which means the mass must keep static. Further, from Fig. 8(a), in order to lower the starting frequency, we need to increase the mass of the resonator as much as possible, which means that the whole structure will be heavy. In addition, as seen in Fig. 8(b), in order to widen the band gap, the spring stiffness requires to be added but not to make ‘F2,1 ’ over ‘F3 ’, however, which will render the starting frequency to rise. Thus, lowering the starting frequency and widening the band gap with the limitation of quality are difficult to realize simultaneously just by adjusting the parameters of the resonator. 3.3. The bandgap properties of the infinite n system Considering the narrow band gap and the difficulty in lowering the starting frequency and widening the band gap with the limitation of quality simultaneously, a new design idea is introduced to the original structure. Based on the known fact that the frequency ‘F2,1 ’ is proportional to the spring stiffness as shown in Fig. 8(b), an additional spring is connected to the unit cell to control it, as shown in Fig. 1(b.2).
Fig. 9. The band structure of the system with an additional spring kS = kR attached between two plates at the point with the displacement (−a/4, 0) to the resonator. Table 1 The minimum total stiffness of n springs located on the circle with the resonator as the center and the distance x = a/4 as the radius in a unit cell satisfying F2,1 ≥ F3 when n takes different values and the minimum stiffness of one spring with the displacement (−x, 0) to the resonator satisfying F2,1 ≥ F3 when x takes different values, respectively. x = a/4 Number of springs n n · kS / × 105 N m−1
1 1.88
2 1.64
3 1.58
4 1.56
5 1.55
6 1.55
8 1.55
12 1.54
n=1 x kS / × 105 N m−1
a/12 2.55
a/6 2.19
a/4 1.88
a/3 1.66
5a/12 1.54
For the unit cell of the example shown in Fig. 5, an additional spring with the stiffness kS = kR is considered to be attached between two plates at the point with the displacement (−a/4, 0) to the resonator and the band structure is shown in Fig. 9. By comparing Fig. 9 with Fig. 7, what can be found is that only the symmetric vibration mode is affected, which can be easily understood that the two nodes on the upper plate and the lower plate connected to the additional spring keep relatively static in the antisymmetric mode. What’s important, the rising of the bands in symmetric vibration mode plays a significant role in widening the band gap. In order to illustrate the effects of different arrangements of the additional springs on the frequency ‘F2,1 ’, Table 1 is shown. The table displays the minimum total stiffness of n springs located on
JID:PLA
6
AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.6 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
Fig. 10. (a) The relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the spring stiffness when frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the quality factor but with the restricted condition F2,2 = F3 .
the circle with the resonator as the center and the distance x = a/4 as the radius in a unit cell satisfying F2,1 ≥ F3 when n takes different values, and the minimum stiffness of one spring with the displacement (−x, 0) to the resonator satisfying F2,1 ≥ F3 when x takes different values, respectively. Here, n springs are ranged equally spaced. From the table, we can see that the value of the minimum total stiffness of n springs reduces firstly and stabilizes gradually with the increase of n when x is certain and the value of the minimum stiffness of one spring reduces continuously with the increase of x. The results provide a good basis for the arrangement of the elastic material chosen as the function of spring in the actual structure. In order to discuss the regulation rules based on the criteria of widening the band gap and lowering the starting frequency in the need of appropriate total mass, a scale factor γmp = mR /(2ρ a2 h) called quality factor is introduced to measure the ratio between the mass of the resonator and the quality of the two plates in a unit cell. Fig. 10(a) shows the relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the spring stiffness when γmp = 0.5. Here, in the calculation, mR is obtained from γmp . What should be pointed is that there is no need for concerning the effect on the frequency ‘F2,1 ’ as it can be adjusted by the additional springs easily. From it, we can see that the width of the band gap rises first and falls later with the increasing of the spring stiffness, and when ‘F2,2 ’ equals to ‘F3 ’, the band gap reaches the widest. Based on this, the relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the quality factor but with the restricted condition F2,2 = F3 , are shown in Fig. 10(b). Here, mR is obtained from γmp and kR is got from F2,2 = F3 . It shows that the width of the band gap is proportional to the quality factor and the starting frequency is inversely proportional to it. Besides, as shown in Fig. 10(a), the starting frequency of the band gap can be lowered by reducing the spring stiffness appropriately when the quality factor is certain, but which will make the width of the band gap narrower. To further study the influences of the geometric parameter on the band gap, the relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the lattice constant with the restricted conditions F2,2 = F3 and γmp = 0.5, is displayed in Fig. 11. Here, the mR can be obtained by substituting the value of the lattice constant to γmp . As shown, the starting frequency and the ending frequency decrease with the increasing of the lattice constant, and obviously the descent rate of the ending frequency is larger, which results in the narrower and narrower width of the band gap directly.
γmp = 0.5. (b) The relation curves between the
Fig. 11. The relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the lattice constant a but with the restricted conditions F2,2 = F3 and γmp = 0.5.
By means of combining the discussions above, we can conclude as follows: in the premise of the given γmp , the band gap reaches the widest when ‘F2,2 ’ equals to ‘F3 ’ and the width can be raised by reducing the lattice constant, which will lead to the lifting of the starting frequency. In addition, lowering the spring stiffness, increasing the quality factor or reducing the lattice constant appropriately can move the band gap to the lower frequency region with the width considerable. 4. Conclusions In this letter, we show the existence of band gap in a twolayer uniform thin plate with periodically attached spring–mass resonators between the upper and lower plates. Based on the observation of the steady-state wave profiles of the finite structure under some specific frequencies and imposing additional constraints on the computing matrix, the band structures are divided into two types: the band structures vibrating in the symmetric mode and the antisymmetric mode. Studies on regulation rules show that: by increasing the mass of the resonator or lowering the spring stiffness, the starting frequency is increased; but by increasing the spring stiffness appropriately, the band gap can be adjusted to the widest. Based on the last rule, the design thought of adding springs around the resonator in the unit cell is proposed in the need of widening the band gap and lowering the starting frequency. The band structure show that the bands corresponding to the symmetric vibration mode can be tuned to the
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.7 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
high frequencies easily, which makes the band gap wider, from the original 50 Hz to 200 Hz almost. Besides, adding the number of additional springs and increasing the distance between the additional springs and the original resonator can lower the total stiffness of the additional springs. Considering the limitation of the quality of the whole structure, the need of the wide band gap and the low starting frequency comprehensively, lowering the spring stiffness, increasing the quality factor and reducing the lattice constant appropriately can be treated as the regulation rules of the n = 0 system. The investigation of the present paper, whose a wide band gap can be generated and tuned to control the spread of the waves and isolate the vibration in the specific low frequency region, could provide some useful information on the theoretic significances and engineering application prospects in the field of vibration and noise reduction. Acknowledgements This research is supported by the National Natural Science Foundation of China through Grant No. 11172131 and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant No. 0515G01). References [1] B.R. Mace, Sound radiation from fluid loaded orthogonally stiffened plates, J. Sound Vib. 79 (3) (1981) 439–452. [2] B.R. Mace, The vibration of plates on two-dimensionally periodic point supports, J. Sound Vib. 192 (3) (1996) 629–643. [3] J.H. Lee, J. Kim, Analysis of sound transmission through periodically stiffened panels by space-harmonic expansion method, J. Sound Vib. 251 (2) (2002) 349–366. [4] L. Dozio, M. Ricciardi, Free vibration analysis of ribbed plates by a combined analytical–numerical method, J. Sound Vib. 319 (1) (2009) 681–697. [5] M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure, J. Sound Vib. 158 (2) (1992) 377–382.
7
[6] X. Zhang, Z. Liu, Y. Liu, et al., Elastic wave band gaps for three-dimensional phononic crystals with two structural units, Phys. Lett. A 313 (5) (2003) 455–460. [7] J.O. Vasseur, P.A. Deymier, A. Khelif, et al., Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: a theoretical and experimental study, Phys. Rev. E 65 (5) (2002) 056608. [8] R. Sainidou, N. Stefanou, I.E. Psarobas, et al., Scattering of elastic waves by a periodic monolayer of spheres, Phys. Rev. B 66 (2) (2002) 024303. [9] Z. Liu, X. Zhang, Y. Mao, et al., Locally resonant sonic materials, Science 289 (5485) (2000) 1734–1736. [10] C. Goffaux, J. Sánchez-Dehesa, P. Lambin, Comparison of the sound attenuation efficiency of locally resonant materials and elastic band-gap structures, Phys. Rev. B 70 (18) (2004) 184302. [11] K.M. Ho, C.K. Cheng, Z. Yang, et al., Broadband locally resonant sonic shields, Appl. Phys. Lett. 83 (26) (2003) 5566–5568. [12] M. Hirsekorn, P.P. Delsanto, N.K. Batra, et al., Modelling and simulation of acoustic wave propagation in locally resonant sonic materials, Ultrasonics 42 (1) (2004) 231–235. [13] J.C. Hsu, Local resonances-induced low-frequency band gaps in twodimensional phononic crystal slabs with periodic stepped resonators, J. Phys. D, Appl. Phys. 44 (5) (2011) 55401. [14] M. Oudich, M. Senesi, M.B. Assouar, et al., Experimental evidence of locally resonant sonic band gap in two-dimensional phononic stubbed plates, Phys. Rev. B 84 (16) (2011) 667. [15] Y. Xiao, J. Wen, X. Wen, Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators, J. Phys. D, Appl. Phys. 45 (19) (2012) 195401. [16] G. Wang, X. Wen, J. Wen, et al., Quasi-one-dimensional periodic structure with locally resonant band gap, J. Appl. Mech. (2006) 167–170. [17] D. Yu, Y. Liu, G. Wang, et al., Flexural vibration band gaps in Timoshenko beams with locally resonant structures, J. Appl. Phys. 100 (12) (2006) 124901. [18] J.C. Hsu, T.T. Wu, Lamb waves in binary locally resonant phononic plates with two-dimensional lattices, Appl. Phys. Lett. 90 (20) (2007) 201904. [19] W. Xiao, G.W. Zeng, Y.S. Cheng, Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs, Appl. Acoust. 69 (3) (2008) 255–261. [20] M. Oudich, Y. Li, B.M. Assouar, et al., A sonic band gap based on the locally resonant phononic plates with stubs, New J. Phys. 12 (2) (2010) 201. [21] J. Ma, Z. Hou, B.M. Assouar, Opening a large full phononic band gap in thin elastic plate with resonant units, J. Appl. Phys. 115 (9) (2014) 093508. [22] Y. Li, T. Chen, X. Wang, et al., Enlargement of locally resonant sonic band gap by using composite plate-type acoustic metamaterial, Phys. Lett. A 379 (5) (2015) 412–416. [23] X.F. Liu, Y.F. Wang, Y.S. Wang, et al., Wave propagation in a sandwich plate with a periodic composite core, J. Sandw. Struct. Mater. 16 (3) (2014) 319–338.
[m5G; v1.185; Prn:18/08/2016; 10:42] P.1 (1-7)
Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring–mass resonators Denghui Qian, Zhiyu Shi ∗ State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street No. 29, Nanjing, Jiangsu, 210016, China
a r t i c l e
i n f o
Article history: Received 15 June 2016 Received in revised form 16 July 2016 Accepted 30 July 2016 Available online xxxx Communicated by M. Wu Keywords: Bandgap property Phononic crystal Double panel Symmetric and antisymmetric modes Quality factor
a b s t r a c t Bandgap properties of the locally resonant phononic crystal double panel structure made of a twodimensional periodic array of a spring–mass resonator surrounded by n springs (n equals to zero at the beginning of the study) connected between the upper and lower plates are investigated in this paper. The finite element method is applied to calculate the band structure, of which the accuracy is confirmed in comparison with the one calculated by the extended plane wave expansion (PWE) method and the transmission spectrum. Numerical results and further analysis demonstrate that two bands corresponding to the antisymmetric vibration mode open a wide band gap but is cut narrower by a band corresponding to the symmetric mode. One of the regulation rules shows that the lowest frequency on the symmetric mode band is proportional to the spring stiffness. Then, a new design idea of adding springs around the resonator in a unit cell (n is not equal to zero now) is proposed in the need of widening the bandwidth and lowering the starting frequency. Results show that the bandwidth of the band gap increases from 50 Hz to nearly 200 Hz. By introducing the quality factor, the regulation rules with the comprehensive consideration of the whole structure quality limitation, the wide band gap and the low starting frequency are also discussed. © 2016 Elsevier B.V. All rights reserved.
1. Introduction As is well-known, how to decrease the vibration and reduce the noise is the hot issue that researchers have long been concerning and making efforts to solve. Existing studies show that vibrations mostly propagate along the surrounding structures from the vibration sources of the existing industrial products such as the aircraft, submarine, automobile and so on, by which the noise radiated exactly constitutes the important part of the cabin noise. In addition, the complex sandwich plate structures with the multifunction properties such as light-quality, large-stiffness, excellent shock resistance, good thermal diffusivity and so on are widely applied to the containment structures of the aircraft, submarine and automobile [1–4]. Thus, the researches on the vibration transmission control of the complex sandwich plate structures will play an important role in restraining the structure vibration and reducing the noise in the cabin. The bringing forward of the phononic crystal concept provides a new idea for the study of the theory of the vibration damping and noise reduction propagated along the structure. Phononic crys-
*
Corresponding author. E-mail addresses: [email protected] (D. Qian), [email protected] (Z. Shi).
http://dx.doi.org/10.1016/j.physleta.2016.07.068 0375-9601/© 2016 Elsevier B.V. All rights reserved.
tal is a kind of periodic composite material with the existence of acoustic/elastic wave band gap which promises a broad application prospect in the field of vibration and noise control. Over the past two decades, the propagation of elastic wave in phononic crystal has attracted a lot of attention which is mainly focused on calculation methods and properties of the band gap such as formation mechanisms and regulation rules, but the application researches particularly on the field of vibration and noise control are still immature. Presently, Brag scattering mechanism [5–8] and locally resonant mechanism [9–12] are developed, and the frequency range of the band gap based on the first mechanism is almost two orders of magnitude higher than that based on the second mechanism [9]. Hence, the studies on the complex sandwich plate structures with the design idea of the traditional locally resonant phononic crystal introduced will provide a new idea for restraining the structure vibration and reducing the noise in the unmanageable low frequency region [13–15] of the cabin of some industrial products. Presently, bandgap properties of the complex sandwich plate structures with an array of resonant elements periodically mounted have rarely been studied. However, such an idea has been widely implemented in some basic elastic structures such as rods, beams and single plates in recent years. Wang et al. [16] studied the propagation of longitudinal elastic waves in quasi-one-dimensional
JID:PLA
AID:23995 /SCO Doctopic: General physics
2
[m5G; v1.185; Prn:18/08/2016; 10:42] P.2 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
Fig. 1. (a) An infinitely double panel structure with periodically attached spring–mass resonators. (b) The unit cell of the structure (the upper plate is ignored) with n springs surrounded in the case of n = 0, 1, 2, 4. (c) The first Brillouin zone.
structure consisting of harmonic oscillators periodically jointed on a slender beam and Yu et al. [17] investigated the flexural vibration in Timoshenko beams with periodically attached local resonators theoretically and experimentally. As for the locally resonant phononic crystal single plate, the so-called filled-in system and stubbed-on system, which are formed by etching holes periodically in a solid matrix plate and then filling them with scatterers as well as stubbing resonant units periodically onto the free surfaces of the plate respectively, were investigated. In Ref. [18] and Ref. [19], the plates with filled-in rubber resonant units and filled-in rubber-coated heavy mass resonant units were researched separately. Similarly to the filled-in structure, the locally resonant phononic crystal single plates constructed by periodically depositing rubber stubs with and without Pb capped on the surface of the plate were studied by Oudich et al. [20]. Besides, Xiao et al. [15] researched the propagation of flexural waves in a locally resonant thin plate made of a two-dimensional periodic array of spring– mass resonators attached on a thin homogeneous plate, which can be regarded as the simplified model of the stubbed-on system. All the papers show that a band gap in the low frequency region can be opened up by the resonant responses of the resonant units and the bands corresponding to the in-plane and the out-of-plane modes are staggered, based on which, a new structure with the three-layered spherical resonant units was proposed and a large sub-wavelength full band gap was opened in Ref. [21]. Based on the researches on the filled-in system and stubbed-on system, Li et al. [22] investigated the propagation characteristics of Lamb waves in a locally resonant phononic crystal single plate whose resonant unit was combined by the filled-in and stubbed-on units. In addition, the formation mechanisms and regulation rules of the band gap in a sandwich plate with a periodic composite core constituted by a square array of elastic cylinders embedded in a solid matrix was researched by Liu et al. [23]. In this paper, we investigate the propagation characteristics of flexural waves in a locally resonant double panel system consisting of a two-layer uniform thin plate with periodically attached spring–mass resonators in the cavity, each of which is surrounded by n springs. At first, for the proposed locally resonant phononic crystal double panel structure without spring surrounded (i.e. n = 0), the accurate band structure is studied. Then, the formation mechanisms and regulation rules of the band gap of the n = 0 system are researched in detail, based on which the improved system with n springs surrounded is proposed in the need of widening the band gap and lowering the starting frequency. The new bandgap properties are discussed and presented. Besides, the relations between the minimum value of the total stiffness and the number, the location of the additional springs are researched. All the results are expected to be of theoretic significances and engineering application prospects in the field of vibration and noise reduction. 2. Model and method As the model shown in Fig. 1(a), an infinitely double panel structure with periodically attached resonators is taken into consideration. What should be noted in this paper is that the effect of
the air between the two plates is ignored. The parameters of the isotropic plates are separately defined as follows: the mass density, Young’s modulus, Poisson’s ratio and thickness of the lower plate are ρ1 , E 1 , μ1 and h1 while the ones of the upper plate are ρ2 , E 2 , μ2 and h2 . In addition, each resonator of the structure consists of two springs with the same stiffness kR and a mass mR surrounded by n springs with the uniform stiffness kS (Fig. 1(b) shows the unit cell of the structure (the upper plate is ignored) in the case of n = 0, 1, 2, 4). In this paper, the rectangular lattice, whose basis vectors are a1 = (a1 , 0) and a2 = (0, a2 ), is adopted for numerical calculation. So, as shown in Fig. 1(a), we define the neutral surface of the lower thin plate as the X Y plane which is used as the basis of building the coordinate system, and array each resonator between two plates at the point denoted by the basis vectors a 1 and a2 :
R = ma1 + na2 ,
(1)
where m and n are integers. For the structure proposed in this paper, the resulting finite element model with the damping term not considered can be represented as follows:
K − ω 2 M w = F,
(2)
where K and M are the structural stiffness and mass matrices, ω represents the circular frequency of structural vibration, w and F denote the displacement and external load vectors. It is well known that the periodic boundary conditions according to the Bloch–Floquet theorem [20] should be used for the interfaces between the nearest unit cells:
w(x + a1 , y + a2 ) = exp i (kx a1 + k y a2 ) w(x, y ),
(3)
where (x, y ) denotes the position vector, kx and k y are the components of the Bloch wave vector limited in the irreducible first Brillouin zone (1BZ), as shown in Fig. 1(c). In the present work, based on all the formulas presented above, the band structure of the infinite system and the vibration transmittance of the corresponding finite system with countable periodic array of the unit cell are calculated, respectively. Here, the four node quadrilateral plate element is adopted as the discussed finite element. Besides, the spring element and mass element are used to describe the resonator. Besides, the size of the plate meshing must be adapted to the variation behavior of the solution. As is well-known, the smaller the mesh size is used, the better the convergence of the computation can be obtained, but the longer the calculation time has to be used. So it is important to choose an adaptive mesh that allows us to have sufficiently good convergence in an acceptable calculation time. Fig. 2 shows the finite element mesh division of a unit cell. For the divided elements, according to equation (3), the nodes on the right and up boundaries of the unit cell can be replaced by the ones on the left and down boundaries. As a consequence, the node number and the matrix dimension are reduced in equation (2).
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.3 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
3
3. Numerical results and analyses In this section, the band gap of the proposed locally resonant phononic crystal double panel structure with no spring surrounded (i.e. n = 0) is studied firstly. What should be indicated is that, the upper plate and the lower plate are considered as completely uniform with the material aluminum. In addition, all the parameters are as follows: E 1 = E 2 = E = 77.6 GPa, μ1 = μ2 = μ = 0.35, ρ1 = ρ2 = ρ = 2730 kg m−3 , h1 = h2 = h = 0.002 m, a1 = a2 = a = 0.1 m, mR = 0.1 kg and kR = 4 × 105 N m−1 unless special illustrations are added. Based on the researches on the bandgap properties of the n = 0 system, the original structure is improved and the n = 0 system is proposed and investigated to gain the band gap with the lower starting frequency and wider band gap.
Fig. 2. The finite element mesh division of a unit cell.
3.1. The accuracy of the band structure calculated by the FE method
Fig. 3. The finite element model of the finite structure with countable periodic array of the unit cell.
The band structure of the proposed system can be calculated by using the condition that the equation (2) added periodic boundary condition has non-zero solution when the external load vector F equals zero, that is:
K − ω2 M w = 0.
(4)
Particularly noted, in equation (4), the elements in the matrices K and M are coupled with the items containing the Bloch wave vector such as e ikx a1 , e ik y a2 and so on, which are not the original structural stiffness matrix and mass matrix anymore. Equation (4) is the typical generalized eigenvalue problem and the corresponding characteristic frequencies ω can be get by solving it for the given Bloch wave vector k, based on which the band structure will be obtained. Fig. 3 shows the finite element model of the finite structure with countable periodic array of the unit cell. In this work, the finite double panel structure is made of 12 × 12 unit cells, the model is vibrated without any boundary constraints, the excitation point is picked at the midpoint of the lower plate to avoid the free boundary effects mentioned in Ref. [15] and the response point is picked at the corner of the upper plate. In addition, the finite element software MSC/PATRAN is applied to draw the transmission spectrum and the steady-state wave profile graphs to help study the bandgap properties.
In this subsection, the band structure of the infinite system with no springs surrounded and the transmission spectrum of the corresponding finite system are calculated respectively. Moreover, the band structure is also calculated by using the extended PWE method [15]. Fig. 4(a) shows the band structures of the infinite n = 0 system calculated by the FE method and the PWE method as a comparison, and Fig. 4(b) displays the transmission spectrum of the corresponding finite system. From Fig. 4(a), within the frequency 800 Hz, the band structures calculated by the FE method and the PWE method are coincident very well, which can be attributed to the thin thickness of the plates. In addition, by comparing Fig. 4(a) with (b), what can be found is that there indeed exists a big attenuation in the transmission spectrum, of which the frequency range matches very well with the one of the band gap. In general, the band structure calculated by the FE method is accurate. 3.2. The bandgap properties of the infinite n = 0 system At first, the band structure of the locally resonant double panel is compared to the one of the locally resonant single plate proposed in Ref. [15]. Here, the single plate is formed by taking away the upper plate and the upper spring and the parameters of the two structures are identical. Fig. 5 shows the band structures of the locally resonant double panel and the locally resonant single plate with the same parameters as a comparison. For the band structure of the single plate, a complete band gap is formed between ‘g1 ’ and ‘g2 , g3 , g4 ’. But for the double panel structure, some changes can be seen obviously. For example, ‘g1 ’ is offset upward to ‘G1 ’; ‘g2 ’ and ‘g4 ’ are
Fig. 4. (a) The band structures of the infinite n = 0 system calculated by the FE method and the PWE method. (b) The transmission spectrum of the corresponding finite system.
JID:PLA
4
AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.4 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
Fig. 5. Band structure of the locally resonant double panel. As a comparison, the band structure of the locally resonant single plate with the same parameters is also shown in the figure.
Fig. 6. The steady-state wave profiles of the finite structure under six specific frequencies in the band structure.
divided into ‘G2,1 ’, ‘G2,2 ’ and ‘G4,1 ’, ‘G4,2 ’; but the band ‘g3 ’ is kept still to ‘G3 ’, which is called the Bragg frequency band and just decided by the characteristics of the plate [15]. Eventually, a narrow band gap between ‘G1 ’ and ‘G2,1 , G3 , G4,1 ’ comes into being. Then, some steady-state wave profiles of the finite double panel structure are considered to help study the formation mechanisms
of the band gap. Fig. 6 shows the steady-state wave profiles of the finite structure under six specific frequencies. In the figure, (a), (d) and (e) are corresponding to the wave profiles of the frequencies ‘280 Hz’, ‘400 Hz’ and ‘535 Hz’ located inside the pass bands ‘G1 ’, ‘G2,1 ’ and ‘G2,2 ’ respectively, and (b), (c) and (f) display the wave profiles of the frequencies ‘310 Hz’, ‘385 Hz’ and ‘550 Hz’ lain around the boundary frequencies ‘F1 ’, ‘F2,1 ’ and ‘F2,2 ’ separately. Observing carefully and estimating roughly, Figs. 6(a) and (b) show that in the pass band ‘G1 ’, the upper plate and the lower plate vibrate in phase with the same vibration mode and the frequency ‘F1 ’ is the ending frequency to vibrate in this type, which can be called as the antisymmetric mode of the two plates. Besides, Figs. 6(c) and (d) show that in the pass band ‘G1, 2 ’, the two plates vibrate with the same vibration mode but in the opposite phase and the starting frequency of this vibration form is ‘F2,1 ’, which can be called as the symmetric mode of the two plates. What can be seen from Fig. 6(e) is that when the frequency is between ‘F3 ’ and ‘F2,2 ’, the vibration form is the coupling of the above two vibration modes. Fig. 6(f) shows that when the frequency is close to ‘F2,2 ’, the vibration form appears the antisymmetric vibration mode, which is on account of that the antisymmetric mode plays the leading role. What’s more, by comparing Fig. 6(a) with (f), we can find that though in two wave profiles, the two plates vibrate in phase with the same vibration mode, but the vibrating phases of the masses in the resonators are quite different (labeled by the circles in Fig. 6(a) and (f)), which Fig. 6(a) exhibits the masses vibrate in phase with the two plates while Fig. 6(f) displays the completely opposite phenomenon. To further confirm the vibration forms of the two plates, the band structures corresponding to the symmetric and antisymmetric vibration modes divided from Fig. 6 are calculated, separately. In detail, we firstly divide the displacement vector w to wl , wu and wr , which represent the displacement vectors of the lower plate, the upper plate and the mass in a resonator separately, and then consider wl = −wu and wl = wu in equation (4) respectively. What should be noted is that when the two plates vibrate in symmetric mode, the masses in the resonators keep still without any vibration apparently, that is wr = 0. Substituting the equations above into it, equation (4) can be reduced to calculate the band structures of the symmetric and antisymmetric vibration modes of the system, respectively. Fig. 7 shows the band structures of the locally resonant double panel structure vibrating in the forms of symmetric vibration (Fig. 7(a)) and antisymmetric vibration (Fig. 7(b)). By comparing Fig. 7 with Fig. 5, we can find that ‘G1 ’, ‘G2,2 ’ and ‘G4,2 ’ represent the bands of the antisymmetric vibration mode while ‘G2,1 ’ and ‘G4,1 ’ are opposite. Besides, what can be seen is that both the vi-
Fig. 7. The band structures of the locally resonant doubling plates vibrating in the forms of symmetric vibration (a) and antisymmetric vibration (b).
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.5 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
5
Fig. 8. The relation curves between the frequencies ‘F1 ’, ‘F2,1 ’, ‘F2,2 ’, ‘F3 ’ and (a) the mass mR as well as (b) the spring stiffness kR , respectively.
bration forms share the Bragg frequency band ‘G3 ’, which can be used to explain the coupling of the two vibration modes displayed in Fig. 6(e). In general, all the phenomena are fairly consistent with which described by Fig. 6. In summary, the formation mechanisms of the first band gap can be described as: the bands ‘G1 ’, ‘G2,2 ’, ‘G3 ’ and ‘G4,2 ’ vibrating in antisymmetric mode open a wide band gap, but the common band ‘G3 ’ and the bands ‘G2,1 ’, ‘G4,1 ’ in symmetric vibration mode cut the formative gap to narrower. In addition, the frequency ‘F1 ’ refers to the starting frequency of the band gap and the minimum of the frequencies ‘F2,1 ’, ‘F2,2 ’ and ‘F3 ’ represents the ending frequency. Therefore, in order to satisfy the need of widening the band gap and lowering the starting frequency, the regulation rules of the specific frequencies should be discussed. Fig. 8(a) shows the relation curves between the frequencies ‘F1 ’, ‘F2,1 ’, ‘F2,2 ’, ‘F3 ’ and the mass mR , and Fig. 8(b) displays the relation curves between the frequencies mentioned above and the spring stiffness kR . During the study, the rest parameters are same as the ones of the example shown in Fig. 4(a) while considering the influences of one parameter on the specific frequencies. From the figure, some phenomena displayed agree very well with the formation mechanisms analysed before. For example, the frequency ‘F3 ’ has nothing to do with the mass and the spring stiffness because it lies in the Bragg frequency band ‘G3 ’ just decided by the characteristics of the plate; the frequency ‘F2,1 ’ is not relevant to the mass on account of that it represents the symmetric vibration mode, which means the mass must keep static. Further, from Fig. 8(a), in order to lower the starting frequency, we need to increase the mass of the resonator as much as possible, which means that the whole structure will be heavy. In addition, as seen in Fig. 8(b), in order to widen the band gap, the spring stiffness requires to be added but not to make ‘F2,1 ’ over ‘F3 ’, however, which will render the starting frequency to rise. Thus, lowering the starting frequency and widening the band gap with the limitation of quality are difficult to realize simultaneously just by adjusting the parameters of the resonator. 3.3. The bandgap properties of the infinite n system Considering the narrow band gap and the difficulty in lowering the starting frequency and widening the band gap with the limitation of quality simultaneously, a new design idea is introduced to the original structure. Based on the known fact that the frequency ‘F2,1 ’ is proportional to the spring stiffness as shown in Fig. 8(b), an additional spring is connected to the unit cell to control it, as shown in Fig. 1(b.2).
Fig. 9. The band structure of the system with an additional spring kS = kR attached between two plates at the point with the displacement (−a/4, 0) to the resonator. Table 1 The minimum total stiffness of n springs located on the circle with the resonator as the center and the distance x = a/4 as the radius in a unit cell satisfying F2,1 ≥ F3 when n takes different values and the minimum stiffness of one spring with the displacement (−x, 0) to the resonator satisfying F2,1 ≥ F3 when x takes different values, respectively. x = a/4 Number of springs n n · kS / × 105 N m−1
1 1.88
2 1.64
3 1.58
4 1.56
5 1.55
6 1.55
8 1.55
12 1.54
n=1 x kS / × 105 N m−1
a/12 2.55
a/6 2.19
a/4 1.88
a/3 1.66
5a/12 1.54
For the unit cell of the example shown in Fig. 5, an additional spring with the stiffness kS = kR is considered to be attached between two plates at the point with the displacement (−a/4, 0) to the resonator and the band structure is shown in Fig. 9. By comparing Fig. 9 with Fig. 7, what can be found is that only the symmetric vibration mode is affected, which can be easily understood that the two nodes on the upper plate and the lower plate connected to the additional spring keep relatively static in the antisymmetric mode. What’s important, the rising of the bands in symmetric vibration mode plays a significant role in widening the band gap. In order to illustrate the effects of different arrangements of the additional springs on the frequency ‘F2,1 ’, Table 1 is shown. The table displays the minimum total stiffness of n springs located on
JID:PLA
6
AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.6 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
Fig. 10. (a) The relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the spring stiffness when frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the quality factor but with the restricted condition F2,2 = F3 .
the circle with the resonator as the center and the distance x = a/4 as the radius in a unit cell satisfying F2,1 ≥ F3 when n takes different values, and the minimum stiffness of one spring with the displacement (−x, 0) to the resonator satisfying F2,1 ≥ F3 when x takes different values, respectively. Here, n springs are ranged equally spaced. From the table, we can see that the value of the minimum total stiffness of n springs reduces firstly and stabilizes gradually with the increase of n when x is certain and the value of the minimum stiffness of one spring reduces continuously with the increase of x. The results provide a good basis for the arrangement of the elastic material chosen as the function of spring in the actual structure. In order to discuss the regulation rules based on the criteria of widening the band gap and lowering the starting frequency in the need of appropriate total mass, a scale factor γmp = mR /(2ρ a2 h) called quality factor is introduced to measure the ratio between the mass of the resonator and the quality of the two plates in a unit cell. Fig. 10(a) shows the relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the spring stiffness when γmp = 0.5. Here, in the calculation, mR is obtained from γmp . What should be pointed is that there is no need for concerning the effect on the frequency ‘F2,1 ’ as it can be adjusted by the additional springs easily. From it, we can see that the width of the band gap rises first and falls later with the increasing of the spring stiffness, and when ‘F2,2 ’ equals to ‘F3 ’, the band gap reaches the widest. Based on this, the relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the quality factor but with the restricted condition F2,2 = F3 , are shown in Fig. 10(b). Here, mR is obtained from γmp and kR is got from F2,2 = F3 . It shows that the width of the band gap is proportional to the quality factor and the starting frequency is inversely proportional to it. Besides, as shown in Fig. 10(a), the starting frequency of the band gap can be lowered by reducing the spring stiffness appropriately when the quality factor is certain, but which will make the width of the band gap narrower. To further study the influences of the geometric parameter on the band gap, the relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the lattice constant with the restricted conditions F2,2 = F3 and γmp = 0.5, is displayed in Fig. 11. Here, the mR can be obtained by substituting the value of the lattice constant to γmp . As shown, the starting frequency and the ending frequency decrease with the increasing of the lattice constant, and obviously the descent rate of the ending frequency is larger, which results in the narrower and narrower width of the band gap directly.
γmp = 0.5. (b) The relation curves between the
Fig. 11. The relation curves between the frequencies ‘F1 ’, ‘F2,2 ’, ‘F3 ’, band gap width and the lattice constant a but with the restricted conditions F2,2 = F3 and γmp = 0.5.
By means of combining the discussions above, we can conclude as follows: in the premise of the given γmp , the band gap reaches the widest when ‘F2,2 ’ equals to ‘F3 ’ and the width can be raised by reducing the lattice constant, which will lead to the lifting of the starting frequency. In addition, lowering the spring stiffness, increasing the quality factor or reducing the lattice constant appropriately can move the band gap to the lower frequency region with the width considerable. 4. Conclusions In this letter, we show the existence of band gap in a twolayer uniform thin plate with periodically attached spring–mass resonators between the upper and lower plates. Based on the observation of the steady-state wave profiles of the finite structure under some specific frequencies and imposing additional constraints on the computing matrix, the band structures are divided into two types: the band structures vibrating in the symmetric mode and the antisymmetric mode. Studies on regulation rules show that: by increasing the mass of the resonator or lowering the spring stiffness, the starting frequency is increased; but by increasing the spring stiffness appropriately, the band gap can be adjusted to the widest. Based on the last rule, the design thought of adding springs around the resonator in the unit cell is proposed in the need of widening the band gap and lowering the starting frequency. The band structure show that the bands corresponding to the symmetric vibration mode can be tuned to the
JID:PLA AID:23995 /SCO Doctopic: General physics
[m5G; v1.185; Prn:18/08/2016; 10:42] P.7 (1-7)
D. Qian, Z. Shi / Physics Letters A ••• (••••) •••–•••
high frequencies easily, which makes the band gap wider, from the original 50 Hz to 200 Hz almost. Besides, adding the number of additional springs and increasing the distance between the additional springs and the original resonator can lower the total stiffness of the additional springs. Considering the limitation of the quality of the whole structure, the need of the wide band gap and the low starting frequency comprehensively, lowering the spring stiffness, increasing the quality factor and reducing the lattice constant appropriately can be treated as the regulation rules of the n = 0 system. The investigation of the present paper, whose a wide band gap can be generated and tuned to control the spread of the waves and isolate the vibration in the specific low frequency region, could provide some useful information on the theoretic significances and engineering application prospects in the field of vibration and noise reduction. Acknowledgements This research is supported by the National Natural Science Foundation of China through Grant No. 11172131 and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant No. 0515G01). References [1] B.R. Mace, Sound radiation from fluid loaded orthogonally stiffened plates, J. Sound Vib. 79 (3) (1981) 439–452. [2] B.R. Mace, The vibration of plates on two-dimensionally periodic point supports, J. Sound Vib. 192 (3) (1996) 629–643. [3] J.H. Lee, J. Kim, Analysis of sound transmission through periodically stiffened panels by space-harmonic expansion method, J. Sound Vib. 251 (2) (2002) 349–366. [4] L. Dozio, M. Ricciardi, Free vibration analysis of ribbed plates by a combined analytical–numerical method, J. Sound Vib. 319 (1) (2009) 681–697. [5] M.M. Sigalas, E.N. Economou, Elastic and acoustic wave band structure, J. Sound Vib. 158 (2) (1992) 377–382.
7
[6] X. Zhang, Z. Liu, Y. Liu, et al., Elastic wave band gaps for three-dimensional phononic crystals with two structural units, Phys. Lett. A 313 (5) (2003) 455–460. [7] J.O. Vasseur, P.A. Deymier, A. Khelif, et al., Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: a theoretical and experimental study, Phys. Rev. E 65 (5) (2002) 056608. [8] R. Sainidou, N. Stefanou, I.E. Psarobas, et al., Scattering of elastic waves by a periodic monolayer of spheres, Phys. Rev. B 66 (2) (2002) 024303. [9] Z. Liu, X. Zhang, Y. Mao, et al., Locally resonant sonic materials, Science 289 (5485) (2000) 1734–1736. [10] C. Goffaux, J. Sánchez-Dehesa, P. Lambin, Comparison of the sound attenuation efficiency of locally resonant materials and elastic band-gap structures, Phys. Rev. B 70 (18) (2004) 184302. [11] K.M. Ho, C.K. Cheng, Z. Yang, et al., Broadband locally resonant sonic shields, Appl. Phys. Lett. 83 (26) (2003) 5566–5568. [12] M. Hirsekorn, P.P. Delsanto, N.K. Batra, et al., Modelling and simulation of acoustic wave propagation in locally resonant sonic materials, Ultrasonics 42 (1) (2004) 231–235. [13] J.C. Hsu, Local resonances-induced low-frequency band gaps in twodimensional phononic crystal slabs with periodic stepped resonators, J. Phys. D, Appl. Phys. 44 (5) (2011) 55401. [14] M. Oudich, M. Senesi, M.B. Assouar, et al., Experimental evidence of locally resonant sonic band gap in two-dimensional phononic stubbed plates, Phys. Rev. B 84 (16) (2011) 667. [15] Y. Xiao, J. Wen, X. Wen, Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators, J. Phys. D, Appl. Phys. 45 (19) (2012) 195401. [16] G. Wang, X. Wen, J. Wen, et al., Quasi-one-dimensional periodic structure with locally resonant band gap, J. Appl. Mech. (2006) 167–170. [17] D. Yu, Y. Liu, G. Wang, et al., Flexural vibration band gaps in Timoshenko beams with locally resonant structures, J. Appl. Phys. 100 (12) (2006) 124901. [18] J.C. Hsu, T.T. Wu, Lamb waves in binary locally resonant phononic plates with two-dimensional lattices, Appl. Phys. Lett. 90 (20) (2007) 201904. [19] W. Xiao, G.W. Zeng, Y.S. Cheng, Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs, Appl. Acoust. 69 (3) (2008) 255–261. [20] M. Oudich, Y. Li, B.M. Assouar, et al., A sonic band gap based on the locally resonant phononic plates with stubs, New J. Phys. 12 (2) (2010) 201. [21] J. Ma, Z. Hou, B.M. Assouar, Opening a large full phononic band gap in thin elastic plate with resonant units, J. Appl. Phys. 115 (9) (2014) 093508. [22] Y. Li, T. Chen, X. Wang, et al., Enlargement of locally resonant sonic band gap by using composite plate-type acoustic metamaterial, Phys. Lett. A 379 (5) (2015) 412–416. [23] X.F. Liu, Y.F. Wang, Y.S. Wang, et al., Wave propagation in a sandwich plate with a periodic composite core, J. Sandw. Struct. Mater. 16 (3) (2014) 319–338.