Issue 
Eur. Phys. J. Appl. Phys.
Volume 94, Number 3, June 2021



Article Number  30902  
Number of page(s)  15  
Section  Physics of Energy Transfer, Conversion and Storage  
DOI  https://doi.org/10.1051/epjap/2021210036  
Published online  21 June 2021 
https://doi.org/10.1051/epjap/2021210036
Regular Article
Vibration and acoustic insulation properties of generalized phononic crystals
^{1}
College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, P.R. China
^{2}
College of Mechanical and Electrical Engineering, Qiqihar University, Qiqihar 161006, P.R. China
^{*} email: shujs@hrbeu.edu.cn
Received:
10
February
2021
Received in final form:
10
May
2021
Accepted:
19
May
2021
Published online: 21 June 2021
Based on the previous studies, the concept of generalized phononic crystals (GPCs) is further introduced into the cylindrical shell structures, and a type of cylindrical shells of generalized phononic crystals (CSGPCs) is constructed. Subsequently, the structure field and acousticstructural coupled field of that composite cylindrical shells are examined respectively in this paper. Considering the Bloch theorem is not capable of explaining the generalized periodic situation existing in this structure field, a new analysis method involving transferring matrix eigenvalue based on the mechanical state vector is proposed to calculate the energy band structure. Through observing the energy band structure, an obvious wave band gap is obtained when the elastic wave propagates in the CSGPCs for modes with different order, whose forming mechanism includes two aspects, i.e., the wave front expansion effect and the Bragg scattering effect. In addition, we further explore the related influences of the longitudinal wave mode and shear wave mode in structure on these band gaps, and some conclusions are illustrated. For acousticstructural coupled field, the expressions of the acoustic transmission coefficients for different modes are built, and the frequency responses are numerically calculated to verify the band gap characteristics of the CSGPCs. Furthermore, the acoustic pressure distribution of the internal and external acoustic fields is also analyzed in detail, and the influence laws of the parameters (offset distance and frequency) of the line source on acoustical pressure distribution and its directivity are explored.
Note to the reader: Further to the publication of an erratum, the citation of the article was modified on 19 July 2021.
© EDP Sciences, 2021
1 Introduction
As an artificial periodical and functional structure, phononic crystals (PCs) have caught widespread attention by the ability of controlling the propagative characteristics of different elastic waves [1–5]. PC models adhere to the crystal structure, namely construct PC models in rectangular coordinate system, while for the circular plate, cylindrical shell and spherical shell structure, it is difficult to construct the PC structure with similar vibration reduction performance. Due to the limited types of PCs under rectangular coordinate system, we shifted to focus on the PC structure under orthogonal curvilinear coordinate system, and put forward the concept of “generalized phononic crystals” [6].
At present, some research similar to the GPCs have also been conducted some scholars. For example, Yao Lingyun presented a single cylindrical shells phononic crystal model. It indicates that, similar to the plate structure, cylindrical shell phonon crystals also have quite good band gap and total forbidden band gap along the axial line and direction of cylinder ring [7]. Daniel Torrent proposed a class of radial sonic crystals (RSCs) made by fluidlike materials arranged periodically along the radial direction and the energy band diagram of the different modes of acoustic waves was obtained by calculating the different band gaps [8,9]. It should be noted that with the purpose to transform the aperiodic coefficients of the wave equations into periodic ones so as to meet the requirement of the Bloch theorem, they introduced acoustic metamaterials with mass density anisotropy. In practice, the design and manufacture of the RSCs seemed to be rather difficult. Xu et al. [10] also considered a kind of twodimensional arcshaped phononic crystals (APCs) with radial periodicity under cylindrical coordinate system. On the premise of the sufficiently large inner diameter of the arc (thus the wave equations could be treated approximately as having periodical coefficients, so the Bloch theorem is applicable), they investigated the wave propagation in the far field, and discussed the transmission properties of acoustic waves in different modes by using the Bloch theorem. Moreover, Li et al. [11] and Ma et al. [12] studied the propagation of Lamb wave in two kinds of RSCs plate. However, the rationality of directly using the Bloch theorem in the analysis process is doubtful since the wave equation is incompletely periodic (i.e., not all coefficients are periodic). The main reason why their simulation results can agree well with the energy band curves is based on that the Lamb wave band gap they obtained is located in rather high frequency region, which is actually the far field case. Seyyed [13] explored the planar acoustic wave scattering problem of functionally graded cylindrical shells, and calculated numerically the backscattering spectra. In addition, Cai et al. [14] took advantage of the particularity of functionally graded materials to investigative the twodimensional propagation of elastic waves in the radial PCs and discuss the influence of gradient distribution on the band gaps. Zhen and Pi [15] examined the transmission and reflection coefficients in the case of point and line sound source located at the center of composite spherical and cylindrical shells composed by two kinds of fluid. He pointed out the composite shells have different pass bands and band gaps comparing to periodic structures, however, the more general cases of solid medium and sound source with offset have not be taken into consideration.
Based on the above understanding, as a further study of our previous work, the concept of GPCs is further introduced into the cylindrical shell structure, and a class of cylindrical shells of generalized phononic crystals (CSGPCs) with radial periodical characteristics are constructed in this paper. Here, it should be noted that many scholars have done some research on laminated cylindrical shell structure and obtained some research results, but from the perspective of analysis, it is not the same as this paper. Our work focus on following two aspects: firstly, the energy band analysis and band gap discussion are conducted aiming at the structure field of the CSGPCs. As a substitute of the Bloch theorem and localization factor method, a kind of transfer matrix eigenvalue analysis method based on the mechanical state vector is presented, which can not only make a unified analysis for both the near and far fields, but also make the functions of the wave front effect and the Bragg scattering effect observed more easily (especially for the far field), then the deficiencies of the Bragg theorem and localization factor analysis can be avoid; secondly, the acoustic propagation properties and the soundproof effect of the CSGPCs under the excitation of internal line sound source with offset are investigated. We attempt to reveal some meaningful phenomena and its formation mechanism by this research so as to add some new contents into the theoretical study of PCs/GPCs and the soundproof research of cylindrical shell structure.
2 Model and energy band structure of CSGPCs
The conceptual layout is sketched in Figure 1. The CSGPCs is constructed alternately with two kinds of coaxial cylindrical shells made by different homogeneous materials. In order to emphasize the essence, two layers are assumed to be connected as ideal bonding and the side effects of the adhesive layer are ignored here. Along the radial direction, materials 1 and 2 are arranged alternately. The adjacent 1 and 2 can be regarded as “periodic” unit cell. The width of the unit cell is denoted by the symbol a. a_{1} and a_{2} represent the radial width of two adjacent cylindrical shells. It should also be mentioned that, compared with the existing material with anisotropy and the functional gradient materials proposed in previous studies, the CSGPCs is composed of two kinds of different homogeneous materials of which the design and preparation are much more convenient and practical.
Fig. 1 Sketch of the CSGPCs. 
2.1 Equation of elastic wave field
The wave motion in an isotropic elastic medium is governed by the classical Navier's equation [16]:(1)
Here, ρ is the solid material density, μ, λ are the Lame constants, and u is the displacement vector that can advantageously be expressed as sum of the gradient of a scalar potential and the curl of a vector potential:(2)
Combining equations (1) and (2) yields:(3)where k_{p} and k_{s} are complex wave numbers, known as:(4)
Equation (3) is a decoupling wave equation.
In cylindrical coordinates, each component expression of vibration displacement can be written from equation (2):(5)
The potential functions of longitudinal wave and transverse wave motivated in each unit of CSGPCs can be expressed respectively as:(7)where and are the morder Hankle function of the first and second kind, respectively,C_{m}, D_{m}, E_{m} and G_{m} are undetermined coefficients.
Combining equations (5)–(7), the expressions of the displacement and stress components of CSGPCs in arbitrary layer can be expressed as:(8)where , , and represent the stress and displacement components corresponding to the morder intrinsic mode of the jth layer in the nth unit, whose relationship with the undetermined coefficients C_{m}, D_{m}, E_{m}, G_{m} is presented as follows:(9) where , , are the 4 × 4 matrix, whose expressions of the specific elements are shown in Appendix A.
2.2 Energy band structure
The relation of the state vectors between the (n + 1)th and the nth unit is [17]:(10)
where , T_{n} is the transfer matrix between two adjacent units. Assume that the GPC model has N units, the total transfer matrix is:(11)
where V is the total transfer matrix, its eigenvalue can be written as:(12)
The real part λ represents the space attenuation rate of the elastic wave, the imaginary part κ represents the phase change of the wave.
We take the CSGPCs consisting of four units as the calculation model and the material and structural parameters have been listed in Tables 1 and 2, respectively.
The dispersion curves of the eigenvalues (take the natural logarithm) in 0order mode of CSGPCs and homogeneous cylindrical shells (HCS), i.e., energy band curves, are as shown in Figure 2. It should be noted that, for facilitate comparison, the structural parameters of the two types of cylindrical shells are consistent, namely, the size of the inner and outer diameters of the two types of cylindrical shells are the same.
Known from Figure 2, for the whole frequency range, the real part of energy band structure in 0order mode of CSGPCs and HCS λ is greater than 0, which shows that the amplitude of elastic wave in the process of propagating in shells is always attenuated. The curves of λ for HCS (epoxy) and HCS (steel) decrease rapidly with the increase of the frequency within 0.1–1.3 and 0.1–3.2 kHz respectively; when in higher frequency ranges, the value of λ is always steady around 0.2939. Actually, it is easy to understand, when cylindrical wave propagates from inside to outside (regardless of the reflection), the radial displacement of the particle can be expressed as follows:(13)where is the Hankle function of the first kind, known from Literature [18], when kr is very small, its asymptotic expression is:(14)when kr is rather large, its asymptotic expression is:(15)
Since the asymptotic expression of is changing with kr, the attenuation rate of the elastic wave also changes. It is clearly observed from equation (15) that, when kr is large sufficiently, the attenuation of cylindrical wave can be characterized as (r represents curvature), so we have:(16)where r_{0} = 0.2 m and r_{2N} = 0.36 m, thus λ = λ_{min} = 0.2939 can be solved. Accordingly, when the elastic wave propagates from inside to outside in HCS, the elastic wave attenuation is caused only by the expansion of wave front due to no reflection effect existing.
When the elastic wave of 0order mode propagates in the CSGPCs, λ > 0.2939, and the wave has a relatively strong attenuation within the frequency ranges of 0.1–3.0, 15.5–61.5, 66.5–70 kHz, we call these three frequency ranges as elastic wave band gaps; in the rest of the frequency ranges, λ = 0.2939, the attenuation is weak which is only caused by the expansion of the wave front yet, and we call them as nonbandgap. It is worth noting that the CSGPCs and HCS have the same value of λ_{min} (i.e., 0.2939), meaning that the wave attenuation in CSGPCs is caused by both the wave front and Bragg scattering. However, at the vicinity of 0.1 Hz, the CSGPCs' real part λ = 1.98 and that of HCS (Epoxy) and HCS (Steel) are 0.78, 0.84, respectively, showing that the Bragg scattering effect can also enhance the attenuation of the wave of 0order mode in low frequency band, here we call it as lowfrequency band gap.
In order to further analyze the influence of the wave front, we present the energy band structure aiming at those models with different sizes of the inner diameter. It can be seen from Figure 3 that, with the inner diameter increasing gradually, the influence of the wave front gets smaller and smaller, the value of the curve λ has slight decrease entirely, λ_{min} reduced from 0.2939–0.0507. In addition, the bandwidth and attenuation of the lowfrequency band gap also decreases, while the position of the band gap in midhigh frequency region has no obvious change, only has the attenuation slightly reduced, which shows that the formation of the band gap in midhigh frequency region is mainly caused by the Bragg scattering effect, while the function of wave front is just further increasing the band gap attenuation. When r_{0} increases to infinity, the cylindrical wave will evolve into plane wave, the influence of the wave front will disappear, and then the lowfrequency band gap disappears, however, the band gaps in high frequency region continue to exist. In this case the band gaps are just the traditional plane longitudinal ones. Obviously, the phenomenon illustrates from another perspective that the generation of the lowfrequency band gap for the 0order mode is caused by both the wave front and Bragg scattering effect, where the wave front effect should be the main factor, which directly determines the existence of the lowfrequency band gap, while the Bragg scattering effect has obvious enhancing effect on its attenuation.
From the above analysis, it can be seen that only longitudinal waves will be excited in the case of 0order mode, which is equivalent to the axisymmetric propagation of longitudinal waves in CSGPCs, which belongs to a special case. Here, we further do the calculation for other modes (m = 1, 2, 3), the results are shown in Figure 4. Since the shear wave and longitudinal wave exist at the same time in the nonzeroorder modes, λ and κ become more complex, however, some obvious regular phenomena still can be observed.
It can be observed from Figure 4 that, for nonzeroorder modes, the energy band curves show some peculiarities different from the zeroorder mode. For example, at low frequencies, with the increase of the order, the number of the small band gaps increases progressively, and move to the midhigh frequencies gradually; While at midhigh frequencies, the nonzeroorder modes no longer show a complete band gap, but break the large band gap in zeroorder mode (15.5–61.5 kHz) into three small band gaps, corresponding to 15.5–28.5, 30.5–56, 58.4–61.5 kHz. However, the values of the maximum attenuation of band gaps in each order mode are basically the same. In addition, for the whole frequency range examined, the real parts of the energy band structure in the first four order modes of CSGPCs λ are all greater than zero, and the minimum values of the energy band are the same λ_{min} (0.2939) , which shows that when the frequency is large sufficiently (or kr is large sufficiently), the functions of the wave front in each order mode are the same; For the lowfrequency band gap near 0 Hz in nonzeroorder modes, with the increase of the order, both the bandwidth and the attenuation of the band gaps will increase gradually.
According to the above research, for the zeroorder mode, only exist the longitudinal wave in cylindrical shells; while for the nonzeroorder modes, both the longitudinal and shear wave will exist in solid cylindrical shell at the same time, thus we need to further analyze the effects of the longitudinal wave and shear wave on the band gap properties of the CSGPCs. Since there is only longitudinal wave in ideal fluid medium, here we aim at the analysis of the characteristics of wave band gaps of the CSGPCs (fluidsolid) composed by the fluid and solid mediums and the CSGPCs (fluidfluid) composed by two kinds of fluid mediums. The two kinds of shells will be compared to explore the impact of the longitudinal wave and shear wave on the band gap properties.
Firstly, we consider the propagation properties of the CSGPCs (fluidsolid). For comparison with the CSGPCs (solidsolid) expediently, the fluid material 1 is assumed to have the same density and wave velocity with the steel material. The energy band structures of the wave for 0–3 order modes of the CSGPCs (fluidsolid) are described in Figure 5.
It can be seen that, no matter the waves in any mode, λ_{min} is always 0.2939; In high frequencies, λ and κ have no obvious change basically in different modes, and the waves in different modes have almost the same band gap range (15.3–61.5 kHz); however, in low frequencies, there is a big difference between λ and κ. In the frequency range of 0.1∼5 kHz, the wave band gap range is 0.1–0.6 kHz in the case of m = 0; the wave band gap ranges are 0.1–0.5, 1–1.2, 1.4–1.7 kHz in the case of m = 1; the band gap ranges are 0.1–1.0, 1.8–2.2, 2.3–3.3 kHz in the case of m = 2; the band gap ranges are 0.1–1.5, 2.6–2.9, 3–3.8, 3.9–4.8 kHz in the case of m=3. With the increase of the order m, there have more lowfrequency band gaps, and the width is larger which can even cover the low frequency region completely. But it is worth noting that, the phase mutation phenomenon exists in the lowfrequency band gaps, accordingly, λ reaches the peak. The frequency where the phase mutation occurs is getting larger and larger with the increase of the order m. In addition, the phase mutation phenomenon also exists in high frequency region, despite that the frequency point increases slightly with the increase of m.
Now we consider the band gap properties of the CSGPCs (fluidfluid) composed of two kinds of fluid mediums. Similarly, for facilitate comparison with CSGPCs (fluidsolid), the density and wave velocity of fluid material 1 are also assumed the same as steel, likewise, fluid material 2 is considered to have the same density and wave velocity with epoxy. Besides, the same structural size of the two types of cylindrical shells should be guaranteed. The energy band structures of the CSGPCs (fluidfluid) for the 0–3 order modes are depicted in Figure 6.
It can be found that, with the increase of the mode order, the band gaps in low frequencies have obvious changes relatively, i.e., the cutoff frequency gradually increases. However, the start and cutoff frequencies and the attenuation in the midhigh band gaps remain the same basically, and the wave band gap ranges are the same as the fluid/solid system. Therefore, in midhigh frequencies, the formation of wave band gaps for each order is mainly caused by the Bragg scattering effect on the longitudinal waves of CSGPCs. By comparing Figures 5 and 6, it can be seen that, within the scope of the band gap, there is no phase mutation phenomenon in the real part of z for the fluid/fluid system and the curves are very smooth with no aiguille, implying that the existence of the shear wave mode is the reason of the appearance of the phase mutation in the CSGPCs (fluidsolid) system.
Linear elastic material properties of the constituents.
Structure parameters.
Fig. 2 Energy band curves of the 0order mode of CSGPCs and HCS. 
Fig. 3 Energy band carves with different inner diameters of CSGPCs in 0order mode. 
Fig. 4 Energy band curves of CSGPCs for different modes. 
Fig. 5 Energy band curves of CSGPCs(fluidsolid) in different modes. 
Fig. 6 Energy band carves of CSGPCs(fluidfluid) in different modes. 
3 Coupled acousticstructural analysis
3.1 Acoustic pressure transmission coefficient
Firstly, let us analyze the behavior of the CSGPCs when a line sound source p^{0} is put inside the cavity defined by the shell (see Fig. 1). In this region, the general expression for the exciting field is [19–21]:(17)where (e, θ_{0}) is the location of the line source, is the Hankle function, J_{m} is the Bessel function, , and c_{0} is the sound speed inside the cavity. For e < r < r_{0}, the source field can be compactly written as:(18)with(19)
For brevity, let θ_{0} = 0^{∘}.
In the presence of a cylindrical periodic structure, the emitted wave will be reflected back and forth by the structure. Therefore, the total wave include both the wave directly from the source and the reflected one. In the neighborhood of the source (r < r_{0}), the total wave can be written as:(20)
Clearly p^{r} must be finite inside the structure and satisfy the Helmholtz wave equation:(21)
The general solution to this equation in the region can be written as:(22)
The coefficients a_{m} and b_{m} are determined from the boundary conditions in the source region. The boundary conditions require that the total field cannot be divergent at the origin. As a result, we have . Equation (22) then can be written as:(23)
The transmitted field p^{t} will appear at the region outside the shells (r > r_{2N}):(24)
The vibration velocities of the particle of the medium in the inner and outer acoustic fields of CSGPCs are:(25)
Since the amplitude of the emitted wave in the active field for the mthorder mode is C_{m} = J_{m} (k_{0}e), the amplitude of the transmission acoustic wave is B_{m}, for convenience, the transmission coefficient of acoustic waves in the mthorder mode is defined as , and the reflection coefficient can be defined as: .
Known from the analysis above, the acoustic pressure transmission coefficients and reflection coefficients in each order have nothing to do with the size of the angle θ, but related to the order of the mode m and the offset distance e.
The unknown coefficients A_{m} and B_{m} can be determined from the appropriate boundary conditions imposed at the inner (r_{0}) and the outer (r_{2N}) surfaces of the CSGPCs. Thus, assuming continuity of normal fluid and solid velocities, normal stress and fluid pressure, and vanishing of tangential stress at r_{0} and r_{2N} implies that [22]:(26)
Finally, making use of equations (8), (20), (24) and (25) in equaion (26), we obtain: (27)
where, V_{(b,d)} (b, d = 1, 2, 3, 4) represents the elements in the total transfer matrix V.
Combine the equations above, the coefficients A_{m}, B_{m} can then be calculated.
3.2 Finite order approximation
Equations (18) and (19) demonstrate clearly the nature of the source field emitted by the line sound source. For r > e, the field is composed of a sum of individual, outward going azimuthally waves of order m, whose strengths are given by the amplitudes C_{m}. Even though the sum is infinite, for a finite source position e the amplitudes C_{m} are appreciable only for a finite number of waves. Therefore the infinite sum can be approximated well by a finite number of terms.
Known from equation (19), the amplitude C_{m} of the acoustic pressure of each order is related to the frequency f and the offset e. First of all, the influence of the offset distance on C_{m} is discussed. We assume the fluid mediums inside and outside of the CSGPCs are water (ρ = 1000kg/m^{3}, c = 1480m/s), when the calculated frequency is 70 kHz, the behaviors of the acoustic amplitude of each order with different offset distance with the change of m are shown in Figure 7.
Figure 7 shows that, for different offset distance e, the behaviors of C_{m} are basically the same, namely, with the increase of the order, the curves swing up and down violently, then the acoustic amplitude reduced to 0 rapidly after C_{m} reaches the maximum. In addition, with the increase of offset distance, in order to guarantee sufficient approximation accuracy, the number of the finite order m needing to be considered should also increase gradually. For instance: when e = 0.02 m, we have C_{12} < 0.001, so 11 is taken as the number of the finite order; when e = 0.05 m, C_{23} < 0.001, 22 should be taken; similarly, when e = 0.1 m, C_{39} < 0.001, 38 is rational. It needs to be pointed out particularly that C_{1} = 0 in the case of e = 0 (i.e, the sound source being located at the center), that is, there exists only the acoustic wave of 0order mode.
Next, the effects of frequency on the amplitude C_{m} of the acoustic pressure of each order are discussed. When the offset distance e = 0.1 m, the behaviors of the acoustic amplitude of each order at different frequencies are shown in Figure 8.
Similar to the influence of the offset distance on the acoustic amplitude, for different frequencies, the variations of C_{m} are the same basically. Namely, with the increase of the order, the curves swing up and down violently, then the acoustic amplitude reduced to 0 rapidly after C_{m} reaches the maximum. With the increase of the frequency, the number of the finite order required also increases gradually. When f = 20kHz, we have C_{15} < 0.001, so 14 is taken as the number of the finite order; similarly, 24, 33, 42 should be adopted respectively in the cases of C_{25} < 0.001, C_{34} < 0.001, C_{43} < 0.001 corresponding the frequencies at 40, 60 and 80 kHz.
Known from the analysis above, when the offset distance of the sound source is not zero, the internal acoustic field of the CSGPCs is the superposition of infinite subitems, for the sound source in limited location, we can use finite number of orders to approximate. The finite number is influenced by the frequency and offset distance: the higher the frequency, the larger the number, and so does the offset distance.
Fig. 7 Acoustic amplitude of different mode at certain frequency. 
Fig. 8 Acoustic amplitude of different modes under certain offset distance. 
3.3 Numerical example
Here, we do the numerical calculation aiming at the offset line sound source excitation (the offset distance e = 0.1 m) to get the frequency response curves of the acoustic pressure transmission coefficients of CSGPCs (solidsolid) for the 0–3 order modes.
As shown in Figure 9, in terms of the frequency response curve of the acoustic pressure transmission coefficient in 0order mode, the acoustic band gaps do exist in 0.1–2.9, 15.3–62, 67–70 kHz obviously. While for nonzeroorder modes, in low frequencies, with the increase of the order, the number of the small band gaps increases progressively, and move to the midhigh frequencies gradually; while in midhigh frequencies (15.3–62 kHz), the nonzeroorder modes show three band gaps, which basically corresponds to 15.3–28.8, 30.4–56.3, 58.3–62 kHz, and with the rising of the mode order, the band gap structure become more obvious; in addition, in high frequencies (67–70 kHz), we can observe the obvious acoustic band gaps, due to the limitation of the calculated frequency, the complete band gap structure isn't shown here. It is worth noting that the minimum of acoustic pressure transmission coefficients (with the exception of some antiresonance points) in 0–3 order modes are basically the same.
In summary, through the analysis of the numerical simulation results of acoustic pressure transmission coefficients of CSGPCs (solidsolid) for each order mode, we can see that the acoustic attenuation characteristics shown here and the results of energy band analysis given in Section 2 are consistent, which can also verify the feasibility of the method of transfer matrix eigenvalue analysis based on the mechanical state vector presented in Section 2.
Fig. 9 Different modes of acoustic wave TR frequency response curve. 
3.4 Directivity of acoustic pressure distribution
The finite order needs to be used to approximate in calculating the internal and external acoustic fields of the CSGPCs, while the finite order will be affected by the offset of the line sound source and the acoustic frequency. Therefore, in this section, we further explore the influence of the offset distance and the frequency on the internal and external acoustic fields mainly by means of numerical calculation and finite element simulation method.
For the offset distance of the line sound source e = 0.1 m, corresponding to Figure 4, several typical frequencies are selected to be analyzed, i.e., the lowfrequency wave within the range of nonbandgap in each order mode (4 kHz), the highfrequency wave within the range of nonbandgap in nonzero mode (30 kHz), the highfrequency wave within the range of band gaps in each order mode (40 kHz).
The results of the finite element simulation and the numerical calculation of the acoustic pressure distribution inside and outside acoustic fields of the CSGPCs at three frequencies (4, 30 and 40 kHz) are shown in Figure 10. In the case of f = 4 kHz, the directivity of the acoustic pressure is relatively apparent, and the positions of the peak and valley are basically the same. As for 30, 40 kHz, it can be seen that, with the increase of the frequency, the acoustic pressure distributions become more and more complex, and the peak and valley change alternately, which means the directivity is extremely obvious. Actually, it is easy to understand since that the number of the finite orders required for rational approximation is decided by both the frequency and offset distance, namely, the higher the frequency, the higher the acoustic mode inspired, i.e. the increasing number of the finite orders will lead to an increase in the summation items of acoustic pressure. In addition, according to the results of the numerical calculation, the maximum acoustic pressure of the external field corresponds to the frequency of 4 kHz while the minimum corresponds to the 40 kHz. Besides, the values of the acoustic pressures corresponding to 30 kHz is between the maximum and minimum. Figure 4 can be used to well explain the reasons, i.e., the values of the real parts for each order mode corresponding to f = 4 kHz are all 0.2939; the values corresponding to 40 kHz are all 9.851; and the frequency f = 30 kHz is located within the band gaps of the zeroorder mode and simultaneously within the nonbandgap of the nonzeroorder modes, whose corresponding values of the real parts are 8.989 and 0.2939 respectively. By comparison, we can deduce that the attenuation effects arranged from strong to weak are as follows: f = 40 kHz, f = 30 kHz, f = 4 kHz, therefore, it is correct that the corresponding acoustic pressures of the external field of the CSGPCs are in ascending order. It should be noted that, since the superposition of the finite terms is adopted to get the results of numerical calculation, while the infinity terms are used for computation in simulation, thus, there is some deviation in the two calculation results. However, it can be seen from Figure 10, the results of numerical calculation and finite element simulation are basically consistent, which further shows the correctness of the theoretical analysis above.
Since the number of the finite orders is also affected by the offset distance, the effects of the offset distance on the acoustic pressure distribution at particular frequencies are further discussed then. For convenience, here we only present the results of the finite element simulation. Figure 11 shows the simulation nephograms of nine cases: the frequencies are 25, 30, 60 kHz respectively, and the offset distances are taken as e = 0.02 m, e = 0.05 m, e = 0.1 m respectively.
It is not difficult to see that, when the offset distance is relatively small, the acoustic pressure distributions inside and outside of the CSGPCs are relatively uniform. With the increase of the offset distance, the peaks and valleys of the acoustic pressure increase gradually, and the directivity is more and more complex. Similarly, we can still do the same interpretation, that is, when the offset distance is small, the number of finite orders is small, thus the acoustic pressure distribution is relatively uniform; when the offset distance is relatively large, the number of finite orders should be increased, leading to the increase of the summation items of the acoustic pressure in the inner and outer acoustic fields, therefore, the peaks and valleys increase gradually and the directivity becomes much more complex.
Fig. 10 Numerical results and finite element simulation nephograms of acoustic pressure distribution at the innermost and outermost surfaces of the CSGPCs with different frequencies; (a) frequency f = 4 kHz; (b) f = 30 kHz; (c) f = 40 kHz. 
Fig. 11 Acoustic pressure distribution internal and external of the CSGPCs with different offset distances. 
4 Conclusion
On the basis of previous studies, the concept of generalized phononic crystals (GPCs) is introduced into the cylindrical shell structure, and a type of cylindrical shells of generalized phononic crystals (CSGPCs) is constructed in this paper. With the method of establishing the transfer matrix of the mechanical state vector combined with eigenvalue analysis, the energy band structure of CSGPCs is obtained and used to reveal the phenomenon and formation mechanism of the wave band gaps of this kind of structure. Further, we explore the acoustic characteristics of the CSGPCs through the offset line sound source excitation, the acoustic transmission coefficients and the acoustic pressure distribution in the inner and outer acoustic fields are given by the coupled acousticstructural analysis. Our results show that:
the band gaps of elastic wave or acoustic wave for each order mode propagating in the CSGPCs can be accurately calculated by the method of total transfer matrix eignvalue analysis based on the mechanical state vector, and the effects of the wave front and the Bragg scattering are more easily to be observed. Thereby, on one hand, it can avoid the inapplicability of the Bloch theorem, on the other hand, it can also avoid the deficiency of the localization factor analysis presented in our previous work. In fact, the method is also applicable to the energy band analysis of other similar GPC structures.
the analysis of the structure field of the CSGPCs shows that, there exists significant band gap phenomenon as elastic wave propagates in the CSGPCs, and the formation of the band gaps is mainly caused by two aspects, namely, the wave front effect and the Bragg scattering effect. In the lowfrequency band gaps, the wave front effect is dominant and the Bragg scattering effect plays a role as enhancing the attenuation; while in the midhigh frequency band gaps, the Bragg scattering effect is decisive, and the effect of the wave front is relatively small. From the perspective of the elastic wave mode in structure, the formation of the band gaps mainly depends on the longitudinal wave mode, and the existence of the shear mode will lead to the phase mutation both in low frequencies and midhigh frequencies. In addition, it should also be mentioned that, compared with the existing material with anisotropy and the functional gradient materials proposed in previous studies, the CSGPCs is composed of two kinds of different homogeneous materials of which the design and preparation are much more convenient and practical.
the coupled acousticstructural analysis shows that, for the line sound source with offset, we can use finite orders to approximate despite the internal acoustic field of the CSGPCs is the superposition of infinite items. The quantity of the finite orders is mainly determined by the frequency and offset distance: the higher the frequency, the larger the number, and so does the offset distance. The results from the numerical calculation of the acoustic pressure transmission coefficients show that the CSGPCs have a significant soundproof effect, which is consistent with the results of the energy band analysis of structure field. In addition, under the excitation of the offset line sound source, the acoustic fields, both inside and outside of the CSGPCs, exhibit significant directivity, which is directly related to the finite order number. Hence the directivity is mainly affected by the offset distance and frequency: the larger the offset distance and frequency, the more the peaks and valleys of the acoustic pressure.
This paper provides a new solution in the field of vibration and noise reduction. Follow this line, to take advantage of the acoustic band gap properties of the CSGPCs combining with the principles of existing sound insulation covers (such as attaching various kinds of sound absorption materials), a new type of sound insulation cover with better performance is likely to be achieved. Considering the research is still limited to the passive structure, whose band gap properties are untunable once the structure and material are determined, which is not ideal for many engineering applications. Therefore, the future study should focus on the active structure. For example, the active adjustment ability of the external control on the band gaps can be investigated intensively by introducing the piezoelectric material (PVDF) to construct the CSGPCs in order to obtain more excellent wave band gap properties.
Author contribution statement
All the authors have participated in the preparation of the manuscript. They have read and approved the final manuscript.
Acknowledgments
This work was funded by the project (grant number 51875112) supported by the National Natural Science Foundation of China.
Appendix A
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Cite this article as: Wang Xingguo, Shu Haisheng, Zhang Lei, Vibration and acoustic insulation properties of generalized phononic crystals, Eur. Phys. J. Appl. Phys. 94, 30902 (2021)
All Tables
All Figures
Fig. 1 Sketch of the CSGPCs. 

In the text 
Fig. 2 Energy band curves of the 0order mode of CSGPCs and HCS. 

In the text 
Fig. 3 Energy band carves with different inner diameters of CSGPCs in 0order mode. 

In the text 
Fig. 4 Energy band curves of CSGPCs for different modes. 

In the text 
Fig. 5 Energy band curves of CSGPCs(fluidsolid) in different modes. 

In the text 
Fig. 6 Energy band carves of CSGPCs(fluidfluid) in different modes. 

In the text 
Fig. 7 Acoustic amplitude of different mode at certain frequency. 

In the text 
Fig. 8 Acoustic amplitude of different modes under certain offset distance. 

In the text 
Fig. 9 Different modes of acoustic wave TR frequency response curve. 

In the text 
Fig. 10 Numerical results and finite element simulation nephograms of acoustic pressure distribution at the innermost and outermost surfaces of the CSGPCs with different frequencies; (a) frequency f = 4 kHz; (b) f = 30 kHz; (c) f = 40 kHz. 

In the text 
Fig. 11 Acoustic pressure distribution internal and external of the CSGPCs with different offset distances. 

In the text 
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