**2. Vibration characteristics of infinite periodic ABH shells**

### **2.1 Gaussian expansion for the vibration of infinite periodic ABH shells**

The goal of this section is that of developing a semi-analytical model for characterizing the vibration of the infinite periodic ABH shell. Let us consider three variables, *u*, *v*, and *w*, which are the displacements in the axial, circumferential, and radial directions, respectively. They can be decomposed by

$$\mu(\mathbf{x}, \boldsymbol{y}, t) = \sum\_{i} a\_i(t)\boldsymbol{\nu}\_i(\mathbf{x}, \boldsymbol{y}) = \boldsymbol{a}^\top \boldsymbol{\Psi} = \boldsymbol{\Psi}^\top \boldsymbol{a},\tag{1}$$

$$\nu(\mathbf{x}, \mathbf{y}, t) = \sum\_{i} b\_{i}(t)\xi\_{i}(\mathbf{x}, \mathbf{y}) = \mathbf{b}^{\top}\boldsymbol{\xi} = \boldsymbol{\xi}^{\top}\boldsymbol{b},\tag{2}$$

$$\mathfrak{w}(\mathfrak{x}, \mathfrak{y}, t) = \sum\_{i} c\_{i}(t) \rho\_{i}(\mathfrak{x}, \mathfrak{y}) = \mathfrak{c}^{\mathsf{T}} \mathfrak{o} = \mathfrak{o}^{\mathsf{T}} \mathfrak{c},\tag{3}$$

where

$$\mathbf{a} = \hat{\mathbf{A}} \exp\left(\mathrm{j}at\right), \mathbf{b} = \hat{\mathbf{B}} \exp\left(\mathrm{j}at\right), \mathbf{c} = \hat{\mathbf{C}} \exp\left(\mathrm{j}at\right), \tag{4}$$

are the coefficient vectors to be determined, while *ψ*, *ξ*, and *φ* are the shape function vectors with entries *ψi*ð Þ *x*, *y* , *ξi*ð Þ *x*, *y* , and *φi*ð Þ *x*, *y* , respectively. With the aid of Kronecker product, the vectors *ψ*, *ξ*, and *φ* can be factorized as

$$
\mathfrak{w}(\mathfrak{x}, \mathfrak{y}) = \mathfrak{a}^{\mathfrak{w}}(\mathfrak{x}) \otimes \mathfrak{f}^{\mathfrak{w}}(\mathfrak{y}),
\tag{5}
$$

$$\mathfrak{f}(\mathfrak{x}, \mathfrak{y}) = \mathfrak{a}^{\xi}(\mathfrak{x}) \otimes \mathfrak{f}^{\xi}(\mathfrak{y}),\tag{6}$$

$$
\mathfrak{g}(\mathfrak{x}, \mathfrak{y}) = \mathfrak{a}^{\mathfrak{p}}(\mathfrak{x}) \otimes \mathfrak{f}^{\mathfrak{p}}(\mathfrak{y}),
\tag{7}
$$

where *<sup>α</sup>*ð Þ*<sup>i</sup>* ð Þ *<sup>i</sup>* <sup>¼</sup> *<sup>ψ</sup>*, *<sup>ξ</sup>*, *<sup>φ</sup>* are column vectors containing basis functions depending on the *<sup>x</sup>* direction yet *<sup>β</sup>*ð Þ*<sup>i</sup>* ð Þ *<sup>i</sup>* <sup>¼</sup> *<sup>ψ</sup>*, *<sup>ξ</sup>*, *<sup>φ</sup>* the ones on the *<sup>y</sup>* direction. To accurately capture the localized displacements in the ABH portion, the entries of *α* and *β* are selected as Gaussian functions

$$a\_i(\mathbf{x}) = \mathfrak{Z}^{\epsilon\_x/2} \exp\left[-\left(\mathfrak{Z}^{\epsilon\_x}\mathfrak{x} - q\_{\mathrm{xi}}\right)^2/2\right],\tag{8}$$

$$\beta\_i(\mathbf{y}) = \mathfrak{Z}^{\epsilon\_p/2} \exp\left[-\left(\mathfrak{Z}^{\epsilon\_p}\mathbf{y} - q\_{ji}\right)^2/2\right],\tag{9}$$

in which *sx* and *sy* are the scaling parameters, while *qx* and *qy* are the translational parameters, in the *x* and *y* directions, respectively. For brevity, readers are referred to our previous works [4, 26] to thoroughly comprehend the detailed process of how to produce acceptable basis.

Unlike finite and flat plates, bear in mind that the displacements, on the one hand, must be continuous in the circumferential direction (<sup>0</sup> and <sup>1</sup> are enough, see [22]), which requires

$$\mathbb{C}^{0}: u(\mathbf{x}, -\pi R) = u(\mathbf{x}, \pi R), \ v(\mathbf{x}, -\pi R) = v(\mathbf{x}, \pi R), \ w(\mathbf{x}, -\pi R) = w(\mathbf{x}, \pi R), \tag{10}$$

$$\mathbb{C}^{1}: \frac{\partial u}{\partial \mathbf{y}}(\mathbf{x}, -\pi R) = \frac{\partial u}{\partial \mathbf{y}}(\mathbf{x}, \pi R), \quad \frac{\partial v}{\partial \mathbf{y}}(\mathbf{x}, -\pi R) = \frac{\partial v}{\partial \mathbf{y}}(\mathbf{x}, \pi R), \quad \frac{\partial w}{\partial \mathbf{y}}(\mathbf{x}, -\pi R) = \frac{\partial w}{\partial \mathbf{y}}(\mathbf{x}, \pi R). \tag{11}$$

On the other hand, Bloch-Floquet periodic boundary conditions must be imposed in the axial direction for a unit cell (<sup>0</sup> and <sup>1</sup> , too), namely

$$\mathbb{C}^{0}: u(0, y) = u(L\_{\text{cell}}, y)\lambda, \ v(0, y) = v(L\_{\text{cell}}, y)\lambda, \ w(0, y) = w(L\_{\text{cell}}, y)\lambda,\tag{12}$$

$$\mathbb{C}^{1}: \frac{\partial u}{\partial \mathbf{x}}(0, y) = \frac{\partial u}{\partial \mathbf{x}}(L\_{\text{cell}}, y)\lambda, \ \frac{\partial v}{\partial \mathbf{x}}(0, y) = \frac{\partial v}{\partial \mathbf{x}}(L\_{\text{cell}}, y)\lambda, \ \frac{\partial w}{\partial \mathbf{x}}(0, y) = \frac{\partial w}{\partial \mathbf{x}}(L\_{\text{cell}}, y)\lambda,\tag{13}$$

where *λ* ¼ exp jð Þ *kxLcell* , *kx* is the axial wavenumber in the irreducible Bernouin zone [18]. Via implementing the reconstruction process in [18], the continuity in the circumferential direction (Eqs. (10) and (11)) and the periodicity in the axial one (Eqs. (12) and (13)) can be satisfied.

Provided the kinetic energy, *K*, and the potential one, *U*, are presented in terms of *u*, *v*, and *w* [27], the Lagrangian of the whole system can be built

$$L = K - U = \frac{1}{2}\dot{q}^{\top}\mathbf{M}\dot{q} - \frac{1}{2}q^{\top}\mathbf{K}q,\tag{14}$$

where

$$\boldsymbol{q} = \left[\hat{\mathbf{A}}^{\top}, \hat{\mathbf{B}}^{\top}, \hat{\mathbf{C}}^{\top}\right]^{\top} \exp\left(\mathbf{j}\boldsymbol{\alpha}t\right) \equiv \hat{\mathbf{Q}} \exp\left(\mathbf{j}\boldsymbol{\alpha}t\right),\tag{15}$$

represents the assembled undetermined time-dependent vector related to admissible shape functions. *M* represents the mass matrix and *K* the stiffness one. *Periodic Acoustic Black Holes to Mitigate Sound Radiation from Cylindrical Structures DOI: http://dx.doi.org/10.5772/intechopen.101959*

Finally, applying the Euler–Lagrange equations *<sup>∂</sup><sup>t</sup> <sup>∂</sup>q*\_*<sup>L</sup>* � *<sup>∂</sup>q<sup>L</sup>* <sup>¼</sup> **<sup>0</sup>** to Eq. (14) yields the equations of motion in the frequency domain,

$$\left(-\boldsymbol{\alpha}^{2}\mathbf{M} + \mathbf{K}\right)\hat{\mathbf{Q}} = \mathbf{0},\tag{16}$$

whose solution permits calculating the dispersion curves and eigenmodes for infinite periodic ABH shells.

## **2.2 Numerical results**

### *2.2.1 Dispersion curves and band gaps*

The dispersion curves of an ABH cell, whose geometry and material are detailed in **Table 1**, have been carried out and plotted in **Figure 3a**. For the purpose of validation, the result from a reference FEM model has also been included in **Figure 3a**. From the figure, it is seen that the two results are very close at each


*ρ, shell density; ρv, damping layer density; E, shell young modulus; Ev, damping layer young modulus; η, shell loss factor; ηv, damping layer loss factor; ν, shell Poisson ratio; νv, damping layer Poisson ratio.*

#### **Table 1.**

*Geometry and material parameters of the ABH cylindrical shell.*

#### **Figure 3.**

*(a) Dispersion curves together with band gaps calculated with GEM and FEM. (b) Transmission from the left end to the right for a finite shell having five cells with and without damping layers, carried out with the GEM.*

wavenumber, indicating the correctness of the present GEM model. Most importantly, four-band gaps (BGs) are observed within 1000 Hz, in which the first one is very small. Using the GEM without axial periodic boundary conditions for a finite shell having five cells, we can compute the transmission from one end to the other. As shown in **Figure 3b**, the transmission is very low at BG frequencies. For the shell without damping layers (undamped), the transmission is strong in the passbands. However, this situation can be ameliorated after implementing the damping layers, with a maximum reduction of up to �25 dB.

To reveal the mechanism of the BGs, we have computed the first six eigenmodes of the unit cell, at wavenumber *kx* ¼ 0 for *λ*. As illustrated in **Figure 4**, for most orders, the vibration is very strong in the ABH area. While this is not the case for the 1-st order (see **Figure 4a**). This is because of the ring frequency (173 Hz), below which the cylindrical shell is almost not vibrating in the radial direction. However, for the 2-nd to 6-th orders (see **Figure 4b**–**f**), the wave is gradually concentrating in the ABH portion, belonging to the locally resonant effect. That is, the BGs shown in **Figure 3** are locally resonant ones, similar to the periodic ABH beams reported in [28].

### *2.2.2 Parametric analysis: effects of the ABH order, central thickness, and radius*

The ABH profile is generally controlled by three parameters, *m*, *hc*, and *rabh*, which represent the ABH order, central thickness, and radius, respectively. It is worthwhile testing how these parameters affect the BGs.

Let us first look at the influence of the ABH order. As shown in **Figure 5a**, the 2-nd BG starts to gradually decrease as *m* increases. The 3-rd BG is however distinctive because the BG inverses near *m* ¼ 3, then the upper bound almost keeps still but its lower bound drops. For the 4-th one, the width of the BG first becomes larger then turns out smaller as *m* goes up. In general, the total width of the four BGs almost remains the same, but they will be more compact and converge to lower frequencies. Note that we have also included the changes of the BGs for the ABH shell with a damping layer (damped). It is seen that the phenomenon is close to the

#### **Figure 4.**

*The first 6 eigenmodes for infinite periodic cells at wavenumber kx* ¼ *0. The shaded area stands for the ABH portion. (a)-(f) respectively correspond to the 1-st to the 6-th orders.*

*Periodic Acoustic Black Holes to Mitigate Sound Radiation from Cylindrical Structures DOI: http://dx.doi.org/10.5772/intechopen.101959*

**Figure 5.** *Band gaps changing with (a) the ABH order, (b) the central thickness, and (c) the ABH radius.*

former case. Due to the added stiffness of the layer, the BGs however occur at higher frequencies.

Different from the ABH order, the BGs are very sensitive to the central thickness. As graphed in **Figure 5b**, generally, the BGs become very narrow as the central thickness increases from 0<sup>þ</sup> to *huni* ¼ 0*:*03 m. It seems that the central thickness of the ABH needs to be small enough for wide BGs, whose local resonance is more significant. Howbeit, it is not realistic because truncation always exists. Moreover, we observe that for the 4-th BG there is an inflection point near *hc* ¼ 0*:*002 m, indicating that the optimal thickness can be found in this range. For comparison, the effects of the damping layer are also characterized in **Figure 5b**, but it seems not important to the BGs. However, for very small *rabh* the added mass of the damping layer is more dominant than its added stiffness. This is why the BGs for the damped shell are lower than those for the undamped shell.

Finally, the effects of the ABH radius are computed and illustrated in **Figure 5c**. We consider the ABH radius varying from 0 to *Lcell=*2 ¼ 0*:*5 m. From the figure, we can see that the BGs move to low-frequency range very fast as *rabh* grows. Particularly, the BGs are prone to inverse when the radius is not very large (*rabh* <0*:*4 m). The 2-nd BG inverses at *rabh* ¼ 0*:*14 m, while the 3-rd BG has double inversions respectively at *rabh* ¼ 0*:*1 m and *rabh* ¼ 0*:*32 m, whereas the 4-th BG contains at least two inversions at *rabh* ¼ 0*:*22 m and *rabh* ¼ 0*:*37 m. When *rabh* >0*:*4 m, the BGs become stable and progressively move to lower frequencies. Again, the influence of the damping layer is also characterized in **Figure 5c**. On the one hand, it is inspected that the damping layer has significance merely for a large ABH radius. On the other, the BGs locate at higher frequencies because of added stiffness, as reported in **Figure 5a**.
