Acoustic performance prediction of a multilayered finite cylinder equipped with porous foam media

Abstract

This paper presents an analytical model to embed porous materials in a finite cylindrical shell in order to obtain the sound transmission loss coefficient. Although the circumferential modes are considered only for calculating the amount of the transmitted noise through an infinitely long cylinder, the present study employs the longitudinal modes in addition to circumferential ones to analyze the vibroacoustic performance of a simply supported cylinder subjected to the porous core based on the first order shear deformation theory. To achieve this goal, the structure is immersed in a fluid and excited by an acoustic wave. In addition, the acoustic pressures and the displacements are developed in the form of double Fourier series. Since these series consist of infinite modes, it is essential to terminate this process by considering adequate modes. Hence, the convergence checking algorithm is employed in the form of some three-dimensional configurations with respect to length, frequency and radius. Afterwards, some figures are plotted to confirm the accuracy of the present formulation. In these configurations, the obtained sound transmission loss from the present study is compared with that of the infinite one. It is shown that by increasing the length of the structure, the results are approached to sound transmission loss of the infinite shells. Moreover, a new approach is proposed to show the transverse displacement of a finite poroelastic cylinder at different frequencies. Based on the outcomes, it is found that by enhancing the length of the poroelastic cylinder, the amount of the transmitted sound into the structure is reduced at the high frequency domain. However, the sound insulation property of the structure is improved at the low frequency region when the radius of the shell is decreased.

Keywords

Sound transmission loss acoustic analysis finite length cylinder porous material wave propagation

1. Introduction

In recent years, the application of the sandwich structures in various engineering industries including aerospace, marine and automotive has been increased. Hence, many authors have modeled these structures by adding a porous material as an intermediate layer because of its light weight and its excellent sound absorption capability. These materials consist of two phases involving a solid and a fluid. Accordingly, although various works have been presented on the acoustic analysis of an infinitely long cylinder, modeling the porous material as a core in a finite shell causes to present more accurate and reliable results. Then, a new approach is proposed in this study to inspect the vibroacoustic performance of a finite cylinder subjected to a porous material.

In order to analyze the wave propagation on the poroelastic materials, some theories with various approximate calculations can be found in the literature. Biot (1956a, 1956b) is known as the researcher who proposed a perfect model of wave propagation through porous materials. In Biot’s model, although the shear wave propagates, no deformation in the solid phase is created. Biot’s theory was then extended by Bolton et al. (1996) to obtain the sound transmission loss (STL) of an infinite sandwich panel subjected to the porous core. Following the past researches, Biot’s theory was investigated by Ahmed Shah (2011) to study the STL of a poroelastic cylinder. Son and Kang (2012) also used this method to show the effect of shear waves in a transversely isotropic layer lined with a porous material, regarding the geometry as well as the boundary condition. Guo et al. (2017) analyzed the far-field acoustic radiation through a finite length cylinder submerged at a finite depth from the water surface. For this purpose, the structure was excited by pressure fluctuation because of the turbulent boundary layer (TBL). Talebitooti et al. (2016b) determined the STL of an infinitely long composite cylinder. They also added a porous material into an infinite cylinder under various boundary configurations (Talebitooti et al., 2018a). In other works (Talebitooti et al., 2018d; Talebitooti and Zarastvand, 2018c), the focus was specially placed on the vibroacoustic behavior of the infinite shells based on the first and third order shear deformation theories. They also inspected the effect of a porous material on the sound insulation property of the infinite and finite doubly curved shells (Talebitooti and Zarastvand, 2018b; Talebitooti et al., 2018e). In addition, in other works (Talebitooti et al., 2017; 2019), an optimization method based on the genetic algorithm (GA) was used to optimize the STL of the poroelastic structures. Ghassabi et al. (2019a) achieved the acoustic response of a shell structure considering the three-dimensional (3D) theory. They also inspected the acoustic performance of the nanocomposite doubly curved shells (Ghassabi et al., 2019b).

Study of the previous works clearly shows that although a lot of researchers calculated the STL of various infinitely long sandwich structures including the plate and cylinder, this issue on the finite cylindrical shells is very restricted. Furthermore, the response of a finite length cylinder under incident sound waves has not been thoroughly explored yet. For instance, as one of the few papers in this field, a finite length cylinder interlayered with porous materials and excited by pressure fluctuation due to the exterior TBL has been studied by Zhou et al. (2015) based on the simplified method, which involves some disadvantages. In this technique, the poroelastic material is replaced by an equivalent fluid. In addition, it is a complicated method with the rather low punctuality particularly in the small radius cases. As another consequence, the authors (Zhou et al., 2015) considered neither the STL of the structure nor the effect of the length on the structural response. On the contrary, a new approach is presented in this study to obtain the amount of the transmitted noise into a simply supported poroelastic cylinder based on the Full method. In order to prove the reliability of the outcomes, they are validated by enlarging the length of the poroelastic structure. Finally, the effects of various parameters on the STL of a finite cylinder are discussed.

2. Formulation

2.1. System modeling

As depicted in Figure 1(a), a simply supported cylinder subjected to the porous core encounters an acoustic wave having angle $α$ with the longitudinal axis of the structure. In addition, another configuration is plotted in Figure 1(b), wherein the side view of the acoustic media is displayed. Due to the incident of this acoustic wave, a part of the incident wave transmits into the shell, while the remaining reflects from the shell. It is noteworthy that $θ$ is the arc angle, considered as an axis in the cylindrical coordinate to extract the equations. Besides, the shell’s interior wall is assumed to be completely absorbent.

2.2. Wave propagation through the porous medium

Since Biot’s theory is considered for modeling the poroelastic material, the following equations can be presented (Biot, 1956a)

\nabla^{2} φ_{1,2}^{s} + δ_{1,2}^{2} φ_{1,2}^{s} = 0

(1a)

\nabla^{2} ψ^{s} + δ_{3}^{2} ψ^{s} = 0

(1b)

In equations (1a) and (1b),

\nabla^{2}

is the Laplacian operator in the cylindrical coordinate as below

\nabla^{2} φ_{1,2}^{s} = \frac{\partial^{2} φ_{1,2}^{s}}{\partial r^{2}} + \frac{1 \partial φ_{1,2}^{s}}{r \partial r} + \frac{1 \partial^{2} φ_{1,2}^{s}}{r^{2} \partial θ^{2}} + \frac{\partial^{2} φ_{1,2}^{s}}{\partial z^{2}}

(2)

\nabla^{2} ψ^{s} = (Δ ψ_{r}^{s} - \frac{ψ_{r}^{s}}{r^{2}} - \frac{2}{r^{2}} \frac{\partial ψ_{θ}^{s}}{\partial θ}) \hat{r} + (Δ ψ_{θ}^{s} - \frac{ψ_{θ}^{s}}{r^{2}} + \frac{2}{r^{2}} \frac{\partial ψ_{r}^{s}}{\partial θ}) \hat{θ} + (Δ ψ_{z}^{s}) \hat{z}

(3)

In order to solve equations (1a) and (b), the following assumptions are taken into account

φ_{1}^{s} = f_{1} (r) cos (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(4)

φ_{2}^{s} = f_{2} (r) cos (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(5)

ψ_{r}^{s} = g_{r} (r) sin (n θ) \cos (\frac{m π}{L} z) e^{i ω t}

(6)

ψ_{θ}^{s} = g_{θ} (r) cos (n θ) cos (\frac{m π}{L} z) e^{i ω t}

(7)

ψ_{z}^{s} = g_{z} (r) sin (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(8)

In equations (4)–(8),

f_{i (i = 1,2)} (r)

and

g_{i (i = r, θ, z)} (r)

represent the unknown amplitudes as below

f_{1} (r) = C_{1} J_{n} (a_{1 m} r) + C_{2} Y_{n} (a_{1 m} r)

(9)

f_{2} (r) = C_{3} J_{n} (a_{2 m} r) + C_{4} Y_{n} (a_{2 m} r)

(10)

g_{z} (r) = A_{3} J_{n} (a_{3 m} r) + B_{3} Y_{n} (a_{3 m} r)

(11)

g_{r} (r) = A_{1} J_{n + 1} (a_{3 m} r) + B_{1} Y_{n + 1} (a_{3 m} r)

(12)

g_{θ} (r) = - A_{1} J_{n + 1} (a_{3 m} r) - B_{1} Y_{n + 1} (a_{3 m} r)

(13)

In equations (9)–(13),

C_{i (i = 1,2,3,4)}

A_{i (i = 1,3)}

and

B_{i (i = 1,3)}

as constant coefficients are investigated, extracted due to the solution of the Bessel equations. It is useful to note that the detail description of the solution process is presented in Appendix 1. Furthermore, the below equations are used

{\begin{matrix} a_{1 m} = \sqrt{δ_{1}^{2} - {(\frac{m π}{L})}^{2}} \\ a_{2 m} = \sqrt{δ_{2}^{2} - {(\frac{m π}{L})}^{2}} \\ a_{3 m} = \sqrt{δ_{3}^{2} - {(\frac{m π}{L})}^{2}} \end{matrix}

(14)

In equation (14), the squared complex wavenumbers are presented below

\begin{array}{l} δ_{i (i = 1,2)}^{2} = {(\frac{ω}{V_{i}})}^{2} = \frac{ω^{2}}{2 (P R - Q^{2})} [P k_{22} + R k_{11} - 2 Q k_{12} - \sqrt{Δ}] \\ δ_{3}^{2} = \frac{ω^{2}}{V_{3}^{2}} = \frac{ω^{2}}{N} ((k_{11} k_{22} - k_{12}^{2}) / k_{22}) \end{array}

(15)

where

Δ = [P k_{22} + R k_{11} - 2 Q k_{12}] - 4 (P R - Q^{2}) (k_{11} k_{22} - k_{12}^{2})

. In addition, the following equations are considered

\begin{array}{l} k_{12} = ρ_{12} + \frac{i b}{ω} k_{22} = ρ_{22} - \frac{i b}{ω} \\ Q = (1 - h) E_{f} R = h E_{f} \end{array}

(16)

In equation (16),

ρ_{11} = (1 - h) ρ_{s} + ρ_{a}

and

ρ_{22} = h ρ_{f} + ρ_{a}

. However,

P = A + 2 N

, wherein

N = E_{s} / 2 (1 + ν)

and

A = ν E_{s} (1 - 2 ν) / (1 + ν)

The displacements of the solid and fluid phases can be expressed as below

u = \nabla φ^{s} + \nabla \times ψ^{s}

(17)

U = \nabla φ^{f} + \nabla \times ψ^{f}

(18)

Consequently, by expanding equations (17) and (18) in three

directions, the following equations can be derived

u_{r} (r, θ, z, t) = \frac{\partial φ^{s}}{\partial r} + \frac{1 \partial ψ_{z}^{s}}{r \partial θ} - \frac{\partial ψ_{θ}^{s}}{\partial z} = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} u_{r}^{n m} (r) cos (n θ) \sin (\frac{m π}{L} z) e^{i ω t}

(19)

u_{θ} (r, θ, z, t) = \frac{1 \partial φ^{s}}{r \partial θ} + \frac{\partial ψ_{r}^{s}}{\partial z} - \frac{\partial ψ_{z}^{s}}{\partial r} = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} u_{θ}^{n m} (r) sin (n θ) \sin (\frac{m π}{L} z) e^{i ω t}

(20)

u_{z} (r, θ, z, t) = \frac{\partial φ^{s}}{\partial z} + \frac{1}{r} (\frac{\partial (r ψ_{θ}^{s})}{\partial r} - \frac{\partial ψ_{r}^{s}}{\partial θ}) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} u_{z}^{n m} (r) cos (n θ) cos (\frac{m π}{L} z) e^{i ω t}

(21)

U_{r} (r, θ, z, t) = \frac{\partial φ^{f}}{\partial r} + \frac{1 \partial ψ_{z}^{f}}{r \partial θ} - \frac{\partial ψ_{θ}^{f}}{\partial z} = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} U_{r}^{n m} (r) cos (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(22)

U_{θ} (r, θ, z, t) = \frac{1 \partial φ^{f}}{r \partial θ} + \frac{\partial ψ_{r}^{f}}{\partial z} - \frac{\partial ψ_{z}^{f}}{\partial r} = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} U_{θ}^{n m} (r) sin (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(23)

U_{z} (r, θ, z, t) = \frac{\partial φ^{f}}{\partial z} + \frac{1}{r} (\frac{\partial (r ψ_{θ}^{f})}{\partial r} - \frac{\partial ψ_{r}^{f}}{\partial θ}) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} U_{z}^{n m} (r) cos (n θ) cos (\frac{m π}{L} z) e^{i ω t}

(24)

In equations (19)–(24),

φ^{i}

presents the scalar potential for the solid

(i = s)

and fluid

(i = f)

phases as

φ^{s} = φ_{1}^{s} + φ_{2}^{s}

and

φ^{f} = μ_{1} φ_{1}^{s} + μ_{2} φ_{2}^{s}

, wherein

μ_{i (i = 1,2)} = φ_{i}^{f} / φ_{i}^{s} = (P δ_{i}^{2} - ω^{2} k_{11}) / (ω^{2} k_{12} - Q δ_{i}^{2})

and

μ_{3} = \frac{ψ^{f}}{ψ^{s}}

. In addition,

u_{i (i = r, θ, z)}^{n} (r)

and

U_{i (i = r, θ, z)}^{n} (r)

are the displacement amplitudes for the solid and fluid phases, as listed in Appendix 2. However, the below equations for the solid stresses in three directions, and the fluid pressure is considered as

σ_{i j (i j = r, θ, z)} = 2 N e_{i j} + (A e + Q ε) δ_{i j}

(25)

s = Q e + R ε

(26)

where

δ_{i j}

is the Kronecker delta function, which has zero

(i \neq j)

and unique

(i = j)

values, respectively. Moreover,

s

presents the fluid pressure, which has a minus value in the compression mode. It is noteworthy that

e = \nabla \cdot u

and

ε = \nabla \cdot U

demonstrate the dilatations of the solid and fluid phases. As a result, the below equations are considered

σ_{r r} (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} σ_{r r}^{n} (r) cos (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(27)

σ_{r z} (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} σ_{r z}^{n} (r) cos (n θ) cos (\frac{m π}{L} z) e^{i ω t}

(28)

σ_{r θ} (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} σ_{r θ}^{n} (r) sin (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(29)

s (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} s^{n} (r) cos (n θ) sin (\frac{m π}{L} z) e^{i ω t}

(30)

The detail description of

σ_{i j (i j = r, θ, z)}^{n}

and

s^{n} (r)

can be found in Appendix 3.

2.3. Shell equations

Since first order shear deformation theory (FSDT) is considered to analyze the dynamic behavior of a finite cylinder, the below equations for the outer $(j = o)$ and inner $(j = i)$ shells are presented (Qatu, 2004)

\frac{\partial N_{z}}{\partial z} + \frac{\partial N_{θ z}}{R_{j} \partial θ} + q_{z} = ({\bar{I}}_{1} \ddot{u} + {\bar{I}}_{2} {\ddot{ψ}}_{z})

(31)

\frac{\partial N_{θ}}{R_{j} \partial θ} + \frac{\partial N_{z θ}}{\partial z} + \frac{Q_{θ}}{R_{j}} + q_{θ} = ({\bar{I}}_{1} \ddot{v} + {\bar{I}}_{2} {\ddot{ψ}}_{θ})

(32)

- \frac{N_{θ}}{R_{j}} + \frac{\partial Q_{z}}{\partial z} + \frac{\partial Q_{θ}}{R_{j} \partial θ} + q_{r} = ({\bar{I}}_{1} \ddot{w})

(33)

\frac{\partial M_{z}}{\partial z} + \frac{\partial M_{θ z}}{R_{j} \partial θ} = ({\bar{I}}_{2} \ddot{u} + {\bar{I}}_{3} {\ddot{ψ}}_{z})

(34)

\frac{\partial M_{θ}}{R_{j} \partial θ} + \frac{\partial M_{z θ}}{\partial z} = ({\bar{I}}_{2} v + {\bar{I}}_{3} {\ddot{ψ}}_{θ})

(35)

In equations (31)–(35), the superscript (double dots) presents the second derivate with respect to time. Moreover, the external forces in three directions for the inner

(r = R_{p}^{i})

and outer

(r = R_{p}^{o})

shells are considered as below

{\begin{array}{l} q_{z} = - σ_{r z} \\ q_{θ} = - σ_{r θ} \\ q_{r} = σ_{r r} + s - p_{1}^{I} - p_{1}^{R} \end{array}

(36)

{\begin{array}{l} q_{z} = σ_{r z} \\ q_{θ} = σ_{r θ} \\ q_{r} = - σ_{r r} - s + p_{2}^{T} \end{array}

(37)

In addition, the forces and moments can be determined in the following forms

[\begin{matrix} N_{α} \\ N_{β} \\ N_{α β} \\ N_{β α} \\ M_{α} \\ M_{β} \\ M_{α β} \\ M_{β α} \end{matrix}] = [\begin{matrix} {\bar{A}}_{11} & A_{12} & {\bar{A}}_{16} & A_{16} & {\bar{B}}_{11} & B_{12} & {\bar{B}}_{16} & B_{16} \\ A_{12} & {\hat{A}}_{22} & A_{26} & {\hat{A}}_{26} & B_{12} & {\hat{B}}_{22} & B_{26} & {\hat{B}}_{26} \\ {\bar{A}}_{16} & A_{26} & {\bar{A}}_{66} & A_{66} & {\bar{B}}_{16} & B_{26} & {\bar{B}}_{66} & B_{66} \\ A_{16} & {\hat{A}}_{26} & A_{66} & {\hat{A}}_{66} & B_{16} & {\hat{B}}_{26} & B_{66} & {\hat{B}}_{66} \\ {\bar{B}}_{11} & B_{12} & {\bar{B}}_{16} & B_{16} & {\bar{D}}_{11} & D_{12} & {\bar{D}}_{16} & D_{16} \\ B_{12} & {\hat{B}}_{22} & B_{26} & {\hat{B}}_{26} & D_{12} & {\hat{D}}_{22} & D_{26} & {\hat{D}}_{26} \\ {\bar{B}}_{16} & B_{26} & {\bar{B}}_{66} & {\bar{B}}_{66} & {\bar{D}}_{16} & D_{26} & {\bar{D}}_{66} & D_{66} \\ B_{16} & {\hat{B}}_{26} & B_{66} & {\hat{B}}_{66} & D_{16} & D_{26} & D_{66} & {\hat{D}}_{66} \end{matrix}] [\begin{matrix} ε_{0 α} \\ ε_{0 β} \\ ε_{α β} \\ ε_{β α} \\ k_{α} \\ k_{β} \\ k_{α β} \\ k_{β α} \end{matrix}]

(38)

[\begin{matrix} Q_{α} \\ Q_{β} \end{matrix}] = [\begin{matrix} {\bar{A}}_{55} & A_{45} \\ A_{45} & {\hat{A}}_{44} \end{matrix}] [\begin{matrix} γ_{0 α ξ} \\ γ_{0 β ξ} \end{matrix}]

(39)

In equations (38) and (39),

A_{i j}, B_{i j} and D_{i j}

present the stiffnesses, considered with the same formulas from Qatu (2004). Besides, the strains and curvatures are formulated as

\begin{array}{l} γ_{0 α β} = \frac{\partial w_{0}}{\partial α} + ψ_{α} γ_{0 β ξ} = \frac{\partial w_{0}}{\partial β} - \frac{v_{0}}{R} + ψ ε_{0 α} = \frac{\partial u_{0}}{\partial α} \\ ε_{0 β} = \frac{\partial v_{0}}{\partial β} + \frac{w_{0}}{R} ε_{0 α β} = \frac{\partial v_{0}}{\partial α} ε_{0 β α} = \frac{\partial u_{0}}{\partial β} \\ k_{α} = \frac{\partial ψ_{α}}{\partial α} k_{β} = \frac{\partial ψ_{β}}{\partial β} k_{α β} = \frac{\partial ψ_{β}}{\partial α} \\ k_{β α} = \frac{\partial ψ_{α}}{\partial β} \end{array}

(40)

2.4. Boundary conditions

By developing equations (31)–(35) for the inner and outer shells, 10 equations are obtained. However, the number of unknown parameters is higher than that of equations, so it is necessary to get some new equations. Accordingly, employing the geometrical boundary conditions, as well as the acoustic boundary conditions, which take into account coupling of the solid–fluid interactions, along with the interlayered boundary conditions, which consider the porous material–shell interfaces are required. It is noteworthy that since in this study a finite poroelastic cylinder is acoustically analyzed, the structure is modeled in such a way that the both ends of the cylinder are under the simply supported boundary condition.

2.4.1. Geometrical boundary condition

Since in this study a simply supported poroelastic cylinder is excited by an acoustic wave, the displacements and rotations of the outer and inner shells with specified geometric boundary conditions can be shown as a sum of the longitudinal and circumferential modes as below

{\begin{matrix} u^{j} \\ v^{j} \\ w^{j} \\ ψ_{z}^{j} \\ ψ_{θ}^{j} \end{matrix}} = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} {\begin{matrix} u_{0 n m}^{j} cos (\frac{m π}{L} z) cos (n θ) \\ v_{0 n m}^{j} sin (\frac{m π}{L} z) sin (n θ) \\ w_{0 n m}^{j} sin (\frac{m π}{L} z) cos (n θ) \\ ψ_{0 z n m}^{j} cos (\frac{m π}{L} z) cos (n θ) \\ ψ_{0 θ n m}^{j} sin (\frac{m π}{L} z) sin (n θ) \end{matrix}} e^{i (ω t)}, j = o, i

(41)

2.4.2. Acoustic boundary condition

As obviously defined, it is essential to satisfy the continuity condition between fluid particle and inner $(r = R_{i})$ and outer $(r = R_{o})$ shells; likewise, the following equations can be considered

\frac{\partial (p_{1}^{I} + p_{1}^{R})}{\partial r} = - ρ_{o} \frac{\partial^{2} w^{o}}{\partial t^{2}}

(42)

\frac{\partial p_{2}^{T}}{\partial r} = - ρ_{i} \frac{\partial^{2} w^{i}}{\partial t^{2}}

(43)

2.4.3. Interlayered boundary condition

At the outer $(r = R_{p}^{o})$ and inner $(r = R_{i})$ interfaces between the porous core and the shell, the below equations should be satisfied

{\begin{array}{l} u_{z} (r, θ, z) = u^{o} (z, θ, - \frac{h_{o}}{2}) = u_{0}^{o} (z, θ) - \frac{h_{o}}{2} ψ_{z}^{o} (z, θ) \\ v_{θ} (r, θ, z) = v^{o} (z, θ, - \frac{h_{o}}{2}) = v_{0}^{o} (z, θ) - \frac{h_{o}}{2} ψ_{θ}^{o} (z, θ) \\ u_{r} (r, θ, z) = w^{o} (z, θ, - \frac{h_{o}}{2}) = w_{0}^{o} (z, θ) \\ U_{r} (r, θ, z) = w^{o} (z, θ, - \frac{h_{o}}{2}) = w_{0}^{o} (z, θ) \end{array}

(44)

{\begin{array}{l} u_{z} (r, θ, z) = u^{i} (z, θ, \frac{h_{i}}{2}) = u_{o}^{i} (z, θ) + \frac{h_{i}}{2} ψ_{z}^{i} (z, θ) \\ v_{θ} (r, θ, z) = v^{i} (z, θ, \frac{h_{i}}{2}) = v_{0}^{i} (z, θ) + \frac{h_{i}}{2} ψ_{θ}^{i} (z, θ) \\ u_{r} (r, θ, z) = w^{i} (z, θ, \frac{h_{i}}{2}) = w_{0}^{i} (z, θ) \\ U_{r} (r, θ, z) = w^{i} (z, θ, \frac{h_{i}}{2}) = w_{0}^{i} (z, θ) \end{array}

(45)

2.5. Acoustic pressures

In this section, the procedure is followed to determine the acoustic pressures for the special case of a simply supported cylinder. Herewith, the below wave equation is applied

(\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1 \partial^{2}}{r^{2} \partial θ^{2}} + \frac{\partial^{2}}{\partial z^{2}}) p = \frac{1 \partial^{2} p}{c^{2} \partial t^{2}}

(46)

Consequently, for solving equation (46), the following equation is used

p (r, θ, z, t) = R (r) Θ (θ) Z (z) e^{i ω t}

(47)

In the next step, by replacing equation (47) into (46), the below equations can be deduced

\frac{\partial^{2} R}{\partial r^{2}} + \frac{1 \partial R}{r \partial r} + (k_{r}^{2} - \frac{n^{2}}{r^{2}}) R = 0

(48)

\frac{d^{2} Z}{d z^{2}} + k_{z}^{2} Z = 0

(49)

\frac{d^{2} Θ}{d θ^{2}} + n^{2} Θ = 0

(50)

As discussed in equations (42) and (43), the Eulerian boundary conditions should be satisfied for the inner and outer shells. Accordingly, the acoustic modes in the longitudinal axis should be equal to the lateral vibration modes as

Z (z) = \sin (\frac{m π}{L} z)

(51)

As a result, by applying the Fourier expansion, the incident wave can be described in terms of the lateral vibration modes as below

p_{1}^{I} (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} P_{m}^{I} ε_{n} {(- i)}^{n} J_{n} (k_{1 r} r) cos (n θ) sin (\frac{m π}{L} z) e^{i (ω t)}

(52)

In equation (52),

ε_{n}

presents the Neumann factor as

ε_{n} = {\begin{matrix} 1 & n = 0 \\ 2 & n > 1 \end{matrix}

. In addition, the above equations involving

k_{1 r} = k_{1} \cos α and k_{1 z} = k_{1} \sin α

are considered as the wavenumbers, wherein

k_{1} = (ω / c_{1}) (1 / 1 + M sin (α))

shows the wavenumber of the incident wave and

M

denotes the Mach number of external flow. Furthermore,

P_{m}^{I}

is defined as

P_{m}^{I} = \frac{2 P_{0}}{L} \int_{0}^{L} e^{i (- k_{1 z} z)} \sin (\frac{m π}{L} z) d z

(53)

As depicted in Figure 1, the structure is under excitation of an acoustic wave; therefore, some waves are transmitted and the remaining ones are reflected as below

p_{1}^{R} (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} P_{1 n m}^{R} H_{n}^{2} (k_{1 m} r) cos (n θ) \sin (\frac{m π}{L} z) e^{i (ω t)}

(54)

p_{2}^{T} (r, θ, z, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} P_{2 n m}^{T} H_{n}^{1} (k_{2 m} r) cos (n θ) \sin (\frac{m π}{L} z) e^{i (ω t)}

(55)

In equations (54) and (55),

k_{j m (j = 1,2)}

are obtained as

k_{j m} = \sqrt{{(ω / c_{j})}^{2} - {(m π / L)}^{2}}

(56)

2.6. STL

In order to determine the amount of the transmitted noise through a finite poroelastic cylinder, firstly it is essential to solve the following matrix equation, obtained by developing equations (31)–(35) for the outer and inner shells along with equations (42)–(45) as

{[L]}_{20 \times 20} {[g]}_{20 \times 1} = {[F]}_{20 \times 1}

(57)

In equation (57),

[F]

is a vector, which presents the acoustic forces, and

[L]

is a coefficient matrix. In addition,

[g]

illustrates the unknown variables in the following form

[g] = [C_{1} C_{2} C_{3} C_{4} A_{3} B_{3} A_{1} B_{1} u_{0 n}^{i} v_{0 n}^{i} w_{0 n}^{i} ψ_{z n}^{i} ψ_{θ n}^{i} u_{0 n}^{o} v_{0 n}^{o} w_{0 n}^{o} ψ_{z n}^{o} ψ_{θ n}^{o} {P_{1 n}^{T} P_{2 n}^{T}]}^{T}

(58)

It is noteworthy that the unknown constants in each mode can be found by solving equation (57). Consequently, by obtaining these constants, the STL coefficient is achieved because of dividing the input sound power

(W^{I})

over the transmitted sound power

(W^{T})

per unit length of the cylinder as below

STL = 10 \log_{10} (\frac{W^{I}}{W^{T}}) = 10 \log_{10} (\frac{W^{I}}{\sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} W_{m n}^{T}})

(59)

In equation (59), the transmitted sound power is presented as

W_{m n}^{T} = \frac{1}{2} R e {\int_{0}^{L} \int_{0}^{2 π} p_{2}^{T} \frac{\partial}{\partial t} {(w_{0}^{j})}^{*} R d θ d z}

(60)

where

R e {}

and

{(\dots)}^{*}

show the real part and the complex conjugate of the argument, respectively. Furthermore, the below equation as the incident sound power is investigated

W^{I} = \frac{\cos α P_{0}^{2}}{ρ_{1} c_{1}} \times R_{o} \times L

(61)

Finally, by substituting equations (60) and (61) into (59), STL of a simply supported poroelastic cylinder can be defined as

STL = - 10 \log_{10} \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} \frac{1}{2} \frac{R e {P_{2 n m}^{T} H_{n}^{1} (k_{2 m} R) {(i ω w_{0 n m}^{j})}^{*}} ρ_{1} c_{1} π}{ε_{n} \cos (α) P_{0}^{2}}

(62)

3. Results

3.1. Convergence checking

In order to meet enough accuracy in the calculation of the STL, sufficient number of modes should be employed in the solution process. Figure 2 shows an algorithm for this convergence procedure. Herewith, in each frequency, two convergence loops for the longitudinal and circumferential modes, together with a convergence condition for each loop, are necessary. Thus, when these conditions are met, the loops are terminated and then STL is calculated at this corresponding frequency. Accordingly, in the following, some various 3D configurations are plotted in Figures 3–5 to, respectively, show the effects of the excitation frequency, length and radius of the structure on the convergence. For this purpose, the geometrical properties of the shell and the porous core are listed in Tables 1 and 2.

Table 1.

Geometrical and acoustical properties of the used material.

Material (fluid)	Cylindrical shell	Cavity	Ambient
Material (fluid)	Aluminum	Cavity	Ambient
Density (kg/m³)	2700	1.21	1.21
Young’s modulus (GPa)	70	—	—
Poisson’s ratio	0.33	—	—
Sound speed (m/s)	—	343	343
Inner radius R_i(m)	1.8	—	—
Outer radius R₀(m)	1.84	—	—
Incidence angle	60°	—	—

Table.2.

Poroelastic core properties.

Parameters	Poroelastic shell
Material	Polyurethane
Porous thickness (mm)	38
Inner radius $R_{p}^{i} (m)$	1.801
Outer radius $R_{p}^{o} (m)$	1.839
Porosity (h)	0.9
Bulk density of the solid phase $ρ_{1} (kg / m^{3})$	30
Bulk Young’s modulus E_m (Pa)	8 × 10⁵
Bulk Poisson’s coefficient ν	0.4
Tortuosity a_∞	7.9
Flow resistivity σ	25 × 10³
Loss factor η	0.265

As illustrated in Figure 3, further modes in both of the circumferential and longitudinal directions should be involved in this procedure to converge the STL at higher frequencies. This is predictable because once the solution is converged at a given frequency, it should be converged at all frequencies lower than that frequency.

Figure 4 shows the influence of increasing the length on the STL convergence of the structure. Although by increasing this parameter the number of the essential longitudinal modes is increased, no change is created in the number of circumferential modes. On the contrary, by enhancing the radius of the shell, the number of the required circumferential modes is increased, whereas the number of the longitudinal one remains constant, as displayed in Figure 5.

In Figure 6, the transverse displacement of the structure is plotted at various frequencies. The achieved outcomes indicate that by enhancing the frequency, the number of the required longitudinal and circumferential modes is increased.

3.2. Model verification

As obviously defined in literature, although various works can be found through power transmission of the infinitely long cylindrical shells, this study analyzes the acoustic behavior of a finite length cylinder subjected to the porous core. Herewith, in order to validate the formulation, by increasing the length of the finite configuration, the corresponding outcomes are compared with those of infinitely long ones.

As depicted in Figure 7, the obtained results from the present study are compared with those of Liu and He (2015), Talebitooti et al. (2016a) and Zhou et al. (2014) for a poroelastic cylinder with a radius of $R_{o} = 1.84 (m)$ . The trend of curves demonstrates that by enhancing the length to infinite, the STL of the present study is approached to that of an infinite cylinder. Herewith, although there is an admissible agreement between present outcomes and those of Talebitooti et al. (2016a), some discrepancies can be observed. It is noteworthy that Talebitooti et al. (2016a) have offered their results based on the extended full method (EFM) against the Full method, used in this study. Although the EFM considers only the circumferential modes to analyze the acoustic behavior of an infinitely long cylinder, this study applies the longitudinal modes in addition to circumferential ones to obtain the STL of a finite cylinder. Furthermore, in STL comparison between present results and those of Liu and He (2015), although the trend of both curves is the same, the results are not completely adapted to each other due to the simplified method, applied by Liu and He (2015). Besides, it is possible to attribute these discrepancies to the differences in the mass of the two configurations and in derivation of the finite and infinite shell equations. Therefore, the more of the observed discrepancies in this figure are expected to be due to the difference between the Full method and the simplified method (Liu and He, 2015). In fact, the simplified method neglects some of the weakened sound waves in the structure. In this configuration, the results are also compared with Zhou et al. (2014). Since the authors (Zhou et al., 2014) have offered their outcomes based on the simplified method similar to those presented by Liu and He (2015), the same trends can be observed from both results. It is noteworthy that in this figure, the locations of the ring and coincidence frequencies of the STL curve are also highlighted.

In order to ascertain the accuracy of the current formulation for structure with other radii, Figure 8 is presented to show a new comparison between the present results and those of infinite shells (Talebitooti et al., 2016a) with a radius of $R_{o} = 0.2 (m)$ . To achieve this end, the length of the poroelastic structure is extended to infinite. As illustrated, in the low frequency range where a few longitudinal modes are excited, two curves are overlapped. On the contrary, at the mid and high frequency ranges, where more longitudinal modes are included, STL of the finite cylinder appears in a lower level than the infinite one because of considering more transmitting energy involving longitudinal wave resonances.

3.3. The influence of length of structure

As shown in Figure 9, the effect of the length of the structure on the amount of the transmitted noise through a simply supported cylinder subjected to the porous core is presented and discussed. For this purpose, the initial length of the structure $(L_{i})$ is assumed to be $40 (m)$ . Although by increasing the length no remarkable changes can be seen at low frequencies, STL of the structure is increased at the high frequency domain. In fact, there is a direct effect between the length of the structure and STL curve at the high frequency domain over $10 (kHz)$ so that by increasing the length, it is significantly enhanced because of reducing the amplitude of vibration. On the contrary, at the high frequency domain, the structure is not able to react corresponding to excitation. Although in this region the ratio of length to wavelength has a remarkable value, by increasing the length, it is more enhanced and leads to increase in the amount of the STL. Besides, it is noteworthy that increasing the length causes the high frequency domain to occur at the lower frequencies.

3.4. The influence of shell radius

Figure 10 presents the influence of various radii on the STL of a finite cylinder. As obviously defined in this configuration, by varying the radius, the STL of the structure is changed in stiffness and mass control regions, whereas it remains constant in the coincidence control domain. In fact, by reducing the radius, the stiffness of structure is enhanced and leads to decrease in the vibrational acceleration of the shell and the amount of the transmitted sound into the structure. Hence, the STL is enhanced in this domain (stiffness control) below the ring frequency. In addition, at the mid frequency region between the ring and coincidence frequencies, which is known as mass control, the mass of the structure is considered as a key parameter. Since there is a direct effect between the radius and the mass, then by reducing the radius, it is decreased. This issue enhances the amount of the transmitted sound into the structure. On the contrary, the STL of a finite poroelastic cylinder is not sensitive to radius at the high frequency region over $5 kHz$ . In description of the fact of this trend, it is noteworthy that the frequency region over coincidence frequency is nominated as the coincidence control domain, wherein the damping effect plays an important role. By varying the radius, the damping of the structure is not varied; therefore, no remarkable change is created on the STL curves.

3.5. The effect of the porous thickness

As indicated in Figure 11, the vibroacoustic performance of a finite structure can be improved by varying the porous thickness. For this purpose, the initial porous thickness of the structure $(h_{p_{i}})$ is assumed to be $10 mm$ . Generally, the positive effect of the poroelastic material is highlighted at the high frequency domain, wherein the wavelengths are short enough. Based on the results, this issue is easily confirmed. It is shown that although the acoustic characteristic of the structure against the transmitted noise is not enhanced by varying the porous thickness at the frequency domain below $1500 Hz$ , wherein the wavelengths are long enough, the significant improvement can be seen over this region. It is noteworthy that the rate of this variation is very impressive in this range of frequency. On the contrary, at the low frequency domain, the increment of the thickness may decrease the stiffness of the structure and leads to transfer some acoustic waves through the structure.

4. Concluding remarks

In this study, the vibroacoustic performance of a simply supported cylinder interlayered with porous materials was presented and discussed. In fact, a new approach was presented to model the poroelastic core in a finite cylinder based on the Full method. The main aim was devoted to inspect the effect of the length of the structure in the formulation, contrary to the infinite cylindrical shells. For this purpose, an analytical solution was proposed, wherein the displacements and the acoustic pressures were developed based on the circumferential and longitudinal modes. Subsequently, the model was validated with the obtained results of an infinite cylinder. It was shown that the STL curve of a finite length cylinder is approached to an infinite one when the length increases. This fact confirmed the accuracy of the formulation and the numerical code, applied in this study. Furthermore, the results indicated that changing the length and radius of the structure would affect the required modes for convergence in the longitudinal and circumferential directions. As other consequences, enhancing the length of the shell leads to improvement in the STL at the high frequency domain. However, by increasing the radius, the STL curve behaves in the opposite way particularly at the low frequency region. In addition, the porous thickness was considered as the parameter which improved the behavior of the STL curve in regions where wavelengths are short enough.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Roohollah Talebitooti

Appendix 1

Generally, equations (1a) and (1b) are considered to analyze the wave propagation through the poroelastic material. Since these equations include a series of parameters involving scalar and vector potentials, equations (4)–(8) are assumed based on the characteristics of a finite cylinder subjected to the porous core. Subsequently, by inserting the Laplacian operator from equations (2) and (3) into (1a) and (1b), the following equations can be derived (63)

f_{1}^{″} + \frac{f_{1}^{'}}{r} + ((\frac{ω^{2}}{V_{1}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{n^{2}}{r^{2}}) f_{1} = 0

(64)

f_{2}^{″} + \frac{f_{2}^{'}}{r} + ((\frac{ω^{2}}{V_{1}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{n^{2}}{r^{2}}) f_{2} = 0

(65)

g_{z}^{″} + \frac{g_{z}^{'}}{r} + ((\frac{ω^{2}}{V_{3}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{n^{2}}{r^{2}}) g_{z} = 0

(66)

g_{r}^{″} + \frac{g_{r}^{'}}{r} + ((\frac{ω^{2}}{V_{3}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{{(n + 1)}^{2}}{r^{2}}) g_{r} - \frac{2 n}{r^{2}} g_{θ} = 0

(67)

g_{θ}^{″} + \frac{g_{θ}^{'}}{r} + ((\frac{ω^{2}}{V_{3}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{{(n + 1)}^{2}}{r^{2}}) g_{θ} + \frac{2 n}{r^{2}} g_{r} = 0

As obviously defined, equations (63)–(67) present the five Bessel differential equations. For instance, one of these equations (equation (63)) is solved, in the following form.

In order to summarize equation (63), not only it is considered that $δ_{1}^{2} = ω^{2} / V_{1}^{2}$ but also the following equation $a_{1 m}^{2} = [δ_{1}^{2} - {(m π / L)}^{2}]$ is taken into account. It is useful to note that this phrase for other equations is considered as (68)

a_{2 m}^{2} = [δ_{2}^{2} - {(\frac{m π}{L})}^{2}] and a_{3 m}^{2} = [δ_{3}^{2} - {(\frac{m π}{L})}^{2}]

Therefore, equation (63) can be rewritten as below (69)

\frac{\partial^{2} f_{1}}{\partial r^{2}} + \frac{1}{r} \frac{\partial f_{1}}{\partial r} + (a_{1 m}^{2} - \frac{n^{2}}{r^{2}}) f_{1} = 0

It is noteworthy that equation (69) has a solution as below (70)

f_{1} = C_{1} J_{n} (a_{1 m} r) + C_{2} Y_{n} (a_{1 m} r)

Similarity, the other equations can be solved. Additionally, since there is a coupling between equations (66) and (67), the following variable changes as and are considered. As a result, the below equations can be deduced (71)

g_{1}^{″} + \frac{g_{1}^{'}}{r} + ((\frac{ω^{2}}{V_{3}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{{(n + 1)}^{2}}{r^{2}}) g_{1} = 0

(72)

g_{2}^{″} + \frac{g_{1}^{'}}{r} + ((\frac{ω^{2}}{V_{3}^{2}} - {(\frac{m π}{L})}^{2}) - \frac{{(n - 1)}^{2}}{r^{2}}) g_{2} = 0

Finally, by solving these two equations when $| m π V_{1} / L | < ω$ , equations (9)–(13) can be obtained.

Appendix 2

In order to extract the displacements of the solid and fluid phases of the porous core, the below equations are, respectively, considered (73)

u = \nabla φ^{s} + \nabla \times ψ^{s}

(74)

U = \nabla φ^{f} + \nabla \times ψ^{f}

where

\nabla φ^{i (i = s, f)} = \frac{\partial φ^{i}}{\partial r} r + \frac{1}{r} \frac{\partial φ^{i}}{\partial θ} θ + \frac{\partial φ^{i}}{\partial z} z

and

\nabla \times ψ^{i (i = s, f)} = (\frac{1 \partial ψ_{z}}{r \partial θ} - \frac{\partial ψ_{θ}}{\partial z}) r + (\frac{\partial ψ_{r}}{\partial z} - \frac{\partial ψ_{z}}{\partial r}) θ + (\frac{\partial ψ_{θ}}{\partial r} - \frac{1}{r} \frac{\partial ψ_{r}}{\partial θ}) z

Since these displacements should be developed in $r, θ, and z$ directions, equations (19)–(24) can be deduced. To clear this issue, the solution procedure is followed to analyze one of these equations as below.

To develop equation (73) for the solid phase in the $r$ direction, it is essential to sum the first terms of the gradient and curl operators in the following form (75)

u_{r} (r, θ, z) = \frac{\partial φ^{s}}{\partial r} + (\frac{1 \partial ψ_{z}^{s}}{r \partial θ} - \frac{\partial ψ_{θ}^{s}}{\partial z})

Consequently, by replacing the assumed values for scalar and vector potentials based on equations (4)–(8) into (75), the above relationship is obtained (76)

\begin{array}{l} u_{r}^{n m} (r) = C_{1} \frac{\partial}{\partial r} J_{n} (a_{1 m} r) + C_{2} \frac{\partial}{\partial r} Y_{n} (a_{1 m} r) + C_{3} \frac{\partial}{\partial r} J_{n} (a_{2 m} r) + C_{4} \frac{\partial}{\partial r} Y_{n} (a_{2 m} r) + A_{3} \frac{n}{r} J_{n} (a_{3 m} r) \\ + B_{3} \frac{n}{r} Y_{n} (a_{3 m} r) - \frac{m π}{L} A_{1} J_{n + 1} (a_{3 m} r) - \frac{m π}{L} B_{1} Y_{n + 1} (a_{3 m} r) \end{array}

Similar to the applied technique, the other displacements are determined as below (77)

\begin{array}{l} u_{θ}^{n m} (r) = - [C_{1} \frac{n}{r} J_{n} (a_{1 m} r) + C_{2} \frac{n}{r} Y_{n} (a_{1 m} r) + C_{3} \frac{n}{r} J_{n} (a_{2 m} r) + C_{4} \frac{n}{r} Y_{n} (a_{2 m} r) \\ + A_{3} \frac{\partial}{\partial r} J_{n} (a_{3 m} r) + B_{3} \frac{\partial}{\partial r} Y_{n} (a_{3 m} r) + \frac{m π}{L} A_{1} J_{n + 1} (a_{3 m} r) + \frac{m π}{L} B_{1} Y_{n + 1} (a_{3 m} r)] \end{array}

(78)

\begin{array}{l} u_{z}^{n m} (r) = [\frac{m π}{L} C_{1} J_{n} (a_{1 m} r) + \frac{m π}{L} C_{2} Y_{n} (a_{1 m} r) + \frac{m π}{L} C_{3} J_{n} (a_{2 m} r) + \frac{m π}{L} C_{4} Y_{n} (a_{2 m} r) \\ - A_{1} (\frac{(n + 1)}{r} + \frac{\partial}{\partial r}) J_{n + 1} (a_{3 m} r) - B_{1} (\frac{(n + 1)}{r} + \frac{\partial}{\partial r}) Y_{n + 1} (a_{3 m} r)] \end{array}

(79)

\begin{array}{l} U_{r}^{n m} (r) = C_{1} μ_{1} \frac{\partial}{\partial r} J_{n} (a_{1 m} r) + C_{2} μ_{1} \frac{\partial}{\partial r} Y_{n} (a_{1 m} r) + C_{3} μ_{2} \frac{\partial}{\partial r} J_{n} (a_{2 m} r) + C_{4} μ_{2} \frac{\partial}{\partial r} Y_{n} (a_{2 m} r) \\ + A_{3} μ_{3} \frac{n}{r} J_{n} (a_{3 m} r) + B_{3} μ_{3} \frac{n}{r} Y_{n} (a_{3 m} r) - \frac{m π}{L} A_{1} μ_{3} J_{n + 1} (a_{3 m} r) - \frac{m π}{L} B_{1} μ_{3} Y_{n + 1} (a_{3 m} r) \end{array}

(80)

\begin{array}{l} U_{θ}^{n m} (r) = - [C_{1} μ_{1} \frac{n}{r} J_{n} (a_{1 m} r) + C_{2} μ_{1} \frac{n}{r} Y_{n} (a_{1 m} r) + C_{3} μ_{2} \frac{n}{r} J_{n} (a_{2 m} r) + C_{4} μ_{2} \frac{n}{r} Y_{n} (a_{2 m} r) \\ + A_{3} μ_{3} \frac{\partial}{\partial r} J_{n} (a_{3 m} r) + B_{3} μ_{3} \frac{\partial}{\partial r} Y_{n} (a_{3 m} r) + \frac{m π}{L} A_{1} μ_{3} J_{n + 1} (a_{3 m} r) + \frac{m π}{L} B_{1} μ_{3} Y_{n + 1} (a_{3 m} r)] \end{array}

(81)

\begin{array}{l} U_{z}^{n m} (r) = [\frac{m π}{L} C_{1} μ_{1} J_{n} (a_{1 m} r) + \frac{m π}{L} C_{2} μ_{1} Y_{n} (a_{1 m} r) + \frac{m π}{L} C_{3} μ_{2} J_{n} (a_{2 m} r) + \frac{m π}{L} C_{4} μ_{2} Y_{n} (a_{2 m} r) \\ - A_{1} μ_{3} (\frac{(n + 1)}{r} + \frac{\partial}{\partial r}) J_{n + 1} (a_{3 m} r) - B_{1} μ_{3} (\frac{(n + 1)}{r} + \frac{\partial}{\partial r}) Y_{n + 1} (a_{3 m} r)] \end{array}

Appendix 3

(82)

\begin{array}{l} σ_{r r}^{n m} (r) = [2 N {C_{1} \frac{\partial^{2}}{\partial r^{2}} J_{n} (a_{1 m} r) + C_{2} \frac{\partial^{2}}{\partial r^{2}} Y_{n} (a_{1 m} r) + C_{3} \frac{\partial^{2}}{\partial r^{2}} J_{n} (a_{2 m} r) + C_{4} \frac{\partial^{2}}{\partial r^{2}} Y_{n} (a_{2 m} r) \\ + A_{3} (- \frac{n}{r^{2}} + \frac{n}{r} \frac{\partial}{\partial r}) J_{n} (a_{3 m} r) + B_{3} (- \frac{n}{r^{2}} + \frac{n}{r} \frac{\partial}{\partial r}) Y_{n} (a_{3 m} r) - \frac{m π}{L} A_{1} \frac{\partial}{\partial r} J_{n + 1} (a_{3 m} r) - \frac{m π}{L} B_{1} \frac{\partial}{\partial r} Y_{n + 1} (a_{3 m} r)} \\ - (A + μ_{1} Q) δ_{1}^{2} (C_{1} J_{n} (a_{1 m} r) + C_{2} Y_{n} (a_{1 m} r)) - (A + μ_{2} Q) δ_{2}^{2} (C_{3} J_{n} (a_{2 m} r) + C_{4} Y_{n} (a_{2 m} r))] \end{array}

(83)

\begin{array}{l} σ_{r z}^{n m} (r) = [N {2 \frac{m π}{L} C_{1} \frac{\partial}{\partial r} J_{n} (a_{1 m} r) + 2 \frac{m π}{L} C_{2} \frac{\partial}{\partial r} Y_{n} (a_{1 m} r) + 2 \frac{m π}{L} C_{3} \frac{\partial}{\partial r} J_{n} (a_{2 m} r) \\ + 2 \frac{m π}{L} C_{4} \frac{\partial}{\partial r} Y_{n} (a_{2 m} r) + \frac{m π}{L} A_{3} \frac{n}{r} J_{n} (a_{3 m} r) + \frac{m π}{L} B_{3} \frac{n}{r} Y_{n} (a_{3 m} r) \\ + A_{1} (- {(\frac{m π}{L})}^{2} + \frac{(n + 1)}{r^{2}} - \frac{(n + 1)}{r} \frac{\partial}{\partial r} - \frac{\partial^{2}}{\partial r^{2}}) J_{n + 1} (a_{3 m} r) + B_{1} (- {(\frac{m π}{L})}^{2} + \frac{(n + 1)}{r^{2}} - \frac{(n + 1)}{r} \frac{\partial}{\partial r} - \frac{\partial^{2}}{\partial r^{2}}) Y_{n + 1} (a_{3 m} r)] \end{array}

(84)

\begin{array}{l} σ_{r θ}^{n m} (r) = [N {C_{1} 2 n (\frac{1}{r^{2}} - \frac{1}{r} \frac{\partial}{\partial r}) J_{n} (a_{1 m} r) + C_{2} 2 n (\frac{1}{r^{2}} - \frac{1}{r} \frac{\partial}{\partial r}) Y_{n} (a_{1 m} r) + C_{3} 2 n (\frac{1}{r^{2}} - \frac{1}{r} \frac{\partial}{\partial r}) J_{n} (a_{2 m} r) + C_{4} 2 n (\frac{1}{r^{2}} - \frac{1}{r} \frac{\partial}{\partial r}) Y_{n} (a_{2 m} r) \\ + A_{3} (- \frac{n^{2}}{r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} - \frac{\partial^{2}}{\partial r^{2}}) J_{n} (a_{3 m} r) + B_{3} (- \frac{n^{2}}{r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} - \frac{\partial^{2}}{\partial r^{2}}) Y_{n} (a_{3 m} r) + \frac{m π}{L} A_{1} (\frac{(n + 1)}{r} - \frac{\partial}{\partial r}) J_{n + 1} (a_{3 m} r) + \frac{m π}{L} B_{1} (\frac{(n + 1)}{r} - \frac{\partial}{\partial r}) Y_{n + 1} (a_{3 m} r)}] \end{array}

(85)

\begin{array}{l} s^{n m} = - [C_{1} δ_{1}^{2} (Q + R μ_{1}) J_{n} (a_{1 m} r) + C_{2} δ_{1}^{2} (Q + R μ_{1}) Y_{n} (a_{1 m} r) + \\ C_{3} δ_{2}^{2} (Q + R μ_{2}) J_{n} (a_{2 m} r) + C_{4} δ_{2}^{2} (Q + R μ_{2}) Y_{n} (a_{2 m} r)] \end{array}

References

Ahmed Shah

(2011) Flexural wave propagation in coated poroelastic cylinders with reference to fretting fatigue. Journal of Vibration and Control 17: 1049–1064.

Biot

(1956a) Theory of elastic waves in a fluid-saturated porous solid. I. Low frequency range. Journal of Acoustic Society of America 28: 168–178.

Biot

(1956b) Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. Journal of Acoustic Society of America 28(2): 179–191.

Bolton

Shiau

Kang

(1996) Sound transmission through multi-panel structures lined with elastic porous materials. Journal of Sound and Vibration 191(3): 317–347.

Guo

Zhu

, et al. (2017) Vibration and acoustic radiation of a finite cylindrical shell submerged at finite depth from the free surface. Journal of Sound and Vibration 393: 338–352.

Ghassabi

Talebitooti

Zarastvand

(2019a) State vector computational technique for three-dimensional acoustic sound propagation through doubly curved thick structure. Computer Methods in Applied Mechanics and Engineering 352: 324–344.

Ghassabi

Zarastvand

Talebitooti

(2019b) Investigation of state vector computational solution on modeling of wave propagation through functionally graded nanocomposite doubly curved thick structures. Engineering with Computers 1–7.

Liu

(2015) On sound transmission through double-walled cylindrical shells lined with poroelastic material: comparison with Zhou’s results and further effect of external mean flow. Journal of Sound and Vibration 358: 192–198.

Qatu

(2004) Vibration of Laminated Shells and Plates. Elsevier.

10.

Son

Kang

(2012) Shear wave propagation in a layered poroelastic structure. Wave Motion 49: 490–500.

11.

Talebitooti

Daneshjou

Kornokar

(2016a) Three dimensional sound transmission through poroelastic cylindrical shells in the presence of subsonic flow. Journal of Sound and Vibration 363: 380–406.

12.

Talebitooti

Zarastvand

Gheibi

(2016b) Acoustic transmission through laminated composite cylindrical shell employing third order shear deformation theory in the presence of subsonic flow. Composite Structures 157: 95–110.

13.

Talebitooti

Gohari

Zarastvand

(2017) Multi objective optimization of sound transmission across laminated composite cylindrical shell lined with porous core investigating non-dominated sorting genetic algorithm. Aerospace Science and Technology 69: 269–280.

14.

Talebitooti

Choudari Khameneh

Zarastvand

, et al (2018a) Investigation of three-dimensional theory on sound transmission through compressed poroelastic sandwich cylindrical shell in various boundary configurations. Journal of Sandwich Structures & Materials 21: 2313–2357.

15.

Talebitooti

Zarastvand

(2018b) The effect of nature of porous material on diffuse field acoustic transmission of the sandwich aerospace composite doubly curved shell. Aerospace Science and Technology 78: 157–170.

16.

Talebitooti

Zarastvand

(2018c) Vibroacoustic behavior of orthotropic aerospace composite structure in the subsonic flow considering the third order shear deformation theory. Aerospace Science and Technology 75: 227–236.

17.

Talebitooti

Zarastvand

Gohari

(2018d) Investigation of power transmission across laminated composite doubly curved shell in the presence of external flow considering shear deformation shallow shell theory. Journal of Vibration and Control 24: 4492–4504.

18.

Talebitooti

Zarastvand

Gohari

(2018e) The influence of boundaries on sound insulation of the multilayered aerospace poroelastic composite structure. Aerospace Science and Technology 80: 452–471.

19.

Talebitooti

Zarastvand

Darvishgohari

(2019) Multi-objective optimization approach on diffuse sound transmission through poroelastic composite sandwich structure. Journal of Sandwich Structures & Materials p.1099636219854748.

20.

Zhou

Bhaskar

Zhang

(2014) The effect of external mean flow on sound transmission through double-walled cylindrical shells lined with poroelastic material. Journal of Sound and Vibration 333: 1972–1990.

21.

Zhou

Bhaskar

Zhang

(2015) Sound transmission through double cylindrical shells lined with porous material under turbulent boundary layer excitation. Journal of Sound and Vibration 357: 253–268.