**5. Applying full method to two-dimensional problem**

38 Wave Processes in Classical and New Solids

Figure 1 shows a schematic of the cylindrical double shell of infinite length subjected to a plane

which the subscripts *i* and *e* represent the inner and outer shells. A concentric layer of porous material is installed between the shells. The acoustic media in the outside and the inside of the

**Figure 1.** Schematic diagram of the double-walled cylindrical composite shell lined with porous materials

shell are represented by density and speed of sound: 1 1 *s c*, outside and 3 3 *s c*, inside.

. The radii and the thicknesses of the shells are *Ri e*, and *i e*, *h* in

**4. Model specification** 

wave with an incidence angle

For a two-dimensional problem as shown in the *x y* plane of Fig. 2, the potential of the incident wave can be expressed as [1]:

$$\mathfrak{N}\_{i} = e^{-j(\xi\_{x}x + \xi\_{1y}y)} \tag{12}$$

where 1 sin *<sup>x</sup>* , 1 1 cos *<sup>y</sup>* , 1 1 / *c* , 1*c* is the speed of sound in incident Medium, is the angle of incidence.

**Figure 2.** Illustration of wave propagation in the porous layer

Three kinds of the waves propagate in porous material, therefore six traveling waves, which have the same trace wave numbers, are induced by an oblique incident wave in a finite depth layer of porous material, as shown in Fig. 2. The *x* and *y* direction components of the displacements and stresses of the solid and fluid phases were derived by Bolton et. al. [1]. The displacements in the solid phase are:

$$
\hat{\mu}\_{\mathbf{x}} = j\tilde{\varepsilon}\_{\mathbf{x}} e^{-j\tilde{\varepsilon}\_{\mathbf{x}} \mathbf{x}} \left[ \frac{D\_1}{\tilde{\varepsilon}\_{\alpha}^2} e^{-j\tilde{\varepsilon}\_{\alpha y} y} + \frac{D\_2}{\tilde{\varepsilon}\_{\alpha}^2} e^{j\tilde{\varepsilon}\_{\alpha y} y} + \frac{D\_3}{\tilde{\varepsilon}\_{\beta}^2} e^{-j\tilde{\varepsilon}\_{\beta y} y} + \frac{D\_4}{\tilde{\varepsilon}\_{\beta}^2} e^{j\tilde{\varepsilon}\_{\beta y} y} \right]
$$

$$
$$

$$\hat{\boldsymbol{u}}\_y = j \mathbf{e}^{-j\boldsymbol{\xi}\_x \mathbf{x}} \left[ \frac{\tilde{\boldsymbol{\xi}}\_{\alpha y}}{\tilde{\boldsymbol{\xi}}\_{\alpha}^2} \mathbf{D}\_1 \mathbf{e}^{-j\boldsymbol{\xi}\_{xy} \mathbf{y}} - \frac{\tilde{\boldsymbol{\xi}}\_{\alpha y}}{\tilde{\boldsymbol{\xi}}\_{\alpha}^2} \mathbf{D}\_2 \mathbf{e}^{j\boldsymbol{\xi}\_{xy} \mathbf{y}} + \frac{\tilde{\boldsymbol{\xi}}\_{\beta y}}{\tilde{\boldsymbol{\xi}}\_{\beta}^2} \mathbf{D}\_3 \mathbf{e}^{-j\boldsymbol{\xi}\_{\beta y} \mathbf{y}} - \frac{\tilde{\boldsymbol{\xi}}\_{\beta y}}{\tilde{\boldsymbol{\xi}}\_{\beta}^2} \mathbf{D}\_4 \mathbf{e}^{j\boldsymbol{\xi}\_{\beta y} \mathbf{y}} \right]$$

$$+ j \frac{\tilde{\boldsymbol{\xi}}\_x}{\tilde{\boldsymbol{\xi}}\_t^2} e^{-j\boldsymbol{\xi}\_x \mathbf{x}} \left[ \mathbf{D}\_5 e^{-j\boldsymbol{\xi}\_y \mathbf{y}} + \mathbf{D}\_6 e^{j\boldsymbol{\xi}\_{yy} \mathbf{y}} \right] \tag{14}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 41

> 

 

, and

2

(20)

*p t*

*dx* (21)

(22)

. The Inertia term, *I* ,

, *j* 1 ,

, 2 2

 

 

2 2

 

*ty t x*

11 12 2

22 12 22 12

 

*<sup>h</sup> dW x u Wx*

 

(24)

(25)

(26)

*I W B jW* (23)

where 2 2 

> 

 

 

> 12 22

 .

*b*

*g*

1 2

 *y x*

11 12 2

 

 

2

,

( , ) ( ) and *j t w xt W xe t t*

 , 2 2 

 *y x*

These complex relations and six constants *D D* 1 6 have to be determined by applying boundary conditions (BCs). When the elastic porous material is bonded directly to a panel, there exist six BCs. Also, the transverse displacement and in-plane displacement at the

( ,) ( ) *j t w xt W xe p p*

<sup>ˆ</sup> ( ), *y t u Wx* <sup>ˆ</sup> ( ) and *U Wx y t* ( ) <sup>ˆ</sup> ( )( / ) <sup>2</sup>

*amb P x xy x t xP*

 

1

1

1

( )

*l l*

( ) 2 2

*l l*

*l l*

where *Ph* is the panel thickness and *Vy* is the normal acoustic particle velocity. Two BCs

ˆ ˆ 42 3 (/) (/) <sup>ˆ</sup> <sup>ˆ</sup> ( ) <sup>2</sup> *P*

> ˆ ˆ 22 3 (/) ( ) ˆ *yx x A P xt*

ˆ*amb p* is the acoustic pressures applied on the panel, *Pq* is the normal force per unit panel

( )

( )

*A Qyy* 

ˆ ˆ ( )

*I yy* 

1

*L l*

1

*L l*

1 <sup>1</sup> <sup>ˆ</sup> <sup>ˆ</sup> ( ) <sup>2</sup>

*l B Qy y* 

*l*

*l*

*L l*

 

*h p q j D I W B jW*

 

area exerted on the panel by the elastic porous material ˆ ˆ *<sup>s</sup> <sup>f</sup>*

and the extensional, coupling and bending stiffness, *A*ˆ , *B*ˆ and *D*ˆ are [8]:

*x p*

 

*P y q* 

( ) *W x <sup>t</sup>* and ( ) *W x <sup>p</sup>* are transverse and in-plane displacements. Four BCs are obtained from

 

*b*

 

22 12 22 12

neutral axis can be followed as [1]:

( ), *V jWx y t* 

are obtained from the equations of motion, followed as:

the interface compatibility:

 

The displacements in the fluid phase are:

$$
\hat{\mathcal{U}}\_x = j\xi\_x e^{-j\xi\_x x} \left[ b\_1 \frac{D\_1}{\xi\_\alpha^2} e^{-j\xi\_\alpha y} + b\_1 \frac{D\_2}{\xi\_\alpha^2} e^{j\xi\_\alpha y} + b\_2 \frac{D\_3}{\xi\_\beta^2} e^{-j\xi\_\beta y} + b\_2 \frac{D\_4}{\xi\_\beta^2} e^{j\xi\_\beta y} \right]
$$

$$
$$

$$\hat{H}\_y = j e^{-j\xi\_x x} \left[ b\_1 \frac{\xi\_{\alpha y}}{\xi\_\alpha^2} D\_1 e^{-j\xi\_{\alpha y} y} - b\_1 \frac{\xi\_{\alpha y}}{\xi\_\alpha^2} D\_2 e^{j\xi\_{\alpha y} y} + b\_1 \frac{\xi\_{\beta y}}{\xi\_\beta^2} D\_3 e^{-j\xi\_{\beta y} y} - b\_1 \frac{\xi\_{\beta y}}{\xi\_\beta^2} D\_4 e^{j\xi\_{\beta y} y} \right]$$

$$+ j \xi \frac{\xi\_x}{\xi\_t^2} e^{-j\xi\_x x} \left[ D\_5 e^{-j\xi\_{\alpha y} y} + D\_6 e^{j\xi\_{\alpha y} y} \right] \tag{16}$$

The stresses in the solid phase are:

$$\begin{aligned} \hat{\sigma}\_{y}^{s} &= e^{-j\tilde{\varepsilon}\_{x}y} \begin{bmatrix} 2\delta \frac{\tilde{\varepsilon}\_{xy}^{2}}{\tilde{\varepsilon}\_{a}^{2}} + A + b\_{1}\mathcal{K} \\ 2\delta \frac{\tilde{\varepsilon}\_{xy}^{2}}{\tilde{\varepsilon}\_{a}^{2}} + A + b\_{1}\mathcal{K} \end{bmatrix} D\_{2}e^{-j\tilde{\varepsilon}\_{ay}y} + \left(2\delta \frac{\tilde{\varepsilon}\_{ay}^{2}}{\tilde{\varepsilon}\_{a}^{2}} + A + b\_{1}\mathcal{K}\right) D\_{2}e^{j\tilde{\varepsilon}\_{ay}y} \\ + \left(2\delta \frac{\tilde{\varepsilon}\_{\beta y}^{2}}{\tilde{\varepsilon}\_{\beta}^{2}} + A + b\_{2}\mathcal{K}\right) D\_{3}e^{-j\tilde{\varepsilon}\_{\beta y}y} + \left(2\delta \frac{\tilde{\varepsilon}\_{\beta y}^{2}}{\tilde{\varepsilon}\_{\beta}^{2}} + A + b\_{2}\mathcal{K}\right) D\_{4}e^{j\tilde{\varepsilon}\_{\beta y}y} \\ + 2\delta \frac{\tilde{\varepsilon}\_{x}\tilde{\varepsilon}\_{ty}}{\tilde{\varepsilon}\_{t}^{2}} \left(D\_{5}e^{-j\tilde{\varepsilon}\_{y}y} - D\_{6}e^{j\tilde{\varepsilon}\_{y}y}\right) \end{bmatrix} \tag{17}$$

$$\begin{aligned} \hat{\boldsymbol{\tau}}\_{xy} &= e^{-j\boldsymbol{\xi}\_{xy}\boldsymbol{y}} \mathcal{S} \begin{bmatrix} 2\underline{\boldsymbol{\xi}}\_{x}\underline{\boldsymbol{\xi}}\_{ay} \left( D\_{1}e^{-j\boldsymbol{\xi}\_{ay}\boldsymbol{y}} - D\_{2}e^{j\boldsymbol{\xi}\_{ay}\boldsymbol{y}} \right) + \frac{2\underline{\boldsymbol{\xi}}\_{x}\underline{\boldsymbol{\xi}}\_{by}}{\underline{\boldsymbol{\xi}}\_{y}^{2}} \left( D\_{3}e^{-j\boldsymbol{\xi}\_{by}\boldsymbol{y}} - D\_{4}e^{j\boldsymbol{\xi}\_{by}\boldsymbol{y}} \right) \\ + \frac{\left( \underline{\boldsymbol{\xi}}\_{x}^{2} - \underline{\boldsymbol{\xi}}\_{by}^{2} \right)}{\underline{\boldsymbol{\xi}}\_{t}^{2}} \left( D\_{5}e^{-j\boldsymbol{\xi}\_{by}\boldsymbol{y}} + D\_{6}e^{j\boldsymbol{\xi}\_{by}\boldsymbol{y}} \right) \end{bmatrix} \tag{18}$$

The stresses in the fluid phase are:

$$\hat{\sigma}^f = e^{-|\xi\_z|^2} \left[ \left( \chi + b\_1 \mu \right) \mathbf{D}\_1 e^{-|\xi\_{xy} y|} + \left( \chi + b\_1 \mu \right) \mathbf{D}\_2 e^{|\xi\_{xy} y|} + \left( \chi + b\_2 \mu \right) \mathbf{D}\_3 e^{-|\xi\_{\theta \beta} y|} + \left( \chi + b\_2 \mu \right) \mathbf{D}\_4 e^{|\xi\_{\theta \beta} y|} \right] \tag{19}$$

#### Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 41

40 Wave Processes in Classical and New Solids

The displacements in the fluid phase are:

2

  *t*

2 2

*t*

 

 

*x x*

The stresses in the solid phase are:

*x*

 

*j x*

*e*

The stresses in the fluid phase are:

*y*

ˆ

*xy*

 

 2 22 2 1 23 4 <sup>ˆ</sup> *yy yy <sup>x</sup> j x y yy y jy jy jy jy <sup>y</sup> u je D e D e D e D e* 

> 2 5 6 *ty ty <sup>x</sup> j x jy jy <sup>x</sup>*

ˆ *yy y y <sup>x</sup> j x jy jy jy jy*

2 5 6 *ty ty <sup>x</sup> ty j x jy jy*

*jg e D e D e* 

*D D <sup>D</sup> <sup>D</sup> U je b e b e b e b e* 

2 2

*ty ty*

2 5 6

*x ty jy jy*

2 2

*De De*

*x ty jy jy*

*De De*

1 1 1 2 2 3 2 4 <sup>ˆ</sup> *yy y y <sup>x</sup> <sup>f</sup> j x j y j y j y j y e b De b De b De b De*

2 5 6

 

*j e De De*

 

1 2 3 4 1 12 2 2 22 2

11 12 13 14 2 22 2 <sup>ˆ</sup> *yy yy <sup>x</sup> j x y yy y j y j y j y j y U je b D e b D e b D e b D e <sup>y</sup>*

 

2 2 1 1 1 2

*y y x*

 

*ty ty*

 

  2 2 2 3 2 4

2 2 1 2 3 4

 

> 

*x y jy jy x y jy jy*

*De De De De*

*y y j y j y*

 

 

> 

> >

> > >

 

 

 

> 

*y y y y*

 

 

> 

*A b De A b De*

2 5 6 *ty ty <sup>x</sup> j x jy jy <sup>x</sup>*

*jg e D e D e*

2 2

2 2

*j x y y j y j y s*

*e A b De A b De*

 

*t*

*t*

*t*

ˆ 2 2

  

> 

> >

 

 

*y y*

 

 

 

   

 

> 

 

 

> 

> >

 

 

 

 

(14)

 

(15)

(16)

 

 

> 

 

  (17)

(18)

  (19)

 

   

$$\begin{aligned} \text{where} & \qquad \boldsymbol{\xi}\_{\alpha y} = \sqrt{\boldsymbol{\xi}\_{\alpha}^{2} - \boldsymbol{\xi}\_{x}^{2}}, & \qquad \boldsymbol{\xi}\_{\beta y} = \sqrt{\boldsymbol{\xi}\_{\beta}^{2} - \boldsymbol{\xi}\_{x}^{2}}, & \qquad \boldsymbol{\xi}\_{y} = \sqrt{\boldsymbol{\xi}\_{t}^{2} - \boldsymbol{\xi}\_{x}^{2}}, & \qquad j = \sqrt{-1}, \\\ \boldsymbol{b}\_{1} &= \frac{\left(\hat{\boldsymbol{\rho}}\_{11}\mu - \hat{\boldsymbol{\rho}}\_{12}\boldsymbol{\chi}\right)}{\left(\hat{\boldsymbol{\rho}}\_{22}\boldsymbol{\chi} - \hat{\boldsymbol{\rho}}\_{12}\boldsymbol{\mu}\right)} - \frac{\left(\boldsymbol{\phi}\mu - \boldsymbol{\chi}^{2}\right)}{\left[\boldsymbol{\alpha}^{2}\left(\hat{\boldsymbol{\rho}}\_{22}\boldsymbol{\chi} - \hat{\boldsymbol{\rho}}\_{12}\boldsymbol{\mu}\right)\right]} \boldsymbol{\xi}\_{\alpha}^{2}, & \boldsymbol{b}\_{2} = \frac{\left(\hat{\boldsymbol{\rho}}\_{11}\mu - \hat{\boldsymbol{\rho}}\_{12}\boldsymbol{\chi}\right)}{\left(\hat{\boldsymbol{\rho}}\_{22}\boldsymbol{\chi} - \hat{\boldsymbol{\rho}}\_{12}\boldsymbol{\mu}\right)} - \frac{\left(\boldsymbol{\phi}\mu - \boldsymbol{\chi}^{2}\right)}{\left[\boldsymbol{\alpha}^{2}\left(\hat{\boldsymbol{\rho}}\_{22}\boldsymbol{\chi} - \hat{\boldsymbol{\rho}}\_{12}\boldsymbol{\mu}\right)\right]} \boldsymbol{\xi}\_{\beta}^{2}, & \quad \text{and} \\\ \boldsymbol{g} = -\frac{\hat{\boldsymbol{\rho}}\_{12}}{\hat{\boldsymbol{\rho}}\_{22}}. \end{aligned}$$

These complex relations and six constants *D D* 1 6 have to be determined by applying boundary conditions (BCs). When the elastic porous material is bonded directly to a panel, there exist six BCs. Also, the transverse displacement and in-plane displacement at the neutral axis can be followed as [1]:

$$w\_p(\mathbf{x}, t) = \mathcal{W}\_t(\mathbf{x}) e^{j\alpha t} \text{ and } \quad w\_p(\mathbf{x}, t) = \mathcal{W}\_p(\mathbf{x}) e^{j\alpha t} \tag{20}$$

( ) *W x <sup>t</sup>* and ( ) *W x <sup>p</sup>* are transverse and in-plane displacements. Four BCs are obtained from the interface compatibility:

$$\hat{W}\_y = j a \mathcal{W}\_t(\mathbf{x})\_\prime \ \hat{u}\_y = \mathcal{W}\_t(\mathbf{x})\_\prime \ \hat{\mathcal{U}}\_y = \mathcal{W}\_t(\mathbf{x}) \ \text{and} \ \hat{u}\_x = \mathcal{W}\_p(\mathbf{x})(- / +) \frac{h\_p}{2} \frac{d\mathcal{W}\_t(\mathbf{x})}{d\mathbf{x}} \tag{21}$$

where *Ph* is the panel thickness and *Vy* is the normal acoustic particle velocity. Two BCs are obtained from the equations of motion, followed as:

$$(\mathbf{j} + \boldsymbol{\slash} - \mathbf{j}) \hat{\mathbf{p}}\_{amb} (-\boldsymbol{\slash} + \mathbf{j}) \boldsymbol{\epsilon}\_{\mathbf{p}} - \mathbf{j} \boldsymbol{\xi}\_{\mathbf{x}} \frac{\mathbf{h}\_{\mathbf{p}}}{2} \hat{\boldsymbol{\tau}}\_{xy} = (\hat{\mathbf{D}} \boldsymbol{\xi}\_{\mathbf{x}}^{\mathbf{4}} - \alpha^2 \mathbf{I}) \mathbf{W}\_{\mathbf{t}} - \hat{\mathbf{B}} \boldsymbol{\xi}\_{\mathbf{x}}^{\mathbf{3}} \mathbf{j} \mathbf{W}\_{\mathbf{p}} \tag{22}$$

$$(\mathbf{j} + \mathbf{/} - \mathbf{)}\hat{\mathbf{r}}\_{yx} = (\hat{A}\xi\_x^2 - \alpha^2 I)\mathbf{W}\_P - \hat{B}\xi\_x^3 j\mathbf{W}\_t \tag{23}$$

ˆ*amb p* is the acoustic pressures applied on the panel, *Pq* is the normal force per unit panel area exerted on the panel by the elastic porous material ˆ ˆ *<sup>s</sup> <sup>f</sup> P y q* . The Inertia term, *I* , and the extensional, coupling and bending stiffness, *A*ˆ , *B*ˆ and *D*ˆ are [8]:

$$I = \sum\_{l=1}^{L} \rho^{(l)} (y\_l - y\_{l-1}) \tag{24}$$

$$\hat{A} = \sum\_{l=1}^{L} \hat{\mathcal{Q}}^{(l)} (y\_l - y\_{l-1}) \tag{25}$$

$$\hat{B} = \frac{1}{2} \sum\_{l=1}^{L} \hat{\mathbb{Q}}^{(l)} (y\_l^2 - y\_{l-1}^2) \tag{26}$$

$$\hat{D} = \frac{1}{3} \sum\_{l=1}^{L} \hat{\mathbb{Q}}^{(l)} (y\_l^3 - y\_{l-1}^3) \tag{27}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 43

(33)

(34)

0

0

 , *q*

(37)

(35)

(36)

, and *<sup>z</sup> q* are

2

and the

where the subscripts *f* and *s* represents the fluid and solid phases, respectively.

2( ) <sup>2</sup> <sup>1</sup> () 0 1 1 <sup>2</sup> *I R p p I R c pp*

the Laplacian operator in the cylindrical coordinate system, and 1*c* is the speed of sound of outside medium. The wave equations in the fluid phase of the porous layer and internal

The shell motions are described by classic theory, fully considering the displacements in all

normal direction to the middle surface of the shell be *r* . Equations of motion in the axial, circumferential and radial directions of a laminated composite thin cylindrical shell in

> *N N u i e q Ii e z R i e <sup>t</sup> i e*

*NNMM v i e q I zR R zR <sup>t</sup>*

<sup>2</sup> <sup>2</sup> 2 2

 

<sup>2</sup> 2 2 , <sup>2</sup> , , ,

external pressure components. 0*<sup>u</sup>* , 0*<sup>v</sup>* , and <sup>0</sup> *<sup>w</sup>* are the displacements of the shell at the

neutral surface in the axial, circumferential, and radial directions respectively. The inertia

1 2

*I I*

*<sup>M</sup> N MM wi e q I z Rt RR z*

 

 

*i e i e i e*

*ie ie i e*

*R* is cylindrical radius. Mid-surface strain and curvature can be expressed as:

In which the subscripts *i* and *e* represent the inner and outer shells. *q*

1 , , <sup>2</sup> , ,

, , <sup>2</sup> ,, ,

11 1 ,

 

1 1 , <sup>2</sup> *i e z ie*

three directions. Let the axial coordinate be *z* , the circumferential direction be

*t*

*Rp* are the acoustic pressures of the incident and reflected waves and <sup>2</sup> is

2 0

*ie ie*

,

1 ( )

*z dz*

1 , 1, *<sup>l</sup> l <sup>L</sup> <sup>y</sup> <sup>l</sup> <sup>y</sup> <sup>l</sup>*

 

 

**7. Formulation of the problem** 

where *<sup>I</sup> <sup>p</sup>* and 1

In the external space, the wave equation becomes [3]:

space are the same as Eq. (33) with different variable names.

1 2 <sup>1</sup> *II I* where *R* 

cylindrical coordinate can be written as below [7]:

terms are followed as:

where ( )*<sup>l</sup>* is the mass density of the *l* th layer of the shell per unit midsurface area and *<sup>l</sup> y* is the distance from the midsurface to the surface of the *l* th layer having the farthest *y*

coordinate. The material constant ˆ ( )*<sup>l</sup> Q* is defined as ( ) ( ) 1 12 21 ˆ 1 *l <sup>l</sup> <sup>E</sup> <sup>Q</sup> v v* where ( ) 1 *<sup>l</sup> E* is module of

elasticity in the direction 1, and ( ) 12 *<sup>l</sup> v* and ( ) 21 *<sup>l</sup> v* are Poisson's ratios in the directions 1 and 2 of the *l* th ply, respectively. The fiber coordinates of ply is described, as 1 and 2, where direction 1 is parallel to the fibers and 2 is perpendicular to them. In BCs, the first signs are appropriate when the porous material is attached to the positive *y* facing surface of the panel, and the second signs when the porous material is attached to the negative *y* facing surface.

#### **6. Prediction of ratios of the energy**

The energy related to the waves in the fluid phase and solid phase are descript as follows [2].

The airborne wave:

$$E\_{1f} = \frac{1}{2} \left| \phi \left| \left( \mathcal{X} + b\_1 \mu \right) \cdot b\_1 \frac{\xi\_{\alpha y}^2}{\xi\_{\alpha}^2} D\_1^2 \right| \right| \tag{28}$$

$$E\_{1s} = \frac{1}{2} \left[ (1 - \phi) . \left| \left( 2\mathcal{S} \frac{\xi\_{\alpha y}^2}{\xi\_{\alpha}^2} + A + b\_1 \mathcal{Y} \right) \frac{\xi\_{\alpha y}^2}{\xi\_{\alpha}^2} D\_1^2 \right| \right] \tag{29}$$

And the frame wave:

$$E\_{3f} = \frac{1}{2} \left[ \phi \left| \left( \mathcal{X} + b\_2 \mu \right) \cdot b\_2 \frac{\xi\_{\beta y}^2}{\xi\_{\beta}^2} D\_3^2 \right| \right] \tag{30}$$

$$E\_{3s} = \frac{1}{2} \left[ (1 - \phi) \left. \left( 2\delta \frac{\xi\_{\beta \chi}^2}{\xi\_{\beta}^{\prime 2}} + A + b\_2 \chi \right) \frac{\xi\_{\beta \chi}^2}{\xi\_{\beta}^{\prime 2}} D\_3^2 \right] \right] \tag{31}$$

$$E\_{5s} = \frac{1}{2} \left[ (1 - \phi) . \middle| \begin{matrix} \mathcal{L}\mathcal{S} \begin{pmatrix} \mathcal{L}^2 \\ \mathcal{S}\_t^2 \end{pmatrix}^2 D\_5^2 \\\ \end{pmatrix} \right] \tag{32}$$

where the subscripts *f* and *s* represents the fluid and solid phases, respectively.

## **7. Formulation of the problem**

42 Wave Processes in Classical and New Solids

elasticity in the direction 1, and ( )

**6. Prediction of ratios of the energy** 

coordinate. The material constant ˆ ( )*<sup>l</sup> Q* is defined as

12 *<sup>l</sup> v* and ( )

where ( )*<sup>l</sup>* 

surface.

[2].

The airborne wave:

And the frame wave:

( ) 3 3

*l l*

is the mass density of the *l* th layer of the shell per unit midsurface area and *<sup>l</sup> y*

ˆ

1 <sup>1</sup> <sup>ˆ</sup> <sup>ˆ</sup> ( ) <sup>3</sup> *L l*

21

the *l* th ply, respectively. The fiber coordinates of ply is described, as 1 and 2, where direction 1 is parallel to the fibers and 2 is perpendicular to them. In BCs, the first signs are appropriate when the porous material is attached to the positive *y* facing surface of the panel, and the second signs when the porous material is attached to the negative *y* facing

The energy related to the waves in the fluid phase and solid phase are descript as follows

*<sup>f</sup> E bb D*

1 . 2

2

<sup>1</sup> 1 .2

1 . 2

2

<sup>1</sup> 1 .2

2

*s*

1 11 1 2

1 2 2 1 1

*<sup>f</sup> E bb D*

3 22 3 2

3 2 2 2 3

5 5 2 <sup>1</sup> 1 .2

*E D*

 

*<sup>s</sup> E Ab D* 

 

 

*<sup>s</sup> E Ab D* 

 

 

2 2

2

2

 

 

> 

2

 

 

 

> 

 

2 2

*y y*

 

2 2

2 2

*y y*

<sup>2</sup> <sup>2</sup>

*ty*

*t*

 

*y*

*y*

is the distance from the midsurface to the surface of the *l* th layer having the farthest *y*

*l D Qy y* 

1

(27)

( )

*l*

12 21

*v v* where ( )

*<sup>l</sup> v* are Poisson's ratios in the directions 1 and 2 of

1

(28)

(30)

(32)

(31)

(29)

*<sup>l</sup> E* is module of

( ) 1

*<sup>l</sup> <sup>E</sup> <sup>Q</sup>*

1

In the external space, the wave equation becomes [3]:

$$c\_1 \nabla^2 (p^I + p\_1^R) + \frac{\hat{c}^2 (p^I + p\_1^R)}{\hat{c} t^2} = 0 \tag{33}$$

where *<sup>I</sup> <sup>p</sup>* and 1 *Rp* are the acoustic pressures of the incident and reflected waves and <sup>2</sup> is the Laplacian operator in the cylindrical coordinate system, and 1*c* is the speed of sound of outside medium. The wave equations in the fluid phase of the porous layer and internal space are the same as Eq. (33) with different variable names.

The shell motions are described by classic theory, fully considering the displacements in all three directions. Let the axial coordinate be *z* , the circumferential direction be and the normal direction to the middle surface of the shell be *r* . Equations of motion in the axial, circumferential and radial directions of a laminated composite thin cylindrical shell in cylindrical coordinate can be written as below [7]:

$$\frac{\partial \mathcal{N}\_{\alpha}}{\partial z} + \frac{1}{R\_{\dot{\mathbf{i}}\_{\prime},\mathcal{E}}} \frac{\partial \mathcal{N}\_{\alpha\beta}}{\partial \theta} + q\_{\alpha\_{\dot{\mathbf{i}}\_{\prime},\mathcal{E}}} = -\hat{I}\_{\dot{\mathbf{i}}\_{\prime},\mathcal{E}} \frac{\partial^{2} u^{0}\_{\dot{\mathbf{i}}\_{\prime},\mathcal{E}}}{\partial t^{2}} \tag{34}$$

$$\frac{\partial \mathbf{N}\_{\alpha\beta}}{\partial \mathbf{z}} + \frac{1}{\mathbf{R}\_{i,\varepsilon}} \frac{\partial \mathbf{N}\_{\beta}}{\partial \boldsymbol{\theta}} + \frac{1}{\mathbf{R}\_{i,\varepsilon}} \left[ \frac{\partial \mathbf{M}\_{\alpha\beta}}{\partial \mathbf{z}} + \frac{1}{\mathbf{R}\_{i,\varepsilon}} \frac{\partial \mathbf{M}\_{\beta}}{\partial \boldsymbol{\beta}} \right] + q\_{\beta i,\varepsilon} = -\hat{\mathbf{I}}\_{i,\varepsilon} \frac{\partial^{2} \mathbf{v}\_{i,\varepsilon}^{0}}{\partial t^{2}} \tag{35}$$

$$\frac{\partial^2 M\_{\alpha}}{\partial z^2} - \frac{\partial N\_{\beta}}{R\_{i,\varepsilon}} + 2\frac{1}{R\_{i,\varepsilon}} \frac{\partial^2 M\_{\alpha\beta}}{\partial \epsilon \partial z} + \frac{1}{R\_{i,\varepsilon}^2} \frac{\partial^2 M\_{\beta}}{\partial \theta^2} + q\_{z\_{i,\varepsilon}} = -\hat{I}\_{i,\varepsilon} \frac{\partial^2 w\_{\hat{I}\_{\varepsilon},\mathcal{E}}^0}{\partial t^2} \tag{36}$$

In which the subscripts *i* and *e* represent the inner and outer shells. *q* , *q* , and *<sup>z</sup> q* are external pressure components. 0*<sup>u</sup>* , 0*<sup>v</sup>* , and <sup>0</sup> *<sup>w</sup>* are the displacements of the shell at the neutral surface in the axial, circumferential, and radial directions respectively. The inertia terms are followed as:

$$\hat{I} = \hat{I}\_1 + \hat{I}\_2 \left(\frac{1}{R}\right) \\ \text{where } \left[\hat{I}\_1, \hat{I}\_2\right] = \sum\_{l=1}^{L} \int\_{y\_{l-1}}^{y\_l} \rho^{(l)} \left[1, z\right] dz \tag{37}$$

*R* is cylindrical radius. Mid-surface strain and curvature can be expressed as:

$$\varepsilon\_{0\alpha} = \frac{\partial u^0}{\partial z}, \quad \varepsilon\_{0\beta} = \frac{1}{R} \left\{ \frac{\partial v^0}{\partial \theta} + w^0 \right\} \text{ and } \quad \gamma\_{0\alpha\beta} = \frac{\partial v^0}{\partial z} + \frac{1}{R} \frac{\partial u^0}{\partial \theta} \tag{38}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 45

(49)

, wave number in the internal cavity, is

*<sup>e</sup>* (50)

*<sup>e</sup>* (51)

@*r Ri* (52)

@*r Ri* (53)

(54)

1*c* ,

3

3

(47)

(48)

 

 

> 

, wave number in the external medium, is defined as 1

2 0

2 0

2 0

2 0

<sup>1</sup>*s* and 3*s* are densities of outside and inside acoustic media, respectively. 2*s* is the equivalent density of the porous material and can be obtained from the Simplified method as suggested in [2]. The harmonic plane incident wave *<sup>I</sup> p* can be expressed in cylindrical

> ( ) (, , ,) ( ) ( )cos( ) <sup>1</sup> <sup>0</sup> *<sup>z</sup> I n jt z p rz t pe jJ r n n nr n*

where <sup>0</sup> *p* is the amplitude of the incident wave, *n* 0,1,2,3,... indicates the circumferential

for 0 *n* and 2 for *n* 1,2,3,... , and *nJ* is the Bessel function of the

 

@*r R*

@*r R*

0 0

*n*

0

0 0

*n*

*n*

In Eqs. (44 - 46) 1

 

1 1 sin *<sup>z</sup>*

expressed as 3

coordinates as [2]:

mode number, 1 *<sup>n</sup>*

first kind of order *n* .

   , 3 1 *z z* 

 

 , 1 1 cos *<sup>r</sup>* 

> 3*c*

and in Eqs. (47 - 49) 3

 .

The boundary conditions at the two interfaces between the shells and fluid are [2]:

( ) 1 1 <sup>2</sup> *<sup>w</sup> I R <sup>e</sup> pp s r t* 

( ) 22 2 <sup>2</sup> *<sup>w</sup> T R <sup>e</sup> pp s r t* 

( ) 22 2 <sup>2</sup> *<sup>w</sup> T R <sup>i</sup> pp s r t* 

> ( ) 3 3 <sup>2</sup> *<sup>w</sup> <sup>T</sup> <sup>i</sup> p s r t*

> > 1

0

  

 , 2 2 3 33 *r z*

0

cos( )exp[ ( )] *i ni <sup>z</sup>*

sin( )exp[ ( )] *i ni <sup>z</sup>*

0 0 cos( )exp[ ( )] <sup>3</sup> <sup>0</sup> *w w n jt z i ni <sup>z</sup>*

*v v n jt z* 

*u u n jt z* 

$$\kappa\_{\alpha} = -\frac{\hat{\sigma}^2 w^0}{\hat{\sigma} z^2}, \quad \kappa\_{\beta} = \frac{1}{R} \left\{ \frac{\partial v^0}{\partial \theta} - \frac{\hat{\sigma}^2 w^0}{\hat{\sigma} \theta^2} \right\}
\text{and} \quad \kappa = \frac{1}{R} \left\{ \frac{\partial v^0}{\partial z} - 2 \frac{\hat{\sigma}^2 w^0}{\hat{\sigma} z \partial \theta} \right\} \tag{39}$$

The forces and moments are:

$$
\begin{bmatrix} N\_{\alpha} \\ N\_{\beta} \\ N\_{\alpha\beta} \\ M\_{\alpha} \\ M\_{\alpha} \\ M\_{\beta} \\ M\_{\alpha\theta} \end{bmatrix} = \begin{bmatrix} A\_{11} & A\_{12} & A\_{16} & B\_{11} & B\_{12} & B\_{16} \\ A\_{21} & A\_{22} & A\_{26} & B\_{21} & B\_{22} & B\_{26} \\ A\_{61} & A\_{62} & A\_{66} & B\_{61} & B\_{62} & B\_{66} \\ B\_{11} & B\_{12} & B\_{16} & D\_{11} & D\_{12} & D\_{16} \\ B\_{21} & B\_{22} & B\_{26} & D\_{21} & D\_{22} & D\_{26} \\ B\_{61} & B\_{62} & B\_{66} & D\_{61} & D\_{62} & D\_{66} \end{bmatrix} \begin{bmatrix} \kappa\_{0\alpha} \\ \gamma\_{0\alpha\beta} \\ \kappa\_{\alpha} \\ \kappa\_{\beta} \\ \kappa\_{\beta} \\ \kappa\_{\gamma} \end{bmatrix} \tag{40}
$$

where the extensional, coupling and bending stiffness, *Ap <sup>q</sup>* , *pq B* and *Dp q* are:

$$A\_{\vec{p}\vec{q}} = \sum\_{l=1}^{L} \mathcal{Q}\_{\vec{p}\vec{q}}^{(l)} (y\_l - y\_{l-1}) \quad \vec{p}, \tilde{q} = 1, 2, 3 \tag{41}$$

$$B\_{\tilde{p}\tilde{q}} = \frac{1}{2} \sum\_{l=1}^{L} \mathbb{Q}\_{\tilde{p}\tilde{q}}^{(l)} (y\_l^2 - y\_{l-1}^2) \quad \tilde{p}, \tilde{q} = 1, 2, 3 \tag{42}$$

$$D\_{\vec{p}\vec{q}} = \frac{1}{\mathfrak{Z}} \sum\_{l=1}^{L} \mathbb{Q}\_{\vec{p}\vec{q}}^{(l)} (y\_l^3 - y\_{l-1}^3) \quad \tilde{p}, \tilde{q} = 1, 2, 3 \tag{43}$$

( )*<sup>l</sup> Qp <sup>q</sup>* , material constant, is the function of physical properties of each ply. The displacement components of the inner and outer shell at an arbitrary distance *r* from the midsurface along the axial, the circumferential and the radial directions are [8]:

$$\mu\_{\epsilon}^{0} = \sum\_{n=0}^{\infty} \mu\_{n\epsilon}^{0} \cos(n\theta) \exp[j(\alpha t - \xi\_{1z} z)] \tag{44}$$

$$v\_{\varepsilon}^{0} = \sum\_{n=0}^{\infty} v\_{n\varepsilon}^{0} \sin(n\theta) \exp[j(\alpha t - \underline{\xi}\_{1z}\underline{z})] \tag{45}$$

$$w\_{\varepsilon}^{0} = \sum\_{n=0}^{\alpha} w\_{n\epsilon}^{0} \cos(n\theta) \exp[j(\alpha t - \xi\_{1z} z)] \tag{46}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 45

$$\mu\_i^0 = \sum\_{n=0}^{\infty} \mu\_{ni}^0 \cos(n\theta) \exp[j(\alpha t - \xi\_{3z} z)] \tag{47}$$

$$v\_i^0 = \sum\_{n=0}^{\alpha} v\_{ni}^0 \sin(n\theta) \exp[j(\alpha t - \xi\_{3z} z)] \tag{48}$$

$$w\_{\stackrel{\circ}{1}}^{0} = \underset{n=0}{\overset{\infty}{\sum}}{\overset{\infty}{w}} w\_{\stackrel{\circ}{1}\stackrel{0}{\underline{\omega}}}^{0} \cos(n\theta) \exp[j(\alpha t - \underline{\xi}\_{3\underline{z}} z)] \tag{49}$$

In Eqs. (44 - 46) 1 , wave number in the external medium, is defined as 1 1*c* , 1 1 sin *<sup>z</sup>* , 1 1 cos *<sup>r</sup>* and in Eqs. (47 - 49) 3 , wave number in the internal cavity, is expressed as 3 3*c* , 3 1 *z z* , 2 2 3 33 *r z* .

The boundary conditions at the two interfaces between the shells and fluid are [2]:

44 Wave Processes in Classical and New Solids

 

The forces and moments are:

0 <sup>0</sup> , *<sup>u</sup> z*

0

*R* 

*R* 

0

 

*N AAAB B B N AAAB B B N AAAB B B M B B BDDD M B B BDDD M B B BDDD*

where the extensional, coupling and bending stiffness, *Ap <sup>q</sup>* , *pq B* and *Dp q* are:

( )

1

*L l pq pq l l l B Qy y* 

1

*L l pq pq l l l D Qy y* 

1

along the axial, the circumferential and the radial directions are [8]:

0 0

*n*

0

0 0

*n*

0

0 0

*n*

0

*L l pq pq l l l A Qyy* 

0

0

 

0 0

(38)

(39)

(40)

*v u* 1 *z R*

0 20 <sup>1</sup> <sup>2</sup> *v w Rz z*

*p q* , 1,2,3 (41)

*p q* , 1,2,3 (42)

*p q* , 1,2,3 (43)

1

1

1

(44)

(45)

(46)

 

 

> 

 

<sup>1</sup> and *<sup>v</sup> w*

0 20 2 2 <sup>1</sup> and *v w*

0 11 12 16 11 12 16 0 21 22 26 21 22 26 61 62 66 61 62 66 0 11 12 16 11 12 16 21 22 26 21 22 26 61 62 66 61 62 66

1

1

1

( )

() 2 2

() 3 3

( )*<sup>l</sup> Qp <sup>q</sup>* , material constant, is the function of physical properties of each ply. The displacement components of the inner and outer shell at an arbitrary distance *r* from the midsurface

cos( )exp[ ( )] *e ne <sup>z</sup>*

sin( )exp[ ( )] *e ne <sup>z</sup>*

cos( )exp[ ( )] *e ne <sup>z</sup>*

*w w n jt z* 

*v v n jt z* 

*u u n jt z* 

<sup>1</sup> ( ) <sup>3</sup>

<sup>1</sup> ( ) <sup>2</sup>

 

 

2 0 2 , *w z*

$$\frac{\partial}{\partial r}(p^I + p\_1^R) = -s\_1 \frac{\partial^2 w\_e^0}{\partial t^2} \quad \text{ @ } r = R\_e \tag{50}$$

$$\frac{\partial}{\partial r}(p\_2^T + p\_2^R) = -s\_2 \frac{\partial^2 w\_e^0}{\partial t^2} \quad \text{ @ } r = R\_e \tag{51}$$

$$\frac{\partial}{\partial r}(p\_2^T + p\_2^R) = -s\_2 \frac{\partial^2 w\_{\text{j}}^0}{\partial t^2} \quad \text{ @ } r = R\_{\text{j}} \tag{52}$$

$$\frac{\partial}{\partial r}(p\_3^T) = -s\_3 \frac{\partial^2 w\_j^0}{\partial t^2} \quad \text{ @ } r = R\_j \tag{53}$$

<sup>1</sup>*s* and 3*s* are densities of outside and inside acoustic media, respectively. 2*s* is the equivalent density of the porous material and can be obtained from the Simplified method as suggested in [2]. The harmonic plane incident wave *<sup>I</sup> p* can be expressed in cylindrical coordinates as [2]:

$$p^{\rm I}(r,z,\theta,t) = p\_0 e^{j\{\alpha t - \xi\_{1z}^{\prime} z\}} \sum\_{\substack{\mathcal{D} \\ n=0}}^{\infty} \varepsilon\_{\mathcal{N}}(-j)^{\mathcal{N}} J\_n(\xi\_{1r} r) \cos(n\theta) \tag{54}$$

where <sup>0</sup> *p* is the amplitude of the incident wave, *n* 0,1,2,3,... indicates the circumferential mode number, 1 *<sup>n</sup>* for 0 *n* and 2 for *n* 1,2,3,... , and *nJ* is the Bessel function of the first kind of order *n* .

Considering the circular cylindrical geometry, the pressures are expanded as:

$$p\_1^R(r, z, \theta, t) = e^{j(\alpha t - \underline{\xi}\_{1Z} z)} \sum\_{\substack{\Sigma \\ n = 0}}^{\infty} p\_{n1}^R H\_n^2(\underline{\xi}\_{1r} r) \cos(n\theta) \tag{55}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 47

 1 0 3 3 1 11

0 0

2 ( )sin cos

In the following, the averaged TLs of the structure is calculated in terms of the 1/3 octave

Eqs. (44) to (49) and (54) to (58) are obtained in series form. Therefore, enough numbers of modes should be included in the analysis to make the solution converge. Therefore, an iterative procedure is constructed in each frequency, considering the maximum iteration number. Unless the convergence condition is met, it iterates again. When the TLs calculated at two successive calculations are within a pre-set error bound, the solution is considered to have converged. An algorithm for the calculation of TLs at each frequency is followed as:

Parametric numerical studies of transmission loss (TLs) are conducted for a double-walled composite laminated cylindrical shell lined with porous material specified as follows, considering 1/3 octave band frequency. Table 1 presents the geometrical and environmental properties of a sandwich cylindrical structure. Each layer of the laminated composite shells

 

is the maximum incident angle. Integration of Eq. (62) is conducted numerically

( ) cos( ) *T*

0

*m*

 

*n n e*

 ( ) 

the real part and the complex conjugate of the argument.

by Simpson's rule. Finally, the average TLs is obtained as:

 

for 0 *n* and 2 *<sup>n</sup>*

To consider the random incidences,

band for random incidences.

**9. Convergence algorithm** 

where 1 *<sup>n</sup>* 

where *m*

REPEAT

<sup>1</sup> 10log *<sup>n</sup> TL*

Set 1 *n n*

UNTIL <sup>6</sup> <sup>1</sup> <sup>10</sup> *n n TL TL* 

**10. Numerical results** 

Re ( )( )

2

*d*

 

can be averaged according to the Paris formula [9]:

**<sup>τ</sup>** (62)

<sup>1</sup> 10log *avg TL* **<sup>τ</sup>** (63)

(61)

for *n* 1,2,3,... . Re and the superscript represents

*n n ri n i*

 

*p H R j w sc R R p*

$$p\_{\mathbf{2}}^{T}(r,z,\theta,t) = e^{j\{\alpha t - \xi\_{2}z\}} \sum\_{\substack{\Sigma \\ n=0}}^{\infty} p\_{n\Sigma}^{T} H\_{n}^{1}(\xi\_{2r}r) \cos(n\theta) \tag{56}$$

$$p\_{\mathbf{2}}^{\mathcal{R}}(r,z,\theta,t) = e^{j\{\alpha t - \xi\_{\mathbf{2}}z\}} \sum\_{\substack{\Sigma \\ n=0}}^{\infty} p\_{n\mathbf{2}}^{\mathcal{R}} H\_{n}^{2}(\xi\_{\mathbf{2}r}r) \cos(n\theta) \tag{57}$$

$$p\_{\mathbf{3}}^{T}(r,z,\theta,t) = e^{j\{\alpha t - \frac{\varphi}{3}\mathbf{3}\_{\mathbf{3}}z\}} \sum\_{\substack{\Sigma \\ n=0}}^{\infty} p\_{n\mathbf{3}}^{T} H\_{n}^{1}(\xi\_{\mathbf{3}r}r) \cos(n\theta) \tag{58}$$

where <sup>1</sup> *Hn* and <sup>2</sup> *Hn* are the Hankel functions of the first and second kind of order *<sup>n</sup>* , 231 *zzz* , and 2 2 2 2 *r z* . The wave number, 2 , in the porous core can be obtained from the simplified method as suggested in [2].

Substitution of the expressions in the displacements of the shell (Eq. (44)-Eq.(49)) and the acoustic pressures (Eq. (54)-Eq.(58)) equations into six shell equations (Eq. (34)-Eq.(36)) and four boundary conditions (Eq. (50)-Eq.(53)) yields ten equations, which can be decoupled for each mode if the orthogonality between the trigonometric functions is utilized. These ten equations can be sorted into a form of a matrix equation as follow:

$$\left[\begin{array}{c} \textbf{Pr} \end{array}\right] \left< u\_{ni'}^{0} v\_{ni'}^{0} w\_{ni'}^{0} u\_{ne'}^{0} v\_{ne'}^{0} w\_{ne'}^{0} p\_{n1'}^{R} p\_{n2'}^{T} p\_{n2'}^{R} p\_{n3}^{T} \right>^{T} = \left\{ \textbf{Pr} \right\} \tag{59}$$

where [ ] **h** is a 10 10 matrix with components given in Appendix and **λ** is:

$$\mathbf{x}\begin{Bmatrix} \mathbf{x} \end{Bmatrix} = \left\{ 0, 0, 0, 0, 0, -p\_0 \, \varepsilon\_n (-\mathbf{j})^n J\_n(\xi\_{1r} \, \mathbf{R}\_\varepsilon) \xi\_{1r'} - p\_0 \varepsilon\_n (-\mathbf{j})^n \frac{dJ\_n(\xi\_{1r} \, \mathbf{R}\_\varepsilon)}{dr} \xi\_{1r'}, 0, 0, 0 \right\}^T \tag{60}$$

The ten unknown coefficients 1 *R <sup>n</sup> p* , 2 *T np* , 2 *R np* , 3 *T np* , <sup>0</sup> *ne <sup>u</sup>* , <sup>0</sup> *ne <sup>v</sup>* , <sup>0</sup> *wne* , <sup>0</sup> *ni <sup>u</sup>* , <sup>0</sup> *ni <sup>v</sup>* and <sup>0</sup> *wni* are obtained in terms of 0 *p* with solving Eq. (59) for each mode *n* , which then can be substituted back into Eqs. (44) to (49) and (54) to (58) to find the displacements of the shell and the acoustic pressures in series forms.

#### **8. Calculation of transmission losses (TLs)**

The transmission coefficient, ( ) , is the ratio of the amplitudes of the incident and transmitted waves. ( ) is obviously a function of the incidence angle defined by [2]:

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 47

$$\tau(\boldsymbol{\gamma}) = \sum\_{n=0}^{\infty} \frac{\mathrm{Re}\left\{ \boldsymbol{p}\_{n3}^{\mathrm{T}} \boldsymbol{H}\_{n}^{1} (\boldsymbol{\xi}\_{3r} \boldsymbol{R}\_{i}) (jow} \boldsymbol{\zeta}\_{1n}^{0})^{\*} \right\} \mathbf{s}\_{1} \mathbf{c}\_{1} \pi \boldsymbol{R}\_{i} \tag{61}$$

where 1 *<sup>n</sup>* for 0 *n* and 2 *<sup>n</sup>* for *n* 1,2,3,... . Re and the superscript represents the real part and the complex conjugate of the argument.

To consider the random incidences, ( ) can be averaged according to the Paris formula [9]:

$$\overline{\mathbf{r}} = 2 \int\_0^{\gamma\_m} \tau(\gamma) \sin \gamma \cos \gamma d\gamma \tag{62}$$

where *m* is the maximum incident angle. Integration of Eq. (62) is conducted numerically by Simpson's rule. Finally, the average TLs is obtained as:

$$\text{TL}\_{avg} = 10 \log \frac{1}{\overline{\mathbf{\pi}}} \tag{63}$$

In the following, the averaged TLs of the structure is calculated in terms of the 1/3 octave band for random incidences.

#### **9. Convergence algorithm**

Eqs. (44) to (49) and (54) to (58) are obtained in series form. Therefore, enough numbers of modes should be included in the analysis to make the solution converge. Therefore, an iterative procedure is constructed in each frequency, considering the maximum iteration number. Unless the convergence condition is met, it iterates again. When the TLs calculated at two successive calculations are within a pre-set error bound, the solution is considered to have converged. An algorithm for the calculation of TLs at each frequency is followed as:

REPEAT

46 Wave Processes in Classical and New Solids

231 *zzz*

 

Considering the circular cylindrical geometry, the pressures are expanded as:

 

 

 

 

2 2 *r z*

obtained from the simplified method as suggested in [2].

equations can be sorted into a form of a matrix equation as follow:

*R <sup>n</sup> p* , 2 *T np* , 2 *R np* , 3 *T np* , <sup>0</sup>

, and 2 2

 

The ten unknown coefficients 1

The transmission coefficient,

transmitted waves.

and the acoustic pressures in series forms.

 ( ) 

**8. Calculation of transmission losses (TLs)** 

 ( ) 

( ) <sup>1</sup> <sup>2</sup> (, , ,) ( )cos( ) <sup>1</sup> 1 1 <sup>0</sup> *jt z R R <sup>z</sup> p rz t e pH r n*

( ) <sup>2</sup> <sup>1</sup> (, , ,) ( )cos( ) <sup>2</sup> 2 2 <sup>0</sup> *jt z T T <sup>z</sup> p rz t e pH r n*

*n*

( ) <sup>2</sup> <sup>2</sup> (, , ,) ( )cos( ) <sup>2</sup> 2 2 <sup>0</sup> *jt z R R <sup>z</sup> p rz t e pH r n*

*n*

( ) <sup>3</sup> <sup>1</sup> (, , ,) ( )cos( ) <sup>3</sup> 3 3 <sup>0</sup> *jt z T T <sup>z</sup> p rz t e pH r n*

*n*

where <sup>1</sup> *Hn* and <sup>2</sup> *Hn* are the Hankel functions of the first and second kind of order *<sup>n</sup>* ,

. The wave number, 2

Substitution of the expressions in the displacements of the shell (Eq. (44)-Eq.(49)) and the acoustic pressures (Eq. (54)-Eq.(58)) equations into six shell equations (Eq. (34)-Eq.(36)) and four boundary conditions (Eq. (50)-Eq.(53)) yields ten equations, which can be decoupled for each mode if the orthogonality between the trigonometric functions is utilized. These ten

> 00 000 0 <sup>1223</sup> ,, ,,, ,, , , *<sup>T</sup> RT RT*

 **λ** (60)

obtained in terms of 0 *p* with solving Eq. (59) for each mode *n* , which then can be substituted back into Eqs. (44) to (49) and (54) to (58) to find the displacements of the shell

is obviously a function of the incidence angle

0 1 10 1 ( ) 0,0,0,0,0, ( ) ( ) , ( ) ,0,0,0

*n n n re n n re r n r dJ R p jJ R p j dr*

 

*ne <sup>u</sup>* , <sup>0</sup>

where [ ] **h** is a 10 10 matrix with components given in Appendix and **λ** is:

 

<sup>1</sup>

*n*

 

 

 

 

*nnr*

*nnr*

*nnr*

*nnr*

*ni ni ni ne ne ne n n n n uvwuvw p p p p* **h λ** (59)

*ne <sup>v</sup>* , <sup>0</sup> *wne* , <sup>0</sup>

, is the ratio of the amplitudes of the incident and

*ni <sup>u</sup>* , <sup>0</sup>

 

> 

 

  (55)

(56)

(57)

(58)

*T*

defined by [2]:

*ni <sup>v</sup>* and <sup>0</sup> *wni* are

, in the porous core can be

$$TL\_n = 10\log\frac{1}{\overline{\tau}}$$

Set 1 *n n*

UNTIL <sup>6</sup> <sup>1</sup> <sup>10</sup> *n n TL TL* 

#### **10. Numerical results**

Parametric numerical studies of transmission loss (TLs) are conducted for a double-walled composite laminated cylindrical shell lined with porous material specified as follows, considering 1/3 octave band frequency. Table 1 presents the geometrical and environmental properties of a sandwich cylindrical structure. Each layer of the laminated composite shells

are made of graphite/epoxy, see Table 2. The plies were arranged in a 0 ,45 ,90 , 45 ,0 *<sup>s</sup>*

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 49

Graphite/epoxy Glass/epoxy

Axial Modulus (GPa) 137.9 38.6 Circumferential Modulus (GPa) 8.96 8.2 Shear Modulus (GPa) 7.1 4.2 Density (kg/m3) 1600 1900 Major Poisson's Ratio (-) 0.3 0.26

Figure 6 indicates that whenever the radius of the shell descends, the TL of the shell is ascending in low frequency region. It is due to the fact that the flexural rigidity of the cylinder will be increased with reduction of shells radii. In addition, decreasing the radius of the shell leads to weight reduction and then in high frequency especially in Mass-controlled

Figure 7 shows the effect of the composite material on TL. Materials chosen for the comparison are graphite/epoxy and glass/epoxy (Table 2). The figure shows that material must be chosen properly to enhance TL at Stiffness-control zone. The results represent a desirable level of TL at Stiffness-control zone (Lower frequencies) for graphite/epoxy. It is readily seen that, in higher frequency, as a result of density of materials, the TL curves are ascending. Therefore, the TL of glass/epoxy is of the highest condition in the Mass-

It is well anticipated that increase of porous layer thickness leads to increase of TL. As illustrated in Fig. 8, a considerable increase due to thickening the porous layer is obtained. As it is well obvious from this figure, the weight increase of about 12% ( 60 *<sup>c</sup> h* mm) and 25% ( 100 *<sup>c</sup> h* mm), the averaged TL values are properly increased about 35% and 60%, respectively in broadband frequency. It is a very interesting result that can encourage

Figure 9 shows a comparison between the transmission loss for a ten-layered composite shell and an aluminum shell with the same radius and thickness. Since, the composite shell is stiffer than the aluminum one, its TL is upper than that of aluminum shell in the Stiffnesscontrolled region. However, as a result of lower density of composite shell, it does not

The effect of stacking sequence is shown in Fig. 10. Two patterns 0 ,90 ,0 ,90 ,0 *<sup>s</sup>* and

90 ,0 ,90 ,0 ,90 *<sup>s</sup>* are defined to designate stacking sequence of plies. The arrangements of layers are so effective on TLs curve, especially on Stiffness-controlled region. However,

appear to be effective as an aluminum shell in Mass-controlled region.

no clear discrepancy is depicted in Mass-controlled region.

**Table 2.** Orthotropic properties [3]

controlled region.

region the power transmission into the structure increases.

engineers to use these structures in industries.

pattern.

The results are verified by those investigated by the authors' previous work for an especial case in which the porous material properties go into fluid phase (In other world, the porosity is close to 1). The comparison of these results shown in Fig. 3, indicates a good agreement.

The calculated transmission loss for the laminated composite shell is compared with those of other authors for a special case of isotropic materials. In other word, in this model the mechanical properties of the lamina in all directions are chosen the same as an isotropic material such as Aluminum, and then the fiber angles are approached into zero. Fig. 4 compares the TL values of the special case of laminated composite walls obtained from present model and those of aluminum walls from Lee's study [2]. The results show an excellent agreement.

We are going to verify the model in behavior comparing the results of cylindrical shell in a case where the radius of the cylindrical shell becomes large or the curvature becomes negligible with the results of the flat plate done by Bolton [1] (See Fig. 5). It should be also noted that both structures sandwich a porous layer and have the same thickness. Although it is not expected to achieve the same results as the derivation of the shell equations is quite different comparing with derivation of plate equations, however, the comparison between the two curves indicate that they behave in a same trend in the broadband frequencies.


\* *in vacuo.*

**Table 1.** Geometrical and environmental properties [2]


**Table 2.** Orthotropic properties [3]

48 Wave Processes in Classical and New Solids

pattern.

agreement.

excellent agreement.

broadband frequencies.

Bulk Density of Solid

Bulk Young's Modulus\*

**Table 1.** Geometrical and environmental properties [2]

Phase\*

\* *in vacuo.*

are made of graphite/epoxy, see Table 2. The plies were arranged in a 0 ,45 ,90 , 45 ,0 *<sup>s</sup>*

The results are verified by those investigated by the authors' previous work for an especial case in which the porous material properties go into fluid phase (In other world, the porosity is close to 1). The comparison of these results shown in Fig. 3, indicates a good

The calculated transmission loss for the laminated composite shell is compared with those of other authors for a special case of isotropic materials. In other word, in this model the mechanical properties of the lamina in all directions are chosen the same as an isotropic material such as Aluminum, and then the fiber angles are approached into zero. Fig. 4 compares the TL values of the special case of laminated composite walls obtained from present model and those of aluminum walls from Lee's study [2]. The results show an

We are going to verify the model in behavior comparing the results of cylindrical shell in a case where the radius of the cylindrical shell becomes large or the curvature becomes negligible with the results of the flat plate done by Bolton [1] (See Fig. 5). It should be also noted that both structures sandwich a porous layer and have the same thickness. Although it is not expected to achieve the same results as the derivation of the shell equations is quite different comparing with derivation of plate equations, however, the comparison between the two curves indicate that they behave in a same trend in the

Material Air Composite Porous Material Composite Air Density (kg/m3) 1.21 - - - 0.94 Speed of Sound (m/s) 343 - - - 389 Radius (mm) - 172.5 - 150 - Thickness (mm) - 2 20 3 -

(kg/m3) - - 30 - -

(kPa) - - 800 - - Bulk Poisson's Ratio (-) - - 0.4 - - Flow Resistivity (MKs) - - 25000 - - Tortuosity (-) - - 7.8 - - Porosity (-) - - 0.9 - - Loss Factor (-) - - 0.265 - -

Ambient Outer Shell Porous Core Inner Shell Cavity

Figure 6 indicates that whenever the radius of the shell descends, the TL of the shell is ascending in low frequency region. It is due to the fact that the flexural rigidity of the cylinder will be increased with reduction of shells radii. In addition, decreasing the radius of the shell leads to weight reduction and then in high frequency especially in Mass-controlled region the power transmission into the structure increases.

Figure 7 shows the effect of the composite material on TL. Materials chosen for the comparison are graphite/epoxy and glass/epoxy (Table 2). The figure shows that material must be chosen properly to enhance TL at Stiffness-control zone. The results represent a desirable level of TL at Stiffness-control zone (Lower frequencies) for graphite/epoxy. It is readily seen that, in higher frequency, as a result of density of materials, the TL curves are ascending. Therefore, the TL of glass/epoxy is of the highest condition in the Masscontrolled region.

It is well anticipated that increase of porous layer thickness leads to increase of TL. As illustrated in Fig. 8, a considerable increase due to thickening the porous layer is obtained. As it is well obvious from this figure, the weight increase of about 12% ( 60 *<sup>c</sup> h* mm) and 25% ( 100 *<sup>c</sup> h* mm), the averaged TL values are properly increased about 35% and 60%, respectively in broadband frequency. It is a very interesting result that can encourage engineers to use these structures in industries.

Figure 9 shows a comparison between the transmission loss for a ten-layered composite shell and an aluminum shell with the same radius and thickness. Since, the composite shell is stiffer than the aluminum one, its TL is upper than that of aluminum shell in the Stiffnesscontrolled region. However, as a result of lower density of composite shell, it does not appear to be effective as an aluminum shell in Mass-controlled region.

The effect of stacking sequence is shown in Fig. 10. Two patterns 0 ,90 ,0 ,90 ,0 *<sup>s</sup>* and 90 ,0 ,90 ,0 ,90 *<sup>s</sup>* are defined to designate stacking sequence of plies. The arrangements of layers are so effective on TLs curve, especially on Stiffness-controlled region. However, no clear discrepancy is depicted in Mass-controlled region.

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 51

**Figure 5.** Comparison of an isotropic cylindrical double-walled shell with a negligible curvature and a

**Figure 6.** Comparison of a cylindrical double-walled shell lined with porous material with different

double-panel structure

inner shell's radius

**Figure 3.** Comparison of cylindrical double-walled shell with and without foam where porosity close to 1

**Figure 4.** Comparison of cylindrical isotropic double-walled shell with special case of laminated composite double-walled shell ( 150 *Ri* mm, 200 *Re* mm, 3 *<sup>i</sup> h* mm, 2 *<sup>e</sup> h* mm, and 47.5 *<sup>c</sup> h* mm)

**Figure 3.** Comparison of cylindrical double-walled shell with and without foam where porosity close to 1

**Figure 4.** Comparison of cylindrical isotropic double-walled shell with special case of laminated composite

double-walled shell ( 150 *Ri* mm, 200 *Re* mm, 3 *<sup>i</sup> h* mm, 2 *<sup>e</sup> h* mm, and 47.5 *<sup>c</sup> h* mm)

**Figure 5.** Comparison of an isotropic cylindrical double-walled shell with a negligible curvature and a double-panel structure

**Figure 6.** Comparison of a cylindrical double-walled shell lined with porous material with different inner shell's radius

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 53

**Figure 9.** Comparison between Aluminum and ten-layer laminated composite shell

**Figure 10.** TL curves for the ten-layered composite shell with respect to stacking sequence

Transmission losses (TLs) of double-walled composite laminated shells sandwiching a layer of porous material were calculated. It is also considered the acoustic-structural coupling effect as well as the effect of the multi-waves in the porous layer. In order to make the

**11. Conclusions** 

**Figure 7.** Comparison of a TL curves for ten-layer laminated composite shell with different material

**Figure 8.** Comparison of a cylindrical double-walled shell lined with porous material with different core thickness

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 53

**Figure 9.** Comparison between Aluminum and ten-layer laminated composite shell

**Figure 10.** TL curves for the ten-layered composite shell with respect to stacking sequence

#### **11. Conclusions**

52 Wave Processes in Classical and New Solids

core thickness

**Figure 7.** Comparison of a TL curves for ten-layer laminated composite shell with different material

**Figure 8.** Comparison of a cylindrical double-walled shell lined with porous material with different

Transmission losses (TLs) of double-walled composite laminated shells sandwiching a layer of porous material were calculated. It is also considered the acoustic-structural coupling effect as well as the effect of the multi-waves in the porous layer. In order to make the problem solvable, one dominant wave was used to model the porous layer. In general the comparisons indicated the benefits of porous materials. Also, a considerable increase due to thickening the porous layer was obtained. For example, the weight increase of about 12% and 25% may respectively lead to an increase of 35% or 60% in amount of averaged TL values in broadband frequency. In addition, it was shown that increasing the axial modulus of plies made the TL be increased in low frequency range. Moreover, the comparison of double-walled shell with a gap and the one sandwiched with porous materials (where the porosity is close to 1) indicated a good agreement. Eventually the arrangement of layers in laminated composite can be so effective in Stiffness-controlled region. Therefore, optimizing the arrangement of layers can be useful in future study.

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 55

, *W WI T* The incident and transmitted power flow per unit length

<sup>2</sup> *c* Speed of sound in the fluid phase of porous materials

*Tp* Incident, Reflected and transmitted pressures in

<sup>1</sup>*s* , <sup>2</sup>*s* , <sup>3</sup>*s* Density of external medium, equivalent density of the

<sup>0</sup> *u* , <sup>0</sup> *v* , <sup>0</sup> *w* Displacements of the shell in the axial, circumferential, and radial directions

*<sup>l</sup> y* Distance from the midsurface to the surface of the *l* th

external, porous layer and cavity media

porous material and internal medium

layer having the farthest *y* coordinate

of the shell

1 3 *c c*, Speed of sound in external and cavity media

, *i e h h* Shell wall thickness of inner and outer shell

( ) *W x <sup>t</sup>* , ( ) *W x <sup>p</sup>* Transverse and in-plane displacements

*se* Solid volumetric strain

*n* Circumferential mode number

**u** Displacement vector of the solid

Shear modulus of the porous material

Bulk density of the solid phase

Inertial coupling term

Flow resistivity

Porosity

 Equivalent masses Viscous characteristic length

Densities of the fluid parts of the porous material

 Mass density of the *l* th layer of the shell per unit midsurface area

<sup>2</sup> Laplacian operator in the cylindrical coordinate system

, *r* Cylindrical coordinate

Neumann factor

Air viscosity

Tortuosity

, *<sup>z</sup> q* External pressure components

ˆ*amb p* Acoustic pressures applied on the panel <sup>0</sup> *p* Pressure amplitude of the incident wave

*Ph* Thickness of Panel

*q* , *q*

1 *Rp* , <sup>2</sup> *Tp* , <sup>2</sup> *Rp* , <sup>3</sup>

*z* ,

*n* 

*v* 

0 

1 

*a* 

11 

*r* 

 , <sup>12</sup> 

 , <sup>22</sup> 

( )*l* 
