**Nomenclature**


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


54 Wave Processes in Classical and New Solids

**Author details** 

K. Daneshjou

H. Ramezani

R. Talebitooti

( ) 1

*I* , *I* , 1*I* , 2*<sup>I</sup>*

*N* , *N* , *N* , *M* , *M* , *M*

**Nomenclature** 

the arrangement of layers can be useful in future study.

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

*Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran* 

*Department of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran* 

*<sup>l</sup> E* Module of elasticity in the direction 1 of the *l* th ply

*nJ* Bessel function of the first kind of order *n*

<sup>1</sup> *<sup>f</sup> E* , <sup>1</sup>*<sup>s</sup> E* , <sup>3</sup>*<sup>s</sup> E* , <sup>3</sup> *<sup>f</sup> E* , <sup>5</sup>*<sup>s</sup> E* The strain energy associated with the displacement in the solid and fluid phases

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

Forces and Moments

*Young Researchers Club, Islamic Azad University - Shahrerey branch, Tehran, Iran* 

*A*<sup>1</sup> , *A*2 Physical and geometrical factors

*Ap <sup>q</sup>* , *pq <sup>B</sup>* , *Dp <sup>q</sup>* , *A*<sup>ˆ</sup> , *B*<sup>ˆ</sup> , *D*ˆ Stiffness coefficients

*D D* 1 6 Wave equations constants *E* Bulk Young's modulus

Inertia terms

*N*Pr Prandtl number

<sup>ˆ</sup> ( )*<sup>l</sup> <sup>Q</sup>* , ( )*<sup>l</sup> Qp q* Material constant

, *R Ri e* Radii of inner and outer shell *Vy* Normal acoustic particle velocity


Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 57

2 2 2

*z z z*

 

*z z*

)

2 4 2

  3,10 3 ( ) *H R n ri* 

2 3 2

*<sup>i</sup> <sup>z</sup>*

2

*z z z*

2 3 2

 

 

*<sup>j</sup> <sup>n</sup> <sup>R</sup> <sup>j</sup> <sup>n</sup> <sup>B</sup>*

16

2 2 2

 

)

*z*

2

 

> 

 

*zz z z i i i i*

> 

2 2 2 2

3 2 2 3

1 11 1 2 () ( ) () ()

*i ii i*

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

2

*i*

*R*

2 12

11 1 1 1 () () ()3 ( )

*A nB n j B n B j n A j <sup>R</sup> R R R R*

*i i i i i*

3 2 2

*A n A j n B n B j nA j <sup>R</sup> <sup>R</sup> R R*

11 11 ( ) 2 () 2 ( ) 4 ()

*A j nA j B j nB j A n R R R R*

<sup>1</sup> 1 11 ( ) ( ) ( )

 

2,1 12 16 12 2 2 16 62

2,2 22 2 26 22 3 2 26 66

*z z z i*

*B j D n D j nD j I <sup>R</sup> RR R*

*i i i i*

2,3 22 2 12 22 3 2 23 62

61 66 22 32 4 21 22

*B j B n j B nD j nD n <sup>R</sup> RR R*

*z zz z*

*D j nB j D j D j n R R <sup>R</sup> <sup>R</sup>*

3

*A j nB n B j n <sup>R</sup> R R*

1 11 () ( ) ( )

*z z*

66 62 3 2 66

*i*

3,1 <sup>12</sup> <sup>11</sup>

3 62 2 62

*R A*

*<sup>R</sup> <sup>n</sup> <sup>B</sup>*

*R A*

*i i i*

2

66 22 43 2 23 33

*i ii i*

1 11 1 3 () 2 (

*<sup>R</sup> <sup>n</sup> <sup>B</sup>*

*i i i i*

62 61 66 22 3

*A j B j B nj B n R R <sup>R</sup>*

*z z z i i i*

1 11 ( ) ()3 ( )

*z z*

3,2 22 2 12 22 3 2 23

2 66

2 3 2

2 3 2 3

*zz z*

 

3,9 2 ( ), *H R n ri* 

3,3 22 2 21 22 3 2 23 11

11 1 1 <sup>2</sup> 2 ( )4 ( )

1 11 2 () ( )

3 2

*A nB n j B n B j n <sup>R</sup> <sup>R</sup> R R*

*i i i i*

11 1 1 () () ()3 ( )

*n j*

 

*<sup>z</sup>*

   

*z z z*

<sup>1</sup>

3 2

*A B j B n B j nDj R RR <sup>R</sup>*

*B jD j D j n R R <sup>R</sup>*

*zz z*

1 11 2 () 4 ( )4 ( )

*D n j D j nD jn R R <sup>R</sup>*

*i i i i*

, <sup>2</sup>

11 1 2 (

*D j nD n D j n R RR*

21 2 43 22 23

*i i i*

*i ii*

12 2 3 16 62

*i i i*

2 2 42

*z i*

32 2 2 61 33

1 3,8 2 ( ), *H R n ri* 

33 2 4 22

*i i*

*D j nD nI R R*

1 1 4 ()

*i i i*

*z z z*

23 3 2 32 61 33 2

*z z*

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

3

*j B j B*

1 11 1 <sup>2</sup> ( )2 ( )
