Applied Mathematics and Mechanics Wave

Appl. Math. Mech. -Engl. Ed., 32(6), 701–718 (2011) DOI 10.1007/s10483-011-1450-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Wave transmission through laminated composite double-walled cylindrical shell lined with porous materials∗ K. DANESHJOU1 , H. RAMEZANI1 , R. TALEBITOOTI2 (1. Department of Mechanical Engineering, Iran University of Science and Technology, Tehran 16844, Iran; 2. Department of Automotive Engineering, Iran University of Science and Technology, Tehran 16844, Iran)

Abstract A study on free harmonic wave propagation in a double-walled cylindrical shell, whose walls sandwich a layer of porous materials, is presented within the framework of the classic theory for laminated composite shells. One of the most effective components of the wave propagation through the porous core is estimated with the aid of a flat panel with the same geometrical properties. By considering the effective wave component, the porous layer is modeled as a fluid with equivalent properties. Thus, the model is simplified as a double-walled cylindrical shell trapping the fluid media. Finally, the transmission loss (TL) of the structure is estimated in a broadband frequency, and then the results are compared. Key words

transmission loss, porous media, cylindrical laminated composite shell

Chinese Library Classification O347 2010 Mathematics Subject Classification

74F10, 74J20

Nomenclature E, Ri , Re , c1 , c2 , c3 , hi , he , n, α∞ , δ, ρ0 ,

bulk Young’s modulus; radius of the inner shell; radius of the outer shell; speed of the sound in the external medium; speed of the sound in the fluid phase of porous materials; speed of the sound in the cavity medium; shell wall thickness of the inner shell; shell wall thickness of the outer shell; circumferential mode number; tortuosity; shear modulus of the porous material; density of the fluid parts of the porous

ρ1 , Λ, σr , φ, ξ1 , ξ3 , ξα , ξβ , ξ4 , νˆ, ηˆ, ς,

material; bulk density of the solid phase; viscous characteristic length; flow resistivity; porosity; wave number in the external medium; wave number in the cavity medium; complex wave numbers of the compression wave; complex wave number of the shear wave; bulk Poisson’s ratio; loss factor; ratio of specific heats.

∗ Received Jun. 1, 2010 / Revised Feb. 14, 2011 Corresponding author R. TALEBITOOTI, E-mail: [email protected]

702

1

K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

Introduction

In recent years, the use of composite structures in many fields of engineering has been increasing due to their light weight and high stiffness. However, the noise transmission through these structures especially at a high frequency is a critical issue, because it suffers from a low density. To reduce the noise transmission, engineers try to apply the porous layer as a core. A porous medium consists of a frame (solid part) surrounded by a fluid (liquid or gas). Generally, the frame and the fluid parts are referred to as the frame borne and the airborne, respectively. The porous materials are usually sorted in two brands. The first brand is made of metals and ceramics, which possesses high stiffness and moderate sound absorption. These materials are used in many engineering fields. The second one includes sponges and polymeric foams, which are soft and flexible. There are a lot of theories that model porous materials with different approximate calculation. In some of those theories, the solid phase is supposed to be rigid. Thus, a single longitudinal wave propagates in materials. In the other theories, the elasticity of the solid phase is considered. However, the shear deformation is ignored. Therefore, just two longitudinal waves can propagate in materials. These models do not allow oblique incident cases to be handled. Bolton et al.[1] developed an applicable theory founded by Biot[2] to model the porous material. It is considered as a homogeneous material of the elastic frame and the fluid contained in the pores. The theory (full method) allows three waves in the porous material, two longitudinal waves and one rotational wave. Therefore, it can properly handle the oblique waves. One of the longitudinal waves propagates in the elastic frame, and the other propagates in the fluid medium. The shear wave propagated in the elastic frame is the rotational wave. The full method was applied to calculate the transmission losses (TLs) of the elastic double flat shells of infinite extents lined with elastic porous liners[1] . Lee and Kim[3] developed a simplified method to analyze the sound transmission through isotropic curved structures lined with the porous material. The foundation of this approximate method considers the strongest wave between those ones. Hundal and Kumar[4] investigated the symmetric wave propagation in the porous medium. Furthermore, the sound transmission through different kinds of shells and panels has been attended by many researchers. Koval[5] and Blaise et al.[6] investigated the sound transmission through isotropic and orthotropic shells. Blaise et al.[6] extended Koval’s work to obtain the transmission coefficient of an orthotropic shell subjected to an oblique plane sound wave with two independent incident angles. Tang et al.[7] investigated the sound transmission through a cylindrical sandwich shell with the honeycomb core. Liu et al.[8] studied the sound transmission through curved panels. Daneshjou et al.[9–11] has recently investigated the acoustic behavior of orthotropic and laminated composite cylindrical shells as well as doublewalled cylindrical shells. Literature shows that no work has been done on the context of the sound transmission through composite cylindrical shells sandwiching a layer of the porous material. In this paper, the authors investigate the sound transmission through the sandwich structure, which includes the porous material core between the two laminated composite cylindrical shells. It is typical to assume that the cylinder is infinitely long, the input to the system is a plane wave incident from outside, and the cavity is anechoic. Under these assumptions, the analytical solutions can be obtained without ignoring any vibro-acoustic coupling effects[3] . The composite shells have been modeled by the classic laminated shell theory. The numerical results are illustrated to properly study the geometrical and physical properties of the composite and porous material.

2

Propagation of sound in porous media

If the porous material is assumed to be a homogeneous aggregate of the elastic frame and the fluid trapped in pores, its acoustic behavior can be considered by the following two wave

Wave transmission through double-walled cylindrical shell

703

equations[1] : ∇4 es + A1 ∇2 es + A2 es = 0,

(1)

∇2  + ξ42  = 0.

(2)

Equations (1) and (2) determine two elastic longitudinal waves and one rotational wave, ¯ is the solid volumetric strain, u ¯ is the displacement respectively. In Eqs. (1) and (2), es = ∇ · u ¯ is the rotational strain in the solid phase, and vector of the solid,  = ∇ × u 





ω 2 (ϕ ρ 22 + μ ρ 11 − 2χ ρ 12 ) , ϕμ − χ2

A2 =

ω 4 ( ρ 22 ρ 11 − ρ 12 ) . ϕμ − χ2







A1 =

2





ρ 11 , ρ 12 , and ρ 22 are equivalent masses (the mass coefficients that account for the effects of the non-uniform relative fluid flow through pores[1] ) given by 

ρ 11 = ρ1 + ρa − jσr φ2



ρ 12 = −ρa + jσr φ2



1 ω

1 ω +

ρ 22 = φρ0 + ρa − jσr φ2

+

4jα2∞ κv ρ0  , σr2 Λ2 φ2

(3)

4jα2∞ κv ρ0  , σr2 Λ2 φ2

1 ω

+

(4)

4jα2∞ κv ρ0  , σr2 Λ2 φ2

(5)

where ρ1 and ρ0 are the densities of the solid and fluid parts of the porous material. Moreover, the parameters α∞ , κv , Λ, σr , φ, and ω are the tortuosity, the air viscosity, the viscous characteristic length, the flow resistivity, the porosity, and the angular frequency, respectively. ρa is the inertial coupling term, i.e., ρa = φρ0 (α∞ − 1).

(6)

χ, ϕ, and μ represent the material properties that include the bulk modulus of the frame and fluid, Poisson’s ratio, the shear modulus, the porosity, the ratio of the specific heats, the atmospheric pressure, the characteristic length, the air viscosity, and the Prandtl number. The details of χ, ϕ, and μ are described in Ref. [12]. The complex wave numbers of the two compression (longitudinal) waves ξα and ξβ are[12] ξα2 =

ω2 √    (ϕ ρ 22 + μ ρ 11 − 2χ ρ 12 + ℘), 2 2(ϕμ − χ )

(7)

ξβ2 =

ω2 √    (ϕ ρ 22 + μ ρ 11 − 2χ ρ 12 − ℘), 2(ϕμ − χ2 )

(8)

where 









2

℘ = (ϕ ρ 22 + μ ρ 11 − 2χ ρ 12 )2 − 4(ϕμ − χ2 )( ρ 22 ρ 11 − ρ 12 ).

(9)

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K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

The wave number of the shear (rotational) wave is ω 2  ρ 22 ρ 11 − ρ 12  ,  δ ρ 22 

ξ42 =

2



(10)

where δ is the shear modulus of the porous material.

3

Simplified method

As the full method is too complicated to model the porous layer in the curved sandwich structures, a simplified method is expanded for this category of structures[3] . The foundation of this approximate method considers the strongest wave between those ones. It includes two steps. At the first step, a flat double laminated composite with infinite extents with the same cross sectional construction is considered using the full method. Then, only the strongest wave number is chosen from the results, and the material is modeled using the wave number and its corresponding equivalent density. Thus, the material is modeled as an equivalent fluid. The strain energy which is related to the displacement in the solid and fluid phases can be defined for each wave component. The energy terms can be represented as follows: E1s and E1f are for the airborne wave, E3s and E3f are for the frame wave, and E5s is for the shear wave, in which the subscripts s and f represent the solid and fluid phases, respectively. For each new problem, comparing the ratios of the energy carried by the frame wave and the shear wave to the airborne wave in the fluid and solid phases, i.e., E1f , E1f

E1s , E1f

E3f , E1f

E3s , E1f

E5s , E1f

yields the strongest wave component in the entire frequency range[3].

4

Model specification

Figure 1 shows a schematic of the cylindrical double shell of the infinite length subjected to a plane wave with an incidence angle γ. The radii and the thickness of the shells are Ri,e and hi,e in which the subscripts i and e represent the inner and outer shells. A concentric layer of the porous material is installed between the shells. The acoustic media in the outside and the inside of the shell are represented by the density and the speed of the sound, i.e., (s1 , c1 ) is inside, and (s3 , c3 ) is outside.

5

Applying full method to two-dimensional problem

For a two-dimensional problem shown in the x-y plane in Fig. 2, the potential in the incident wave can be expressed as[1] i = e−j(ξx x+ξ1y y) ,

(11)

where ξx = ξ1 sin γ,

ξ1y = ξ1 cos γ,

ξ1 =

ω , c1

c1 is the speed of sound in the incident medium, γ is the angle of incidence, and j is the imaginary unit j2 = −1. Three kinds of waves propagate in the porous material. Therefore, six traveling waves, which have the same trace wave number, are induced by an oblique incident wave in a finite depth layer of the porous material, as shown in Fig. 2. The x- and y-direction components of the

Wave transmission through double-walled cylindrical shell

Fig. 1

705

Schematic diagram of double-walled cylindrical composite shell lined with porous materials

Fig. 2

Illustration of wave propagation in porous layer

displacements and stresses of the solid and fluid phases were derived by Bolton et al.[1] . The displacements in the solid phase are D2 jξαy y D3 −jξβy y D4 jξβy y  e + 2e + 2e ξα2 ξβ ξβ   ξ4y − j 2 e−jξx x D5 e−jξ4y y − D6 ejξ4y y , ξ4

u ˆx = jξx e−jξx x

D

1 −jξαy y e ξα2

 ξαy ξβy ξβy jξαy y −jξβy y jξβy y D e + D e − D e 2 3 4 ξα2 ξβ2 ξβ2   ξx + j 2 e−jξx x D5 e−jξ4y y + D6 ejξ4y y . ξ4

u ˆy = je−jξx x

ξ

+

αy D1 e−jξαy y ξα2

(12)

−

(13)

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K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

The displacements in the fluid phase are   x = jξx e−jξx x b1 D1 e−jξαy y + b1 D2 ejξαy y + b2 D3 e−jξβy y + b2 D4 ejξβy y U 2 2 2 2 ξα ξα ξβ ξβ − jg

 ξ4y −jξx x  D5 e−jξ4y y − D6 ejξ4y y , e 2 ξ4

(14)

  y = je−jξx x b1 ξαy D1 e−jξαy y − b1 ξαy D2 ejξαy y + b1 ξβy D3 e−jξβy y − b1 ξβy D4 ejξβy y U ξα2 ξα2 ξβ2 ξβ2 + jg

 ξx −jξx x  D5 e−jξ4y y + D6 ejξ4y y . e 2 ξ4

(15)

The stresses in the solid phase are  ξ 2   ξ2  αy αy 2δ 2 + A + b1 Q D1 e−jξαy y + 2δ 2 + A + b1 Q D2 ejξαy y ξα ξα  ξ2   ξ2  βy βy + 2δ 2 + A + b2 Q D3 e−jξβy y + 2δ 2 + A + b2 Q D4 ejξβy y ξβ ξβ   ξx ξ4y  + 2δ 2 D5 e−jξ4y y − D6 ejξ4y y , ξ4

σ ˆys = e−jξx x

τˆxy = e−jξx x δ +

(16)

 2ξ ξ   2ξx ξβy   x αy D1 e−jξαy y − D2 ejξαy y + D3 e−jξβy y − D4 ejξβy y 2 2 ξα ξβ

2   (ξx2 − ξ4y ) −jξ4y y jξ4y y D . e + D e 5 6 ξ42

(17)

The stress in the fluid phase is σ ˆ f = e−jξx x (Q + b1 R)D1 e−jξαy y + (Q + b1 R)D2 ejξαy y

+ (Q + b2 R)D3 e−jξβy y + (Q + b2 R)D4 ejξβy y , where ξαy =

 ξα2 − ξx2 ,

ξβy =

ξ4y =

ξ42 − ξx2 ,

j=

b1 =

b2 =

















ρ 11 R − ρ 12 Q ρ 22 Q − ρ 12 R

ρ 11 R − ρ 12 Q ρ 22 Q − ρ 12 R 

ρ g = − 12 , ρ 22

δ=

−

−

ξβ2 − ξx2 ,

√ −1, P R − Q2 

ω2( ρ

22 Q



− ρ 12 R)

P R − Q2 



ω 2 ( ρ 22 Q − ρ 12 R)

E , 2(1 + ν)

ξα2 ,

ξβ2 ,

Q = (1 − φ)G,

(18)

Wave transmission through double-walled cylindrical shell

R = φG,

P = A + 2δ,

A=

707

νE . (1 + ν)(1 − 2ν)

Here, E and ν are Young’s modulus in vacuo and Poisson’s ratio of the bulk solid phase, respectively. Assuming that the pores are shaped in a cylindrical form, and an expression for G is 1 −2ωρ0 α∞ j √ 2  J (2P r ) −1 1 (ς − 1) φσ φσr r 2 G = ρ0 c2 1 + , (19) √ 1 P r 2 −2jωρ0 α∞ J0 (2P r 12 −2ωρ0 α∞ j ) φσr where ς is the ratio of the specific heat, c2 is the speed of sound in the fluid phase of the porous materials, P r represents the Prandtl number, and J0 (0) and J1 (0) are the Bessel functions of the first kind with zero and first orders, respectively. These complex relations and six constants D1 , D2 , · · · , D6 have to be determined by applying the boundary conditions (BCs). When the elastic porous material is directly bonded to a panel, there exist six BCs. Also, the transverse displacement and the in-plane displacement at the neutral axis can be defined as follows[1] : wt (x, t) = Wt (x)ejωt ,

wp (x, t) = Wp (x)ejωt .

(20)

Four BCs are obtained from the interface compatibility[1] , i.e., ⎧ y = Wt , ⎨ Vy = jωWt , uˆy = Wt , U ⎩

u ˆx = Wp (∓)

hp dWt 2 dx .

(21)

Two BCs are obtained from the equations of motion as follows[1]: (∓)p(∓)qP − jξx

hp  4 − ω 2 I)Wt − Bξ  3 jWp , τxy = (Dξ x x 2

 x2 − ω 2 I)Wp − Bξ  x3 jWt , (∓)ˆ τyx = (Aξ

(22) (23)

where hp is the panel thickness, pˆamb is the acoustic pressure applied on the panel, and Vy is the normal acoustic particle velocity. The inertia term, I, and the extensional, the coupling,  B,  and D,  are[13] and the bending stiffness, A, I=

L 

ρ(l) (yl − yl−1 ),

(24)

l=1

= A

L 

 (l) (yl − yl−1 ), Q

(25)

l=1 L

 2 = 1  (l) (yl2 − yl−1 B Q ), 2

(26)

l=1 L

  (l) (y 3 − y 3 ), = 1 Q D l l−1 3 l=1

(27)

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K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

in which ρ(l) is the mass density of the lth layer of the shell per unit midsurface area, and yl is the distance from the midsurface to the surface of the lth layer having the farthest y coordinate.  (l) is defined as The material constant Q  (l) = Q

(l)

E1 , 1 − ν12 ν21

(l)

(l)

(l)

where E1 is the module of the elasticity in the direction 1, and ν12 and ν21 are Poisson’s ratios in the directions 1 and 2 of the lth ply, respectively. The fiber coordinates of the ply are described as 1 and 2, where the direction 1 is parallel to the fibers, and the direction 2 is perpendicular to them. qP is the normal force per unit panel area exerted on the panel by the elastic porous material, and qP = −ˆ σys − σ ˆ f . In the BCs, the first and second signs are appropriate when the porous material is attached to the positive and negative y-facing surfaces of the panel, respectively.

6

Prediction of ratios of energy

The energies related to the waves in the fluid phase and the solid phase are described as follows[3] : (i) The airborne wave is E1f =

 2 ξαy 1    φ(Q + b1 R)b1 2 D12  , 2 ξα

(28)

E1s =

 ξ 2   ξ2 1   αy αy (1 − φ) 2δ 2 + A + b1 Q 2 D12  . 2 ξα ξα

(29)

E3f =

2  ξβy 1    φ(Q + b2 R)b2 2 D32  , 2 ξβ

(30)

E3s =

 ξ 2   ξ2 1   βy βy (1 − φ) 2δ 2 + A + b2 Q 2 D32  , 2 ξβ ξβ

(31)

E5s =

  ξ 2 2  1   4y (1 − φ)2δ 2 D52  . 2 ξ4

(32)

(ii) The frame wave is

Here, the subscripts f and s represent the fluid and solid phases, respectively.

7

Formulation of problem In the external space, the wave equation becomes[9] c1 ∇2 (pI + pR 1)+

∂ 2 (pI + pR 1) = 0, ∂t2

(33)

2 where pI and pR 1 are the acoustic pressures of the incident and reflected waves, and ∇ is the Laplacian operator in the cylindrical coordinate system. The wave equations in the fluid phase of the porous layer and the internal space are the same as Eq. (33) with different variable names. The shell motion is described by the classic theory, fully considering the displacements in all three directions. Let the axial coordinate be z, the circumferential direction be θ, and the

Wave transmission through double-walled cylindrical shell

709

normal direction to the middle surface of the shell be r. The equations[13] of the motion in the axial, circumferential, and radial directions of a laminated composite thin cylindrical shell in the cylindrical coordinate can be written as follows:  ∂ 2 u0i,e 1 ∂Nαβ ∂Nα + + qαi,e = − I i,e , ∂z Ri,e ∂θ ∂t2

(34)

0  ∂ 2 vi,e 1 ∂Nβ 1  ∂Mαβ 1 ∂Mβ  ∂Nαβ + + + + qβi,e = − I i,e , ∂z Ri,e ∂θ Ri,e ∂z Ri,e ∂β ∂t2

(35)

0  ∂ 2 wi,e ∂Nβ 1 ∂ 2 Mαβ 1 ∂ 2 Mβ ∂ 2 Mα − +2 + qzi,e = − I i,e , + 2 2 2 2 ∂z Ri,e Ri,e ∂θ∂z Ri,e ∂θ ∂t

(36)

in which the subscripts i and e represent the inner and outer shells, and qα , qβ , and qz are the external pressure components. The inertia terms are as follows: 





I = I1 + I2

1 , R

where L      I 1, I 2 = l=1

yl

ρ(l) [1, z] dz.

(37)

yl−1

The midsurface strain and curvature can be expressed as[13] ε0α =

∂u0 , ∂z

κα = −

ε0β =

∂ 2 w0 , ∂z 2

 1  ∂v 0 + w0 , R ∂θ

γ0αβ =

∂ 2 w0  1  ∂v 0 , − R2 ∂θ ∂θ2

κβ =

In Eqs. (38) and (39), R is the cylindrical The forces and moments are[13] ⎡ ⎤ ⎡ Nα A11 A12 ⎢ Nβ ⎥ ⎢ A21 A22 ⎢ ⎥ ⎢ ⎢ Nαβ ⎥ ⎢ A61 A62 ⎢ ⎥ ⎢ ⎢ Mα ⎥ = ⎢ B11 B12 ⎢ ⎥ ⎢ ⎣ Mβ ⎦ ⎣ B21 B22 Mαβ B61 B62

1 ∂u0 ∂v 0 + , ∂z R ∂θ

κ=

(38)

1  ∂v 0 ∂ 2 w0  −2 . R ∂z ∂z∂θ

(39)

radius. A16 A26 A66 B16 B26 B66

B11 B21 B61 D11 D21 D61

B12 B22 B62 D12 D22 D62

B16 B26 B66 D16 D26 D66

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

ε0α ε0β γ0αβ κα κβ κ

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(40)

[13] where the extensional, coupling, and bending stiffness, Ap˜ ˜q , Bp˜ ˜q , and Dp˜ ˜q , are

Ap˜ ˜q =

L 

(l)

Qp˜ ˜q (yl − yl−1 ),

p˜, q˜ = 1, 2, 3,

(41)

l=1 L

Bp˜ ˜q =

1  (l) 2 2 Qp˜ ˜q (yl − yl−1 ), 2

p˜, q˜ = 1, 2, 3,

(42)

p˜, q˜ = 1, 2, 3,

(43)

l=1 L

Dp˜ ˜q =

1  (l) 3 3 Qp˜ ˜q (yl − yl−1 ), 3 l=1

710

K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI (l)

and Qp˜ ˜q , the material constant, is the function of the physical properties of each ply. The displacement components of the inner and outer shells at an arbitrary distance r from the midsurface along the axial, the circumferential, and the radial directions are[9,13–14] u0e =

∞ 

u0ne cos(nθ) exp[j(ωt − ξ1z z)],

(44)

0 vne sin(nθ) exp[j(ωt − ξ1z z)],

(45)

0 wne cos(nθ) exp[j(ωt − ξ1z z)],

(46)

u0ni cos(nθ) exp[j(ωt − ξ3z z)],

(47)

0 vni sin(nθ) exp[j(ωt − ξ3z z)],

(48)

0 wni cos(nθ) exp[j(ωt − ξ3z z)].

(49)

n=0

ve0

=

∞  n=0

we0 =

∞  n=0

u0i =

∞  n=0

vi0 =

∞  n=0

wi0 =

∞  n=0

In Eqs. (44)–(46), ξ1 , the wave number in the external medium, is defined as ξ1 =

ω , c1

ξ1z = ξ1 sin γ,

ξ1r = ξ1 cos γ,

and in Eqs. (47)–(49), ξ3 , the wave number in the internal cavity, is expressed as ω 2 . ξ3 = , ξ3z = ξ1z , ξ3r = ξ32 − ξ3z c3 The boundary conditions at the two interfaces between the shells and the fluid are[3,11] ∂ I ∂ 2 we0 (p + pR 1 ) = −s1 ∂r ∂t2

at r = Re ,

(50)

∂ T ∂ 2 we0 (p2 + pR 2 ) = −s2 ∂r ∂t2

at r = Re ,

(51)

∂ 2 wi0 ∂ T (p2 + pR 2 ) = −s2 ∂r ∂t2

at r = Ri ,

(52)

∂ 2 wi0 ∂ T (p3 ) = −s3 ∂r ∂t2

at r = Ri ,

(53)

and s2 is the equivalent density of the porous material and can be obtained from the simplified method as suggested. The harmonic plane incident wave pI can be expressed in the cylindrical coordinates as[3,11] pI (r, z, θ, t) = p0 ej(wt−ξ1z z)

∞  n=0

εn (−j)n Jn (ξ1r r) cos(nθ),

(54)

Wave transmission through double-walled cylindrical shell

711

where p0 is the amplitude of the incident wave, n (= 0, 1, 2, · · · ) indicates the circumferential mode number, εn = 1 for n = 0 and εn = 2 for n = 0, 1, 2, · · · , and Jn is the Bessel function of the first kind of order n. Considering the circular cylindrical geometry, the pressures are expanded as[3,11] j(wt−ξ1z z) pR 1 (r, z, θ, t) = e

∞ 

2 pR n1 Hn (ξ1r r) cos(nθ),

(55)

1 pT n2 Hn (ξ2r r) cos(nθ),

(56)

2 pR n2 Hn (ξ2r r) cos(nθ),

(57)

1 pT n3 Hn (ξ3r r) cos(nθ),

(58)

n=0

j(wt−ξ2z z) pT 2 (r, z, θ, t) = e

∞  n=0

j(wt−ξ2z z) pR 2 (r, z, θ, t) = e

∞  n=0

j(wt−ξ3z z) pT 3 (r, z, θ, t) = e

∞  n=0

where H1n and H2n are the Hankel functions of the first and second kinds of order n, and the wave number, ξ2 , in the porous core can be obtained from the simplified method as suggested. The substitution of the expressions in the displacements of the shell and the acoustic pressure equations into six shell equations and four boundary conditions 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 in the form of a matrix equation as follows:     0 0 0 0 T R T T ¯ , , wni , u0ne , vne , wne , pR = λ [] u0ni , vni n1 , pn2 , pn2 , pn3

(59)

  ¯ is where [] is a 10 × 10 matrix with components given in Appendix, and λ   ¯ = {0, 0, 0, 0, 0, −p0 εn (−j)n Jn (ξ1r Re ) ξ1r , −p0 εn (−j)n J (ξ1r Re )ξ1r , 0, 0, 0}T . λ n

(60)

T R T 0 0 0 0 0 0 The ten unknown coefficients pR n1 , pn2 , pn2 , pn3 , une , vne , wne , uni , vni , and wni are obtained in terms of p0 with solving Eq. (59) for each mode n, which then can be substituted back into Eqs. (44)–(49) and (54)–(58) to find the displacements of the shell and the acoustic pressures in series forms.

8

Calculation of TLs

The transmission coefficient, τ (γ), is the ratio of the amplitudes of the incident and transmitted waves. τ (γ) is obviously a function of the incidence angle γ defined by[3]   ∞ 1 0 ∗  s1 c1 πRi Re pT n3 Hn (ξ3r Ri )(jωw1n ) τ (γ) = , 2 ε R cos γp n e 0 n=0

(61)

where εn = 1 for n = 0 and εn = 2 for n = 1, 2, · · · . Re{} and the superscript ∗ represent the real part and the complex conjugate of the argument, respectively. To consider the random incidence, τ (γ) can be averaged according to the Paris formula[15] as  τ¯ = 2 0

γm

τ (γ) sin γ cos γdγ,

(62)

712

K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

where γm is the maximum incident angle. The integration of Eq. (62) is conducted numerically by Simpson’s rule. Finally, the average TL is obtained as 1 L = 10 lg . Tavg τ¯

(63)

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

9

Convergence algorithm

Equations (44)–(49) and (54)–(58) are obtained in the 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 that 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 be convergent. An algorithm for the calculation of TLs at each frequency is as follows: Repeat 1 TnL = 10 lg , τ¯ and set n=n+1 until

10

 L   L Tn+1  − Tn  < 1E − 6.

Numerical results

The parametric numerical studies of TLs are conducted for a double-walled laminated composite cylindrical shell lined with porous materials specified below, considering the 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 made of graphite/epoxy (see Table 2). The plies are arranged in a [0◦ , 45◦ , 90◦ , −45◦, 0◦ ]s pattern. Table 1

Geometrical and environmental properties[3] Ambient

Outer shell

Porous Core

Inner shell

Material

Air

Composite

Porous material

Composite

Cavity Air

Density ρ/(kg·m−3 )

1.21

–

–

–

0.94

Speed of sound c/(m·s−1 )

343

–

–

–

389

Radius R/(mm)

–

172.5

–

150

–

Thickness h/(mm)

–

2

20

3

–

Bulk density of solid phase∗ ρb /(kg·m−3 )

–

–

30

–

–

Bulk Young’s

modulus∗

E/(kPa)

–

–

800

–

–

Bulk Poisson’s ratio (–) v Flow resistivity σr /(MKs)

– –

– –

0.4 25 000

– –

– –

Tortuosity (–)

–

–

7.8

–

–

Porosity (–)

–

–

0.9

–

–

Loss factor (–)

–

–

0.265

–

–

∗

in vacuo.

Wave transmission through double-walled cylindrical shell Table 2

713

Orthotropic properties[10] Graphite/epoxy

Glass/epoxy

Axial modulus δa /(GPa)

137.9

38.6

Circumferential modulus δc /(GPa)

8.96

8.2

Shear modulus δ/(GPa)

7.1

4.2

1 600

1 900

0.3

0.26

Density

ρ/(kg·m−3 )

Major Poisson’s ratio (–)

The results are verified by the authors’ previous work for an especial case in which the porous material properties go into a fluid phase (in other words, the porosity is close to 1). The comparison of these results shown in Fig. 3 indicates good agreement. The calculated TLs for the laminated composite shell are compared with those of other authors for a special case of isotropic materials. In other words, in this model, the mechanical properties of the lamina in all directions are chosen to be the same as those of an isotropic material such as Aluminum. Then, the fiber angles approach zero. Figure 4 compares the TL values of the special case of the laminated composite walls obtained from the present model and those of the aluminum walls from Ref. [3]. The results show excellent agreement.

Fig. 3

Comparison of cylindrical doublewalled shell without and with foam where porosity is close to 1

Fig. 4

Comparison of cylindrical isotropic double-walled shell with special case of laminated composite doublewalled shell

We are going to verify the model in behavior comparing the results of the 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 in Ref. [1] (see Fig. 5). It should also be noted that both structures sandwich a porous layer and have the same thickness. Although it is not expected to achieve the same result as the derivation of the shell equations is quite different comparing with the derivation of the plate equations, the comparison between the two curves indicates that they behave in a same trend in the broadband frequencies. Figure 6 indicates that whenever the radius of the shell descends, the TL of the shell is ascends in a low frequency region. It is due to the fact that the flexural rigidity of the cylinder will be increased with the reduction of the shell radii. In addition, decreasing the radius of the shell leads to the weight reduction, and then in a high frequency especially in the mass-controlled region, the power transmission into the structure increases. Figure 7 shows the effect of the composite material on the TL. The materials chosen for the comparison are graphite/epoxy and glass/epoxy (see Table 2). The figure shows that the

714

Fig. 5

K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

Comparison of isotropic cylindrical double-walled shell with negligible curvature and double-panel structure

Fig. 6

Comparison of cylindrical doublewalled shell lined with porous material with different inner shell radii

material must properly be chosen the toenhance TL at the stiffness-control zone. The results represent an adesirable level of the TL at the stiffness-control zone (lower frequencies) for graphite/epoxy. It is readily seen that, in a higher frequency, as a result of the density of the materials, the TL curves are ascending. Therefore, the TL of glass/epoxy is of the highest condition in the mass-controlled region. It is well anticipated that the increase of the porous layer thickness leads to the increase of the 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 increases of about 12% (hc = 60 mm) and 25% (hc = 100 mm), the averaged TL values are properly increased about 35% or 60%, respectively, in a broadband frequency. It is a very interesting result that can encourage engineers to use these structures in industries.

Fig. 7

Comparison of TL curves for tenlayered laminated composite shell with different materials

Fig. 8

Comparison of cylindrical doublewalled shell lined with porous material with different core thickness

Figure 9 shows a comparison between the TLs 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 the aluminum shell in the stiffness-controlled region. However, as a result of the lower density of the composite shell, it does not appear to be effective as an aluminum shell in the mass-controlled region. The effect of stacking a sequence is shown in Fig. 10. Two patterns [0◦ , 90◦ , 0◦ , 90◦ , 0◦ ]s and [90◦ , 0◦ , 90◦ , 0◦ , 90◦ ]s are defined to designate stacking the sequence of the plies. The

Wave transmission through double-walled cylindrical shell

715

arrangements of layers are so effective on the TL curves, especially on the stiffness-controlled region. However, no clear discrepancy is depicted in the mass-controlled region.

Fig. 9

11

Comparison between aluminum and ten-layered laminated composite shell

Fig. 10

TL curves for ten-layered composite shell with respect to stacking sequence

Conclusions

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

References [1] Bolton, J. S., Shiau, N. M., and Kang, Y. J. Sound transmission through multi-panel structures lined with elastic porous materials. Journal of Sound and Vibration, 191(3-4), 317–347 (1996) [2] Biot, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid — I: lowfrequency range. Journal of the Acoustical Society of America, 28(2), 168–191 (1956) [3] Lee, J. H. and Kim, J. Simplified method to solve sound transmission through structures lined with elastic porous material. Journal of the Acoustical Society of Amer ica, 110(5), 2282–2294 (2001) [4] Hundal, B. S. and Kumar, R. Symmetric wave propagation in a fluid-saturated incompressible porous medium. Journal of Sound and Vibration, 288(1-2), 361–373 (2005) [5] Koval, L. R. On sound transmission into an orthotropic shell. Journal of Sound and Vibration, 63(1), 51–59 (1979) [6] Blaise, A., Lesuer, C., Gotteland, M., and Barbe, M. On sound transmission into an orthotropic infinite shell: comparison with Koval’s results and understanding of phenomena. Journal of Sound and Vibration, 150(2), 233–243 (1991)

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[7] Tang, Y. Y., Robinson, J. H., and Silcox, R. J. Sound transmission through a cylindrical sandwich shell with honeycomb core. 34th AIAA Aerospace Science Meeting and Exhibit (AIAA-96-0877), Reno, Nevada, USA, 1, 877–886 (1996) [8] Liu, B., Feng, L., and Nilsson, A. Influence of overpressure on sound transmission through curved panels. Journal of Sound and Vibration, 302(4-5), 760–776 (2007) [9] Daneshjou, K., Nouri, A., and Talebitooti, R. Sound transmission through laminated composite cylindrical shells using analytical model. Archive of Applied Mechanics, 77(6), 363–379 (2007) [10] Daneshjou, K., Talebitooti, R., and Nouri, A. Analytical model of sound transmission through orthotropic double walled cylindrical shells. CSME Transaction, 32(1), 43–66 (2007) [11] Daneshjou, K., Nouri, A., and Talebitooti, R. Analytical model of sound transmission through orthotropic cylindrical shells with subsonic external flow. Aerospace Science and Technology, 13(1), 18–26 (2009) [12] Allard, J. F. Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Chapman and Hall, London (1993) [13] Qatu, M. S. Vibration of Laminated Shells and Plates, Elsevier Academic, Amsterdam (2004) [14] Leissa, A. W. Vibration of Shells, Scientific and Technical Information Center, NASA, Washington, D. C. (1973) [15] Pierce, A. D. Acoustics, McGraw-Hill, New York (1981) Appendix The non-zero components of the matrix [] appearing in Eq. (59) are as follows: 1,1 = A11 (−jξz )2 + 2A16

1,2 = A12

1 1 1 1 (−jξz ) n + A16 (−jξz )2 + B12 2 (−jξz ) n + B16 (−jξz )2 + A62 2 (−n2 ) Ri Ri Ri Ri

+ A66

1,3 = A12

1 1 1 (−n) − B62 3 n3 − 2B66 2 (−n2 ) (−jξz ) , Ri2 Ri Ri

1 1 1 1 (−jξz ) n + A16 (−jξz )2 + B12 2 (−jξz ) n + B16 (−jξz )2 + A62 2 (−n2 ) Ri Ri Ri Ri

+ A66

2,2 = A22

1 1 1 (−jξz ) n + B62 3 (−n2 ) + B66 2 (−jξz ) n, Ri Ri Ri

1 1 1 (−jξz ) − B11 (−jξz )3 − B12 2 (−jξz ) (−n2 ) − 3B16 (−jξz )2 (−n) Ri Ri Ri

+ A62

2,1 = A12

1 1 1 (−jξz ) n + B62 3 (−n2 ) + B66 2 (−jξz ) n, Ri Ri Ri

1 1 1 1 (−n2 ) + 2A26 (−jξz ) n + 2B22 3 (−n2 ) + 4B26 2 (−jξz ) n + A66 (−jξz )2 Ri2 Ri Ri Ri

+ 2B66

2,3 = A22

1 1 (−jξz ) (−n) + A66 2 (−n2 ) − Ii ω 2 , Ri Ri

1 1 1 1 (−jξz )2 + D22 4 (−n2 ) + 2D23 3 (−jξz ) n + D33 2 (−jξz )2 − Ii ω 2 , Ri Ri Ri Ri

1 1 1 1 1 (−n) − B12 (−n) (−jξz )2 − B22 3 n3 − 3B23 2 (−jξz ) (−n2 ) + A62 (−jξz ) Ri2 Ri Ri Ri Ri

− B61 (−jξz )3 − 2B66 − 3D23

1 1 1 1 (−n) (−jξz )2 + B22 3 (−n) − D21 2 (−jξz )2 (−n) − D22 4 n3 Ri Ri Ri Ri

1 1 1 1 (−jξz ) (−n2 ) + B32 2 (−jξz ) − D61 (−jξz )3 − 2D33 2 (−jξz )2 (−n), Ri3 Ri Ri Ri

Wave transmission through double-walled cylindrical shell 1 1 1 (−jξz ) − B11 (−jξz )3 − B12 2 (−jξz ) (−n2 ) − 3B16 (−jξz )2 (−n) Ri Ri Ri 1 1 1 + A62 2 (−n) − B62 3 n3 − 2B66 2 (−n2 ) (−jξz ) , Ri Ri Ri

3,1 = A12

1 1 1 1 1 (−n) − B12 (−n) (−jξz )2 − B22 3 n3 − 3B23 2 (−jξz ) (−n2 ) + A62 (−jξz ) Ri2 Ri Ri Ri Ri 1 1 1 1 − B61 (−jξz )3 − 2B66 (−n) (−jξz )2 + B22 3 (−n) − D21 2 (−jξz )2 (−n) − D22 4 n3 Ri Ri Ri Ri 1 1 1 1 − 3D23 3 (−jξz ) (−n2 ) + B32 2 (−jξz ) − D61 (−jξz )3 − 2D33 2 (−jξz )2 (−n), Ri Ri Ri Ri

3,2 = A22

1 1 1 1 − 2B21 (−jξz )2 − 2B22 3 (−n2 ) − 4B23 2 (−jξz ) (−n) + D11 (−jξz )4 Ri2 Ri Ri Ri 1 1 1 − 2D12 2 (−n2 ) (−jξz )2 + 4D16 (−jξz )3 (−n) + 4D62 3 (−jξz ) n3 Ri Ri Ri 1 1 ` ´ + 4D33 2 (−jξz )2 (−n2 ) + D22 4 n4 − Ii ω 2 , Ri Ri

3,3 = A22

3,8 = H1n (ξ2r Ri ),

3,9 = H2n (ξ2r Ri ),

4,4 = A11 (−jξz )2 + 2A16

3,10 = H1n (ξ3r Ri ),

1 1 (−jξz ) (−n) + A66 2 (−n2 ) − Ie ω 2 , Re Re

1 1 1 1 (−jξz ) n + A16 (−jξz )2 + B12 2 (−jξz ) n + B16 (−jξz )2 + A62 2 (−n2 ) Re Re Re Re 1 1 1 (−jξz ) n + B62 3 (−n2 ) + B66 2 (−jξz ) n, + A66 Re Re Re

4,5 = A12

1 1 1 (−jξz ) − B11 (−jξz )3 − B12 2 (−jξz ) (−n2 ) − 3B16 (−jξz )2 (−n) Re Re Re 1 1 1 + A62 2 (−n) − B62 3 n3 − 2B66 2 (−n2 ) (−jξz ) , Re Re Re

4,6 = A12

1 1 1 1 (−jξz ) n + A16 (−jξz )2 + B12 2 (−jξz ) n + B16 (−jξz )2 + A62 2 (−n2 ) Re Re Re Re 1 1 1 (−jξz ) n + B62 3 (−n2 ) + B66 2 (−jξz ) n, + A66 Re Re Re

5,4 = A12

1 1 1 1 (−n2 ) + 2A26 (−jξz ) n + 2B22 3 (−n2 ) + 4B26 2 (−jξz ) n + A66 (−jξz )2 Re2 Re Re Re 1 1 1 1 + 2B66 (−jξz )2 + D22 4 (−n2 ) + 2D23 3 (−jξz ) n + D33 2 (−jξz )2 − Ie ω 2 , Re Re Re Re

5,5 = A22

1 1 1 1 1 (−n) − B12 (−n) (−jξz )2 − B22 3 n3 − 3B23 2 (−jξz ) (−n2 ) + A62 (−jξz ) Re2 Re Re Re Re 1 1 1 1 (−n) (−jξz )2 + B22 3 (−n) − D21 2 (−jξz )2 (−n) − D22 4 n3 − B61 (−jξz )3 − 2B66 Re Re Re Re 1 1 1 1 − 3D23 3 (−jξz ) (−n2 ) + B32 2 (−jξz ) − D61 (−jξz )3 − 2D33 2 (−jξz )2 (−n), Re Re Re Re

5,6 = A22

1 1 1 (−jξz ) − B11 (−jξz )3 − B12 2 (−jξz ) (−n2 ) − 3B16 (−jξz )2 (−n) Re Re Re 1 1 1 + A62 2 (−n) − B62 3 n3 − 2B66 2 (−n2 ) (−jξz ) , Re Re Re

6,4 = A12

717

718

K. DANESHJOU, H. RAMEZANI, and R. TALEBITOOTI

6,5 = A22

1 1 1 1 1 (−n) − B12 (−n) (−jξz )2 − B22 3 n3 − 3B23 2 (−jξz ) (−n2 ) + A62 (−jξz ) Re2 Re Re Re Re

− B61 (−jξz )3 − 2B66 − 3D23 6,6 = A22

1 1 1 1 (−n) (−jξz )2 + B22 3 (−n) − D21 2 (−jξz )2 (−n) − D22 4 n3 Re Re Re Re

1 1 1 1 (−jξz ) (−n2 ) + B32 2 (−jξz ) − D61 (−jξz )3 − 2D33 2 (−jξz )2 (−n), Re3 Re Re Re

1 1 1 1 − 2B21 (−jξz )2 − 2B22 3 (−n2 ) − 4B23 2 (−jξz ) (−n) Re2 Re Re Re

1 1 (−n2 ) (−jξz )2 + 4D16 (−jξz )3 (−n) Re2 Re 1 1 1 ` ´ + 4D62 3 (−jξz ) n3 + 4D33 2 (−jξz )2 (−n2 ) + D22 4 n4 − Ie ω 2 , Re Re Re

+ D11 (−jξz )4 − 2D12

6,7 = H2n (ξ1r Re ), 8,6 = − s2 ω 2 , 

6,8 = H1n (ξ2r Re ), 

8,8 = H1n (ξ2r Re )ξ2r ,

9,8 = H1n (ξ2r Ri )ξ2r ,



6,9 = −H2n (ξ2r Re ), 

2 8,9 = pR n2 Hn (ξ2r Re )ξ2r ,

9,9 = H2n (ξ2r Ri )ξ2r ,

10,3 = −s3 ω 2 ,

d , dr

ξz = ξ1z = ξ2z = ξ3z ,

ξ2r =

7,7 = H2n (ξ1r Re )ξ1r ,

9,3 = −s2 ω 2 , 

10,10 = H1n (ξ3r Ri )ξ3r .

Here, () =



7,6 = −s1 ω 2 ,

q ξ22 − ξz2 .