An analytical study of sound transmission through stiffened double laminated composite sandwich plates

Aerospace Science and Technology 82–83 (2018) 92–104

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An analytical study of sound transmission through stiffened double laminated composite sandwich plates Tao Fu a , Zhaobo Chen a,∗ , Hongying Yu a , Zhonglong Wang a , Xiaoxiang Liu b,∗ a b

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, PR China Beijing Institute of Control Engineering, Beijing 100094, PR China

a r t i c l e

i n f o

Article history: Received 6 April 2018 Received in revised form 28 August 2018 Accepted 12 September 2018 Available online 13 September 2018 Keywords: Laminated sandwich plate FSDT Vibroacoustic response Carbon nanotubes Periodic structures

a b s t r a c t The main objective of this research work is focused on sound transmission loss analysis of stiffened double laminated composite sandwich plate structures subjected to plane sound wave excitation, wherein the laminated composite plates are composed of perfectly bonded functionally graded carbon nanotubes (CNTs) reinforced composite layers, and in each layer, the carbon nanotubes is uniform or functionally graded along with the thickness direction of structures, and three types of the CNTs distributions are studied. The extended rule of mixture is employed to determine the properties of the composite material. The compatibility of displacements on the interface between the plate and the stiffeners is employed to derive the governing equation of each stiffener. Then fluid–structure coupling is considered by imposing velocity continuity condition at fluid–structure interfaces. By using the space harmonic approach and virtual work principle, the sound transmission loss is described analytically. Since no existing results of sound insulation can be found for such composite material plate structure, comparison studies can only be made with the isotropic and laminated case. Good agreement is found from these comparison studies. Based on the developed theoretical mode, the influences of the volume fractions of CNTs, distribution type of CNTs, structural damping, lamination angle and the number of layers on sound transmission loss are subsequently presented. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction Sandwich structures are multi-layered structures which result from the assembly of two face sheets and a core, which are extensively used in many engineering applications such as building constructions, cabin skin of aircrafts, ship hulls and underwater [1,2]. The greatest advantage of sandwich structures is that optimal designs can be obtained for different applications by choosing different materials and geometric configurations of the face sheets and cores [3]. The type of core can be of any material or architecture but three types are most generally used, which include stiffened core, honeycomb core and soft or solid core. It is noteworthy that the most generally used sandwich structure in aerospace industry is not only subjected to various mechanical loads, but also exposed to the thermal and noise environment caused by aerodynamic heating and aerodynamics [4,5]. Hence, apart from structural strength and stiffness requirements, an increasing amount of attention is being paid to the sound radiation and sound transmission loss characteristics of sandwich structures.

*

Corresponding authors. E-mail addresses: [email protected] (Z. Chen), monkeyfi[email protected] (X. Liu).

https://doi.org/10.1016/j.ast.2018.09.012 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

In order to understand and characterize the vibroacoustic behaviors of the sandwich structures, many studies about the behaviors of the sandwich structures are made. For honeycomb core sandwich structures, Ng and Hui [6] experimentally studied transmission loss of honeycomb panels by using a new test specimen and presented a new honeycomb core design to improve the noise transmission loss at frequencies between 100 and 200 Hz. Wang et al. [7] researched the sound transmission loss in symmetric unidirectional sandwich panels with isotropic skins, and either honeycomb or isotropic core. Arunkumar et al. [8] used the 2D equivalent finite element model to investigate the sound transmission loss characteristics of isotropic honeycomb sandwich panels and it is concluded that the effect of core geometry on sound transmission loss is significant. Subsequently, Arunkumar et al. [9] extended the method to a honeycomb panel with composite facing, and from the results it is demonstrated that FRP panel can be used to replace the aluminum panel without losing acoustic comfort with nearly 40 percent weight reduction. The vibroacoustic bending properties of a finite sandwich panel with honeycomb core and composite faces also have been investigated by Guillaumie [10]. Koch et al. [11] investigated the vibration and noise reduction capability of a honeycomb structure filled with granular

T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

material. And the results showed lower sound pressure levels than the original empty honeycomb structure. Moreover, for periodic stiffened core sandwich structure, Lee and Kim [12] replaced the rib-stiffener as translational springs, rotational springs and lumped mass when they studied the sound transmission performance of a single stiffened plate subjected to a plane wave excitation. Similarly, Alba and Ramis [13] studied the effect of rib-stiffener on sound transmission characteristics of stiffened panel structures. Subsequently, Wang et al. [14] extended the method to double-leaf partitions connected through vertical resilient studs and investigated the physical mechanisms determining sound transmission. Following the work of Wang et al., Xin and Lu [15] proposed a more accurate theoretical model for sound transmission from an orthogonally rib-stiffened sandwich structure. In his study, the effects of rib-stiffeners were included by introducing the tensional forces, bending moments and torsional moments as well as the corresponding inertial terms into the governing equations of the two face panels. Yin and Cui [16] modeled the rib-stiffener as Bernoulli–Euler beams and studied the acoustic radiation from an infinite stiffened laminated composite cylindrical shell based on the classical laminated composite plate theory. The same approach was adopted by Cao et al. [17] to investigate the acoustic characteristics of stiffened symmetric and anti-symmetric laminated plates. Considering only the bending moments of rib-stiffeners, they used the first order shear deformation theory to describe the equations of motion for the laminated composite plate. A refined theoretical model was proposed later by Xin and Lu [18] for sound transmission from an orthogonally rib-stiffened sandwich plate excited by acoustic wave, in which the inertial effects, bending and torsional moments were considered. On the other hand, with the ongoing development of hightech industry, demand for advanced materials has led to the development of substitutes of traditional engineering materials like aluminum, steel, etc. Meanwhile it is also very important to analyze vibration and acoustic response characteristics of a structural member made up of a new advanced material. Compare to traditional materials, laminate composite offers definite advantages such as high strength to weight ratios and excellent sound insulation properties. However, there is an inconsistency in the mechanical properties at interface of the two different materials which constitute the traditional laminated composite structures [19]. This leads to the stress concentration at the interfaces. Known as a newborn class of composite materials, functionally graded materials (FGM) have continuous variation of material properties from one face to another and thus eliminate the abrupt changes in the stresses. A significant advantage of FGM is the capability of having both metal and ceramic properties. What’s more the performance of structural components made of FGM can be further enhanced with the addition of a new type of material, namely, carbon nanotubes (CNTs) [20]. It is reported that Young modulus of CNTs may be more than 1 TPa where its density is only 1.3 g/cm3 [21]. The concept of functionally graded carbon nanotube reinforced composites (FG-CNTRC) was first introduced by Shen [22]. Rich literature exists on the theoretical modeling and analysis of FG-CNT reinforced composite material. Approaches such as using shear deformation plate theory, finite element method and analytical solutions were carried out to study this kind of material. Regarding the classification of plate deformation theories reported in previous studies, there are mainly three major theories which are usually known as: the classical plate theory (CPT), the firstorder shear deformation theory (FSDT) and the higher-order shear deformation theory (HSDT). Using these theories, numerous studies have been reported on the free vibration, bending and buckling analyses of FG-CNT reinforced composite structures. Among those, Zghal et al. [23–26] investigated the free vibration, static and dynamic response, mechanical buckling of FG-CNT reinforced

93

composite plates and shells using finite elements. Besides, nonlinear deflections analyses of FG-CNT reinforced composite plate and shell structures [27–29] were also studied by separately using nonlinear double directors shell model and discrete form of Kirchhoff finite element model. Trabelsi et al. [30] used a modified first-order shear deformation theory (FSDT) to investigate the thermal post-buckling of functionally graded material structures. Similarly, Shen and Zhang [31] studied the thermal buckling and postbuckling of FG-CNT reinforced composite rectangular plates with symmetric distribution of CNTs across the plate thickness. Wang and Shen [32] analyzed the nonlinear large amplitude vibration of FG nanocomposite plates reinforced by single-walled CNTs resting on an elastic foundation in thermal environments. The material properties of single-walled CNTs were assumed to be temperaturedependent and graded along the thickness direction. Zhu et al. [33] adopted the finite element method and first order shear deformation theory to study the static and free vibration of FG-CNT reinforced composite plates. Also, they proposed a meshless local Petrov–Galerkin approach based on the moving Kriging interpolation technique to investigate the geometric nonlinear thermoelastic of functionally graded plates in thermal environments [34]. Zhang et al. [35] employed the IMLS-Ritz method to study the free vibration of FG-CNT reinforced composite straight-sided quadrilateral plates resting on elastic foundations. Following the work of Zhang et al., Selim et al. [36] used the Reddy’s higher-order shear deformation theory (HSDT) and the element-free kp-Ritz method to investigate the free vibration behaviors of FG-CNT reinforced composite plates in a thermal environment. Similarly, the vibrations of FG-CNT reinforced composite thick plates with elastically restrained edges are also studied by Zhang et al. [37]. From above studies, it can be seen that for periodic stiffened core sandwich plate structure, although the effects of structure geometric parameters on the property of sound insulation have been described very completely, the application of new composite material in sound insulation, especially for functionally graded or laminated FG-CNT reinforced composite materials with outstanding advantages of mechanical, is relatively rare. Meanwhile, for functionally graded or laminated FG-CNT reinforced composite materials, previous researches generally focused mainly on relatively simple plate or shell constructions and studied the structure static and dynamic characteristics analysis, which indicates it’s application is not meant to be exhaustive. Therefore, there is a need for adopting those composite materials to investigate sound transmission issues of complex structures. The new contribution of the article is that an attempt is made to use the laminated FG-CNT reinforced composite materials with outstanding advantages of mechanics to investigate the property of sound insulation of stiffened double laminated composite sandwich plates structures subjected to plane sound wave excitation, wherein the laminated composite plates are composed of perfectly bonded functionally graded carbon nanotubes (CNTs) reinforced composite layers, and in each layer, the carbon nanotubes is uniform or functionally graded along the thickness direction of structures, and three types of the CNT distributions are studied. The extended rule of mixture is employed to determine the properties of the composite material. The compatibility of displacements on the interface between the plate and the stiffeners is employed to derive the governing equation of each stiffener. Then fluid– structure coupling is considered by imposing velocity continuity condition at fluid–structure interfaces. By using the space harmonic approach and virtual work principle, the sound transmission loss is described analytically. Since no existing results of sound insulation can be found for such composite material plate structure, comparison studies can only be made with the isotropic and laminated case. Good agreement is found from these comparison studies. Based on the developed theoretical mode, the influences of

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T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

shown in Fig. 1. The mathematical expression of the effective volume fractions of each type of FG-CNT distribution can be expressed as:

⎧ ∗ V cnt ⎪ ⎪ ⎪ ⎪ ⎨ (1 +

V cnt ( z) =

Fig. 1. Pressure wave impinging upon the stiffened double laminated plates. (a) Schematic diagram of interaction between incident acoustic wave. (b) Different distributions of CNTs along the plate thickness.

the volume fractions of CNTs, distribution type of CNTs, structural damping, lamination angle and the number of layers on sound transmission loss are subsequently presented. Through numerical results, we try to give a further comprehension of the underlying mechanism of sound transmission through stiffened double laminated FG-CNT reinforced composite sandwich plates. 2. Theoretical model 2.1. Problem statement and material properties definition The stiffened double laminated FG-CNTRC sandwich plate shown in Fig. 1 is composed of FG-CNT reinforced composite layers with identical material properties and different plies orientations. The plate geometry and dimensions are defined with respect to a Cartesian coordinate system, introduced such that the bottom and top surfaces of the plate lie in the plane z = −h/2 and z = h/2, and the coordinate system is located in the mid plane of the plate. The stiffeners periodically located at x = {0, ±l x , ±2l x , · · ·} and y = {0, ±l y , ±2l y , ±3l y , · · ·} along the x and y directions respectively are uniformly distributed on the plate surface. On the source side, the structure is impinged by a plane sound wave P 1 of angular frequency ω and the sound speed in air is noted as c 0 . The wave makes an incident angle ϕ with the z axis and its projection in the xy plane makes an azimuth angle θ with the x axis. The wave has an amplitude I and a wave number k0 (= ω/c 0 ). In addition, for each layer, distribution of CNTs across the thickness of the plate may be uniform or functionally graded, it is usually referred to as functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate. From the mathematical point of view, various dispersion profiles may be considered for the CNTs across the thickness of the panel, however, linearly graded patterns of CNTs are more observed in the researches due to their consistency with the fabrication processes. As a result, four different types of distributions of CNTs are considered in this study: (1) UD represents the uniform distribution of CNTs, in which they are uniformly distributed along the thickness direction; (2) FG-V, FG-O and FG-X represent three different types of functionally graded distributions, wherein FG-V features CNTs concentrated at the top region, FG-O features CNTs concentrated at the mid-plane, and FG-X features CNTs concentrated at both the top and the bottom regions, as

⎪ ⎪ ⎪ ⎪ ⎩

UD

2z ∗ ) V cnt h 2| z | ∗ 2(1 − h ) V cnt 2| z | ∗ 2( h ) V cnt

FG-V FG-O

  h h − ≤z≤ 2

2

(1)

FG- X

∗ denotes for the CNTs volume fractions, z is the local where V cnt thickness coordinate variable for a typical layer. The effective material properties of the CNTRC structure can be determined based on the well-known Eshelby–Mori–Tanaka-scheme model and extended rule of mixture model. As stated in Zghal et al. [25] and Frikha et al. [26], these two models are efficient in the estimation of mechanical properties of CNTRC structure, but the extended rule of mixture is much simpler and convenient. For that, the extended rule of mixture is adopted in this paper and the expressions are as follows: cnt E 11 ( z) = η1 V cnt ( z) E 11 + V m ( z) E m

η2 E 22 ( z)

η3 G 12 ( z)

V cnt ( z)

=

+

cnt E 22

=

V cnt ( z)

+

G cnt 12

(2)

V m ( z)

(3)

Em V m ( z)

(4)

Gm

cnt m where E 11 , G cnt and G m are Young’s modulus and shear mod12 , E ulus of CNTs and matrix, η1 , η2 and η3 are CNT efficiency parameters, V m is the volume fraction of matrix. In addition, the total volume of the composite is taken as the sum of the individual constituent of the carbon nanotubes and the polymer matrix volume fractions and can be conceded as:

V m ( z) = 1 − V cnt ( z)

(5)

Also, the effective Poisson’s ratio and the mass density of FGCNTRC material can also be obtained using the rule of mixture as follows: cnt ν12 (z) = V cnt (z)ν12 + V m ( z)ν m ν12 (z) ν21 (z) = E 22 ( z)

(6)

ρ (z) = V cnt (z)ρ cnt + V m (z)ρ m

(8)

(7)

E 11 ( z)

cnt 11

α11 (z) = V cnt (z)α 

cnt 12

α22 (z) = 1 + ν



+ V m ( z)α cnt 22

V cnt ( z)α

m



+ 1+ν

m



(9) m

V m ( z)α − ν12 α11 ( z) (10)

cnt cnt cnt where ν12 and ν m are Poisson’s rations and α11 , α22 and αm are the thermal expansion coefficients of the carbon nanotube and matrix, respectively.

2.2. General formulations Considering moderately thick laminated composite plates, the first order shear deformation theory (FSDT) is used to account for displacement field [38]:

U 1 (x, y , z, t ) = u 1 (x, y , t ) + zφx1 (x, y , t ) V 1 (x, y , z, t ) = v 1 (x, y , t ) + zφ y1 (x, y , t ) W 1 (x, y , z, t ) = w 1 (x, y , t )

(11)

T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

where (u 1 , v 1 , w 1 , φx1 , φ y1 ) are unknown functions to be determined, t denotes the time and (u 1 , v 1 , w 1 ) denotes the displacements of the mid plane (z = 0). φx1 and φ y1 are the rotations of a transverse normal about the y-axis and x-axis, respectively. The dynamic equilibrium relations of the stiffened top panel are given by integrating the stress continuity relation through the thickness of the panel [38]:

N x,x + N xy , y = I 1 u¨ 1 + I 2 φ¨ x1 N xy ,x + N y , y = I 1 v¨ 1 + I 2 φ¨ y1



¨ 1 + P 1 x, y , − Q x,x + Q y , y = I 1 w +

F 1x

+

M 1x

+

y F1

h1

+



2

 − P 2 x, y ,

h1

⎡

L 13 L 23 L 33 L 43 L 53

L 14 L 24 L 34 L 44 L 54

⎤⎡

⎤

L 15 u1 L 25 ⎥ ⎢ v 1 ⎥ ⎥⎢ ⎥ L 35 ⎥ ⎢ w 1 ⎥ ⎦ ⎣ L 45 φx1 ⎦ L 55 φ y1

⎡

L 14 L 24 L 34 L 44 L 54

P 1 (x, y , z; t )

= Ie − j (kx x+k y y +kz z−ωt ) +∞ +∞   + A mn e − j [(kx +2mπ /lx )x+(k y +2nπ /l y ) y −k z,mn z−ωt ]

⎤

m=−∞ n=−∞

m=−∞ n=−∞

(14) y

∞

m x m=−∞ Q yi δ(x − ml x ), M i = ∞ y n n=−∞ Q xi δ( y − nl y ), M i = n=−∞

m  m=−∞ M T yi δ (x − ml x ), n  M T xi δ ( y − nl y ), and δ(·)

Fi = is the Dirac delta function. The compatibility of displacements on the interface between the plate and the stiffeners is employed to derive the governing equation of each stiffener along the x- or y-direction [39].

∂4 w ∂2 w + ρ Am 2 = Q m yi , 4 ∂x ∂t i = 1, 2

E Im

E In

−G J n

(15)

2

∂ w ∂ w − ω2 ρ I n0 = M nT xi ∂y ∂ y ∂ x2

 B mn e − j [(kx +2mπ /lx )x+(k y +2nπ /l y ) y −k z,mn z−ωt ]

(20)

+∞ 

=

+∞ 

C mn e − j [(kx +2mπ /lx )x+(k y +2nπ /l y ) y +k z,mn z−ωt ]

m=−∞ n=−∞

(21)

where

kx = k0 sin ϕ cos θ,

k y = k0 sin ϕ sin θ,

k z = k0 cos ϕ

The k z,mn is the (m, n)th space harmonic wave number in the z-direction, and the corresponding acoustic pressure should satisfy the scalar Helmholtz equation



⎧ ⎫  ⎨ P1 ⎬ ∂2 ∂2 ∂2 P2 = 0 + + + k20 ⎩ ⎭ ∂ x2 ∂ y 2 ∂ z2 P

(22)

3

Substituting Eqs. (19)–(21) into (22), the k z,mn is given by

∂4 w ∂2 w + ρ An 2 = Q xin , 4 ∂x ∂t

2 ∂2 w 2 m∂ w −G J m − ω ρ I = Mm 0 T yi , ∂x ∂ x∂ y 2 2

+∞ 

+∞ 

+

⎤

0 0

∞

B mn e − j [(kx +2mπ /lx )x+(k y +2nπ /l y ) y +k z,mn z−ωt ]

P 3 (x, y , z; t )

0

∞

+∞ 

m=−∞ n=−∞

0 ⎥ ⎢ y y h1 h1 =⎢ P ( x , y , + d ) − P ( x , y , + d + h2 ) − F 2x − M 2x − F 2 − M 2 ⎥ 2 3 2 2 ⎦ ⎣

where F ix =

+∞ 

⎤

L 15 u2 L 25 ⎥ ⎢ v 2 ⎥ ⎥⎢ ⎥ L 35 ⎥ ⎢ w 2 ⎥ ⎦ ⎣ L 45 φx2 ⎦ L 55 φ y2

(19)

P 2 (x, y , z; t )

=

⎤⎡

(18)

where α1,mn and α2,mn are the displacement amplitudes of top and bottom plate, respectively. Similarly, the sound pressures inside and outside the cavity can be represented by space harmonic series [41,42]:

Similarly, the governing equations of bottom panel in terms displacements are given as:

L 13 L 23 L 33 L 43 L 53

α2,mn e− j[(kx +2mπ /lx )x+(k y +2nπ /l y ) y−ωt ]

m=−∞ n=−∞

0

L 12 L 22 L 32 L 42 L 52

+∞ 

+∞ 

=

(12)

⎥ ⎢ ⎥ ⎢ y y ⎥ h1 h1 x x (13) =⎢ P ( x , y , − ) − P ( x , y , ) + F + M + F + M 2 ⎢ 1 1 1 1 1 ⎥ 2 2 ⎦ ⎣ 0

L 11 ⎢ L 21 ⎢ ⎢ L 31 ⎣L 41 L 51

(17)

w 2 (x, y ; t )

0 0

⎡

α1,mn e− j[(kx +2mπ /lx )x+(k y +2nπ /l y ) y−ωt ]

m=−∞ n=−∞

y M1

The (N x , N y , N xy ) denote the in-plane force resultants, (M x , M y , M xy ) denote the moment resultants, ( Q x , Q y ) denote the shear resultants. The final governing equations of upper panel in terms of displacements can be obtained as a concise matrix form:

L 12 L 22 L 32 L 42 L 52

+∞ 

+∞ 

=

2

M y , y + M xy ,x − Q y = I 2 v¨ 1 + I 3 φ¨ y1

L 11 ⎢ L 21 ⎢ ⎢ L 31 ⎣L 41 L 51

where (E I m , E I n ), (G J m , G J n ), ( Q m , Q xin ) and (M m , M nT xi ) are yi T yi the flexural stiffness, torsional stiffness, equivalent line force and equivalent line moments for the x- and y-wise stiffeners, respecn tively. ( A m , A n ), (I m , I n ) and (I m 0 , I 0 ) are the cross-sectional area, moment of inertia and polar moment of inertia for the x- and y-wise stiffeners, respectively. Since the sandwich structure is periodic in the xy plane and excited by a harmonic plane sound wave (see Fig. 1), the panel responses can be expressed using space harmonic expansion [40]:

w 1 (x, y , t )



M x,x + M xy , y − Q x = I 2 u¨ 1 + I 3 φ¨ x1

⎡

95

(16)

  k z,mn =

ω c0

2

2  2  2mπ 2nπ − kx + − ky + lx

ly

(23)

When (ω/c 0 )2 < (kx + 2mπ /l x )2 + (k y + 2nπ /l y )2 , the pressure waves become evanescent waves so that k z,mn should be taken as [41,42]:

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T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104





k z,mn = j

kx +

2mπ

2

lx

2  2  2nπ ω + ky + − ly

c0

δ

As well, continuity conditions at fluid-panel interfaces require that

 ∂ P 1  = ω2 ρ0 w 1 , ∂ z z=−h1 /2  ∂ P 2  = ω2 ρ0 w 2 , ∂ z z=h1 /2+d

 ∂ P 2  = ω2 ρ0 w 1 , ∂ z z=h1 /2  ∂ P 2  = ω2 ρ0 w 2 ∂ z z=h1 /2+h2 +d

Finally, the virtual work principle requires that

(24)

 w1

A mn = B mn = 

B mn =

ω2 ρ0 α1,00 jk z

ω2 ρ0 α1,mn jk z,mn



(25)

−

+ 

,

2k z,mn sin(k z,mn d) e jk z,mn (h1 +h2 +d)

= (27) (28)

The displacement amplitudes α1,mn and α2,mn can be derived by using the virtual work principle, which are then used to calcu and C mn . Once late the sound pressure amplitudes A mn , B mn , B mn coefficient C mn is known, calculating the sound transmission loss is straightforward (see Eqs. (36) and (37)). The forces in the face sheet make contributions to the virtual work as:

δ



l y lx  =

w1

0



K 1∗ w 1 (x, y ) − P 1 x, y , −

0

 + P 2 x, y ,

δ



l y lx  =

w2

0

0







h1



2

h1





2

K 2∗ w 2 (x, y ) − P 2 x, y ,



=−

x1

δ



=

x2

δ



h1 2

 +d

(30)

 y2





M 2x (x, 0)

δ α2,kl e

l y =−



y

j αk x

dx

(31)



y

F 1 (0, y ) + M 1 (0, y ) δ α1,kl e j αl y dy ,

0

l y = 0



y

y





w2

+δ

x2



=0

(33)

y2

ω2 ρ0

+

jk z,kl

ω2 ρ0 cos(k z,kl d) k z,kl sin(k z,kl d)

ω2 ρ0 k z,kl sin(k z,kl d) +∞ 

l xl y α2,kl +

l xl y α1,kl +∞ 

( R 1 + R 2 βl βn )l x α1,kn

n=−∞

(− Q 1 − Q 2 βl βn )l x α2,kn

+∞ 

( R 3 + R 4 αk αm )l y α1,ml

+

+∞ 

(− Q 3 − Q 4 αk αm )l y α2,ml

m=−∞

when k = 0 and l = 0 when k = 0 or l = 0

2Il xl y 0

ω2 ρ0

+

jk z,kl

ω2 ρ0 cos(k z,kl d) k z,kl sin(k z,kl d)

ω2 ρ0 k z,kl sin(k z,kl d) +∞ 

(34)

l xl y α2,kl

l xl y α1,kl

(− R 1 − R 2 βl βn )l x α1,kn

n=−∞

+

+∞ 

( Q 1 + Q 2 βl βn )l x α2,kn

+∞ 

(− R 3 − R 4 αk αm )l y α1,ml

F 2 (0, y ) + M 2 (0, y ) δ α2,kl e j αl y dy

+∞ 

( Q 3 + Q 4 αk αm )l y α2,ml = 0

(35)

where

F 1x (x, 0) + M 1x (x, 0) δ α1,kl e j αk x dx,

F 2x (x, 0) +

+δ

m=−∞

0

y1

δ



−

+

0

lx

K 2∗ −

(29)

  h1 + h2 + d δ w ∗2 dxdy + P 3 x, y ,





m=−∞

The contributions to the virtual work by the stiffener along the x and y-direction are equal to

δ



+

2

lx

y1

δ

n=−∞

δ w ∗1 dxdy 

= 0,

n=−∞

(26)

ω2 ρ0 [α1,mn e− jkz,mn (h1 +d) − α2,mn e− jkz,mn h1 ]

jk z,mn



m=−∞

2k z,mn sin(k z,mn d)

C mn = −

+ +

ω2 ρ0 [α1,mn e jkz,mn (h1 +d) − α2,mn e jkz,mn h1 ]

ω2 ρ0 α2,mn

+δ

x1

K 1∗ −

, m = 0 or n = 0

,



Substituting Eqs. (17)–(21), (29)–(32) into (33) and noting that the virtual displacement is arbitrary, we obtain

Substituting Eqs. (17)–(21) into Eq. (25) and utilizing the fact that the sums must be true for all values of x and y, the pressure coefficients and displacement amplitude coefficients are related for each combination (m, n) by

A 00 = I +

+δ

(32)

R 1 = ρ A n1 ω2 − E I n1 αk4 ,

R 2 = −G J 1n αk2 + ρ I n01 ω2 ,

2 m 4 R 3 = ρ Am 1 ω − E I 1 βl ,

2 R 4 = −G J 1m αk2 + ρ I m 01 ω ,

Q 1 = ρ A n2 ω2 − E I n2 αk4 ,

Q 2 = −G J 2n αk2 + ρ I n02 ω2 ,

2 m 4 Q 3 = ρ Am 2 ω − E I 2 βl ,

2 Q 4 = −G J 2m αk2 + ρ I m 02 ω ,

αk = kx + 2kπ /lx ,

αm = kx + 2mπ /lx ,

βl = k y + 2lπ /l y ,

βn = k y + 2nπ /l y

For an isotropic thin plate in bending, the dynamic stiffness K i∗ is given by D i (αk2 + βl2 )2 − mi ω2 , where D i = Eh3i /12(1 − ν 2 ), in which D i is the bending stiffness, E is Young’s modulus, ν is the Poisson’s ratio, respectively. For the composite panel, however, this formulation is not strictly rigorous. In existing available literature, this feature has been studied by Cao’s model [17], Yin and Cui [16] and they have showed how to calculate the dynamic stiffness of the composite panel. The same approach is used in this paper but with a plane sound wave instead of a point force excitation of the composite panel. Therefore, the dynamic stiffness can be derived as shown in Appendix A.

T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

According to the knowledge of convergence, the infinite linear algebraic Eqs. (34) and (35) should be truncated to a finite but sufficient large number of terms, i.e., m = −kˆ to kˆ and n = −ˆl to ˆl (both kˆ and ˆl are positive integers). The values of kˆ and ˆl which are used in further calculation have to be determined by the convergence analysis of the solution, and hence can be numerically solved. Lee and Kim [12] argued that once the solution converges at a given frequency, it converges in the range lower than the given frequency. For convergence criteria, Lee and Kim also assumed that once the difference between the STL results calculated at two successive calculations is less than a preset error band, the solution is considered to have converged. Therefore, to calculate the STL, the highest frequency of interest (i.e. 10 kHz) and the error band (0.01 dB) are chosen, and it is found that the number of terms, 21, is sufficient for the convergence of STL calculations results within the error band of 0.01 dB. Solving Eqs. (34) and (35) one can obtain the vibration displacements of the two plates, and then the coefficient C mn of the transmitted pressure amplitudes is obtained through Eq. (28) and used to calculate the transmission coefficient of the periodic model. Since the transmission coefficient is a function of sound incident angles ϕ and θ , the transmission coefficient is defined here as the ratio of the transmitted sound power to the incident sound power [44], as

+∞

τ (ϕ , θ) =

m=−∞

+∞

n=−∞ |C mn | | I |2 k z

2

Re(k z,mnn )

(36)

Then, the sound transmission loss (STL) expressed in decibel scale (dB) [42] is obtained, as

 ! 2π ! ϕmax STL = −10 log10

0

0 ! 2π 0

τ (ϕ , θ) sin θ cos θ dθ dϕ

! ϕmax 0



sin θ cos θ dθ dϕ

(37)

In addition, to describe the vibration intensity of the plate structure, the averaged quadratic velocity L u expressed in dB is introduced as follows:

⎧ ! 2 ⎪ L u1 = 10 log10 ( A ωu w 1 w ∗1 d A ) ⎪ A1 ⎪ 1 ref ⎪  ⎪ ⎨ = 10 log (ω2 +∞ +∞ α1,mn /u ) ref 10 m=−∞ n=−∞ Lu = ! 2 ω ⎪ ∗ ⎪ L = 10 log10 ( A u w 2 w 2d A ) ⎪ A 2 ref ⎪ u2 ⎪ +∞ 2 +∞ ⎩ 2 = 10 log10 (ω m=−∞ n=−∞ α2,mn /u ref )

(38)

where the subscripts 1 and 2 denote the top panel and bottom panel, respectively, and the u ref is reference quadratic velocity (u ref = 1.0 × 10−5 m/s). The asterisk denotes complex conjugate. 3. Validation study 3.1. Key resonance parameters It is important to bringing forward some important resonance frequencies before studying noise transmission into such structure, wherein the first is the mass–air–mass resonance where the two panels move in opposite phase labeled by the symbol of filled circles, which can be derived as follows [45,46]:

 fm =

1

k(ms1 + ms2 )

2π

ms1 ms2

(39)

The standing-wave resonance occurs at f s,n when the gap depth matches integer numbers of the half-wavelength of the incident sound, which is labeled by the symbol of filled squares. The last resonance is coincident resonance, which occurs at f coin when the

97

velocity of the bending wave matches the trace velocity of the incident wave on the face sheet.

f coin =

c 02 2π sin ϕ



meff D eff

,

f s,n =

nc 0 2d

,

n = 1, 2, 3...

(40)

where k is the equivalent stiffness for double plate with stiffeners and those of double plate with air cavity. meff and D eff are the surface mass density and the bending stiffness of the plate, respectively, and they are defined as [47]: 0.5h 

meff = −0.5h

0.5h 

ρ (z)dz,

D eff = −0.5h

E ( z) 1 − ν12 ( z)ν21 ( z)

z2 dz

(41)

For a homogeneous isotropic " plate, the coincident resonance frequency in Eq. (40) is f coin = (c 02 12ρ (1 − ν 2 )/ E )/2π h sin ϕ . 3.2. Various characteristic frequency ranges According to these resonance frequencies mentioned above, the noise transmission behavior can be divided into three basic regions, which include stiffness controlled, mass controlled and coincidence controlled region [47]. The stiffness controlled part corresponds to the portion of the curve from zero to first resonance frequency, and due to internal damping in the plate structure, resonances can also occur which dramatically decrease the STL. The portion of the curve between the first " resonance frequency and critical coincidence frequency ( f cr = (c 02 meff / D eff )/2π ) corresponds to mass controlled region, and the STL curves follow a 6 dB increase caused by doubling of mass. In addition the curve after critical coincidence frequency corresponds to coincidence controlled region. 3.3. Validation of the acoustic model and influence of stiffeners To verify the validity of the present theoretical model and numerical code, two sets of cases are considered in this study. Firstly, since there are no available results for sound transmission loss from laminated FG-CNT reinforced composite plate, to check the applicability of the present model, the numerical calculations are compared with published theoretical results obtained by Wang and Lu [14] and experiment results obtained by Hongisto et al. [41] for unidirectional stiffened double panel. The size of the 2 mm thick experiment stiffened steel plate is 1.105 m by 2.25 m, and the height of empty space of the sandwich structure is 45 mm. The relevant geometrical dimensions and material property parameters can be found in Fig. 11(b) of Ref. [41] and Table 2 of Ref. [14]. Furthermore, to make the comparison possible, the proposed complete theoretical models are simplified as one set of parallels unidirectional stiffened, and the one-third octave bands of sound transmission of model results are also given to compare with experimental results. It can be seen from Fig. 2(b) that the present model predictions follow a similar trend of the experimental data especially for significant resonance dips occurred at approximately 6300 Hz. The discernible discrepancies between the theory and experiment can be attributed to a number of factors, such as the imperfect normal plane sound wave, the measuring errors by microphones, and the inevitable structural flanking transmission paths. But the discrepancies between present theory model and analysis data of Wang and Lu [14] are expected, which can be attributed to the difference in vibration modeling of the ribstiffeners between the present model and Wang’s theory. In Wang and Lu’s work [14], the stiffeners were approximated as a series of lumped mass, translational springs and rotational springs to link the sandwich structure. In contrast, in the present theory model,

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T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

Fig. 3. Comparison between present model predictions and those by Xin and Lu [18] for sound transmission loss of orthogonal stiffened isotropic plates. The locations of three basic regions are marked only for double plate with air cavity structure in the figure for clarity.

Fig. 2. Comparison between present model predictions and published theoretical results obtained by Wang and Lu [14] and experiment results obtained by Hongisto et al. [41] for unidirectional stiffened double panel: (a) the full octave bands of sound transmission; (b) the one-third octave bands of sound transmission. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

the stiffeners are modeled as beams, which include the inertial effects, bending and torsional moments. Secondly, to validate the proposed complete theoretical model with bidirectional stiffened, the model predictions are also compared with published results of Xin and Lu [18] for sound transmission of orthogonal stiffened plates, and the relevant geometrical dimensions and material property parameters are identical as those of Xin and Lu [18]. To indicate the advantage of the laminated composite model, the predictions for the structures with laminated FG-CNT reinforced composite materials are also included in Fig. 3. In addition, to assess the effect of stiffeners, the predictions are also compared with double-leaf partition with air cavity, which is a special case of the stiffened sandwich structure when the stiffness of the stiffener is converted to the air stiffness kair = ρ0 c 02 /d. The results of homogeneous materials (denoted model 1), illustrated in Fig. 3, show a good agreement between the present results and that of reported in [18]. For the laminated FG-CNT reinforced composite materials (denoted model 2), it shows better sound transmission performance compared to homogeneous material. This improvement of sound insulation can be

attributed to the fact that the increases in surface mass density and stiffness of composite plate are higher compared to homogeneous aluminum plates in wide frequency band range. In addition, as depicted in those figure, the mass–air–mass resonance, the standing-wave and the coincident frequencies obtained from present study are similar to those calculated by Eqs. (39)–(40). It is noteworthy that in stiffness control region, the stiffened structure does better in sound insulation performance than double plate with air cavity structure as the stiffeners increase the stiffness of plate structure and thus leading to the increase of corresponding STL values. But in mass control and coincidence control regions, the sound insulation performance of stiffened structure is poorer than that of double plate with air cavity when the frequency increases, which mainly due to the coupling of complex periodical resonance caused by the periodically placed stiffeners, and the stiffeners transmit more sound energy than air cavity. Moreover, for the stiffened structure, the panel responses and sound pressures expressed using space harmonic expansion, which consist a series of wave numbers (kx + 2mπ / L x and k y + 2mπ / L x ) and additional produce a series of resonance peaks and dips, and thus lead to weak of sound insulation performance in resonant frequencies. 4. Results and discussion 4.1. Simulation parameters In this section, numerical calculations based on theoretical formulations presented above are performed to explore the vibration and acoustic response of stiffened double laminated FG-CNT reinforced composite sandwich plates. The stiffeners dimensions are chosen with depth d = 0.08 m, width b = 0.001 m, and stiffener spacing l x = l y = 0.2 m. The density of air and the speed of sound in air are set to 1.21 kg/m3 and 343 m/s. The top and bottom FG-CNT reinforced composite laminate plates are component of five layers with equal thickness h1 = h2 = 0.0012 m and the orientations of the layer are [30◦ /−30◦ /30◦ /−30◦ /30◦ ] for the following detailed investigations. Unless specified otherwise, the matrix material properties considered by Han and Elliot [48] are used in preparing the numerical results, which are ν m = 0.34, ρ m = 1150 kg/m3 , E m = 2.5 GPa. The CNTs selected in this paper are assumed to be of the type of armchair (10, 10) single-walled carbon nanotube (SWCNT) with the following material proper-

T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

Fig. 4. Influence of CNTs volume fraction on sound transmission loss for the FG-X type stiffened laminated sandwich plate.

Fig. 5. Influence of CNTs volume fraction on averaged quadratic velocity for the top panel. cnt cnt ties [49]: E 11 = 5.6466 TPa, E 22 = 7.08 TPa, G cnt 12 = 1.9455 TPa. Eqs. (2)–(4) are used to calculate E 11 , E 22 and G 12 , and then for each case, we assume that G 13 = G 23 = G 12 . The CNTs efficiency parameters ηi (i = 1, 2, 3) used in Eqs. (2)–(4) are estimated by matching Young’s modulus and the shear modulus of FG-CNT reinforced composite material obtained by the extend rule of mixture to molecular simulation results, and these parameters are provided ∗ = 0.11; η = as: η1 = 0.149, η3 = η2 = 0.934 for the case of V cnt 1 ∗ 0.15, η3 = η2 = 0.941 for the case of V cnt = 0.14, and η1 = 0.149, ∗ = 0.17. It should be noted that η3 = η2 = 1.381 for the case of V cnt since the effects of structure geometric parameters on the property of sound insulation have been described very completely (see Section 1), numerical investigations are conducted specially focusing on the influence of laminated FG-CNT reinforced composite materials on the property of sound insulation of stiffened double laminated composite sandwich plates structures.

4.2. Influence of CNTs volume fraction on sound transmission loss Fig. 4 compares the effect of the variation of CNTs volume fraction on sound transmission loss, the STL frequency curves plotted by one-third octave bands are also shown in Fig. 8 to eliminate

99

Fig. 6. Influence of CNTs volume fraction on averaged quadratic velocity for the bottom panel.

Fig. 7. Averaged quadratic velocity of top panel and bottom panel when CNTs volume fraction set as 0.11.

the fluctuations of dense peaks and dips (see Fig. 4) in medium and high frequency regions caused by the periodically placed stiffeners. As can be seen in Fig. 4, the STL peaks and dips of three curves are shifted to higher frequency as the CNTs volume fraction increases, particularly in the stiffness controlled range below 600 Hz. In Figs. 5–6, the averaged quadratic velocities of the top panel and the bottom panel, directly indicated the total vibration energy of the panel, also showed the similar tendency as the STL response, especially for the first resonance peak and dip. Another noteworthy aspect is that the averaged quadratic velocity of the top panel is higher than the bottom panel, as shown in Fig. 7. This is because the top panel is directly impinged by a plane sound wave and thus transmits more vibration energy. Upon comparing Fig. 4 with Figs. 5–6, we observe that the effect of the variation of CNTs volume fraction on acoustic performance of the structure is the most obvious in the stiffness controlled region, which is due to the fact that higher CNTs volume fraction increases the stiffness of the stiffened laminated sandwich plate, and thus leading to the increase of corresponding resonance frequencies. In the medium and high frequency regions, the coincidence frequencies f coin for CNTs volume fraction with 0.11, 0.14 and 0.17, are 4123.1 Hz, 3664.9 Hz and 3344.7 Hz respectively. Although the CNTs volume fraction in-

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T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

Fig. 8. Influence of CNTs volume fraction on sound transmission loss (a) and averaged quadratic velocity (b).

Fig. 9. Influence of CNTs distributions on sound transmission loss when CNTs volume fraction was set as 0.11.

crease brings the coincident frequency diminish and lets the sound insulation seem perform better in high frequency range, the tendencies of the STL and averaged quadratic velocity curves are not significantly regular altered. The one-third octave bands of STL and average quadratic velocity of bottom panel are separately compared in Fig. 8, and the correlation between them can be clearly observed. It should be noted that the noticeable STL minimum at low frequency in Fig. 4 is associated with the maximum vibration energy in Figs. 5–6. It is understandable that the stronger vibrations of top panel and the bottom panel will deteriorate the sound insulation characteristics. 4.3. Influence of CNTs distributions on sound transmission loss In order to investigate the influence of CNTs distributions on the transmission sound performance, four different CNTs distributions (UD, FG-X, FG-O and FG-V) are chosen in the numerical calculation. The results are obtained for the CNTs volume fraction of V cnt = 0.11. As can be observed in Fig. 9, among the four curves the one representing the FG-O shows the lowest resonance frequency values in the stiffness controlled region, while that of

Fig. 10. Influence of CNTs distributions on averaged quadratic velocity for the top panel.

Fig. 11. Influence of CNTs distributions on averaged quadratic velocity for the bottom panel.

FG-X type is the highest. The averaged quadratic velocities of the top panel and the bottom panel are illustrated in Figs. 10–11, and the first resonance peak and dip frequency values of averaged quadratic velocity are also observed minimum and maximum for the FG-X and FG-O type stiffened laminated sandwich plate, respectively. This is reasonable to the fact that the CNT volume fractions are maximum at the top and bottom surfaces of the FG-X type stiffened laminated sandwich plate whereas minimum for FG-O type (see Fig. 1). Similar phenomenon has been reported previously, the CNT reinforcements distributed closer to the top and bottom are more efficient than those distributed nearer the mid-plane for increasing the stiffness of CNTs reinforced composite plates [50]. The one-third octave bands of average quadratic velocity of top panel and bottom panel for UD distribution are separately compared in Fig. 12. As shown in the figure, a similar phenomenon can be also seen that since the plane sound wave directly impinged, the averaged quadratic velocity of the top panel is higher than the bottom panel. Taking an overall view of Figs. 9–11, the transmission loss and averaged quadratic velocity curve at medium and high frequencies show more fluctuations compared to low frequency, and the coincidence frequencies f coin for the UD, FG-X, FG-O and FG-V distributions type, are 5015.9 Hz, 4123.1 Hz,

T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

Fig. 12. Averaged quadratic velocity of top panel and bottom panel for the UD type distribution.

101

Fig. 14. Influence of structural damping on sound transmission loss when CNTs volume fraction was set as 0.11.

factor values have no obvious differences. The discrepancies appear in the medium and high frequency regions above 300 Hz, particularly the resonance dip in the frequency of 400 Hz and the resonance peak in the frequency of 600 Hz and 1030 Hz. Moreover, the values of peaks and dips on the curve increase with damping loss factor except for several individual peaks and dips. In other words, the laminated sandwich plates with higher damping loss factor tend to be better sound transmission performance. This is because the increment of damping loss factor leads to the substantial decrease of plate vibration. 4.5. Influence of lamination angle and the number of layers on sound transmission loss

Fig. 13. Influence of CNTs distributions on sound transmission loss (a) and averaged quadratic velocity (b).

6953.3 Hz and 5015.2 Hz respectively. The corresponding one-third octave bands curves of STL and average quadratic velocity of bottom panel are plotted in Fig. 13. As shown, some parts curves seem move a slight distance and the property of sound insulation rises a little when the CNT distributions are varied from FG-O to FG-X type. 4.4. Influence of structural damping loss factor on sound transmission loss This subsection discusses the influence of structural damping loss factor on the sound transmission loss of UD type sandwich plate. In order to better understand the effect of structural damping, three different damping loss factors (η = 0.01, 0.05, 0.09) are chosen in the numerical calculation. The results are obtained for the CNTs volume fraction of V cnt = 0.11. As can be seen in Fig. 14, at the range below 300 Hz, the sound transmission performances of laminated sandwich plates with three different damping loss

As shown in Fig. 1, the laminated composite plates are composed of FG-CNT reinforced composite layers with different lamination angle, wherein the laminated angle represents the laying direction of the single layer material in the composite layer and plays an important role in adjusting the structural property. Therefore in this subsection the effects of lamination angle on the sound transmission performance of FG-X type stiffened laminated sandwich plate with two typical lamination schemes: symmetric [β/−β/β/−β/β ] and anti-symmetric [β/−β/β/β/−β ] plies, are also studied. Five different lamination angles (β = 15◦ , 30◦ , 45◦ , 60◦ and 75◦ ) are chosen in the numerical calculation. Other conditions are consistent with those in Section 4.2. It is interesting to note that since the orientations of stiffened laminated sandwich plate have axial symmetry about 45◦ , the curves of 15◦ and 30◦ overlap the angles of 60◦ and 75◦ , and thus only three different curves are shown in Fig. 15. Similarly, the influence of lamination angle of anti-symmetric laminates is consistent with symmetric laminates, as shown in Fig. 16. In addition, it can be seen from Figs. 15 and 16 that the STL peaks and dips of three curves are gradually shifted to low frequency as the lamination angle increases from 15◦ to 45◦ , and increase when the lamination angle changes from 45◦ to 75◦ . And the effect of lamination angle for symmetric laminates is more obvious, especially for the first resonance dip. Those differences are probably because the different lamination schemes lead to the difference of bending and extension coupling effect, wherein for symmetric laminates, the bending and extension coupling stiffness are equal to zero, while for antisymmetric laminates, the coupling stiffness is not zero. What’s more the effects of lamination angle and the number of layers on coincidence frequency f coin are also illustrated in Fig. 17, wherein

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T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

Fig. 15. Influence of lamination angle on sound transmission loss of the symmetric laminates.

Fig. 17. Influence of lamination angle and number of layer on coincidence frequency: (a) symmetric laminates; (b) anti-symmetric laminates. Fig. 16. Influence of lamination angle on sound transmission loss of the antisymmetric laminates.

the number of layers is changing from four layers to eight layers, and the lamination angle is varied in steps of ten degree. From Fig. 17, it can obviously be seen that with the persisting increase of lamination angle, the coincidence frequency dramatically increases. It is also observed that, for a certain value of lamination angle, the frequencies are found to increase with an increase in the number of layer. Such behavior is due to that the stiffness of stiffened laminated sandwich plate decreases by increasing the lamination angle and the number of layer, and thus causes a corresponding increase of the coincidence frequency. 5. Conclusions In this study, the acoustic behavior of stiffened double laminated functionally graded carbon nanotubes (CNTs) reinforced composite sandwich plate structures subjected to plane sound wave excitation was discussed based on the first order shear deformation theory. Four types of CNTs distributions along the plate thickness are considered, which include uniformly distributed (UD) and three other functionally graded distributions. The extended rule of mixture is employed to determine the properties of the composite material. The compatibility of displacements on the in-

terface between the plate and the stiffeners is employed to derive the governing equation of each stiffener. Then fluid–structure coupling is considered by imposing velocity continuity condition at fluid–structure interfaces. By using the space harmonic approach and virtual work principle, the sound transmission loss is described analytically. The excellent accuracy of the established analytical model is demonstrated by numerical examples and comparison of the present results with those available in the literature. Furthermore, effects of the CNTs volume fraction, lamination angle and CNTs distribution types on the structural and acoustic response of the stiffened double laminated sandwich plate are also investigated. Numerical results show that the noticeable STL minimum at low frequency is associated with the maximum vibration energy of panel, and the peaks and dips of STL curves are found to shift towards higher frequency range with increase of the CNTs volume fraction since the stiffness of stiffened laminated sandwich plate is larger when the CNTs volume fraction is higher. We also found that due to the CNTs volume fractions are maximum at the top and bottom surfaces of the FG-X type stiffened laminated sandwich plate whereas minimum for FG-O type, the first resonance peak and dip frequency of STL curves are observed minimum and maximum for the FG-X and FG-O type stiffened laminated sandwich plates, respectively. In addition, due to the increment of damping loss factor leads to the substantial decrease

T. Fu et al. / Aerospace Science and Technology 82–83 (2018) 92–104

of plate vibration, the sandwich structure tends to be better sound transmission performance. For both symmetric and anti-symmetric angle ply, changing the lamination angle has pronounced influence on the acoustic response of stiffened laminated sandwich plate, and since the orientations of stiffened laminated sandwich plate have axial symmetry about 45◦ , the changing of acoustic response is symmetric with the lamination angle β = 45◦ . The lamination angle and the number of layer also strongly affect the coincidence frequency of the structure, and increasing value of lamination angle and the number of layer result in the increasing of coincidence frequency, which can be favorably tuned to achieve optimal acoustical performances in terms of factual application requirements. Conflict of interest statement The authors have no conflict of interest. Acknowledgement The authors are grateful to the referees for their valuable suggestions. This work presented here were supported by National Key R&D Program of China under the contract number 2017YFB1300600, and by the National Natural Science Foundation of China under the contract number 11772103 and 61304037. Appendix A For a plate composed of different layers materials, the equation of motion for the composite plates subjected a plane sound wave with amplitude P excitation can be obtained as a concise matrix form [16,17]

⎡

L 11 ⎢ L 21 ⎢ ⎢ L 31 ⎣L 41 L 51

L 12 L 22 L 32 L 42 L 52

L 13 L 23 L 33 L 43 L 53

L 14 L 24 L 34 L 44 L 54

⎤⎡

L 15 u L 25 ⎥ ⎢ v ⎥⎢ L 35 ⎥ ⎢ w ⎦⎣ φ L 45 x L 55 φy

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣

0 0 P 0 0

⎤ (A.1)

2 L 11 = − A 11 αm − 2 A 16 αm βn − A 66 βn2 + I 1 ω2 , 2 L 12 = − A 16 αm − A 26 βn2 − ( A 12 + A 66 )αm βn 2 L 14 = − B 11 αm − 2B 16 αm βn − B 66 βn2 + I 2 ω2





Q¯ ikj 1, z, z2 dz,

k =1 t k

A isj = κ

N 

t k +1 

Q¯ ikj dz

(A.2)

k =1 t k

where Q¯ ikj are the transformed components of the kth layer stiffness which are defined in Reddy [38] as follows:





k k k k Q¯ 11 ( z) = Q 11 ( z) cos4 θ + 2 Q 12 ( z) + 2Q 66 ( z) sin2 θ cos2 θ k ( z) sin4 θ + Q 22  k k k k Q¯ 12 ( z) = Q 11 ( z) + Q 22 ( z) − 4Q 66 ( z) sin2 θ cos2 θ  k ( z) sin4 θ + cos4 θ + Q 12  k k k k Q¯ 16 ( z) = Q 11 ( z) − Q 12 ( z) − 2Q 66 ( z) sin θ cos3 θ  k k k ( z) − Q 22 ( z) + 2Q 66 ( z) sin3 θ cos θ + Q 12  k k k k Q¯ 22 ( z) = Q 11 ( z) sin4 θ + 2 Q 12 ( z) + 2Q 66 ( z) sin2 θ cos2 θ k ( z) cos4 θ + Q 22   k k k k k Q¯ 26 ( z) = Q 11 ( z) − Q 12 ( z) − 2Q 66 ( z) sin3 θ cos θ + Q 12 ( z) k k 3 − Q 22 ( z) + 2Q 66 ( z) sin θ cos θ  k k k k k Q¯ 66 ( z) = Q 11 ( z) + Q 22 ( z) − 2Q 12 ( z) − 2Q 66 ( z) sin2 θ cos2 θ  k ( z) sin4 θ + cos4 θ + Q 66 k k k Q¯ 44 ( z) = Q 44 ( z) cos2 θ + Q 55 ( z) sin2 θ  k k k Q¯ 45 ( z) = Q 55 ( z) − Q 44 ( z) cos θ sin θ k k 2 k ¯ Q 55 ( z) = Q 55 ( z) cos θ + Q 44 ( z) sin2 θ (A.3)

in

⎥ ⎥ ⎥ ⎦

where L i j are the differential operators, The detailed expression of L i j in Eq. (A.1) are listed in the following [17]:

L 13 = 0,

( Ai j , B i j , D i j ) =

t N k+1 

103

which

k Q 11 (z) = E k11 (z)/(1 −

k k ν12 (z)ν21 (z)),

k Q 12 ( z) =

k k k k k k E k12 ( z)ν12 (z)/(1 − ν12 (z)ν21 (z)), Q 22 (z) = E k22 (z)/(1 − ν12 (z)ν21 (z)),

k k k Q 66 (z) = G k12 (z), Q 44 (z) = G k23 (z), Q 55 (z) = G k13 (z). In Mindlin’s model [43], the shear correction factor κ is π 2 /12 and the same value is used in this paper. Substituting coefficients L i j into Eq. (A.1) and solving the matrix in Eq. (A.1), the transverse displacement w is obtained and the dynamic stiffness of the panel associated with the wave propagating in the structure can be derived:

K∗ =

P w

.

(A.4)

2 L 15 = − B 16 αm − B 26 βn2 − ( B 12 + B 66 )αm βn , 2 L 22 = − A 66 αm − A 22 βn2 − 2 A 26 αm βn + I 1 ω2

L 23 = L 25 = L 33 = L 34 =

References

2 0, L 24 = − B 16 m − B 26 βn2 − ( B 12 + B 66 ) m βn 2 − B 66 m − B 22 βn2 − 2B 26 m βn + I 2 2 s 2 s s 2 2 A 55 − 0 2 /k z,mn , m + 2 A 45 m βn + A 44 βn − I 1 s s j A 55 m + j A 45 βn

α

α

α

α

α

α

ω

ω

ρ ω

α

L 45 =

s s j A 45 m + j A 44 βn , 2 s − D 11 m − D 66 βn2 − 2D 16 m βn − A 55 + I3 2 2 s − D 16 m − ( D 12 + D 66 ) m βn − D 26 βn2 − A 45

L 55 =

2 − D 66 m

L 35 = L 44 =

α

α

α

α

L i j = L ji ,

α −

ω

α

D 22 βn2

− 2D 26 αm βn − A 44 + I 3 ω2

( i , j = 1, 2, 3, 4, 5)

where αm = kx + 2mπ /l x and βn = k y + 2nπ /l y . The extensional A i j , coupling B i j , bending D i j and transverse shear A isj stiffness are given by

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