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Sound transmission loss of laminated composite sandwich structures with pyramidal truss cores

T

⁎

Dong-Wei Wanga, Li Maa, , Xin-Tao Wanga, Zhi-Hui Wena, Christ Glorieuxb a b

Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China Laboratory of Acoustics, Division of Soft Matter and Biophysics, Department of Physics and Astronomy, KU Leuven, B3001 Heverlee, Belgium

A R T I C LE I N FO

A B S T R A C T

Keywords: Composite Sandwich structure Pyramidal truss cores Sound transmission loss First order shear deformation theory

This paper presents a theoretical model that allows to calculate the acoustic transmission loss through laminated composite pyramidal sandwich structure consisting of two parallel plates connected by trusses. First-order shear deformation theory is adopted to model the vibration of such a structure, accounting for in-plane motion and coupling between extension and ﬂexure of the diﬀerent components, and taking into account elastic anisotropy. The interaction between the structure and the surrounding ﬂuid is taken into account by imposing a velocity continuity condition at the interfaces. The displacement and stress ﬁelds are calculated in Fourier domain by solving the set of boundary condition equations at the connecting points. The theoretical predictions for the acoustic transmission through the structure show satisfactory agreement with experimental results of a standing wave tube experiment on specimens that were fabricated by cutting trusses from carbon ﬁber reinforced composite plates and snap-ﬁtting them to plates made of the same material. Numerical simulations are used to verify, the inﬂuence of the stacking geometry and material parameters on the acoustic transmission for diﬀerent frequencies and angles of incidence. Conclusions are presented that are helpful for practical design.

1. Introduction Laminated ﬁber-reinforced composite sandwich structures oﬀer a great prospects for a broad range of application in the ﬁelds of aviation [1], shipping [2] and transportation [3]. Compared to classical structures, they have superior properties in terms of speciﬁc strength and stiﬀness [4,5]. Their properties are also suitable for applications in heat transfer, impact resistance [6] and energy absorption [7]. The core of a sandwich structure is conventionally made of foam and honeycomb [5]. Compared to foam-ﬁlled structures, sandwich structures with a lattice geometry encompass additional potential by virtue of the presence of open space in the core, which can be used for placing components, thus allowing for multi-functional applications that require the structure to carry loads. Till now, most of the research on these structures has focused on the manufacturing process [8], the determination of their static properties [9]. Their outstanding dynamic performance has been assessed in terms of vibration and shock loading [10–12], but little work was done on the determination of their acoustic performance [13]. This paper tackles the question of acoustic transmission through and radiation from lattice sandwich plates, starting from existing studies on classic sandwich and stiﬀened plates. Many authors discussed the wave

⁎

propagation in layered media by the method of vibration theory of plate [14–16], ﬁnite element analysis [17–19] and energy method [20–22]. In the classic vibro-acoustic method, a straightforward method to investigate the wave propagation in a sandwich structure of plates connected by trusses is to make a coarse approximation of the core region, considering it as a homogenous layer in a multilayered plate [23]. However, even in the case where the wavelength is longer than the distance between nearby trusses, this approximation yields a poor prediction of the sound transmission loss (STL). Mead and Pujara [24] proposed a spatial harmonic model to study the structural response of plate that was stiﬀened by a periodical array of beams. Lee [25,26] extended the model and studied acoustic wave propagation through periodically rib-stiﬀened plate and cylindrical shell, and investigated the eﬀect of relevant design parameters. Wang [7] applied the model on lightweight double-leaf partitions and compared it to results of a method that made use of homogenization. Mace [27] utilized a spatial Fourier transform (SFT) to solve the vibration response of an orthogonally stiﬀened plate, which was expressed by two-dimensional model. Zhang [28] optimized the radiation of a rib-stiﬀened sandwich structure by making use of active control via a periodic array of shunted piezoelectric patches. Zhou [29,30] and Liu [31] extended the method to cylinder and investigate its sound insulation property with

Corresponding author. E-mail address: [email protected] (L. Ma).

https://doi.org/10.1016/j.compstruct.2019.03.077 Received 2 December 2018; Received in revised form 15 February 2019; Accepted 21 March 2019 Available online 26 March 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

Composite Structures 220 (2019) 19–30

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Nomenclature

mp mass density of the plate αt elevation angle of truss azimuthal angle of truss ϑj Lt length of truss l length of unit cell b width of truss W amplitude of displacement θ elevation angle of incident wave φ azimuthal angle of incident wave ω angular frequency kx, ky (α, β) wavenumbers in x- and y- direction γ wavenumber in z-direction ρ0 air density c0 sound speed in air Iin intensity of incident wave Itr intensity of transmitted wave τ sound transmission coeﬃcient STL sound transmission loss E11,E22,E33 modulus of the ﬁber-reinforced composite layer G12,G13,G23 shear modulus of the ﬁber-reinforced composite layer N boundary of numbers of components ν Poisson’s ratio k transformed plane stress-reduced stiﬀness of kth layer Q¯ x Ki equivalent stiﬀness fco coincidence frequency fst standing-wave frequency

u, v, w displacements of plate u0, v0, w0 midplane displacements of plate δ Dirac delta function rotational angles ϕx ,ϕy Nxx, Nxy, Nyy in-plane force resultants Mxx, Mxy, Myy moment resultants Qx , Qy transverse force resultant I0, I1, I2 mass moments of inertia h thickness of plate d height of core ρ density of the structure p acoustic pressure perturbation P amplitude of pressure perturbation Fx, Fy, Fz forces applied on the plates by truss F1, F2 axial force of truss Mrx, Mry moments applied on the plates by truss Q1, Q2 bending force of truss M1, M2 moments acting on truss Aij extensional stiﬀness Dij bending stiﬀness Bij bending-extensional coupling stiﬀness Kr rotational stiﬀness of truss K equivalent spring stiﬀness of a half truss KQ shear stiﬀness of truss D bending stiﬀness of the plate

transmission loss of an anisotropic composite laminated sandwich structure with pyramidal truss cores. The present work employs FSDT to establish 3D periodic model immersed in acoustic ﬁeld and uses SFT to solve the governing equations with periodic constraints. Numerical simulations are made to investigate the inﬂuence of the wave parameters, the material properties and the geometry on the sound transmission, which the goal of aiding the design process of this kind of structures while taking into account acoustic requirements.

poroelastic core. The plate materials considered in references [14,15,23–28] were elastically isotropic. When ﬁber reinforced composite plates are considered, anisotropy of the elastic behavior of the plate material has to be taken into account. Yin [32] used classic laminated composite plate theory (CLPT) to study a laminated composite plate that was reinforced by doubly periodic parallel stiﬀeners. A more accurate method, ﬁrst order shear deformation theory (FSDT), was recently explored. Based on this theory, Shen [33,34] studied sound transmission through laminated composite plates reinforced by two sets of orthogonal stiffeners. The in-plane motion, extensional, ﬂexural and their coupling eﬀects were accounted for. Cao [35] investigated the acoustic response of composite sandwich plates connected by rods to noise generated by a subsonic turbulent boundary layer. Talebitooti [36] calculated the sound transmission loss across laminated composite cylindrical shells separated by an air gap by means of beams made of composite material. The used governing equations for the wave propagation in the beams did not taken into account the elastic anisotropy of the beams. In the aforementioned investigations, there is no mention of

2. Theoretical formulations The considered composite lattice sandwich structure (Fig. 1) consists of two parallel laminated face plates that are lined with composite pyramidal truss cores. The origin of the coordinates system is chosen on the top surface of top plate. The z-axis is perpendicular to the plate, pointing towards the core. The local coordinate systems for the plate and for the trusses are deﬁned on the symmetry plane in their middle. The structure is immersed in ﬂuid (e.g. air). In the following, the article describes the interactions between

Fig. 1. Models of (a) pyramidal lattice structure and (b) unit cell. 20

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D.-W. Wang, et al.

~ε δ tFxt ⎧ u0 ⎫ ⎧ ⎫ 0 ⎫ ⎪ v~0ε ⎪ ⎪ ⎪ ⎧ δ tF yt 0 ⎪ ⎪ ~ε⎪ ⎪ ⎪ ⎪ M0· w0 + + p~int + p~ret ∓ p~coε − p~trb = 0 δ tFzt ⎨ ⎨ ~ε ⎬ ⎬ ⎨ ⎬ 0 ⎪ ϕx ⎪ ⎪ hεδ tFxt /2 + δ tMrxt ⎪ ⎪ ⎪ ε ~ ⎪ϕ ⎪ ⎪ ε t t ⎪ ⎩ 0 ⎭ h δ F y /2 + δ tMryt y ⎩ ⎭ ⎩ ⎭

(5)

where M0 shown in Appendix B is the matrix consisting of extensional stiﬀnesses Aij, bending-extensional coupling stiﬀnesses Bij and bending stiﬀnesses Dij [37]. Operator denotes the operation of spatial Fourier transformation. Fig. 2. Model of a truss showing the diﬀerent axial (F) and bending (Q) forces, and moments (M). K and Kr are the axial and bending stiﬀnesses respectively.

2.1. Forces and moments between trusses and plates

forces and displacements in the diﬀerent components of the structure, in order to determine the acoustic transmission of an incident airborne wave, and from that the acoustic insulation of the composite plate structure. The calculations take into account both the structure borne and airborne transmission paths. Using ﬁrst-order shear deformation theory (FSDT) [37], the displacements of composite face sheets in a pyramidal lattice sandwich structure can be expressed as:

The trusses and plates are interacting via forces and moments at their connection points, which are periodically spaced both in the x and y directions. The main deformations of the trusses, i.e. their bending and compression, are depicted in Fig. 2. The displacements of two ends along truss are deﬁned as (u1tr , u2tr ) respectively, and the corresponding rotational angles are (ϕxtr1, ϕxtr2) . The axial forces F1, F2 and bending forces Q1, Q2 along the truss can then be expressed as [38]:

uε (x , y, z , t ) = u0ε (x , y, t ) + zϕxε (x , y, t )

F1 = −

K (K − mω2) tr u1 2K − mω2

v ε (x , y, z , t ) = v0ε (x , y, t ) + zϕyε (x , y, t )

F2 = −

K2 u tr 2K − mω2 1

Q1 = −

KQ (KQ − mω2) tr v1 2KQ − mω2

+

Q2 = −

KQ2 v tr 2KQ − mω2 1

KQ (KQ − mω2) tr v2 2KQ − mω2

w ε (x ,

w0ε (x ,

y, z, t ) =

y, t )

+

−

I0ε v0,ε tt

−

I1ε ϕyε, tt

+

δ εFy

M1 = −

=0

Qxε, x + Qyε, y − I0i w0,ε tt + δ εFz + pint + pret ∓ pcoε − ptrb = 0 ε ε ε ε ε ε ε ε ε Mxε, x + Mxy , y − Q x − I1 u 0, tt − I2 ϕx , tt + h δ Fx /2 + δ Mrx = 0 ε ε ε ε ε ε ε ε ε ε Mxy , x + My, y − Qy − I1 v0, tt − I2 ϕy, tt + h δ Fy /2 + δ Mry = 0

−∞

n∈Z

1 l

2inπx ⎞ ⎝ l ⎠

∑ exp ⎛ n∈Z

Kr (Kr − ρI2tr ω2) ∂ϕxtr1 2Kr − ρI2tr ω2

∂x 1

∂ϕx1 Kr2 2Kr − ρI2tr ω2 ∂x 1

htr /2

+

tr

+

∂ϕx 2 Kr2 2Kr − ρI2tr ω2 ∂x 1

Kr (Kr − ρI2tr ω2) ∂ϕxtr2 2Kr − ρI2tr ω2

(8)

∂x 1

I2tr

in the laminate structure of htr /2

k

k

tr

tr

h /2 h /2 k k k I2tr = ∫−htr /2 z 2dz , K Q = 4b ∫−htr /2 Q¯ x (1 − v12 v21 ) z 2/(Lt /2)3dz

(9) k where Q¯ x

is the transformed plane stress-reduced stiﬀnesses of kth layer along x direction, Lt is the length of truss, htr is the thickness of truss, b is the width of truss, νij is Poisson’s ratio deﬁned as the ratio of transverse strain in the jth direction to the axial strain in the ith direction when stressed in the ith direction. The relation between the forces on the end of a truss and the associate reactive forces on the connecting plate can be written as: (ε ) 0 ⎤ ⎡ cαc ϑj sαc ϑj ⎧ Fx ⎫ ⎢ cαs ϑj sαs ϑj 0 ⎥ Fi ⎪ Fy ⎪ ⎧ Fi ⎫ ⎪ ⎪ ⎥⎧ ⎫ ⎢ 0 ⎥ Qi = R Qi Fz = ⎢ sαt − cαt ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎢ 0 0 (c ϑj )2 ⎥ ⎩ Mi ⎭ ⎩ Mi ⎭ ⎪ Mrx ⎪ ⎥ ⎢ ⎪ Mry ⎪ 2 s 0 0 ( ϑ ) ⎢ j ⎥ ⎩ ⎭ ⎦ ⎣

(3)

According to Poisson summation formula, notice that [38]:

∑ δ (x − nl) =

(7)

k k k k K = b ∫−htr /2 Q¯ x (1 − v12 v21 ) dz / Lt /2, Kr = b ∫−htr /2 Q¯ x (1 − v12 v21 ) z 2dz

hε /2

−∞

KQ2 v tr 2K − mω2 2

The stiﬀnesses and the rotary inertia interest can be obtained by [37]:

(2)

delta function, (I0ε , I1ε , I2ε ) = ∫−hε /2 ρ ε (1, z , z 2) dz are the mass moments of inertia. pin and pre denote the incident and radiated pressures in the incident acoustic ﬁeld. pco is the pressure in the core and ptr represents the pressure in the transmitted ﬁeld. A comma in subscripts denotes partial diﬀerentiation with respect to the subscript u0,x=∂u0/∂x, and so on. The associated stress forces and moments are deﬁned in Appendix A. In order to solve Eq. (2), a spatial Fourier transform is carried out where α and β are the transformed wavenumbers in the x and y-direction:

∼ (α, β ) = ∫+∞ ∫+∞ w (x , y ) e j (αx + βy) dxdy w −∞ −∞ +∞ +∞ ∼ 1 w (x , y ) = (α, β ) e−j (αx + βy) dαdβ ∫ ∫ w

+

tr

M2 = −

where Fx, Fy, Fz, Mrx, Mry are forces and moments applied on the plate by a truss, h is the thickness of the top or bottom plate and δ is the Dirac

4π 2

(6)

where K is the equivalent spring stiﬀness of a half truss, Kr is the rotational stiﬀness, KQ is the equivalent shear stiﬀness and m is the mass. The moments indicated in Fig. 2 take the form:

ε ε ε ε ε ε ε Nxx , x + Nxy, y − I0 u 0, tt − I1 ϕx , tt + δ Fx = 0 ε N yy ,y

K (K − mω2) tr u2 2K − mω2

+

(1)

where the superscript ε = t or b refers to the top or bottom plate, u, v and w are the displacements along the x-, y- and z-axis, u0, v0 and w0 are the midplane displacements, ϕx and -ϕy are the rotational angles about yand x- axis, respectively. The Euler-Lagrange equations of the plate incorporating the reaction forces due to the truss in the core can then be written as [37]:

ε Nxy ,x

K2 u tr , 2K − mω2 2

+

(10)

where αt, ϑj are the elevation and azimuthal angles of trusses respectively as shown in Fig. 1(b), sαt and cαt are short notations for sine and cosine of αt. For pyramidal cores, ϑj = (2j − 1) π /4 , j = 1,2,3,4. The displacements of trusses and plates are connected by:

(4)

Combining Eqs. (2)–(4), the preliminary equations of motion in wavenumber domain are established as:

{uitr , vitr , ϕxitr }T = RT {u0ε , v0ε , w0ε , ϕxε , ϕyε }T 21

(11)

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D.-W. Wang, et al.

The locations of the connecting joints in the unit cell are [δ(x − nl), δ(y − ml)] for the top plate and [δ(x − l/2 − nl), δ(y − l/2 − ml)], [δ(x + l/2 − nl), δ(y − l/2 − ml)] [δ(x + l/2 − nl), δ(y − l/2 − ml)], [δ(x + l/2 − nl), δ(y + l/2 − ml)] for the bottom where n,m are integers. Incorporating Eqs. (3),(4),(6)-(11) and taking the discrete connecting joints into account, the reactive forces and moments on two plates can be expressed in terms of the displacements:

wave in the air The ﬂuid-structure coupling must be characterized by continuity of the velocity component normal to the plate at the diﬀerent interfaces, leading to:

δ tFxt = 4cos2 ϑ(K1cos2 αt + K5sin2 α )·u0t δ1 + (K2 [U b] + K 6 [V b])[δ2 ]

vz |z = ht + d = − jωρ

δ tF yt

=

4sin2 ϑ(K1cos2 αt

+ K5

sin2 α

1

vz |z = 0 = − jωρ

1

vz |z = ht = − jωρ

=

4(K1sin2 αt

+ K5

cos2 α

t t )· v0 δ1

t t )· w0 δ1

+

δ bFxb = 4cos2 ϑ(K2cos2 αt + K 6sin2 αt )·u0t δ1 + (K1 [U b] + K5 [V b])[δ2 ] δ bFyb = 4sin2 ϑ(K2cos2 αt + K 6sin2 αt )·v0t δ1 + (K1 [U b] + K5 [V b])[δ2 ] δ bFzb = 4(K2sin2 αt + K 6cos2 αt )·w0t δ3 + (K1 [U b tan αt /cos ϑj]

= jωw0t ,

z = ht + d

∂ptr 0 ∂z

= jωw0b,

z = ht + hb + d

= jωw0b,

(17)

∼ pin (α, β , 0) = 4π 2P1 δ (α − k 0 sin θ cos φ) δ (β − k 0 sin θ sin φ) ∼ ∼t (α, β )/ jγ + ∼ p (α, β , 0) = ω2ρ w p (α, β, 0)

t /cos ϑj ])[δ2]

0

re

(13)

∼ pco (α, β , z ) =

Similarly,

0

in

ρ0 ω 2 jγ (e jγd − e−jγd)

∼b (α, β ) e jγht − w ∼t (α, β ) e jγ (ht + d) ] e−jγz {[w 0 0

∼b (α, β ) e−jγht − w ∼t (α, β ) e−jγ (ht + d) ] e jγz } + [w 0 0 t b ∼ ∼b (α, β )/ jγ ptr (α, β , ht + hb + d ) = −ρ0 ω2e jγ (h + h + d) w 0

δ tMrx = 4K3cos2 ϑ·ϕxt , x δ1 + K 4 cos2 ϑ{Φbx , x }{δ2 }T δ tMry = 4K3sin2 ϑ·ϕyt , y δ1 + K 4sin2 ϑ{Φby, y }{δ2 }T

(18)

δ bMrx = 4K 4 cos2 ϑ·ϕxt , x δ1 + K 4 cos2 ϑ{Φbx , x }{δ2 }T

where γ = (14)

k 02

−

α2

−

β2 .

2.3. Solution

where Ki (i = 1 ∼ 6) and δj (j = 1 ∼ 4), are equivalent stiﬀnesses and combinations of Dirac function, respectively. Operating Eqs. (12)-(14) with Fourier transformation, the transformed forces and moments can be rewritten by the sum of the displacements in the form as: n + m∼ ∼t u 0b (α n, βm ) ⎫ ⎧ ∑ (−1) ε ε ⎧ ∑ u 0 (α n, βm ) ⎫ ⎧ δ Fx ⎫ ⎪ ⎪ ⎪∑∼ n + m∼ v 0b (α n, βm ) ⎪ v t0 (α n, βm ) ⎪ ⎪ δ ε F εy ⎪ ⎪ ∑ (−1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∼t ∼ b (α , β ) ⎪ n + mw n 0 m F δ ε F zε = M1ε ∑ w 0 (α n, βm ) + M2ε ∑ (−1) ⎬ ⎨ ⎬ ⎨ ∼t ⎨ ε ε ⎬ ∼b n m + δ M ϕ ( α , β ) ∑ ϕx (α n, βm ) ⎪ n ⎪∑ (−1) ⎪ ⎪ rx ⎪ m ⎪ x ε ⎪ ⎪ ⎪ ⎪ ∼t ⎪δεM ⎪ ∼b ry ⎭ ⎪ ∑ ϕy (α n, βm ) ⎪ ⎩ ⎪ ∑ (−1) n + mϕy (α n, βm ) ⎪ ⎭ ⎩ ⎭ ⎩

∂pco 0 ∂z

= jωw0t ,

where ω is angular frequency, ρ0 is the density of air and d is the thickness of the core. Notice that pco is the wave in the core including positive and negative going waves, expressed as pco (x , y, z ) = pco+ (x , y ) e−jγz + pco− (x , y ) e jγz Solving the second order differential equation (16) with the boundary equation and performing a 2D Fourier transformation, the wave pressures are obtained in wavenumber domain [33,39]:

(K2 [U b tan α /cos ϑj]

δ bMry = 4K 4sin2 ϑ·ϕyt , y δ1 + K3sin2 ϑ{Φby, y }{δ2 }T

z = ht

1

(12)

+ K5

∂pco 0 ∂z

vz |z = ht + hb + d = − jωρ

+ K 6 [V b tan α /cos ϑj ])[δ2]

[V b tan α

∂ (pin + pre ) ∂z z=0

1

+ (K2 [U b tan ϑj] + K 6 [V b tan ϑj ])[δ2 ] δ tFzt

0

Incorporating Eqs. (5), (15) and (18), the governing equation could be written as:

Mt ·W t (α, β ) + Mt1· ∑ W t (αn, βm) + Mb1· ∑ W b (αn, βm) = P Mb·W b (α, β ) + Mt 2· ∑ W t (αn, βm) + Mb2· ∑ W b (αn, βm) = 0

(19)

ε

where W , ε = t,b are two sets of Fourier domain displacement components, deﬁned as:

W ε (αn, βm) = {u0ε (αn, βm), v0ε (αn, βm), w0ε (αn, βm), ϕxε (αn, βm), ϕyε (αn, βm )}T

(15)

(20)

where αn = α + 2πn/l, βm = β + 2πm/l with l the size of a unit cell. M2ε are coeﬃcient matrices. The details about the equivalent stiﬀnesses and the transformed expression can be found in Appendix C.

The intensity of incident wave above and transmitted wave in the air below the bottom plate can be obtained from the displacements of the bottom plate via [33]:

2.2. Acoustic loading

Iin =

M1ε ,

In view of obtaining the sound insulation of the structure, an obliquely incident plane wave impinges on the top plate and generates a reﬂected wave, a transmitted wave and structural deformations. Both the transmitted wave in air in the core and the trusses in the core act upon the bottom plate, which is then re-radiating into the air. Omitting the harmonic time dependence ejωt for the sake of conciseness, the incident wave is given by pin = P1 e−jk 0 (sin θ cos φx + sin θ sin φy + cos θz ) , where k0 is the wavenumber, θ and φ are elevation and azimuthal angles of incident plane wave. The spatial part of the acoustic waves must satisfy the Helmholtz equation 2 ⎛⎜∇2 − 1 ∂ ⎟⎞ p (x , y, z , t ) = 0 2 c0 ∂t 2 ⎠ ⎝

P12 cos θ , 2ρ0 c0

Itr =

ρ0 ω3 8π 2

|w0b (αn, βm )|2

∑ 2

k 02 − αn2 − βm2

(21)

where, the angularly averaged acoustic transmission coeﬃcient of the structure is given by 2π

τ (θ , φ) = |Itr |/|Iin|,

τ¯ =

θmax

∫0 ∫0

2π

τ (θ , φ) sin θ cos θdθdφ

θmax 0

∫0 ∫

sin θ cos θdθdφ

. (22)

The sound transmission loss (STL) is then found as:

STL = −10log10 τ¯

(23)

2.4. Convergence check of numerical simulations The parameters of the structure listed in Table 1 are used to investigate its acoustic properties in this paper. In the table, E11, E22 and E33 are the modulus of the ﬁber-reinforced composite layer with material coordinate system (x1, x2, x3) as shown in Fig. 3(b) [37]. E11 is the

(16)

where p stands for the pressure in the incoming air region, pin and pre, in the core, pco and in the outgoing air region, ptr. c0 is the speed of sound 22

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3. Experimental validation

Table 1 Mechanical and geometry properties of the unidirectional carbon/epoxy (T700/epoxy) laminate and sandwich structure. Property

Value

Longitudinal modulus E11 Transverse modulus E22, E33 Poisson’s ratio ν12 Shear modulus G12, G13, G23 Density ρ Thickness of plate h Thickness of core d Elevation angle of truss αt Azimuth angle of trussϑt Stacking angles Distance between joint l

122 GPa 8.5 GPa 0.28 4 GPa 1560 kg/m3 1.2 mm 13.9 mm 45 degrees 45 degrees (0/90/0/90/0/90)s 18 mm

In order to validate the theoretical model in this paper, a sound transmission measurement was conducted on a sandwich structure with pyramidal truss cores, which was manufactured by a slitting and interlocking assembly method using carbon ﬁber reinforced polymer (CFRP) laminates sheets. The composite laminate sheets were made from unidirectional carbon ﬁber/epoxy prepregs (T700/epoxy composite, Shanghai Kangzhan Composites Co., Ltd, China). The detailed properties of the unidirectional carbon/epoxy composite laminate are listed in the Table 1. These properties were determined according to ASTM D3039/D3039M-08, D6641/D6641M-09 and D3518/D3518M13 [40]. The manufacturing process of sandwich structure with pyramidal truss cores as shown in Fig. 5 contains 3 steps:

longitudinal Young’s modulus along the x1-axis. The ﬁber direction is oriented along the x1-axis. E22 and E33 are the transverse modulus of the layer along corresponding axis. The cross section of the cross-ply composite plate is shown in Fig. 3(c). The shadow parts are the layers where the ﬁbers are perpendicular to x-axis while others are those where the ﬁbers are parallel to the x-axis. The ﬁbers in truss are set as parallel or perpendicular to the longitudinal axis of the truss, shown in Fig. 3(a). The component number of the governing equations with harmonic waves are inﬁnite. Although the accuracy rises as the number increases, the complexity and computation time increase tremendously. Therefore, the waves must be truncated to have reasonable limit boundary n,m = [−N, N] insofar as its convergence. The well acknowledged criterion [25] is employed which shows that once the results converge at a bound, they will converge for all lower frequencies. The highest frequency of interest of the present paper is 20 kHz. The convergence algorithm as shown in Fig. 4(a) is implemented to ensure that the value of STL with inﬁnite wave components has converged at an appropriate number. The criterion of the convergence is deﬁned as at diﬀerence of STL lower than 0.1 dB. Fig. 4(b) depicts the the convergence of the numerical model prediction for STL as a function of the number of spatial frequency components per direction N at a frequency of 20 kHz. The rate of convergence gradually improves and the STL value change from N = 12 to N = 13 is as small as 0.01 dB and therefore the result converges in the concerning frequency range at N = 12 and each displacement are 25th order expansion functions.

a) Manufacturing CFRP laminate sheets from unidirectional carbon ﬁber/epoxy prepregs; b) Utilizing computer numerical control (CNC) milling to fabricate plates and two dimensional (2D) truss patterns from CFRP laminate sheets; c) Interlocking assembly 2D truss patterns and pasting to the face sheets. The laminated composite sheet used in the experiment was a 12-ply laminate. An impedance tube (Model SW477, BSWA Ltd., CHN) shown in Fig. 6, was used to determine the sound transmission loss. A loudspeaker at the leftmost end of the tube generated a random noise signal over the frequency range 60–1600 Hz. The source loudspeaker is connected to the ampliﬁer and then the signal analyzer. The analyzer collects the data from four microphones at measuring positions and sends to the computer to handle the sound signals with the software to get the sound insulation value. The specimen with a diameter 100 mm is put in the tube. Each side there are two microphones measuring the positive and negative travelling waves. The experiments were accomplished by two independent testing measurements, conducted separately with open termination and tube termination. The software handled the data of the two measurements and obtained the insulation value. During the measurement, the silicone grease was used at the edge of the specimens to achieve free boundary as well as sealing. Fig. 7 shows that the experimental results are reasonably well predicted by the theoretical model. The values rise as frequency increases

Fig. 3. The sketch of (a) the truss and (b) ﬁber-reinforced layer and (c) composite plate. 23

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Fig. 4. (a) Iterative algorithm for ﬁnding the maximum boundary number N. (b) Convergence of STL versus boundary number N at 20 kHz.

Fig. 5. Illustration of manufacturing process of composite sandwich structure with pyramidal cores. (a) Patterns cut from the laminate sheets. (b) Interlocking assembly of truss cores. (c) Bonding truss cores with the face sheets. (d) Specimen.

truss core are shown below:

following the same trend. The discrepancy varies from 0 to 4 dB. Except in the range 500–580 Hz, the discrepancy is no more than 10% which is considerable. The structural mode are widely aﬀected by the edge boundary condition when the bending wavelength is larger than the distance between stiﬀened connections [41]. The boundary condition between specimen and the hard-wall tube is not ideal free. On the other hand, the model is used to predict the sound insulation of inﬁnite structure whilst the specimen is ﬁnite in experiment. Besides the experimental error, these are the main reasons that cause discrepancy especially in low frequency range.

4.1. Eﬀect of plate theory approximation The used ﬁrst order shear deformation theory can be simpliﬁed to classic laminated plate theory (CLPT) by setting ϕxε = −∂w0ε / ∂x , ϕyε = −∂w0ε / ∂y . Fig. 8 shows a comparison between CLPT and FSDT. The two curves coincide well in the mass-controlled low frequency region. The at higher frequencies, small deviation exists. The dips representing acoustic resonances are keep still because the size and the external conditions are same. In the theory of FSDT, the transverse normals do not remain perpendicular to the midsurface after deformation and the transverse shear strains in taken into account. FSDT is more accurate at the price of more computational complexity.

4. Parameter study In the following, the eﬀects of the diﬀerent structural and material parameters on the sound transmission properties of a composite pyramidal sandwich structure are investigated with the aim of ﬁnding design rules to maximize its sound transmission loss. The base conﬁguration is the one of Fig. 1, with parameters in Table 1. The ﬁbers in the truss core follow the directions of the trusses. The geometrical ﬁgure of

4.2. Dependence on the angle of incidence Fig. 9(a) shows the frequency dependence of the sound transmission loss for diﬀerent elevation angles θ. It is obvious that the elevation 24

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Fig. 6. (a) Schematic of the experimental system. (b) Photograph of the impedance tube.

Fig. 8. STL versus frequency curves based on FSDT and CLPT.

Fig. 7. Experimental results and model prediction for the sound transmission loss of the structure described in Fig. 1 and Table 1.

constraints bring about discrete wavenumber and produce many symmetric resonances. The classic resonances in a plate structure occur at the coincidence frequency and at standing-wave resonance frequencies. The former occurs when the velocity of the bending wave matches the projection of the propagation velocity vector of the incident wave on the plane of the plate [7], while the latter happens when the projection of half of the wavelength along the direction of wave in the core onto the normal to the plate ﬁts an integer number of times into the height d of the core region:

angle plays a signiﬁcant role on STL and denser resonance dips appear on STL versus frequency curves in oblique incident case than those in the normal case. Apart from several individual resonant regions, the averaged STL values turn smaller as θ rises, particularly in low range. It is mainly because the possibility of the interference between sound wave and structural bending wave. In other words, the normal incident meets best acoustic insulation in general. To be more speciﬁc, the curves show a similar trend of increasing STL with increasing frequency at low frequencies, known as mass-law, and contain several peaks and dips at high frequencies, which strongly vary with changing angle of incidence. The STL curves of a simple plate can be divided into three regions: stiﬀness-controlled, resonance-controlled, mass-controlled and damping-controlled regions [42]. For the considered pyramidal sandwich structure the behavior is more complicated. The periodic

fco =

c02 2π sin2 θ

mp D

,

fst =

c0 n , 2d cos θ

n = 1, 2, 3. ..

(24)

where D and mp denote the bending stiﬀness and the surface density respectively. The ﬁrst order standing-wave resonances (denoted by “●”) found 25

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Fig. 9. Dependence of the STL curves on (a) elevation angle θ and (b) azimuthal angle φ.

the spacing between peaks becomes larger as αt rises and the match between the wave and spacing happen at higher frequency. Comparing to the periodic structures [7,33,44], the spacing of discrete joints in this paper is around 10 times smaller than those. That is the reason why the curves seems coincide in low frequency range and discrepancies mainly exist in high frequency range.

from the curves in Fig. 9 are 12.3, 14.7 and 17.4 (kHz) for θ = 0, 30, 45 degrees respectively. These values match pretty well with the values predicted by Eq. (24), respectively 12.2, 14.1 and 17.3 (kHz). As the dominant motion of the plate is bending, the relevant bending stiﬀness component to calculate the coincidence frequency is D11 [23,37]. Inserting D11 in Eq. (24), the coincidence frequencies are obtained as 18.5 and 12.3 (kHz) (denoted by “ ”) for θ = 30, 45 degrees angle of incidence respectively, which is reasonably close to the respective peak frequencies, 18.9 and 13.0 (kHz). Other peaks and dips in the STL spectrum can be related to the dispersion relation bands structure that is evoked by the periodicity of the model. The peaks appear when the nodal points of the vibrational plate exposed to the incident wave happen at the periodic connecting points with trusses. That is no vibrations are transmitted through the truss which becomes perfect reﬂector. The contrary situation is that the wavenumber of incident wave matches that of periodic structure, i.e. large displacement occurs at connecting points [7]. Moreover, the value drops down generally as the incident angle increases. Vertical incident wave meets best sound insolation performance. For a ﬁxed elevation angle, the STL curve is shown for azimuthal angle φ varying from 0 to 45 degrees in steps of 15 degrees in Fig. 9(b). The curves coincide below 9 kHz. The ﬁrst order standing-wave resonance is found to be 17.4 kHz regardless of the azimuthal angle because of the same height of the core. Some dips and peaks related to the periodic property however vary with φ reﬂecting the anisotropy of the structure. The reason is similar to that mentioned above. The distance of the projections of the connecting points on the direction of the incident wave varies with φ. The peaks determined by the situation of the points are aﬀected and so do the dips.

4.4. Inﬂuence of stacking geometry When ﬁxing the structural dimensions, then the stacking geometry is the most important parameter. The designability of composite structure enables adjusting the laminate layout to optimize the acoustic properties. For simplicity, the identical top and bottom plates and constant truss are assumed. The main terms that aﬀect the results are bending stiﬀness and bending-extensional coupling and are listed in Table 2 to elaborate the results. For convenience, the subscript “s” stands for symmetric and “as” for anti-symmetric. The STL curves of composite plates with diﬀerent layouts in Fig. 11 coincide well in the low frequency range since the dominant coeﬃcient are mass density which is independent of layout. Discrepancies appear as the frequency increases. Eﬀects of stacking can be expected via changes in the mechanical properties, bending stiﬀness and bending-extensional coupling which aﬀect the coincident resonance and the transformation between in-plane and transverse deformations. The red curves in Fig. 11(a) and (b) are analogous for their same stiﬀness and coincident frequency. Moreover, the coincident resonances and distribution of dips due to classic resonance and pass band in frequency range inﬂuence the results

4.3. Dependence on truss distribution The truss elevation angle, αt, is one of the most crucial geometric parameters. It equals to the length of the trusses and the dimension of unit cell, which aﬀects the discrete wavenumbers αn and βm within the context of constant thickness. Fig. 10 shows the STL curves for diﬀerent truss elevation angles. The curves coincide in the mass-controlled region and deviate more and more as the frequency increases. The increase of αt brings about the decrease of dimension of unit cell l and increase of spacing 2π/l between wavenumbers. The homogenous model is much close to the periodic model when the wavelength is two times larger than the unit cell, i.e. the results of the two theory are same at low frequencies [43]. Then the former makes the behavior of the core tend to the one of a uniform layer in low frequency range, leading that the curve turns smoother and the general isolating property turns better overall. The latter one leads to the shifting of peak values. It is because

Fig. 10. Frequency dependence of the STL for diﬀerent truss elevation angles αt. 26

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Table 2 The mechanical properties of laminate composite plate with diﬀerent layouts. Layout

D11 (Pa)

D16 (Pa)

B11 (kPa)

B16 (kPa)

fco (kHz)

[45,−45,45,−45,45,−45]s [90,0]6 [90,0,90,0,90,0]s [0,45,90,90,45,0]s [90,90,0,0,90,90]s [45,0,45,0,45,0]as [45,45,0,0,45,45]as [90,0,90,0,90,0]as [90,90,90,90,90,90]as

5.47 9.45 7.39 12.49 5.49 10.04 8.63 7.39 9.44

1.03 0 0 0 0 0 0 0 0

0 3.42 0 0 0 0 0 0 20.54

0 0 0 0 0 −5.99 −6.84 0 0

21.52 16.37 18.52 14.25 21.48 15.89 17.13 18.52 16.38

results because the main deformation of truss is bending and compression which are little inﬂuenced by shear modulus. So the dashed red line coincides with the original one. Increasing the longitudinal and transverse modulus enhances the forces and moments restraining the vibration of plates and results in the rising of the values in dips and depression of the peaks. The change does not aﬀect the plates and other external conditions, then no inﬂuence is observed on the frequencies of dips and peaks. Combined with the conclusion in Fig. 10, it is concluded that the modulus of truss aﬀects the amplitude of the STL results whilst the size inﬂuencing the size of unit cell aﬀects the frequency of dips and peaks. In a word, the sizes of the structure and the mechanics of the plate aﬀect the resonant frequencies. The modulus of truss acting like a damper aﬀect the amplitude of the extreme values.

to a large extent, which can be observed in Fig. 11(a). By comparing the layout pair ([90,0,90,0,90,0]s and [45,−45,45,−45,45,−45]s), distinct diﬀerences are observed in the high frequency region when the plies are rotated by 45 degrees about the z-axis. That is attributed to the change of the angles between the ﬁbers in the plate and trusses and that the associated change of angle between the orientation of the material coordinates and periodic unit direction. The mechanic properties of the whole structure change. The coupling eﬀect of bending-extensional stiﬀness is shown in Fig. 11(b). The energy transformation between the transverse and inplane displacements alters the energy transmitted through the structure. The large value of coupling leads to good averaged insulating property in high frequency range. The similar conclusion has been obtained by Zhang [45] with theoretical analysis that the “ordered” ply performs better in resonance-controlled region for anti-symmetric laminate plate.

5. Conclusions In the paper, an analysis of composite sandwich plate with pyramidal truss core is developed by adopting ﬁrst order shear deformation theory and Fourier transform. The experiment is conducted to validate the theory using standing wave tube. The samples are fabricated by slitting and interlocking assembly method. After the validation, the eﬀect of parameters including incident angle, geometry parameter and material properties are quantiﬁed and some conclusions are reached. FSDT is more accurate in some degree. Vertical incident wave and larger elevation angle of truss meet best sound insolation performance and classic resonant frequencies are well predicted. The increase of elevation angle of truss leads to the smoother STL curve and the better isolating performance in general. The stacking geometry has little inﬂuence in low frequency but plays signiﬁcant role in high frequency. It determines the stiﬀness of plate and aﬀects insulating property to a large extent. Moreover, altering the mechanic properties of the plate or the sizes of the structure will shift and depress the resonant frequencies. Changing modulus of truss leads to the depression

4.5. Inﬂuence of the elastic modulus of the prepreg The modulus of unidirectional carbon ﬁber/epoxy prepreg is one of the most important parameters that aﬀect the stiﬀness of structure. The modulus of plate and truss are investigated separately shown in Fig. 12. Fig. 12(a) depicts the eﬀect of plate. The STL curves almost coincide below 8 kHz. Increasing longitudinal, transverse or shear modulus could enlarge the stiﬀness of plate and consequential resonant frequency resulting in dips moving to higher frequency. Meanwhile, the amplitudes of the peaks drop down and the dips rise up which are due to the decrease of vibrational amplitude. The increases of diﬀerent modulus play diﬀerent weight in variation of STL. Considering the practical application, the enhancement of transverse and shear modulus which are weaker in comparison with longitudinal modulus leads to remarkable improvement. As for the modulus of truss shown in Fig. 12(b), enhancing the shear modulus has little inﬂuence on the

Fig. 11. STL versus frequency against stacking geometry: (a) symmetric and (b) anti-symmetric ply. 27

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Fig. 12. Frequency dependence of the STL for diﬀerent values of the elastic modulus of (a) plate and (b) truss.

Acknowledgement

in extreme values only. Diﬀerent directions of the modulus of prepreg inﬂuence the result in diﬀerent degree. Increasing the longitudinal one seems more eﬀective than the others but maybe diﬃcult in practice. Based on practical application, making up for the shortcoming in shear and transverse modulus and ordered ply are more eﬃcient in isolation design. In general, the present work shows the theoretical eﬀective investigation of laminated composite sandwich structure with pyramidal truss cores.

The present work is supported by National Natural Science Foundation of China under Grant Nos. 11672085 and 11432004. Data availability The raw/processed data required to reproduce these ﬁndings will be made available on email request.

Appendix A. Governing equations of the plate model. The in-plane force resultants N’s, the moment resultants M’s and transverse force resultants Q’s can be expressed in the terms of the displacements by the relations respectively [37]:

ϕx , x ⎫ ⎧ Nxx ⎫ ⎡ A11 A12 A16 ⎤ ⎧ u 0, x ⎫ ⎡ B11 B12 B16 ⎤ ⎧ ⎪ ⎪ v0, y Nyy = ⎢ A12 A22 A26 ⎥ ϕy, y + ⎢ B12 B22 B26 ⎥ ⎨u + v ⎬ ⎢ ⎨ ⎨ ⎬ ⎢ ⎬ ⎥ ⎥ 0, y 0, x ⎭ ϕ + ϕy, x ⎪ ⎣ B16 B26 B66 ⎦ ⎪ ⎩ Nxy ⎭ ⎣ A16 A26 A66 ⎦ ⎩ ⎩ x,y ⎭

(A-1)

ϕx , x ⎫ ⎧ Mxx ⎫ ⎡ B11 B12 B16 ⎤ ⎧ u 0, x ⎫ ⎡ D11 D12 D16 ⎤ ⎧ ⎪ ⎪ v0, y Myy = ⎢ B12 B22 B26 ⎥ ϕy, y + ⎢ D12 D22 D26 ⎥ ⎨ ⎬ ⎢ ⎨ ⎬ ⎨ ⎥ u 0, y + v0, x ⎢ ⎥ ϕ +ϕ ⎬ ⎭ ⎣ D16 D26 D66 ⎦ ⎪ y, x ⎪ ⎩ Mxy ⎭ ⎣ B16 B26 B66 ⎦ ⎩ ⎩ x,y ⎭

(A-2)

w + ϕy ⎫ ⎧Q y ⎫ = Ks ⎡ A 44 A 45 ⎤ ⎧ 0, y A A ⎢ ⎥ ⎨ ⎬ ⎨ Q w 45 55 x ⎣ ⎦ ⎩ 0, x + ϕx ⎬ ⎩ ⎭ ⎭

(A-3)

where Ks is the shear correction coeﬃcient, The elastic stiﬀnesses Aij , Bij and Dij can be deﬁned in terms of lamina stiﬀness Qijk as: N

(Aij , Bij , Dij ) =

∑ Q¯ijk (1, z, z 2) dz.

(A-4)

k=1

where

k Q¯ij

are the plane stress-reduced stiﬀnesses of kth layer.

k k k k k + 2Q66 Q¯ 11 = Q11 cos4 θ k + 2(Q12 )sin2 θ k cos2 θ k + Q22 sin4 θ k k k k k k + Q22 − 4Q66 Q¯ 12 = (Q11 )sin2 θ k cos2 θ k + Q12 (sin4 θ k + cos4 θ k ) k k k k k + 2Q66 Q¯ 22 = Q11 sin4 θ k + 2(Q12 )sin2 θ k cos2 θ k + Q22 cos4 θ k k k k k k k k − Q12 − 2Q66 − Q22 + 2Q66 Q¯ 16 = (Q11 ) sin θ k cos3 θ k + (Q12 )sin3 θ k cos θ k k k k k k k k − Q12 − 2Q66 − Q22 + 2Q66 Q¯ 26 = (Q11 )sin3 θ k cos θ k + (Q12 ) sin θ k cos3 θ k k k k Q¯ 44 = Q44 cos2 θ k + Q55 sin2 θ k k k k − Q44 Q¯ 45 = (Q55 ) cos θ k sin θ k k k k Q¯55 = Q55 cos2 θ k + Q44 sin2 θ k

k = In Eq. (A-5), Q11

loss factor.

k (1 + jη) E11 k vk 1 − v12 21

(A-5) k ,Q12 =

k E k (1 + jη) v12 22 k vk 1 − v12 21

k ,Q22 =

k (1 + jη) E22 k vk 1 − v12 21

k k k k k k = G23 (1 + jη),Q55 = G12 (1 + jη), where η = 0.01 is the material = G13 (1 + jη),Q66 ,Q44

28

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Appendix B M0 is a ﬁve dimensional symmetric matrix. M0 (1, 1) = I0 ω2 − A11 α 2 − 2A16 αβ − A66 β 2, M0 (1, 2) = −A16 α 2 − (A12 + A66 ) αβ − A26 β 2, M0 (1, 3) = 0 M0 (1, 4) = I1 ω2 − B11 α 2 − 2B16 αβ − B66 β 2, M0 (1, 5) = −B16 α 2 − (B12 + B66 ) αβ − B26 β 2, M0 (2, 3) = 0 M0 (2, 2) = I0 ω2 − A66 α 2 − 2A26 αβ − A22 β 2, M0 (2, 4) = M0 (1, 5), M0 (2, 5) = I1 ω2 − B66 α 2 − 2B26 αβ − B22 β 2 M0 (3, 3) = −Ks A55 α 2 − 2Ks A 45 αβ − Ks A 44 β 2 − I0 ω2 , M0 (3, 4) = −Ks A55 α − Ks A 45 β M0 (3, 5) = −Ks A 45 α − Ks A 44 β , M0 (4, 4) = I2 ω2 − D11 α 2 − 2D16 αβ − D66 β 2 − Ks A55 M0 (4, 5) = −D16 α 2 − (D12 + D66 ) αβ − D26 β 2 − Ks A 45 , M0 (5, 5) = I2 ω2 − D66 α 2 − 2D26 αβ − D22 β 2 − Ks A 44 . Appendix C. The expressions of forces and moments applied on the plates. The symbols in Eqs. (12)–(14) are: Kr (Kr − ρI2tr ω2) K (K − mω2) Kr2 K (K − mω2) K2 , K2 = , K5 = − Q Q 2 , K3 = , K4 = − , K6 2KQ − mω 2Kr − ρI2tr ω2 2Kr − ρI2tr ω2 2K − mω2 2K − mω2 b b b b b b b 2 2 [U ] = [u1 , ⋯, u4 ] where uj = cos αt (cos ϑj u0 + cos ϑj sin ϑi v0 ) + cos ϑj sin αt cos αt w0 , j = 1 ∼ 4. [V b] = [v1b, ⋯, v4b] where v jb = sin2 αt (cos2 ϑj u0b + cos ϑj sin ϑj v0b) − cos ϑj sin αt cos αt w0b , j = 1 ∼ 4.

K1 = −

δ1 =

=

KQ2 2KQ − mω2

; [Φiε, i] = ϕiε, i [1, 1, 1, 1],

i = x , y;

∑ δ (x − nl) δ (y − ml)

[δ2] = [∑ δ (x − l/2 − nl) δ (y − l/2 − ml),

T

∑ δ (x + l/2 − nl) δ (y − l/2 − ml), ∑ δ (x + l/2 − nl) δ (y + l/2 − ml), ∑ δ (x − l/2 − nl) δ (y + l/2 − ml)] +∞

+∞

where the notation ∑ is shorten for ∑n =−∞ ∑m =−∞. Eqs. (12) and (13) are transformed and rewritten as

(δ t Ftx) = (δ t Fty) = (δ t Ftz) = (δ bFbx) = (δ bFby) = (δ bFzb) =

4 cos2 ϑ[(K1cos2 α t + K5sin2 α t ) ∑ ∼ u t0 (α n, βm ) + (K2cos2 α t + K 6sin2 α t ) ∑ (−1) n + m∼ u 0b (α n, βm )] l2 4 sin2 ϑ[(K1cos2 α t + K5sin2 α t ) ∑ ∼ v t0 (α n, βm ) + (K2cos2 α t + K 6sin2 α t ) ∑ (−1) n + m∼ v 0b (α n, βm )] l2 4 t b ∼ (α , β ) + (K sin2 α + K cos2 α ) ∑ (−1) n + mw ∼ (α , β )] [(K1sin2 α t + K5cos2 α t ) ∑ w n 2 t 6 t n 0 0 m m l2

(C-1)

4 cos2 ϑ[(K2cos2 α t + K 6sin2 α t ) ∑ ∼ u t0 (α n, βm ) + (K1cos2 α t + K5sin2 α t ) ∑ (−1) n + m∼ u 0b (α n, βm )] l2 4 sin2 ϑ[(K2cos2 α t + K 6sin2 α t ) ∑ ∼ v t0 (α n, βm ) + (K1cos2 α t + K5sin2 α t ) ∑ (−1) n + m∼ v 0b (α n, βm )] l2 4 ∼ t (α , β ) + (K sin2 α + K cos2 α ) ∑ (−1) n + mw ∼ b (α , β )] [(K2sin2 α t + K 6cos2 α t ) ∑ w n 1 t 5 t n 0 0 m m l2

(C-2)

∼b βm ) + K 4 ∑ (−1) n + mϕx (α n, βm ) ⎤ ⎦ ∼t ∼b 4 2 n + m t (δ Mry ) = 2 sin ϑ(−iβ ) ⎡K3 ∑ ϕy (α n, βm ) + K 4 ∑ (−1) ϕy (α n, βm ) ⎤ l ⎣ ⎦ ∼t ∼b 4 2 b n m + (δ Mrx ) = 2 cos ϑ(−iα ) ⎡K 4 ∑ ϕx (α n, βm ) + K3 ∑ (−1) ϕx (α n, βm ) ⎤ l ⎦ ⎣ ∼t ∼b 4 2 t n + m (δ Mry ) = 2 sin ϑ(−iβ ) ⎡K 4 ∑ ϕy (α n, βm ) + K3 ∑ (−1) ϕy (α n, βm ) ⎤ l ⎣ ⎦ (δ t Mrx ) =

∼t 4 cos2 ϑ(−iα ) ⎡K3 ∑ ϕx (α n, l2 ⎣

The matrices

M1ε ,

M2ε

(C-3)

in Eq. (15) are obtained by combing Eqs. (C-1)–(C-3).

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