Modeling and vibration analysis of sectional-laminated cylindrical thin shells with arbitrary boundary conditions

Applied Acoustics 162 (2020) 107184

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Modeling and vibration analysis of sectional-laminated cylindrical thin shells with arbitrary boundary conditions Bingfu Zhong a,c, Chaofeng Li a,b,⇑, Peiyong Li a a

School of Mechanical Engineering and Automation, Northeastern University, 110819 Shenyang, China Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, 110819 Shenyang, China c School of Mechanical Engineering, Shanghai Jiao Tong University, 200240 Shanghai, China b

a r t i c l e

i n f o

Article history: Received 8 January 2019 Received in revised form 6 November 2019 Accepted 9 December 2019

Keywords: Sectional-laminated cylindrical thin shells Free vibration Continuity condition Arbitrary boundary conditions

a b s t r a c t The continuity condition is proposed for modeling and analyzing the vibration characteristics of sectional-laminated cylindrical thin shells with arbitrary boundary conditions. The orthogonal polynomials are used to describe the vibration displacements of the sectional-laminated cylindrical shell. Four sets of springs are applied to simulate arbitrary boundary conditions. The Donnell thin shell theory is employed to formulate the theoretical model. To test the convergence and accuracy of the present modeling method, free vibrations of sectional-laminated cylindrical shells are examined with clampedclamped and simply-simply boundary conditions. The numerical results are compared with those obtained by using the finite element software ANSYS. The effects of the pre-assigned weighted parameters, the length ratio of the non-laminated cylindrical shell and the thickness ratio of inner and outer layers on natural frequencies and mode shapes are presented with arbitrary boundary conditions. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Sectional-laminated cylindrical thin shells are designed for where the stiffness of some sections of the shell is required to change like sectional coating. There would have been a variety of engineering applications, such as the automobile, aerospace, and ship. The dynamic study of these shells involves some new characteristics caused by sectional-laminated structures. The new vibration characteristics of these shells need to be analyzed for providing the designers and engineers to avoid structural damage and dangerous vibrations. Therefore, many scholars in this field are interested and have carried out a lot of research. The research about dynamic characteristics of the combined cylindrical shell structure started from the 1980s. Irie et al. [1] studied the free vibration of joined conical-cylindrical shells. The authors used the transfer matrix of the shell to obtain the governing equations of vibration. The natural frequencies and the mode shapes of vibration calculated were presented. Patel et al. [2–4] used the finite element method for analyzing the free vibration of laminated anisotropic composite conical-cylindrical shell structures. The natural frequencies and the mode shapes were obtained

⇑ Corresponding author at: School of Mechanical Engineering and Automation, Northeastern University, 110819 Shenyang, China. E-mail address: [email protected] (C. Li). https://doi.org/10.1016/j.apacoust.2019.107184 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.

based on field consistency approach. Then, authors studied the nonlinear thermo-elastic buckling/post-buckling characteristics of laminated circular conical-cylindrical/conical-cylindrical-conical joined shells. Considering geometric nonlinearity, the influences of semi-cone angle, material properties and number of circumferential waves was investigated. Kamat et al. [5] analyzed the dynamic instability analysis of a joined conical and cylindrical shell subjected to periodic in-plane load based on first-order shear deformation theory. The influence of various parameters was brought out. Caresta et al. [6,7] firstly present the free vibrational characteristics of isotropic coupled conical-cylindrical shells. They authors took a wave solution and a power series solution to describe the displacements. Then, the authors presented the structural and acoustic responses of a submarine under harmonic force excitation. Lee [8] improved a pseudospectral method to study the axisymmetric free vibration analysis of a hermetic capsule. The author took the continuity conditions of deformations and stress resultants at the junctions as the constraints of the expansion coefficients. Qu et al. [9–11] used a modified variational principle to predict the free, steady-state and transient vibration of composite laminate shells, and performed different combinations of classical and non-classical boundary conditions. Then, based on the firstorder shear deformation theory, the authors analyzed vibration characteristics of conical-cylindrical-spherical shells. Lee et al. [12] applied the Rayleigh-Ritz method to study the free vibration

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B. Zhong et al. / Applied Acoustics 162 (2020) 107184

Nomenclature u, v, w

Displacements in x, h, z directions of the sectionallaminated cylindrical shell uL, vL, wL Displacements in x, h, z directions of the laminated cylindrical shell uS, vS, wS Displacements in x, h, z directions of the non-laminated cylindrical shell HL Thickness of the laminated shell LL Length of the laminated shell RL Radius of the laminated shell qL Density of the laminated shell b Fiber orientation Aij, Bij, Dij the extensional, coupling and bending stiffness matrices ðLÞ ðLÞ E11 , E22 Young’s modulus of the laminated shell

lð12LÞ , lð21LÞ Poisson’s ratios of the laminated shell HS LS RS E(S)

l(S) H L R

Thickness of the non-laminated shell Length of the non-laminated shell Radius of the non-laminated shell Young’s modulus of the non-laminated shell Poisson’s ratios of the non-laminated shell Thickness of the sectional-laminated shell Length of the sectional-laminated shell Radius of the sectional-laminated shell

characteristics of a joined spherical-cylindrical shell with three classical boundary conditions. In his research, the effects of the shallowness of the spherical shell and length of the cylindrical shell to the free vibrational behavior of joined shell structure are investigated. Chen et al. [13] presented an analytic method to analyze free and forced vibration characteristics of ring-stiffened combined conical-cylindrical shells with arbitrary boundary conditions. The boundary conditions and continuity conditions between adjacent substructures are used to assemble the final governing equation. The influences of boundary conditions and ring stiffeners on the free vibration are studied. The effects of direction of external force and bulkheads on the forced vibration are also discussed. Ma et al. [14] used the modified Fourier-Ritz method to take a free and forced vibration analysis of coupled conical-cylindrical shells with arbitrary boundary conditions. The effects of dimensional and elastic restraint parameters on the free vibrations are reported. Li et al. [15] used the Donnell’s shell theory, the Chebyshev polynomials and Lagrange equation to investigate the free vibration of two kind of non-continuous elastic supported laminated composited cylindrical shell, in which the arc supported and point supported boundary condition were simulated by using the artificial springs. Xie et al. [16] used the wave based method to research the elastically coupled structres by using the Flugge shell theory and thin plate theory, in which the thin annular plate and cylindrical shell are coupled. Then, the influence of annular plates, boundary condition and elastic coupling condition for free and forced vibration were discussed. Sarkheil et al. [17] investigated the free vibration of a rotating joined cylindrical-conical shell. The governing equations were solved by employing the power series method. The effects of different parameters such as rotation speed, cone angle, circumferential wave number, length to radius ratio, and shell thickness on the frequencies of forward and backward waves were investigated. Li et al. [18] used the first-order shear shell theory, the Chebyshev polynomials and the Lagrange equation to investigate the vibration control of discontinuous piezoelectric laminated shell with point supported elastic boundary conditions, in which the point supported boundary condition was simulated by using

Nx, Mx Qxh, Qx

Force and moment resultant vectors Circumferential and lateral Kevlin-Kirchhoff shear force resultants n Dimensionless scale in the x direction ku, kv, kw, kh Stiffness of axial, circumferential, radial, rotational spring Un(n), Vn(n), Wn(n) Admissible displacement functions NT The number of terms for orthogonal polynomials n The circumferential wave number m The axial wave number x Natural frequency t Time ML, KL Mass matrix, stiffness matrix of the laminated shell MS, KS Mass matrix, stiffness matrix of the non-laminated shell M Mass matrix of the sectional-laminated cylindrical shell K Stiffness matrix of the sectional-laminated cylindrical shell KBK Boundary stiffness matrix of the sectional-laminated cylindrical shell KCK Connection stiffness matrix of the sectional-laminated cylindrical shell KCk Continuous stiffness matrix of the sectional-laminated cylindrical shell

the artificial springs. Qin et al. [19] comparied the different admissible displacement function, the modified Fourier series, the Orthogonal polynomials, and Chebyshev polynomials, and proved the accurary, covergence reate and computational efficiency of kind of admissoble displacement function. According the paper, the Orthogonal ploynomials is used in the free vibration analysis of laminated cylindrical shell with arbitrary boundary conditions. Li et al. [20–23] studied the forced vibration and free vibration characteristics of two kinds of coupled structures. Firstly, authors analyzed the effect of the supporting stiffness, length-radius and thickness-radius ratios on the nonlinear characteristics. Then, they presented the influence on natural frequency parameters and mode shapes of bolted joined cylindrical-cylindrical shell. Qin et al. [24] studied the vibration characteristic of a rotating cylindrical shell coupled with an annular plate by providing a general approach. It can be seen from the above literature that most of the previous combined or coupled vibration problem studies are centered on the single-layered shells. However, few works have been presented for the vibration characteristics of sectional-laminated cylindrical shells with arbitrary boundary conditions. And the layer coupling is a new research which is lacking in considering it. Therefore, the study apply the interface potentials to analyze the continuity condition for modeling method and vibration analysis of sectional-laminated cylindrical shells with arbitrary boundary conditions. In this paper, based on Donnell’s thin-shell theory and the orthogonal polynomials, the vibration differential equations of sectional-laminated cylindrical shell are derived. The force and moment resultant are used to introduce to assemble the continuity condition between non-laminated and laminated shells. At first, several comparisons are carried out to validate the approach modeling method in present study with ANSYS results. Then, the following work studies the effect of the pre-assigned weighted parameters, the length ratio of the non-laminated cylindrical shell and the thickness ratio of inner and outer layers on natural frequencies and mode shapes of sectional-laminated cylindrical shells with arbitrary boundary conditions.

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discussion. For the isotropic non-laminated cylindrical thin shell, n 2 ½0; LS =L. The strain energy of the isotropic non-laminated cylindrical thin shell is defined as

2. Theoretical formulations 2.1. Description of the model A sectional-laminated cylindrical shell with the length L, radius R, thickness H, the fiber orientation is defined by b is illustrated in Fig. 1(a). A fixed coordinate system (x, h and z) is taken as the reference along directions of the middle surface of the shell. The displacements of the shell in x, h and z directions are denoted by u, v and w, respectively. Prior to developing the model, the following assumptions will be made for simplifying the computational effort: The displacements and rotary angles of the interfaces for the shell satisfy continuity condition. The Fig. 1(b) presents that the left side of the shell is an isotropic non-laminated cylindrical shell, and the other side is a composite laminated cylindrical shell. The length of the non-laminated shell is LS. The length of the laminated shell is LL. The radius of the non-laminated cylindrical shell is RS. The radius of the laminated shell is RL. The thickness of nonlaminated shell is HS. The thickness of the laminated shell is HL. The vibration displacements of the sectional-laminated shell are expressed asud , v d and wd ðd ¼ S; LÞ, which are axial, circumferential and radial displacements. The subscripts S and L present an isotropic non-laminated cylindrical shell and a composite laminated cylindrical shell, respectively. Meanwhile, as shown in Fig. 1(b), along each end of the shell, three sets of translational springs (ku , kv and kw ) and one set of rotational springs (kh ) are distributed uniformly to simulate arbitrary boundary conditions which are expressed in the form of axial force resultant, tangential force resultant, transverse force resultant, and the flexural moment resultant, respectively. Specifically, 0

0

0

0

ku , kv , kw and kh denote the four sets springs distributed along 1 1 ku , kv , (P)

1 kw

1 kh

and represent another four sets the edge x/L = 0 while along the edge x/L = 1. H is the thickness of the pth layer of the laminated shell (p = 1, 2, 3). 2.2. Expressions of the sectional-laminated cylindrical thin shell’s energy 2.2.1. Expressions of the isotropic non-laminated cylindrical thin shell’s energy The kinetic energy of the isotropic non-laminated cylindrical thin shell is given by

Z Z LS qS HS RS L 2p L  _ 2 _ 2 _ 2  TS ¼ uS þ v S þ wS dndh 2 0 0

ð1Þ

where qs is the density of the isotropic non-laminated cylindrical thin shell. In this paper, the length x is replaced by a dimensionless length n defined by n = x/L for convenience of the computation and

US ¼

Z

L 2

Z 2p Z

HS 2

H  2S

0

LS L

n

o

rðxSÞ eðxSÞ þ rðhSÞ eðhSÞ þ sðxhSÞ cðxhSÞ RS dndhdz

0

ð2Þ

rðxSÞ , rðhSÞ and sðxhSÞ are the stresses of a point at the isotropic ðSÞ ðSÞ ðSÞ non-laminated cylindrical thin shell. ex , eh and cxh are the strains where

of a point at the non-laminated shell. They are defined as

8 > > > <

eðxSÞ ¼ eðxSð0Þ Þ þ zkðxSÞ ðSÞ

ðS Þ

ðS Þ

ð3Þ

eh ¼ ehð0Þ þ zkh > > > : cðSÞ ¼ cðSÞ þ zvðSÞ xh xh xhð0Þ 8 9ðSÞ > < rx > =

8 > <

9

ðSÞ ex > = ðSÞ eh rh ¼Q > > > > : : cxh ; sxh ;

ð4Þ

where 0 6 z 6 HS . According to Donnell’s thin shell theory, the strain-displacement relation of the neutral surface of the nonlaminated shell can be written as follows

8 > eðSÞ ¼ 1 @uS > > < xð0Þ L @n eðhSð0Þ Þ ¼ R@Sv@hS þ wRSS > > > ðS Þ : cxhð0Þ ¼ R@uS @hS þ 1L @@nv S

ð5Þ

where the footnote (0) denotes the middle surface of the isotropic non-laminated cylindrical thin shell. ðSÞ

ðSÞ

The middle surface curvature kx , kh follows

8 > > > > <

ðS Þ

kx ¼  L12

and

vðxhSÞ are defined as

@ 2 wS @n2

vS wS kh ¼ R12 @@h  R@2 @h 2 > S S > > > : vðSÞ ¼ 1 @ v S  2 @ 2 wS ðS Þ

2

RS L @n

xh

ð6Þ

L RS @n@h

The strain stiffness matrix Q ðSÞ is calculated as

2

EðSÞ

6 1ðlðSÞ Þ 6 6 ðSÞ ¼ 6 lðSÞ E 2 6 1ðlðSÞ Þ 4 2

Q

ðS Þ

0

lðSÞ EðSÞ 2 1ðlðSÞ Þ EðSÞ 1ðlðSÞ Þ

2

0

3

0

7 7 7 0 7 7 5 ðSÞ G12

ð7Þ

where EðSÞ is elastic moduli. lðSÞ is the Poisson’s ratios. GðSÞ is shear   ðSÞ modulus and G12 ¼ EðSÞ =2 1 þ lðSÞ .

Fig. 1. Schematic diagram of sectional-laminated cylindrical thin shell.

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2.2.2. Expressions of the composite laminated cylindrical thin shell The kinetic energy of the composite laminated cylindrical thin shell is given by

qL HL RL L

TL ¼

Z 2p Z

2

1

LS L

0

  2 2 _ 2L dndh u_ L þ v_ L þ w

LRL UL ¼ 2

1

Z 2p

LS L

0

eT S edhdn

n

eðxLð0Þ Þ eðhLðÞ0Þ cðxhLÞð0Þ kðxLð0Þ Þ kðhLðÞ0Þ vðxhLÞð0Þ

o

ðLÞ

rðhLÞ and sðxhLÞ are the stresses of a point at the composite ðLÞ ðLÞ ðLÞ laminated cylindrical thin shel. ex , eh and cxh are the strains of a point at the laminated shell. According to Donnell’s thin shell theory, the strain-displacement relation of the neutral surface of the laminated shell can be written as follows

8 > eðLÞ ¼ 1 @uL > > < xð0Þ L @n eðhLðÞ0Þ ¼ R@Lv@hL þ wRLL > > > : cðLÞ ¼ @uL þ 1 @ v L xhð0Þ RL @h L @n

ð11Þ

ðLÞ

and

ð12Þ

A11 6 6 A21 6 6 6 A61 S¼6 6B 6 11 6 6B 4 21 B61

A12

A16

B11

B12

A22

A26

B11

B22

A62

A66

B61

B62

B12

B16

D11

D12

B22

B26

D21

D22

B62

B66

D61

D62

B16

3

7 B26 7 7 7 B66 7 7 D16 7 7 7 D26 7 5 D66

ð13Þ

Aij ¼

Q kij ðhkþ1  hk Þ;

Bij ¼

k¼1

Dij ¼

1 2

N X

  2 2 Q kij h kþ1  h k ;

k¼1

N   1X 3 3 Q kij h kþ1  h k 3 k¼1

ð14Þ

in which N is the amount of the laminated layers. For the laminated shell, the distance between the outer and inner sides of the kth layer and the middle surface of the shell is hk and hk+1(k = 1, 2, 3). Q ij is elements of the matrix Q

Q

ðLÞ

ðLÞ E11

3

ð16Þ

lð12LÞ Eð22LÞ ðLÞ ðLÞ 1l12 l21 ðLÞ

E22

ðLÞ ðLÞ

1l12 l21

0

3 0 7 7 7 0 7 7 5 ðLÞ G12

ð17Þ

ðLÞ E22

ðLÞ

ðLÞ

where and are elastic moduli. l12 and l21 are the Poisson’s ratios. The relationship between them can be calculated by using ðLÞ

ðLÞ

ðLÞ

ðLÞ

E11 l21 ¼ E22 l12 . G12 is shear modulus. 2.2.3. Boundary conditions To obtain the potential energy associated with the sectionallaminated cylindrical thin shell subjected to arbitrary boundary conditions (as shown in Fig. 1(b)), four sets of elastic springs are applied. The potential energy of the boundary springs can be expressed by

U spr ¼

1 2

Z 2p 0

v

k0 u2S ð0Þ þ k0 v 2S ð0Þ þ k0 w2S ð0Þ u

w

 2 Z @wS ð0Þ 1 2p u 2 R dh þ k1 uL ð1Þ S @n 2 0 L2  2 @wL ð1Þ v w h 1 RL dh þ k1 v 2L ð1Þ þ k1 w2L ð1Þ þ k1 2 @n L 1

ð18Þ

Arbitrary boundary conditions are achieved by adjusting stiffness of the springs in proper values, according to Ref. [21,22].

ðLÞ

  1 1 T ¼ T ðLÞ  Q ðLÞ  T ðLÞ

The basic essence of the present method is to construct the interface potentials UC in Eq. (18). There exists a rich body of literature on the establishment of modified variational functionals. In this work, a combination of a modified variational principle (MVP) and least-squares weighted residual method (LSWRM) is proposed to obtain the interface potentials, found in Qu [9]. In doing so, the potentials UC are written as

UC ¼

where Aij, Bij and Dij (i, j = 1, 2, 6) are the extensional, coupling and bending stiffness matrices. For an arbitrary laminated cylindrical shell, the stiffness components can be defined as N X

2sinbcosb

2.3. Continuity condition

L RL @n@h

Meanwhile, the transformation stiffness matrix S is given by

2

Q ðLÞ

ðLÞ

E11

6 1lð12LÞ lð21LÞ 6 6 ðLÞ ðLÞ ¼ 6 l12 E22 6 1lðLÞ lðLÞ 4 12 21 0

vðxhLÞ are defined as

8 2 ðLÞ > kx ¼  L12 @@nw2L > > > < 2 ðLÞ vL wL kh ¼ R12 @@h  R@2 @h 2 > L L > > > ð L Þ : v ¼ 1 @ v L  2 @ 2 wL RL L @n

2

h

ðLÞ

sin b

7 2 sin b cos2 b 2sinbcosb 5 2 2 sinbcosb sinbcosb cos b  sin b

þ k0

The middle surface curvature kx , kh follows

2

cos2 b

The matrix Q ðLÞ is the constants matrix relating stresses with strains, which is defined as

ðLÞ

ð10Þ

whererx ,

xh

6 T ðLÞ ¼ 4

ð9Þ

where S is transformation stiffness matrix. eT is strain vector, which is given by

eT ¼

2

ð8Þ

where qs is the density of the composite laminated cylindrical thin shell. For the composite laminated cylindrical thin shell, n 2 ½LS =L; 1. The strain energy of the composite laminated cylindrical thin shell is defined as

Z

where T ðLÞ is the transformation matrix, which is defined as

, which is calculated as

ð15Þ

Z 



1 2



ðLÞ

ðLÞ

1u NðxLÞ Hu;SL þ 1v Nxh 1u Hv ;SL þ 1w Q x Hw;SL  1r MðxLÞ Hh;SL dl Z 











1u ku H2u;SL þ 1v kv H2v ;SL þ 1w kw H2w;SL þ 1r kh H2h;SL dl ð19Þ

where the first integral expression is derived by means of MVP to relax the enforcement of the interface constraints. The second integral expression is obtained from LSWRM, and the existence of this term is twofold: to ensure a numerically stable operation for continuity condition. The integrations in Eq. (18) are carried out over the interfaces. N ðxLÞ and MðxLÞ are defined in the aforementioned force and ðLÞ

ðLÞ

moment resultant vectors. Nxh and Q x are, respectively, the circumferential and lateral Kevlin–Kirchhoff shear force resultants. These force and moment resultants are obtained by integrating the stresses over the shell thickness (see in Fig. 2). Hu;SL , Hv ;IL , Hw;IL and Hh;IL are the relevant continuity equations on the common Hu;SL ¼ uL  uS , Hv ;SL ¼ v L  v S , interfaces, defined as 







Hw;SL ¼ wL  wS , Hh;SL ¼ @wL =@x  @wS =@x. ku , kv , kw and kh are

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B. Zhong et al. / Applied Acoustics 162 (2020) 107184

where n is the circumferential wave number, x is the circular frequency of the shell. sinðnhÞ and cosðnhÞ are circumferential mode functions. U n ðnÞ, V n ðnÞ and W n ðnÞ are axial mode functions. In present study, we choose orthogonal polynomials as the admissible displacement functions, which are introduced in Ref. [21]. So, the displacements of the sectional-laminated composite cylindrical shell can be expressed as follows

8 PNT T u > > < uðn; h; t Þ ¼ qU U ¼ P i¼1 qU ui ðnÞcosðnhÞ v v ðn; h; tÞ ¼ qV V T ¼ NT i¼1 qV ui ðnÞsinðnhÞ > > PNT : T wðn; h; t Þ ¼ qW W ¼ i¼1 qW uw i ðnÞcosðnhÞ

where qU , qV and qW are the time of terms. NT is the number of terms in calculation. uui ðnÞ, uvi ðnÞ and uw i ðnÞ are the admissible displacement functions.

Fig. 2. Force and moment diagram of the combined cylindrical shell.

pre-assigned weighted parameters. For the case of two adjacent shell segments, 1u ¼ 1v ¼ 1w ¼ 1r ¼ 1.

(

ðLÞ Nxh

!  ðJÞ ðJÞ ðJÞ ðJÞ @uL A16 @uL B16 @ v L A12 B12 @ v L ¼ A11 þ 2 þ þ A16 þ þ @x RL @h RL @x RL @h RL ) 2 ðJÞ 2 ðJÞ 2 ðJÞ A12 ðJÞ @ wL B12 @ wL 2B16 @ wL w  B11  2  þ ð20Þ 2 RL L @x2 RL @x@h RL @h ( ! ðJÞ ðJÞ B16 @uL A26 B26 @uL ¼ A16 þ þ 2 þ RL @x RL @h RL ! ! ðJÞ ðJÞ 2B66 D66 @ v L A26 2B26 D26 @ v L þ 2 þ 2 þ 3 þ A66 þ þ RL @x RL @h RL RL RL ! !  2 ðJÞ ðJÞ A26 B26 D16 @ wL B26 D26 @ 2 wL ðJÞ þ 2 wL  B16 þ  þ þ 2 3 2 2 RL RL @x @h RL RL RL ) ! 2 ðJÞ B26 D26 @ wL ð21Þ 2 þ 2 RL @x@h RL (

ðLÞ Qx

 ðJÞ ðJÞ ðJÞ ðJÞ @ 2 uL 2B66 @ 2 uL 3B16 @ 2 uL D16 @ 2 v L ¼ B11 þ þ þ þ B 16 @x2 RL @x@h RL @x2 R2 @h2 ! ! ð J Þ ðJÞ 2B26 2D26 @ 2 v L B12 D12 2B66 2D66 @ 2 v L þ þ þ þ þ þ RL RL @x@h @h2 R2L R3L R2L R2L þ

ðJÞ B12 @wL

RL

@x

 ðJÞ



ðJÞ @ 3 wL D11 @x3

4D16 @ 3 wL  RL @x2 @h

ðJÞ 2B26 @wL 2D26 þ 2  3 @h RL R ! ) L ðJÞ D12 4D66 @ 3 wL þ 2 @x@h2 R2L RL

ðJÞ @ 3 wL 3

@h

ð22Þ

!  ðJÞ ðJÞ ðJÞ ðJÞ @uL B13 @uL D13 @ v L B12 D12 @ v L ¼ B11 þ 2 þ þ B13 þ þ @x RL @h RL @x RL @h RL ) 2 ðJÞ 2 ðJÞ 2 ðJÞ B12 ðJÞ @ wL D12 @ wL 2D16 @ wL ð23Þ þ w  D11  2  RL L @x2 RL @x@h RL @h2

2.4. Admissible displacement functions The vibration displacements of the sectional-laminated cylindrical thin shell are Galerkin discretized. For the shell under vibration mode, the admissible displacement functions can be written as follows

8 > < uðn; h; t Þ ¼ U n ðnÞcosðnhÞsinxt v ðn; h; tÞ ¼ V n ðnÞsinðnhÞsinxt > : wðn; h; t Þ ¼ W n ðnÞcosðnhÞsinxt

3.1. Model procedure Substituting Eq. (25) into Eq. (1) and Eq. (2), the kinetic energy of the sectional-laminated shell can be obtained as follow

T ¼ T S þ T L ¼ q_ M q_ T

ð24Þ

ð26Þ

Also substituting Eq. (25) into Eq. (2), Eq. (9), Eq. (18) and Eq. (23), and the strain energy U e can be obtained as follow

U e ¼ U S þ U L þ U C þ U spr

ð27Þ

¼ 12 qT ðK þ K BK þ K CK  K Ck Þq

Substituting the kinetic energy (Eq. (26)) and potential energy (Eq. (27)) into the Lagrange equation (Eq. (28)),

8 > > > > <

d dt



@T @ q_ U







@U e @T þ @q ¼0  @q U U

d @T @T e þ @U ¼0  @q dt @ q_ V @qV V > >   > > : d @T  @T þ @Ue ¼ 0 dt

@ q_ W

@qW

ð28Þ

@qW

vibration differential equations are obtained, that is

€ þ ðK þ K BK þ K CK  K Ck Þq ¼ 0 Mq

ð29Þ

where

 M¼

(

M ðxLÞ

3. The model procedure and validation

ðLÞ

ðLÞ

N ðxLÞ , M ðxLÞ , N xh and Q x are calculated as

NðxLÞ

ð25Þ

MS 0 "

K CK ¼

  KS 0 ;K¼ ML 0

K CKS

K TCKSL

K CKSL

K CKL

  K BKS 0 ; K BK ¼ KL 0

# ; K Ck ¼

"

0

K TCkSL

K CkSL

K CkL

0 K BKL

 ð30Þ

# ; q ¼ ½ qS

qL 

T

ð31Þ M is the mass matrix of the sectional-laminated shell. K is the structure stiffness matrix of the sectional-laminated shell. K BK is the boundary stiffness matrix of the sectional-laminated shell. K CK is the connection stiffness matrix of the sectional-laminated shell. K Ck is continuity stiffness matrix of the sectional-laminated shell. The detail expressions for M, K and K BK can be sourced in Ref. [20,21]. The detail expressions for K CK and K Ck are given in Appendix A, B, respectively. 3.2. Model validation In this paper, to verify the accuracy of the present model, two boundary conditions are simulated by present analytical formulation and the results are compared with those obtained by the finite element software ANSYS. Case I and II are respectively introduced to simulate the clamped-clamped and simply-simply boundary

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v

u

w

v

u

w

conditions: Case I: k0 ¼ k0 ¼ k0 ¼ k0 ¼ 1012 , k1 ¼ k1 ¼ k1 ¼ k1 ¼ v

12

w k0

v

w k1

h

12

u k1

u k0

h k1

h

h k0

10 . Case II: k0 ¼ ¼ k1 ¼ ¼ 10 , ¼ ¼ ¼ ¼ 0. In next analyze, we use the Case I and Case II to replace the clamped-clamped and simply-simply boundary conditions. The geometric parameters and material properties of the isotropic non-laminated cylindrical shell at the left side are listed in Table 1. The geometric parameters and material properties of the composite laminated cylindrical thin shell at the other side are shown in Table 2. In present study, here chooses the pre-assigned weighted ðLÞ

parameters of the sectional-laminated shell as 105E22 , according to Ref. [9]. The dynamic model in the ANSYS model as shown in Fig. 3, SHELL281 is used. The comparison of the natural frequencies for the lowest six circumferential mode numbers are shown in Fig. 4 (a) and Fig. 4(b). For generality and convenience, the following error is defined Error ¼ jxPresent  xANSYS j=xANSYS  100%, where xPresent represents the frequency with respect to NT terms of admission functions, and xANSYS denotes the ANSYS frequency results.

Table 1 Geometric parameters and material properties of the non-laminated cylindrical thin shell. Geometric parameters

Material properties

LS =RS ¼ 1

EðSÞ = 7.6GPa lðSÞ = 0.26 qS = 1643 kg/m3

HS =RS ¼ 0:02 RS = 0.1 m

Table 2 Geometric parameters and material properties of the laminated cylindrical thin shell. Geometric parameters

Material properties

LL =RL ¼ 1

E22 = 7.6GPa

ðLÞ

HL =RL ¼ 0:02 RL = 0.1 m 





[0 =90 =0 ] H(1) = H(2) = H(3)

ðLÞ

ðLÞ

E11 =E22 ¼ 2:5 ðLÞ

G12 ¼ 4:1GPa

lð12LÞ ¼ 0:26 qL ¼ 1643kg=m3

It is indicated that the present results agree well with the results obtained by ANSYS with two boundary conditions. Moreover, the convergence of the present method is examined at the same time since NT terms of orthogonal polynomials. From Fig. 4, it can be observed that the error of the frequency decreases gradually with the increasing of the numbers of polynomials NT. It is easily to be checked that the use of 7 terms of admissible displacement functions here can be good accuracy in this section. In this paper, the number of polynomial terms NT is chosen as 7 for following investigations. 4. Numerical results and discussions In this section, the natural frequencies and mode shapes of the sectional-laminated cylindrical thin shell are investigated. The effects on vibration characteristics of the pre-assigned weighted parameters, the length ratio of the non-laminated cylindrical shell and the thickness ratio of inner and outer layers are studied. The geometrical parameters and material properties of the shell are the same as those listed in Section 3.2. 4.1. The effect of pre-assigned weighted parameters In engineering applications, the material properties of inner and outer layers may be different from those of the middle layer for the laminated shell. Therefore, the effect of pre-assigned weighted parameters could change. This section discusses three types of elastic moduli of inner and outer layers related to those of the middle layer, which divides into elastic moduli of inner and outer n o n o ðLÞ ðLÞ layers equal to the middle layer’s ( E22 ¼ E22 ¼ inner outer n o ðLÞ E22 ), elastic moduli of inner and outer layers much larger middle n o n o n o ðLÞ ðLÞ ðLÞ than the middle layer’s ( E22 ¼ E22 ¼ 103 E22 ), inner

outer

middle

and elastic moduli of inner and outer layers much smaller than n o n o n o ðLÞ ðLÞ ðLÞ ¼ E22 ¼ 103 E22 ). The the middle layer’s ( E22 inner

outer

middle

error is based on the same calculated principle in Section 3.2. 4.1.1. Elastic moduli of inner and outer layers equal to the middle layer’s The effect of pre-assigned weighted parameters on natural frequencies with two boundary conditions is shown in Fig. 5, where the elastic moduli of inner and outer layers is equal to the middle layer’s. Meanwhile, This paper assumes that the pre-assigned weighted parameters in the four directions are the same, i.e. 

Fig. 3. Finite element model of the sectional-laminated cylindrical thin shell in ANSYS.







kk ¼ ku ¼ kv ¼ kw ¼ kh . From Fig. 5(a) and (b), pre-assigned weighted parameters have an obvious influence on the natural frequencies of the sectional-laminated cylindrical thin shell with two boundary conditions at the axial wave number m = 1 and the lowest six circumferential mode numbers n = 1 ~ 6. When the preassigned weighted parameter of the shell is in the range of 108 ~ 1015Pa, the error is mostly close to 0%. However, the error will suddenly become unstable at some circumferential wave numbers when the value is about 1010Pa. For example, with Case I, the error will increase to about 0.2% when n = 6. Meanwhile, the pre-assigned weighted parameters of the shell will present different favorable range at different circumferential wave numbers n. For example, with Case II, the favorable range of the pre-assigned weighted parameter of the shell at n = 2 is much smaller than at n = 1. From the figures, it can be indicated that pre-assigned weighted parameters of the sectional-laminated cylindrical thin shell with different boundary conditions show different favorable ranges to calculate

B. Zhong et al. / Applied Acoustics 162 (2020) 107184

7

Fig. 4. Comparisons of the frequency of the sectional-laminated shell.

Fig. 5. Effect of pre-assigned weighted parameters (m = 1).

natural frequencies and the favorable range is 1012 ~ 1014Pa n o n o ðLÞ ðLÞ (103 E22 ~ 105 E22 ) for the modeling method promiddle

middle

posed in this paper, where the elastic moduli of inner and outer layers is equal to the middle layer’s. 4.1.2. Elastic moduli of inner and outer layers much larger than the middle layer’s The effect of pre-assigned weighted parameters on natural frequency with two boundary conditions is presented in Fig. 6, where the elastic moduli of inner and outer layers is much larger than the middle layer’s. Meanwhile, this section assumes that the preassigned weighted parameters in the four directions are the same, 







i.e. kk ¼ ku ¼ kv ¼ kw ¼ kh . From Fig. 6(a) and (b), pre-assigned weighted parameters indicate an obvious influence on the natural frequencies of the

sectional-laminated cylindrical thin shell with two boundary conditions at the axial wave number m = 1 and the lowest six circumferential mode numbers n = 1 ~ 6. When the pre-assigned weighted parameter of the shell is in the range of 108 ~ 1014Pa, the error is mostly close to 0%. Meanwhile, the pre-assigned weighted parameter of the shell will show different favorable range at different circumferential wave numbers n. For example, with Case II, the favorable range of the pre-assigned weighted parameter of the shell at n = 2 is much smaller than at n = 1. From the figures, it can be indicated that pre-assigned weighted parameters of the sectional-laminated cylindrical thin shell with different boundary conditions have a favorable range to calculate natural frequencies n o ðLÞ and the favorable range is 108 ~ 1014Pa (10-1 E22 ~ middle n o ðLÞ 5 10 E22 ) for this modeling method, where the elastic moduli middle

of inner and outer layers is much larger than the middle layer’s.

Fig. 6. Influence of pre-assigned weighted parameters (m = 1).

8

B. Zhong et al. / Applied Acoustics 162 (2020) 107184

4.1.3. Elastic moduli of inner and outer layers much smaller than the middle layer’s The effect of pre-assigned weighted parameters on natural frequency with two boundary conditions is shown in Fig. 7, where the elastic moduli of inner and outer layers is much smaller than the middle layer’s. Meanwhile, this section assumes that the preassigned weighted parameters in the four directions are the same, 







i.e. kk ¼ ku ¼ kv ¼ kw ¼ kh . From Fig. 7(a) and (b), pre-assigned weighted parameters present an obvious influence on the natural frequencies of the sectional-laminated cylindrical thin shell with two boundary conditions at the axial wave number m = 1 and the lowest six circumferential mode numbers n = 1 ~ 6. When the preassigned weighted parameter of the shell is in the range of 108 ~ 1014Pa, the error is mostly close to 0%. However, the error will suddenly become unstable at some circumferential wave numbers when the value is about 1010Pa. For example, with Case I, the error will increase to about 0.2% when n = 3. Meanwhile, the pre-assigned weighted parameter of the shell will show different favorable range at different circumferential wave numbers n. For example, with Case II, the favorable range of the pre-assigned weighted parameter of the shell at n = 2 is much smaller than at n = 1. From the figures, it can be indicated that pre-assigned weighted parameters of the isotropiclaminated cylindrical thin shell with different boundary conditions indicate different favorable ranges to calculate natural frequencies and the favorable range is 1011 ~ 1014Pa n o n o ðLÞ ðLÞ (102 E22 ~ 105 E22 ), where the elastic moduli of middle

middle

inner and outer layers is much smaller than the middle layer’s.

4.2. Effect of the length ratio of the non-laminated shell Based on Section 4.1, the following section chooses the preassigned weighted parameters of the sectional-laminated cylindriðLÞ

cal thin shell as 104E22 and the elastic moduli of inner and outer layers is equal to the middle layer’s. The effect of the length ratio of the non-laminated cylindrical shell is studied in this section. As shown in Fig. 8, the effect of the length ratio of the nonlaminated shell (LS/L) on the natural frequencies of the sectionallaminated cylindrical thin shell is presented. With the increase of the length ratio of the non-laminated cylindrical shell, the natural frequencies of the combined cylindrical shell exist some changes (n = 1 ~ 6, m = 1). At different circumferential wave numbers, the trends of the change are not same. For example, the natural frequency (m = 1) of the combined shell firstly decreases, then increases, and finally decreases with Case I when n = 1, 2 and 3 or with Case II when n = 2 and 3. The natural frequency (m = 1) of the sectional-laminated shell firstly decreases and finally increases with Case II when n = 1. Meanwhile, comparing with Fig. 8(a) and (b), the boundary condition could change the effect of the length ratio of the non-laminated shell. For example, when n = 1, the natural frequency (m = 1) of the sectional-laminated shell firstly decreases, then increases and finally decreases with Case I and the natural frequency (m = 1) of the sectional-laminated shell decreases firstly and increases finally with Case II. Furthermore, the change of the length ratio of the non-laminated shell presents the greatest influence on the natural frequency of the sectionallaminated shell at n = 6, m = 1. The influence of the length ratio of the non-laminated shell on the mode shapes of the sectional-laminated shell is analyzed, as

Fig. 7. Effect of pre-assigned weighted parameters (m = 1).

Fig. 8. Effect of the length ratio of the single shell.

B. Zhong et al. / Applied Acoustics 162 (2020) 107184

9

Fig. 9. Effect of the length ratio (boundary conditions: Case I, n = 6, m = 1).

Fig. 10. Effect of the length ratio (boundary conditions: Case II, n = 6, m = 1).

Table 3 MAC values relative to the LS/L = 0.3 cylindrical thin shell (n = 6, m = 1). LS =L MAC values

Case I Case II

0.3

0.5

0.7

1 1

0.970 0.968

0.929 0.922

shown in Figs. 9 and 10 with MATLAB software. Table 3 presented for the MAC values of radial model shape based on LS/L = 0.3 with two boundary conditions. As the length ratio increase from 0.3 to 0.7, the MAC decrease. It is observed that the mode shapes are mainly presented of the composite laminated cylindrical thin shell. 4.3. Effect of the thickness ratio of inner and outer layers Based on Section 4.2, the following section chooses the length ratio of the non-laminated shell as 0.5, i.e. LS/L = 0.5. This section assumes that the thickness ratio of inner and outer layers of the laminated shell is the same and the thickness of the middle layer is constant.

As shown in Fig. 11, the effect of the thickness ratio of inner and outer layers (H(1) + H(3))/H(2) on the natural frequencies of the sectional-laminated cylindrical thin shell is presented. With the increase of the thickness ratio of inner and outer layers, the natural frequencies of the sectional-laminated shell exist some changes (n = 1 ~ 6, m = 1). At different circumferential wave numbers, the trends of the change are not same. For example, the natural frequency (m = 1) of the sectional-laminated shell decreases with Case II when n = 1. The natural frequency (m = 1) of the sectional-laminated shell firstly increases and finally decreases with Case II when n = 2. Meanwhile, comparing with Fig. 11(a) and (b), the boundary condition could change the effect of the thickness ratio of inner and outer layers. For example, when n = 1, the natural frequency (m = 1) of the sectional-laminated shell increases with Case I and the natural frequency (m = 1) of the sectional-laminated shell decreases firstly and increases finally with Case II. Furthermore, the change of the thickness ratio of inner and outer layers indicates the greatest influence on the natural frequency of the sectional-laminated shell at n = 1, m = 1. The influence of the thickness ratio of inner and outer layers on the mode shapes of the sectional-laminated shell is investigated, as

Fig. 11. Effect of the thickness ratio of inner and outer layers.

10

B. Zhong et al. / Applied Acoustics 162 (2020) 107184

Fig. 12. Effect of the thickness ratio (boundary conditions: Case I and n = 6, m = 1).

Fig. 13. Effect of the thickness ratio (boundary conditions: Case II and n = 6, m = 1).

Table 4 MAC values based on the (H(1) + H(3))/H(2) = 0 cylindrical thin shell (n = 6, m = 1). (H(1) + H(3))/H(2) MAC values

Case I Case II

0

1

2

1 1

0.880 0.890

0.657 0.700

shown in Figs. 12 and 13 with MATLAB software. Based on the Section 4.2, when LS =L ¼ 0:5, the mode shapes are mainly presented of the composite laminated cylindrical thin shell. Then, the MAC values of radial model shape based on the (H(1) + H(3))/H(2) = 0 are given in Table 4. According the Figs and Tables, with the increase of thickness ratio of inner and outer layers of the composite laminated cylindrical thin shell, the deformation of mode shapes of the laminated shell part become less obvious, at the same time, the MAC related to (H(1) + H(3))/H(2) = 0 decrease.

(2) The length ratio of the non-laminated cylindrical thin shell presents an influence on the natural frequencies and mode shapes of sectional-laminated cylindrical thin shells with arbitrary boundary conditions. The greatest influence exists at n = 6 and m = 1 when the length ratio of the nonlaminated shell is 0.5. (3) The thickness ratio of inner and outer layers shows an influence on the natural frequencies and mode shapes of sectional-laminated cylindrical thin shells. The influence is affected by boundary conditions and the circumferential wave numbers. These conclusions can provide a new application based on MVP and LSWRM for more complex sectional-laminated composite cylindrical thin shells with arbitrary boundary conditions, such as combined laminated-single-laminated shells. Declaration of Competing Interest

5. Conclusion The vibration characteristics of sectional-laminated cylindrical thin shells with arbitrary boundary conditions are studied by using the interface potentials. With comparison and validation, the modeling method proposed in this paper can effectively deal with the analysis of dynamic characteristics of the sectional-laminated cylindrical thin shell. By applying this method, the influences of the pre-assigned weighted parameters, the length ratio of the non-laminated cylindrical thin shell and the thickness ratio of inner and outer layers are investigated and we have obtained the following conclusions: (1) The pre-assigned weighted parameters show a great influence on calculating natural characteristics of sectionallaminated cylindrical thin shells. The influence is affected by boundary conditions and the material properties of inner and outer layers. The favorable range of the pre-assigned n o n o ðLÞ ðLÞ weighted parameters is 103 E22 ~ 105 E22 . middle

middle

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The project is supported by the National Natural Science Foundation of China (No.51575093) and the Fundamental Research Funds for the Central Universities of China (Nos. N180313008, N182410007-06 and N170302001). Appendix A. Detailed expressions for the connection stiffness matrix

2 K CKS

6 ¼4

K CKS 0 0

11

0 K CKS 0

22

3

0 0 K CKS

7 5 33

ðA1Þ

11

B. Zhong et al. / Applied Acoustics 162 (2020) 107184

K CKS

K CKS

K CKS

¼

11

¼

22

2

K CKSL

K CKSL

K CKSL

6 ¼4

11

Z 2p  h iT kv V ðJÞ V ðJÞ dh

RL 2

22

22

¼

ðA3Þ

8 " # 9  Z 2p <  h iT k @WðJÞ @WðJÞ T = h ðJÞ ðJÞ k W W þ 2 dh : w ; @n L @n 0 0

11

0

K CKSL

0

0 Z 2p

RL 2

Z 2p 0

Z 2p

RL ¼ 2

0

K CKL

11

7 5

0 K CKSL

ðA5Þ K CkSL3 1 ¼ RL

33

ðA6Þ

h iT  2kv V ðJÞ V ðJÞ dh

ðA7Þ

K CKL

0 K CKL

K CKL

K CKL

11

22

22

22

K CKL

0

RL ¼ 2

K CkSL3 3

3

33

ðA10Þ

ðA11Þ

8 " # 9  Z 2p <  h iT k @WðJÞ @WðJÞ T = h ðJÞ ðJÞ k W W þ 2 dh : w ; @n L @n 0

K CkL

Appendix B. Detailed expressions for the continuity stiffness matrix

K CkSL K CkSL1 1 ¼ RL

K CkSL1 2 ¼ RL

K CkSL

12

K CkSL

13

K CkSL

22

K CkSL

23

31

K CkSL

32

K CkSL

33

Z 2p ( A 0

) @U ðJÞ h ðJÞ iT A16 @U ðJÞ h ðJÞ iT U U þ dh L @n RL @h

Z 2p ( A 0

7 5

11

) h iT A @UðJÞ h iT 16 @U 66 ðJÞ ðJÞ dh þ V V L @n RL @h ðJÞ

K CkSL1 3 ¼ 0 K CkSL2 1

) Z 2p ( A12 @V ðJÞ h ðJÞ iT A16 @VðJÞ h ðJÞ iT dh ¼ RL þ U U RL @h L @n 0

K CkL1 1 6 T 6 ¼ 6 K CkL1 2 4 K TCkL1 3

K CkL1 3 ¼ RL

K CkL1 2 K CkL2 2 K TCkL2 3

K CkL1 3

3

7 K CkL2 3 7 7 5 K CkL3 3

ðB10Þ

Z 2p ( A

11

K CkL2 2 ¼ RL

16

Z 2p A12 ðJÞ h ðJÞ iT U W dh RL 0

8 Z 2p > < AL66 0

ðB12Þ

ðB13Þ

h iT h iT h iT h i9 ðJÞ ðJÞ ðJÞ V ðJÞ þ DLR662 @V@n VðJÞ þ AR26L @V@h VðJÞ þ DR263 @V@h VðJÞ > = L L h ðJÞ iT h ðJÞ iT h ðJÞ iT h ðJÞ i >dh > : þ A66 V ðJÞ @V þ DLR662 VðJÞ @V@n þ AR26L VðJÞ @V@h þ DR263 VðJÞ @V@h ; L @n @V ðJÞ @n

L

ðB14Þ

ðB2Þ

ðB4Þ

ðB11Þ

Z 2p ( A

L

ðB3Þ

ðB8Þ

@n

@U ðJÞ h ðJÞ iT A66 @UðJÞ h ðJÞ iT þ V V L @n RL @h 0 9 " #T " #T A12 ðJÞ @VðJÞ A16 ðJÞ @VðJÞ = dh U þ U þ RL @h L @n ;

K CkL1 2 ¼ RL

ðB1Þ

ðB1Þ

@n@h

@U ðJÞ h ðJÞ iT A16 @U ðJÞ h ðJÞ iT þ U U L @n RL @h 0 " #T " #T 9 A11 ðJÞ @U ðJÞ A16 ðJÞ @U ðJÞ = þ U U þ dh ; L @n RL @h

3

21

26

8 h iT h iT h iT 9 3 ðJÞ 3 ðJÞ > > @ 3 W ðJÞ 16 @ W > >  DL11 W ðJÞ  2DR326 @ @hW3 WðJÞ  4D WðJÞ > > 3 > > @n3 L2 RL @n2 @h > > L > > > > > Z 2p < = h iT h ðJÞ iT h ðJÞ iT > ð J Þ ð J Þ ð J Þ 3 2 2 ðJÞ þ4D66 Þ @ W D11 @ W D12 @ W @W @W ¼ RL dh  ðD12LR W   2 2 3 2 2 2 @n @n @n@h @n @h L LR > > L L 0 > > > > > > > > h ðJÞ iT > > > > ðJÞ 2 > > : ; þ 2D2 16 @ W @W

K CkL1 1 ¼ RL

ðA12Þ

11

ðB7Þ

Z 2p ( A

2

0

K CkSL 6 K CkSL ¼ 4 K CkSL

Z 2p A12 ðJÞ h ðJÞ iT W U dh RL 0

ðB9Þ

ðA9Þ

Z 2p  h iT ku U ðJÞ U ðJÞ dh

2

L

L RL

7 5

Z RL 2p  ðJÞ h ðJÞ iT ¼ kv V V dh 2 0 RL ¼ 2

h iT h iT h iT 9 2 ðJÞ 2 ðJÞ W ðJÞ þ DLR162 @ @nV2 WðJÞ þ 2DR326 @ @hV2 WðJÞ > = L L dh h iT h iT > D12 @V ðJÞ @W ðJÞ D16 @VðJÞ @WðJÞ ; þ LR2 @h þ L2 R @n @n @n

@ 2 V ðJÞ @n@h

h iT D @ 2 WðJÞ h iT 16 W ðJÞ V ðJÞ  2 V ðJÞ RL L RL @n2 0 ) D26 @ 2 WðJÞ h ðJÞ iT 2D66 @ 2 WðJÞ h ðJÞ iT dh V  V  3 RL @h2 LR2L @n@h

K CkSL3 2 ¼ RL

8 " # 9  < h iT k @WðJÞ @WðJÞ T = h ðJÞ ðJÞ 2 kw W W þ 2 dh : ; @n L @n

0 0

> :

0

ðB5Þ

ðB6Þ

h iT  2ku U ðJÞ U ðJÞ dh

0

L

L

ðA8Þ 2

K CkSL2 3 ¼ RL

8 þ2D66 Z 2p > < D12LR 2

3

0 22

ðA4Þ

0

RL 2

K CKL 6 ¼4 0

K CkSL2 2

0

K CKSL

¼

ðA2Þ

0

RL ¼ 2

22

K CKSL

Z 2p  h iT ku U ðJÞ U ðJÞ dh

RL 2

Z 2p ( A66 @V ðJÞ h ðJÞ iT D66 @VðJÞ h ðJÞ iT ¼ RL þ 2 V V L @n LRL @n 0 ) A26 @VðJÞ h ðJÞ iT D26 @VðJÞ h ðJÞ iT þ 3 V V þ dh RL @h RL @h

K CkL2 3

8 h iT h iT h iT 9 > > ðJÞ ðJÞ ðJÞ D12 þ2D66 @ 2 V ðJÞ D16 @ 2 VðJÞ 2D26 @ 2 VðJÞ > > W þ W þ W > > 2 2 2 3 2 @n@h > > LRL @h LRL @n RL > > > Z 2p > = < h i h i h i T T T ð J Þ ð J Þ ð J Þ ð J Þ ð J Þ ð J Þ D16 @V A26 @W dh ¼ RL þ DLR122 @V@h @W þ þ V W 2 @n @n @n RL L R > > L L 0 > > > h 2 ðJÞ iT h 2 ðJÞ iT h 2 ðJÞ iT > > > > > > > W 66 ; :  D216 V ðJÞ @ W2  DR263 VðJÞ @ @hW2  2D VðJÞ @@n@h @n L R LR2 L

L

L

ðB15Þ

12

K CkL3 3

B. Zhong et al. / Applied Acoustics 162 (2020) 107184

8 h iT h iT h iT 9 3 ðJÞ 3 ðJÞ > @ 3 W ðJÞ > 16 @ W > >  DL11 W ðJÞ  2DR326 @ @hW3 WðJÞ  4D WðJÞ 3 2 2 > > @n3 @n @h L R > > L L > > > > h i h i h i > > T T T > > > > ðJÞ ðD12 þ4D66 Þ @ 3 W ðJÞ D11 @ 2 WðJÞ @WðJÞ D12 @ 2 WðJÞ @WðJÞ > >  LR2 W  L3 @n2  LR2 @h2 > > 2 @n @n > > @n@h > > L L > Z 2p > < = h i h i h i T T T ðJÞ @ 3 WðJÞ ðJÞ @ 3 WðJÞ 2D16 @ 2 W ðJÞ @W ðJÞ D11 2D26 ¼ RL dh þ L2 R @n@h @n  L3 W  3 3 W 3 @n @h RL > > L 0 > > > > > > 4D16 ðJÞ h@3 W ðJÞ iT ðD12 þ4D66 Þ ðJÞ h@3 WðJÞ iT D11 @WðJÞ h@ 2 WðJÞ iT > > > > > > W @n@h2  L3 @n > > >  L2 RL W @n2 @h  LR2L > @n2 > > > > > > h 2 ðJÞ iT h 2 ðJÞ iT > > ð J Þ ð J Þ > > 2D16 @W @ W @ W : ;  DLR122 @W þ @n @n @n@h @h2 L2 R L

L

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