Dynamics properties of composite sandwich open circular cylindrical shells

Composite Structures 189 (2018) 148–159

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Dynamics properties of composite sandwich open circular cylindrical shells a,b

Yanchun Zhai a b

a

a

, Mengjiang Chai , Jianmin Su , Sen Liang

b,⁎

T

School of Mechanical Engineering, Weifang University of Science and Technology, Weifang 262700, China School of Mechanical Engineering, Qingdao University of Technology, 266520 Qingdao, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Open circular cylindrical shells Composite material Constrained viscoelastic core First-order shear deformation shell theory Dynamic properties

This paper deals with the dynamic properties of three-layered composite sandwich open circular cylindrical shells (CSOCCS). First, the equations of motion that govern the free vibrations of CSOCCS are derived by applying Hamilton’s principle based on the first-order shear deformation shell theory. Owing to considering the effect of rotary inertias and shear deformation, thin-to-moderately thick shells can be analyzed. Next, these equations are solved by means of the closed-form Navier method. The calculated results are compared with the findings of previous studies and those obtained by the finite element method, and a good agreement is observed. The variation of modal loss factor and frequency with system parameters is evaluated and presented graphically. It is the first time to study the dynamic properties of composite open circular cylindrical shells with constrained viscoelastic core.

1. Introduction A typical sandwich structure consists of two stiff face layers that carry the great portion of the bending load separated by a light inner core that has energy dissipating property [1]. Composite sandwich structures are very effective in sound insulation and reducing vibration response of lightweight and flexible structures, where the soft core is strongly deformed in shear, due to the adjacent stiff layers [2]. CSOCCS is a special form of closed circular cylindrical sandwich shells, and often used as structural components of pressure vessels, roof structures, open space buildings, and marine structures. A huge amount of research efforts have been devoted to the vibration analysis of sandwich structures, such as beams, plates and shells. Since 1958 and the work of Kerwin and Edwar [3], later, Ross et al. [4], DiTaranto [5], and Mead and Markus [6] extended Kerwin’s work, and various kinds of shell theories have been proposed and developed. Among them, thin shell theories neglecting the shear deformation [7–12]; first order shear deformation theories [13,14]; and recently, the high-order shear deformation theory [15,16] widely applied to dynamic behaviors, optimal design, bending, impacting and vibration. And, by making some different assumptions and simplifications, various sub-series thin shell theories were developed, such as Goldenveizer–Novozhilov’s theory [7,8], Reissner–Naghdi’s theory [9,10],

⁎

and Donnel–Mushtari's theory [11,12], etc, some review articles described these theories in detail [17–19]. On the basis of these theories, many relevant studies of open circular cylindrical shells have been carried out, quite extensively, by both analytical and numerical methods in the subsequent decades. Suzuki and Leissa [20] developed an exact solution procedure for determining the free vibration frequencies and mode shapes of open cylindrical shells. Yu et al. [21] obtained the exact solutions using the generalized Navier method for the free vibration analysis of open circular cylindrical shells with different combinations of boundary conditions based on the Donnell–Mushtari’s theory. Selmane and Lakis [22] presented a method for the dynamic and static analysis of thin, elastic, anisotropic and non-uniform open cylindrical shells. Lim et al. [23] presented a three-dimensional elastic analysis of the vibration of open cylindrical shells, and obtained the natural frequencies and vibration mode shapes via a three-dimensional displacement-based extremum energy principle. Zhang and Xiang [24] developed an analytical procedure for determining the free vibration frequencies of open circular cylindrical shells with intermediate ring supports based on the Flügge thin shell theory. Kandasamy and Singh presented different numerical methods based on the Rayleigh-Ritz method for the forced vibration of open cylindrical shells [25] and the free vibration of skewed open circular cylindrical deep shells [26]. L.K. Abbas et al. [27] used the transfer

Corresponding author. E-mail address: [email protected] (S. Liang).

https://doi.org/10.1016/j.compstruct.2018.01.076 Received 30 August 2017; Received in revised form 28 December 2017; Accepted 22 January 2018 0263-8223/ © 2018 Published by Elsevier Ltd.

Composite Structures 189 (2018) 148–159

Y. Zhai et al.

and constraining layers, 1, 2, 3 represent the principal coordinate system of the composite material. 1 and 2 represent the direction parallel to and perpendicular to fiber orientation respectively. And 3 denote the direction perpendicular to 1–2 plane. The angle between the principal coordinate of the composite material in ith layer and the x-axis of the shell is named angle of fiber and denoted by θi . In order to derive the governing equations, assumptions are as follows: 1) shear modulus and elasticity modulus of the viscoelastic material adopt the constant complex modulus, 2) normal strain is negligible, 3) transverse displacement of all points on a normal to the shell is constant, 4) there is no slip between the layers.

matrix method to analyze natural frequencies and mode shapes of openvariable thickness circular cylindrical shells in a high-temperature field, and employed the fourth-order Runge-Kutta method to solve the matrix equation. S. Joniak et al. [28] proposed an analytical formula of critical stresses, and investigated the elastic buckling and limit load of open circular cylindrical thin shells in pure bending state with the use of the finite element method. Ye T. et al. [29,30] presented a unified formulation to investigate the vibrations of composite laminated deep open shells with various shell curvatures and arbitrary restraints, including cylindrical, conical and spherical ones. Z. Su et al. [31] studied the free vibration analysis of moderately thick functionally graded open shells with general boundary conditions with Rayleigh-Ritz method. X.L. Yao et al. [32] employed an analytical solution of the traveling wave form to study the free vibration analysis of open circular cylindrical shells based on the Donnell-Mushtari-Vlasov thin shell theory. Punera and Kant [33] presented the free vibration analysis of functionally graded open cylindrical shells by using various refined higher order theories. Q. Wang et al. [34] presented a new three-dimensional exact solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions. Through the correlative literature reviews, we can see that most of the existing works of the laminated composite open circular cylindrical shells (LCOCCS) were restricted to the undamped open cylindrical shells, and concretely concentrated on the following aspects: free and force vibration, solution methods, dynamic analysis, bucking and bending, and so on. To the authors’ knowledge, there is no work reported on the dynamic properties of CSOCCS. However, in practice, CSOCCS have been widely applied to the field of engineering and technology; therefore, it becomes very important and necessary to study the dynamics properties of CSOCCS, thus providing some useful numerical results for the practical engineering applications of CSOCCS.

2.1. Kinematic relations Based on the first-order shear deformation shell theory and the above assumptions, the displacements in the middle surface of each layer are expressed by:

Ui (x ,θ,z ,t ) = ui (x ,θ,t ) + z (i) αi (x ,θ,t ) Vi (x ,θ,z ,t ) = vi (x ,θ,t ) + z (i) βi (x ,θ,t ) Wi (x ,θ,z ,t ) = w (x ,θ,t )

(1)

where, i = 1,2,3. t is time variable; Ui , Vi and Wi are the generalized displacements of ith layer along the axial, circumferential and radial coordinates, respectively. ui,vi are the axial and circumferential displacements of the middle surface of ith layer, respectively. αi,βi are the rotations of transverse normal to center plane with respect to the circumferential and axial coordinate, respectively. The z (i) is measured with respect to the mid-plane of the ith layer, −hi /2 ≤ z (i) ≤ hi /2, i = 1,2,3. The linear strain–displacement relations of ith layer space are written as follows:

2. Vibration equation of CSOCCS

(i) εxx = ∂ui / ∂x + z (i) ∂αi/ ∂x (i) εθθ = ∂vi/ Ri ∂θ + w / Ri + z (i) ∂βi / Ri ∂θ

The geometry and dimension of the CSOCCS with total thickness h and subtended angle θ0 is shown in Fig. 1, and axial (x), circumferential (y) and radial (z) coordinate system are defined. The symbols Ri and hi (i = 1, 2, 3) denote radius and thickness of the ith layer, and the subscripts 1, 2, and 3 in the following derivation are designated for base shell, viscoelastic core and constraining layer, respectively. The length of open cylindrical shells is a. In the base layers

(i) γxθ = ∂ui / Ri ∂θ + ∂vi/ ∂x + z (i) (∂αi/ Ri ∂θ + ∂βi / ∂x )

γθz(i) = kc (∂w / Ri ∂θ + βi−vi/ Ri ) (i) γxz = kc (∂w / ∂x + αi )

(2) (i) (i) (i) εxx , εθθ , γxθ

where, kc is the shear correction factor. represent ith normal strain components in the middle surface of the shell along coordinate (i) represent ith shear strain components in the middle surface axis; γθz(i) , γxz along coordinate axis. According to the generalized Hooke's law, the stress–strain relation of the ith layer can be expressed, as follows: (i)

(i) (i) (i) 0 0 ⎞ ⎛ ε11 ⎞ ⎛ σ11 ⎞ ⎛Q11 Q12 0 (i) ( i ) ( i ) ( i) ⎜ σ ⎟ ⎜Q Q 0 0 0 ⎟ ⎜ ε22 ⎟ 22 12 22 ⎜ ⎜ (i) ⎟ ⎜ ⎟ (i) ⎟ (i) = 0 Q66 0 0 ⎟ ⎜ γ12 ⎟ ⎜ σ12 ⎟ ⎜ 0 (i) ⎟ (i) (i) ⎜ τ23 ⎟ ⎜ 0 0 0 Q44 0 ⎟ ⎜ γ23 ⎜ τ (i) ⎟ ⎜ 0 (i) ⎟ ⎜ (i) ⎟ 0 0 0 Q55 ⎠ γ13 ⎝ 13 ⎠ ⎝ ⎝ ⎠

(3) (i) (i) τ23 ,τ13

(i) (i) (i) σ11 ,σ22 ,σ12

where, represent ith normal stress components; represent ith shear stress components. The reduced stiffness components (i) Qmn of ith layer are given, as follows, (i) Q11 =

E1(i) (i) (i) ν21 1−ν12

(i) = ,Q12

(i) (i) ν12 E2 (i) (i) ν21 1−ν12

(i) = ,Q22

E2(i) (i) (i) ν21 1−ν12

(i) (i) (i) = G23 ,Q44 ,Q55

(i) (i) (i) = G13 = G12 ,Q66 ,i = 1,2,3

(i) (i) (i) (i) (i) Among them, E1(i) , E2(i) , G12 , G13 , G23 , ν12 , ν21 represent elasticity modulus, shear modulus, and Poisson’s ratio of ith layer, respectively. By coordinate transformation, the stress–strain relations in the coordinate system can be obtained as

Fig. 1. Geometry and coordinate system of the CSOCCS.

149

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Y. Zhai et al.

(1) (1) (1) R1 A11 u1,xx + A12 (v1,xθ + w ,x ) + A16 (u1,xθ + R1 v1,xx + u1,xθ )

(i) (i) (i) (i) (i) 0 ⎞ ⎛ εxx ⎞ ⎛ σxx ⎞ ⎛Q11 Q12 Q16 0 (i) ⎜ σ (i) ⎟ ⎜Q (i) Q (i) Q (i) 0 0 ⎟ ⎜ εθθ ⎟ θθ 12 22 26 ⎟ ⎜ (i) ⎟ ⎜ (i) ⎟ ⎜ (i) (i) (i) 0 ⎟ ⎜ γxθ ⎟ ⎜ τxθ ⎟ = ⎜Q16 Q26 Q66 0 (i) (i) ⎟ (i) (i) ⎜ 0 ⎜ τθz 0 0 Q44 Q45 ⎟ ⎜ γθz ⎟ ⎟⎜ ⎟ ⎜ (i) ⎟ ⎜⎜ (i) (i) ⎟ ⎜ γ (i) ⎟ 0 0 Q45 Q55 ⎠ ⎝ xz ⎠ ⎝ τxz ⎠ ⎝ 0

+

(1) A26 (1) ⎛ 1 u1,θθ + v1,xθ⎞ (v1,θθ + w ,θ ) + A66 R1 ⎝ R1 ⎠

+

1⎡ (2) (2) (2) R2 A11 u2,xx + A12 (v2,xθ + w ,x ) + A16 (u2,xθ + R2 v2,xx + u2,xθ ) 2⎢ ⎣

+

(2) A26 (2) ⎛ 1 u2,θθ + v2,xθ⎞ ⎤ (v2,θθ + w ,θ ) + A66 ⎥ R2 ⎝ R2 ⎠⎦

(4)

(i)

(i) (i) (i) (i) cos4 θi + 2(Q12 )sin2 θi cos2 θi + Q22 sin4 θi Q11 = Q11 + 2Q66 (i)

(i) (i) (i) (i) )sin2 θi cos2 θi + Q12 (sin4 θi + cos4 θi ) Q12 = (Q11 + Q22 −4Q66 (i) Q22 (i) Q16 (i) Q26 (i) Q66 (i) Q44 (i) Q45 (i) Q55

=

(i) Q11 sin4 θi

+

(i) 2(Q12

+

(i) 2Q66 )sin2 θi cos2 θi

+

−

(i) cos4 θi Q22

(i) (i) (i) (i) (i) (i) )sinθi cos3 θi + (Q12 )sin3 θi cosθi = (Q11 −Q12 −2Q66 −Q22 + 2Q66

⎟

⎜

⎟

1 ⎡ (2) (2) (2) R2 D11 α2,xx + D12 β2,xθ + D16 (α2,xθ + R2 β2,xx + α2,xθ ) h2 ⎢ ⎣

+

(2) D26 1 (2) (2) ⎛ 1 β α2,θθ + β2,xθ⎞ ⎤ + D66 ⎥ + h2 C45 (w ,θ + R2 β2−v2) R2 2,θθ ⎝ R2 ⎠⎦

+

R2 ρ2 h2 ∂2u2 R2 ρ2 h 22 ∂2α2 1 (2) ∂ 2u C55 R2 (w ,x + α2) = R1 ρ1 h1 21 + − h2 2 12 ∂t ∂t 2 ∂t 2

(i) (i) (i) (i) (i) (i) )sin3 θi cosθi + (Q12 )sinθi cos3 θi = (Q11 −Q12 −2Q66 −Q22 + 2Q66 (i) (i) (i) (i) (i) )sin2θi cos2 θi + Q66 (sin4 θi + cos4 θi ) = (Q11 + Q22 −2Q12 −2Q66

⎜

(i) (i) cos2 θi + Q55 sin2 θi = Q44

⎜

⎟

(9-a)

(i) (i) )sinθi cosθi = (Q55 −Q44 (i) (i) cos2 θi + Q44 sin2 θi = Q55

(3) (3) (3) R3 A11 u3,xx + A12 (v3,xθ + w ,x ) + A16 (u3,xθ + R3 v3,xx + u3,xθ )

where, θi is the angle of fiber in ith layer.

+

(3) A26 (3) ⎛ 1 u3,θθ + v3,xθ⎞ (v3,θθ + w ,θ ) + A66 R3 ⎝ R3 ⎠

2.2. Equations of motion

+

1⎡ (2) (2) (2) R2 A11 u2,xx + A12 (v2,xθ + w ,x ) + A16 (u2,xθ + R2 v2,xx + u2,xθ ) 2⎢ ⎣

The equations of the motion of CSOCCS based on the Hamilton principle could be obtained:

+

(2) A26 (2) ⎛ 1 u2,θθ + v2,xθ⎞ ⎤ (v2,θθ + w ,θ ) + A66 ⎥ R2 ⎝ R2 ⎠⎦

(5)

+

1 ⎡ (2) (2) (2) R2 D11 α2,xx + D12 β2,xθ + D16 (α2,xθ + R2 β2,xx + α2,xθ ) h2 ⎢ ⎣

where, T is the kinetic energy of sandwich plate, U is the deformation energy. The deformation energy’s expression is

+

(2) D26 1 (2) (2) ⎛ 1 β α2,θθ + β2,xθ⎞ ⎤ + D66 ⎥− C45 (w ,θ + R2 β2−v2) R2 2,θθ ⎝ R2 ⎠ ⎦ h2

∫t

t1

(δT −δU ) = 0

0

U=

3

−

1 2

∑ ∫0 ∫0 ∫−h /2 (εxx(i) σxx(i) + εθθ(i) σθθ(i) + εxθ(i) σxθ(i) + γθz(i) τθz(i) i

+

(i) (i) γxz τxz ) Ri dzdxdθ

θ0

hi /2

a

⎜

3

2

+

hi2 ⎡ ⎛ ∂αi ⎞2 ⎛ ∂βi ⎞ ⎤ ⎤ + ⎥ ⎥ dθdx 12 ⎢ ⎝ ∂t ⎠ ⎦ ⎦ ⎣ ⎝ ∂t ⎠

R2 ρ2 h2 ∂2u2 R2 ρ2 h 22 ∂2α2 1 (2) ∂ 2u C55 R2 (w ,x + α2) = R3 ρ3 h3 23 + + h2 2 12 ∂t 2 ∂t 2 ∂t

(1) A12 u1,xθ +

θ0

i=1

a

2

⎜

(1) A22 (1) u1,xx (v1,θθ + w ,θ ) + R1 A16 R1

1 (1) ⎛ (1) v1,xθ + w ,x + u1,θθ + v1,xθ⎞ + A66 (u1,xθ + R1 v1,xx ) + A26 R1 ⎝ ⎠ v (1) (1) ⎛ 1 w ,θ + β1− 1 ⎞ + C45 (w ,x + α1) + C44 R1 ⎠ ⎝ R1

2

⎜

2

⎟

⎜

⎟

(7)

+

where, ρi is the mass density of ith layer. According to the continuity relationship between layers, the displacements of the core layer are yielded:

⎟

A (2) 1 ⎡ (2) (2) ⎛ 1 A12 u2,xθ + 22 (v2,θθ + w ,θ ) + A26 u2,θθ + v2,xθ⎞ R 2⎢ 2 ⎝ R2 ⎠ ⎣ ⎜

β2 =

1 (u 2 1 1 (v 2 1

+ u3) + + v3 ) +

v (2) (2) ⎛ 1 w ,θ + β2− 2 ⎞ + C45 (w ,x + α2) ⎤ + C44 ⎥ R2 ⎠ ⎝ R2 ⎦

1 (h α −h3 α3 ) 4 1 1 1 (h β −h3 β3 ) 4 1 1

1 1 (u −u1)− 2h (h1 α1 h2 3 2 1 1 ( ) (h1 β1 v v − − 3 1 2h h 2

2

−

+ h3 α3 ) + h3 β3)

⎟

(2) (2) (2) u2,xx + A26 + R2 A16 (v2,xθ + w ,x ) + A66 (u2,xθ + R2 v2,xx )

⎜

α2 =

⎟

(9-b)

∂u ∂v ⎡ ∂w Ri ρi hi ⎢ ⎛ ⎞ + ⎛ i ⎞ + ⎛ i ⎞ ∂t ⎠ ∂t ⎠ ∂t ⎠ ⎝ ⎝ ⎝ ⎣

∑ ∫0 ∫0

v2 =

⎟

⎜

(6)

1 2

u2 =

⎟

i=1

The kinetic energy’s expression is

T=

⎜

⎟

D (2) 1 ⎡ (2) (2) ⎛ 1 (2) D12 α2,xθ + 22 β2,θθ + D26 α2,θθ + β2,xθ⎞ + R2 D16 α2,xx h2 ⎢ R 2 ⎝ R2 ⎠ ⎣ ⎜

⎟

(2) (2) + D26 (α2,xθ + R2 β2,xx ) ⎤ β2,xθ + D66 ⎥ ⎦

(8)

1 ⎡ v (2) ⎛ 1 (2) (w ,x + α2) ⎤ R2 C44 w ,θ + β2− 2 ⎞ + R2 C45 ⎥ h2 ⎢ R2 ⎠ ⎝ R2 ⎣ ⎦ 2 2 R2 ρ2 h2 ∂2v2 R2 ρ2 h2 ∂ β2 ∂ 2v 1 = R1 ρ1 h1 2 + − 2 12 ∂t ∂t 2 ∂t 2

Thus can reduce the number of variables to 9, they are u1, u3, v1, v3, α1, α3 , β1, β3 and w , so the kinetic energy and deformation energy’s formula could be rewritten by bringing the Eq. (8) into Eqs. (6) and (7), then combining the Eq. (5), equilibrium equations of free vibration are obtained and listed in the appendix A. By bringing the Eqs. (A4–A6) into Eq. (A1), we can obtain the equations of motion.

+

150

⎜

⎟

(9-c)

Composite Structures 189 (2018) 148–159

Y. Zhai et al.

Table 1 Comparison of natural frequencies of LCOCCS. m

Contrast

1

Frequencies (Hz)

Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present Ref. [32] Present

2 3 4 5 6 7 8 9 10

(3) A12 u3,xθ +

n=1

n=2

n=3

n=4

n=5

n=6

n=7

n=8

n=9

n = 10

10.683 10.675 9.182 9.125 19.590 19.534 34.671 34.619 54.099 54.0483 77.848 77.795 105.917 105.853 138.304 138.220 175.010 174.889 216.034 215.858

35.112 35.124 13.998 13.954 20.578 20.513 35.217 35.154 54.570 54.508 78.301 78.236 106.364 106.289 138.750 138.65 175.455 175.322 216.478 216.290

62.317 62.344 24.121 24.094 23.272 23.198 36.365 36.285 55.422 55.3411 79.079 78.995 107.119 107.025 139.496 139.380 176.198 176.045 217.220 217.011

85.709 85.747 37.061 37.046 28.184 28.105 38.369 38.267 56.739 56.6336 80.214 80.104 108.195 108.074 140.550 140.407 177.242 177.062 218.260 218.023

103.796 103.843 50.864 50.858 35.052 34.973 41.410 41.286 58.608 58.473 81.741 81.599 109.608 109.45 141.919 141.741 178.593 178.377 219.602 219.329

117.261 117.313 64.336 64.336 43.239 43.160 45.529 45.385 61.097 60.929 83.695 83.515 111.375 111.180 143.612 143.393 180.253 179.995 221.247 220.931

127.240 127.296 76.826 76.8302 52.131 52.052 50.627 50.464 64.239 64.036 86.105 85.883 113.511 113.271 145.637 145.370 182.229 181.922 223.199 222.833

134.726 134.785 88.060 88.067 61.261 61.182 56.519 56.339 68.029 67.791 88.989 88.722 116.030 115.740 148.002 147.683 184.525 184.162 225.491 225.037

140.462 140.523 97.998 98.007 70.311 70.229 62.993 62.798 72.428 72.154 92.351 92.037 118.942 118.598 150.715 150.337 187.144 186.721 228.035 227.548

144.974 145.036 106.726 106.735 79.082 78.997 69.855 69.646 77.369 77.062 96.184 95.821 122.250 121.849 153.780 153.340 190.092 189.603 230.923 230.367

(3) A22 (3) u3,xx (v3,θθ + w ,θ ) + R3 A16 R3

(3) (3) (3) (α3,xθ + R3 β3,xx + α3,xθ ) + R3 D11 α3,xx + D12 β3,xθ + D16

(3) ⎛ 1 (3) u3,θθ + v3,xθ + v3,xθ + w ,x ⎞ + A66 + A26 (u3,xθ + R3 v3,xx ) ⎝ R3 ⎠ v (3) (3) ⎛ 1 w ,θ + β3− 3 ⎞ + C45 + C44 (w ,x + α3) R3 ⎠ ⎝ R3 ⎜

+

v (3) ⎛ 1 (3) ⎛ 1 (3) (w ,x + α3) ⎤ + D66 α3,θθ + β3,xθ⎞−R3 ⎡C45 w ,θ + β3− 3 ⎞ + C55 ⎥ ⎢ R3 ⎠ ⎝ R3 ⎠ ⎝ R3 ⎦ ⎣

⎟

⎜

⎜

−

⎟

A (2) 1 ⎡ (2) (2) ⎛ 1 A12 u2,xθ + 22 (v2,θθ + w ,θ ) + A26 u2,θθ + v2,xθ⎞ ⎢ R2 2⎣ ⎝ R2 ⎠ ⎜

⎟

+

(2) (2) (2) u2,xx + A26 + R2 A16 (v2,xθ + w ,x ) + A66 (u2,xθ + R2 v2,xx )

+

(2) C44

−

⎛ 1 w ,θ + β − v2 ⎞ + C (2) (w ,x + α2) ⎤ 45 2 ⎥ R2 ⎠ ⎝ R2 ⎦ ⎜

⎟

+

D (2) 1 ⎡ (2) (2) ⎛ 1 (2) D12 α2,xθ + 22 β2,θθ + D26 + α2,θθ + β2,xθ⎞ + R2 D16 α2,xx h2 ⎢ R 2 ⎝ R2 ⎠ ⎣ ⎜

⎟

1 ⎡ (2) ⎛ 1 v (2) (w ,x + α2) ⎤ R2 C44 w ,θ + β2− 2 ⎞ + C45 ⎥ h2 ⎢ R2 ⎠ ⎝ R2 ⎣ ⎦ 2 2 R2 ρ2 h2 ∂2v2 R2 ρ2 h2 ∂ β2 ∂ 2v 3 = R3 ρ3 h3 2 + + ∂t ∂t 2 ∂t 2 2 12 ⎜

⎟

(2) A26 (2) ⎛ 1 (v2,θθ + w ,θ ) + A66 u2,θθ + v2,xθ⎞ ⎤ ⎥ R2 ⎝ R2 ⎠⎦ ⎜

⎟

h3 ⎡ (2) (2) (2) (α2,xθ + R2 β2,xx + α2,xθ ) R2 D11 α2,xx + D12 β2,xθ + D16 2h2 ⎢ ⎣ (2) D26 (2) ⎛ 1 β α2,θθ + β2,xθ⎞ ⎤ + D66 ⎥ R2 2,θθ ⎝ R2 ⎠⎦ ⎜

⎟

h3 ⎡ (2) ⎛ 1 v (2) (w ,x + α2) ⎤ R2 C45 w ,θ + β2− 2 ⎞ + C55 ⎥ 2h2 ⎢ R2 ⎠ ⎝ R2 ⎣ ⎦ 3 2 2 2 2 R3 ρ3 h3 ∂ α3 R2 ρ2 h3 h2 ∂ α2 R2 ρ2 h3 h2 ∂ u2 =− + − 4 12 24 ∂t 2 ∂t 2 ∂t 2

(1) D12 α1,xθ +

(9-d)

⎜

⎟

(9-f)

(1) D22

R1

(1) (1) ⎛ 1 (1) (α1,xθ + R1 β1,xx ) β1,θθ + R1 D16 α1,xx + D26 α1,θθ + β1,xθ + β1,xθ⎞ + D66 ⎝ R1 ⎠ ⎜

⎜

⎜

⎟

v (1) (1) ⎛ 1 (w ,x + α1) ⎤ −R1 ⎡C44 w ,θ + β1− 1 ⎞ + C45 ⎢ ⎥ R1 ⎠ ⎝ R1 ⎣ ⎦

(1) ⎛ 1 (1) (1) (1) R1 D11 α1,xx + D12 β1,xθ + D16 (α1,xθ + R1 β1,xx + α1,xθ ) + β + D66 α1,θθ + β1,xθ⎞ R1 1,θθ ⎝ R1 ⎠ v (1) (1) ⎛ 1 (w ,x + α1) −R1 C45 w ,θ + β1− 1 ⎞−R1 C55 R1 ⎠ ⎝ R1 ⎜

⎟

(2) A22

h ⎡ (2) (2) (2) 1 (v 2,θθ + w ,θ ) + A26 ( u2,θθ + v 2,xθ ) + R2 A16 + 1 ⎢A12 u2,xθ + u2,xx 4 R2 R2 ⎣

⎟

⎟

⎤ v (2) (2) 1 (2) (2) (v 2,xθ + w ,x ) + A66 (u2,xθ + R2 v 2,xx ) + C44 ( w ,θ + β2− 2 ) + C45 (w ,x + α2) ⎥ + A26 R2 R2 ⎦

A (2) h ⎡ (2) (2) (2) + 1 ⎢R2 A11 u2,xx + A12 (v 2,xθ + w ,x ) + A16 (u2,xθ + R2 v 2,xx + u2,xθ ) + 26 (v 2,θθ + w ,θ ) 4 R2 ⎣

D (2) h ⎡ (2) (2) (2) (2) ⎛ 1 α2,xθ + 22 β2,θθ + D26 α2,θθ + β2,xθ⎞ + R2 D16 α2,xx + D26 β2,xθ − 1 ⎢D12 2h2 R2 ⎝ R2 ⎠ ⎣ ⎜

⎤ h ⎡ (2) (2) (2) ⎛ 1 (2) u2,θθ + v 2,xθ⎞ ⎥− 1 ⎢R2 D11 α2,xx + D12 β2,xθ + D16 (α2,xθ + R2 β2,xx + α2,xθ ) + A66 ⎝ R2 ⎠ ⎦ 2h2 ⎣ ⎜

+

⎟

⎟

(1) D26

(2) D26

⎜

h3 ⎡ (2) (2) (2) (v2,xθ + w ,x ) + A16 (u2,xθ + R2 v2,xx + u2,xθ ) R2 A11 u2,xx + A12 4 ⎢ ⎣

+

(2) (2) + D26 (α2,xθ + R2 β2,xx ) ⎤ β2,xθ + D66 ⎥ ⎦

−

(3) D26 β R3 3,θθ

⎟

⎟

⎤ v h (2) (2) ⎛ 1 (2) (α2,xθ + R2 β2,xx ) ⎥ + 1 R2 ⎡C44 (w ,x + α2) ⎤ w ,θ + β2− 2 ⎞ + C45 + D66 ⎥ 2h2 ⎢ R2 ⎠ ⎝ R2 ⎣ ⎦ ⎦ 3 2 2 2 2 R2 ρ2 h1 h2 ∂ v 2 R1 ρ1 h1 ∂ β1 R2 ρ2 h1 h2 ∂ β2 = + − 4 12 24 ∂t 2 ∂t 2 ∂t 2 ⎜

⎤ h v (2) ⎛ 1 (2) ⎛ 1 (2) β α2,θθ + β2,xθ⎞ ⎥ + 1 R2 ⎡C45 w ,θ + β2− 2 ⎞ + C55 (w ,x + α2) ⎤ + D66 ⎢ ⎥ R2 2,θθ 2h2 ⎣ R2 ⎠ ⎝ R2 ⎠⎦ ⎝ R2 ⎦ ⎜

⎟

R2 ρ2 h1 h2 ∂2u2 R1 ρ1 h13 ∂2α1 R2 ρ2 h1 h22 ∂2α2 = + − 4 12 24 ∂t 2 ∂t 2 ∂t 2

⎜

⎟

(9-e)

151

⎟

(9-g)

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41.954 42.000 5.312 5.296

(3) D12 α3,xθ +

(3) D22 (3) (3) ⎛ 1 β α3,xx + D26 α3,θθ + β3,xθ + β3,xθ⎞ + R3 D16 R3 3,θθ ⎝ R3 ⎠ ⎜

⎜

−

⎜

⎟

(2) (2) (2) u2,xx + A26 (v2,xθ + w ,x ) + A66 (u2,xθ + R2 v2,xx ) + R2 A16

v (2) ⎛ 1 (2) w ,θ + β2− 2 ⎞ + C45 (w ,x + α2) ⎤ + C44 ⎥ R R 2⎠ ⎝ 2 ⎦

38.013 37.927 6.239 6.172

−

⎟

D (2) h3 ⎡ (2) (2) ⎛ 1 (2) D12 α2,xθ + 22 β2,θθ + D26 α2,θθ + β2,xθ⎞ + R2 D16 α2,xx ⎢ R2 2h2 ⎣ ⎝ R2 ⎠ ⎜

⎟

(2) (2) β2,xθ + D66 (α2,xθ + R2 β2,xx ) ⎤ + D26 ⎥ ⎦

h3 ⎡ (2) ⎛ 1 v (2) R2 C44 w ,θ + β2− 2 ⎞ + C45 (w ,x + α2) ⎤ ⎥ R2 ⎠ 2h2 ⎢ ⎝ R2 ⎣ ⎦ 3 2 2 2 2 R3 ρ3 h3 ∂ β3 R2 ρ2 h3 h2 ∂ β2 R2 ρ2 h3 h2 ∂ v2 + − =− 4 12 24 ∂t 2 ∂t 2 ∂t 2

35.677 35.607 6.754 6.742

+

⎜

⎟

(9-h)

v1,θ ⎞ (1) (1) ⎛ 1 (2w ,xθ + R1 β1,x −v1,x + α1,θ ) + C45 C44 w ,θθ + β1,θ− R1 ⎠ R 1 ⎝ v3,θ ⎞ (1) (3) ⎛ 1 (w ,xx + α1,x ) + C44 + R1 C55 w ,θθ + β3,θ− R3 ⎠ R ⎝ 3

34.901 34.812 0.681 0.677

⎜

⎟

⎜

⎟

(3) (3) (2w ,xθ + R3 β3,x −v3,x + α3,θ ) + R3 C55 (w ,xx + α3,x ) + C45

v2,θ ⎞ (2) ⎛ 1 (2) (2w ,xθ + R2 β2,x −v2,x + α2,θ ) w ,θθ + β2,θ− + C44 + C45 R2 ⎠ R ⎝ 2

34.710 34.590 7.024 6.979

⎜

⎟

A (1) A (1) A (1) (2) (1) (w ,xx + α2,x )−⎜⎛A12 u1,x + 22 v1,θ + 22 w + 26 u1,θ + R2 C55 R1 R1 R1 ⎝

28.389 28.472 5.116 4.943

A (2) A (2) A (2) (1) (2) (2) v2,x ⎞⎟ v1,x ⎟⎞−⎜⎛A12 u2,x + 22 v2,θ + 22 w + 26 u2,θ + A26 + A26 R2 R2 R2 ⎠ ⎝ ⎠ A (3) A (3) A (3) (3) (3) u3,x + 22 v3,θ + 22 w + 26 u3,θ + A26 v3,x ⎞⎟ −⎛⎜A12 R R R3 3 3 ⎝ ⎠ = (R1 ρ1 h1 + R2 ρ2 h2 + R3 ρ3 h3 ) w ,tt

(9-i)

20.241 20.255 9.015 8.970

where, the subscript ‘comma’ represents the partial derivative, such as (.),x represents differentiation with respect to variable x, and (.),xθ denotes differentiation with respect to variables x and θ , and so on.

16.923 16.898 12.215 12.207

3. Results and discussion 3.1. Solution and numerical examples

11.710 11.607 4.693 4.780

In all numerical examples analyzed in the next section, the following boundary conditions of simple support will be used: At edges x = 0 and x = a:

N x(i) = 0 vi = 0 w = 0 Mx(i) = 0 βi = 0

(10-a)

At edges θ =0 and θ = θ0 :

ANSYS Present ANSYS Present

Nθ(i) = 0 ui = 0 w = 0 Mθ(i) = 0 αi = 0

(10-b)

where i = 1,3. The analytical solutions for the dynamic properties of simply supported CSOCCS will be obtained using Navier’s type solution. The generalized displacement field to satisfy the above boundary conditions is expressed in double trigonometric series as,

Loss factor (%)

Frequencies (Hz)

(m,n) = (3,3) (m,n) = (3,2) (m,n) = (1,2) (m,n) = (3,1) (m,n) = (2,3) (m,n) = (2,2) (m,n) = (2,1) (m,n) = (1,1)

Modes Contrast

⎟

A (2) h3 ⎡ (2) (2) ⎛ 1 A12 u2,xθ + 22 (v2,θθ + w ,θ ) + A26 u2,θθ + v2,xθ⎞ ⎢ R2 4 ⎣ ⎝ R2 ⎠

⎜

Table 2 The first ten frequencies and loss factors of CSOCCS (a = 10m , h1 = h3 = 10mm , h2 = 5mm , θ0 = π/6 ).

⎟

v (3) (3) ⎛ 1 (3) w ,θ + β3− 3 ⎞ + C45 (α3,xθ + R3 β3,xx )−R3 ⎡C44 (w ,x + α3) ⎤ + D66 ⎥ ⎢ R R 3⎠ ⎝ 3 ⎦ ⎣

40.214 40.148 2.751 2.772

(m,n) = (2,4)

(m,n) = (3,4)

Y. Zhai et al.

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Fig. 2. ANSYS model and some mode shapes of the CSOCCS.

Fig. 3. Effect of thickness ratio (h1/h3) on loss factors and frequencies (Hz) (R2 = 5m , h = 25mm , h2 = 5mm ): (a) a/ R2 = 2 , θ0 = π/6 , (b) a/ R2 = 3 , θ0 = π/6 and (c) a/ R2 = 3 , θ0 = π/3 .

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Fig. 4. Frequencies (Hz) and loss factors of CSOCCS versus thickness h2 or h1 (a = 10m , R2 = 5m , θ0 = π/6 ): (a) h1 = h3 = 10mm (b) h2 = 5mm , h1 = h3.

Table 3 Frequencies and loss factors of CSOCCS with various ratios of a/R2, R2/h and h2/h (θ0 = π/6 , a = 10m , n = 1). R2/h

Frequencies (Hz)

10

50

100

Loss factor (%)

10

50

100

∞

ui (x ,θ,t ) =

h2/h

0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8

a/R2=1 m=1

m=2

m=3

m=4

m=5

m=6

m=7

m=8

46.17853 25.81255 17.02539 19.33901 16.61374 14.56449 17.82089 16.23934 14.47378 0.131099 0.445157 1.532402 0.752804 1.057575 2.045507 0.829831 1.058232 1.937221

143.7752 69.47731 31.79825 30.67338 15.43432 9.378605 16.46584 9.625411 7.68251 0.044856 0.217362 1.558223 1.075738 4.387893 17.74686 3.670143 11.13135 25.91766

299.6737 149.2635 68.03423 65.52365 31.22581 15.97606 33.42732 16.79946 10.87166 0.020513 0.102976 0.748578 0.516806 2.358658 13.47533 1.981004 8.099563 28.82582

498.128 257.8257 118.6847 114.5715 54.19189 26.20627 58.10468 28.2468 16.23449 0.011634 0.060476 0.434386 0.29616 1.380738 8.836698 1.159361 5.065665 22.89944

727.4188 392.2309 183.1417 177.2428 83.73544 39.39728 89.85028 43.06573 23.02634 0.007504 0.04038 0.284615 0.190973 0.899939 6.088251 0.75459 3.396315 17.75654

978.2804 549.4267 260.9079 253.207 119.775 55.49933 128.5762 61.17745 31.21727 0.005276 0.02933 0.202098 0.13306 0.63184 4.409899 0.528683 2.419999 13.89955

1243.887 726.3666 351.4263 342.1001 162.2588 74.5062 174.227 82.56751 40.82566 0.003946 0.022606 0.151924 0.097883 0.467829 3.327169 0.390418 1.806613 11.05597

1519.396 920.1679 454.0849 443.5144 211.1323 96.41381 226.7438 107.2281 51.86628 0.003093 0.018203 0.119173 0.074952 0.360421 2.593685 0.299847 1.398175 8.944222

i and Bmn are the coefficients. Substituting Eq. (11) into Eq. (9) gives the equations of motion of i i i i CSOCCS in terms of the unknowns Umn , Vmn , Wmn , Amn and Bmn as

∞

∑ ∑

∗

i cos(nπx / a)sin(mπθ / θ0 ) eiω t , Umn

m=1 n=1 ∞ ∞

vi (x ,θ,t ) =

∑ ∑

∗

i sin(nπx / a)cos(mπθ / θ0 ) eiω t , Vmn

([K ]−(ω∗)2 [M ]){X } = 0

m=1 n=1 ∞ ∞

w (x ,θ,t ) =

∑ ∑

where M is the mass matrix, K is the complex stiffness matrix, and 1 3 1 3 1 3 1 3 X = (Umn ,Umn ,Vmn ,Vmn ,Wmn,Amn ,Amn ,Bmn ,Bmn ) The circular frequency and the loss factors of CSOCCS could be calculated from the formulae (13)

∗

Wmnsin(nπx / a)sin(mπθ / θ0 ) eiω t ,

m=1 n=1 ∞ ∞

αi (x ,θ,t ) =

∑ ∑

∗

i cos(nπx / a)sin(mπθ / θ0 ) eiω t , Amn

m=1 n=1 ∞ ∞

βi (x ,θ,t ) =

∑ ∑ m=1 n=1

ω=

∗

i sin(nπx / a)cos(mπθ / θ0 ) eiω t , Bmn

(12)

(11)

Re(ω∗)2 ,η = Im((ω∗)2)/Re(ω∗)2

(13)

Owing to no relevant research results about damping property of CSOCCS, the calculated results are compared with the findings of

i i i , Vmn , Wmn , Amn where, i = 1, 3. ω∗ is complex circular frequency; Umn

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Table 4 Frequencies and loss factors of CSOCCS for various ratios of a/R2, R2/h and h2/h (θ0 = π/6 , a = 10 m, n = 1). R2/h

Frequencies (Hz)

10

50

100

Loss factor(%)

10

50

100

h2/h

0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8 0.2 0.6 0.8

a/R2=10 m=1

m=2

m=3

m=4

m=5

m=6

m=7

m=8

338.6704 160.3327 75.52248 71.36479 38.32902 29.77406 39.9722 26.55324 26.75753 0.191656 0.899645 6.070404 4.29777 15.44872 37.95206 12.60478 30.42059 43.20401

1351.071 650.6986 295.9399 283.6958 137.9034 79.36815 146.9272 78.85433 61.25943 0.047796 0.231751 1.683747 1.177584 5.146056 23.19752 4.309996 15.51562 38.10962

2917.707 1450.987 660.981 636.1826 302.7426 154.5912 324.0437 162.1176 104.7263 0.021083 0.105628 0.769361 0.532277 2.435656 13.96954 2.045857 8.441232 30.14582

4909.903 2538.064 1167.869 1127.099 533.0542 257.9143 571.5193 277.8019 159.9018 0.011808 0.061276 0.44083 0.300979 1.403153 8.970769 1.178305 5.149555 23.20856

7210.165 3883.713 1812.684 1753.997 828.6101 390.0126 889.1049 426.204 228.1493 0.007572 0.040687 0.287207 0.192951 0.909103 6.145605 0.762361 3.43029 17.89212

9725.442 5457.4 2590.607 2513.812 1189.049 551.0834 1276.411 607.3756 310.156 0.005307 0.029467 0.203315 0.134015 0.63628 4.439092 0.532461 2.436742 13.97518

12387.35 7228.591 3496.083 3402.934 1613.918 741.1703 1732.951 821.2952 406.2763 0.003962 0.022672 0.15256 0.098398 0.470232 3.343576 0.39247 1.815834 11.10233

15147.49 9168.401 4522.99 4417.29 2102.682 960.2556 2258.151 1067.91 516.6987 0.003102 0.018238 0.119531 0.075253 0.36183 2.603578 0.301055 1.403661 8.974175

structure could make the CSOCCS own the optimal dynamics properties, because symmetric structure make the viscoelastic layer produce more shear deformation. The second is only the thickness of the viscoelastic core or the base layer is a design variable, and is shown in Fig. 4. With thickness h2 increasing, frequencies of the open shell tend to descend and change small, however, the loss factor increases significantly and change greatly. This result is expected since the thicker viscoelastic layer experiences more shear deformation. Nevertheless, the opposite phenomenon occurs as the increase of thickness h1. Mainly because the rigidity of the CSOCCS becomes bigger with the increase of h1, leading to weaken the shear deformation of core layer. As we can see from the figures, choosing proper viscoelastic thickness to total thickness ratio (h2/h) can increase loss factors almost four times. Besides, the thicknesses of the base layer and constraining layer have a major effect on the natural frequencies, and the thickness of the viscoelastic core has a main influence on the loss factor. We also notice that the effect of thickness h2 (or h1) on frequencies and loss factors is gradually weakened. Frequencies and loss factor change distinctly with thinner thickness, and almost not vary when the thickness is at some degree. In practice, the dimensional parameters of CSOCCS, including length, radius, and thickness, et al. have a combined contribution on its loss factors and frequencies. So in the following, the combined effect of length to thickness ratio (a/R2), radius to total thickness ratio (R2/h), and viscoelastic thickness to total thickness ratio (h2/h) on the symmetric CSOCCS with constant length will be considered, as shown in Tables 3 and 4. The contrast of Tables 3 and 4 show that no matter how a/R2 and R2/h change, an increase in ratio h2/h always makes the frequencies dramatically drop down, at the same time, has a highly appreciable enhancement on damping performance of CSOCCS. Based on the constant R2/h and h2/h, as the a/R2 ratio increases, it seems that frequencies rise dramatically in all circumferential mode numbers, however, loss factors increase distinctly in low circumferential mode numbers and change small in high circumferential mode numbers. So a thinner CSOCCS tends to own a greater frequency and loss factor than a thicker one. With R2/h ratio increasing, it is obvious that the frequency gradually decreases, inversely, loss factor tends to gradually increase, but change small in low circumferential mode numbers and increase distinctly in high circumferential mode numbers. In addition, comparing the values of frequency and loss factor, it can

previous studies of LCOCCS and those obtained by ANSYS. The material properties and geometrical dimensions used for LCOCCS are given as follows: E = 2.1 × 1011Pa , μ = 0.3, ρ = 7800kg / m3 , h1 = h2 = h3 = 2mm , a = 10m , R2 = 5m , θ0 = π/6, and the contrast results are shown in Table 1. There are some differences between the results owing to adopting different solutions on this vibration problem. However, we can see adequate agreement from the Table 1. The material properties used for the constraining layer and base layer are given as follows: E1 = E3 = 210GPa , ρ1 = ρ3 = 7800kg / m3, and μ1 = μ3 = 0.3; and the material properties of viscoelastic core layer are: G = 2.3(1 + 0.5i) × 106Pa,μ 2 = 0.34 , and ρ2 = 999kg / m3 . The results compared with the ones obtained by ANSYS are listed in Table 2. In the ANSYS model, the SOLID185 element is adopted, and 3 translational degrees of freedom are used. Besides, a total of 18,000 elements (10 radial, 30 circumferential and 60 longitudinal elements) are used during the calculation. ANSYS model and some mode shapes obtained from ANSYS are shown in Fig. 2. What needs illustration is that these data of CSOCCS will be employed throughout the whole passage (unless explicitly stated). In addition, n and m, in Tables 1 and 2, represent the axial mode number and circumferential mode number of the vibration, respectively. And, in all calculations, all inertia terms have been retained and shear correction factor (k = 5/6) has been used.

3.2. Influence of material parameters on dynamics properties of CSOCCS The influence of the geometrical parameters as well as the physical parameters on dynamics properties will be analyzed in this section. In the following, two cases are first considered to study the influence of thickness on dynamic properties of CSOCCS. The first case of this paper assumes that the thickness of core layer and total thickness of the CSOCCS are constant. The thickness ratio (h1/ h3) of bottom and upper composite layer is considered, and ranges from 0.1 to 3. The natural frequencies and loss factors of CSOCCS versus the value of h1/h3 with different length to radius ratios (a/R2) and subtended angles (θ0 ) are plotted in Fig. 3. Obviously, loss factors are strongly influenced by the value of h1/h3 compared with natural frequencies. With h1/h3 ratio varying, the variety scope of frequencies is slight, but the loss factors increase manyfold. Besides, frequencies and loss factors show different patterns of variation. Frequencies decrease initially, then start increasing, meanwhile, the loss factors are successively rising and falling, present a convex parabolic distribution and reach their peak values at h1/h3 = 1. Therefore, the symmetric 155

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Fig. 5. Frequencies and loss factors of CSOCCS versus number of circumferential mode with various subtended angles and ratios of a/R2, R2/h and h2/h (a = 10 m, h1 = h3 (mm), n = 1), (a): a/R2 = 2, R2/h = 50, h2/h = 0.2; (b): a/R2 = 2, R2/h = 100, h2/h = 0.6; (c): a/R2 = 10, R2/h = 100, h2/h = 0.6.

Next, the Variation of natural frequencies and loss factors of the CSOCCS versus shear parameter g ( g = G (1−μ2 )/ E1) [36] with various ratios of a/R2, R2/h and h2/h are plotted in Fig. 6. In order to cover a wide range of shear modulus, a logarithmic scale of g-axis has been used. As can be seen from the Fig. 6, the natural frequencies increase almost continuously, but initially are almost constant with lower shear parameter g for all situations, then increase rapidly with the growth of the g. However, the loss factors show a different pattern of variation: first, climb up and reach a peak value for a particular g, then fall to lower values. Meanwhile, there exists an optimal shear modulus of the core to lead the best damping performance. Thus, the highest damping will be achieved by choosing the proper region of core layer’s sheer modulus.

be observed that high damping is always corresponding to low frequency. Fig. 5 presents the influence of subtended angle on the frequencies and loss factors of CSOCCS. Obviously, no matter how the dimensional parameters of CSOCCS change, the variation tendency of the natural frequencies and modal loss factors almost do not shift at the same subtended angle. You can also see, on one hand, loss factor generally decreases and the frequency gradually increases when the subtended angle is smaller (i. e.θ0 = 10°). On the other hand, with subtended angle increasing, the variation of loss factor evolves into upgrading firstly then descending, meanwhile, frequency declines and then climbs up. Besides, it is clear that there always exists an optimal value of mode number (n, m) to lead the maximized damping performance (i. e.

θ0 = 110° 4. Conclusion

corresponding to (n, m) = (1, 5)), and the greater the subtended angle becomes, the bigger circumferential mode number corresponding to crests of loss factor.

A discrete layer model based on the first-order shear deformation 156

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Fig. 6. Frequencies and loss factors of CSOCCS versus sheer parameter with various subtended angles and ratios of a/R2, R2/h and h2/h (a = 10 m, h1 = h3 (mm), θ0 = π/6 , n = 1), (a): a/ R2 = 1, R2/h = 20; (b): a/R2 = 5, R2/h = 20; (c): a/R2 = 5, R2/h = 100.

shell theory has been made use of to study the dynamic properties of three-layered composite open circular cylindrical shells with a constrained damping treatment for different system parameters, i.e. different thickness ratios, length to radius ratios, subtended angles and sheer parameters, of which the effect of rotary inertias and shear deformation is considered. The major results obtained in the present paper can be concluded as follows:

likely obtain a greater loss factor. 3. The loss factor generally decreases with a smaller subtended angle, and evolves into first rising and descending latter with subtended angle increasing, and vice versa for frequency. 4. Choosing the proper region of core layer’s sheer modulus could achieve the highest damping properties, and the longer and thicker CSOCCS is, the bigger sheer parameter g corresponding to the highest damping properties is.

1. The symmetric structure could make the CSOCCS own the optimal dynamics properties. Besides, the thicknesses of the base layer and constraining layer have a major effect on the natural frequencies, and the thickness of the viscoelastic core has a main influence on the loss factor. 2. A taller and thicker CSOCCS tends to have a bigger fundamental frequency most possibly; and a taller and thinner CSOCCS very

Acknowledgment This research was financially supported by the National Natural Science Foundation of China (51375248).Conflict of interests The authors hereby confirm that no conflict of interest exists for this article.

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Appendix A: Equilibrium equations

δu1: R1

∂N x(1) ∂x

(1) ∂N xθ

+

+ 2 R2 ⎛ ⎝

∂N x(2) ∂x

+

+ 2 R2 ⎛ ⎝

∂N x(2) ∂x

+

1

⎜

∂θ

(2) ∂N xθ

(2) ∂Mxθ

(2)

⎞− 1 R2 ⎛ ∂Mx + h ∂x ⎠ 2 ⎝

⎟

R2 ∂θ

⎜

R2 ∂θ

(2) ⎞ −Qxz = ρ1 h1 R1 ⎠ ⎟

∂2u1 ∂t 2

ρ h2 ρ2 h2 ∂2u ∂2α R2 22 − 2122 R2 22 2 ∂t ∂t

+

(A1-a)

δu3: R3

∂N x(3) ∂x

+

(3) ∂N xθ

1

⎜

∂θ

(2) ∂N xθ

R2 ∂θ

⎞+ ⎠

⎟

∂M (2) 1 R ⎜⎛ ∂xx h2 2

(2) ∂Mxθ

+

R2 ∂θ

⎝

(2) ⎞ −Qxz = ρ3 h3 R3 ⎠ ⎟

∂2u3 ∂t 2

+

ρ2 h2 ∂2u R2 22 2 ∂t

ρ2 h22

+

12

R2

∂2α2 ∂t 2

(A1-b)

δv1: ∂Nθ(1) ∂θ

+ R1

(1) ∂N xθ

(2)

(2) ∂N xθ

∂N 1 (1) + Qθz + 2 R2 ⎜⎛ R θ∂θ + 2 ⎝

∂x

(2) Qθz

+

∂x

(2)

⎞− 1 R ⎛ ∂Mθ + h2 2 R2 ∂θ ⎝ ⎠

⎟

R2

⎜

(2) ∂Mxθ

∂x

(2) ⎞ −Qθz = ρ1 h1 R1 ⎠ ⎟

∂2v1 ∂t 2

+

ρ h2 ∂2β ρ2 h2 ∂2v R2 22 − 2122 R2 22 2 ∂t ∂t

(A1-c)

δv3: ∂Nθ(3) ∂θ

+ R3

(3) ∂N xθ

(2)

(2) ∂N xθ

∂N 1 (3) + Qθz + 2 R2 ⎛⎜ R θ∂θ + 2 ⎝

∂x

(2) Qθz

+

∂x

R2

⎞⎟ + ⎠

(2)

∂M 1 R ⎜⎛ θ h2 2 R2 ∂θ

+

(2) ∂Mxθ

⎝

∂x

(2) ⎞ −Qθz = ρ3 h3 R3 ⎠ ⎟

∂2v3 ∂t 2

ρ2 h2 ∂2v R2 22 2 ∂t

+

+

ρ2 h22 12

∂2β2

R2

∂t 2

(A1-d)

δα1: R1

∂Mx(1) ∂x

+ R1

(1) ∂Mxθ

R1 ∂θ

(1) −R1 Qxz +

∂N (2) h1 R ⎜⎛ ∂xx 4 2

⎝

+

(2) ∂N xθ

R2 ∂θ

(2)

⎞− h1 R2 ⎛ ∂Mx + 2h2 ∂x ⎠ ⎝

⎟

⎜

(2) ∂Mxθ

(2) ⎞ −Qxz = ⎠

ρ2 h1 h2 ∂2u R2 22 4 ∂t

+

(2) ⎞ −Qxz =− ⎠

ρ2 h3 h2 ∂2u R2 22 4 ∂t

+

R2 ∂θ

⎟

ρ1 h13

R1

∂2α1 ρ2 h1 h22 ∂2α − 24 R2 22 ∂t 2 ∂t

R3

∂2α3 ρ2 h3 h22 ∂2α − 24 R2 22 ∂t 2 ∂t

12

(A1-e)

δα3: R3

∂Mx(3) ∂x

+

(3) ∂Mxθ

∂θ

(3) −R3 Qxz − 4 h3 R2 ⎛ ⎝ 1

⎜

∂N x(2) ∂x

+

(2) ∂N xθ

R2 ∂θ

(2)

⎞− h3 R2 ⎛ ∂Mx + 2h2 ∂x ⎠ ⎝ ⎟

⎜

(2) ∂Mxθ

R2 ∂θ

⎟

ρ3 h33 12

(A1-f)

δβ1: ∂Mθ(1) ∂θ

+ R1

(1) ∂Mxθ

(1) −R1 Qθz +

∂x

(2)

∂N h1 R ⎛⎜ θ 4 2 R2 ∂θ

+

⎝

(2) ∂N xθ

∂x

+

(2) Qθz

R2

(2)

⎞⎟− h1 R ⎛ ∂Mθ + 2h2 2 R2 ∂θ ⎝ ⎠ ⎜

(2) ∂Mxθ

∂x

(2) ⎞ −Qθz = ⎠ ⎟

ρ2 h1 h2 ∂2v R2 22 4 ∂t

ρ1 h13

+

12

R1

∂2β1 ∂t 2

−

ρ2 h1 h22 24

R2

∂2β2 ∂t 2

(A1-g)

δβ3: ∂Mθ(3) ∂θ

+ R3

(3) ∂Mxθ

∂x

∂N (2)

(3) −R3 Qθz − 4 h3 R2 ⎛⎜ R θ∂θ + 2 ⎝ 1

(2) ∂N xθ

∂x

+

(2) Qθz

R2

(2)

⎞⎟− h3 R ⎛ ∂Mθ + 2h2 2 R2 ∂θ ⎝ ⎠ ⎜

(2) ∂Mxθ

∂x

(2) ⎞ −Qθz =− ⎠ ⎟

ρ2 h3 h2 ∂2v R2 22 4 ∂t

+

ρ3 h33 12

R3

∂2β3 ∂t 2

−

ρ2 h3 h22 24

R2

∂2β2 ∂t 2

(A1-h)

δw: (1) ∂Qθz

∂θ

+ R1

(1) ∂Q xz

∂x

−Nθ(1) +

(2) ∂Qθz

∂θ

+ R2

(2) ∂Q xz

∂x

−Nθ(2) +

(3) ∂Qθz

∂θ

+ R3

(3) ∂Q xz

∂x

−Nθ(3) = (ρ1 h1 R1 + ρ2 h2 R2 + R3 ρ3 h3 )

∂2w ∂t 2

(A1-i)

(i) (i) (i) (i) where, N x(i) ,Nθ(i) , N xθ denote the normal force resultants; Mx(i) , Mθ(i) , Mxθ represent bending and twisting moment resultants; Qxz and Qθz are the shear force resultants (See Fig. 7).

Fig. 7. Force and moment resultants in a shell.

(i)

⎛ Nx ⎞ ⎜ N (i) ⎟ θ ⎜ (i) ⎟ N ⎜ xθ ⎟ = (i) ⎟ ⎜ Qθz ⎜ (i) ⎟ ⎝ Qxz ⎠

(i)

hi

∫− h2

i

2

⎛ σxx ⎞ ⎜ σ (i) ⎟ θθ ⎜ (i) ⎟ τ ⎜ xθ ⎟ dz (i) ⎟ ⎜ τθz ⎜ (i) ⎟ ⎝ τxz ⎠

(A2) 158

Composite Structures 189 (2018) 148–159

Y. Zhai et al.

(i)

⎛ Mx ⎞ ⎜ M (i) ⎟ = θ ⎜⎜ (i) ⎟⎟ M ⎝ xθ ⎠

(i)

⎛ σx ⎞ ⎜ σ (i) ⎟ zdz θ ⎜⎜ (i) ⎟⎟ τ ⎝ xθ ⎠

hi 2 h − i 2

∫

(A3)

Bring the formula (4) into formula (A2) and (A3), they could be rewritten in the following general form [35]: (i)

(i) ⎛ N x ⎞ ⎛ A11 ( ) i ⎜ N ⎟ ⎜ A (i) θ 12 ⎜ (i) ⎟ ⎜ (i) ⎜ N xθ ⎟ = ⎜ A16 (i) ⎜ Mx(i) ⎟ ⎜ B11 ⎜ (i) ⎟ ⎜ (i) M B ⎜ θ ⎟ ⎜ 12 ⎜ M (i) ⎟ ⎜ B (i) ⎝ xθ ⎠ ⎝ 16

(i) (i) (i) (i) (i) A12 A16 B11 B12 B16 ⎞ ∂ui / ∂x ⎞ (i) (i) (i) (i) (i) ⎟ ⎛ A22 A26 B12 B22 B26 ⎜ ∂vi/ Ri ∂θ + w / Ri ⎟ ⎟ (i) (i) (i) (i) (i) A26 A66 B16 B26 B66 ⎟ ⎜ ∂ui / Ri ∂θ + ∂vi/ ∂x ⎟ ⎟ (i) (i) (i) (i) (i) ⎜ ∂αi/ ∂x B12 B16 D11 D12 D16 ⎟ ⎜ ⎟ ∂βi / Ri ∂θ (i) (i) (i) (i) (i) ⎟ B22 B26 D12 D22 D26 ⎜ ⎟ ∂αi/ Ri ∂θ + ∂βi / ∂x ⎟ (i) (i) (i) (i) (i) ⎟ ⎝ ⎠ B26 B66 D16 D26 D66 ⎠

(A4)

(i) (i) (i) ⎛Qθz ⎞ ⎛C44 C45 ⎞ ⎛ ∂w / Ri ∂θ + βi−vi/ Ri ⎞ = ⎜ (i) (i) ⎟ ⎜ ⎟ ⎜Q (i) ⎟ ∂w / ∂x + αi ⎠ ⎝ xz ⎠ ⎝C45 C55 ⎠ ⎝

(i) Amn ,

(i) Bmn ,

(i) Cmn ,

(i) Dmn ,

hi 2 h − 2i

(i) (i) (i) (Amn ,Bmn ,Dmn )=∫ hi

(A5)

are the stiffness coefficients, and

(i)

Qij (1,z ,z 2) dz ,(m,n = 1,2,6)

(i)

(i) Cmn = ∫ 2hi Qij dz ,(m,n = 4,5) −2

(A6)

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