Nội dung

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 4 (2016), pp. 249 – 265 DOI:10.15625/0866-7136/6954 FREE VIBRATION ANALYSIS OF JOINED COMPOSITE CONICAL-CYLINDRICAL-CONICAL SHELLS CONTAINING FLUID Vu Quoc Hien1 , Tran Ich Thinh2,∗ , Nguyen Manh Cuong2 1 Viet Tri University of Industry, Phu Tho, Vietnam 2 Hanoi University of Science and Technology, Vietnam ∗ E-mail: tranichthinh@yahoo.com Received September 12, 2015 Abstract. A new continuous element (CE) formulation has been presented in this paper for the vibration analysis of cross-ply composite joined conical-cylindrical-conical shells containing fluid. Governing equations are obtained using thick shell theory of Midlin, taking into account the shear deflection effects. The velocity potential, Bernoulli’s equation and impermeability condition have been applied to the shell-fluid interface to obtain an explicit expression for fluid pressure. The dynamic stiffness matrix has been built from which natural frequencies have been calculated. The appropriate expressions among stress resultants and deformations are extracted as continuity conditions at the joining section. A matlab program is written using the CE formulation in order to validate our model. Numerical results on natural frequencies are compared to those obtained by the Finite Element Method and validated with the available results in other investigations. This paper emphasizes advantages of CE model, the effects of the fluid filling and shell geometries on the natural frequencies of joined composite conical-cylindrical-conical shells containing fluid. Keywords: Free vibration, cross-ply composite joined conical-cylindricall-conical shells, dynamics stiffness matrix, continuous element method. 1. INTRODUCTION The joined shells filled with fluid of revolution have many applications in various branches of engineering such as mechanical, aeronautical, marine, civil and power engineering. Hence, the comprehension of dynamic behaviours of such structures is of great important in order to design and fabric safer and more economic composite shell structures. There are many computational methods available for the free vibration of the cylindrical and conical and joined cylindrical-conical shells, such as the exact wave solution, c 2016 Vietnam Academy of Science and Technology 250 Vu Quoc Hien, Tran Ich Thinh, Nguyen Manh Cuong Ritz method, Gelerkin method, differential quadrature method, state space method, finite difference method, Finite Element Method (FEM) and Continuous Element Method (CEM) or Dynamic Stiffness Method (DSM). Sivadas and Ganesan [1] investigated the effects of thickness variation on natural frequencies of laminated conical shells by a semianalytical finite element method. Xi, Yam and Leung [2, 3] analyzed the free vibration of a laminated composite circular cylindrical shell partially filled with fluid using a semianalytical finite element technique based on the Reissner-Mindlin theory and compressible fluid equations. Katsutoshi et al. [4] analyzed the free vibrations of a laminated composite circular cylindrical shell partially filled with liquid. Toorani and Lakis [5] studied the effect of shear deformation in the dynamic behaviour of anisotropic laminated open cylindrical shells filled with fluid. Kochupillai, Ganesan and Padmanabhan [6] performed a dynamic analysis of composite shells conveying fluid based on semi-analytical coupled finite element formulation Larbi et al. [7] presented the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid A semi-analytical approach has been utilized by Toorani and Lakis [8] to determine the swelling effect on the dynamic behaviour of composite cylindrical shells conveying fluid. Tong [9, 10] proposed the power series expansion approach to study the free vibration of orthotropic composite laminated conical shells. Shu [11] has employed the differential quadrature method to study the vibration of conical shells. The investigations of vibration analysis for composite cylindrical shells are carried out by using different approaches such as 2D finite element model based on classical thin shell theory [12], 2D analytical method using the cubic spline functions [13], analytical method based on the first-order shear deformation theory (FSDT) [14]. Senthil and Ganesan [15] performed a dynamic analysis of composite conical shells filled with fluid. Kerboua, Lakis and Hmila [16] using a combination of finite element method and classical shell theory to determine the natural frequencies of anisotropic truncated conical shells in interaction with fluid. Irie et al. [17] used the transfer matrix approach to study the free vibration of joined isotropic conical-cyclindrical shells. Patel et al. [18] presented some vibrational results for laminated composite joined conical-cyclindrical shell with first order shear deformation theory using finite element method (FEM). Recently, Caresta and Kessissoglou [19] analyzed the free vibrations of joined truncated conical-cyclindrical shells. The displacements of the conical sections were solved using a power series solution, while a wave solution was used to describe the displacements of the cylindrical sections. Both DonnellMushtari and Flugge equations of motion were used. Kouchakazadeh and Shakouri [20] studied the vibrational behaviour of two joined cross-ply laminated conical shells, joined cylindrical-conical shells. Governing equations are obtained using thin-walled shallow shell theory of Donnell type and Hamilton’ s principle. The appropriate expressions among stress resultants and deformations are extracted as continuity condition at the joining section of the cones. Free vibration analysis of joined composite conical-cylindrical-conical shells containing fluid 251 In those studies, low natural frequencies are generally investigated. The known disadvantage of traditional methods like FEM is the discretization operation of the domain which causes errors in dynamic analysis, especially in medium and high frequencies. For medium and high frequency range, the CEM can be applied with many advantages: high precision, rapid calculating speed, reduction of the model size and of the computing time. Numerous Continuous Elements have been established for metal and composite beams [21, 22] and plates [23]. Nguyen Manh Cuong and Casimir [24] have succeeded in building the DSM for thick isotropic plate and shells of revolution. The CE models for composite cylindrical shells and conical shells presented in works of Tran Ich Thinh and Nguyen Manh Cuong [25, 26] imposed a considerable advancement of the study on CEM for metal and composite structures. Recently, the new research for thick laminated composite joined cylindrical-conical shells by Tran Ich Thinh, Nguyen Manh Cuong and Vu Quoc Hien [27] has emphasized the strong capacity of CEM in assembling complex structure. The main objective of this paper is to present a detailed study on free vibrations of a composite conical-cylindrical-conical shells containing an incompressible and inviscid liquid. Illustrative examples are provided to demonstrate the accuracy and efficiency of the developed numerical procedure. 2. FORMULATION OF JOINED CROSS-PLY COMPOSITE CONICAL-CYLINDRICAL-CONICAL SHELLS CONTAINING FLUID Let’s investigate the joined conical-cylindrical-conical shells containing fluid in Fig. 1. R1 is the radius of the cylinder,R2 is the radius of the larger end of the cone. L1 and L2 are lengths of the cylinder and cone respectively. Conical shell theory using the Reissner-Mindlin assumption will be used to mobilize both conical and cylindrical shells. R2 u3 α w3 R1 L1 u1 L2 w1 w2 H u2 α L2  R2 Fig. 1. Geometry of joined composite conical-cylindrical-conical shells containing fluid 252 Vu Quoc Hien, Tran Ich Thinh, Nguyen Manh Cuong 2.1. Composite conical shell containing fluid formulation 2.1.1. Constitutive relations Consider a laminated composite shell of total thickness h composed by N orthotropic layers. The plane stress-reduced stiffness are calculated as E1 , 1 − υ12 υ21 E2 = , 1 − υ12 υ21 Q11 = Q22 Q12 = υ12 E2 , 1 − υ12 υ21 Q66 = G12 , Q44 = G23 , (1) Q55 = G13 , where Ei , Gij , υ12 , υ21 : elastic constants of the kth layer and the laminate stiffness coefficients (Aij , Bij , Dij , Fij ) are defined by N Aij = ∑ Q̄ijk (zk+1 − zk ), Bij = k =1 1 N k 2 1 N k 3 2 Q̄ ( z − z ) , D = ij ∑ Q̄ij (zk+1 − z3k )(i, j = 1, 2, 6) ij k +1 k 2 k∑ 3 =1 k =1 N Fij = ∑ Qijk (z−k+1 zk )(i, j = 4, 5) k =1 (2) where zk−1 and zk are the boundaries of the kth layer. 2.1.2. Strains, stress and internal forces resultant Following the Reissner-Mindlin assumption, the displacement components are assumed to be u ( x, θ, z, t) = u0 ( x, θ, t) + zϕ x ( x, θ, t) , v ( x, θ, z, t) = v0 ( x, θ, t) + zϕθ ( x, θ, t) , w ( x, θ, z, t) = w0 ( x, θ, t) , (3) a) The strain-displacement relations of conical shell are (with R( x ) = R1 + x.sinα):     ∂u0 ∂ϕ x 1 ∂v0 1 ∂ϕθ εx = , kx = , εθ = u0 sin α + + w0 cos α , k θ = ϕ x sin α + , ∂x ∂x R ∂θ R ∂θ ∂v0 1 ∂u0 sin α 1 ∂ϕ x ∂ϕθ sin α − cos α 1 ∂w0 ε Sθ = + − v0 , k xθ = + − ϕθ , γθZ = υ0 + + ϕθ , ∂s R ∂θ R R ∂θ ∂x R R R ∂θ (4) b) Force resultants–displacement relationships: The force and moment resultants are expressed in terms of deformations for crossply axis-symmetric shell as follows      N A B 0 ε  M  =  B D 0  k . Q 0 0 F γ (5) Free vibration analysis of joined composite conical-cylindrical-conical shells containing fluid 253 Substituting Eqs. (2) and (4) in Eqs. (5), the forces-displacements expressions for laminated composite conical shell are written as follows     ∂u0 ∂ϕ x A12 ∂v0 B12 ∂ϕθ Nx = A11 + u0 sin α + + w0 cos α + B11 + ϕ x sin α + , ∂x R ∂θ ∂x R ∂θ     ∂u0 ∂ϕ x A22 ∂v0 B22 ∂ϕθ Nθ = A12 + u0 sin α + + w0 cos α + B12 + ϕ x sin α + , ∂x R ∂θ ∂x R ∂θ     ∂v0 1 ∂u0 sin α ∂ϕθ sin α 1 ∂ϕ x Nxθ = A66 + − v0 + B66 + − ϕθ , ∂x R ∂θ R R ∂θ ∂x R     ∂u0 ∂ϕ x B ∂v0 w0 cos α D ∂ϕ Mx = B11 + 12 u0 sin α + + + D11 + 12 ϕ x sin α + θ , ∂x R ∂θ R ∂x R ∂θ     B22 ∂v0 ∂ϕ x D22 ∂ϕθ ∂u0 + u0 sin α + + w0 cos α + D12 + ϕ x sin α + , Mθ = B12 ∂x R ∂θ ∂x R ∂θ     ∂u0 sin α ∂ϕθ sin α ∂v0 1 ∂ϕ x Mxθ = B66 + − υ0 + D66 + − ϕθ , ∂x R∂θ R R ∂θ ∂x R     ∂w0 1 ∂w0 − cos α Q x = kF55 + ϕ x , Qθ = kF44 υ0 + + ϕθ , ∂x R R ∂θ (6) where k is the shear correction factor (k = 5/6) 2.1.3. Equations of motion The equations of motion using the FSDT for laminated composite conical shell containing fluid are sin α 1 ∂Nxθ ∂Nx + = I0 ü0 + I1 ϕ̈ x , ( Nx − Nθ ) + ∂x R R ∂θ ∂Nxθ 2 sin α 1 ∂Nθ cos α + Nxθ + + Qθ = I0 v̈0 + I1 ϕ̈θ , ∂x R R ∂θ R ∂Mx sin α 1 ∂Mxθ + − Q x = I1 ü0 + I2 ϕ̈ x , ( M x − Mθ ) + ∂x R R ∂θ 2 sin α 1 ∂Mθ ∂Mxθ + Mxθ + − Qθ = I1 v̈0 + I2 ϕ̈θ , ∂x R R ∂θ ∂Q x 1 ∂Qθ sin α cos α + + Qx − Nθ − P cos α = I0 ẅ0 , ∂x R ∂θ R R (7) where u0 , v0 , w0 : displacement, ϕ x, ϕθ : rotation of tangents along the x and θ. And k +1 N zR Ii = ∑ ρ(k) zi dz, (i = 0, 1, 2), where ρ(k) is the material mass density of the kth layer. k =1 z k Substituting α = 0 in Eqs. (6) and (7), the forces-displacements relations and the equations of motion for laminated composite cylindrical shell containing fluid can be obtained [27]. 254 Vu Quoc Hien, Tran Ich Thinh, Nguyen Manh Cuong 2.1.4. Fluid equations The potential function Φ(r, θ, x, t) satisfies the Laplace equation in cylindrical coordinates (r, θ, x ) ∂2 Φ 1 ∂Φ 1 ∂2 Φ ∂2 Φ + + 2 2 + 2 = 0. (8) 2 ∂r r ∂r r ∂θ ∂x Then, the Bernoulli equation is written by P ∂Φ + = 0. ∂t ρf (9) By linearizing this expression, the pressures on the internal regions are P= −ρ ∂Φ ∂t , (10) Σ where Σ is the portion of the structure’s surface in contact with fluid. The condition of impermeability of the surface of shell in contact with fluid can be expressed as ∂w0 ∂Φ = , (11) vf = ∂r Σ ∂t Σ where w0 is the normal displacement of the shell, v f is the velocity of fluid. The hydrodynamic pressure acting on the cylindrical shell is then defined by [27] P = −ρ f 2 1 ∂ 2 w0 ∗ ∂ w0 = m . m + k n RIm+1 (k n R) /Im (k n R) ∂t2 ∂t2 (12) This value will be introduced in (7) in order to establish the Dynamic Stiffness Matrix for the studied structure. 2.1.5. Continuity conditions The continuity conditions at the conical-cylindrical shell joint can be obtained from Caresta and Kessissoglou [19] as follows ∂w1 ∂w2 = , ∂x1 ∂x2 = Nx2 cos α − Q x2 sin α, Q x1 = Nx2 sin α + Q x2 cos α, Mxθ1 = Mxθ2 , Mx1 = Mx2 . (13) u1 = u2 cos α − w2 sin α, Nx1 v1 = v2 , w1 = u2 sin α + w2 cos α, 3. CONTINUOUS ELEMENT FORMULATION FOR THICK LAMINATED COMPOSITE JOINED CONICAL-CYLINDRICAL-CONICAL SHELLS CONTAINING FLUID 3.1. Strong formulation Here, the same state-vector y = {u0 , v0 , w0 , ϕ x , ϕθ , Nx , Nxθ, Q x , Mx , Mxθ }T . Next, the Lévy series expansion for state variables is written as Free vibration analysis of joined composite conical-cylindrical-conical shells containing fluid 255 {u0 ( x, θ, t), w0 ( x, θ, t), ϕθ ( x, θ, t), Nx ( x, θ, t), Q x ( x, θ, t), Mx ( x, θ, t)}T = ∞ = ∑ {um ( x ), wm ( x ), ϕθm ( x ), Nxm ( x ), Q xm ( x ), Mxm ( x )}T cos mθeiωt , m =1 {v0 ( x, θ, t), ϕ x ( x, θ, t), Nxθ ( x, θ, t), Mxθ ( x, θ, t)}T = ∞ = ∑ (14) {vm ( x ), ϕ xm ( x ), Nxθm ( x ), Mxθm ( x )}T sin mθeiωt , m =1 where m is the number of circumferential wave. Substituting (14) in Eqs. (6) and (7), a system of ordinary differential equations in the x-coordinate for the mth mode can be expressed in the matrix form for each circumferential mode m as [25–28] dum D B = c4 sin α.um + mc4 vm + c4 cos α.wm + c5 sin α.ϕ xm + mc5 ϕθm + 11 Nxm − 11 Mxm , dx c1 c1 m sin α D66 B66 dwm 1 dvm Q xm , = um − vm − Nxθm + Mxθm , = − ϕ xm + dx R R c10 c10 dx kF55 dϕ xm B A = c2 sin α.um + mc2 vm + c2 cos α.wm + c3 sin α.ϕ xm + mc3 ϕθm − 11 Nxm + 11 Mxm , dx c1 c1 dϕθm m sin α B66 A66 = ϕ xm − ϕθm + Nxθm − Mxθm , dx R R c10 c10

dNxm = c6 sin2 α − I0 v 2 um + mc6 sin α.vm + c6 sin α cos α.wm + c7 sin2 α − I1 ω 2 ϕ xm + dx   m 1 Nxm − Nxθm − c2 sin α.Mxm , + mc7 sin α.ϕθm − sin α c4 + R R     dNxθm kF kF cos2 α − I0 ω 2 vm + m cos α c6 + 44 wm + = mc6 sin α.um + m2 c6 + 44 2 dx R R2   kF44 cos α 2 sin α 2 2 + mc7 sin α.ϕ xm + m c7 − − I1 ω ϕθm − mc4 Nxm − Nxθm − mc2 Mxm , R R     m2 kF44 dQ xm kF44 2 2 ∗ 2 − I0 ω − m ω wm+ = c6 sin α cos α.um + m cos α c6 + 2 vm + c6 cos α + dx R R2   mkF44 sin α + c7 sin α cos α.ϕ xm + mc7 cos α − ϕθm − c4 cos α.Nxm − Q xm − c2 cos α.Mxm , R R

dMxm = 2c8 sin2 α − I1 ω 2 um + 2mc8 sin α.vm + 2c8 sin α cos α.wm + 2c9 sin2 α − I2 ω 2 ϕ xm + dx    1 m + 2mc9 sin α.ϕθm − 2c5 sin α.Nxm + Q x − 2 sin α c3 + Mxm − Mxθm , R R     dMxθm kF cos α kF = mc8 sin α.um + m2 c8 − 44 − I1 ω 2 vm + m c8 cos α − 44 wm + dx R R
2 sin α + mc9 sin α.ϕ xm + m2 c9 + kF44 − I2 ω 2 ϕθm − mc5 .Nxm − mc3 Mxm − .Mxθm , R (15) 256 Vu Quoc Hien, Tran Ich Thinh, Nguyen Manh Cuong with 2 c1 = A11 D11 - B11 , c2 = ( A12 B11 − A11 B12 ) /Rc1 , c4 = ( B11 B12 − A12 D11 ) /Rc1 , c3 = ( B11 B12 − A11 D12 ) /Rc1 , c5 = ( B11 D12 − B12 D11 ) /Rc1 , c6 = ( A12 c4 + B12 c2 + A22 /R) /R, c7 = ( A12 c5 + B12 c3 + B22 /R) /R , c8 = ( B12 c4 + D12 c2 + B22 /R) /R, c9 = ( B12 c5 + D12 c3 + D22 /R) /R , c10 = 2 B66 − A66 D66 . Eq. (15) can be expressed in the matrix form for each circumferential mode m dym = Am ym , dx (16) with Am is a 10 × 10 matrix (see Appendix). 3.2. Dynamic transfer matrix, dynamic stiffness matrix K (ω ) RL The dynamic transfer matrix [T]m is given by: Tm (ω ) = e 0 Then [T]m is separated into four blocks   T11 T12 . [T] m = T21 T22 Am ( x,ω )dx . (17) Finally, the dynamic stiffness matrix [K(ω )]m for conical shell containing fluid is determined by   −1 −1 T11 −T12 T12 . (18) [K(ω )]m = −1 −1 T21 − T22 T12 T11 T22 T12 m Similarly, we can obtain the dynamic stiffness matrix [K(ω )]m for cylindrical shell containing fluid [27]. The assembly procedure of the finite element method is used to construct the Dynamic Stiffness Matrix for combined conical-cylindrical-conical shells containing fluid. Kcon+fluid K()m = Kcyl+fluid Kcon+fluid Natural frequencies will be extracted from the harmonic responses of the structure by using the procedure detailed in [25–28]. Free vibration analysis of joined composite conical-cylindrical-conical shells containing fluid 257 4. NUMERICAL RESULTS AND DISCUSSION 4.1. Modal analysis A computer program based on Matlab is developed using DSM to solve a number of numerical examples on free vibration of composite joined conical-cylindrical-conical shells containing fluid with different fluid level and geometries shell. Fundamental frequencies will be validated with respect to those of literature and to the results obtained by FEM. First, natural frequencies are validated for a Free-clamped cross-ply laminated composite cylindrical shells containing fluid having the following dimensions and material: E1 = 206.9 GPa; E2 = 18.62 GPa; ν12 = 0.28, G12 = 4.48 GPa; G13 = 4.48 GPa; G23 = 2.24 GPa; ρ = 2048 kg/m3 , layers [00 /900 /00 /900 ]; h = 9.525 mm; R = 0.1905m; L = 0.381m, ρ f = 1000 kg/m3 . Our results comparing to FEM-ANSYS and Xi et al. values [2] are illustrated in Tab. 1. Table 1. Comparison of natural frequencies of laminated composite cylindrical shells partially filled with fluid by FEM and by CEM in comparison with results of Xi et al. [2] Fluid Mode Xi [2] FEM CEM Error (%) % Reduction levers (n, m) (1) (60 × 19 × 10) (2) (1)-(2) with H/L = 0 1 (1,2) 419,54 454,15 460,3 1,35 - 2 (1,3) 517,2 561,02 566,2 0,92 - 3 (1,1) 645,24 677,56 679,12 0,23 - 1(1,2) 377,2 401,12 402,52 0,35 12,55 2(1,3) 440,64 483,49 482,42 -0,22 14,80 3(1,1) 581,98 606,03 614,61 1,42 9,50 1(1,2) 206,41 236,06 235,6 -0,19 48,80 2(1,3) 280,09 303,28 297,7 -1,84 47,42 3(1,1) 314,85 355,67 372,36 4,69 45,17 H/L = 0 H/L = 0.5 H/L = 1 The next step of our research on composite axis-symmetric joined shell is obviously to validate the presented formulation for this type of structure. Consider the freeclamped (F-C) joined cross-ply laminated conical-cylindrical shells having the following dimensions and material: L/R1 = 1; h/R1 = 0.01; h = 2 mm; Li = L, E1 = 135 Gpa; α = 00 , 300 , 600 ; E2 = 8.8 Gpa; G12 = 4.47 Gpa; υ12 = 0.33; ρ = 1600 kg/m3 . Our results comparing to FEM-ANSYS values with different meshes and with analytical solutions of Kouchakzadeh [20] are illustrated in Tab. 2. The agreement between FEM, CEM and Xi results, Kouchakzadeh results is very good in Tab. 1 and Tab 2. 258 Vu Quoc Hien, Tran Ich Thinh, Nguyen Manh Cuong Table 2. The lowest fundamental frequency parameter Ω1 = ωR1 [ρh/A11 ]1/2 and corresponding circumferential wave numbers (m) for various types lamination sequences of joined cross-ply laminated conical-cylindrical shells (FC boundary condition) No 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Cone angles α = 00 α = 300 α = 600 Kouchakzadeh [20] Ansys Ansys (1) 40 × 10 80 × 20 [0/90] 0.0455(4) 0.0454 0.0454 [90/0] 0.0455(4) 0.0453 0.0453 [0/0/0] 0.0288(5) 0.0286 0.0286 [0/90/0] 0.0365(5) 0.0362 0.0362 [0/0/90] 0.0403(4) 0.0401 0.0401 [0/90/90] 0.0567(4) 0.0565 0.0565 [90/90/0] 0.0577(4) 0.0576 0.0576 [90/90/90] 0.1183(3) 0.1187 0.1188 [0/90]2 0.0533(4) 0.0531 0.0532 [0/90]S 0.0442(4) 0.0438 0.0440 [90/0]2 0.0533(4) 0.0531 0.0532 [90/0]S 0.0593(3) 0.0591 0.0592 [0/90] 0.0366(4) 0.0363 0.0363 [90/0] 0.0367(4) 0.0364 0.0364 [0/0/0] 0.0238(4) 0.0236 0.0236 [0/90/0] 0.0309(4) 0.0306 0.0306 [0/0/90] 0.0313(4) 0.0310 0.0310 [0/90/90] 0.0492(4) 0.0489 0.0489 [90/90/0] 0.0500(4) 0.0498 0.0498 [90/90/90] 0.1389(3) 0.1394 0.1395 [0/90]2 0.0468(4) 0.0466 0.0466 [0/90]S 0.0384(4) 0.0381 0.0381 [90/0]2 0.0469(4) 0.0467 0.0467 [90/0]S 0.0530(3) 0.0528 0.0528 [0/90] 0.0250(3) 0.0248 0.0248 [90/0] 0.0249(3) 0.0247 0.0247 [0/0/0] 0.0181(4) 0.0178 0.0178 [0/90/0] 0.0241(4) 0.0238 0.0238 [0/0/90] 0.0223(3) 0.0222 0.0222 [0/90/90] 0.0315(3) 0.0313 0.0313 [90/90/0] 0.0320(3) 0.0318 0.0318 [90/90/90] 0.0963(3) 0.0967 0.0968 [0/90]2 0.0302(3) 0.0300 0.0300 [0/90]S 0.0285(3) 0.0281 0.0282 [90/0]2 0.0302(3) 0.0300 0.0300 [90/0]S 0.0342(3) 0.0348 0.0248 Layers CEM Errors (%) (2) (2)-(1) 0.0458 0.66 0.0457 0.44 0.0289 0.35 0.0366 0.27 0.0405 0.49 0.0570 0.53 0.0578 0.17 0.1183 0.00 0.0535 0.37 0.0443 0.23 0.0535 0.37 0.0595 0.34 0.0367 0.27 0.0367 0.00 0.0239 0.42 0.0310 0.32 0.0313 0.00 0.0494 0.40 0.0502 0.40 0.1392 0.22 0.0470 0.43 0.0385 0.26 0.0471 0.42 0.0532 0.38 0.0253 1.19 0.0252 1.19 0.0182 0.55 0.0243 0.82 0.0226 1.33 0.0318 0.94 0.0323 0.93 0.0979 1.63 0.0305 0.98 0.0287 0.70 0.0304 0.66 0.0344 0.58