Free Vibration of Laminated Truncated Conical Shell Structure using Spline Approximation

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 12, Number 6 (2016), pp Research India Publications Free Vibration of Laminated Truncated Conical Shell Structure using Spline Approximation K.K. Viswanathan a,b,1 a UTM Centre for Industrial and Applied Mathematics, Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia. b Kuwait College of Science and Technology, Doha Area, 7th Ring Road, P.O.Box 27235, Safat 13133, Kuwait. Jang Hyun Lee Department of Naval Architecture and Ocean Engineering, Inha University, 100 Inharo, Nam-gu, Incheon 22212, South Korea. M.D. Mohamad Hapiz and Z.A. Aziz UTM Centre for Industrial and Applied Mathematics, Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia. Abstract Free vibration of laminated truncated conical shell structures is studied by using spline approximation method. The governing equations of motion for conical shell are formulated in terms of longitudinal, circumferential and transverse displacement components, by using extension of Love s first approximation theory. Assuming the displacement components are in a separable form, a system of coupled equations on three displacement functions for each shell are obtained. Since exact solutions are generally impossible, a numerical solution procedure is adopted in which the displacement functions are approximated by using Bickley-type splines of suitable order, which are cubic and quintic, resulting into a generalised eigenvalue problem. 1 Corresponding author, visu20@yahoo.com

2 The convergence and comparative results are presented. Effects of the circumferential node number, relative thickness ratio, length ratio and semi-cone angle on the frequencies of the laminated truncated conical shell with two kinds of material properties under two different boundary conditions are investigated. AMS subject classification: Keywords: Free vibration; Conical shell; Coupled equations; Spline approximation. 1. Introduction Shell is a curvy thin-walled structure that is made from single or multilayer of isotropic or orthotropic materials. Conical shell structures are a kind of shell that is widely used in the fields of construction engineering such as aerospace, missiles, automotive, and ship building, as well as industrial engineering fields such as chemical industries, petroleum and petrochemical industries, and other industries. The main reason behind their extensive use in recent years is due to having curvature, the most significance feature which provides more strength to the structures. Moreover, the high strength-to-weight and stiffness-to-weight ratios and their ability to be tailored to meet design requirements of strength and stiffness are favoured by many industries. Shell structures made up from composite are ideal as these structures offer more stiffness and strength, better damping, shock absorbing characteristics and high temperature resistance when compared to the structures formed by using homogenous material. The higher specific strength of these composites means that the weight of certain components can be reduced. While the lamination of the structures leads to design with minimum weight and maximum reliability. By controlling the lamination angle and the stacking sequence, these can alter their structural properties, leading to an optimal design. In return, this helps to construct low cost and fuel economic structures. The study of free vibration of laminated conical shells has been analyzed by many researchers. Irie et al. [1, 2] studied free vibration of conical shells with constant and variable thickness using transfer matrix approach and Rayleigh-Ritz method, respectively. While Srinivasan and Krishnan [3] studied free vibration analysis of isotropic conical shell panel with all edges clamped by using an integral equation technique. On the other hand, Kayran and Vinson [4] analysed the free vibration characteristic of isotropic and laminated composite truncated circular conical shells including transverse shear deformation theory using transfer matrix method. Sivadas and Ganesan [5] used Finite Element Method, a versatile numerical method to study the vibration of laminated conical shells with variable thickness. Tong [6, 7] presented an analysis on free vibration of orthotropic conical shells and axisymmetric laminated conical shells including shear deformation theory. Solution in the form of power series is obtained from the governing equations for displacement functions. Khatri [8] presented an analysis on axisymmetric vibration of multi-layered conical shells with core layers of viscoelastic material by using Galerkin method to approximate the solution. While Shu [9] studied on vibration

3 Free Vibration of Laminated Truncated Conical Shell Structure 4639 of composite laminated conical shells for orthotropic materials by applying method of generalised differential quadrature. This method improved the differential quadrature technique for the computation of weighting coefficients. Three dimensional elasticity solutions were used by Wu and Wu [10] to study the free vibration analysis of laminated conical shells by asymptotic approach. Hua and Lam [11] applied differential quadrature method to study the influence on characteristics of a rotating composite laminated conical shell. Meanwhile Wu and Lee [12] studied free vibration analysis of laminated conical shells with variable stiffness by using differential quadrature method. The first order shear deformation theory is used to analyse the effects of transverse shear deformation. Free vibration analysis of thin conical shells under different boundary conditions was studied by Liew et al. [13]. Analysis on the shells was carried out by using element-free kp-ritz method. In the meantime, spline function techniques were used by Viswanathan and Navaneethakrishnan [14] to study conical shells, with only considering special orthotropic shell while neglecting rotary inertia and shear deformation effects. Tripathi et al. [15] presented free vibration of laminated composite conical shells with random material properties, where the sensitivity of the randomness in material parameters on linear free vibration response is considered. Civalek [16] used discrete singular convolution algorithm method to determine the frequencies of free vibration of laminated conical shells, whereas Sofiyev et al. [17] studied vibrations of orthotropic non-homogenous conical shells with free boundary conditions. Ghasemi et al. [18] studied free vibration of composite conical shells under various boundary conditions by using solution of beam function and Galerkin method. Viswanathan et al. [19, 20, 21] presented renowned studies on this field. They have carried out many researches on free vibration of symmetric and anti-symmetric angleply or cross-ply laminated conical shells of constant or variable thickness using classical shell theory and shear deformation theory. Bickley-type spline was adopted on these studies to obtain the solutions for the generalised eigenvalue problem. In the present work, free vibration of laminated truncated conical shell is studied by using spline approximation method. The problem is formulated by using Love s theory to obtain the equations of motion for both conical and cylindrical shells. These governing differential equations are derived in terms of displacement functions by using stress-strain and strain-displacement relations together with other related kinematic relations. General solutions for longitudinal, circumferential and transverse displacements are assumed in separable form of trigonometric equations, which reduced to a system of ordinary differential equations after introducing non-dimensional parameters into equations of motion, which results in three linear equations for each shell. Spline approximation method is adopted to solve the equations which comprise of cubic and quintic splines, assuming the splines agree with the functions used to approximate the nodes which coincide with the knots. Collocation with these splines yields a set of field equations, which together with continuity and boundary conditions reduce the problem to a generalised eigenvalue problem with eigenparameter as the solution. The variations of the natural frequencies with the circumferential node number, the relative thickness ratio, the semi-vertex angle of the cone and the length ratio of the cone under different boundary conditions are analysed.

4 4640 K.K. Viswanathan, et al. 2. Formulation of the Problem The coordinate system and the geometry of the shell are shown in Figure 1. From the figure, r a and r b are the radii at the small and large ends, α is the semi-vertex angle and l = b a is the slanted length of the truncated cone, where a is the slanted length of the tip of the cone that has been removed and b is the original slanted length of the cone. The shell thickness is denoted by h and the coordinates x, θ and z are longitudinal, circumferential and transverse coordinates, respectively, while u, v and w are longitudinal, circumferential and transverse displacements. Figure 1: Geometry of a layered truncated conical shell of constant thickness. The equations of motion in terms of stress resultants N x, N θ, N xθ and moment resultants M x, M θ, M xθ are given as N x x + N x N θ + 1 N xθ 2 u = I 0 x x sin α θ t 2 N xθ x + 2N xθ + 1 N θ x x sin α θ + 1 x tan α 1 M θ 2 v + x 2 = I 0 sin α tan α θ t 2 N θ x tan α + 2 M x x M xθ x sin α x θ + 1 x 2 sin 2 α + 2 M x x x 1 M θ x x = I 0 M xθ x + 2M xθ x 2 tan α 2 M θ θ M xθ x 2 sin α θ 2 w t 2 (2.1)

5 Free Vibration of Laminated Truncated Conical Shell Structure 4641 where N x N θ N xθ =dz M x M θ =dz M xθ z z σ x σ θ σ xθ dz σ x σ θ zdz (2.2) σ xθ and I 0 is the mass inertia, given by I 0 = ρh, where ρ is the mass density, while σ x, σ θ are normal stress and σ xθ is shear stress. According to the first order shear deformation theory, the strain-displacement relations in terms of two independent coordinates x and θ can be simplified as follows [23] ε x =ε x0 + zκ x ε θ =ε θ0 + zκ θ ε xθ =ε xθ0 + zκ xθ (2.3) where ε x and ε θ are normal strains and ε xθ is shear strain at any point in the shells, ε x0 and ε θ0 are strains and ε xθ0 is shear strain of the middle surface, κ x and κ θ are curvature changes and κ xθ is twist of the middle surface. Neglecting the radius of the twist, R xθ, the strain ε and the curvature changes κ of the middle surface are given by [24] ε x0 = u x ε θ0 = u x + 1 ε xθ0 = 1 x sin α x sin α v θ + u θ + v x v x w x tan α κ x = 2 w x 2 1 v κ θ = x 2 sin α tan α θ 1 2 w x 2 sin 2 α θ 2 1 w x x κ xθ = 1 v x tan α x 2v x 2 tan α 2 2 w x sin α x θ + 2 w x 2 sin α θ. (2.4) The stress-strain relations of the k-th layer by neglecting the transverse normal strain and stress, are of the form k σ x Q 11 Q 12 k k Q 16 ε x σ θ = Q 12 Q 22 Q 26 ε θ. (2.5) σ xθ Q 16 Q 26 Q 66 ε xθ

6 4642 K.K. Viswanathan, et al. When the materials are oriented at an angle φ with the x-axis, the transformed stressstrain relations are given by σ x σ θ σ xθ k Q 11 Q 12 Q 16 = Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 ε x ε θ ε xθ k k. (2.6) where Q ij and Q ij (i, j = 1, 2, 6) are the elastic stiffness coefficients given in Appendix A [21]. Substituting Equation (2.4) into Equation (2.3) and then into Equation (2.6) and carrying out the integration over the thickness with respect to z-axis, from layer to layer, yields N x N θ N xθ M x M θ M xθ A 11 A 12 A 16 B 11 B 12 B 16 A 12 A 22 A 26 B 12 B 22 B 26 = A 16 A 26 A 66 B 16 B 26 B 66 B 11 B 12 B 16 D 11 D 12 D 16 B 12 B 22 B 26 D 12 D 22 D 26 B 16 B 26 B 66 D 16 D 26 D 66 ε x0 ε θ0 ε xθ0 κ x κ θ κ xθ (2.7) where A ij, B ij and D ij (i, j = 1, 2, 6) are extensional rigidities, bending-stretching coupling rigidities and bending rigidities coefficients, respectively, arising from the piecewise integration over the shell thickness, given by A ij = B ij = 1 2 D ij = 1 3 N k=1 N k=1 N k=1 Q k ij (z k z k 1 ) Q k ij (z2 k z2 k 1 ) Q k ij (z3 k z3 k 1 ). (2.8) For current study, it is assumed that there are no stretching-shearing, twistingshearing and bending-shearing, hence A 16 = A 26 = B 16 = B 26 = D 16 = D 26 = 0. Applying this condition to Equation (2.7) then substituting into the equilibrium equations, yields the complex differential equations in terms of U, V and W. The displacements are assumed in the separable form given by u(x,θ,t)=u(x)cos nθe iωt v(x, θ, t) =V(x)sin nθe iωt w(x, θ, t) =W(x)cos nθe iωt (2.9) where t is the time, ω is the angular frequency of vibration, n is circumferential node number and U, V and W are the functions of wave amplitudes.

7 Free Vibration of Laminated Truncated Conical Shell Structure 4643 The non-dimensional parameters are introduced as follows U(x) = lū(x), V (x) = l V (X), W(x) = l W(X), X = x a, a x b, X [0, 1], l I 0 λ =ωl, γ = h, γ = h A 11 r a a, β = a (2.10) b where X is distance coordinate, λ is frequency parameter, γ is ratio of thickness to radius, γ is ratio of thickness to length, and β is the length ratio. Substituting Equations (2.9) and (2.10) into the governing differential equations, the resulting equation in the matrix form is written as L 11 L 12 L 13 Ū L 21 L 22 L 23 V = 0. (2.11) L 31 L 32 L 33 W The differential equations in Equation (2.11) contain derivatives of third order in Ū, second order in V and fourth order in W. Hence, these equations are not amenable to the solution procedure as being proposed to adopt. Therefore, these equations need some modifications in order to obtain a new set of equations in which the derivatives of Ū and V are order of two and W is of order of four. The modifications produced a set of modified equations, given by L 11 L 12 L 13 L 21 L 22 L 23 L 31 L 32 L 33 Ū V W = 0. (2.12) where L ij (i, j = 1, 2, 3) are the differential operators and L ij (i, j = 1, 2, 3) are the new differential operators as given in Appendix B. 3. Solution Procedure The displacement functions U(X), V(X) and W(X) are approximated by cubic and quantic splines. Ū(X) = V(X)= W(X) = 2 a i X i + i=0 2 c i X i + i=0 4 e i X i + i=0 N 1 j=0 N 1 j=0 N 1 j=0 b j (X X j ) 3 H(X X j ) d j (X X j ) 3 H(X X j ) f j (X X j ) 5 H(X X j ) (3.1)

8 4644 K.K. Viswanathan, et al. where H(X X j ) is the Heaviside step function, N is the number of subintervals which lies in the range of X[0, 1], and a i, b j, c i, d j, e i and f j (i = 0, 1, 2,...,4, j = 0, 1, 2,...,N 1) are the spline coefficients. The collocation points are the knots of the splines at X = X s = s, where s = 0, 1, 2,...,N. N In this study, boundary conditions chosen to analyse the problem are Clamped- Clamped (C-C), which denotes both ends of the coupled shell are clamped and Simply Supported-Simply Supported (S-S), which denotes both ends of the coupled shell are simply supported. The conditions for each boundary condition are given by [25] u = v = w = w x = 0 (3.2) at X = 0 and X = 1 for C-C boundary condition, and u = v = w = M x = 0 (3.3) at X = 0 and X = 1 for S-S boundary condition. Each of the boundary conditions imposed gives eight more equations on spline coefficients. Combining them with those obtained earlier yields a homogenous system of (3N +11) equations with (3N +11) spline coefficients. These equations can be arranged in matrix form as follows [ P ]{ q } = λ 2 [ Q ]{ q } (3.4) where [ P ] and [ Q ] are square matrices of order (3N + 11) (3N + 11) and { q } is a column matrix of order (3N + 11) 1. This is a generalised eigenvalue problem in which λ 2 is the eigenparameter and { q } is the eigenvector with spline coefficients as the elements. 4. Results and Discussion In this work, free vibration of laminated truncated conical shells is studied. Two layers of lamination are used in this study, in which inner and outer layers of the shell are made up of High Strength Graphite (HSG) and S-Glass Epoxy (SGE) materials, respectively. The effects of circumferential node number (n), relative thickness ratio (δ), semi-vertex angle (α) and length ratio (β)under different boundary conditions are studied. Convergence study is conducted for C-C and S-S boundary conditions in order to establish the number of subintervals N for the range of X[0, 1]. The purpose of determining the precise value of N is to assure high accuracy of the results generated by the program. Since the problem deals with the matrices of large orders, double precision method is used throughout for numerical computation. Extensive trials run by the computer program were carried out, starting from N = 4 onwards. The value of N is raised while the value of the other parameters is fixed until no more significant variations of λ are observed. It can be seen that the choice of N = 14 is adequate since for the next values of N, the percentage of changes in λ values are very small.

9 Free Vibration of Laminated Truncated Conical Shell Structure 4645 Figure 2: Variation of frequency parameter with relative thickness ratio and the effect of coupling under different boundary conditions with α = 30, n = 2, β = 0.5, γ = 0.05 and h = Figures 2 and 3 show the effect of altering the relative thickness ratio d on the frequency parameter λ for α = 30 and α = 60, respectively, under different boundary conditions by fixing the parameters n = 2, β = 0.5, γ = 0.05 and h = The continuous and dashed lines correspond respectively to the inclusion and exclusion of the coupling effect between the longitudinal and flexural deflections, characterised by taking the bending-stretching coupling rigidities coefficient, B ij = 0 and B ij = 0, respectively. When δ = 0, the inner layer vanished, hence the shell becomes homogenous which made up of SGE material; when δ = 1, the outer layer disappeared and the shell becomes homogenous again which made up of HSG material. For 0 <δ<1, both layers are present. Generally, the frequency parameter variation curves corresponding to the lowest meridional mode m = 1 have the least undulations. As m increases, the undulations are getting progressively noticeable. It is also obvious that as m decreases, the differences in λ variation between the inclusion and exclusion of the coupling effect are getting vanished. Values of λ under C-C boundary condition are observed to be higher than those under S-S boundary condition. Meanwhile, comparison of λ variations between α = 30 and α = 60 under respective boundary conditions and respective n shows that λ values when α = 30 are lower than α = 60 for higher value of m. Figures 4 and 5 show the behaviour of the frequency parameter λ with reference to the circumferential node number(2 n 10) for α = 30 and α = 60, respectively, under different boundary conditions by fixing the parameters β = 0.5, γ = 0.05 and h = The continuous and dashed lines correspond respectively to the variations of λ with δ = 0.4 and δ = 0.7. On the whole, the increment in the value of the relative thickness ratio results in decrement of the values of the frequency parameter. It is noticeable that as m decreases,

10 4646 K.K. Viswanathan, et al. Figure 3: Variation of frequency parameter with relative thickness ratio and the effect of coupling under different boundary conditions with α = 60, n = 2, β = 0.5, γ = 0.05 and h = Figure 4: Variation of frequency parameter with circumferential node number under different boundary conditions with α = 30, β = 0.5, γ = 0.05 and h = 0.01.

11 Free Vibration of Laminated Truncated Conical Shell Structure 4647 Figure 5: Variation of frequency parameter with circumferential node number under different boundary conditions with α = 60, β = 0.5, γ = 0.05 and h = the differences in λ variations between δ = 0.4 and δ = 0.7 are getting lessen. Figure 4 shows that the variations of λ with n for α = 30 tend to behave as minimum curves, while for α = 60 in Figure 5, the value of λ rises gradually with n under both C-C and S- S boundary conditions. It is also observed that values of λ under C-C boundary condition are higher than those under S-S boundary condition. By comparing the variations of λ between α = 30 and α = 60 under respective boundary conditions, it can be seen that the chosen value of a does not really affect the outset of λ variations; except for m = 3, where variations with α = 30 are lower than those with α = 60. In Figure 4, value of λ decreases with n up to around n = 4 and n = 5, and then increases at fast pace. The curvature at the turning points seems to be distinct down the meridional mode numbers. The percentage of changes in λ values with respect to m=1, 2 and 3 for δ = 0.4 and δ = 0.7 are 38.29%, 25.37%, 16.90% and 39.33%, 25.80%, 16.95% for C-C boundary condition, 38.48%, 35.15%, 25.14% and 39.93%, 36.71%, 25.47% for S-S boundary condition. In Figure 5, value of λ increases gradually with n. There is a slight variation on λ values in the range of 3 <n<6 for δ = 0.7 under S-S boundary condition. The percentage of changes in λ values with respect to m=1, 2 and 3 for δ = 0.4 and δ = 0.7 are 38.30%, 13.90%, 6.90% and 36.92%, 13.30%, 7.13% for C-C boundary condition, 81.55%, 29.01%, 11.78% and 82.20%, 27.86%, 11.63% for S-S boundary condition. Figures 6 and 7 show the effect of the semi-cone angle a on the frequency parameter λ for n = 2 and n = 4, respectively, under different boundary conditions by fixing the parameters β = 0.5, γ = 0.05 and h = When semi-cone angle a varies, radius at the small end of the conical shell, r a = a sin α is also varies. Hence, ratio of thickness to radius γ = h/r a is no longer a constant. Instead, ratio of thickness to length γ = h/a is used to replace γ as a new constant, where γ = γ csc α. It is significant to observe that the behaviours of λ variation with α are dependent

12 4648 K.K. Viswanathan, et al. on the chosen value of circumferential node number under both C-C and S-S boundary conditions. It is seen as a trending that when the value of the relative thickness ratio is increased, the values of the frequency parameter will be decreased. It is also noted that at lower values of m, the distinctions in λ variations between δ = 0.4 and δ = 0.7 become very small. Values of λ under C-C boundary condition are observed to be higher than those under S-S boundary condition. By comparing the variations of λ between n = 2 and n = 4 under respective boundary conditions, it can be seen that in the range of 20 <α<90, the distinctions between the variations are very small. Figure 6: Variation of frequency parameter with semi-cone angle under different boundary conditions with β = 0.5, γ = 0.05, n = 2 and h = In Figure 6, value of λ decreases with α, and the rate of decrease reduces when approaching α = 90, where the cone shells turn to annular plates. It is observed that value of λ decreases gradually with α under C-C boundary condition compared to those under S-S boundary condition, where λ value decreases at much faster pace with α. There is also a slight variation on α values from the outset until α = 50 for δ = 0.7 under C-C boundary condition. In Figure 7, the frequency parameter value decreases rapidly and almost constantly up to α = 20 under both C-C and S-S boundary conditions. There is a slight variation on λ values in the range of 30 <α<60 for δ = 0.7 under both boundary conditions. The rate of decrease in λ value reduces for higher values of α. Figures 8 and 9 show the variation of angular frequency ω (in Hz) with respect to length ratio of the cone β for α = 30 and α = 60, respectively, under different boundary conditions by fixing the parameters γ = 0.05, n = 2 and h = Here, angular frequency ω is shown instead of frequency parameter λ since both λ and β are in terms of l = b a. It is seen as a common feature that the variations of ω with β tend to behave as exponential curves under both C-C and S-S boundary conditions, where the values of the angular frequency increase with the length ratio of the cone. There is no variation

13 Free Vibration of Laminated Truncated Conical Shell Structure 4649 Figure 7: Variation of frequency parameter with semi-cone angle under different boundary conditions with β = 0.5, γ = 0.05, n = 4 and h = Figure 8: Variation of angular frequency with length ratio of the cone under different boundary conditions with α = 30, γ = 0.05, n = 2 and h = 0.01.

14 4650 K.K. Viswanathan, et al. in ω from the outset when the cone is almost complete in its shape (β = 0.1) until the length of the shell is approaching medium length (β = 0.5). The change takes place smoothly in the interval 0.4 <β<0.8 and then increase steeply as β approaching to 0.9 (very short shell). With decreasing in semi-cone angle, the range of nearly linear part of the curves increases. The rate of gradual change is higher, and the rapid increase in ω starts earlier for higher modes. As m decreases, the distinctions in ω variations between δ = 0.4 and δ = 0.7 become almost insignificant. It is also noticeable that the values of ω under C-C boundary condition are higher than those under S-S boundary condition. Figure 9: Variation of angular frequency with length ratio of the cone under different boundary conditions with α = 60, γ = 0.05, n = 2 and h = By judging the variations of ω between α = 30 and α = 60 under respective boundary conditions,it can be seen that these two values of a do not seem to cause any significant difference qualitatively, except for the cases of S-S boundary condition, where the steepness of the curves decreased as α increased from 30 to 60. Unlike the curves under C-C boundary condition where the rapid increment observed to begin at β = 0.8, the increment commonly starts at β = 0.7 under S-S boundary condition. 5. Conclusion From this study, it is found that the natural frequencies of laminated conical shell structures vary with the relative thickness ratio of the layers, the circumferential node number, the semi-cone angle, the length ratio of the cone and the boundary conditions. Values of circumferential node number do not seem to cause any significant difference qualitatively while the other parameters are fixed. The increment in the value of the relative thickness ratio results in decrement of the frequencies. The difference is less pronounced for lower mode. In general, the effect of neglecting the coupling effect between the longitudinal

15 Free Vibration of Laminated Truncated Conical Shell Structure 4651 and flexural deflections usually results in the raise of the frequencies. The effect is more prominence with higher mode. Variations of frequency with circumferential node number tend to behave as minimum curves. Frequency value decreases with semi-cone angle, however the rate of decrease reduces when the cone shells turn to annular plates. While shortening of the length of the cone results in increment of the frequencies. The effect is more obvious for higher mode. All of these behaviours are observed under both C-C and S-S boundary conditions. Despite of these similarities, values of the frequencies under C-C boundary condition are seen to be remarkably higher than those under S-S boundary condition. This finding could also be significant especially to the industrial designers. Appendix Appendix A Elastic stiffness coefficients Q ij and Q ij (i, j = 1, 2, 6) appearing in Equations (2.5) and (2.6) are defined by Q 11 =Q 11 cos 4 φ + Q 22 sin 4 φ + 2(Q Q 66 ) sin 2 φ cos 2 φ (A.1) Q 12 =(Q 11 + Q 22 Q 66 ) sin 2 φ cos 2 φ + Q 12 (cos 4 φ + sin 4 φ) (A.2) Q 16 =(Q 11 Q 12 2Q 66 ) cos 3 φ sin φ (Q 22 Q 12 2Q 66 ) sin 3 φ cos φ (A.3) Q 22 =Q 11 sin 4 φ + Q 22 cos 4 φ + 2(Q Q 66 ) sin 2 φ cos 2 φ (A.4) Q 26 =(Q 11 Q 12 2Q 66 ) cos φ sin 3 φ (Q 22 Q 12 2Q 66 ) sin φ cos 3 φ (A.5) Q 66 =(Q 11 + Q 22 2Q 12 2Q 66 ) sin 2 φ cos 2 φ + Q 66 (cos 4 φ + sin 4 φ) (A.6) E x Q 11 = (A.7) 1 v xθ v θx v xθe x Q 12 = = v θxe x (A.8) 1 v xθ v θx 1 v xθ v θx E θ Q 22 = 1 v xθ v θx (A.9) Q 66 =G xθ (A.10) where E x and E θ areyoung s modulus in respective directions, v xθ and v θx are Poisson s ratios and G xθ is shear modulus.

16 4652 K.K. Viswanathan, et al. Appendix B Differential operators L ij and L ij (i, j = 1, 2, 3) appearing in Equations (2.11) and (2.12) are defined by L 11 = d2 dx 2 + τ d dx ( s 3 + s 10 n 2 csc 2 α)τ 2 + λ 2, (B.1) L 12 =[ s 2 + s 10 + ( s 5 + s 11 )τ cot α]nτ csc α d dx [ s 3 + s 10 + ( s 5 + s s 11 )τ cot α]nτ 2 csc α, (B.2) d 3 L 13 = s 4 dx 3 s 4τ d2 dx 2 +[ s 2 cot α + s 6 τ + ( s s 11 )n 2 τ csc 2 α]τ d dx [ s 3 cot α + ( s 5 + s s 11 )n 2 τ csc 2 α]τ 2, (B.3) L 21 = [ s 2 + s 10 + ( s 5 + s 11 )τ cot α]nτ csc α d dx [ s 3 + s 10 + ( s 6 + s 11 )τ cot α]nτ 2 csc α, (B.4) L 22 =( s s 11 τ cot α + s 12 τ 2 cot 2 α) d2 dx 2 + ( s 10 s 12 τ 2 cot 2 α)τ d dx {[ s 3 + (2 s 6 + s 9 τ cot α)τ cot α]n 2 csc 2 α + s 10 + s 11 τ cot α}τ 2 + λ 2, (B.5) L 23 =[ s s 11 + ( s s 12 )τ cot α]nτ csc α d2 dx 2 + ( s 6 + s 9 τ cot α)nτ 2 csc α d dx [( s 3 + s 6 τ cot α) cot α + ( s 6 + s 9 τ cot α)n 2 τ csc 2 α]nτ 2 csc α, (B.6) L 31 = s 4 τ d2 dx 2 {[ s 2 cot α ( s 3 s 4 s 6 )τ ( s 4 s 10 s 5 2 s 11 )n 2 τ csc 2 α]τ + s 4 λ 2 } d dx [ s 3 cot α s 6 τ(1 n 2 csc 2 α)]τ 2, (B.7) L 32 = { s 4 ( s 2 + s 10 ) s 5 2 s 11 +[ s 4 ( s 5 + s 11 ) s 8 2 s 12 ]τ cot α}nτ csc α d2 dx 2 +{ s 4 ( s 3 + s 10 ) s 6 +[ s 4 ( s 5 + s s 11 ) 2 s 8 s 9 4 s 12 ]τ cot α}nτ 2 csc α d dx +{ s 6 τ(1 cot 2 α) [ s 3 2( s 8 + s s 12 )τ 2 ] cot α ( s 6 + s 9 τ cot α)n 2 τ csc 2 α}nτ 2 csc α, (B.8) L 33 =( s 4 2 s 7) d4 dx 4 + ( s2 4 2 s 7)τ d3 dx 3 {( s 2 s 4 2 s 5 ) cot α + ( s 4 s 6 s 9 )τ +[ s 4 ( s s 11 ) 2( s s 12 )]n 2 τ csc 2 α}τ d2 dx 2 +{ s 3 s 4 cot α +[ s 4 ( s 5 + s s 11 ) 2( s s 12 )]n 2 τ csc 2 α s 9 τ}τ 2 d dx { s 3 cot α s 6 τ)cot α +[2 s 6 cot α 2( s 8 + s s 12 )τ + s 9 n 2 τ csc 2 α]n 2 τ csc 2 α}τ 2 + λ 2. (B.9)

17 Free Vibration of Laminated Truncated Conical Shell Structure 4653 where s 2 = A 12, s 3 = A 22, s 4 = B 11, s 5 = B 12, s 6 = B 22, s 7 = D 11, A 11 A 11 A 11 A 11 A 11 A 11 s 8 = D 12, s 9 = D 22, s 10 = A 66, s 11 = B 66, s 12 = D 66, (B.10) A 11 A 11 A 11 A 11 A 11 s 2 =s 2, s 3 = s 3, s 4 = s 4 l, s 5 = s 5 l, s 6 = s 6 l, s 7 = s 7 l 2, s 8 = s 8 l 2, s 9 = s 9 τ = l a + lx. Acknowledgment l 2, s 10 = s 10, s 11 = s 11 l, s 12 = s 12 l 2, (B.11) (B.12) This work was supported by a Special Education Program for Offshore Plant by the Ministry of Trade, Industry and Energy Affairs (MOTIE), Korea. References [1] Irie, T., Yamada, G. and Kaneko, Y., 1982, Free Vibration of a Conical Shell with Variable Thickness, Journal of Sound and Vibration, 82, pp [2] Irie, T., Yamada, G. and Tanaka, K., 1984, Natural Frequencies of Truncated Conical Shells, Journal of Sound and Vibration, 92, pp [3] Srinivasan, R. S. and Krishnan, P.A., 1987, FreeVibration of Conical Shell Panels, Journal of Sound and Vibration, 117, pp [4] Kayran, A. and Vinson, J. R., 1990, Free Vibration Analysis of Laminated Composite Truncated Circular Conical Shells, AIAA Journal, 28, pp [5] Sivadas, K. R. and Ganesan, N., 1991, Vibration Analysis of Laminated Conical Shells with Variable Thickness, Journal of Sound and Vibration, 148, pp [6] Tong, L., 1993, Free Vibration of Orthotropic Conical Shells, International Journal of Engineering Science, 31, pp [7] Tong, L., 1994, Free Vibration of Laminated Conical Shells including Transverse Shear Deformation, International Journal of Solids and Structures, 31, pp [8] Khatri, K. N., 1994, Axisymmetric Vibration of Multilayered Conical Shells with Core Layers of Viscoelastic Material, Computers and Structures, 58, pp [9] Shu, C., 1996, Free Vibration Analysis of Composite Laminated Conical Shells by Generalized Differential Quadrature, Journal of Sound and Vibration, 194, pp

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