Free vibration of symmetric angle-ply laminated circular cylindrical shells

Free vibration of symmetric angle-ply laminated circular cylindrical shells is studied using Spline approximation. The equations of motions in longitudinal, circumferential and transverse displacement components, are derived using Love's first approximation theory. The coupled differential equations are solved using Spline approximation to obtain the generalized eigenvalue problem. Parametric studies are performed to analyse the frequency response of the shell with reference to the material properties, number of layers, ply orientation, length and circumferential node number and different boundary conditions.


Introduction
Shell structures are widely used as structural components in engineering, aerospace, naval and chemical industry. They are often used as load-bearing structures for aircrafts, rockets, submarines and missile bodies. The cylindrical shells used in those designs are stiffened to achieve better strength, stiffness, temperature resistant and light weight characteristics. Recently, ' [1]' studied laminated shells. Free vibration of layered cylindrical shells of variable thickness was examined and the frequencies were analyzed for specially orthotropic materials using collocation with splines ' [2]'. The paper on angle-ply laminated cylindrical shells based on the Love-type version of a unified shear deformable shell theory was studied by ' [3]'. A refined high-order global-local theory was used to analyse the laminated composite beams using the finite element method including shear deformation ' [4]'. Recently, Topal and Uzman ' [5]' analyzed thermal buckling load optimization of angle-ply laminated cylindrical shells. Recently, ' [6]' investigated the free vibration of symmetric angle-ply cylindrical shells of variable thickness. Most of the researchers analyzed the problems of free vibration for angle-ply laminated shells without using spline approximation. Thus, there may not be available sufficient work on angle-ply laminated cylindrical shells with orthotropic materials using spline approximation technique in the past literature. This paper studies the free vibrations of angle-ply layered circular cylindrical shells using spline approximation where the problem is formulated by extending Love's first approximation theory on homogenous shell. The shell is made up of uniform layers of isotropic or specially orthotropic materials.
Studies are carried out for cylindrical shells with clamped-clamped and simply supported boundary conditions along the axial direction and the layers of the material are considered to be thin .The layers are perfectly bounded together to move without interface slip. The effects of the angle-ply, different materials and geometric parameters on the frequencies of angle-ply laminated cylindrical shells under different boundary conditions are analysed. In the analysis, spline approximation technique will be adopted to approximate the displacement functions to analyse the frequencies.

Formulation of the Problem
Consider a thin laminated circular cylindrical shell of having the radius r, length ℓ and thickness h. Each layer in the laminated composite is assumed to be homogeneous, linearly elastic, and specially orthotropic. The layers are perfectly bonded at the interfaces. The coordinate system (x, θ, z) is defined at the mid surface of the shell and u, v and w are the displacements in the directions of x, θ and z respectively.
The equations of motion for thin circular cylindrical shells are given by the following (Viswanathan et al., 2010): The stress resultants and stress couples are given by , , , , , , (4) where N x , N θ , N xθ are stress resultants, M x , M θ , M xθ are moment resultants and σ x , σ θ , σ xθ , are stresses in respective directions. For a thin laminated cylindrical shell, the stress and strain relation of the kth layer is defined as where ( ) k Q are elastic coefficients and ( ) k are normal and shear strains of the k-th layer.
When the materials are oriented at an angle θ with the x-axis, the transformed stress-strain relations are given by Substituting (6) into (4), we get the equations of stress resultants and moment resultants as follows. Here u, v and w are the displacement functions of the mid plane in longitudinal, circumferential and transverse directions and A ij , B ij and D ij are the laminate stiffnesses defined by Here z k is the distance of the k-th layer from the reference surface.
It is assumed that there are no stretching-shearing, twisting-shearing and symmetric angle-ply lamination, so that A 16 = A 26 = D 16 = D 26 = 0 and all B ij =0. Substituting ' equation (7)' in to the 'equations (1-3)' one can obtain the differential equations in terms of displacement functions u,v and w.
The displacement components u,v and w are assumed in the separable form given by where x and θ are the coordinates defined in the longitudinal and circumferential directions respectively, is the angular frequency of vibration, t is the time and n is the circumferential node number.
Substituting the 'equation (10)' in the coupled differential equations, one can obtain the equations in terms U, V and W and can be written in the matrix form as 11 12 13 21 22 23 where ij L are linear differential operators in X. The differential equations on the displacement function contain derivatives of third order in U, second order in V and forth order in W. As such they are not amenable to the solution procedure. Hence the equations are combined within themselves and a modified set of equations are derived. The procedure adopted to this end is to differentiate the first of 'equation (11)' with respect to x once and to use it to eliminate U ''' in the third equation. The modified set of equations are of order 2 in U, order 2 in V and order 4 in W.

Non-Dimensionlization
The following non dimensional parameters are introduced: λ=ℓ ( R 0 /A 11 ) 1/2 , a frequency parameter; δ k =h k /h, the relative thickness of the k-th layer; H = h/r, the thickness parameter; L = ℓ /r, a length parameter; X = x / ℓ , a distance co-ordinate; R = r / ℓ, a radius parameter (12) where is the length of the cylinder, r is its radius, k h is the thickness of the k -th layer, h is the total thickness of the shell, 0 R is the inertial coefficient and 11 A is a standard extensional rigidity coefficient.
Clearly the range of X is lies between 0 and 1.

Method of Solution
The displacement functions U(X), V(X) and W(X) are approximated by the cubic and quantic splines Here H(X-X j ) is the Heaviside step function and N is the number of intervals into which the range [0,1] of X is divided. The points X=X s =s/N (s=0,1,2,…N) are chosen as the knots of the splines, as well as the collocation points. By applying the collocation points to the differential equations given in (11)  Combining any one of the boundary conditions, resulting a generalized eigenvalue problem as follows: [ where [ M ] and [P ] are matrices of order (3N+7) (3N+7) , {q} is a column matrix of order (3N+7)×1 and λ is the eigenparameter.

Results and Discussions
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Conclusions
Layering of the shell wall influences the natural frequencies of the vibration of the cylindrical shell. The relative thickness, as well as the angle ply orientation between them affects the frequencies. A desired frequency of vibration, within a range of frequencies, may be obtained by a proper choice of the relative thickness of layers and the angle-ply rotation among the chosen materials of the layers. We can choose the desired frequency of vibration from the results by a proper choice of the coefficient of thickness variations and arrangements of ply-angles.

Acknowledgement
The authors thankfully acknowledge the financial support from the UTM-Flagship Research Grant Vote No. 01G40, Research management Centre (RMC), Universiti Teknologi Malaysia, Malaysia for completion of this research work.