Free vibration analysis of laminated composite conical, cylindrical shell and annular plate with variable thickness and general boundary conditions

This paper presents a unified solution method to investigate the free vibration behaviors of laminated composite conical shell, cylindrical shell and annular plate with variable thickness and arbitrary boundary conditions using the Haar wavelet discretization method (HWDM). Theoretical formulation is established based on the first order shear deformation theory(FSDT) and displacement components are extended Haar wavelet series in the axis direction and trigonometric series in the circumferential direction. The constants generating by the integrating process are disposed by boundary conditions, and thus the equations of motion of total system including the boundary condition are transformed into an algebraic equations. Then natural frequencies of the laminated composite structures are directly obtained by solving these algebraic equations. Stability and accuracy of the present method are verified through convergence and validation studies. Effects of some material properties and geometric parameters on the free vibration of laminated composite shells are discussed and some related mode shapes are given. Some new results for laminated composite conical shell, cylindrical shell and annular plate with variable thickness and arbitrary boundary conditions are presented, which may serve as benchmark solutions.


Introduction
Conical shell, cylindrical shell and annular plate are widely used in a variety of engineering fields such as mechanical, architectural, aerospace, marine and other industries. With the development of science and manufacturing technology, various composite materials have emerged, and many studies have conducted to the dynamic characteristics of them. So far, many researches have proposed to investigate the free vibration behavior of laminated composite structures with variable thickness through several theories and methods. However, the structures with variable thickness are present in the actual engineering applications, and it is needed to analyze the vibration characteristic of them accurately.
Based on the classical thin shell theory, Irie, et al. [1] carried out the analysis on the free vibration of a truncated conical shell with variable thickness by using the transfer matrix approach and calculated the natural frequencies and mode shapes numerically. Sivadas and Ganesan [2][3][4][5][6] investigated asymmetric free vibration behavior of isotropic cantilever conical shell, circular cylindrical shell and laminated cylindrical, conical shells with variable thickness using Love's first approximation thin shell theory and finite element method and Gautham and Ganesan [7] analyzed the axisymmetric free vibration if thick orthotropic spherical shells with linearly varying thickness along the meridian.
Sankaranarayanan et al. [8,9] analyzed the free vibration of the laminated conical shells of variable thickness using the classical thin shell theory and energy method based on the Rayleigh-Ritz procedure.
Jiang and Redekop [10] developed a solution method based on the Sanders-Budiansky shell equations for analyzing the static and free vibration characteristics of linear elastic orthotropic toroidal shells of variable thickness. In this study, the thickness of shell are changed in circumferential direction. Duan and Koh [11] derived analytical solutions for axisymmetric transverse vibration of cylindrical shells with variable thickness for the first time, in which solutions are derived in terms of generalized hypergeometric function in which function. Based on the classical Donnell's theory, Chen et al. [12] performed the buckling analysis the cylindrical shells with variable thickness using the perturbation technique, in which the variation of the thickness are according to in the axial direction. Liu et al. [13] presented an analytical method based on the Flügge theory and equivalent method of ring-stiffeners for the free vibration of a fluid loaded ring-stiffened conical shell with variable thickness in the low frequency range. Tran et al. [14] studied the vibration characteristics of functionally graded cylindrical shells with the variable thickness, in which the thickness of the shell varies linearly along the longitudinal direction, using the FSDT and Hamilton's principle. Nihal and David [15] formulated the dynamic stiffness equation for variable thickness cylindrical shells based on the Donnell, Timoshenko and Flugge theories and obtained the natural frequencies using the Wittrick-Williams algorithm. Afonso and Hinton [16,17] studied the free vibration characteristics of plates and shells with arbitrary thickness variation and boundary conditions using finite element method(FEM). Based on the classical Donnell's and Love's shell theories, Taati et al. [18] investigate the free vibration characteristics of thin cylindrical shells with variable thickness and a constant angular velocity. Efraim and Eisenberger [19] obtained the free vibration frequencies and mode shapes of thick spherical shell segments with variable thickness and different boundary conditions using a dynamic stiffness method. Zheng et al. [20] applied energy method based on Donnell-Mushtari shell theory to investigate the vibration characteristics of cylindrical shell with arbitrary variable thickness and general boundary conditions. The references related to vibration analysis of shells with variable thickness can be found Tornabene's studies [21][22][23][24][25][26] and Kang's studies [27][28][29][30][31][32][33][34]. There are many literatures on the free vibration analysis of shells and plates with variable thickness, however the free vibration analysis of laminated composite structures with variable thickness is almost impossible to find. Therefore, the focus of this paper is on the free vibration analysis of laminated composite conical shell, cylindrical shell and annular plate with variable thickness. In the vibration analysis of structures, to select a reasonable solution method is very important to satisfy the accuracy and efficiency of calculation.
Recently, the Haar wavelets, first introduced by Alfred Haar in 1910, have attracted considerable attention of researchers because it is mathematically the simplest orthogonal compactly supported wavelets of all wavelet families and the solution procedure is simple and direct. The Haar wavelet method has been proven to be an effective tool for solving various problems, such as differential and integral equations [35][36][37][38][39][40][41][42][43], biharmonic equations and Poisson equations. In addition, the Haar wavelet has been also proven to be an effective tool for solving the static and dynamic problems of various structures, such as beams [44][45][46][47][48], plates [49][50][51] and shells [51][52][53][54][55][56][57][58][59][60]. Therefore, in this paper, the Haar wavelet discretization method, whose effectiveness has been verified in the vibration analysis of various laminated composite shell structures, is selected to investigate the free vibration characteristics of laminated composite conical, cylindrical and annular plate. The natural frequency obtained by this method is compared with that by previous literatures and FEM. The effects of some parameters on the free vibration of considered structures are discussed, and new results of the frequency parameters and mode shapes are given.

Theoretical formulations 2.1 Description of the model
The geometric relations and coordinate system of the laminated composite conical shell, cylindrical shell and annular plate are shown in Fig. 1. The reference surface is defined by the geometric middle surface. The fiber orientation angle of the k th layer is represented by ϕf. The coordinate system x, θ, z of the laminated composite structures is introduced, the displacements in the axial, circumferential and normal directions are denoted by u, v and w, respectively. For generalize of the boundary conditions, the elastic spring technique is introduced, and the linear elastic springs in the axial, circumferential and normal directions are represented as ku, kv and kw, the rotation springs in the θ and x axis direction are denoted as kφ and kθ, respectively. The cone length and cone semi-vertex angle of the conical shell are denoted by L and φ, respectively. R1 and R2 are the small and large radius and the radius R is a function of axial coordinate x. The semi-vertex angle φ is denoted by the angle between the x-axis and the rotating axis.
In Fig. 1(c), it is worth noting that, by setting the semi-vertex angle φ=0, we can reduce the formulation of conical shells to that of cylindrical shells. In addition, by setting the semi-vertex angle φ=π/2, we can reduce the formulation of conical shells to that of annular plate with outer radius R2 and inner radius R1.
The thicknesses at origin and end of the shell are represented by h1 and h2, respectively, and generalized equation of thickness depends on following as where α and λ are thickness variation parameters.   the thickness of structure in α=0 is uniform, and it is increased or decreased according to the changes of α. Then, according to the changes of λ, the thickness profile is linearly or nonlinearly changed in Fig.   3. In specially, when λ=1, whatever α is, the thickness of structure is linearly increased or decreased.

Formulation for analysis
In the present study, the FSDT is employed for driving the equation of motion for the considered structures. According to FSDT, for any point within the shell, the displacement and rotation components of the reference plane can be written as follows [51,63,64,65]: where u0, v0 and w0 is the displacements at a point of the middle surface, along axial, circumferential and normal directions, respectively; ϕx and ϕθ represent the rotations of the reference surface about the θ-and x-axis. t is the time variable. The strain-displacement relationships can be written as [51,63,64]:  0  11  12  16  11  12  16  0  12  22  26  12  22  26  0  16  26  66  16  26  66   11  12  16  11  12  16   12  22  26  12  22  26   16  26  66  16 26 66 55 45 45 44 x xz where Nx, Nθ and Nxθ denote the in-plane force resultants, Mx, Mθ and Mxθ represent the bending and twisting moment resultants. Qx and Qθ are the transverse shear force resultants. κ is the shear correction factor and in this paper is set as κ= 5/6. The stiffness coefficients Aij, Bij and Cij are defined as: where Nk is the number of layers. Zk+1 and Zk denote distances from the shell reference surface to the outer and inner surfaces of the k th layer. The coordinate Zk of the bottom surface of the k th layer is expressed as a function of x.
On the other hand, because the thickness of the shell are changed in x-axis direction, the stiffness coefficients Aij, Bij and Dij are functions of x, therefore, partial derivatives of the stiffness coefficients are appeared, can be written as follows.

Governing equations
In the paper, the Hamilton's principle is adopted for driving the equilibrium equations of motion of the laminated composite conical, cylindrical shells and annular plate with variable thickness [51].   (14) where ρk is the density of the k th layer, and the cone radius at any point along its length is given by where Lij(i, j=1-5) are the differential operators, the detailed expressions can be found Appendix A.
It is obvious that each of the displacement and rotation components at most has second-order derivatives.
For the certain circumferential wave number n, the displacements of the Eq. (16) are expressed as: where U(x), V(x), W(x), Φ(x) and Θ(x) are unknown variable functions to be determined, ω is the angular frequency. Substituting Eq. (17) into Eq. (16) and then multiplying the governing equations with R 2 , and multiplying cos(nθ) or sin(nθ) in the governing equations, then integrating s from 0 to 2π.
According to: The detailed expressions of the coefficients items Lij k can be found Appendix B.
Disposing the boundary condition in vibration problems has always been one of the most challenging and important issues. In this paper, for generalizing the boundary conditions, artificial springs technique have been employed. Therefore, the boundary conditions are modeled using three kinds of linear springs (ku, kv, kw) and two kinds of rotational springs (kφ, kθ) by assigning that these springs are at proper stiffness. The boundary condition equations for laminated composite conical, cylindrical shells and annular plate with variable thickness can be expressed as follows: Similarly, the boundary condition equations of right boundary can be obtained by applying x=L in the Eq. (21). Therefore, the generalized boundary equations of shell according to the spring stiffness can be obtained, the various boundary conditions can be modeled with setting appropriate values of the spring stiffness.

Implementation of the HWDM
In current study, Haar wavelet series are employed for discretization of the derivatives in governing equations of whole system including boundary conditions. The basic theories of Haar wavelet have been introduced in Xie's studies [52][53][54][55][56] and author's studies [57][58][59][60]. Therefore, in this paper, the explanation for Haar wavelet is downplayed.
In the HWDM, the highest order derivatives of the displacement components are defined by the Haar wavelet series and the lower order derivatives can be obtained by integrating Haar wavelet series.
The highest order derivative of the displacements in the governing equations of shell is second order, which can be expressed by means of the Haar wavelet series as follows; where ai, bi, ci and di are unknown coefficients of the Haar wavelets. The first order derivatives of displacements are obtained by integrating Eq. (22) and the displacement functions can be obtained by integrating the above result again. The first order derivatives of displacements and displacement functions are expressed as follows.
Eqs. (22) and (23) can be expressed in the discretized matrix form as follows: where Haar wavelet H and its integrals P1, P2 are defined in matrix form as 11 21 1  P (27) and notations are defined as follows.
where, f, g, h, k and l indicate the integral constants, which can be obtained by applying the boundary condition. The highest order of the displacements of boundary condition equations is first order, and the first order derivatives and displacements in Eq (23) are calculated when ξ=0 and ξ=1. The discretization of the boundary condition equation can be manipulated in the same way as that of the displacement, and it can be written in the matrix form as follows.
, , where notations are defined as follows.
Therefore, the equations of motion of the total systems of the laminated composite structures including boundary condition are discretized using the HWDM and can be expressed in the matrix form as: ,,,,, , , ,,,, where subscripts d and b indicate the discrete equilibrium equations of motion and the boundary conditions. Ab at both sides of Eq. (33) can be eliminated by performing some algebraic manipulations, and the standard characteristic equation is expressed as follows.
Through the further simplification of the above equation, the following matrix expressions can be obtained.
where, K and M are stiffness and mass matrixes of the structure respectively

Numerical example and discussion
In this section, some numerical examples are presented to analyze the free vibration of conical, cylindrical shells and annular plate. In order to verify the accuracy and reliability of the proposed method, different boundary conditions, thickness profiles and geometric parameters are considered. The material properties in this research are as follows: E22=10Gpa, E11=open, G12=G13=0.6E2, G23=5Gpa, μ12=0.25, ρ=1500kg/m 3 . Then, for convenience, the frequency parameter is defined as follows. Ω=ωR1(ρh/E22) 1/2 .

Convergence
In theory, Haar wavelet series can be infinitely expanded. However, for the accuracy of solution and efficiency of calculation, it must be truncated at an appropriate finite number. Therefore, convergence studies are needed to establish the number of terms that should be used to obtain accurate results. The convergence criterion of the present method for a four-layered, cross-ply [0°/90°/0°/90°] C-C conical shell with respect to different the maximal level of resolution J is examined in Fig. 4. The geometric parameters are following as; L=2m, R1=1m, φ=30°, h1=0.05m, and thickness variation parameters are same as α=0.5, λ=1. Hence, according to Fig. 4, it can be deduced that the present solution provides rapid convergence high accuracy even at low level of the resolution J, so that the computational cost in the HWDM is significantly reduced. It is also observed that when the resolution J reaches a certain value, the results remain almost unchanged. Hence, the maximal level of resolution J for the following numerical examples is uniformly chosen as J=7. As mentioned in theoretical formulation, the artificial spring technique is introduced for the generalization of boundary conditions in this paper, and the boundary conditions are changed according to the stiffness values of artificial spring such as ku, kv, kw, kφ and kθ. Therefore, the stiffness values must be selected to determine the classical and elastic boundary conditions.   Table 1. For convenience of the presentation, classical boundary conditions such as the clamped, free, simply-supported and shear-diaphragm are indicated with C, F, SS and SD, respectively. Also three kinds of elastic boundary conditions are considered in this example, which are denoted as E1, E2 and E3.

Validation
In previous subsection, through the convergence study, the  Table 2-4, the result by the proposed method are very consistent with that of literatures. Table 5 shows the frequency parameters of laminated composite conical shell according to different semi-vertex angles in comparison with the results of literatures. In a conical shell, if the semi-vertex angle is 90 degree or 0 degree, it is thought to be an annular plate or a cylindrical shell. Table 5 shows the frequency parameters of laminated composite conical, cylindrical shells and annular plate in comparison with each other. In the comparison study, the result by the proposed method are also consistent with that of literatures.      Table 5. Comparison of the frequency parameters Ω=ωR1(ρh/A11) 1/2 for three-layered composite shells with different sets of boundary conditions and semi-vertex angles (R1=1m, L=2m, E22=10GPa, E11=150GPa, μ12=0.25,  The main purpose of this paper is to analyze the free vibration of laminated composite structure with the varying thickness. So, the accuracy of the proposed method on these structures must be verified.
Because of the lack of literature studying the free vibration of laminated composite structures with the varying thickness, the comparison study is only conducted in comparison with ABAQUS.  Table 6, R1=0.5m, h1=0.05m, L=3m in Table 7 and R1=0.5m, h1=0.05m in Table 8. For ABAQUS analysis, an element type S4R is used. In cylindrical shells, when L/R=2 and L/R=5, the number of elements is 3150 and 7850, respectively.  In Table 6-8, regardless of the geometric dimensions and boundary conditions, the natural frequencies of laminated composite conical, cylindrical shells and annular plate with the varying thickness by the proposed method are very consistent with those by FEM. From this, the proposed method is a reasonable method, which can ensure a high accuracy in the free vibration analysis of conical, cylindrical shells and annular plate with the varying thickness.

Parametric studies
In the last section, parametric studies are carried out to investigation the effects of material properties, geometrical parameter and boundary conditions on free vibration of laminated composite conical shell, cylindrical shell and annular plate with variable thickness. Table 9 in conical shell, R1=1m, L=2m, φ=45°, h1=0.05m in cylindrical shell and R1=1m, R2=3m, h1=0.05m in annular plate. In Table 9-11, the change of boundary condition affects the frequency parameters of laminated composite structure with the varying thickness explicitly. In particular, regardless of the structure, the frequency parameter is the largest in C-C boundary condition. Then, the frequency parameter of conical shell is the smallest in E3-E3 boundary condition, that of cylindrical shell in E3-C boundary condition, and that of annular plate in C-F boundary condition. Then, regardless of the cylindrical shell with even thickness, the frequency parameters of cylindrical shell with varying thickness have different values in the opposite boundary conditions (C-F and F-C). This is because of the changes in boundary condition caused by different thicknesses.   Table 11. The frequency parameters for laminated composite annular plate with variable thickness and different boundary conditions.
BCs  Table 12-14 show the frequency parameters of laminated composite structure with varying thickness according to the fiber angle. The material properties and geometric dimensions are the same as Table 9-11. The differential point is that all structures are laminated by 3 layers, and h1=0.1m in all structures.
In Table 12-14, the fiber angle affects the frequency parameters of laminated composite structure with the varying thickness explicitly.  In order to study the effect of fiber angle on the frequency parameters more specifically, while the fiber angle increases from 0° to 180°, the change of frequency parameter is considered. Fig. 6    Then, the effect of α and λ on the frequency parameter is investigated. Fig. 9 shows the change of frequency parameter of laminated composite structure according to the changes of α and λ. The material properties and geometric dimensions are the same as Fig. 6-8. In Fig. 9, when λ increases, the frequency parameter converges to a certain value. Then, when α is less than 0, the frequency parameter decreases gradually and converges to a certain value. In addition, when α is more than 0, it increases gradually and converges to a certain value.  In C-C and SS-SS boundary, the frequency parameter in α=-0.5 is more than that in α=0. The value in α=0.5 is less than that in α=0. In C-F boundary condition, the value in α=-0.5 is less than that in α=0 at the start point. However, when the circumferential wave number reaches a certain value, the value increases again. In addition, the value in α=0.5 is more than that in α=0 at start point. However, when the circumferential wave number reaches a certain value, the value decreases again. In E3-E3 boundary condition, the value changes irregularly at the start point. However, when the circumferential wave number exceeds a certain value, the value increases.
As the next example, the effect of the number of layer on the free vibration of laminated composite structures will be investigated.  In the last example, Fig. 12 shows the variations of the first three frequency parameters of the [0°/90°/0°/90°] composite laminated composite structures with variable thickness according to the elastic modulus ratios. In here, the parameters for thickness profile α=-0.5, 0 and 0.5, λ=1. The elastic modulus ratios E11/E22 vary from 1 to 200Gpa. From the Fig.12, it is obvious that the frequency parameters increase as the elastic modulus ratios increase.

Conclusion
In this study, a simple and accurate numerical solution method on the basis of the HWDM is presented conditions, which can be used as reference results for subsequent researches in this field.