Free Vibration Analysis of Closed Moderately Thick Cross-Ply Composite Laminated Cylindrical Shell with Arbitrary Boundary Conditions

A semi-analytic method is adopted to analyze the free vibration characteristics of the moderately thick composite laminated cylindrical shell with arbitrary classical and elastic boundary conditions. By Hamilton’s principle and first-order shear deformation theory, the governing equation of the composite shell can be established. The displacement variables are transformed into the wave function forms to ensure the correctness of the governing equation. Based on the kinetic relationship between the displacement variables and force resultants, the final equation associated with arbitrary boundary conditions is established. The dichotomy method is conducted to calculate the natural frequencies of the composite shell. For verifying the correctness of the present method, the results by the present method are compared with those in the pieces of literatures with various boundary conditions. Furthermore, some numerical examples are calculated to investigate the effect of several parameters on the composite shell, such as length to radius ratios, thickness to radius ratios and elastic restrained constants.


Introduction
With the rapid development of the industry, composite laminated materials are increasingly used. The composite laminated cylindrical shell is one of the principal structural components and is widely used in various engineering applications, such as naval equipment, vehicle engineering, aerospace, and basic industries. In the past few decades, the dynamic analysis of composite shells has made considerable progress. People are paying more and more attention to developing more accurate and effective mathematical models and analyzing their dynamic behavior. Some researchers have proposed some of the classical and improved theories, also, different calculation methods are developed. The extensive researches are evolved by Lessia [1], Qatu [2][3][4][5], Reddy [6], Carrera [7,8], Ye [9] and others [10][11][12].

The Description of the Model
Consider the model in Figure 1, the moderately thick composite laminated cylindrical shell with general boundary conditions. L, R, and h denote the length, mean radius and thickness of the shell. The global coordinate (x, θ, z) are set, the x, θ and z axes are taken in the axial circumferential and radial directions. In the k'th layer, the included angle of the composite material and principle direction is defined as β. The distances from the top and bottom surfaces to the middle surface are defined as Zk+1 and Zk. The middle surface displacements of the composite shell are defined as u0, v0, and w0, their directions are set in the x, θ and z axes. The transverse rotations about the θ and x axes are represented as ϕx and ϕθ. There are five groups of linear distribution and rotational springs and each ends.

Kinematic Relations and Stress Resultants
Through the description of the moderately thick composite laminated cylindrical shell, the displacement resultant of the shell is shown by the middle surface displacements and rotation variables, expressed as [2,[49][50][51][52][53][54][55]: where u0, v0, and w0 are the displacements of the middle surface in the axial, circumferential and radial directions, ϕx and ϕθ are the transverse normal rations of the x and θ axis. t represents the time variables. The relationship between the strains and curvature changes of the moderately thick composite laminates shell is defined as:

Kinematic Relations and Stress Resultants
Through the description of the moderately thick composite laminated cylindrical shell, the displacement resultant of the shell is shown by the middle surface displacements and rotation variables, expressed as [2,[49][50][51][52][53][54][55]: u(x, θ, z, t) = u 0 (x, θ, t) + zφ x (x, θ, t) v(x, θ, z, t) = v 0 (x, θ, t) + zφ θ (x, θ, t) w(x, θ, z, t) = w 0 (x, θ, z, t) (1) where u 0 , v 0 , and w 0 are the displacements of the middle surface in the axial, circumferential and radial directions, φ x and φ θ are the transverse normal rations of the x and θ axis. t represents the Materials 2020, 13, 884 4 of 21 time variables. The relationship between the strains and curvature changes of the moderately thick composite laminates shell is defined as: where ε 0 xx , ε 0 θθ , and ε 0 xθ are the strains in the middle surface. χ xx , χ θθ and χ xθ denote the curvature changes. So, the relationship between the strain and displacement of the kth layer is shown: where Z k < z < Z k+1 . Related to the Hooke's law, the relationship between the strains and stresses is given as: where Q k ij (i, j = 1, 2, 4, 5, 6) are the elastic properties of the material. Through the transform matrix, the transformation stiffness matrix of the composite shell is determined as: where Q k ij (i, j = 1, 2, 4, 5, 6) are the transformation stiffness constants associated with the stresses and strains. For the orthotropic material, the constants can be given as: where E 1 and E 2 are Young's modulus of the kth layer in the principal directions. µ 12 and µ 21 are the Poisson's rations. Furthermore, the relationship of the Poisson's rations is governed by the equation µ 12 E 2 = µ 21 E 1 . G 12 , G 13 and G 23 are the rigidity modulus. For the isotropic material, the material relationship of coefficients is E = E 1 = E 2 , G = G 12 = E 1 /(2 + 2µ 12 ) and G 12 = G 13 = G 23 .
In Equation (5), T is the transformation matrix, which is obtained as: where m and n are the direction coefficients in the kth layer. m and n are defined as m = cos(β), n = sin(β) and β is the included angle. The integration of load-bearing stresses in the cross-section and in-plane applies a moment in the thickness direction, the force and moment resultants are shown as: where N is the amount of the layer. Submitting Equations (2)-(4) into Equation (8), the relationship between the force and moment resultants to the strains is obtained as [2,49]: where {N x , N θ , N xθ } are the normal and shear force resultants. {M x , M θ , M xθ } represent the bending and twisting moment resultants. {Q x , Q θ } denote the transverse shear force resultants. K c is the shear correction factor and is taken as 5/6 in this paper. According to [49], the shear correction factor is caused by the true transverse shear stress predicted based on the three-dimensional elastic theory. In Equation (9), A ij , B ij and D ij (i,j = 1,2,4,5,6) are the stretching stiffness coefficients, coupling stiffness coefficients and bending stiffness coefficients, which can be given as: For analysis of the certain cross-ply moderately thick composite laminated cylindrical shell, the coefficients A 16 = A 26 = B 16 =B 26 = D 16 = D 26 = 0.

Governing Equations
Based on the FSDST and Hamilton's principle, the governing equations of moderately thick composite laminated shell can be obtained as [2,49]: where {I 0 , I 1 , Materials 2020, 13, 884 6 of 21 in which ρ k is the density constant. By submitting Equations (2) and (9) into Equation (11), the governing equation of motion for the moderately thick cross-ply composite laminated cylindrical shell can be given as: where L ij (i, j = 1, 2, 3, 4, 5) are the coefficients, which can be obtained as:

Implementation of the WBM
For the general cross-ply moderately thick composite laminated cylindrical shell, the generalized displacements functions are set as in the wave function forms: U n e ik n x cos(nθ)e −iωt V n e ik n x sin(nθ)e −iωt W n e ik n x cos(nθ)e −iωt Φ xn e ik n x cos(nθ)e −iωt Φ θn e ik n x sin(nθ)e −iωt where k n is the characteristics wave number in the axial directions. U n , V n , W n , Φ xn , Φ θn are the displacement amplitudes that are associated with the circumferential mode number n. ω is the circular frequency and t is the time variable. Submitting Equation (14) into Equation (13), the governing equations are: where T ij (i, j = 1, 2, 3, 4, 5) is the coefficient elements of the matrix T which can be shown as: To ensure the equation has a non-trivial solution, it is necessary to eliminate the determinant of the coefficient matrix T. So, the governing equation of the axial wave number k n can be reduced as a tenth order polynomial equation, which can be shown as: Equation (17) is a fifth-order equation of k 2 n and b 10 , b 8 , b 6 , b 4 , b 2 and b 0 are the coefficients which are determined by the coefficient matrix T. The detailed expression of the coefficients is too complex and it is not at the core of the theoretical part of this article. So, the authors ignored it to make the paper leaner. The roots of the equation are solved with ten characteristics roots, ±k n,1 , ±k n,2 , ±k n,3 , ±k n,4 , ±k n,5 . Based on the characteristics roots, there is one set of basic solution resultants {ξ n,i , η n,i , 1, χ n,i , ψ n,i } T for the corresponding characteristics wave number ±k n,i (i = 1-5), which are defined as: where ∆, ∆ i (i = 1, 2, 4, 5) are given as: So, the generalized displacement functions can be transformed as: Materials 2020, 13, 884 , sin(nθ), cos(nθ), cos(nθ), sin(nθ)} is the modal matrix in the circumferential direction. P n (x) = diag {exp(jk n,1 ), exp(jk n,2 ), . . . , exp(jk n,ns )} is the wave number matrix and n s is the number of the characteristics roots of Equation (17) and the value of it is 10. W n = {W n,1 , W n,2 , . . . , W n,ns } T is the wave contribution factor resultant. D n is the displacement coefficient matrix, which can be shown as: ξ n,1 ξ n,2 · · · ξ n,n s −1 ξ n,n s η n,1 η n,2 · · · η n,n s −1 η n,n s 1 1 · · · 1 1 χ n,1 χ n,2 · · · χ n,n s −1 χ n,n s ψ n,1 ψ n,2 · · · ψ n,n s −1 ψ n,n s The generalized force and moment resultant (9) and (20) as: where F n is the force and moment coefficient matrix and the elements F n,ji (j = 1-5, i = 1-ns) are shown as: For the classical boundary conditions, some boundary conditions are introduced as: Free edge (F): Clamped edge (C): Simply-supported edge (SS): Shear-diaphragm edge (SD): Also, the elastic boundary conditions can be given in some forms as: when the elastic restrained with the stiffness constant K u in the axial direction, the corresponding boundary equation can be shown as: u : where K v , K w , K φx , K φθ are the corresponding stiffness constants in different displacements. For the combination of elastic boundary conditions, the boundary equations can refer to Equation (28).
The total matrix K of the whole structure depends on the generalized displacement resultants, force resultants and boundary conditions. The expression of the total matrix K is: where D n and F n are the displacement and force coefficient matrix; P n is the wave number matrix and the positions are set as x = 0 and x = L. B 1 (x) and B 2 (x) are the boundary matrix which is related to the boundary conditions. For the classical boundary conditions, the boundary matrix B 1 (x) and B 2 (x) are set as: where T δ and T f are the transform matrices of the boundary matrix and the detailed expression of the transform vectors are: Free edge (F): Clamped edge (C): Simply-supported edge (SS): Shear-diaphragm edge (SD): For the elastic boundary conditions, the boundary matrix B 1 (x) and B 2 (x) are given as: where K δ is the stiffness transform matrix and the detailed expression is: when the elastic restrained with the stiffness constant K u in the axial direction, the stiffness transform matrix is given as: When the other directions are under elastic restrained, the stiffness matrices K δ are given with different stiffness constants as: v : When the composite shell is under the combination of elastic restrained, the boundary matrix B 1 (x) and B 2 (x) can refer to the Equations (36) and (37). To calculate the natural frequencies, the external force resultant F should vanish, and by searching the zero position of the total matrix K using the dichotomy method. In each of the circumferential mode numbers n, a series of determinant values of the total matrix K are calculated. The value of the experimental value is generated until the sign change occurs, and then the dichotomy method iteratively interpolates to locate the zero of the determinant.

Numerical Examples and Discussion
In this section, some examples are calculated to investigate the free vibration characteristics of the composite shell with classical, elastic, and their combination boundary conditions. Several numerical examples are accepted to verify the correctness of the present method.

Composite Laminated Cylindrical Shell with Classical Boundary Conditions
The composite shell under the classical boundary conditions is widely used in some engineering field applications and is also the focal point of many researchers. In this part, the dynamic analysis of this topic is analyzed.
First, in Table 1, the three layered [0 • /90 • /0 • ] composite shell under some classical boundary conditions is considered (i.e., F-F, S-S, C-C). The material properties and geometric parameters are given as: R = 1 m, L/R = 5, h/R = 0.05, E 2 = 1 GPa, E 1 /E 2 = 25, µ 12 = 0.25, G 12 = 0.5E 2 , G 13 = 0.5E 2 , G 23 = 0.2E 2 , ρ = 1700 kg/m 3 . The comparison of the frequency parameter Ω = ωL 2 ρ/E 2 /h is studied. The first four circumferential wave numbers (i.e., n = 1, 2, 3, 4) and the first longitudinal mode (i.e., m = 1) are calculated. The frequency parameters are compared with the results by Messia and Soldatos [56] and Jin et al. [57], from Table 1, the differences between the results by the present method and reported literatures are small, the maximum error is 3.01%. The differences are caused by different solution program methods. Furthermore, in each circumferential wave number, the maximum frequency parameters are under the boundary condition C-C, especially, when n = 1, the maximum frequency parameter is fixed under the boundary condition F-F. The reason is that the boundary conditions have a significant effect on the frequency parameters. In order to further investigate the free vibration characteristics of composite laminated cylindrical shells with arbitrary boundary conditions, some mode shapes (n, m) of the composite laminated cylindrical shell are shown in Figure 2. conditions, some mode shapes (n, m) of the composite laminated cylindrical shell are shown in Figure 2.    The numerical examples in the previous studies considered the thin composite shell with various classical boundary conditions. To verify the correctness of the present method, more numerical examples are considered. In Table 2, the fundamental frequency parameter  Table 2, the results of the present method agree well with the results in the literatures, the small differences are related to different shell theory and numerical methods. For solving the vibration characteristics of the moderately thick composite laminated cylindrical shell, the vibration characteristics of the whole system can be solved by the elastic equation: (K−ω 2 × M) = 0, where K is the stiffness matrix for the shallow shell and M is the mass matrix, ω is the natural frequency for the moderately thick composite laminated cylindrical shell. Different boundary conditions cause the stiffness matrix to change. For the simply-supported (S-S) boundary condition, the determinant of the stiffness matrix becomes smaller compared to the clamped (C-C) boundary condition, and when the mass matrix remains unchanged, the natural frequency decreases. When the length to radius value changes from 1 to 2, the length quadratic variable in the frequency parameter 2 2 /100 L E h ω ρ Ω = will be four times larger, and the frequency parameters are also increased. So, the effect of the length to radius ratios on the free vibration characteristics cannot be expressed. The numerical examples in the previous studies considered the thin composite shell with various classical boundary conditions. To verify the correctness of the present method, more numerical examples are considered. In Table 2, the fundamental frequency parameter Ω = ωL 2 ρ/E 2 /100h of the moderately thick composite shell with the different length to radius ratios under four types of classical boundary conditions (i.e., S-S, S-C, C-C, C-F) are shown. There are two types of cross-ply laminated schemes (i.e., [0 • /90 • ] and [0 • /90 • /0 • ]) and two kinds of length to radius ratios (i.e., L/R = 1, 2) are discussed. The results of the present method are compared with the results by Khdeir et al. [58], Thinh and Nguyen [59] and Jin et al. [57]. The geometric and material parameters are given as: R = 1 m, h/R = 0.2, E 2 = 1 GPa, E 1 /E 2 = 40, µ 12 = 0.25, G 12 = 0.6E 2 , G 13 = 0.5E 2 , G 13 = 0.5E 2 , ρ = 1600 kg/m 3 . From Table 2, the results of the present method agree well with the results in the literatures, the small differences are related to different shell theory and numerical methods. For solving the vibration characteristics of the moderately thick composite laminated cylindrical shell, the vibration characteristics of the whole system can be solved by the elastic equation: (K−ω 2 × M) = 0, where K is the stiffness matrix for the shallow shell and M is the mass matrix, ω is the natural frequency for the moderately thick composite laminated cylindrical shell. Different boundary conditions cause the stiffness matrix to change. For the simply-supported (S-S) boundary condition, the determinant of the stiffness matrix becomes smaller compared to the clamped (C-C) boundary condition, and when the mass matrix remains unchanged, the natural frequency decreases. When the length to radius value changes from 1 to 2, the length quadratic variable in the frequency parameter Ω = ωL 2 ρ/E 2 /100h will be four times larger, and the frequency parameters are also increased. So, the effect of the length to radius ratios on the free vibration characteristics cannot be expressed. Table 2. Frequency parameters Ω = ωL 2 ρ/E 2 /100h for two types of cross-ply composite laminated cylindrical shell with different length to radius ratios and boundary conditions (R = 1 m, h/R = 0.2, E 2 = 1 GPa, E 1 /E 2 = 40, µ 12 = 0.25, G 12 = 0.6E 2, G 13 = 0.5E 2 , G 13 = 0.5E 2 , ρ = 1600 kg/m 3 ). Next, the effect of thickness to radius ratios on the frequency parameter is considered, the boundary condition is set as simply-supported. Two types of cross-ply laminated schemes (i.e., [0 • /90 • /90 • /0 • ] and [0 • /90 • /90 • /0 • ]) and three kinds of thickness to radius ratios (i.e., h/R = 0.1, 0.2, 0.3) are discussed. The material parameters and geometric constants are same as the previous example, the ratio of length to radius is given as L/R = 1. The frequency parameters of the three lowest natural frequencies Ω = ωh ρ/G 12 /π are compared with the results in the literature that were investigated by Thinh [59] and Jin et al. [57]. From Table 3, the differences between the results of the present method and other results in the literature are small, and the differences are related to a variety of numerical methods and shell theories. Table 3. Frequency parameters Ω = ωh ρ/G 12 /π for two types of cross-ply composite laminated cylindrical shells with different thickness to radius ratios under simply-supported boundary conditions For analysis of the effect of length to radius ratios and thickness to radius ratios, one type of three-layered cross-ply [0 • /90 • /0 • ] composite laminated cylindrical shell with simply-supported and clamped boundary conditions is considered. The first longitudinal modal (i.e., m = 1) frequency parameter Ω = ωR ρ/E 2 is calculated for different circumferential numbers (i.e., n = 1, 2, 3) with various thickness to radius ratios (i.e., h/R = 0.05-0.1), and length to radius ratios (i.e., L/R = 1-4) are calculated in Tables 4 and 5. The material properties are given as: E 2 = 2 GPa, E 1 /E 2 = 25, µ 12 = 0.25, G 12 = 0.5E 2 , G 13 = 0.5E 2 , G 23 = 0.2E 2 , ρ = 1600 kg/m 3 . When studying the effect of the length to radius ratios, keeping material parameters and radius constant, the frequency parameters are only related to the natural frequency of the moderately thick composite laminated cylindrical shell. It can be seen from Tables 4 and 5, with the growth of the length to the radius ratios L/R, the frequency parameter is generally decreased. Furthermore, the frequency parameter generally grows with the thickness to radius ratio increase. So, the effects of length to radius ratio and thickness to radius ratio are different from the frequency parameter of the moderately thick composite laminated cylindrical shell with simply-supported and clamped boundary conditions.

Composite Laminated Cylindrical Shell with Elastic Boundary Conditions
It is necessary and significant to study the vibration analysis of the composite laminated cylindrical shell under elastic restrained. Through the introducing of the elastic boundary conditions, the stiffness transform matrix is established by different elastic boundary conditions, in this paper, four types of typical elastic boundary conditions are considered: Type 1 (EC1): axial displacement is under elastic restrained and the corresponding stiffness transform matrix K δ is given as: Type 2 (EC2): circumferential displacement is under elastic restrained and the corresponding stiffness transform matrix K δ is given as: K v = 10 7 , K δ = diag 0, 10 7 , 0, 0, 0 .

Layer-Type
n SD-SD S-S C-C    Next, the effect of the stiffness constants is investigated. A three-layered cross-ply [90 • /0 • /90 • ] composite shell with complicated elastic boundary conditions is considered. The composite shell is under elastic restrained with one kind of spring stiffness in each displacement direction at one end; on the other end, the composite shell is under the simply-supported boundary condition. The first longitudinal mode (i.e., m = 1) frequency parameter Ω = ωL 2 ρ/E 2 /h is calculated for various circumferential numbers (i.e., n = 1, 2, 3, 4) with different elastic restrained K u , K v , K w , K φx , K φθ , which are calculated with various stiffness constants (i.e., 0-10 12 ). The material parameters and geometric properties are given as: Table 8, the frequency parameters are almost all in one certain value when the composite shell is only restrained by the rotation spring K φx and K φθ . When the composite shell is only restrained by the circumferential K v and radial spring K w , the frequency parameters generally increase with the changing of the stiffness constant. When the composite shell is only restrained by the axial spring K u , the frequency parameters have smaller growth with the increasing of the stiffness constants. It can be founded that the effect of circumferential spring K v and radial spring K w are more obvious than the other direction springs. When the circumferential wave number n = 1, the increase of the frequency parameters is larger than n = 2, 3. So, when the composite shell is under the S-elastic boundary condition, the effects of circumferential K v and radial spring K w are more obvious than the other direction springs. Table 8. The frequency parameters Ω = ωL 2 ρ/E 2 /h for a three-layered cross-ply [0 • /90 • /0 • ] composite laminated cylindrical shell with S-elastic boundary conditions, one displacement is under elastic restrained and others are free (L/R = 4, h/R = 0.1, E 2 = 2 GPa, E 1 /E 2 = 25, µ 12 = 0.25, G 12 = 0.5E 2 , G 13 = 0.5E 2 , G 23 = 0.2E 2 , ρ = 1500 kg/m 3 ).  Furthermore, the composite shell is considered under the S-elastic boundary condition in which only one displacement is under elastic restrained and other displacements are fixed. The frequency parameter, material constants and geometric properties are the same as the previous example. In Table 8, the frequency parameter Ω = ωL 2 ρ/E 2 /h is calculated. The expression of boundary matrix B 1 (x) and B 2 (x) are reduced as:

Spring Stiffness
For different elastic boundary conditions, the corresponding stiffness transform matrices K δ are given as: In Table 9, the frequency parameters with different elastic restrained stiffness constants are calculated. It is obvious that with the changing of the stiffness constants from 0 to 10 12 , the frequency parameters are almost unchanged and remain in a certain range. So the effect of the elastic restrained stiffness constants for the S-elastic boundary condition, which is set as one displacement restrained and others are fixed of the composite shell, are small and the frequency parameters are almost all remaining in a stable range. So, for various elastic boundary condition combinations, the effects of the elastic spring restrained on the free vibration characteristics of moderately thick composite laminated cylindrical shells are different. In some cases, the effect of the elastic restrained springs is obvious. Also, the effect of the elastic restrained spring is not obvious in some numerical cases. Table 9. The frequency parameters Ω = ωL 2 ρ/E 2 /h for a three-layered cross-ply [0 • /90 • /0 • ] composite laminated cylindrical shell with S-elastic boundary conditions, one displacement is under elastic restrained and others are free (L/R = 4, h/R = 0.1, E 2 = 2 GPa, E 1 /E 2 = 25, µ 12 = 0.25, G 12 = 0.5E 2 , G 13 = 0.5E 2 , G 23 = 0.2E 2 , ρ = 1500 kg/m 3 ).

Conclusions
The wave base method is conducted to analyze the free vibration characteristics of moderately thick composite laminated cylindrical shells with arbitrary classical and elastic boundary conditions. According to the first-order shear deformation shell theory and Hamilton principle, the governing equation of the composite laminated shell is established. The displacement variables are transformed into wave function forms. Related to different boundary conditions, the boundary matrices are obtained to establish the total matrix. The natural frequencies are solved by the dichotomy method to experiment with the zero location of the total matrix determinant. For the wave based method, the advantage is that the boundary conditions are easy to replace. If the boundary conditions need to be changed, only the boundary condition matrix B 1 and B 2 need to be changed, including classical boundaries, elastic boundaries and their combined forms. To analyze the free vibration characteristics of moderately thick composite laminated shells, the solutions are easy to obtain in the wave function forms, and the shell structure does not need to be divided into shell segments. For the free vibration characteristics of the moderately thick composite laminated cylindrical shell with arbitrary boundary conditions, the solutions by the present method have better precision than the results in some reported literatures. Furthermore, some numerical examples are shown and the conclusions follow as: First, the frequency parameters of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions are calculated. Through the comparison of the results, it can be seen that the method proposed in this paper is more accurate for the calculation of the shell.
Second, the effect of the geometric constants, such as length to radius ratios and thickness to thickness ratios, on the frequency parameters are discussed. It is seen that different geometric constants have various effects on the frequency parameters.
Third, the influence of the boundary elastic restrained stiffness constants on the natural frequency parameters is discussed. The changing ranges of the elastic restrained stiffness constants in various directions are from 0-10 12 . From the variations of the natural frequency parameters, it can be concluded that the effect of the elastic restrained stiffness on the natural frequency parameters is not obvious. With the growth of the stiffness constants in various directions, the natural frequencies have a small range of fluctuations and are basically stable within a range.