#### Abstract

This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.

#### 1. Introduction

The functionally graded (FG) spherical shells have been widely used in many engineering fields such as marine and aviation due to their unique mechanical properties. These structures are usually taking a variety of excitations which leads to structure vibration or even damage in the course of usage. Therefore, the investigations of dynamic features of FG spherical shell structures are meaningful, and it is really necessary to establish a unified method for vibration solutions of FG spherical shell structures to improve applications in engineering.

The vibration analysis of various shell structures has obtained a great deal of attention with many researchers in the past few decades; the separation of variables method, classical finite element method, generalized differential quadrature (GDQ) methods, etc., are widely applied in prediction of the vibration behaviors for spherical shells. Polyakov et al. [1] and Zaera et al. [2] investigated the free vibration characteristics of a closed spherical shell by the separation of variables method. At the same time, Hosseini-Hashemi and Fadaee [3] solved the governing equations of the vibrated spherical shell panel by introducing the new auxiliary and potential functions in a similar way. Based on a combination of thin shell theory and classical finite element method, Menaa and Lakis [4] carried out free vibration equations using the hybrid finite element formulation. According to higher-order shear deformation theory (HSDT), Ram and Babu [5] carried out the vibration characteristics of composite spherical shell cap by using the finite element method (FEM), in which nine degrees of freedom at each node is considered. In order to deduce the vibration behaviors of thin spherical shell subject to large amplitude transverse displacement, Thomas et al. [6, 7] assessed the validity range of the approximations by comparing the analytical modal with a numerical solution. Tornabene et al. [8] analyzed the free vibration of spherical shell by comparing the results between FEM, GDQ methods, and exact three-dimensional solution. Li et al. [9–11] applied the improved Fourier series method to solve the accurate solution for the vibration behaviors of Mindlin plates, FG Reissner–Mindlin plate, and sector plates with general boundary conditions. In addition to aforementioned solutions, Buchanan and Rich et al. [12] formulated a nine-node Lagrangian finite element in spherical coordinates to analyze the vibration of thick isotropic spherical shells. The Green quasifunction method is clarified by Li and Yuan [13], in which the free vibration problem of simply supported trapezoidal thin spherical shell is considered. According to Flugge's thin shell theory, Pang et al. [14] used the Ritz Method to present free and forced vibration analysis of airtight cylindrical vessels consisting of elliptical, paraboloidal, and cylindrical shells. Panda and Singh [15, 16] applied a direct iterative method to investigate nonlinear free vibration behavior of laminated composite spherical shell panel. More detailed descriptions with respect to free vibration of related structure can be found in several studies [17–21].

In the field of FG spherical shells, based on first-order shear deformation theory (FSDT), Li et al. [22] presented a unified approach to analyze free vibration of FG shell structures with general end restraints by the Ritz method. Kiani [23] dealt with vibration characteristics of carbon nanotube-reinforced composite spherical panels according to Hamilton's principle and Ritz method. Li et al. [24] carried out the free vibration of four-parameter FG moderately thick spherical panels with general boundary conditions in modified Fourier–Ritz approach. Pang et al. [25] proposed a unified vibration analysis approach on the basis of variational operation for FG shell of revolution with general boundary conditions. The Haar wavelet discretization (HWD) method-based solution was presented by Xie et al. [26] to investigate free vibration of FG spherical shells with arbitrary boundary conditions. Tornabene and Viola [27] focused on the dynamic behavior of moderately thick FG shells of revolution, in which the GDQ method has been used to discretize the system equations. Fantuzzi et al. [28] presented typical 2D finite elements and advanced GDQ numerical approaches for free vibration investigation of simply supported FG Spherical shells. Qu et al. [29] proposed a general formulation to carry out free vibration of FG shells of revolution subject to arbitrary boundary conditions according to the modified variational principle. Neves et al. [30] investigated free vibration of FG spherical shells by radial basis functions collocation according to HSDT. By using the potential functions and new auxiliary, Fadaee et al. [31] solved the exact frequencies of FG spherical shell panels on the basis of strain-displacement relations of Donnell and Sanders theories, as well as compared with the existing literature and FEM. Considering shear deformation theory and geometric nonlinearity of model, Ganapathi [32] studied the dynamic stability characteristics of FG spherical shell subject to external pressure by means of modified Newton–Raphson iteration scheme.

A review of literature reveals that the free vibration behaviors of spherical shells have been well investigated. However, the published references focused on vibration characteristics of FG spherical shells structures subject to general boundary restraints are limited. Especially, the displacement functions are inconvenient to choose, and little study has been compared with experimental results. Thus, it is of great significance to establish a unified method to analyze free vibration characteristics of FG spherical shell structures with general boundary restraints.

#### 2. Theoretical Formulations

##### 2.1. Description of the FG Spherical Shell

The model of FG spherical shell is described in Figure 1. *h* is the thickness of the structure. *R*_{0} and *R*_{1}, respectively, denote the radius on the top and bottom of FG spherical shell. *C*_{s} represents the center of the FG spherical shell. *φ*_{0} and *φ*_{1} are the control variables of structural scale, which represent the center angle corresponding to *R*_{0} and *R*_{1}. *R* denotes radius of the FG spherical shell. It is obvious that the spherical shell can be described by coordinate system (*φ*, *θ*, *z*), in which *φ*, *θ*, and *z*, respectively, denote the axial, circumferential, and normal directions. The displacements in the direction of *φ*, *θ*, and *z* are regarded as *u*, , and , respectively. The displacement components of FG spherical shell are defined as *U*, *V*, and *W*. In order to increase the accuracy of high-order responses, it can be seen from Figure 1 that the FG spherical shells are divided into *H*_{ζ} segments along axial direction, and each segment is connected by springs to ensure continuity. Figure 2 describes the partial view of FG spherical shell with boundaries, in which two group of rotational springs (*k*_{φ}, *k*_{θ}) and three groups of linear springs (*k*_{u}, *k*_{v}, *k*_{w}) are restrained at every segment to simulate the given boundary restraints.

Two types of the FG model are considered in the present study, according to the literature [33, 34]. Young’s modulus (*E*), mass density (*ρ*), and Poisson’s ratios (*µ*) of the FG model are, respectively, expressed aswhere the subscripts *c* and *m* are the ceramic and metallic constituents, respectively, and *V*_{c} is the volume fraction which follows two general four-parameter power-law distributions:where *z* and *p* represent the thickness coordinate and the power-law exponent that takes only positive values. The variation profile of the functionally graded shell thickness can be determined by parameters *a*, *b*, and *c*. Figure 3 shows the variations of the volume fraction through the structure thickness for different values of power-law exponent *p*. It is easy to see that the isotropic material can be achieved as a particular case of functionally graded material when the value of *p* levels off to zero or infinity, and the volume fraction of all constituent materials should satisfy the sum of one. In addition, it should be noted that the distributions of two grading laws are mirror reflections. At the same time, the different power-law distributions can be obtained by setting different value of parameters *a*, *b*, *c*, and *p*.

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##### 2.2. Energy Functional of Each Shell Component

On the basis of FSDT, the displacement fields of the *i*th segment of FG spherical shell are expressed as follows [35]:where , , and , respectively, represent the middle surface displacements of the FG shell in *φ*, *θ*, and *z* directions and the rotation of transverse normal about *θ*- and *φ*-axis is represented by and , accordingly.

The strains of the FG shell at any point can be defined aswhere , , and denote the membrane strains; and are the transverse shear strains; and , , and indicate the curvature changes, which are expressed aswhere the parameters *A* and *B* are the Lamé parameters. Actually, the above symbols can be represented as follows:

On the basis of general Hooke’s law, the corresponding stresses can be achieved as follows:where the normal stresses are represented by and ; the shear stresses are , ; and . *Q*_{ij}(*z*) (*i*, *j* = 1, 2, 6) are the elastic constants, where the values are defined as

By integrating the stresses on the cross section, the force and moment resultants can be expressed as follows:where is the shear correction factor and the value is assumed as = *5*/6 in the study. *A*_{ij}, *B*_{ij}, and *D*_{ij}, respectively, represent extensional, extensional-bending coupling, and bending stiffness, which are shown below:

The strain energy of the *i*th segment of FG spherical shells can be obtained as follows:

By substituting equations (5a)–(5h) and (9a)–(9c) into equation (11), the strain energy expression of the *i*th segment can be expressed by middle surface displacements and rotations. For the sake of description, the strain energy expression can be written as , where , , and are clarified by *A*_{ij}, *D*_{ij}, and *B*_{ij} according to equation (10), accordingly.

The maximum kinetic energy of the select segment can be given aswhere the dot on the symbols represents differentiation about time.

The boundary coupling spring technique is applied to simulate the general boundary restraints and continuity constraints on segment interfaces in the study. The energy in two sides of the boundary springs can be expressed aswhere and represent the spring stiffness value at boundary margin of FG spherical shell, respectively. The energy in connective springs with regard to neighbor segments can be shown as follows:where , , , , and represent the stiffness of springs, respectively.

The total energy of the constraint conditions can be shown in the following equation:

##### 2.3. Admissible Displacements and Solution Procedure

The admissible displacement functions are key factors for the accuracy of solution [36–39], which are inconvenient to select as a result of the continuity and boundary conditions in previous research studies. However, in this work, the penalty method of spring stiffness technique makes displacement functions convenient to choose when conducting the edge restraints and continuity relations of FG spherical shell.

From the literatures reviewed, it is clear that classical Jacobi polynomials [40] valued in the range of , and typical Jacobi polynomial recursion formulas of degree are shown as follows:where and

The displacement functions of segments are generalized by Jacobi polynomial, which are shown as follows:where , , , , and represent the Jacobi expansion coefficients and *n* and *m* denote the semi-wave number in axial and circumferential direction, respectively. *M*_{ζ} is the highest degree of semi-wave number *m*. The ultimate Lagrangian energy functions *L* can be obtained as equation (19a)–(19e):

Minimizing the Lagrangian energy functional in regard to undetermined coefficients by using the Ritz method,

By substituting equations (9a)–(9c), (11), (13), (17), (19a)–(19e), and (20) into equation (21), the matrix form can be summarized aswhere *K* and *M* of equation (22) represent stiffness matrix and mass matrix. And the matrix of symbol *Q* denotes unknown Jacobi coefficients. The frequencies and eigenvectors of FG spherical shells can be obtained by solving equation (22).

#### 3. Numerical Results and Discussion

In the present study, the general boundary restraints clamped, free, shear support, shear diaphragm and elastic edge are represented by C, F, SS, SD, and Ei (*i* = 1, 2, 3), respectively. Unless otherwise specified, material properties, geometrical dimensions, and the porosity coefficient are chosen as: , , , , , *M*_{ζ} = 8, *α* = 0, *β* = −0.5, *H*_{ζ} = 5; *R* = 1 m, *φ*_{0} = 0, *φ*_{1} = *π*/2, and *h* = 0.04 m. The nondimensional frequency is expressed as .

##### 3.1. Convergence Study

The main purpose of this section is to study the convergence of the proposed method. The continuity and boundary conditions are dependent on the parameters of boundary springs. In addition, the accuracy of algorithm is decided by the number of shell segments and Jacobi parameters. Therefore, it is very indispensable to investigate the convergence of above parameters.

Figure 4 displays the frequency parameter of FG spherical shell with diverse boundary parameters. It is remarkable that the boundary condition is changed from free to clamped case with the spring value in the range of 10^{−10}*E*_{c}∼10^{10}*E*_{c}*.* It is obvious that the frequency parameter increases rapidly with the increase of spring value varying from 10^{−3}*E*_{c}∼10^{1}*E*_{c}. It can be known that the frequency parameter tends to be stable when the values of stiffness are in range of 10^{2}*E*_{c}∼10^{10}*E*_{c}*.* That is to say the spring stiffness can be assigned within the range of 10^{2}*E*_{c}∼10^{10}*E*_{c} for clamped boundary condition, and the spring values apparently tend to zero for free boundary condition. In addition, all the general edge restraints of FG spherical shell used in the study are shown in Table 1.

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Figure 5 shows the frequency parameter Ω of FG spherical shell with different number of segments. It is clear that there is little influence on the accuracy of algorithm when the number of segments is no less than 3; in other words, it is easy to get great convergence when *H*_{ζ} is more than 2.

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When the Jacobi parameters *α* and *β* are different, the relative percentage error results of FG spherical shell are exhibited in Figure 6. The results of *α* = 0 and *β* = 0 are selected as the reference values. It is clear that the calculation results change little regardless of the value of Jacobi parameters *α* and *β* in the present method when *n* is a definite value. The maximum percentage error is less than 5 × 10^{−7}. That is to say the choice of polynomial can be various in Jacobi systems.

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Figure 7 illustrates the variation of frequency parameter of FG spherical shell with highest degree of Jacobi polynomial. The stable results can be obtained with the increase in the value of *M*_{ζ}. In other words, the convergence on vibration characteristics of FG spherical shell will be guaranteed when *M*_{ζ} is no less than 4.

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##### 3.2. Free Vibration Behavior of FG Spherical Shell Structure

Table 2 exhibits the comparison of frequency for FG spherical shell with those mentioned in the published literature [29]; it should be noted that the results are in great agreement. That is to say the present method has accurate precision to solve the vibration behaviors of FG spherical shell.

Table 3 displays the frequency parameter Ω of FG spherical shell subject to general edge restraints, and all the results made a comparison with references mentioned above. It is clear that the results have great accuracy by the current method. Thus, the conclusion can be drawn that the proposed approach is accurate to analyze the free vibration characteristics of FG spherical shell structure with general boundary conditions from the comparative study.

The experiment about free vibration of isotropic spherical shell structure was conducted to verify the feasibility of present method; the material properties are selected as *E* = 210 GPa, *ρ* = 7850 kg/m^{3}, *ν* = 0.3, *R* = 0.06 m, and *h* = 0.005 m. The modal test was carried out by loading on multiple excitation points; meanwhile, the responding signals were collected from a single point. Table 4 displays the comparison of natural frequencies which are obtained by the present method, experiment, and FEM for the isotropic spherical shell. It is easy to find that the results achieved by three different approaches are almost infallible, and the maximum error is less than 1%. Figure 8 displays the test system and model. In order to better understand the vibration characteristics of spherical shell, the mode shapes obtained by the present method, FEM, and experiment are shown in Figure 9. It is obvious that the same mode shapes of the present method have great agreement with FEM and experiment.

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Table 5 and Figure 10 show the frequency parameters of FG spherical shell with general boundary conditions. It is clear to find that the boundary conditions have an obvious influence on the vibration characteristics of FG spherical shell, and the values of type FGM_{II} are usually less than type FGM_{I}.

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The frequency parameters of FG spherical shell in regard to *h*/*R* ratio are exhibited in Table 6 and Figure 11, in which four different kinds of *h*/*R* are included. It should be noted that the frequency parameters tend to increase with increasing *h*/*R*, and the results can be apparently affected by boundary conditions.

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#### 4. Conclusions

This paper introduces a unified semianalytical method for the free vibration solutions of FG spherical shell with general edge restraints on the basis of first-order shear deformation theory. The material varies uniformly along the thickness of FG spherical shell, and two types of FG model are considered in present study. The unified displacement functions are simulated by Fourier series and Jacobi polynomial. In addition, the penalty method of spring stiffness technique is applied to obtain the general boundary restraints of FG spherical shell. The vibration solutions of FG spherical shell can be obtained according to the Rayleigh–Ritz method. The unified Jacobi polynomials make the displacement functions easier to choose, which is the most important discovery of the present method. The numbers of shell segments, stiffness of springs, and Jacobi parameters have been studied to investigate the convergence. In addition, the accuracy and feasibility of current method have been verified by comparing the results with the published literatures, experiment, and FEM. For characteristic analysis, the effects of geometric dimensions and boundary conditions of FG spherical shell have also been investigated. The results of the paper can provide the reference data for future research studies.

#### Appendix

The generalized mass and stiffness matrices of FG spherical shell structure used in equation (22) are given as

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This study was funded by the National Natural Science Foundation of China (51209052), National key Research and Development Program (2016YFC0303406), Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (HEUGIP201801), Fundamental Research Funds for the Central University (HEUCFD1515), Assembly Advanced Research Fund of China (6140210020105), and Naval Preresearch Project, China Postdoctoral Science Foundation (2014M552661).