Research Article  Open Access
Lijie Li, Haichao Li, Fuzhen Pang, Xueren Wang, Yuan Du, Shuo Li, "The Modified FourierRitz Approach for the Free Vibration of Functionally Graded Cylindrical, Conical, Spherical Panels and Shells of Revolution with General Boundary Condition", Mathematical Problems in Engineering, vol. 2017, Article ID 9183924, 32 pages, 2017. https://doi.org/10.1155/2017/9183924
The Modified FourierRitz Approach for the Free Vibration of Functionally Graded Cylindrical, Conical, Spherical Panels and Shells of Revolution with General Boundary Condition
Abstract
The aim of this paper is to extend the modified FourierRitz approach to evaluate the free vibration of fourparameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The firstorder shear deformation theory is employed to formulate the theoretical model. In the modified FourierRitz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of twodimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.
1. Introduction
As the based structure elements, functionally graded cylindrical, conical, spherical panels and shells have a wide application in various engineering constructions, such as aircraft, space vehicles, and structures of military industries. Therefore, the knowledge of vibration characteristics of the functionally graded cylindrical, conical, spherical panels and shells is of particular importance for the predesign of the engineering structures.
So far, a huge amount of researches focused on the vibration analysis of functionally graded cylindrical, conical, spherical panels and shells by using different methods and shell theories. Tornabene and Viola [1–3] studied the free vibration of the moderately thick fourparameter functionally graded conical, cylindrical shells and annular plates under some classical boundary conditions by means of the generalized differential quadrature method on the basis of the firstorder shear deformation theory. Zhao and Liew [4, 5] presented the elementfree kpRitz method to investigate the free vibration analysis of mechanical and thermal buckling of functionally graded conical shell panels with classical boundary conditions. Aragh and Hedayati [6] dealt with the free vibration and static response of a twodimensional functionally graded (2D FGM) metal/ceramic open cylindrical shell with classical boundary conditions by using 2D generalized differential quadrature method. Su et al. [7–9] applied the modified Fourier series and Rayleigh–Ritz method to analyze the free vibrations of functionally graded open and closed shells including cylindrical, conical, and spherical ones with general boundary conditions based on firstorder shear deformation theory. Sofiyev and Kuruoglu [10–13] presented a theoretical approach on the basis of the Galerkin method to solve vibration problems of functionally graded (FG) truncated and complete conical shells under mixed classical boundary conditions and resting on elastic foundations. Kim [14] used an analytical method to investigate the free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge. Jin et al. [15, 16] presented a Haar Wavelet Discretization (HWD) methodbased solution approach for the free vibration analysis of functionally graded (FG) cylindrical, spherical, and parabolic shells of revolution with arbitrary boundary conditions. HosseiniHashemi et al. [17] used the state space approach to investigate the free vibration of Levytype thick functionally graded (FG) circular cylindrical shell panels to identify the validity range of two common shell theories namely Donnell and Sanders theories. Iqbal et al. [18] applied the wave propagation approach to study the vibration characteristics of functionally graded material circular cylindrical shells with classical boundary conditions. Shah et al. [19] employed the wave propagation method to study the vibrations of functionally graded cylindrical shells based on the Winkler and Pasternak foundations. Loy et al. [20, 21] studied the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel by means of the Rayleigh–Ritz method under some selected boundary conditions. Najafizadeh and Isvandzibaei [22] used the Rayleigh–Ritz method and third order shear deformation shell theory to analyze the free vibrations of the thin cylindrical shells under ring supports and classical boundary conditions. Naeem et al. [23] applied the Ritz formulation to study the vibration frequencies of functionally graded circular cylindrical shells with classical boundary conditions. Taghvaeipour et al. [24] presented a new finite element formulation to study the structural analysis of functionally graded hollow cylinders. Pradyumna and Bandyopadhyay [25] employed the finite element method to investigate the free vibration analysis of functionally graded curved panels including the cylindrical and spherical panels based on the higherorder shear deformation theory. Neves et al. [26] dealt with free vibration problems of functionally graded shells with classical boundary conditions based on the higherorder shear deformation theory and the radial basis functions collocation. Qu et al. [27] described a general formulation for free, steadystate, and transient vibration analyses of functionally graded shells of revolution subjected to arbitrary boundary conditions by means of a modified variational principle in conjunction with a multisegment partitioning procedure. Santos et al. [28, 29] developed a semianalytical axisymmetric finite element model using the 3D linear elastic theory to study the free vibrations of functionally graded cylindrical shells made up of isotropic properties. Vel [30] presented an exact elasticity solution for the free and forced vibration of functionally graded cylindrical shells with classical boundary conditions.
The above review indicates that there exist some literatures on the free vibration of a functionally graded cylindrical, conical, spherical panels and shells, and those numerical results are very useful for the practical project application. However, from the literatures review, we also know that the most existing numerical solutions for the title problem usually take account of a onefold computational model instead of unified computational model. For instance, Su et al. [7–9] divided the open and closed functionally graded cylindrical, conical, spherical shells to study and resort different admissible functions by means of the modified Fourier series technology. However, in practical project application, the structural forms are generally unknown. Thus, the establishment of a unified, efficient, and accurate computational model for free vibration analysis of functionally graded cylindrical, conical, spherical panels and shells with general boundary conditions is necessary and significant.
In previous study, an improved Fourier series method was proposed by Li [31, 32] to determine the vibration of a single beam under arbitrary boundary conditions. The method has been subsequently exploited to determine the vibration behaviors of structures [33–48]. Recently, a new method of FourierRitz approach was proposed by Jin et al. [49], based on the improved Fourier series method to analyze the free vibration of laminated functionally graded shallow shells with general boundary conditions. In contrast to most existing techniques, the modified FourierRitz method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints, and their combinations without the need of making any changes of the solution procedure. The purpose of this paper is to extend a modified FourierRitz approach to presents a unified computational model for the free vibration of fourparameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions on the basis of the firstorder shear deformation theory. Applying the modified FourierRitz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of twodimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solving the natural frequencies by means of the Ritz method. In order to obtain the unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution, as one merit of this paper, the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian, if a complete shell of revolution needs considering. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.
2. Theoretical Formulations
2.1. Description of the Model
As shown in Figure 1, the basic configuration of the problem considered here is an FG shell. An orthogonal curvilinear coordinate system composed of coordinates , , and is located on the reference surface of the shell. and denote the values of the principal radii of curvature of the reference surface along and axes, respectively. and denote the lengths in and directions, respectively. The thickness of shell element is represented by . As shown in Figure 2, the geometry of panels and shells considered in this paper is a surface of revolution with a cylindrical, conical, and spherical meridian. Based on Figure 2, the coordinate systems and quantities , , , and are given as follows: for cylindrical meridian, , , , , , and ; for conical meridian, , , , , , and ; for spherical meridian, , , , , and . The details of the geometric configuration of the FG panels and shells of revolution structures for verifying the accuracy and versatility of the proposed approach are shown in Figure 3.
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2.2. Kinematic Relations and Stress Resultants
According to the FSDT assumptions, the displacement components of an arbitrary point in the FGM panels and shells are expressed in terms of the displacements and rotations of the reference surface, as given as follows: where , , and denote the displacements of corresponding point on reference surface in the , , and directions, respectively. and are the rotations of the normal to the reference surface about the and direction, respectively, and is the time. Relationships between strains and displacements along the shell reference (middle) surface () are represented by the following:where the symbol and are the Lamé parameters. According to Figure 2, the Lamé parameters of the above types panels and shells can be defined as follows: for case of the cylindrical meridian, and ; for case of the conical meridian, and ; for case of the spherical meridian, and .
The constitutive equations related the force and moment resultants to strains and curvatures of the reference surface are given in the matrix form:where , , and are the inplane force resultants, , , and are moment resultants, and and are transverse shear force resultants. The shear correction factorκ is computed such that the strain energy due to transverse shear stresses in (3) equals the strain energy due to the true transverse stresses predicted by the threedimensional elasticity theory. In application, the shear correction factor does not have to be the same in different geometric directions and depended on many factors such as the geometric and material parameters [50]. However, the aim of in this paper is to study the vibration analysis of vibration analysis of functionally graded cylindrical, conical, spherical panels and shells with general boundary conditions. Thus, in order simplify this study and based on existing literature [40, 51], the shear correction factor selected a generic parameter in the next calculation. , , and ( and 6) are the extensional, extensionalbending coupling and bending stiffness, and they are, respectively, expressed aswhere the elastic constants are functions of thickness coordinate , which are defined as
Typically, the functionally graded materials are made of two or more constituent phases which have smooth and continuous mechanical behavior in one or more directions. In this paper, it is assumed that the FG panels and shells are made of a mixture consisting of ceramic and metal. Young’s modulus , density , and Poisson’s ratio are assumed to vary continuously through the panels thickness and can be expressed as a linear combination:in which the subscripts and represent the ceramic and metallic constituents, respectively. And the volume fraction follows the two general fourparameter powerlaw distributions [1]:where is the powerlaw exponent and takes only positive values, , , and are the material parameters, and their values determine the volume fraction through the general fourparameter powerlaw distributions. In addition, for (7), the difference between the two general fourparameter powerlaw distributions is decided by and . In order to get more intuitive description, Figure 4 shows the variations of the volume fraction for different distributions and values of the powerlaw exponent. Form Figure 4, we can know that the volume fraction with and is symmetric with reference to . The volume fraction of all the constituent materials should add up to one; that is,
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When the value of equals zero or infinity, the homogeneous isotropic material can be obtained as a special case of the functionally graded material.
The strain energy () of the moderately thick functionally graded panels and shells of revolution can be defined as
Substituting (2) and (3) into (9), the strain energy expression of the structure can be written in terms of middle surface displacements and rotations. For convenience, the strain energy expression is divided into three components; that is, , where , , and indicate Stretching, Bending, and BendingStretching coupling energy expressions, respectively.
The corresponding kinetic energy function of the moderately thick FG panels and shells of revolution can be given aswhere
Since the main focus of this paper is to develop a unified computational model to study the vibration of moderately thick FG panels and shells of revolution with general boundary conditions, in order to satisfy the request, the artificial spring boundary technique is adopted here. In this technique, five groups of boundary restraining springs are arranged at all sides of the FG panels and shells of revolution to separately simulate the general boundary conditions. Then the equations describing general elastic supported moderately thick FG panels and shells of revolution can be written as follows:
Thus, as one of the merits of the present study, the unified treatment in dealing with the panels under general boundary conditions can be achieved by assigning the stiffness of the boundary springs with various values. For example, the free boundary condition can be readily obtained by setting the spring coefficients to zero and the clamped boundary can be obtained by assigning the springs’ stiffness to infinity. Table 1 gives the corresponding spring stiffness values for the considered boundaries in which the symbol represents the bending stiffness . Therefore, the potential energy stored in the boundary springs is given as

As mentioned before, we can know that the main focus and merits of this paper are to present a unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution. However, the previous studies are most confined to the signer geometric configuration, that is, panels and shells, the difficulty of which is that the admissible functions of the panels do not fit to the shells. As we all know, in addition to the external boundary conditions, the kinematic and physical compatibility should be satisfied at the common meridian of and , if a complete shell of revolution needs considering [3, 53–55]. The kinematic compatibility conditions include the continuity of displacements. The physical compatibility conditions can only be the five continuous conditions for the generalized stress resultants. Thus, to consider the complete shell of revolution characterized by , it is necessary to implement the kinematic and physical compatibility conditions between the two computational meridians of and 2π. The kinematic compatibility conditions are as follows: The physical compatibility conditions are as follows:
In order to achieve the above requirements, the authors draw on the experience of two elastically coupled rectangular plates [56] to present a coupling spring technique. As one innovation point of present study, three groups of linear and two groups of coupling springs are applied to imitate the kinematic compatibility and physical compatibility conditions of functionally graded panels at the common meridian of and when a complete shell of revolution is considered. Therefore, the potential energies () stored in the five types of coupling springs can be defined as follows:
It should be noted that the stiffness of corresponding springs used at the common meridian of and of functionally graded panels is revalued to be zero automatically.
2.3. The Modified FourierRitz Approach
The modified FourierRitz approach was proposed by Jin et al. [49] to solve the free vibrations of laminated functionally graded shallow shells with general boundary conditions. The presented method consists of two main steps: firstly, the admissible functions of the structure elements are expanded into the modified Fourier series which consist of the twodimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions; secondly, the natural frequencies of the structure elements are obtained by using the variational operation (Ritz method) in terms of the unknown the expanded coefficient. In contrast to most existing techniques, the modified FourierRitz method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints, and their combinations without the need of making any changes to the solution procedure. Therefore, each of the referenced surface displacements or rotations of the normal of a shell can be written as an improved Fourier series as follows:where , , and , , , , and are the Fourier coefficients of twodimensional Fourier series expansions for the displacements functions, respectively. , , , , , , , , , and are the supplemented coefficients of the auxiliary functions and , where . The two types of auxiliary functions and are selected to remove all the discontinuities potentially associated with the firstorder derivatives at the boundaries. Thus, the function sets are capable of representing any free vibration motion of the shell. The two types of auxiliary functions are given as
It is easy to verify that
The Lagrangian energy function () of the moderately thick functionally graded panels and shells of revolution can be written as
Substituting (9), (11), (14), and (17) into (21), then the Lagrangian expression is minimized by taking its derivatives with respect to these coefficients:
Since the displacements and rotation components of the moderately thick functionally graded panels and shells of revolution are chosen as and to obtain the results with acceptable accuracy, a total of equations are obtained. They can be summed up in a matrix form:where , , and , respectively, represent the stiffness matrix, mass matrix, and vector of the unknown coefficients for the panel and shell. The detailed expressions for above matrices are given in Appendix. By solving (23), the frequencies (or eigenvalues) of moderately thick functionally graded panels and shells of revolution can be readily obtained and the mode shapes can be yielded by substituting the corresponding eigenvectors into series representations of displacement components.
3. Numerical Results and Discussion
In this section, some results and discussions about the free vibration of fourparameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution are presented to verify the accuracy and flexibility of the proposed unified approach. Unless mentioned otherwise, the panels and shells under consideration are assumed to be made of a mixture of ZnO_{2} ( GPa, , kg/m^{3}) and Al ( GPa, , kg/m^{3}). In the following frequencies results, the nondimensional frequency parameters of the cylindrical, conical, and spherical panels and shells are expressed as follows: cylindrical meridian: ; conical meridian: ; spherical meridian: .
3.1. Convergence Study and Formulation Validation
From the theoretical formulations, we can know that the computational accuracy relies on a limited number of terms in the displacement expressions in actual calculation. In addition, when considering the shells of revolution, the results strongly rely on the coupling parameters. Thus, one important issue is the choice of the appropriate truncation terms and coupling parameters. The nondimensional frequency parameter convergence of the clamped shells with respect to different coupling parameters is examined in Table 2. The geometrical parameters and powerlaw exponents for the functionally graded shells used in the study are as follows: cylindrical shell: m, m, m, and ; conical shell: m, , m, , and ; spherical shell: m, , , m, and . All coupling parameters are defined with the same value and defined as follows: . It is obvious that the frequency parameters rapidly converge as the coupling parameter increases. In the absence of the coupling parameters or for small coupling parameters, pseudovibration characteristic may appear in the present solution. This implies that small coupling parameters cannot impose the interface continuity conditions in a proper manner. Thus, based on the existed analysis, has been adopted to present all the results in the following analysis.

Next, the convergence studies of the cylindrical, conical, and spherical panels and shells will be discussed. Figure 5 shows the convergence studies of the first three frequency parameters for the structure elements for complete clamped FG panels and shells with different truncated numbers and . The geometrical dimensions of the shells are the same as Table 2 and the panels are also in accordance with Table 2 except the circumferential angle . From the figure, we can know that the proposed unified approach has fast convergence and good stability. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as in the following calculations.
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The validation of the unified computational model versus about the different geometric parameters is shown in Tables 3–8. The details regarding the geometry of the structures considered are indicated as follows:(1)Cylindrical panels (FFCF): m, m, m, and (Table 3).(2)Cylindrical shells (FC): m, m, and m (Table 4).(3)Conical panels (SCSC): m, m, m, , and (Table 5).(4)Conical shells (FC): m, m, m, and (Table 6).(5)Spherical panels (FFCF): m, m, , , and (Table 7).(6)Spherical shells: m, m, , and (Table 8).






In order to check the present method, the numerical results reported by Su et al. [7–9], Qu et al. [27], and Tornabene et al. [1, 3, 52] are also given in the above tables for comparison. From the comparisons, we can see a consistent agreement of present results taken from the current proposed unified approach and referential data. Besides, Tables 3–8 also show that it is appropriate to define the classical boundary conditions in terms of the boundary spring rigidities. Based on the formulation validation, some new results of the FG cylindrical, conical, and spherical panels and shells with classicalelastic and elastic restrain boundary condition are shown in Tables 9–14. Also, for any given modal frequency, the corresponding mode shape can be easily determined by submitting the solved eigenvector into (22). Thus, some select mode shapes of that are performed in Figures 6–8.
