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Free vibration of FGM conical–spherical shells

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Highlights

  • Free vibrations of a joined conical–spherical shell system is investigated using the semi-analytical Fourier-GDQ method.

  • Properties are graded through the thickness where shell is made from a functionally graded material.

  • Two different types of joined shells are assumed which are the C1-continuous shells and the hemispherical-conical shells.

  • Minimum frequencies of the joined shell system are associated to he higher circumferential mode numbers.

Abstract

Natural frequencies of a conical–spherical functionally graded material (FGM) shell are obtained in this study. It is assumed that the conical and spherical shell components have identical thickness. The system of joined shell is made from FGMs, where properties of the shell are graded through the thickness direction. The first order shear deformation theory of shells is used to investigate the effects of shear strains and rotary inertia. The Donnel type of kinematic assumptions are adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations are discretized using the semi-analytical generalized differential quadrature (GDQ) method. Considering various types of boundary conditions for the shell ends and intersection continuity conditions, an eigenvalue problem is established to examine the vibration frequencies. After proving the efficiency and validity of the present method for the case of thin isotropic homogeneous joined shells with the data of conventional finite element software, parametric studies are carried out for the system of combined moderately thick conical–spherical joined shells made of FGMs and various types of end supports.

Introduction

Any combination of cylindrical, conical, and spherical shells plays an important role in a wide range of applications in the pressure vessel industry. Many researchers have contributed their work to the analysis of the vibration characteristics of these joined shells [1], [2], [3], [4]. Irie et al. [5] presented a strategy to analyze the vibration of a joined cylindrical–conical shell element. The transfer matrix of the shell is expressed conveniently by the power series method and the frequency equations are derived for a given set of boundary conditions at the edges. Bagheri et al. [6] investigated the free vibration response of a joined shell system including two conical shells. In another research, they considered the free vibration characteristics of a joined shell system made of two conical shells at the ends and a cylindrical shell at the middle [7]. The first order shear deformation theory of shells is accompanied with the Donnell type of kinematic assumptions to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations are discretized using the semi-analytical generalized differential quadrature (GDQ) method.

Qu et al. [8] analyzed the free vibration of joined cylindrical–conical shell system with classical or non-classical boundary conditions. The thin shell assumptions of Reissner–Naghdi theory are used as the fundamental theoretical assumptions. The interface continuity and boundary conditions are approximately enforced by means of a modified variational principle and least-squares weighted residual method. In a recent work, Soureshjani et al. [9] investigated the free vibration behavior of joined composite sandwich conical–conical shells under external lateral pressure. The corresponding equations are derived based on the first order shear deformation theory (FSDT). Herewith, free vibration equations are extracted with application of Hamilton’s principle. Initial mechanical stresses are obtained by static equilibrium equations. To establish the continuity of two conical shells, compatibility of displacements and stress resultants are satisfied at the junctions. The generalized differential quadrature (GDQ) method is adopted to discretize the governing equations for each conical segment, together with related boundary and continuity conditions in the meridian direction.

The free vibration characteristics of the joined spherical–cylindrical shell with various boundary conditions are investigated by Lee [10]. The Flugge shell theory and Rayleigh’s energy method are applied in order to analyze the free vibration characteristics of the joined shell structure and individual shell components. The natural frequencies and mode shapes are calculated numerically and are compared with those of the FEM and modal test to confirm the reliability of the analytical solution. Using the Reissner–Naghdi–Berry shell theory, Wu et al. [11] applied the domain decomposition method (DDM) to investigate the vibration characteristics of the combined cylindrical-spherical shell with different boundary conditions. In another study, Wu et al. [12] concentrated on the free vibration of a joined cylindrical-spherical shell with elastic support type of boundary conditions using the domain decomposition method. Using the Flugge shell theory and Rayleigh-Ritz energy method, the free vibration characteristics of the pre-stressed joined spherical–cylindrical shell with free-free boundary conditions are investigated by Yosefzad et al. [13]. Qu and his co-authors also applied their previous method [8] to the free vibration analysis of ring-stiffened joined conical-cylindrical shell systems [14], joined conical–cylindrical–spherical shell systems [15], joined cylindrical-spherical shell with elastic-support boundary conditions [16], and spherical–cylindrical–spherical shells [17]. In a series of works, Kang [18], [19], [20] examined the free vibration response of cylindrical shells that are closed by various types of shells of revolution within the framework of three dimensional elasticity theory. The total strain and kinetic energies of the joined shell system are established and the Ritz method with the classical polynomial functions are used to establish the eigenvalue problem and extract the natural frequencies. As a special case, free vibration characteristics of an annular plate–cylindrical shell system is also analyzed. Saunders and Paslay [21] proposed the analytical solution for the natural frequencies of the joined conical and spherical shells by the Rayleigh–Ritz method which shows good agreement with the modal test using an exciting method.

Li et al. [22] obtained the frequencies of a stepped cylindrical shells using the first order shear deformation shell theory. The analytical model is established based on multi-segment partitioning strategy. The solutions about free vibration behavior of functionally graded circular cylindrical shells are obtained by approach of Rayleigh–Ritz method. A hermetic capsule is formulated within the framework of the Flugge shell theory to study the free vibrations of spherical–cylindrical–spherical shell system using the Ritz and FSDT formulations [23]. Arbitrary combinations of boundary conditions are included in this research. Vibrations of composite cylindrical and spherical shells in composite laminated scheme are obtained by Pang et al. [24]. The analytical model is established on base of multi-segment partitioning strategy and first-order shear deformation theory. The displacement functions are made up of the Jacobi polynomials along the axial direction and Fourier series along the circumferential direction. An analysis is done by Li et al. [25] to obtain the natural frequencies of a compound shell system containing cylindrical and spherical segments with different thickness. The energy method and first-order shear deformation theory are adopted to derive the formulas. Ritz method is applied to extract the frequencies of the shell system. Combined paraboloidal, cylindrical and spherical shells with arbitrary boundary conditions are analyzed by Li et al. [26] within the framework of first order shear deformation shell theory. The analytical model is established on the base of multi-segment partitioning strategy and Flugge thin shell theory. The admissible displacement functions are handled by unified Jacobi polynomials and Fourier series. Free vibration characteristics of conical-cylindrical-spherical shell combinations with ring stiffeners are investigated by using a modified variational method [27]. Reissner–Naghdi thin shell theory in conjunction with a multilevel partition technique, stiffened shell combination, shell component and shell segment is employed to formulate the theoretical model. A Fourier spectral element method is applied in [28] to analyze the free vibration of conical–cylindrical–spherical shells with arbitrary boundary conditions. Cylindrical-conical and cylindrical-spherical shells as special cases are also considered. First order shear deformation shell theory is applied to formulate each segment of the compound shell system. A semi-analytic method is presented to analyze the free and also forced vibrations of combined conical–cylindrical–spherical shells with ring stiffeners and bulkheads by Xie et al. [29]. The shell is divided into a spherical shell, a number of conical shells and also stiffeners and bulkhead. The spherical shell is also divided into a number of elements where each one is treated as a conical shell.

Jin et al. [30] performed an investigation on the free vibrations of FGM doubly-curved shells of revolution with arbitrary boundary conditions. The analysis of this work is based on modified Fourier series method and the first order shear deformation shell theory considering the effects of the deepness terms. A two-dimensional generalized differential quadrature method is developed by Tornabene et al. [31] to obtain the natural frequencies of FGM and laminated doubly curved shells and panels of revolution with a free-form meridian. For functionally graded material shells of revolution, a semi-analytical method is developed for studying the free vibrations by Xie et al. [32]. Independent and coupled shells of revolution with uniform and stepped thickness are analyzed in this work. The shell is firstly divided into several shell segments along axial direction, and these segments are treated as conical shells. Torabi and Absari performed an investigation on the free vibration of FGM shells of revolution by Ansari and Torabi [33]. In this work a new quadratic isoparametric superelement is proposed. A unified formulation is developed for any type of shell of revolution.

Present study aims to analyze the free vibrations of a joined shell which is composed of a spherical shell and a conical shell. While this shell may be studies as a special case of shell of revolution, which is well-discussed in the open literature, but in this study it is decomposed into two basic shells. Using the first order shear deformation theory of shell, the Donnell type of kinematic assumptions, and the Hamilton principle the governing dynamic equations of the spherical and conical shells made of FGMs are obtained. The GDQ method is applied to discretize the governing, boundary, and matching conditions of the joined shell system and establish an eigenvalue problem. Results are devoted to explore the effects of power law index, geometrical characteristics, and edge support of the shell.

Section snippets

Material properties of FGMs

The material properties of the ceramic and metal constituents of the joined shell system are assumed to be graded in thickness direction based on the power law function. The ceramic volume fraction Vc and metal volume fraction Vm are assumed to obey the following form Vc=12+zhk,Vm=1VcIn Eq. (1), k is the power law index and dictates the distribution of material properties across the thickness. It is obvious that the surface z=+h2 is ceramic rich and the surface z=h2 is metal rich. Following

Governing equations of the shell system

Consider a joined conical–spherical shell made of functionally graded material of uniform thickness h. Sphere radius is R and its angle is β. Minimum radius of the conical shell is rc=Rsin(β), cone length is L, and the semi-vertex angle of the cone is assumed as α. The system is shown in Fig. 1. The (x,θ,z) system is applied to the conical shell system whereas the (ϕ,θ,z) system is applied to the spherical shell. The coordinate systems and geometrical characteristics are also shown in Fig. 1.

To

Boundary and matching conditions

At the end of conical shell, various types of boundary conditions may be defined. The edge x=L may be clamped (C) free (F) or simply supported (S). Mathematical expression of edge supports on the end of the conical shell takes the form C:u0c=v0c=w0c=φxc=φθc=0F:Nxxc=Nxθc=Qxzc=Mxxc=Mxθc=0S:Nxxc=v0c=w0c=Mxxc=φθc=0 Furthermore, in view of the shell theory adopted in the present study, particular conditions (apex compatibility conditions) need to be enforced to avoid the divisions by zero arising in

Solution procedure

Referring to the definition of normal force and bending moment resultants from Eq. (12) and the motion Eqs. (17), (18), the following separation of variables satisfies the periodicity conditions of the field variables and is also compatible with the motion Eqs. (17), (18) and matching conditions (22), (23) u0i(ζ,θ,t)v0i(ζ,θ,t)w0i(ζ,θ,t)φζi(ζ,θ,t)φθi(ζ,θ,t)=cos(ωt+ψ)sin(nθ)00000cos(nθ)00000sin(nθ)00000sin(nθ)00000cos(nθ)×Ui(ζ)Vi(ζ)Wi(ζ)Φζi(ζ)Φθi(ζ) where in the above equation n, as mentioned

Numerical results and discussion

Current investigation deals with the free vibration response of a joined conical–spherical shell made of functionally graded materials. In the present section, first comparison studies are provided to assure the validity and correctness of the developed formulation. After that, new numerical results are given. The developed formulation may be used for arbitrary combination of conical and spherical shell. However, for development of the numerical results two especial cases of joined shells are

Conclusion

In the current research, the free vibration response of a spherical-conical shell system is evaluated. Shell is assumed to be made from an FGM where properties are graded in thickness. A simple power law function is used to describe the volume fraction of constituents and the simple Voigt rule of mixtures is implemented to evaluate the properties. To establish the governing equations of the shell system the first order shear deformation shell theory and the Donnell type of kinematic assumptions

CRediT authorship contribution statement

H. Bagheri: Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing. Y. Kiani: Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing. M.R. Eslami: Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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