Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions

Accepted Manuscript Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions Xiang Xie, Hui Zheng, Guoyong Jin PII:

S1359-8368(15)00135-3

DOI:

10.1016/j.compositesb.2015.03.016

Reference:

JCOMB 3450

To appear in:

Composites Part B

Received Date: 28 September 2014 Revised Date:

2 December 2014

Accepted Date: 3 March 2015

Please cite this article as: Xie X, Zheng H, Jin G, Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.03.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions Xiang Xie1 , Hui Zheng1,*, Guoyong Jin2 1 State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China

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2 College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China

Abstract

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The objective of this work is to present a Haar Wavelet Differential Quadrature (HWDQ) method-based solution approach for the free vibration analysis of functionally graded (FG) spherical

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and parabolic shells of revolution with arbitrary boundary conditions. The first-order shear deformation theory is adopted to account for the transverse shear effect and rotary inertia of the shell structures. Haar wavelet and their integral and Fourier series are selected as the basis functions for the variables and their derivatives in the meridional and circumferential directions, respectively. The

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constants appearing in the integrating process are determined by boundary conditions, and thus the equations of motion as well as the boundary condition equations are transformed into a set of

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algebraic equations. The proposed approach directly deals with nodal values and does not require

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special formula for evaluating system matrices. Also, the convenience of the approach is shown in handling general boundary conditions. Numerical examples are given for the free vibrations of FG shells with different combinations of classical and elastic boundary conditions. Effects of spring stiffness values and the material power-law distributions on the natural frequencies of shells are also discussed. Some new results for the considered shell structures are presented, which may serve as benchmark solutions.

*Corresponding author, Tel: +86-21-34205910. E-mail address: [email protected] (H. Zheng)

ACCEPTED MANUSCRIPT Keywords: B. Vibration; C. Numerical analysis; Functionally graded shell 1. Introduction With the increasing requirements for high performance materials in industry, the study on

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functionally graded materials (FGMs) has been receiving more and more attention. A typical FGM with a high bending-stretching coupling effect is an inhomogeneous composite made from different

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phases of material constituents (usually ceramic and metal). By gradually varying the volume fraction of constituent materials, their material properties exhibit a smooth and continuous variation

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from one surface to another, thus eliminating interface problems and mitigating stress concentrations. However, the analysis of functionally graded structures is more complicated than that of homogeneous material structures owing to the spatial variations of material properties. In order to investigate the stress and displacement fields of FG structures, one needs to solve the partial

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differential equations with variable coefficients. This class of problem is challenging and thus there exists considerable literature on the mechanical behavior of FG structures [1].

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Spherical and parabolic shells are largely used in many engineering structures due to their

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special geometric shapes. Knowing their natural frequencies and corresponding mode shapes is of considerable importance, because it allows the designers and engineers to provide optimal design and avoid unpleasant, inefficient and structurally damaging resonant. However, it is obvious that the analysis of spherical and parabolic shells has difficulty related to the curvature, which is also the reason for the carrying load capacity of these kinds of structures. Therefore, it remains a challenging task to develop efficient modeling and computational techniques for the vibration analysis of FG spherical and parabolic shells.

ACCEPTED MANUSCRIPT In the last years, a large quantity of research efforts have been devoted to the vibration analysis of shell structures in the literature. As one of the best known of completely closed shell structures, the vibration characteristics of spherical shells are attractive because of their enhanced behavior in a

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variety of engineering applications. Research on the free vibrations of spherical shell can largely be divided into two groups: the shell theories and solution methods. The development of accurate shell

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theories has always been one of the most prominent challenges in solid mechanics for many years, and great efforts have been made on the topic [2-4]. Physically, shells are three-dimensional (3D)

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bodies bounded by two, relatively close, curved surfaces [2, 5-7]. However, the 3D equations of elasticity are very complicated when written in curvilinear shell coordinates. Most researchers who dealt with shells tried to simplify such shell equations by making certain approximations and assumptions for particular applications. Therefore, various two-dimensional (2-D) shell theories were

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proposed, and extensive studies have been carried out based on these theories [2-4, 8-27]. In this way, a careful selection of the proper shell theory is decisive for free vibration analysis of shell structures.

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In order to avoid complicated mathematical presentations and predict the global behaviors of

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moderately thick shells, the first-order shear deformation theory (FSDT) is just employed in the present analysis. Apart from the aforementioned shell theories, it has also been of great interest for researchers to develop an accurate and efficient method which can be used to determine the vibration behaviors of FG spherical shells. So far, a number of methods for the analysis of spherical shells have been reported in literature, such as Differential Quadrature method [11-14], pseudospectral method [15], domain decomposition approach [16], modified Fourier method [17], Finite element method [19-22], etc. There is comparatively limited literature available for parabolic shells,

ACCEPTED MANUSCRIPT compared to those on the free vibrations of spherical shells. A few of the outstanding contributions on the free vibration analyses of parabolic shells include Kang and Leissa [6], Al-Khatiband Buchanan [7], Tornabene et al. [13, 26-29], Chun et al. [23]. From the review of the literature, it

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appears that most of the previous researches were limited to shells with classical boundary conditions. However, the shells with non-classical boundaries such as elastically constrained are widely

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encountered in the engineering practices. Unlike existing weak formulation methods (potential energy formulations) [16, 17], the present work provides direct treatment in dealing with general

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restrained of FG spherical shells, parabolic shells and their mutative shapes.

As a powerful mathematical tool, wavelet analysis has been widely used in signal processing, image compression, numerical analysis and many other fields in recent years. The current wavelets-based approaches include wavelet-Galerkin method [30, 31], wavelet-collocation method

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[32, 33], wavelet-finite elements method [34], etc. In most wavelet-based methods, the calculation of wavelet connection coefficients is a complicated problem. Obviously, attempts to simplify solutions

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based on wavelet methods are required. Recently, the Haar wavelet which is originally introduced by

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Alfred Haar in 1910 has drawn considerable attention. It is the simplest orthonormal wavelet with compact support. However, it should be noticed that Haar wavelet is not continuous. Since its derivatives don’t exist, it is impossible to apply the Haar wavelet directly for solving differential equations. One possible way for employing Haar wavelet is using integral method, as discussed in Refs. [35-37]. Then this simple and direct procedure has been applied to various problems [38-47]. In this work, by virtue of geometrical shape of considered structures and using the Haar wavelet method, the partial differential governing equations with variable coefficients can be transformed

ACCEPTED MANUSCRIPT into a set of linear algebraic equations. Imposing the given arbitrary boundary condition, the integral constants can be obtained and the numerical eigenvalue equations for free vibrations of the FG shells can be derived and then solved routinely. Based on the eigensolution, the natural frequencies and

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mode shapes analysis of these shell structures for various parameters can be easily implemented by using MATLAB program.

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The main aim of this current paper is to present a simple yet efficient approach based on the Haar wavelet collocation for the modeling and vibration analysis of moderately thick FG spherical

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and parabolic shells subjected to arbitrary boundary conditions. Naturally, attempts to solve directly equilibrium equations of motion can be considered as a special differential quadrature method, namely Haar Wavelet Differential Quadrature (HWDQ) method. Also, the collocation algorithm can be viewed as a very effective meshless technique. In order to validate the efficiency and accuracy of

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the proposed method, as well as to explore the limits of its applicability, numerical examples are presented for the free vibrations of FG shells. Effects of some geometrical and material parameters

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on the natural frequencies are also discussed and some representative mode shapes are given for

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illustrative purposes. Differing from the authors' previous works [45-47], the proposed solution approach can also be readily applied to not only the singly-curved shells generated by a straight line revolves about an axis, but also the doubly-curved shells appeared by a curved arc whirls about an axis of rotation.

2. Theoretical formulations 2.1. Description of the model Spherical and parabolic shells are two special cases of shells of revolution. For these shells, a

ACCEPTED MANUSCRIPT circular and parabolic arc, rather than a straight line, revolves about an axis of rotation to generate the completely closed surface. Consider a FG shell of revolution with uniform thickness h. The geometric parameters and coordinate system of a differential element of FG spherical and parabolic

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shells are shown in Fig. 1(a) and 1(c), respectively. From Fig. 1(b), it can be seen that the reference surface of the shell is taken to be at its middle surface where an orthogonal curvilinear coordinate

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system (ϕ, s, z) is fixed. The displacement of the shell in the meridional ϕ, circumferential s and radial z directions are denoted by u, v and w, respectively. The angle formed by the external normal n

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to the reference surface and the axis of rotation oz, or the geometric axis o1z1 of the meridian curve, is defined as the meridional angle ϕ and the angle between the radius of the parallel circle and the x axis is defined as the circumferential angle θ. The position of an arbitrary point within the shell is decided by 0≤ϕ≤ϕ1-ϕ0, 0≤θ≤2π and -h/2≤z≤h/2. The horizontal radius is designated as R0 and the

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curvilinear abscissa s of a generic parallel is determined by the equation s=θR0. The radii of curvature in the meridional and circumferential directions are represented by Rϕ, Rs, respectively. For

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a surface of revolution with a circular curved meridian, they are respectively expressed as: R0 = R sin ϕ + Rb , Rs = R0 / sin ϕ ,

(1)

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Rϕ = R

where R is the constant radius of the circular meridian of the shell. Rb is the distance of the axis of rotation oz and the geometric axis of the meridian o1z1. For a shell with parabolic meridian assumes the following forms: R0 =

k tan ϕ + Rb 2

Rs = R0 / sin ϕ

Rϕ =

k 2 cos3 ϕ

3k sin ϕ = dϕ 2 cos 4 ϕ

dRϕ

(2)

ACCEPTED MANUSCRIPT where k is a characteristic parameter of the parabolic curve. It is determined by the equation k=(a2-d2)/b. More explicit descriptions of some parameters can be found in Ref. [13]. Fig. 2 shows the examined FG shell structures for verifying the accuracy and versatility of the proposed approach.

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It is worth noting that, by setting the Rb≠0, we can obtain the shell structures to that of circular toroid, as shown in Figs. 2(c) (Rb<0) and 2(d) (Rb>0). The spherical shell is generated as a special case of

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the considered shell structures, when Rb=0, as diagramed in Fig. 2(a) and by setting ϕ0=0, the spherical cap is obtained as illustrated in Fig. 2(b). Fig. 2(e) and 2(f) present the parabolic toroid and

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parabolic dome, respectively. 2.2. Kinematic relations and stress resultants

The FSDT of shells is used to derive the equilibrium equations and the related boundary conditions. Based on this theory, the displacement components of an arbitrary point in the FG shell

below [4, 8, 11, 13]:

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are expressed in terms of the displacements and rotation components of the middle surface as given

(3)

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 u (ϕ , s, z, t )   u0 (ϕ , s, t )  φϕ (ϕ , s, t )         v(ϕ , s, z, t )  =  v0 (ϕ , s, t )  + z φs (ϕ , s, t )   w(ϕ , s, z, t )   w (ϕ , s, t )    0    0   

where u0, v0 and w0 are the displacement components of points lying on the middle surface (z=0) of the shell in the meridional, circumferential and radial coordinates, respectively; φϕ and φs represent the transverse normal rotations of the reference surface about the s- and ϕ-axis. t is the time variable. The relationships between strains and displacements in the middle surface for a moderately thick shell are represented by the following [13, 14]:

ACCEPTED MANUSCRIPT    1  ∂u0 +w     Rϕ  ∂ϕ     ∂v u cos ϕ w sin ϕ  + 0  0+ 0  0 R0 R0   ε ϕϕ   ∂s  0   ∂v ∂u v cos ϕ  0  ε ss    + 0− 0 R0  γ ϕ0s   Rϕ ∂ϕ ∂s   =  ∂φϕ  χϕϕ     χ ss    Rϕ ∂ϕ     ∂φs φϕ cos ϕ  χϕ s    +   ∂s R0    ∂φϕ ∂φs φs cos ϕ  + −   R0  ∂s Rϕ ∂ϕ 

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(4)

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0 where ε ϕϕ , ε ss0 and γ ϕ0s are the in-plane meridional, circumferential and shearing components, χϕϕ ,

χss and χϕs are the curvature changes of the shell. The strains-displacement relations are given as follows:

(5)

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 1  ∂w0   0   χϕϕ  ε ϕϕ  ε ϕϕ − u0  + φϕ       0    γ ϕ z   Rϕ  ∂ϕ    ε ss  =  ε ss  + z  χ ss  ,   =   γ  γ 0   χ   γ sz   ∂w0 − v0 sin ϕ + φ  s   ϕs   ϕs   ϕs   ∂s R0 

where γϕz and γsz indicate the transverse shearing strains. The constitutive equations relating the force

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and moment resultants to strains and curvatures of the reference surface are given in the matrix form: A12

0

B11

B12

A11

0

B12

B11

0 B12

A66 0

0 D11

0 D12

B11

0

D12

D11

0

B66

0

0

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 Nϕ   A11 N    s   A12  Nϕ s   0  =  M ϕ   B11  M s   B12     M ϕ s   0

0  0   ε ϕϕ  0   0   ε ss  B66   γ ϕ0s   Qs   A66    = Kc   0   χϕϕ  Qϕ   0   0  χ ss   D66   χϕ s 

0   γ sz    A66  γ ϕ z 

(6)

where Nϕ, Ns and Nϕs are the in-plane force resultants. Mϕ, Ms and Mϕs denote the bending and twisting moment resultants. Qs and Qϕ are the transverse shear force resultants. The parameter Kc is the shear correction factor, which is usually selected as Kc=5/6. The structures materials employed in

ACCEPTED MANUSCRIPT the following study are assumed to be functionally graded and linearly elastic. So, the extensional stiffness Aij, the bending stiffness Dij, and the extensional-bending coupling stiffness Bij are expressed as:

Bij

ij

Dij ) = ∫

h /2

− h /2

(1

z

z 2 ) Qij ( z )dz , i, j=1,2,6

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(A

(7)

where the elastic constant Qij(z) are functions of thickness coordinate z and defined as:

E( z) µ ( z)E( z) E( z) Q12 ( z ) = Q66 ( z) = 2 2 1 − µ ( z) 1 − µ ( z) 2 [1 + µ ( z)]

(8)

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Q11 ( z ) =

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For a functionally graded shell is made of a mixture of two material constituents M1 and M2 (usually ceramic and metal), the material properties of the shell along the thickness direction can be expressed as:

M ( z ) = ( M 1 − M 2 ) V1 + M 2

(9)

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where M(z) represents the material properties including Young's modulus E(z), Poisson's ratio µ(z) and mass density ρ(z). V1 is the volume fraction of M1. In this paper, the volume fraction V1 follows two simple four-parameter power-law distributions [27]:

p

FGMII(a/b/c/p):

c  1 z 1 z  V1 = 1 − a  −  + b  −    2 h   2 h   

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p

(a/b/c/p):

c  1 z 1 z  V1 = 1 − a  +  + b  +    2 h   2 h   

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FGMI

(10a)

(10b)

in which the power-law exponent p is a positive real number (0≤p≤∞) and the parameters a, b, c dictate the material variation profile through the functionally graded shell thickness. It is assumed that the sum of the volume fractions of the two basis components is equal to unity, i.e., V1+V2=1. Therefore, according to the relations defined in Eq. (10), when the power-law exponent p is set equal to zero (i.e., p=0) or equal to infinity (i.e., p=∞), the FG material becomes the homogeneous

ACCEPTED MANUSCRIPT isotropic material, expressed as: p=0→V1=1, V2=0→E(z)=E1, µ(z)=µ 1, ρ(z)=ρ1.

(11a)

p=∞→V1=0, V2=1→E(z)=E2, µ(z)=µ 2, ρ(z)=ρ2.

(11b)

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By setting appropriate values of the parameters p, a, b and c, the different power-law distributions can be obtained.

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2.3. Equations of motion

By means of Hamilton’s principle, the equations of motion of general shell structures with the

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effects of transverse shear and rotary inertia taken into account can be obtained. Then the equations of the considered shell structures can be derived by substituting the proper Lamé parameters of such shells into general shell equations [4, 12, 13, 26, 27]:

∂ 2Φϕ ∂ 2u0 cos ϕ Qϕ + + ( Nϕ − N s ) + = I 0 2 + I1 2 Rϕ ∂ϕ ∂s R0 Rϕ ∂t ∂t ∂Nϕ

Rϕ ∂ϕ ∂Qϕ Rϕ ∂ϕ

+

∂N s ∂ 2v ∂ 2Φ sin ϕ cos ϕ + Qs + 2 Nϕ s = I 0 20 + I1 2 s ∂s R0 R0 ∂t ∂t

+

∂Qs ∂2w cos ϕ Nϕ sin ϕ + Qϕ − − Ns = I 0 20 ∂s R0 Rϕ R0 ∂t

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∂Nϕ s

∂Nϕ s

(12)

∂ 2Φϕ ∂ 2 u0 cos ϕ + + (M ϕ − M s ) − Qϕ = I1 2 + I 2 ∂t 2 Rϕ ∂ϕ ∂s R0 ∂t ∂M ϕ s

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∂M ϕ

Rϕ ∂ϕ

where

+

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∂M ϕ s

∂M s ∂ 2v ∂ 2Φ cos ϕ + 2M ϕ s − Qs = I1 20 + I 2 2 s ∂s R0 ∂t ∂t



( I 0 , I1 , I 2 ) = ∫− h/2 ρ ( z ) (1, z1 , z 2 ) 1 + h /2



z Rϕ

 z  1 +   Rs

  dz 

(13)

in which I0, I1 and I2 are the mass inertias. The above equation includes two terms (1+z/Rϕ) and (1+z/Rs). These terms were ignored in most thin shell theories because they are close to unity. In this

ACCEPTED MANUSCRIPT present paper, for some numerical examples, these terms can also be neglected with respect to unity (i.e. Tables 2, 3, 6, 7 and corresponding mode shapes). By taking advantage of geometrical shape of these complete shell structures, we perform the separation of variables and expand every dependent

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variable of displacement and rotation components of the shells using Fourier (theoretically infinite) series with respect to the circumferential direction. By this way, it simplified the free vibration

shear rotations ϕϕ, ϕs can be expressed in the following form:

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problems of closed shells considerably. Accordingly, the displacement components u0, v0, w0 and

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u0 (ϕ , s, t ) = U (ϕ ) cos( ns )eiωt v0 (ϕ , s, t ) = V (ϕ ) sin( ns )eiωt

w0 (ϕ , s, t ) = W (ϕ ) cos( ns )eiωt

(14)

φϕ (ϕ , s, t ) = Φϕ (ϕ ) cos(ns )eiωt φs (ϕ , s, t ) = Φs (ϕ ) sin(ns )eiωt

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where ω is the angular frequency of vibration and non-negative integer n is the circumferential wave number of the corresponding mode shape. U(ϕ), V(ϕ), W(ϕ), Φϕ(ϕ) and Φs(ϕ) are unknown functions, describing the distribution of vibrational amplitude in the meridional ϕ direction. By

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interchanging the selection of trigonometric terms cosns and sinns, another set of free vibration

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modes can be easily obtained. Also, these expansions automatically satisfy the boundary condition equations on the circumferential edges. For the considered shell structures, the relationship between the curvilinear abscissa s of a generic parallel and the circumferential angle θ assumes the form: ds = R0 dθ

(15)

The Fourier series expansions of Eq. (14) and the relationship of Eqs. (4)-(6) and (15) are next substituted into the Eq. (12), then performing some algebraic manipulation, a two-dimension problem is transformed into a set of uncoupled one-dimension problems, resulting in:

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+ G143

G242

dΦϕ dϕ

G342

dϕ

G442

dϕ

G542

dϕ

dU dW d 2W + G321V + G331W + G332 + G333 + G341Φϕ + dϕ dϕ dϕ 2

+ G351Φs + I 0ω W = 0

dU d 2U dV dW + G413 + G421V + G422 + G431W + G432 + G441Φϕ + 2 dϕ dϕ dϕ dϕ

+ G443

G511U + G512 dΦϕ

dΦs d 2Φs + G253 + I 0ω 2V + I1ω 2Φs = 0 2 dϕ dϕ

2

G411U + G412 dΦϕ

dU dV d 2V + G221V + G222 + G223 + G231W + G241Φϕ + dϕ dϕ dϕ 2

+ G251Φs + G252

G311U + G312 dΦϕ

+ G151Φs + G152

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G211U + G212

dϕ 2

dΦs + I 0ω 2U + I1ω 2Φϕ = 0 dϕ

d 2Φϕ dϕ 2

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dϕ

d 2Φϕ

+ G451Φs + G452

dΦs + I1ω 2U + I 2ω 2Φϕ = 0 dϕ

dU dV d 2V + G521V + G522 + G523 + G531W + G541Φϕ + dϕ dϕ dϕ 2 dΦs d 2Φs + G553 + I1ω 2V + I 2ω 2Φs = 0 2 dϕ dϕ

(16a)

(16b)

(16c)

(16d)

(16e)

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G142

dΦϕ

dU d 2U dV dW + G113 + G121V + G122 + G131W + G132 + G141Φϕ + 2 dϕ dϕ dϕ dϕ

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G111U + G112

+ G551Φs + G552

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where the Gijk are the coefficients defined in Appendix A. After separation of variables with the help of Fourier series, the equations will be discretized by using the present method.

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The treatment of arbitrary boundary conditions can be considered as an extension of the authors’ previous works [45-47]. Three sets of translational springs (ku, kv, kw) and two kinds of rotational springs (kφ, ks) are added along the edges of the shell to imitate the elastic constraints. For the purpose of simplifying the presentation, SD, SS, C, F, E1, E2 and E3 represent shear-diaphragm, simply-supported, clamped, free and three kinds of elastic edge boundaries, respectively. Then the equations describing general elastic supported FGM shells can be written as follows [2, 4]

ACCEPTED MANUSCRIPT ku 0U = Nϕ ,

kv 0V = Nϕ s ,

k w0W = Qϕ ,

kφ 0Φϕ = M ϕ , ks 0Φs = M ϕ s , ku1U = − Nϕ ,

kv1V = − Nϕ s ,

kw1W = −Qϕ ,

kφ 1Φϕ = − M ϕ , ks1Φs = − M ϕ s ,

at ϕ=ϕ0

(17)

at ϕ=ϕ1

(18)

setting kinds of spring system for every separate degree of freedom.

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Taking edge ϕ=ϕ0 for example, the general boundary conditions for shells can be achieved by

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A11 dU A12 cos ϕ A n A A sin ϕ B dΦϕ B12 cos ϕ B n U + 12 V + 11 W + 12 W + 11 + + Φϕ + 12 Φs − ku 0U = 0 Rϕ dϕ R0 R0 Rϕ R0 Rϕ dϕ R0 R0

A66 n A dV A66 cos ϕ B n B dΦs B66 cos ϕ U + 66 − V − 66 Φϕ + 66 − Φs − kv 0V = 0 R0 Rϕ dϕ R0 R0 Rϕ dϕ R0

−

K c A66 K A dW U + c 66 + K c A66Φϕ − kw0W = 0 Rϕ Rϕ dϕ

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−

(19)

B11 dU B12 cos ϕ B n B B sin ϕ D dΦϕ D12 cos ϕ D n U + 12 V + 11 W + 12 W + 11 + + Φϕ + 12 Φs − kφ 0Φϕ = 0 Rϕ dϕ R0 R0 Rϕ R0 Rϕ dϕ R0 R0

B66 n B dV B66 cos ϕ D n D dΦs D66 cos ϕ U + 66 − V − 66 Φϕ + 66 − Φs − ks 0Φs = 0 R0 Rϕ dϕ R0 R0 Rϕ dϕ R0

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−

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By setting proper spring stiffness (from zero to infinity), the different boundary conditions can be easily obtained. Table 1 gives the corresponding spring stiffness values for the considered

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boundaries. The unified treatment in dealing with general boundary condition is one of the merits of the present study.

2.4. The Haar wavelet series and their application For the sake of completeness, clarity and without losing generality, some aspects related to the Haar wavelet will be described. The orthogonal set of Haar wavelet hi(ξ) is a group of square waves with magnitude of ±1 in some intervals and zeros elsewhere. Any highest derivatives appearing in the differential equation, which are square integrable in the interval [0, 1], can be expanded into Haar

ACCEPTED MANUSCRIPT wavelet series of infinite terms. Then this approximation is integrated while the integration constants are determined by the boundary conditions. Thus the boundary condition equations were satisfied exactly for the following analysis. The interval [0, 1] is divided into 2M subintervals of equal length

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∆x=1/2M. If we want to use the Haar wavelets for numerical solutions, it must be put into a discrete form. There are different ways to do it, in this paper, the collocation method is used. The collocation points are given as: (l − 0.5) , l=1, 2, ..., 2M 2M

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ξl =

(20)

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The Haar coefficient matrix H is defined as H (i, l) =hi (ξl). If we want to solve an n-th order PDE, the following integrals are required [43].

ξ ξ ξ ξ 1 pα ,i ( x) = ∫ ∫ L ∫ hi (t )dt α = ( x − t )α −1 hi (t )dt α=1, 2, ...,n, i=1, 2, ..., 2M. ∫ 0 042 4444 0 3 (α − 1)! 0 1444 α −times

(21)

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The case α=0 corresponds to the function hi(t). These integrals can be calculated analytically. For solving boundary value problems, the values pα,i(0) and pα,i(1) should be calculated in order to

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satisfy the boundary conditions. Substituting the collocation points in Eq. (20) into Eq. (21) yields

P (α ) (i, l ) = pα ,i (ξ )

(22)

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where P(a) is a 2M×2M matrix. It should be noted that calculations of the matrices H(i, l) and P(α)(i,l) must be carried out only once.

The Haar wavelet will be used to discretize the derivatives in the governing equations as well as boundary condition equations in terms of displacement and rotation components. The same procedure as in [45] is followed to handle the application of HWDQ method in the present work. Firstly, in order to apply the Haar wavelet method to resolve finite domain problems, the solution domain should be converted into the unit interval [0, 1]. Thus the following normalized variable is

ACCEPTED MANUSCRIPT introduced: ξ =

ϕ − ϕ0 ,in which Ψ=ϕ1-ϕ0. Obviously, ξ ∈ [0,1] . Then by substituting ξ into Eq. Ψ

(16), the equations of motion in terms of a single non-dimensional variable ξ can be obtained. A brief overview of solution process of the Haar wavelet method is introduced in the following description

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for completeness. The basis of the HWDQ method is the representation of the highest derivatives of the function f(ξ) by the truncated Haar wavelet series. Then lower order derivatives and the function

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itself are obtained by integrating. In Eq. (16), each of the displacement and rotation components at most has second-order derivatives. In the interest of simplicity and conciseness, we use f(ξ) to

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represent any one of the unknown functions U(ϕ), V(ϕ), W(ϕ), Φϕ(ϕ) and Φs(ϕ). Then, it is assumed that the solution is sought in the following form (k=2M):

d 2 f (ξ ) k = ∑ ai hi (ξ ) dξ 2 i =1

(23)

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where ai are the unknown wavelet coefficients and hi(ξ) is the Haar wavelet series. k is the number of collocation points in the meridional direction. By integrating Eq. (23) twice, and taking into account Eq. (21), the following forms can be easily obtained.

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df (ξ ) k df (0) = ∑ ai p1,i (ξ ) + dξ dξ i =1

(24)

k

df (0) f (ξ ) = ∑ ai p2,i (ξ ) + ξ + f (0) dξ i =1 T

 df (0)  where p1,i and p2,i see Section 2.4. Let us define a=[a1, ..., an] , b =  , f (0)  . The presences of  dξ  T

integration constants b allow the addition of extra equations. The remaining equations are obtained from the arbitrary boundary conditions. By repeating above solution procedure five times, the whole system of differential equations as well as boundary condition equations can be discretized. The global assembling leads to the following set of linear algebraic equations:

ACCEPTED MANUSCRIPT  K1 K  3

K 2   A M = ω2  1    K4  B  0

M 2   A 0   B 

(25)

where A and B are the unknown wavelet coefficients and the integration constants, respectively. K1,

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K2 and M1, M2 are obtained from the equations of motion described by Eq. (16). K3 and K4 are obtained from discrete boundary condition equations considered in Eqs. (17) and (18). After performing some algebraic manipulation, the above system of equations can be expressed as (26)

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 K1 − K 2 K 4−1 K 3  A = ω 2  M1 − M 2 K 4−1 K 3  A

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The solution method outlined above is discussed in detail in Ref. [47] and will not be repeated here for brevity. It is worth noting that for a well-posed problem, the numbers of equations are identical to the numbers of unknown coefficients. Then the natural frequencies of the considered structures can be determined by solving the standard eigenvalue problem and the mode shape

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corresponding to each eigenvalue can be obtained by substituting eigenvector into Eq. (14). Moreover, it should be noticed that most of the coefficients in Haar wavelet and their integral

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matrices are zero (or near to zero). This merit of sparse representation guarantees rapid convergence and efficiency to the HWDQ technique. In addition, the strong formulation approach for directly

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solving the governing differential equation with variable coefficients, differing from the weak formulation method like FEM, no integration occurs prior the global assembly of the linear system. Also, the current approach is very convenient for solving the boundary value problem, since the boundary conditions are satisfied automatically.

3. Numerical examples and discussion In this section, free vibration analysis of FG spherical and parabolic shells of revolution with arbitrary boundary conditions is presented to verify the reliability, validity and accuracy of the

ACCEPTED MANUSCRIPT proposed method. The symbol such as F-C denotes that the edges ϕ=ϕ0 and ϕ=ϕ1 are free and clamped, respectively. The number of truncated terms that should be used to obtain accurate results and the appropriate value of the maximal level of resolution J for computing were determined in Ref.

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[45]. Otherwise specified, the maximal level of resolution J will be uniformly selected as J=7 in the following discussions.

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3.1. Validation

Several numerical examples for free vibration behavior of FG shells subjected to various

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boundary conditions are presented in order to validate the proposed approach. The functionally graded materials of shells are composed of aluminum (metal) and zirconia (ceramic). The following physical parameters are considered: EM=70 GPa, vM=0.3, ρM=2707 kg/m3 for the aluminum, and Ec=168 GPa, vc=0.3, ρc=5700 kg/m3 for the zirconia, respectively. Firstly, the results for natural

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frequencies of FGM spherical shells, spherical caps and parabolic shells with different boundary conditions are compared with those of Qu et al. who used domain decomposition approach [16], Su

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et al. [17] by modified Fourier method, Tornabene [27] using generalized differential quadrature

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method based on FSDT. The comparisons are listed in Tables 2, 3 and 4 respectively. Note that the definition of symbolism about boundary conditions (such as: C-F) is contrary to that in Ref. [27]. The dimensions and material properties of the structures analyzed are also given along with the tables. Comparing the results obtained, it can be concluded that the present analysis yields accurate results when compared with those different solution approaches. Moreover, various geometric shapes of shell structures and boundary conditions are considered. In fact, it is meaningful that the arbitrary boundary conditions are correctly taken into account in the present approach.

ACCEPTED MANUSCRIPT 3.2. Parameter studies Next, some parameter studies for the FGM spherical and parabolic shells under different boundary conditions and power-law indexes are presented. Since the vibration analysis results for

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these kinds of shells with arbitrary boundary conditions are quite limited in the open literature, some new results for the considered FGM shells are calculated, which can be used for benchmarking by

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researchers as well as the reference data for design engineers. Meantime, effects of some geometrical and material parameters on natural frequencies of the considered structures are also investigated. The

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details regarding the geometry of the studied structures are listed together with the tables. In addition, in the following computations, the zeros frequencies corresponding to the rigid body modes are omitted from the results.

3.2.1. FGM shells with classical boundary conditions

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Table 5 presents the first ten natural frequencies (Hz) of symmetrical circular toroids subjected to C-C and F-F boundary conditions. The following geometrical parameters are used: Rb=1.5 m and

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Rb=-1.5 m. The classical volume fraction profiles (i.e., a=1, b=0) are adopted. It is obvious that for

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different values of p (i.e., p=0.6, 1, 5), the natural frequencies for FGMI shell are higher than FGMII shell. In addition, the frequencies of Rb=-1.5 m circular toroids are always higher than Rb=1.5 m structures. Although not presented here, the same conclusions can be made for other types of boundary conditions.

The effects of boundary conditions on the natural frequencies of the FGM spherical shells are also studied. In general, we are more interested in the lowest natural frequency (i.e. fundamental natural frequency) than others of the system in most engineering practices. Table 6 lists the

ACCEPTED MANUSCRIPT fundamental frequency (Hz) of FGMI(a=1/b=0.5/c=1/p) and FGMII(a=1/b=0.5/c=1/p) spherical shells with different classical boundary conditions (i.e., F-C, F-S, F-SD and F-F) and power-law exponents p (i.e., 0.2, 1, 5, 20). It can be observed that the fundamental natural frequency not always decreases as

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the power-law exponent p increases. These results show that the natural frequencies of the FGM spherical shell could be changed easily through varying the volume fraction and choosing different

effect on the natural frequencies of the shell.

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material configuration. In addition, in all the cases, the boundary conditions have a conspicuous

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Then the proposed approach is applied to explore the influence of the power-law exponent p on the fundamental frequency of the FGM parabolic shells subjected to different boundary conditions. Variations of the frequencies of FGM parabolic shells with different power-law exponent p and boundary conditions are illustrated in Fig. 3 which consists of two parts: the left figure illustrates the

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fundamental frequency versus the power-law exponent p obtained using the FGMI(a/b/c/p) distribution, while the right figure shows the frequencies with different power-law exponent p obtained by using

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the FGMII(a/b/c/p) distribution. The results show high-dependence of the natural frequencies on the

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volume fraction and boundary conditions. For example, for FGMI(a=1/b=0/c/p) and FGMII(a=1/b=0/c/p) distributions, the natural frequencies decrease rapidly within the range of p=1 to p=100 and approach slowly to the values for p=1000. As for other distributions, the variations are more complicated. 3.2.2. FGM shells with elastic restraints Table 7 gives the fundamental frequency (Hz) for FGMI(a=1/b=0.5/c=1/p) and FGMII(a=1/b=0.5/c=1/p) spherical and parabolic shells subjected to different sets of elastic restraints conditions. In order to better explore the effect of the elastic restraint stiffness on the vibration of the considered shells,

ACCEPTED MANUSCRIPT more examples are studied. Fig. 4 gives mode frequencies of the 1st, 3rd and 5th versus the spring stiffness for FGMI(a=1/b=0.5/c=1/p=1) spherical and parabolic shells. The geometric parameters of the shell structures are the same as previous numerical example. The left of Fig. 4 shows the frequencies

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for spherical shells with C-E boundary condition. In other words, the spherical edges are clamped at edge ϕ0=0 while the other edge is restrained by only one type of spring components with different stiffness values. While on the right of Fig.4, figures illustrate the frequencies of parabolic shells

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versus the spring stiffness. The shells are also clamped at edge ϕ0=0, but the other edge is restrained

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by five kinds of springs. By setting four groups of springs stiffness are infinity (1015) and changing the rest one, variations of frequencies with spring stiffness are obtained. For spherical and parabolic shells with elastic boundaries, it can be seen clearly from Fig. 4 that natural frequencies increase rapidly within the range of 106 to 1012 and approach slowly to the values of infinity (1018). But for

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parabolic shells; the frequencies remain in a relatively constant value when the rotational spring stiffness ks changes. This can be attributed to that the effect of Φs on natural frequencies is not so

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significant.

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3.2.3. Modes shapes for FGM shells

For illustrative purposes, some selected mode shapes of the aforementioned structures are depicted in Figs. 5-7. The material constants and geometric parameters are the same as the previous examples. These mode shapes have been reconstructed in three-dimensional view by means of considering the displacement field in Eq. (14) after solving the eigenvalue problem. In fact, for the complete shell structures, by interchanging the selection of trigonometric terms, another set of free vibration modes are readily obtained. In these cases, we have depicted these modes shapes in one

ACCEPTED MANUSCRIPT figure. But for n=0, the torsional modes are appeared. From these figures, the vibration behaviors of the various shells subjected to different boundary conditions can be seen vividly, which enhance our understanding of the vibration characteristics of shell structures of revolution.

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4. Conclusions In this paper, an efficient solution approach based on the Haar wavelet differential quadrature

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method has been proposed to study the free vibration behaviors of FGM spherical and parabolic shells with general boundary conditions. The first-order shear deformation shell theory is adopted to

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formulate the theoretical model. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. The displacement and rotation fields of shells are assumed in the form of the products of trigonometric functions along the circumferential direction

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and Haar wavelet series and their integral for the meridional direction. The integration constants are determined by the boundary conditions. Free vibration results, including natural frequencies and

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mode shapes, for FGM shells with different end conditions are presented and compared with those

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obtained by other methods. The comparisons verify the simplicity and efficiency of the proposed approach in modal analysis of all the moderately thick structures considered. Furthermore, discretizing and programming the numerical procedures are quite easy and straightforward. The proposed approach is advantageous for its less computational effort required for handling the complicated boundary conditions. One reason for this is that the compact Haar wavelet operational matrix is used. Some useful conclusions about the high-dependency of natural frequencies of FGM spherical and parabolic shell upon their volume fraction/geometry parameters

ACCEPTED MANUSCRIPT and boundary condition are made through parametric studies and new modal results for FGM shells are presented. In addition, the present study offers a promising alternative to the analytical and numerical approaches for boundary value problems of FGM shells.

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Acknowledgment

The authors gratefully acknowledge the financial support from the National Natural Science

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Appendix A. Detailed expressions of the coefficients Gijk

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Foundation of China (No. 51275289).

 cos ϕ 1 dRϕ  A66 n 2 A12 sin ϕ A11 cos 2 ϕ A66 G111 = − 2 − − − K G = A − 3   c 112 11 R0 Rϕ R0 R02 Rϕ2  Rϕ R0 Rϕ dϕ 

G113 =

A + A66 A11 A +A n G121 = − 11 2 66 n cos ϕ G122 = 12 2 Rϕ R0 Rϕ R0

B + B66 B11 B +B n G151 = − 11 2 66 n cos ϕ G152 = 12 2 Rϕ Rϕ R0 R0

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G143 =

 cos ϕ 1 dRϕ  B66 n 2 B12 sin ϕ B11 cos2 ϕ A − − + K c 66 G142 = B11  − 2 2  R R R 3 dϕ  R0 Rϕ R0 R0 Rϕ ϕ ϕ 0  

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G141 = −

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 cos ϕ 1 dRϕ  A11 sin ϕ cos ϕ A A  A sin ϕ G131 = A11  − 3 − G132 =  112 + 12 + K c 662   2  R R R dϕ  R R0 Rϕ R0 Rϕ  ϕ  ϕ 0   ϕ

G211 = −

A + A66 A11 + A66 n n cos ϕ G212 = − 12 2 R0 Rϕ R0

G221 = −

 cos 2 ϕ sin ϕ   cos ϕ 1 dRϕ  A11n 2 kc A66 sin 2 ϕ − − A − G = A −   66  222 66  2  R R R 3 dϕ  R02 R02 Rϕ R0  ϕ  R0  ϕ 0 

G223 =

 A  A66 A +k A B +B G231 = −  12 + 11 2 c 66 sin ϕ  n G241 = − 11 2 66 n cos ϕ 2 R R  Rϕ R0 R0  ϕ 0 

G242 = −

 cos 2 ϕ sin ϕ B12 + B66 B n2 n G251 = − 11 2 − B66  −  R2 Rϕ R0 R0 R0 Rϕ 0 

 kc A66 sin ϕ  + R0 

ACCEPTED MANUSCRIPT  cos ϕ 1 dRϕ  B G253 = 662 G252 = B66  − 3   R R R dϕ  Rϕ ϕ  ϕ 0  G311 = −

 cos ϕ 1 dRϕ  A12 cos ϕ A11 sin ϕ cos ϕ K A − − − 3   c 66 Rϕ R0 R02  Rϕ R0 Rϕ dϕ 

A A sin ϕ K c A66  G312 = −  112 + 12 +  R Rϕ R0 Rϕ2   ϕ

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 A  k A n2 A A +k A 2 A sin ϕ A11 sin 2 ϕ G321 = −  12 + 11 2c 66 sin ϕ  n G331 = − c 662 − 112 − 12 − R R  R0 Rϕ Rϕ R0 R02 R0  ϕ 0   cos ϕ 1 dRϕ  k A G333 = c 266 G332 = kc A66  − 3   R R R dϕ  Rϕ ϕ  ϕ 0 

B B sin ϕ K c A66 B12 cos ϕ B11 sin ϕ cos ϕ K c A66 cos ϕ G342 = − 112 − 12 + − + 2 Rϕ Rϕ R0 Rϕ Rϕ R0 R0 R0

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G341 = −

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 B B n 2 B sin ϕ B11 cos2 ϕ A B sin ϕ K A  − + K c 66 G351 =  − 12 − 11 2 + c 66  n G411 = − 66 2 − 12 2  R R R0 Rϕ R0 R0 Rϕ R0 R0   ϕ 0  cos ϕ 1 dRϕ  B B +B G = 11 G = − 11 2 66 n cos ϕ G412 = B11  −  R R R 3 dϕ  413 Rϕ2 421 R0 ϕ  ϕ 0   cos ϕ 1 dRϕ  B11 sin ϕ cos ϕ B12 + B66 n G431 = B11  − −  R R R 3 dϕ  Rϕ R0 R02 ϕ  ϕ 0 

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G422 =

B D n2 A  D sin ϕ D11 B sin ϕ G432 =  112 + 12 − K c 66  G441 = − 662 − kc A66 − 12 − 2 cos 2 ϕ R  R R R R0 R R R 0 ϕ 0 ϕ 0 ϕ   ϕ

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 cos ϕ 1 dRϕ  D D +D G443 = 112 G451 = − 66 2 11 n cos ϕ G442 = D11  − 3   R R R dϕ  Rϕ R0 ϕ  ϕ 0 

D12 + D66 B +B B + B66 n G511 = − 11 2 66 n cos ϕ G512 = − 12 n Rϕ R0 Rϕ R0 R0

G521 = −

G523 =

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G452 =

 cos 2 ϕ sin ϕ B11n 2 − B − 66   R2 R02 R0 Rϕ 0 

 kc A66 sin ϕ  cos ϕ 1 dRϕ  G522 = B66  −  +  R R R 3 dϕ  R0 ϕ   ϕ 0 

 B B66 D +D B sin ϕ K A  G531 =  − 12 − 11 2 + c 66  n G541 = − 66 2 11 n cos ϕ 2  R R Rϕ R0 R0  R0  ϕ 0

G542 = −

 cos 2 ϕ sin ϕ  D12 + D66 D n2 n G551 = − 112 − kc A66 − D66  −   R2 Rϕ R0 R0 Rϕ R0  0 

ACCEPTED MANUSCRIPT  cos ϕ 1 dRϕ  D G553 = 662 G552 = D66  − 3   R R R dϕ  Rϕ ϕ  ϕ 0 

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ACCEPTED MANUSCRIPT List of Tables and Figures Table 1 The corresponding spring stiffness values for the general boundary conditions.

and power-law exponents (ϕ0=π/8, ϕ1=π/2, h/R=0.05, R=1 m, m=1).

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Table 2 Comparison of frequencies (Hz) for FGM spherical shells with different boundary conditions

Table 3 Comparison of the fundamental frequencies (Hz) of FGM spherical caps with different

SC

boundary conditions and power-law exponents (h/R=0.1, Rb=0 m, R=1 m).

Table 4 Comparison of the first ten frequencies (Hz) for C-F FGM parabolic shells with different

M AN U

power-law exponents (Rb=6 m, h=0.3 m, k=4.5, a=0 m, c=-3 m, b=2 m).

Table 5 The first ten frequencies (Hz) for F-F and C-C FGM circular toroids with different power-law exponents (R=3 m, h=0.2 m, ϕ0=π/3, ϕ1=2π/3).

Table 6 The fundamental frequencies (Hz) of various FGM spherical shells with different boundary

TE D

conditions and power-law exponents (h/R=0.05, Rb=0 m, R=1 m). Table 7 The fundamental frequency (Hz) for FGM spherical and parabolic shells subjected to

EP

different sets of elastic restraints conditions (ϕ0=π/4, ϕ1=π/2, h/R=0.05, Rb=0 m, R=1 m for spherical

AC C

shells; h=0.1 m, Rb=0 m, k=8, a=4, c=1, b=2 for parabolic shells). Fig.1. The geometric parameters and coordinate system of a FG shell: (a) the spherical shell coordinate system; (b) a differential element of the FGM shell; (c) the parabolic shell coordinate system; (d) circumferential section of the mid- surface. Fig. 2. FGM shell structures: (a) spherical shell; (b) spherical cap; (c) circular toroid (Rb<0); (d) circular toroid (Rb>0); (e) parabolic toroid; (f) parabolic dome. Fig. 3. Variation of the fundamental frequencies of FGM parabolic shells with different power-law

ACCEPTED MANUSCRIPT exponent p and boundary conditions: (a) FGMI(a=1/b=0/c/p); (b) FGMII(a=1/b=0/c/p); (c) FGMI(a=0/b=-0.5/c=2/p); (d) FGMII(a=0/b=-0.5/c=2/p); (e) FGMI(a=1/b=1/c=2/p); (f) FGMII(a=1/b=1/c=2/p). Fig.4. Variation of frequencies versus the spring stiffness : (a) ku; (b) kv; (c) kw; (d) kφ; (e) ks.

RI PT

Fig. 5. Mode shapes for FGMI(a=1/b=0.5/c=1/p=5) hemispherical caps with F-SD boundary condition. Fig. 6. Mode shapes for FGMI(a=1/b=0.5/c=1/p=1) spherical shells with E3-E3 boundary condition.

AC C

EP

TE D

M AN U

SC

Fig. 7. Mode shapes for FGMI(a=1/b=0.5/c=1/p=1) parabolic shells with E1-E3 boundary condition.

ACCEPTED MANUSCRIPT Table 1 The corresponding spring stiffness values for the general boundary conditions. kv 1e15 1e15 1e15 0 1e15 1e15 1e15

kw 1e15 1e15 1e15 0 1e7 1e15 1e7

M AN U TE D EP

kφ 1e15 0 0 0 1e15 1e7 1e7

RI PT

ku 1e15 1e15 0 0 1e15 1e15 1e15

SC

Essential conditions U=V=W=Φϕ=Φs =0 U=V=W=Mϕ=Φs=0 Nϕ=V=W=Mϕ=Mϕs =0 Nϕ=Nϕs=Qϕ=Mϕ=Mϕs=0 W≠0,U=V=Φϕ=Φs =0 Φϕ≠0, U=V=W=Φs =0 W≠0, Φϕ≠0, U=V=Φs =0

AC C

BC C SS SD F E1 E2 E3

ks 1e15 1e15 0 0 1e15 1e15 1e15

ACCEPTED MANUSCRIPT Table 2 Comparison of frequencies (Hz) for FGM spherical shells with different boundary conditions and power-law exponents (ϕ0=π/8, ϕ1=π/2, h/R=0.05, R=1 m, m=1).

FGMII(a=1/b=0/c/p) 0 777.06 1 771.53 0.6 2 49.58 3 132.82 4 242.05

20

0 1 2 3 4

743.79 738.21 50.72 135.51 246.44

164.93 495.04 778.91 821.72 875.37

164.85 494.98 778.88 821.71 875.39

744.19 738.50 50.86 135.90 247.16

161.07 474.30 746.87 790.26 846.32

777.04 771.53 49.60 132.88 242.16 743.76 738.20 50.73 135.55 246.52

AC C

20

Ref. [16]

786.91 827.89 886.90 881.64 914.97

786.84 827.82 886.85 881.61 914.97

875.70 910.69 894.68 889.45 927.36

875.58 910.57 894.60 889.41 927.34

161.00 474.26 746.84 790.24 846.32

756.34 797.41 851.92 848.53 885.18

756.27 797.35 851.88 848.51 885.17

853.46 887.13 859.37 856.57 897.90

853.32 887.00 859.32 856.53 897.88

164.67 494.05 779.52 822.78 876.35

164.59 494.02 779.46 822.72 876.32

767.98 815.21 887.95 885.56 919.44

767.93 815.16 887.89 885.52 919.43

873.83 908.80 893.81 888.32 925.77

873.71 908.68 893.73 888.28 925.75

160.95 473.80 747.07 790.66 846.63

160.87 473.77 747.02 790.61 846.60

746.69 790.81 852.68 850.59 887.38

746.63 790.75 852.64 850.56 887.37

852.25 885.93 858.76 855.76 896.82

852.12 885.80 858.70 855.73 896.79

RI PT

744.21 738.51 50.85 135.86 247.08

0 1 2 3 4

Ref. [16]

C-C Present

SC

777.61 771.92 49.85 133.53 243.34

FGMI(a=1/b=0/c/p) 0 777.63 1 771.92 0.6 2 49.83 3 133.47 4 243.22

Ref. [16]

SS-SS Present

M AN U

Ref. [16]

SD-SD Present

TE D

n F-F Present

EP

p

ACCEPTED MANUSCRIPT Table 3 Comparison of the fundamental frequencies (Hz) of FGM spherical caps with different boundary conditions and power-law exponents (h/R=0.1, Rb=0 m, R=1 m). ϕ1

p

F-E1

F-C Present

F-E2

Ref.[17]

Present

Ref.[17]

Present

FGMI(a=1/b=0.5/c=1/p)

π/2

879.55

879.53

970.88

873.06

873.04

954.75

856.14

856.12

947.82

850.06

850.04

903.98

904.03

977.25

1

1098.16

1098.11

898.15

898.20

970.91

5

1079.46

1079.41

879.12

879.17

954.78

20

1072.25

1072.20

861.68

861.72

947.84

0.2

528.50

528.48

481.55

481.65

500.30

500.28

480.63

480.70

1

524.92

524.90

478.35

478.45

499.12

499.11

477.85

477.93

5

513.64

513.62

467.70

467.79

492.29

492.28

467.64

467.72

20

504.12

504.10

457.89

457.97

481.22

481.20

457.62

457.69

1105.31

1105.26

903.85

903.91

977.40

977.38

880.62

880.60

1

1095.29

1095.24

897.55

897.60

971.50

971.47

877.59

877.57

5

1071.77

1071.73

877.28

877.33

955.21

955.18

866.02

866.00

20

1066.70

1066.65

860.24

860.28

947.08

947.06

855.78

855.76

0.2

528.44

528.42

481.55

481.65

499.14

499.12

480.33

480.40

1

524.69

524.67

478.35

478.45

494.09

494.07

476.58

476.64

5

513.05

513.03

467.70

467.80

480.33

480.31

464.70

464.75

20

503.70

503.69

457.90

457.98

473.65

473.64

455.78

455.84

TE D

0.2

EP

π/2

977.22

1105.90

AC C

0

π/4

Ref.[17]

1105.95

FGMII(a=1/b=0.5/c=1/p) 0

Present

SC

0

π/4

Ref.[17]

0.2

M AN U

0

F-E3

RI PT

ϕ0

ACCEPTED MANUSCRIPT Table 4 Comparison of the first ten frequencies (Hz) for C-F FGM parabolic shells with different power-law exponents (Rb=6 m, h=0.3 m, k=4.5, a=0 m, c=-3 m, b=2 m). p=0 Present

Ref.[27]

p=0.6

p=1

p=20

p=50

Present

Ref.[27]

Present

Ref.[27]

Present

Ref.[27]

Present

Ref.[27]

FGMI(a=0/b=-0.5/c=2/p)

RI PT

Hz

52.60

52.59

52.38

52.37

52.25

52.24

51.17

51.16

51.00

50.99

f2

52.60

52.59

52.38

52.37

52.25

52.24

51.17

51.16

51.00

50.99

f3

59.28

59.27

58.98

58.97

58.80

58.79

57.75

57.75

57.76

57.75

f4

59.28

59.27

58.98

58.97

58.80

58.79

57.75

57.75

57.76

57.75

f5

59.70

59.69

59.47

59.46

59.32

59.32

58.08

58.08

57.85

57.85

f6

59.70

59.69

59.47

59.46

59.32

59.32

58.08

58.08

57.85

57.85

f7

74.38

74.38

74.05

74.04

73.84

73.83

72.61

72.61

72.53

72.53

f8

74.77

74.76

74.30

74.29

74.00

73.99

73.02

73.01

73.37

73.36

f9

74.77

74.77

74.30

74.29

74.00

74.00

73.02

73.01

73.37

73.36

f10

96.50

96.51

95.77

95.78

95.32

95.33

94.42

94.42

95.25

95.26

52.09

52.09

50.71

50.70

50.62

50.61

52.09

52.09

50.71

50.70

50.62

50.61

58.63

58.62

57.25

57.24

57.35

57.34

M AN U

FGMII(a=0/b=-0.5/c=2/p)

SC

f1

52.60

52.59

52.28

52.28

f2

52.60

52.59

52.28

52.28

f3

59.28

59.27

58.87

58.86

f4

59.28

59.27

58.87

58.86

58.63

58.62

57.25

57.24

57.35

57.34

f5

59.70

59.69

59.34

59.34

59.13

59.12

57.50

57.50

57.37

57.37

f6

59.70

59.69

59.34

59.34

59.13

59.12

57.50

57.50

57.37

57.37

f7

74.38

74.38

73.85

73.84

73.53

73.52

71.66

71.65

71.74

71.74

f8

74.77

74.76

74.16

74.15

73.79

73.78

72.39

72.38

72.86

72.85

f9

74.77

74.77

74.16

74.15

73.79

73.78

72.39

72.38

72.86

72.85

f10

96.50

96.51

95.59

95.60

95.05

95.06

93.61

93.62

94.60

94.60

AC C

EP

TE D

f1

ACCEPTED MANUSCRIPT Table 5 The first ten frequencies (Hz) for F-F and C-C FGM circular toroids with different power-law exponents (R=3 m, h=0.2 m, ϕ0=π/3, ϕ1=2π/3). Rb=1.5 F-F p=0.6

Rb=-1.5 C-C

F-F

p=1

p=5

p=0.6

p=1

p=5

p=0.6

FGMI(a=1/b=0/c/p)

p=1

C-C

RI PT

Hz

p=5

p=0.6

p=1

p=5

9.613

9.574

9.857

307.570

305.977

302.035

90.610

90.417

93.274

351.156

349.556

344.047

f2

9.613

9.574

9.857

307.570

305.977

302.035

90.610

90.417

93.274

351.156

349.556

344.047

f3

15.424

15.361

15.929

310.596

309.009

304.602

124.031

123.722

127.743

377.604

375.719

367.343

f4

15.424

15.361

15.929

310.596

309.009

304.602

124.031

123.722

127.743

377.604

375.719

367.343

f5

27.900

27.785

28.525

310.753

309.126

305.781

249.328

248.720

254.418

383.997

382.497

381.660

f6

27.900

27.785

28.525

310.753

309.126

305.781

249.328

248.720

254.418

383.997

382.497

381.660

f7

43.385

43.206

44.759

319.056

317.452

315.023

296.225

295.541

304.122

434.517

432.138

421.030

f8

43.385

43.206

44.759

319.056

317.452

315.023

296.225

295.541

304.122

480.129

478.586

484.508

f9

53.809

53.582

54.886

321.706

320.063

315.146

426.437

425.311

433.789

480.129

478.586

484.508

f10

53.809

53.582

54.886

321.762

320.063

315.146

426.437

425.311

433.789

522.168

519.786

512.436

M AN U

FGMII(a=1/b=0/c/p)

SC

f1

9.515

9.451

9.735

305.398

303.226

299.335

88.490

87.734

90.582

346.973

344.266

338.903

f2

9.515

9.451

9.735

305.398

303.226

299.335

88.490

87.734

90.582

346.973

344.266

338.903

f3

15.277

15.179

15.759

308.339

306.148

301.969

121.387

120.395

124.473

373.838

371.002

362.985

f4

15.277

15.179

15.759

308.339

306.148

301.969

121.387

120.395

124.473

373.838

371.002

362.985

f5

27.627

27.440

28.186

308.616

306.423

302.956

243.789

241.704

247.403

378.290

375.262

374.409

f6

27.627

27.440

18.186

308.616

306.423

302.956

243.789

241.704

247.403

378.290

375.262

374.409

f7

42.980

42.702

44.283

316.662

314.419

312.133

289.496

287.037

295.641

431.156

428.038

417.667

f8

42.980

42.702

44.283

316.662

314.419

312.133

289.496

287.037

295.641

471.442

467.569

473.356

f9

53.301

52.941

54.257

318.318

316.065

312.394

417.388

413.838

422.249

471.442

467.569

473.356

f10

53.301

52.941

54.257

319.604

317.337

312.394

417.388

413.838

422.249

516.016

512.017

504.937

AC C

EP

TE D

f1

ACCEPTED MANUSCRIPT Table 6 The fundamental frequencies (Hz) of various FGM spherical shells with different boundary conditions and power-law exponents (h/R=0.05, Rb=0 m, R=1 m). F-C

F-S

F-F

0.2 1 5 20

509.226 505.780 494.648 484.874

490.676 488.715 480.208 468.893

52.670 52.179 51.559 52.522

290.729 289.568 287.263 286.775

π/2

0.2 1 5 20

311.213 308.630 303.794 305.046

π/4

3π/4

0.2 1 5 20

105.558 104.667 103.060 103.639

π/2

π/4

3π/4

100.227 99.301 98.003 99.523

75.836 75.152 74.197 75.293

64.293 63.816 62.579 61.966

489.869 485.168 471.499 463.272

52.637 52.040 51.217 52.289

635.168 630.842 616.312 603.373

99.775 99.336 98.584 98.771

0.2 1 5 20

311.089 308.083 302.414 304.083

289.886 285.864 278.192 280.913

46.428 45.908 45.162 46.027

100.178 99.092 97.506 99.201

0.2 1 5 20

105.516 104.481 102.599 103.325

99.490 98.082 95.531 96.820

75.792 74.961 73.717 74.964

64.280 63.761 62.453 61.887

AC C

π/4

509.211 505.716 494.482 484.757

EP

FGMII(a=1/b=0.5/c=1/p) 0 0.2 π/2 1 5 20

635.189 630.952 616.724 603.723

46.456 46.030 45.470 46.239

TE D

π/4

F-SD

RI PT

FGMI(a=1/b=0.5/c=1/p) 0 π/2

p

SC

ϕ1

M AN U

ϕ0

ACCEPTED MANUSCRIPT Table 7 The fundamental frequency (Hz) for FGM spherical and parabolic shells subjected to different sets of

Rb=0 m, k=8, a=4, c=1, b=2 for parabolic shells).

RI PT

elastic restraints conditions (ϕ0=π/4, ϕ1=π/2, h/R=0.05, Rb=0 m, R=1 m for spherical shells; h=0.1 m,

E1-E1

E1-E2

E1-E3

E2-E2

E2-E3

E3-E3

p=0.2 p=1 p=5 p=20 p=0.2 p=1 p=5 p=20

805.03 800.00 783.89 769.71 158.15 157.07 153.83 151.35

911.08 908.06 897.95 888.94 156.93 155.70 152.37 150.53

799.62 794.60 778.82 765.45 156.06 154.79 151.40 149.28

1052.2 1043.0 1020.8 1013.9 156.77 155.51 152.11 150.29

889.08 886.80 879.08 872.28 155.98 154.70 151.18 149.08

795.45 790.40 774.72 761.93 155.93 154.67 151.15 149.06

FGMII(a=1/b=0.5/c=1/p) Spherical shell p=0.2 p=1 p=5 p=20 Parabolic shell p=0.2 p=1 p=5 p=20

804.95 799.59 782.62 768.71 158.14 157.04 153.72 151.27

910.02 903.50 887.21 881.86 157.08 156.32 153.74 151.35

799.83 795.41 779.96 765.86 156.18 155.31 152.56 149.97

1054.0 1050.4 1036.4 1023.2 156.93 156.21 153.67 151.24

888.66 884.84 873.51 868.16 156.11 155.25 152.51 149.88

795.89 792.11 777.76 763.47 156.08 155.24 152.50 149.88

AC C

EP

M AN U

Parabolic shell

TE D

FGMI(a=1/b=0.5/c=1/p) Spherical shell

SC

p

ACCEPTED MANUSCRIPT Rb

Reference surface R0

c1

Rϕ

dϕ

Rs

Rb R0 dϕ

c1

SC

ϕ

R0

Rϕ

Rs

M AN U

c2

Rϕ

Rs

RI PT

c2

ϕ

Fig.1. The geometric parameters and coordinate system of a FG shell: (a) the spherical shell coordinate system; (b) a differential element of the FGM shell; (c) the parabolic shell coordinate system; (d)

AC C

EP

TE D

circumferential section of the mid- surface.

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 2. FGM shell structures: (a) spherical shell; (b) spherical cap; (c) circular toroid (Rb<0); (d) circular

AC C

EP

toroid (Rb>0); (e) parabolic toroid; (f) parabolic dome.

200

200

ACCEPTED MANUSCRIPT 180

160

Frequency (Hz)

C-C S-C S-S F-C F-S

140

120

100 -2 10

160

C-C S-C S-S F-C F-S

140

120

-1

10

0

1

10

10

2

100 -2 10

3

10

10

-1

10

p

2

10

140

160

140

M AN U

C-C S-C S-S F-C F-S

160

SC

180 Frequency (Hz)

180

3

10

C-C S-C S-S F-C F-S

120

-1

10

0

1

10

10

2

10

p

(c) 200

100 -2 10

3

10

-1

10

0

1

10

10

2

10

3

10

p

(d)

TE D

200

180

160

EP

140

C-C S-C S-S F-C F-S

AC C

120

-1

10

0

10

p

(e)

1

10

180 Frequency (Hz)

Frequency (Hz)

200

120

Frequency (Hz)

10

(b)

200

100 -2 10

1

10

p

(a)

100 -2 10

0

RI PT

Frequency (Hz)

180

160

C-C S-C S-S F-C F-S

140

120

2

10

3

10

100 -2 10

-1

10

0

1

10

10

2

10

3

10

p

(f)

Fig. 3. Variation of the fundamental frequencies of FGM parabolic shells with different power-law exponent p and boundary conditions: (a) FGMI(a=1/b=0/c/p); (b) FGMII(a=1/b=0/c/p); (c) FGMI(a=0/b=-0.5/c=2/p); (d) FGMII(a=0/b=-0.5/c=2/p); (e) FGMI(a=1/b=1/c=2/p); (f) FGMII(a=1/b=1/c=2/p).

200

(a) ACCEPTED MANUSCRIPT

800 600 400

1

200

3 5

0 0 10

10

5

10

10

10

st rd th

150 1

100

3 5 50 0 10

15

10

Spring stiffness: ku 200

800 600 400

1 3

200

5 10

5

10

10

10

st rd th

(b)

150

10

10

10

10

rd

th

15

M AN U

Spring stiffness: kv

400

1

200

3 5

10

5

10

10

10

st

rd

th

Frequency (Hz)

600

(c)

(d) 400

AC C

EP

1

100 0 10

10

5

10

10

3 5

10

st

rd th

170 160

10

10

10

(d)

15

1 3

180

5

st rd th

170 160 10

5

10

10

10

Spring stiffness: k

φ

15

φ

600 1 400

3 5

200

10

10

10

Spring stiffness: ks

15

st rd th

Frequency (Hz)

200

(e)

5

10

190

150 0 10

15

800

10

5

200

Spring stiffness: k

0 0 10

th

5

Spring stiffness: kw

Frequency (Hz)

500

rd

3

180

150 0 10

15

st

1

190

Spring stiffness: kw

Frequency (Hz)

5

5

st

200

(c)

0 0 10

Frequency (Hz)

th

15

3

50 0 10

15

TE D

Frequency (Hz)

800

200

10

1

100

Spring stiffness: kv

300

10

SC

(b)

0 0 10

10

rd

Spring stiffness: ku

Frequency (Hz)

Frequency (Hz)

1000

5

st

RI PT

(a)

Frequency (Hz)

Frequency (Hz)

1000

(e) 190 180

1

st

3

rd

5

th

170 160 0 10

10

5

10

10

10

15

Spring stiffness: ks

Fig.4. Variation of frequencies versus the spring stiffness :(a) ku; (b) kv; (c) kw; (d) kφ; (e) ks.

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EP

TE D

Fig. 5. Mode shapes for FGMI(a=1/b=0.5/c=1/p=5) hemispherical caps with F-SD boundary condition.

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TE D

Fig. 6. Mode shapes for FGMI(a=1/b=0.5/c=1/p=1) spherical shells with E3-E3 boundary condition.

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TE D

Fig. 7. Mode shapes for FGMI(a=1/b=0.5/c=1/p=1) parabolic shells with E1-E3 boundary condition.