# A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

Authors: Mariame Nassit; Abderrahmane El Harif; Hassan Berbia; Mourad Taha Janan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.


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## Body

## 1. Introduction

With the fast and continuous advancement of technology, it becomes necessary to have more efficient and effective innovative devices and engineering structures. Many publications have focused on modeling lightweight structures [1–3]. Thin spherical shells are used in civil engineering and a variety of engineering applications such as submarine hulls, aircraft, smart devices, and health monitoring systems [4–7]. It is well known that the spherical shells can be used in complex environmental conditions and can be exposed to a variety of dynamic excitations inducing excessive vibrations [8]. Undesirable vibrations cause structural failure, cracks in parts of machine elements, malfunction of electronic devices, and many other issues [9]. By measuring structure or system frequencies, designers can select speeds remote from resonance and can prevent many adverse effects of vibration [9]. Determining the frequency spectrum of a structure is an essential step of its design. In addition, developing an accurate and unified analytical solution for axisymmetric thin spherical shells with free edge or clamped edge boundary conditions remains an extremely challenging task. This solution is useful to provide reference data for future research in related fields. Hence, a survey of the literature depicts the existence of many studies on free vibrations analysis [10–14]. Besides, the research for curved shells, especially for cylindrical and spherical shells, have attracted much attention recently [7, 15–20].Shell structures' vibrations analysis can be carried out by analytical or numerical methods and can be tackled by different theories and approximations. The linear theories for plates and shells are as follows:(i)
Love–Kirchhoff’s theory that does not take account of transverse shear deformation that generalizes Euler–Bernoulli’s theory for beams [21–23].(ii)
Mindlin–Reissner’s theory for plates and Naghdi–Reissner’s one for shells, taking into account the transverse shear deformation extending Timoshenko’s theory for beams [24–30].They all assume an elastic behavior. They are said to be of the first-order: the displacement fields vary linearly with variablet (Figure 1) in the thickness, while the thickness does not vary.Figure 1
The thin spherical shell’s thickness in the spherical coordinate system.However, higher-order theories have been developed like the theory HSDTs (Reddy) and many others [31].Exact (Naghdi–Kalnins, Kunieda, current method) or approximate (Kunieda) analytical resolution methods are based on Legendre functions [32–34]. One cites among the numerical methods, finite differences, weighted residuals, variational methods (Rayleigh–Ritz), and finite element method [35–38].Using the finite element method, El Harif studied the transverse shear effect in plates and shells of revolution [39]. Batoz et al. focused on structure modeling [40]. Juber investigated the analytical solution for the dynamic analysis of thin composite cylindrical and spherical shells. He presented the fundamental natural frequencies and mode shapes for simply supported composite cylindrical and spherical shells, by applying the third-order theory of Reddy [41]. Moreover, Sammoura et al. developed an analytical solution for curved piezoelectric micromachined ultrasonic transducers by applying Naghdi–Kalnins’ theory [17].The present paper concentrate on a new analytical method inspired by the Naghdi–Kalnins’ theory [32] and the model presented in [39, 40]. This method solves elastic thin spherical shells’ axisymmetric vibrations’ differential equations. For this purpose, we introduce the differential equations of motion in terms of stress, where we take into consideration the terms of the inertial couples in the stress couples’ equations. These terms are omitted in the analytical methods proposed by Naghdi–Kalnins and Kunieda [32, 33]. Then, we introduce a new stress function different from Naghdi–Kalnins’ one and derivate new equations depending on the displacement and the stress function. Moreover, by using the Legendre functions, we analytically solve these differential equations in the specific case of the hemispherical shell’s harmonic axisymmetric vibrations, under the free and clamped edges boundary conditions. To check the validity and accuracy of the current method, we compare the current analytical results with Naghdi–Kalnins’ results [32] and Kunieda’s results [33]. It is worth to mention that Naghdi–Kalnins’ theory is one of the most important references used in different recent studies [17, 42, 43]. After that, we compare the results with the finite element method (FEM) ones. The FEM numerical simulation results confirm the correctness and accuracy of the current analytical method.The remainder of this paper is organized as follows: in Section2, we introduce the analytical model formulation. In Section 3, we propose the general solution for harmonic axisymmetric vibration. After that, we present and discuss the results in Section 4. The conclusions of this study are described in Section 5.


## 2. Analytical Model Formulation

### 2.1. Basic Concept

Inspired by Naghdi–Kalnins’ theory of axisymmetric vibrations of the elastic spherical shell, we introduce here new basic concepts that are used to analyze the axisymmetric vibrations response of the elastic spherical shells as illustrated in Figure2. The main purpose of the approaches presented below is to derivate the expression of the differential equations by applying the motion equations with their stress couples equations. These stress couples equations take into account the terms of the inertial couples. These terms are omitted by Naghdi–Kalnins. In addition, we propose a new stress function that helps to derivate the new equations, different from Naghdi–Kalnins’ equations. On the other side, as the thickness of the shells h is small in comparison with the radius Rh/R≪1, we use Love’s first-order approximation for the dynamic analysis of the thin spherical shell. We assume that the transverse shear and normal strains are negligible. We recall here that the basic equations characterize the elastic deformation of the spherical shell in the weightlessness and the spherical coordinate system ρ,θ,φ.Figure 2
Middle surface element of the thin spherical shell.


### 2.2. Strain-Displacement Relations in Spherical Coordinates

By applying Love-Kirchhoff’s assumption, the strain can be expressed as a function of the displacement vector components and their derivatives in the spherical coordinate system, whereuφ,uθ, and w are the displacement vector components in φ−,θ− , and ρ− directions, respectively.Furthermore, the total strain of a spherical shell can be expressed into two components: membrane strainsεij0 and flexural strains εij1 as mentioned in [44]:(1)εij=εij0+tεij1,where i,j∈φ,θ.As shown in Figure1, t is the radial position measured from the middle surface along the spherical shell thickness, where(2)−h2≤t≤h2.(3)OM⟶=Reρ⟶,OP⟶=Reρ⟶+teρ⟶,where M is a point on the middle surface.The membrane and the flexural strains are given by the following expressions [44]:(4)εφφ0=1Ruφ,φ+w,(5)εθθ0=1Rcotφuφ+1sinφuθ,θ+w,(6)2εθφ0=1Ruθ,φ−cotφuφ+1sinφuφ,θ,(7)εφφ1=1R2uφ,φ−w,φφ,(8)εθθ1=1R2cotφuφ+1sinφuθ,θ−cotφw,φ−1sin2φw,θθ,(9)2εθφ1=1R21sinφuφ,θ−cotφuθ+uθ,φ−2sinφ2w,φθ+2cotφsinφw,θ.


### 2.3. Stress Resultants and Stress Couples in Spherical Coordinate System

The tangential stresses of the spherical shells in spherical coordinates can be expressed as(10)σφφ=E1−ν2εφφ+νεθθ,σθθ=E1−ν2εθθ+νεφφ,σφθ=σθφ=E1−ν2εθφ,where E is Young’s modulus, ν is Poisson’s ratio.As shown in Figure3, it is convenient to introduce the stress resultants Nij, transverse shear stress resultants Qθθ,Qφφ, and the stress couples Mij, where i,j∈θ,φ, by integrating the infinitesimal stresses along the thickness of the spherical shell as [44].Figure 3
The stress resultants, transverse shear stress resultants, and stress couples on a differential element extracted from spherical shell.Moreover, the stress resultantsNφφ,Nθθ,and,Nθφ can be written as(11)NφφNθθNφθ=Nθφ=∫−h/2h/2σφφσθθσθφ1+tRdt,(12)NφφNθθNφθ=Nθφ=Eh1−ν21ν0ν10001−ν2εφφ0εθθ02εθφ0.The stress couples can be expressed by(13)MφφMθθMφθ=Mθφ=∫−h/2h/2σφφσθθσθφ1+tRtdt,MφφMθθMφθ=Mθφ=D1ν0ν10001−ν2εφφ1εθθ12εθφ1,where D is the flexural rigidity defined as [17](14)D=Eh3121−ν2.


### 2.4. Stress Differential Equations of Motion

The stress differential equations of motion in the weightlessness can be expressed as [39, 40](15)sinφNφφ,φ+sinφQφφ−cosφNθθ+Nφθ,θ=mvhRsinφd2uφdt2,(16)sinφNθφ,φ+Nθθ,θ+cosφNφθ+sinφQθθ=mvhRsinφd2uθdt2,(17)sinφQφφ,φ−sinφNφφ−sinφNθθ+Qθθ,θ+Rsinφpr=mvhRsinφd2wdt2,(18)sinφMφφ,φ−RsinφQφφ−cosφMθθ+Mφθ,θ=mvh34sinφd2uφdt2−mvh312sinφd2w,φdt2,(19)sinφMθφ,φ+Mθθ,θ+cosφMφθ−RsinφQθθ=mvh34sinφd2uθdt2,where mv is the mass density of the material and pr is the received pressure perpendicular to the spherical shell.


### 2.5. Derivation of Differential Equations for Axisymmetric Vibrations

According to the torsionless axisymmetry with respect to thez-axis, we have(20)uθ=∂∂θ⋅=0;Nφθ=Nθφ=Nρθ=Qθθ=0;Mφθ=Mθφ=0.The compatibility relation can be deduced from (4), (5), and (12) Nφφ,Nθθ by eliminating uθ and replacing the strains by their expressions according to the stress resultants:(21)tanφNθθ−νNφφ,φ+sec2φ+νNθθ−νsec2φ+1Nφφ=EhRtanφw,φ+tanφw.The expression of the shear stress resultantQφφ as a function of displacements uφ and w can be obtained from (15) and (18):(22)Qφφ=−DR▽2w+2R2w,φ+mv1−ν2RDEd2w,φdt2−2mv1−ν2RDEd2uφdt2.Inspired by [32], we introduce a new stress function Fφ,t through the following new relations:(23)Nφφ=cotφF,φ+F−mv1+νER2d2Fdt2+DR▽2w+2R2w−mv1−ν2RDEd2wdt2,(24)Nθθ=F,φφ+F−mv1+νER2d2Fdt2+DR▽2w+2R2w−mv1−ν2RDEd2wdt2,where, as proposed in [32], we have(25)▽2≡1R2,φφ+cotφ,φ.The tangential displacementuφ can be related to the radial displacement w, by elimination of uφ from (15) as(26)uφ=tanφREhNθθ−νNφφ−w.By using (4), (5), (15), (22), (23), and (24), we get the following differential equation:(27)d2Γdt2=0,where Γ is expressed as(28)Γ=R2▽2F+1−νF−mv1−ν2R2Ed2Fdt2+1−νDR▽2+2R2w−EhRw−mv1−ν1−ν2RDEd2wdt2.By substituting the equations (23) and (24) in the compatibility equation (21), another differential equation related also to F and w can be defined as(29)tanφΓ+Γ,φ=0.The solution of (27) and (29) can be written as Γ+At+B=0, where the coefficients A and B can be set to zero without loss in generality. Then, the first differential equation in F and w can be defined as(30)R2▽2F+1−νF−mv1−ν2R2Ed2Fdt2+1−νDR▽2+2R2w−EhRw−mv1−ν1−ν2RDEd2wdt2=0.To obtain the second differential equation inF and w, the shear stress resultant Qφφ (22) into (17) and using (23) and (24) can be substituted:(31)R2▽2F+2F−2mv1+νR2E1+h212▽2d2Fdt2DR▽2+2R2▽2+2R2w+mvRh−1−ν2DE▽2d2wdt2−Rpr=0.


## 2.1. Basic Concept

Inspired by Naghdi–Kalnins’ theory of axisymmetric vibrations of the elastic spherical shell, we introduce here new basic concepts that are used to analyze the axisymmetric vibrations response of the elastic spherical shells as illustrated in Figure2. The main purpose of the approaches presented below is to derivate the expression of the differential equations by applying the motion equations with their stress couples equations. These stress couples equations take into account the terms of the inertial couples. These terms are omitted by Naghdi–Kalnins. In addition, we propose a new stress function that helps to derivate the new equations, different from Naghdi–Kalnins’ equations. On the other side, as the thickness of the shells h is small in comparison with the radius Rh/R≪1, we use Love’s first-order approximation for the dynamic analysis of the thin spherical shell. We assume that the transverse shear and normal strains are negligible. We recall here that the basic equations characterize the elastic deformation of the spherical shell in the weightlessness and the spherical coordinate system ρ,θ,φ.Figure 2
Middle surface element of the thin spherical shell.


## 2.2. Strain-Displacement Relations in Spherical Coordinates

By applying Love-Kirchhoff’s assumption, the strain can be expressed as a function of the displacement vector components and their derivatives in the spherical coordinate system, whereuφ,uθ, and w are the displacement vector components in φ−,θ− , and ρ− directions, respectively.Furthermore, the total strain of a spherical shell can be expressed into two components: membrane strainsεij0 and flexural strains εij1 as mentioned in [44]:(1)εij=εij0+tεij1,where i,j∈φ,θ.As shown in Figure1, t is the radial position measured from the middle surface along the spherical shell thickness, where(2)−h2≤t≤h2.(3)OM⟶=Reρ⟶,OP⟶=Reρ⟶+teρ⟶,where M is a point on the middle surface.The membrane and the flexural strains are given by the following expressions [44]:(4)εφφ0=1Ruφ,φ+w,(5)εθθ0=1Rcotφuφ+1sinφuθ,θ+w,(6)2εθφ0=1Ruθ,φ−cotφuφ+1sinφuφ,θ,(7)εφφ1=1R2uφ,φ−w,φφ,(8)εθθ1=1R2cotφuφ+1sinφuθ,θ−cotφw,φ−1sin2φw,θθ,(9)2εθφ1=1R21sinφuφ,θ−cotφuθ+uθ,φ−2sinφ2w,φθ+2cotφsinφw,θ.


## 2.3. Stress Resultants and Stress Couples in Spherical Coordinate System

The tangential stresses of the spherical shells in spherical coordinates can be expressed as(10)σφφ=E1−ν2εφφ+νεθθ,σθθ=E1−ν2εθθ+νεφφ,σφθ=σθφ=E1−ν2εθφ,where E is Young’s modulus, ν is Poisson’s ratio.As shown in Figure3, it is convenient to introduce the stress resultants Nij, transverse shear stress resultants Qθθ,Qφφ, and the stress couples Mij, where i,j∈θ,φ, by integrating the infinitesimal stresses along the thickness of the spherical shell as [44].Figure 3
The stress resultants, transverse shear stress resultants, and stress couples on a differential element extracted from spherical shell.Moreover, the stress resultantsNφφ,Nθθ,and,Nθφ can be written as(11)NφφNθθNφθ=Nθφ=∫−h/2h/2σφφσθθσθφ1+tRdt,(12)NφφNθθNφθ=Nθφ=Eh1−ν21ν0ν10001−ν2εφφ0εθθ02εθφ0.The stress couples can be expressed by(13)MφφMθθMφθ=Mθφ=∫−h/2h/2σφφσθθσθφ1+tRtdt,MφφMθθMφθ=Mθφ=D1ν0ν10001−ν2εφφ1εθθ12εθφ1,where D is the flexural rigidity defined as [17](14)D=Eh3121−ν2.


## 2.4. Stress Differential Equations of Motion

The stress differential equations of motion in the weightlessness can be expressed as [39, 40](15)sinφNφφ,φ+sinφQφφ−cosφNθθ+Nφθ,θ=mvhRsinφd2uφdt2,(16)sinφNθφ,φ+Nθθ,θ+cosφNφθ+sinφQθθ=mvhRsinφd2uθdt2,(17)sinφQφφ,φ−sinφNφφ−sinφNθθ+Qθθ,θ+Rsinφpr=mvhRsinφd2wdt2,(18)sinφMφφ,φ−RsinφQφφ−cosφMθθ+Mφθ,θ=mvh34sinφd2uφdt2−mvh312sinφd2w,φdt2,(19)sinφMθφ,φ+Mθθ,θ+cosφMφθ−RsinφQθθ=mvh34sinφd2uθdt2,where mv is the mass density of the material and pr is the received pressure perpendicular to the spherical shell.


## 2.5. Derivation of Differential Equations for Axisymmetric Vibrations

According to the torsionless axisymmetry with respect to thez-axis, we have(20)uθ=∂∂θ⋅=0;Nφθ=Nθφ=Nρθ=Qθθ=0;Mφθ=Mθφ=0.The compatibility relation can be deduced from (4), (5), and (12) Nφφ,Nθθ by eliminating uθ and replacing the strains by their expressions according to the stress resultants:(21)tanφNθθ−νNφφ,φ+sec2φ+νNθθ−νsec2φ+1Nφφ=EhRtanφw,φ+tanφw.The expression of the shear stress resultantQφφ as a function of displacements uφ and w can be obtained from (15) and (18):(22)Qφφ=−DR▽2w+2R2w,φ+mv1−ν2RDEd2w,φdt2−2mv1−ν2RDEd2uφdt2.Inspired by [32], we introduce a new stress function Fφ,t through the following new relations:(23)Nφφ=cotφF,φ+F−mv1+νER2d2Fdt2+DR▽2w+2R2w−mv1−ν2RDEd2wdt2,(24)Nθθ=F,φφ+F−mv1+νER2d2Fdt2+DR▽2w+2R2w−mv1−ν2RDEd2wdt2,where, as proposed in [32], we have(25)▽2≡1R2,φφ+cotφ,φ.The tangential displacementuφ can be related to the radial displacement w, by elimination of uφ from (15) as(26)uφ=tanφREhNθθ−νNφφ−w.By using (4), (5), (15), (22), (23), and (24), we get the following differential equation:(27)d2Γdt2=0,where Γ is expressed as(28)Γ=R2▽2F+1−νF−mv1−ν2R2Ed2Fdt2+1−νDR▽2+2R2w−EhRw−mv1−ν1−ν2RDEd2wdt2.By substituting the equations (23) and (24) in the compatibility equation (21), another differential equation related also to F and w can be defined as(29)tanφΓ+Γ,φ=0.The solution of (27) and (29) can be written as Γ+At+B=0, where the coefficients A and B can be set to zero without loss in generality. Then, the first differential equation in F and w can be defined as(30)R2▽2F+1−νF−mv1−ν2R2Ed2Fdt2+1−νDR▽2+2R2w−EhRw−mv1−ν1−ν2RDEd2wdt2=0.To obtain the second differential equation inF and w, the shear stress resultant Qφφ (22) into (17) and using (23) and (24) can be substituted:(31)R2▽2F+2F−2mv1+νR2E1+h212▽2d2Fdt2DR▽2+2R2▽2+2R2w+mvRh−1−ν2DE▽2d2wdt2−Rpr=0.


## 3. General Solution for Harmonic Axisymmetric Vibrations

The steady-state problem of the spherical shells can be represented by the system of the two differential equations (30) and (31). Thus, the solution of this system in F and w can be written in the following form:(32)Fw=ℜejtωF∗ejtωw∗,where w∗ and F∗ are the magnitudes of the radial displacement w and the stress function F and ω is the circular frequency.The natural frequency can be noted as(33)Ω2=mvω2R2E.By eliminatingF∗ from (30) and (31) and using the notation of the natural frequency, a differential equation in w∗ of the sixth-order can be obtained as(34)▽6w∗+k2▽4w∗+k3▽2w∗+k4w∗+k5Rpr=0,where(35)k2=1R24+2Ω21−ν2,k3=EhR2D1−Ω2,k4=EhDR2+1+3νΩ2−1−ν2Ω4,k5=1D1+ν+Ω21+ν2.In the particular case of the free vibration analysis, the surface loads are equal to zero. Equation (34) is then written as follows:(36)▽6w∗+k2▽4w∗+k3▽2w∗+k4w∗=0.Therefore, the general solution of the free vibration differential equation (36) can be expressed in terms of Legendre functions as(37)w∗=∑α=13wα∗,where each radial displacement wα∗ can be written as(38)wα∗=AαPnαcosφ+BαQnαcosφ,αε1,2,3,where Pnα and Qnα are the Legendre functions of the first and the second kind of order nα, respectively. nα can be defined as(39)nα=λα+141/2−12;αε1,2,3.With the aid of the identities,(40)▽2PncosφQncosφ=−λR2PncosφQncosφ.The characteristic equation of the differential free vibrations equation of spherical shells (34) is(41)λ3−4+2Ω21−ν2λ2+121−ν2Rh21−Ω2λ−121−ν2Rh22+1+3νΩ2+ν2−1Ω4=0.


## 4. Results and Discussion

In this section, we test the validity and accuracy of the method exposed previously by determining the natural frequencies of spherical shells. Then, to demonstrate the correctness of the aforementioned formulation, two appropriate boundary conditions are treated: free edge and clamped edge. The current results are compared, on the one hand, with those available in literature, that is, the analytical results obtained by Naghdi–Kalnins [32] and by Kunieda [33]. It should be noted that these two analytical methods do not take into consideration the terms of the inertial couples in their equations of motion. On the other hand, the current results are also compared with the numerical results obtained by two different axisymmetric finite elements: the first one is SHELL61 that does not take into account transverse shear deformation and the second is SHELL209 that takes it into account. Furthermore, these two shell finite elements are also suitable for the modeling of axisymmetric thin shells. SHELL61 has two nodes and four degrees of freedom per node: displacements in the x (radial), y (axial), and z (circular) directions and rotation around the z-axis. The loading may be axisymmetric or nonaxisymmetric [45, 46]. SHELL209 has three nodes and three degrees of freedom per node: displacements in the x, y directions and rotation about the z-axis [45]. More information on the shape functions is available in [45, 46]. The FEM solutions are computed via ANSYS Mechanical APDL. The mesh is fine enough to guarantee the accuracy of the numerical results. In this study, only the thin shells whose thickness h is less than the twentieth of the radius of curvature R will be considered. The thickness to radius ratio in this work varies in the range 0.0025 to 0.05.


### 4.1. Free Vibration with Free Edge

#### 4.1.1. Frequency Equation

In this section, we focus on solving the problem of axisymmetric vibrations of a thin hemispherical shell with a free edge0≤φ≤φ0 with a free circular boundary at φ0=π/2.In the case of thin hemispherical shells0≤φ≤π/2, the Legendre functions of the second kind Qnα can tend towards infinity for φ=0. To eliminate the singularity of Qnα, the coefficients Bα are set to zero, where αε1,2,3 is as (38).In the free edge boundary condition atφ=π/2, the stress resultant Nφφ, the stress couple Mφφ, and the transverse shear stress resultant Qφφ are set to zero Nφφ=Mφφ=Qφφ=0:(42)∑α=13AαEh1−νR−λαCαDR3Pnα0=0,(43)∑α=13AαDR2λα1−112hR2CαPnα0=0,(44)∑α=13AαDR3λα−2−1−ν2Ω2+161−ν2Ω2h2R2Cαnα+1Pnα+10,where(45)Cα=λα−2+Ω2ν2−1+12R2/h2ν+1ν2−1Ω2+ν+λα−1,and the coefficients of the frequency equation(46)Ciα=0;i,α∈1,2,3.Then, they can be written as(47)C1α=121+ν−h2R2CαλαPnα0,C2α=1+ν−λαPnα0,C3α=λα−2−1−ν2Ω2+161−ν2Ω2h2R2Cαnα+1Pnα+10,where the coefficient C2α was obtained by the linear combination between the two equations (42) and (43).The Legendre function of a complex degree can be calculated by the following integral [28]:(48)Pn0=n+222π∫0π/2cosn+32xcos1/2xdx.To calculate the lowest natural frequency from equation (46), for ν=0.3 and φ0=π/2, we specify the value of h/R and Ω and then evaluate the determinant Ciα by incrementing Ω. This procedure is repeated until the value of the determinant changes sign. Table 1 and Figure 4 present an example of this procedure, where h/R=3/100. As shown in Table 1, the real parts of the natural frequencies are negligible. For this reason, in this case, we consider just the imaginary part. Then, the small box of Figure 4 shows that there are two sign changes between 0.82 and 1.01Hz. Therefore, it means that in this interval, we have two natural frequencies.Table 1
The current analytical value of determinant ath/R = 3/100 and ν=0.3 (free edge boundary condition).
ΩCiαΩCiαΩCiα03.6778e − 06 − 5.2305e + 10i0.60972.8036e − 04 − 3.3459e + 12i0.87387.3712e − 09 + 6.5295e + 07i0.03335.7014e − 12 − 3.8527e + 11i0.6667−1.3833e − 04 − 1.6127e + 12i0.87451.8190e − 08 + 1.9857e + 08i0.0667−2.2297e − 10 − 1.3537e + 12i0.7224−6.1685e − 05 − 6.2522e + 11i0.878−2.6839e − 08 + 7.8168e + 08i0.1667−5.7854e − 04 − 6.8094e + 12i0.74992.7660e − 05 − 3.4821e + 11i0.9−1.0285e − 07 + 2.1516e + 09i0.33332.0823e − 03 − 1.3573e + 13i0.78608.8061e − 06 − 1.3605e + 11i0.9568−2.7725e − 08 + 2.8182e + 08i0.386−6.4884e − 08 − 1.3043e + 13i0.8−7.0441e − 06 − 8.7855e + 10i0.98391.3324e − 08 − 2.0230e + 08i0.4167.4700e − 04 − 1.2178e + 13i0.82215.1090e − 06 − 3.8812e + 10i1.0−7.4506e − 09 − 2.0294e + 08i0.5−3.2127e + 04–7.1348e + 20i0.83332.8874e − 08 − 2.3493e + 10i1.0109−7.4506e − 09 − 8.0483e + 07iFigure 4
The imaginary part of determinant in variation withΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).It is worthy to mention that a determinant is a complex number, as shown in Table2. In this case, the real parts of the determinants are not negligible. For this reason, we used the determinant’s modulus to determine the lowest natural frequencies, which correspond to the minimum values closest to zero. The variation of the determinant’s modulus as a function of natural frequency for h/R=3/100 and ν=0.3 is also plotted in Figure 5. As seen in the zoomed panel, the results indicate the existence of five minimum values, which correspond to the lowest natural frequencies. Their values are presented in Table 3.Table 2
Complex determinant in variation ofΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).
ΩCiαΩCiαΩCiα03.6778e − 06 − 5.2305e + 10i0.878−2.6839e − 08 + 7.8168e + 08i1.12−4.7369e + 08 + 4.4042e − 07i0.03335.7014e − 12 − 3.8527e + 11i0.9−1.0285e − 07 + 2.1516e + 09i1.122−3.2406e + 08 + 2.3935e − 07i0.0667−2.2297e − 10 − 1.3537e + 12i0.9568−2.7725e − 08 + 2.8182e + 08i1.1245−1.0195e + 08 + 1.7176e − 07i0.1667−5.7854e − 04 − 6.8094e + 12i0.98391.3324e − 08 − 2.0230e + 08i1.12594.0270e + 07 + 2.7768e − 07i0.33332.1534e − 08 − 1.3574e + 13i1.0−7.4506e − 09 − 2.0294e + 08i1.12701.6136e + 08 + 1.0339e − 07i0.386−6.4884e − 08 − 1.3043e + 13i1.0109−7.4506e − 09 − 8.0483e + 07i1.12873.6519e + 08 + 7.3535e − 10i0.4167.4700e − 04 − 1.2178e + 13i1.0222−8.9407e − 08 + 1.3505e + 08i1.142.2893e + 09 + 4.1215e − 07i0.5−4.7887e − 04 − 8.4152e + 12i1.0603−2.9802e − 08 + 9.4023e + 08i1.168.6272e + 09 + 5.5198e − 07i0.60972.8036e − 04 − 3.3459e + 12i1.0820−1.1921e − 07 + 6.4678e + 08i1.171.3452e + 10 + 1.5723e − 06i0.6667−1.3833e − 04 − 1.6127e + 12i1.08302.9802e − 08 + 5.9779e + 08i1.23.5002e + 10 + 4.7824e − 06i0.7224−6.1685e − 05 − 6.2522e + 11i1.085−2.0862e − 07 + 4.7795e + 08i1.236.4097e + 10–8.6780e − 06i0.74992.7660e − 05 − 3.4821e + 11i1.089−2.1815e + 08–1.1882e − 08i1.258.2221e + 10 + 5.2564e − 06i0.78608.8061e − 06 − 1.3605e + 11i1.100−8.4835e + 08 + 6.4085e − 08i1.279.2099e + 10 + 1.0384e − 05i0.8−7.0441e − 06 − 8.7855e + 10i1.1100−8.9064e + 08 + 1.2904e − 07i1.298.4653e + 10 + 8.1133e − 06i0.82215.1090e − 06 − 3.8812e + 10i1.1126−8.3109e + 08 + 1.6805e − 07i1.314.8412e + 10–6.7699e − 06i0.83332.8874e − 08 − 2.3493e + 10i1.1162−6.9389e + 08 + 7.3412e − 08i1.4−8.2261e + 11 + 1.3543e − 04i0.87387.3712e − 09 + 6.5295e + 07i1.1187−5.5821e + 08 + 9.8344e − 08i1.5−3.2889e + 12 – 4.0268e − 04iFigure 5
Determinant modulus in variation withΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).Table 3
The lowest natural frequencies for axisymmetric free vibrations, under free edge boundary condition, with the current analytical methods ath/R=3/100 and ν=0.3.
Mode numberΩCurrent10.87320.98331.01141.08951.12661.310


#### 4.1.2. Comparison of Results

In order to verify the accuracy of the current solution, the results are compared with those obtained by Naghdi and Kalnins [32], Kunieda [33], and calculus by the finite elements SHELL61 and SHELL209. The difference estimators ηi,i∈1,2,3,4 are defined as follows:(49)η1=ΩKunieda33−ΩcurrentΩKunieda33×100,η2=ΩKunieda33−ΩNaghdi−Kalnins32ΩKunieda33×100,η3=ΩKunieda33−ΩFE−SHELL61ΩKunieda33×100,η4=ΩKunieda33−ΩFE−SHELL209ΩKunieda33×100,where ΩKunieda33 is the frequency obtained by the Kunieda method, Ωcurrent is the frequency obtained by the current method, ΩNaghdi−Kalnins32 is the frequency obtained by the Naghdi–Kalnins method, ΩFE−SHELL61 is the frequency obtained by the FE-SHELL61, and ΩFE−SHELL209 is the frequency obtained by the FE-SHELL209. It should be noted that the analytical results obtained by the Kunieda method ΩKunieda33 are taken as a reference.Table4 shows a comparison study of the lowest natural frequency for spherical elastic thin shells under free vibration with free edge boundary condition for h/R=1/100 and ν=0.3, calculated by the current method, the Naghdi–Kalnins method [32], and the Kunieda method [33]. We observe that the current results and Kunieda’s ones are identical. It is also seen that the Naghdi–Kalnins results are very close to these results, with η2=0.57%.Table 4
The lowest natural frequency for axisymmetric free vibrations, under free edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
ΩCurrentΩNaghdi−Kalnins32ΩKunieda33η1%η2%0.8700.8750.8700.000.57A comparative study of the first seventeen natural frequencies between the analytical solutions (the current solution and Kunieda’s [33] solution) and numerical solutions (FE-SHELL61 and FE-SHELL209), is carried out in Table 5 for thin spherical shells under free vibration with the free edge boundary condition at h/R=1/100 and ν=0.3.Table 5
Comparison study for natural frequencies for axisymmetric free vibrations, under free edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
Mode number method1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16th17thCurrent0.8700.9520.9831.0011.0291.0421.0781.1511.2541.3891.5591.7611.9962.0702.2622.559Kunieda [33]0.8700.9520.9831.0011.0301.0791.1521.2551.3911.5611.7652.0012.0692.268FE-SHELL610.8700.9520.9821.0001.0291.0771.1511.2541.3891.5581.7611.9611.9952.0702.2612.557FE-SHELL2090.8700.9520.9821.0001.0291.0771.1501.2521.3861.5531.7531.9842.0702.2452.534η1%0.0000.0000.0000.0000.0970.0920.0860.0790.1430.1280.2260.2490.0480.264η3%0.0000.0000.1010.0990.0970.1850.0860.0790.1430.1920.2260.2990.0480.308η4%0.0000.0000.1010.0990.0970.1850.1730.2390.3590.5120.6790.8490.0481.014From Table5, we observe a very good agreement between the different results confirming the accuracy of the current analytical method. On the one hand, we see that the current results are close to the results of Kunieda’s method. The difference between the results of these two analytical methods is very small and does not exceed 0.27%. This closeness is verified even for higher vibration modes. On the other hand, we see also that the FE-SHELL61 results are very close to the SHELL209 results particularly for the first modes. Indeed, when the ratio h/R is very small (here h/R=1/100) the transverse shear becomes negligible and the two finite elements give almost the same results. We observe also that, in order, the current results are closer, first, to the results of the finite element SHELL61, then to those of Kunieda, and finally to those of the finite element SHELL209.The results of Table5 are instructive in more than one way. The current method reveals a new natural frequency 1.042 (6th mode) which does not appear in Kunieda’s method and finite element calculation while the frequency 1.961 (13th mode) obtained by the finite element SHELL61 is nonexistent for the two analytical calculations and the finite element SHELL209 calculation.


### 4.2. Effect of the Thickness to Radius Ratioh/R on the Lowest Natural Frequency

The results in Table6, which are plotted in Figure 6, show the effect of the h/R ratio on the lowest natural frequency of thin spherical shells under free vibrations with the free edge. We observe that the lowest frequency, calculated by the current method and FE methods, is almost independent of the h/R ratio, while that calculated by the Naghdi–Kalnins method is sensitive to this ratio. This strong variation in frequency, calculated by the Naghdi–Kalnins method, as a function of the h/R ratio is not legitimate.Table 6
Variation of the lowest natural frequency for axisymmetric free vibrations, under free edge boundary condition, as a function of ratioh/R with different methods at ν=0.3.
h/R ratioΩNaghdi−Kalnins32ΩCurrentΩFE−SHELL61ΩFE−SHELL2090.010.8750.8700.8700.8700.020.9190.8710.8710.8710.030.9750.8730.8740.8740.051.1000.8790.8790.879Figure 6
Natural frequency as a function of thickness to radius ratioh/R at ν=0.3 (free edge boundary condition).All the methods give almost the same lowest frequency when the ratioh/R is very small.We also observe that the two finite element methods give exactly the same lowest frequency even for high values of the ratioh/R. Which leads us to say that taking into account the transverse shear does not have effect on the lowest frequency and therefore that the current method can be applied in this case even for high h/R ratios.


### 4.3. Free Vibrations with Clamped Edge Boundary Condition

#### 4.3.1. Frequency Equation

For free vibrations with clamped boundary conditions, the edgeφ=φ0 of the shell cannot translate in the ρ and φ directions and cannot undergo any rotation around the eθ axis. The boundary conditions, in this case, can be summarized as(50)w/φ=φ0∗=0,(51)dwdt/φ=φ0∗=0,(52)uφ∗/φ=φ0=0,where φ0=π/2. Also, in this case of clamped boundary conditions, the Legendre functions of the second kind Qnα can tend towards infinity for φ=0. To eliminate the singularity of Qnα, the coefficients Bα are set to zero as mentioned in the free vibrations with free edge condition section.Using the boundary conditions (50) to (52) along with the expression for uφ∗, the following form is obtained:(53)∑α=13AαPnα0=0,∑α=13Aαnα+1Pnα+10=0,∑α=13AαCαnα+1Pnα+10=0.We note that the displacementuφ∗ can be expressed as a function of F,φ (as derived in the appendix).The coefficients of the frequency equation:(54)Ciα=0;,i,α∈1,2,3,and they can be written as(55)C1α=Pnα0,C2α=nα+1Pnα+10,C3α=Cαnα+1Pnα+10.To calculate lowest natural frequency from equation (54)for ν=0.3 and φ0=π/2, we specify the value of h/R and Ω and then evaluate the determinant Ciα by incrementing Ω as done in the section of frequency equation for free vibrations with free edge.Table7 presents the lowest natural frequencies for thin spherical shells at h/R=3/100 and ν=0.3. The results are obtained by the current analytical method under axisymmetric free vibrations with clamped edge boundary condition.Table 7
The lowest natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with current analytical method ath/R=3/100 and ν=0.3.
Mode numberΩCurrent10.78220.97331.08841.099


#### 4.3.2. Comparison of Results

In order to verify the accuracy of the current method, in the case of axisymmetric free vibrations for hemispherical thin shells with clamped edge boundary condition ath/R=1/100 and ν=0.3, the frequencies of the first sixteen modes are compared with those obtained by Kunieda [33] and calculus by the finite elements SHELL61 and SHELL209, as shown in Table 8.Table 8
Comparison of natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
Mode number1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16thCurrent0.7610.9380.9831.0211.0421.0701.1441.2461.3791.5411.6601.7751.9982.2612.552Kunieda [33]0.7600.9380.9841.0201.0711.1441.2471.3811.5421.6601.7772.0022.266FE-SHELL610.7610.9370.9831.0201.0701.1431.2401.2451.3791.5391.6601.7741.9982.2602.552FE-SHELL2090.7610.9370.9831.0191.0691.1421.2431.3751.5341.6571.7671.9852.2422.527η1%0.13100.1010.0980.09300.0800.1440.06400.1120.1990.220η3%0.1310.1060.10100.0930.0870.5610.1440.19400.1680.1990.264η4%0.1310.1060.1010.0980.1860.1740.3200.4340.5180.1800.5620.8491.059The current solutions are in close agreement with the solutions obtained by Kunieda, FE-SHELL61, and FE-SHELL209. The difference between the results obtained by the current method and Kunieda and between the current method and FE-SHELL61 are closely zero. For the higher modes, a negligible difference appears in the results of FE-SHELL209 compared to the others results, probably due to the fact that it takes into account the transverse shear.The results of Table8 are instructive in more than one way. The current method reveals a new natural frequency 1.042 (5th mode) which does not appear in Kunieda’s method and in the the finite element calculation while the frequency 1.240 obtained by the FE-SHELL61 is nonexistent for the three other methods.


#### 4.3.3. Influence of the Thickness to Radius Ratioh/R

The results in Table9 show the effect of the h/R ratio on the four (or five) lowest natural frequencies of thin spherical shells under free vibrations with clamped edge.Table 9
Influence of ratioh/R on the lowest natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with different methods at ν=0.3.
h/R ratioMode numberΩCurrentΩKunieda33ΩSHELL61ΩSHELL2090.002510.7470.7470.7460.74720.9290.9290.9290.92930.9680.9680.9680.96840.9830.9830.9830.9830.00510.7530.7520.7530.75320.9320.9320.9320.93230.9720.9720.9710.97140.9910.9910.9900.9900.0110.7610.7600.7610.76120.9380.9380.9370.93730.9830.9840.9380.98341.0201.0201.0201.0200.0210.7730.7720.7730.77320.9530.9530.9520.95231.0281.0291.0271.02741.06751.1301.1301.129We observe that, for ratioh/R, varying between 0.0025 and 0.02, the natural frequencies obtained by the current method are very close to those obtained by Kunieda’s method and those of the calculation by finite elements.The effect of transverse shear is nonexistent even for the highest ratioh/R=0.02: the element SHELL209 which takes account of the transverse shear gives the same results. Therefore, the current method can be applied in this case even for high h/R ratios and even for high modes, with negligible error as observed above.The effect of theh/R ratio is weak on the lowest frequency. The variation of this frequency is 3.5% when the h/R ratio varies from the value 0.0025 to 0.02 while the influence of the h/R ratio on the frequency of mode 4 is more important. The variation of this frequency is 8.5%, when the h/R ratio varies in the same interval. Kunieda [33] considers that the effect of h/R is not substantial for thin shells.This effect is much greater for thin shells with a clamped edge than for those with a free edge. When theh/R ratio varies from the value 0.01 to 0.02, the percentage variation of the lowest frequency is 1.6% for clamped edge and only 0.1% for free edge.The current method reveals a new natural frequency (1.067, mode 4) which is not obtained by the finite element calculus and which is absent from Kunieda’s results.


## 4.1. Free Vibration with Free Edge

### 4.1.1. Frequency Equation

In this section, we focus on solving the problem of axisymmetric vibrations of a thin hemispherical shell with a free edge0≤φ≤φ0 with a free circular boundary at φ0=π/2.In the case of thin hemispherical shells0≤φ≤π/2, the Legendre functions of the second kind Qnα can tend towards infinity for φ=0. To eliminate the singularity of Qnα, the coefficients Bα are set to zero, where αε1,2,3 is as (38).In the free edge boundary condition atφ=π/2, the stress resultant Nφφ, the stress couple Mφφ, and the transverse shear stress resultant Qφφ are set to zero Nφφ=Mφφ=Qφφ=0:(42)∑α=13AαEh1−νR−λαCαDR3Pnα0=0,(43)∑α=13AαDR2λα1−112hR2CαPnα0=0,(44)∑α=13AαDR3λα−2−1−ν2Ω2+161−ν2Ω2h2R2Cαnα+1Pnα+10,where(45)Cα=λα−2+Ω2ν2−1+12R2/h2ν+1ν2−1Ω2+ν+λα−1,and the coefficients of the frequency equation(46)Ciα=0;i,α∈1,2,3.Then, they can be written as(47)C1α=121+ν−h2R2CαλαPnα0,C2α=1+ν−λαPnα0,C3α=λα−2−1−ν2Ω2+161−ν2Ω2h2R2Cαnα+1Pnα+10,where the coefficient C2α was obtained by the linear combination between the two equations (42) and (43).The Legendre function of a complex degree can be calculated by the following integral [28]:(48)Pn0=n+222π∫0π/2cosn+32xcos1/2xdx.To calculate the lowest natural frequency from equation (46), for ν=0.3 and φ0=π/2, we specify the value of h/R and Ω and then evaluate the determinant Ciα by incrementing Ω. This procedure is repeated until the value of the determinant changes sign. Table 1 and Figure 4 present an example of this procedure, where h/R=3/100. As shown in Table 1, the real parts of the natural frequencies are negligible. For this reason, in this case, we consider just the imaginary part. Then, the small box of Figure 4 shows that there are two sign changes between 0.82 and 1.01Hz. Therefore, it means that in this interval, we have two natural frequencies.Table 1
The current analytical value of determinant ath/R = 3/100 and ν=0.3 (free edge boundary condition).
ΩCiαΩCiαΩCiα03.6778e − 06 − 5.2305e + 10i0.60972.8036e − 04 − 3.3459e + 12i0.87387.3712e − 09 + 6.5295e + 07i0.03335.7014e − 12 − 3.8527e + 11i0.6667−1.3833e − 04 − 1.6127e + 12i0.87451.8190e − 08 + 1.9857e + 08i0.0667−2.2297e − 10 − 1.3537e + 12i0.7224−6.1685e − 05 − 6.2522e + 11i0.878−2.6839e − 08 + 7.8168e + 08i0.1667−5.7854e − 04 − 6.8094e + 12i0.74992.7660e − 05 − 3.4821e + 11i0.9−1.0285e − 07 + 2.1516e + 09i0.33332.0823e − 03 − 1.3573e + 13i0.78608.8061e − 06 − 1.3605e + 11i0.9568−2.7725e − 08 + 2.8182e + 08i0.386−6.4884e − 08 − 1.3043e + 13i0.8−7.0441e − 06 − 8.7855e + 10i0.98391.3324e − 08 − 2.0230e + 08i0.4167.4700e − 04 − 1.2178e + 13i0.82215.1090e − 06 − 3.8812e + 10i1.0−7.4506e − 09 − 2.0294e + 08i0.5−3.2127e + 04–7.1348e + 20i0.83332.8874e − 08 − 2.3493e + 10i1.0109−7.4506e − 09 − 8.0483e + 07iFigure 4
The imaginary part of determinant in variation withΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).It is worthy to mention that a determinant is a complex number, as shown in Table2. In this case, the real parts of the determinants are not negligible. For this reason, we used the determinant’s modulus to determine the lowest natural frequencies, which correspond to the minimum values closest to zero. The variation of the determinant’s modulus as a function of natural frequency for h/R=3/100 and ν=0.3 is also plotted in Figure 5. As seen in the zoomed panel, the results indicate the existence of five minimum values, which correspond to the lowest natural frequencies. Their values are presented in Table 3.Table 2
Complex determinant in variation ofΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).
ΩCiαΩCiαΩCiα03.6778e − 06 − 5.2305e + 10i0.878−2.6839e − 08 + 7.8168e + 08i1.12−4.7369e + 08 + 4.4042e − 07i0.03335.7014e − 12 − 3.8527e + 11i0.9−1.0285e − 07 + 2.1516e + 09i1.122−3.2406e + 08 + 2.3935e − 07i0.0667−2.2297e − 10 − 1.3537e + 12i0.9568−2.7725e − 08 + 2.8182e + 08i1.1245−1.0195e + 08 + 1.7176e − 07i0.1667−5.7854e − 04 − 6.8094e + 12i0.98391.3324e − 08 − 2.0230e + 08i1.12594.0270e + 07 + 2.7768e − 07i0.33332.1534e − 08 − 1.3574e + 13i1.0−7.4506e − 09 − 2.0294e + 08i1.12701.6136e + 08 + 1.0339e − 07i0.386−6.4884e − 08 − 1.3043e + 13i1.0109−7.4506e − 09 − 8.0483e + 07i1.12873.6519e + 08 + 7.3535e − 10i0.4167.4700e − 04 − 1.2178e + 13i1.0222−8.9407e − 08 + 1.3505e + 08i1.142.2893e + 09 + 4.1215e − 07i0.5−4.7887e − 04 − 8.4152e + 12i1.0603−2.9802e − 08 + 9.4023e + 08i1.168.6272e + 09 + 5.5198e − 07i0.60972.8036e − 04 − 3.3459e + 12i1.0820−1.1921e − 07 + 6.4678e + 08i1.171.3452e + 10 + 1.5723e − 06i0.6667−1.3833e − 04 − 1.6127e + 12i1.08302.9802e − 08 + 5.9779e + 08i1.23.5002e + 10 + 4.7824e − 06i0.7224−6.1685e − 05 − 6.2522e + 11i1.085−2.0862e − 07 + 4.7795e + 08i1.236.4097e + 10–8.6780e − 06i0.74992.7660e − 05 − 3.4821e + 11i1.089−2.1815e + 08–1.1882e − 08i1.258.2221e + 10 + 5.2564e − 06i0.78608.8061e − 06 − 1.3605e + 11i1.100−8.4835e + 08 + 6.4085e − 08i1.279.2099e + 10 + 1.0384e − 05i0.8−7.0441e − 06 − 8.7855e + 10i1.1100−8.9064e + 08 + 1.2904e − 07i1.298.4653e + 10 + 8.1133e − 06i0.82215.1090e − 06 − 3.8812e + 10i1.1126−8.3109e + 08 + 1.6805e − 07i1.314.8412e + 10–6.7699e − 06i0.83332.8874e − 08 − 2.3493e + 10i1.1162−6.9389e + 08 + 7.3412e − 08i1.4−8.2261e + 11 + 1.3543e − 04i0.87387.3712e − 09 + 6.5295e + 07i1.1187−5.5821e + 08 + 9.8344e − 08i1.5−3.2889e + 12 – 4.0268e − 04iFigure 5
Determinant modulus in variation withΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).Table 3
The lowest natural frequencies for axisymmetric free vibrations, under free edge boundary condition, with the current analytical methods ath/R=3/100 and ν=0.3.
Mode numberΩCurrent10.87320.98331.01141.08951.12661.310


### 4.1.2. Comparison of Results

In order to verify the accuracy of the current solution, the results are compared with those obtained by Naghdi and Kalnins [32], Kunieda [33], and calculus by the finite elements SHELL61 and SHELL209. The difference estimators ηi,i∈1,2,3,4 are defined as follows:(49)η1=ΩKunieda33−ΩcurrentΩKunieda33×100,η2=ΩKunieda33−ΩNaghdi−Kalnins32ΩKunieda33×100,η3=ΩKunieda33−ΩFE−SHELL61ΩKunieda33×100,η4=ΩKunieda33−ΩFE−SHELL209ΩKunieda33×100,where ΩKunieda33 is the frequency obtained by the Kunieda method, Ωcurrent is the frequency obtained by the current method, ΩNaghdi−Kalnins32 is the frequency obtained by the Naghdi–Kalnins method, ΩFE−SHELL61 is the frequency obtained by the FE-SHELL61, and ΩFE−SHELL209 is the frequency obtained by the FE-SHELL209. It should be noted that the analytical results obtained by the Kunieda method ΩKunieda33 are taken as a reference.Table4 shows a comparison study of the lowest natural frequency for spherical elastic thin shells under free vibration with free edge boundary condition for h/R=1/100 and ν=0.3, calculated by the current method, the Naghdi–Kalnins method [32], and the Kunieda method [33]. We observe that the current results and Kunieda’s ones are identical. It is also seen that the Naghdi–Kalnins results are very close to these results, with η2=0.57%.Table 4
The lowest natural frequency for axisymmetric free vibrations, under free edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
ΩCurrentΩNaghdi−Kalnins32ΩKunieda33η1%η2%0.8700.8750.8700.000.57A comparative study of the first seventeen natural frequencies between the analytical solutions (the current solution and Kunieda’s [33] solution) and numerical solutions (FE-SHELL61 and FE-SHELL209), is carried out in Table 5 for thin spherical shells under free vibration with the free edge boundary condition at h/R=1/100 and ν=0.3.Table 5
Comparison study for natural frequencies for axisymmetric free vibrations, under free edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
Mode number method1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16th17thCurrent0.8700.9520.9831.0011.0291.0421.0781.1511.2541.3891.5591.7611.9962.0702.2622.559Kunieda [33]0.8700.9520.9831.0011.0301.0791.1521.2551.3911.5611.7652.0012.0692.268FE-SHELL610.8700.9520.9821.0001.0291.0771.1511.2541.3891.5581.7611.9611.9952.0702.2612.557FE-SHELL2090.8700.9520.9821.0001.0291.0771.1501.2521.3861.5531.7531.9842.0702.2452.534η1%0.0000.0000.0000.0000.0970.0920.0860.0790.1430.1280.2260.2490.0480.264η3%0.0000.0000.1010.0990.0970.1850.0860.0790.1430.1920.2260.2990.0480.308η4%0.0000.0000.1010.0990.0970.1850.1730.2390.3590.5120.6790.8490.0481.014From Table5, we observe a very good agreement between the different results confirming the accuracy of the current analytical method. On the one hand, we see that the current results are close to the results of Kunieda’s method. The difference between the results of these two analytical methods is very small and does not exceed 0.27%. This closeness is verified even for higher vibration modes. On the other hand, we see also that the FE-SHELL61 results are very close to the SHELL209 results particularly for the first modes. Indeed, when the ratio h/R is very small (here h/R=1/100) the transverse shear becomes negligible and the two finite elements give almost the same results. We observe also that, in order, the current results are closer, first, to the results of the finite element SHELL61, then to those of Kunieda, and finally to those of the finite element SHELL209.The results of Table5 are instructive in more than one way. The current method reveals a new natural frequency 1.042 (6th mode) which does not appear in Kunieda’s method and finite element calculation while the frequency 1.961 (13th mode) obtained by the finite element SHELL61 is nonexistent for the two analytical calculations and the finite element SHELL209 calculation.


## 4.1.1. Frequency Equation

In this section, we focus on solving the problem of axisymmetric vibrations of a thin hemispherical shell with a free edge0≤φ≤φ0 with a free circular boundary at φ0=π/2.In the case of thin hemispherical shells0≤φ≤π/2, the Legendre functions of the second kind Qnα can tend towards infinity for φ=0. To eliminate the singularity of Qnα, the coefficients Bα are set to zero, where αε1,2,3 is as (38).In the free edge boundary condition atφ=π/2, the stress resultant Nφφ, the stress couple Mφφ, and the transverse shear stress resultant Qφφ are set to zero Nφφ=Mφφ=Qφφ=0:(42)∑α=13AαEh1−νR−λαCαDR3Pnα0=0,(43)∑α=13AαDR2λα1−112hR2CαPnα0=0,(44)∑α=13AαDR3λα−2−1−ν2Ω2+161−ν2Ω2h2R2Cαnα+1Pnα+10,where(45)Cα=λα−2+Ω2ν2−1+12R2/h2ν+1ν2−1Ω2+ν+λα−1,and the coefficients of the frequency equation(46)Ciα=0;i,α∈1,2,3.Then, they can be written as(47)C1α=121+ν−h2R2CαλαPnα0,C2α=1+ν−λαPnα0,C3α=λα−2−1−ν2Ω2+161−ν2Ω2h2R2Cαnα+1Pnα+10,where the coefficient C2α was obtained by the linear combination between the two equations (42) and (43).The Legendre function of a complex degree can be calculated by the following integral [28]:(48)Pn0=n+222π∫0π/2cosn+32xcos1/2xdx.To calculate the lowest natural frequency from equation (46), for ν=0.3 and φ0=π/2, we specify the value of h/R and Ω and then evaluate the determinant Ciα by incrementing Ω. This procedure is repeated until the value of the determinant changes sign. Table 1 and Figure 4 present an example of this procedure, where h/R=3/100. As shown in Table 1, the real parts of the natural frequencies are negligible. For this reason, in this case, we consider just the imaginary part. Then, the small box of Figure 4 shows that there are two sign changes between 0.82 and 1.01Hz. Therefore, it means that in this interval, we have two natural frequencies.Table 1
The current analytical value of determinant ath/R = 3/100 and ν=0.3 (free edge boundary condition).
ΩCiαΩCiαΩCiα03.6778e − 06 − 5.2305e + 10i0.60972.8036e − 04 − 3.3459e + 12i0.87387.3712e − 09 + 6.5295e + 07i0.03335.7014e − 12 − 3.8527e + 11i0.6667−1.3833e − 04 − 1.6127e + 12i0.87451.8190e − 08 + 1.9857e + 08i0.0667−2.2297e − 10 − 1.3537e + 12i0.7224−6.1685e − 05 − 6.2522e + 11i0.878−2.6839e − 08 + 7.8168e + 08i0.1667−5.7854e − 04 − 6.8094e + 12i0.74992.7660e − 05 − 3.4821e + 11i0.9−1.0285e − 07 + 2.1516e + 09i0.33332.0823e − 03 − 1.3573e + 13i0.78608.8061e − 06 − 1.3605e + 11i0.9568−2.7725e − 08 + 2.8182e + 08i0.386−6.4884e − 08 − 1.3043e + 13i0.8−7.0441e − 06 − 8.7855e + 10i0.98391.3324e − 08 − 2.0230e + 08i0.4167.4700e − 04 − 1.2178e + 13i0.82215.1090e − 06 − 3.8812e + 10i1.0−7.4506e − 09 − 2.0294e + 08i0.5−3.2127e + 04–7.1348e + 20i0.83332.8874e − 08 − 2.3493e + 10i1.0109−7.4506e − 09 − 8.0483e + 07iFigure 4
The imaginary part of determinant in variation withΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).It is worthy to mention that a determinant is a complex number, as shown in Table2. In this case, the real parts of the determinants are not negligible. For this reason, we used the determinant’s modulus to determine the lowest natural frequencies, which correspond to the minimum values closest to zero. The variation of the determinant’s modulus as a function of natural frequency for h/R=3/100 and ν=0.3 is also plotted in Figure 5. As seen in the zoomed panel, the results indicate the existence of five minimum values, which correspond to the lowest natural frequencies. Their values are presented in Table 3.Table 2
Complex determinant in variation ofΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).
ΩCiαΩCiαΩCiα03.6778e − 06 − 5.2305e + 10i0.878−2.6839e − 08 + 7.8168e + 08i1.12−4.7369e + 08 + 4.4042e − 07i0.03335.7014e − 12 − 3.8527e + 11i0.9−1.0285e − 07 + 2.1516e + 09i1.122−3.2406e + 08 + 2.3935e − 07i0.0667−2.2297e − 10 − 1.3537e + 12i0.9568−2.7725e − 08 + 2.8182e + 08i1.1245−1.0195e + 08 + 1.7176e − 07i0.1667−5.7854e − 04 − 6.8094e + 12i0.98391.3324e − 08 − 2.0230e + 08i1.12594.0270e + 07 + 2.7768e − 07i0.33332.1534e − 08 − 1.3574e + 13i1.0−7.4506e − 09 − 2.0294e + 08i1.12701.6136e + 08 + 1.0339e − 07i0.386−6.4884e − 08 − 1.3043e + 13i1.0109−7.4506e − 09 − 8.0483e + 07i1.12873.6519e + 08 + 7.3535e − 10i0.4167.4700e − 04 − 1.2178e + 13i1.0222−8.9407e − 08 + 1.3505e + 08i1.142.2893e + 09 + 4.1215e − 07i0.5−4.7887e − 04 − 8.4152e + 12i1.0603−2.9802e − 08 + 9.4023e + 08i1.168.6272e + 09 + 5.5198e − 07i0.60972.8036e − 04 − 3.3459e + 12i1.0820−1.1921e − 07 + 6.4678e + 08i1.171.3452e + 10 + 1.5723e − 06i0.6667−1.3833e − 04 − 1.6127e + 12i1.08302.9802e − 08 + 5.9779e + 08i1.23.5002e + 10 + 4.7824e − 06i0.7224−6.1685e − 05 − 6.2522e + 11i1.085−2.0862e − 07 + 4.7795e + 08i1.236.4097e + 10–8.6780e − 06i0.74992.7660e − 05 − 3.4821e + 11i1.089−2.1815e + 08–1.1882e − 08i1.258.2221e + 10 + 5.2564e − 06i0.78608.8061e − 06 − 1.3605e + 11i1.100−8.4835e + 08 + 6.4085e − 08i1.279.2099e + 10 + 1.0384e − 05i0.8−7.0441e − 06 − 8.7855e + 10i1.1100−8.9064e + 08 + 1.2904e − 07i1.298.4653e + 10 + 8.1133e − 06i0.82215.1090e − 06 − 3.8812e + 10i1.1126−8.3109e + 08 + 1.6805e − 07i1.314.8412e + 10–6.7699e − 06i0.83332.8874e − 08 − 2.3493e + 10i1.1162−6.9389e + 08 + 7.3412e − 08i1.4−8.2261e + 11 + 1.3543e − 04i0.87387.3712e − 09 + 6.5295e + 07i1.1187−5.5821e + 08 + 9.8344e − 08i1.5−3.2889e + 12 – 4.0268e − 04iFigure 5
Determinant modulus in variation withΩ at h/R=3/100 and ν=0.3 (free edge boundary condition).Table 3
The lowest natural frequencies for axisymmetric free vibrations, under free edge boundary condition, with the current analytical methods ath/R=3/100 and ν=0.3.
Mode numberΩCurrent10.87320.98331.01141.08951.12661.310


## 4.1.2. Comparison of Results

In order to verify the accuracy of the current solution, the results are compared with those obtained by Naghdi and Kalnins [32], Kunieda [33], and calculus by the finite elements SHELL61 and SHELL209. The difference estimators ηi,i∈1,2,3,4 are defined as follows:(49)η1=ΩKunieda33−ΩcurrentΩKunieda33×100,η2=ΩKunieda33−ΩNaghdi−Kalnins32ΩKunieda33×100,η3=ΩKunieda33−ΩFE−SHELL61ΩKunieda33×100,η4=ΩKunieda33−ΩFE−SHELL209ΩKunieda33×100,where ΩKunieda33 is the frequency obtained by the Kunieda method, Ωcurrent is the frequency obtained by the current method, ΩNaghdi−Kalnins32 is the frequency obtained by the Naghdi–Kalnins method, ΩFE−SHELL61 is the frequency obtained by the FE-SHELL61, and ΩFE−SHELL209 is the frequency obtained by the FE-SHELL209. It should be noted that the analytical results obtained by the Kunieda method ΩKunieda33 are taken as a reference.Table4 shows a comparison study of the lowest natural frequency for spherical elastic thin shells under free vibration with free edge boundary condition for h/R=1/100 and ν=0.3, calculated by the current method, the Naghdi–Kalnins method [32], and the Kunieda method [33]. We observe that the current results and Kunieda’s ones are identical. It is also seen that the Naghdi–Kalnins results are very close to these results, with η2=0.57%.Table 4
The lowest natural frequency for axisymmetric free vibrations, under free edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
ΩCurrentΩNaghdi−Kalnins32ΩKunieda33η1%η2%0.8700.8750.8700.000.57A comparative study of the first seventeen natural frequencies between the analytical solutions (the current solution and Kunieda’s [33] solution) and numerical solutions (FE-SHELL61 and FE-SHELL209), is carried out in Table 5 for thin spherical shells under free vibration with the free edge boundary condition at h/R=1/100 and ν=0.3.Table 5
Comparison study for natural frequencies for axisymmetric free vibrations, under free edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
Mode number method1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16th17thCurrent0.8700.9520.9831.0011.0291.0421.0781.1511.2541.3891.5591.7611.9962.0702.2622.559Kunieda [33]0.8700.9520.9831.0011.0301.0791.1521.2551.3911.5611.7652.0012.0692.268FE-SHELL610.8700.9520.9821.0001.0291.0771.1511.2541.3891.5581.7611.9611.9952.0702.2612.557FE-SHELL2090.8700.9520.9821.0001.0291.0771.1501.2521.3861.5531.7531.9842.0702.2452.534η1%0.0000.0000.0000.0000.0970.0920.0860.0790.1430.1280.2260.2490.0480.264η3%0.0000.0000.1010.0990.0970.1850.0860.0790.1430.1920.2260.2990.0480.308η4%0.0000.0000.1010.0990.0970.1850.1730.2390.3590.5120.6790.8490.0481.014From Table5, we observe a very good agreement between the different results confirming the accuracy of the current analytical method. On the one hand, we see that the current results are close to the results of Kunieda’s method. The difference between the results of these two analytical methods is very small and does not exceed 0.27%. This closeness is verified even for higher vibration modes. On the other hand, we see also that the FE-SHELL61 results are very close to the SHELL209 results particularly for the first modes. Indeed, when the ratio h/R is very small (here h/R=1/100) the transverse shear becomes negligible and the two finite elements give almost the same results. We observe also that, in order, the current results are closer, first, to the results of the finite element SHELL61, then to those of Kunieda, and finally to those of the finite element SHELL209.The results of Table5 are instructive in more than one way. The current method reveals a new natural frequency 1.042 (6th mode) which does not appear in Kunieda’s method and finite element calculation while the frequency 1.961 (13th mode) obtained by the finite element SHELL61 is nonexistent for the two analytical calculations and the finite element SHELL209 calculation.


## 4.2. Effect of the Thickness to Radius Ratioh/R on the Lowest Natural Frequency

The results in Table6, which are plotted in Figure 6, show the effect of the h/R ratio on the lowest natural frequency of thin spherical shells under free vibrations with the free edge. We observe that the lowest frequency, calculated by the current method and FE methods, is almost independent of the h/R ratio, while that calculated by the Naghdi–Kalnins method is sensitive to this ratio. This strong variation in frequency, calculated by the Naghdi–Kalnins method, as a function of the h/R ratio is not legitimate.Table 6
Variation of the lowest natural frequency for axisymmetric free vibrations, under free edge boundary condition, as a function of ratioh/R with different methods at ν=0.3.
h/R ratioΩNaghdi−Kalnins32ΩCurrentΩFE−SHELL61ΩFE−SHELL2090.010.8750.8700.8700.8700.020.9190.8710.8710.8710.030.9750.8730.8740.8740.051.1000.8790.8790.879Figure 6
Natural frequency as a function of thickness to radius ratioh/R at ν=0.3 (free edge boundary condition).All the methods give almost the same lowest frequency when the ratioh/R is very small.We also observe that the two finite element methods give exactly the same lowest frequency even for high values of the ratioh/R. Which leads us to say that taking into account the transverse shear does not have effect on the lowest frequency and therefore that the current method can be applied in this case even for high h/R ratios.


## 4.3. Free Vibrations with Clamped Edge Boundary Condition

### 4.3.1. Frequency Equation

For free vibrations with clamped boundary conditions, the edgeφ=φ0 of the shell cannot translate in the ρ and φ directions and cannot undergo any rotation around the eθ axis. The boundary conditions, in this case, can be summarized as(50)w/φ=φ0∗=0,(51)dwdt/φ=φ0∗=0,(52)uφ∗/φ=φ0=0,where φ0=π/2. Also, in this case of clamped boundary conditions, the Legendre functions of the second kind Qnα can tend towards infinity for φ=0. To eliminate the singularity of Qnα, the coefficients Bα are set to zero as mentioned in the free vibrations with free edge condition section.Using the boundary conditions (50) to (52) along with the expression for uφ∗, the following form is obtained:(53)∑α=13AαPnα0=0,∑α=13Aαnα+1Pnα+10=0,∑α=13AαCαnα+1Pnα+10=0.We note that the displacementuφ∗ can be expressed as a function of F,φ (as derived in the appendix).The coefficients of the frequency equation:(54)Ciα=0;,i,α∈1,2,3,and they can be written as(55)C1α=Pnα0,C2α=nα+1Pnα+10,C3α=Cαnα+1Pnα+10.To calculate lowest natural frequency from equation (54)for ν=0.3 and φ0=π/2, we specify the value of h/R and Ω and then evaluate the determinant Ciα by incrementing Ω as done in the section of frequency equation for free vibrations with free edge.Table7 presents the lowest natural frequencies for thin spherical shells at h/R=3/100 and ν=0.3. The results are obtained by the current analytical method under axisymmetric free vibrations with clamped edge boundary condition.Table 7
The lowest natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with current analytical method ath/R=3/100 and ν=0.3.
Mode numberΩCurrent10.78220.97331.08841.099


### 4.3.2. Comparison of Results

In order to verify the accuracy of the current method, in the case of axisymmetric free vibrations for hemispherical thin shells with clamped edge boundary condition ath/R=1/100 and ν=0.3, the frequencies of the first sixteen modes are compared with those obtained by Kunieda [33] and calculus by the finite elements SHELL61 and SHELL209, as shown in Table 8.Table 8
Comparison of natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
Mode number1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16thCurrent0.7610.9380.9831.0211.0421.0701.1441.2461.3791.5411.6601.7751.9982.2612.552Kunieda [33]0.7600.9380.9841.0201.0711.1441.2471.3811.5421.6601.7772.0022.266FE-SHELL610.7610.9370.9831.0201.0701.1431.2401.2451.3791.5391.6601.7741.9982.2602.552FE-SHELL2090.7610.9370.9831.0191.0691.1421.2431.3751.5341.6571.7671.9852.2422.527η1%0.13100.1010.0980.09300.0800.1440.06400.1120.1990.220η3%0.1310.1060.10100.0930.0870.5610.1440.19400.1680.1990.264η4%0.1310.1060.1010.0980.1860.1740.3200.4340.5180.1800.5620.8491.059The current solutions are in close agreement with the solutions obtained by Kunieda, FE-SHELL61, and FE-SHELL209. The difference between the results obtained by the current method and Kunieda and between the current method and FE-SHELL61 are closely zero. For the higher modes, a negligible difference appears in the results of FE-SHELL209 compared to the others results, probably due to the fact that it takes into account the transverse shear.The results of Table8 are instructive in more than one way. The current method reveals a new natural frequency 1.042 (5th mode) which does not appear in Kunieda’s method and in the the finite element calculation while the frequency 1.240 obtained by the FE-SHELL61 is nonexistent for the three other methods.


### 4.3.3. Influence of the Thickness to Radius Ratioh/R

The results in Table9 show the effect of the h/R ratio on the four (or five) lowest natural frequencies of thin spherical shells under free vibrations with clamped edge.Table 9
Influence of ratioh/R on the lowest natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with different methods at ν=0.3.
h/R ratioMode numberΩCurrentΩKunieda33ΩSHELL61ΩSHELL2090.002510.7470.7470.7460.74720.9290.9290.9290.92930.9680.9680.9680.96840.9830.9830.9830.9830.00510.7530.7520.7530.75320.9320.9320.9320.93230.9720.9720.9710.97140.9910.9910.9900.9900.0110.7610.7600.7610.76120.9380.9380.9370.93730.9830.9840.9380.98341.0201.0201.0201.0200.0210.7730.7720.7730.77320.9530.9530.9520.95231.0281.0291.0271.02741.06751.1301.1301.129We observe that, for ratioh/R, varying between 0.0025 and 0.02, the natural frequencies obtained by the current method are very close to those obtained by Kunieda’s method and those of the calculation by finite elements.The effect of transverse shear is nonexistent even for the highest ratioh/R=0.02: the element SHELL209 which takes account of the transverse shear gives the same results. Therefore, the current method can be applied in this case even for high h/R ratios and even for high modes, with negligible error as observed above.The effect of theh/R ratio is weak on the lowest frequency. The variation of this frequency is 3.5% when the h/R ratio varies from the value 0.0025 to 0.02 while the influence of the h/R ratio on the frequency of mode 4 is more important. The variation of this frequency is 8.5%, when the h/R ratio varies in the same interval. Kunieda [33] considers that the effect of h/R is not substantial for thin shells.This effect is much greater for thin shells with a clamped edge than for those with a free edge. When theh/R ratio varies from the value 0.01 to 0.02, the percentage variation of the lowest frequency is 1.6% for clamped edge and only 0.1% for free edge.The current method reveals a new natural frequency (1.067, mode 4) which is not obtained by the finite element calculus and which is absent from Kunieda’s results.


## 4.3.1. Frequency Equation

For free vibrations with clamped boundary conditions, the edgeφ=φ0 of the shell cannot translate in the ρ and φ directions and cannot undergo any rotation around the eθ axis. The boundary conditions, in this case, can be summarized as(50)w/φ=φ0∗=0,(51)dwdt/φ=φ0∗=0,(52)uφ∗/φ=φ0=0,where φ0=π/2. Also, in this case of clamped boundary conditions, the Legendre functions of the second kind Qnα can tend towards infinity for φ=0. To eliminate the singularity of Qnα, the coefficients Bα are set to zero as mentioned in the free vibrations with free edge condition section.Using the boundary conditions (50) to (52) along with the expression for uφ∗, the following form is obtained:(53)∑α=13AαPnα0=0,∑α=13Aαnα+1Pnα+10=0,∑α=13AαCαnα+1Pnα+10=0.We note that the displacementuφ∗ can be expressed as a function of F,φ (as derived in the appendix).The coefficients of the frequency equation:(54)Ciα=0;,i,α∈1,2,3,and they can be written as(55)C1α=Pnα0,C2α=nα+1Pnα+10,C3α=Cαnα+1Pnα+10.To calculate lowest natural frequency from equation (54)for ν=0.3 and φ0=π/2, we specify the value of h/R and Ω and then evaluate the determinant Ciα by incrementing Ω as done in the section of frequency equation for free vibrations with free edge.Table7 presents the lowest natural frequencies for thin spherical shells at h/R=3/100 and ν=0.3. The results are obtained by the current analytical method under axisymmetric free vibrations with clamped edge boundary condition.Table 7
The lowest natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with current analytical method ath/R=3/100 and ν=0.3.
Mode numberΩCurrent10.78220.97331.08841.099


## 4.3.2. Comparison of Results

In order to verify the accuracy of the current method, in the case of axisymmetric free vibrations for hemispherical thin shells with clamped edge boundary condition ath/R=1/100 and ν=0.3, the frequencies of the first sixteen modes are compared with those obtained by Kunieda [33] and calculus by the finite elements SHELL61 and SHELL209, as shown in Table 8.Table 8
Comparison of natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with different methods ath/R=1/100 and ν=0.3.
Mode number1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16thCurrent0.7610.9380.9831.0211.0421.0701.1441.2461.3791.5411.6601.7751.9982.2612.552Kunieda [33]0.7600.9380.9841.0201.0711.1441.2471.3811.5421.6601.7772.0022.266FE-SHELL610.7610.9370.9831.0201.0701.1431.2401.2451.3791.5391.6601.7741.9982.2602.552FE-SHELL2090.7610.9370.9831.0191.0691.1421.2431.3751.5341.6571.7671.9852.2422.527η1%0.13100.1010.0980.09300.0800.1440.06400.1120.1990.220η3%0.1310.1060.10100.0930.0870.5610.1440.19400.1680.1990.264η4%0.1310.1060.1010.0980.1860.1740.3200.4340.5180.1800.5620.8491.059The current solutions are in close agreement with the solutions obtained by Kunieda, FE-SHELL61, and FE-SHELL209. The difference between the results obtained by the current method and Kunieda and between the current method and FE-SHELL61 are closely zero. For the higher modes, a negligible difference appears in the results of FE-SHELL209 compared to the others results, probably due to the fact that it takes into account the transverse shear.The results of Table8 are instructive in more than one way. The current method reveals a new natural frequency 1.042 (5th mode) which does not appear in Kunieda’s method and in the the finite element calculation while the frequency 1.240 obtained by the FE-SHELL61 is nonexistent for the three other methods.


## 4.3.3. Influence of the Thickness to Radius Ratioh/R

The results in Table9 show the effect of the h/R ratio on the four (or five) lowest natural frequencies of thin spherical shells under free vibrations with clamped edge.Table 9
Influence of ratioh/R on the lowest natural frequencies for axisymmetric free vibrations, under clamped edge boundary condition, with different methods at ν=0.3.
h/R ratioMode numberΩCurrentΩKunieda33ΩSHELL61ΩSHELL2090.002510.7470.7470.7460.74720.9290.9290.9290.92930.9680.9680.9680.96840.9830.9830.9830.9830.00510.7530.7520.7530.75320.9320.9320.9320.93230.9720.9720.9710.97140.9910.9910.9900.9900.0110.7610.7600.7610.76120.9380.9380.9370.93730.9830.9840.9380.98341.0201.0201.0201.0200.0210.7730.7720.7730.77320.9530.9530.9520.95231.0281.0291.0271.02741.06751.1301.1301.129We observe that, for ratioh/R, varying between 0.0025 and 0.02, the natural frequencies obtained by the current method are very close to those obtained by Kunieda’s method and those of the calculation by finite elements.The effect of transverse shear is nonexistent even for the highest ratioh/R=0.02: the element SHELL209 which takes account of the transverse shear gives the same results. Therefore, the current method can be applied in this case even for high h/R ratios and even for high modes, with negligible error as observed above.The effect of theh/R ratio is weak on the lowest frequency. The variation of this frequency is 3.5% when the h/R ratio varies from the value 0.0025 to 0.02 while the influence of the h/R ratio on the frequency of mode 4 is more important. The variation of this frequency is 8.5%, when the h/R ratio varies in the same interval. Kunieda [33] considers that the effect of h/R is not substantial for thin shells.This effect is much greater for thin shells with a clamped edge than for those with a free edge. When theh/R ratio varies from the value 0.01 to 0.02, the percentage variation of the lowest frequency is 1.6% for clamped edge and only 0.1% for free edge.The current method reveals a new natural frequency (1.067, mode 4) which is not obtained by the finite element calculus and which is absent from Kunieda’s results.


## 5. Conclusions

The present method solves the differential equations of the axisymmetric vibrations of elastic thin spherical shells. These differential equations of motion in terms of stress take into account the terms of inertial couples in the equations of stress couples. These terms are omitted in the analytical methods proposed by Naghdi–Kalnins and by Kunieda. A new stress function is introduced and new equations as a function of the radial displacement and the stress function are derived. The harmonic axisymmetric free vibrations are described by a sixth-order differential equation in terms of the radial displacementw, the natural frequency Ω, the thickness to radius ratio of the shell h/R, and Poisson’s ratio ν. The general solution is expressed by the Legendre function of degree nα. Taking into account the boundary conditions makes it possible to reach the analytical solution in the specific case of the axisymmetric harmonic vibrations of the hemispherical shell. In this work, we tested two boundary conditions: free and clamped edges.The results demonstrate the correctness and high accuracy of the current analytical method compared with the Naghdi–Kalnins and Kunieda analytical methods and the finite element calculation.This method is distinguished by taking into account the inertia of rotation (inertial couple in the stress couples’ differential equation of motion) in addition to the two inertias of translation while the methods of Naghdi–Kalnins and Kunieda only take into account the two inertias of translation, which probably explains the evidence by this method of new frequencies. This method can be applied, with good approximation, for the calculation of the first natural frequencies even for moderately high values ofh/R ratios for free edge or clamped edge cases, despite the fact that it does not take into account the effect of the transverse shear.It can also be extremely useful for the validation of numerical methods.



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