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Titre: Title: Hybrid finite element method applied to the analysis of free vibration of spherical shell Auteurs: Authors: Mohamed Menaa et Aouni Lakis Date: 2013 Type: Rapport / Report Référence: Citation: Menaa, M. & Lakis, A. (2013). Hybrid finite element method applied to the analysis of free vibration of spherical shell (Rapport technique n° EPM-RT-2013- 02). Document en libre accès dans PolyPublie Open Access document in PolyPublie URL de PolyPublie: PolyPublie URL: http://publications.polymtl.ca/2800/ Version: Version officielle de l'éditeur / Published version Non révisé par les pairs / Unrefereed Conditions d’utilisation: Terms of Use: Tous droits réservés / All rights reserved Document publié chez l’éditeur officiel Document issued by the official publisher Maison d’édition: Publisher: École Polytechnique de Montréal URL officiel: Official URL: http://publications.polymtl.ca/2800/ Mention légale: Legal notice: Ce fichier a été téléchargé à partir de PolyPublie, le dépôt institutionnel de Polytechnique Montréal This file has been downloaded from PolyPublie, the institutional repository of Polytechnique Montréal http://publications.polymtl.ca
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Page 1: Titre: Hybrid finite element method applied to the ... fileTitre: Hybrid finite element method applied to the ...

Titre:Title:

Hybrid finite element method applied to the analysis of free vibration of spherical shell

Auteurs:Authors: Mohamed Menaa et Aouni Lakis

Date: 2013

Type: Rapport / Report

Référence:Citation:

Menaa, M. & Lakis, A. (2013). Hybrid finite element method applied to the analysis of free vibration of spherical shell (Rapport technique n° EPM-RT-2013-02).

Document en libre accès dans PolyPublieOpen Access document in PolyPublie

URL de PolyPublie:PolyPublie URL: http://publications.polymtl.ca/2800/

Version: Version officielle de l'éditeur / Published versionNon révisé par les pairs / Unrefereed

Conditions d’utilisation:Terms of Use: Tous droits réservés / All rights reserved

Document publié chez l’éditeur officielDocument issued by the official publisher

Maison d’édition:Publisher: École Polytechnique de Montréal

URL officiel:Official URL: http://publications.polymtl.ca/2800/

Mention légale:Legal notice:

Ce fichier a été téléchargé à partir de PolyPublie, le dépôt institutionnel de Polytechnique Montréal

This file has been downloaded from PolyPublie, theinstitutional repository of Polytechnique Montréal

http://publications.polymtl.ca

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EPM–RT–2013-02

HYBRID FINITE ELEMENT METHOD APPLIED TO THE

ANALYSIS OF FREE VIBRATION OF SPHERICAL SHELL

M. Menaa, A.A. Lakis Département de Génie mécanique École Polytechnique de Montréal

Avril 2013

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1

EPM-RT-2013-02

HYBRID FINITE ELEMENT METHOD APPLIED TO THE ANALYSIS OF FREE VIBRATION OF SPHERICAL SHELL

M.Menaa, A.A.Lakis Département de génie mécanique École Polytechnique de Montréal

Avril-2013

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2013 Mohamed Menaa, Aouni A. Lakis Tous droits réservés

Dépôt légal : Bibliothèque nationale du Québec, 2013 Bibliothèque nationale du Canada, 2013

EPM-RT-2013-02 Hybrid finite element method applied to analysis of free vibration of spherical shell par : Mohamed Menaa, Aouni A. Lakis Département génie mécanique École Polytechnique de Montréal Toute reproduction de ce document à des fins d'étude personnelle ou de recherche est autorisée à la condition que la citation ci-dessus y soit mentionnée. Tout autre usage doit faire l'objet d'une autorisation écrite des auteurs. Les demandes peuvent être adressées directement aux auteurs (consulter le bottin sur le site http://www.polymtl.ca/) ou par l'entremise de la Bibliothèque :

École Polytechnique de Montréal Bibliothèque – Service de fourniture de documents Case postale 6079, Succursale «Centre-Ville» Montréal (Québec) Canada H3C 3A7 Téléphone : (514) 340-4846 Télécopie : (514) 340-4026 Courrier électronique : [email protected]

Ce rapport technique peut-être repéré par auteur et par titre dans le catalogue de la Bibliothèque : http://www.polymtl.ca/biblio/catalogue/

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Abstract

In this study, free vibration analysis of spherical shell is carried out. The structural model is based

on a combination of thin shell theory and the classical finite element method. Free vibration equations

using the hybrid finite element formulation are derived and solved numerically. The results are validated

using numerical and theoretical data available in the literature. The analysis is accomplished for

spherical shells of different boundary conditions and radius to thickness ratios. This proposed hybrid

finite element method can be used efficiently for design and analysis of spherical shells employed in high

speed aircraft structures.

1. Introduction

Shells of revolution, particularly spherical shells are one of the primary structural elements in high

speed aircraft. Their applications include the propellant tank or gas-deployed skirt of space crafts. Free

vibration of spherical shell has been investigated by numerous researchers experimentally and

analytically.

Kalnins [1,2], studying analytically free vibrations in shallow spherical shell, selected used two

auxiliary variables for the axial and circumferential displacements while considering the effect of

longitudinal, transverse and rotary inertia as well as transverse shear deformation on the non-asymmetric

vibration of shallow spherical shells. Navaratna [4], Webster [5], Greene et al. [7] used the classical finite

element method to study the free vibration of thin spherical shell. Cohen[3] using a method of iteration

like Stodola’s method determined the natural frequencies and mode shapes of spherical shell method.

Kraus [6] investigated the case of clamped spherical shell using a general theory which included the

effects of transverse shear stress and rotational inertia. Tessler and Spiridigliozzi [8] gave frequencies of

60° clamped spherical shell and hemispherical shell for radius to thickness from 10 to 100 and their

analysis was based upon shell theory. Narasimhan and Alwan [9] analyzed the axisymmetric free

vibration of clamped isotropic spherical shell cap. Thick shell analysis was given by Gautham and

Ganesan [10] for the analysis of a 60° clamped and simply supported spherical shells, the semi-analytical

method was used to reduce the dimension of the problem. The same authors [11] investigated the

analysis of a clamped isotropic hemispherical shell (φ0 =90°). Sai Ram and Sridhar Babu [12] used the

classical finite elements method to study the free vibration of composite spherical shell cap with or

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3

without a cutout. Buchanan and Rich [13] investigated the case of 60° clamped and simply supported

spherical shells using classical finite elements method. Recently, Ventsel et al. [14] used a combined

formulation of the boundary elements method and finite elements method to study the free vibration of an

isotropic simply supported hemispherical shell with different circumferential mode numbers.

The objective of the present study is to develop a general hybrid finite element package for

predicting the dynamic behavior of isotropic spherical shells with boundary conditions which can be

varied as desired. The solution scheme is based on the hybrid finite element method. This method uses

displacements functions derived from the shell theory instead of polynomials in classical finite element

method. The element is a spherical frustum instead of the usual triangular or rectangular shell element.

This developed method demonstrated precise and fast convergence with few elements. On the other hand,

the present theory, because of its usage of shell classical theory for the displacement functions can easily

be adapted to take the hydrodynamic effects into account. Finally, again because of the use of shell

classical theory, we can obtain the high as well as the low frequencies with high accuracy.

2. Finite element formulation

In this study the structure is modeled using hybrid finite element method which is a combination

of spherical shell theory and classical finite element method. In this hybrid finite element method, the

displacement functions are found from exact solution of spherical shell theory rather approximated by

polynomial functions done in classical finite element method. In the spherical coordinate

system(R,θ,φ) shown in Fig. 1, five out of the six equations of equilibrium derived in reference for

spherical shells under external load are written as follows :

( )

( )

1 cot 0sin

1 2 cot 0sin1 cot ( ) 0

sin1 cot 0

sin1 2 cot 0

sin

N NN N Q

N N N Q

Q Q Q N N

M MM M RQ

M M M RQ

φ φθφ θ φ

φθ θφθ θ

φ θφ φ θ

φ φθφ θ φ

φθ θφθ θ

φφ φ θ

φφ φ θ

φφ φ θ

φφ φ θ

φφ φ θ

∂ ∂+ + − + =

∂ ∂∂ ∂

+ + + =∂ ∂

∂ ∂+ + − + =

∂ ∂∂ ∂

+ + − − =∂ ∂

∂ ∂+ + − =

∂ ∂ (1)

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φ

Where Nφ , Nθ, Nφθ are membrane stress resultants; Mφ , Mθ, Mφθ the bending stress resultants and Qφ ,

Qθ the shear forces (Fig. 2). The sixth equation, which is an identity equation for spherical shells, is not

presented here.

W

2X

θ

1X

X3

Fig.1. Geometry of spherical shell

θ

Mθ Mθφ

Mφθ

Nθφ Nθ

Nφθ

φ dφ

Fig.2. Stress resultants and stress couples

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Strain and displacements for three displacements in axialUφ , radial W and circumferential Uθ are related

as follows:

2

2 2

2

2 2 2

2

1

1 1( cot )sin

1 1( cot )sin2

1

2 1 1 1cot cotsin sin

1 1 cotsin

UW

RU

U WR

UUU

RU W

R

U W WUR

UUU

R

φ

θφ

φ

φθθθ

φθ

φφ

θ

φθ θφ

φθθ

φ

φφ θε

ε φφ φ θε

εκ

φκ φκ

φ φφ θ φφ θ

φ φ θ

∂ + ∂

∂+ +

∂ ∂∂ + −

∂ ∂ = = ∂ ∂ − ∂ ∂ ∂ ∂ ∂ + − − ∂ ∂∂

∂∂+ −

∂ ∂

21 12 cot 2sin sin

W Wφ φφ θ φ φ θ

∂ ∂ + − ∂ ∂ ∂

(2)

DisplacementsU , W and V in the global cartesian coordinate system are related to displacements iUφ ,

iW and iUθ indicated in Fig 3. by:

sin cos 0cos sin 0

0 0 1

i i i

i i i

i

U UW WV U

φ

θ

φ φφ φ

− =

(3)

The stress vector σ is expressed as function of strain ε by

[ ] Pσ ε= (4)

Where [ ]P is the elasticity matrix for an anisotropic shell given by

[ ]

11 12 14 15

21 22 24 25

33

41 42 44 45

51 52 54 55

36 66

0 00 0

0 0 0 0 00 00 0

0 0 0 0

P P P PP P P P

PP

P P P PP P P P

P P

=

(5)

Upon substitution of equations (2), (4) and (5) into equations (1), a system of equilibrium equations can

be obtained as a function of displacements:

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( )( )( )

1

2

3

, , , 0

, , , 0

, , , 0

ij

ij

ij

L U W U P

L U W U P

L U W U P

φ θ

φ θ

φ θ

=

=

=

(6)

These three linear partial differentials operators 1L , 2L and 3L are given in the Appendix, and ijP are

elements of the elasticity matrix which, for an isotopic thin shell with thickness h is given by:

[ ]( )

( )

0 0 0 00 0 0 0

10 0 0 0 0

20 0 0 00 0 0 0

10 0 0 0 0

2

D DD D

D

PK KK K

K

νν

ν

νν

ν

− = −

(7)

Where 21EtDν

=−

is the membrane stiffness and ( )

3

212 1EtK

ν=

− is the bending stiffness.

The element is a circumferential spherical frustum shown in Fig. 3. It has two nodal circles with four

degrees of freedom; axial, radial, circumferential and rotation at each node. This element type makes it

possible to use thin shell equations easily to find the exact solution of displacement functions rather than

an approximation with polynomial functions as done in classical finite element method. For motions

associated with the nth circumferential wave number we may write:

( )( )( )

( )( )( )

[ ]( )( )( )

, cos 0 0, 0 cos 0, 0 0 sin

n n

n n

n n

U n u uW n w T wU n u u

φ φ φ

θ θ θ

φ θ θ φ φφ θ θ φ φφ θ θ φ φ

= =

(8)

The transversal displacement ( )nw φ can be expressed as:

3

1( ) n

n ii

w wφ=

= ∑ (9)

Where

( ) ( )cos cosi i

n n ni i iw A P B Qµ µφ φ= + (10)

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iUφ

i

dWdφ

iW jφ

iUθ θ

Fig. 3. Spherical frustum element

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8

And where ( )cosi

nPµ φ , ( )cosi

nQµ φ are the associated Legendre functions of the first and second kinds

respectively of order n and degree iµ .

The expression of the axial displacement uφn (φ) is:

( ) ( )23

1 2sin

ni

n ii

dw nu Edφ φ ψ φ

φ φ=

= −∑ (11)

Where the coefficient Ei is given by:

( )( )(1 ) (1 )

1 1i

ii

kEk

λ ν νλ ν

+ + − −=

+ − + (12)

The auxiliary function ψ is given by the expression:

( ) ( ) ( )4 1 4 1cos cosn nA P B Qψ φ φ φ= + (13)

Finally the circumferential displacement uθn (φ) can be expressed as:

( )3

1

1sin 2

nn i i

i

n du n E wdθψφ

φ φ=

= − +∑ (14)

The degree iµ is obtained from the expression

1 21 1

4 2i iµ λ = + −

(15)

Where iλ is one the roots of the cubic equation:

3 21 2 3 0h h hλ λ λ− + − = (16)

Where

12

22

3

44 (1 )(1 )2(1 )(1 )

hh kh k

ν

ν

=

= + + −

= + − (17)

With 2

212 Rkh

=

The above equation has three roots with one root is real and two other are complex conjugate roots.

The Legendre functions 1

nPµ , 1 1 1

1 1, andn n nP Q Qµ µ µ− − are a real functions whereas

i

nPµ , 1 1, andi i i

n n nP Q Qµ µ µ− −

(i = 2, 3) are complex functions so we can put:

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2 2 2

3 2 2

2 2 2

3 2 2

2 2 2

3 2 2

2 2 2

3 2

1 1 1

1 1 1

1 1 1

1 1

Re( ) Im( )

Re( ) Im( )

Re( ) Im( )

Re( ) Im( )

Re( ) Im( )

Re( ) Im( )

Re( ) Im( )

Re( ) Im

n n n

n n n

n n n

n n n

n n n

n n n

n n n

n n

P P i P

P P i P

Q Q i Q

Q Q i Q

P P i P

P P i P

Q Q i Q

Q P i

µ µ µ

µ µ µ

µ µ µ

µ µ µ

µ µ µ

µ µ µ

µ µ µ

µ µ

− − −

− − −

− − −

− −

= +

= −

= +

= −

= +

= −

= +

= −2

1( )nQµ−

(18)

Setting

( )( )( )

1 1 1

2 2 2 3

3 3 2 3

( 1)

( 1)

( 1)

n n c

n n c ic

n n c ic

µ µ

µ µ

µ µ

− − + =

− − + = +

− − + = −

(19)

1 1

2 2 3

3 2 3

E eE e ieE e ie

== −= +

(20)

Substituting equations (18), (19) and (20) in equations (9), (11) and (14) we have:

( ) ( )1 1

2 2 2 2

2 2 2 2

11 1 1 1

1 12 3 2 2 3 3 3 2 2 3 2 3

1 13 2 3 2 2 3 2 2 3 3

cot

cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( ) ( )

cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( ) (

n nn

n n n n

n n n n

u ne P e c P A

ne P ne P e c e c P e c e c P A A

ne P ne P e c e c P e c e c P i

φ µ µ

µ µ µ µ

µ µ µ µ

φ φ

φ φ

φ φ

− −

− −

= − +

+ − − + + + − + + − − − + +

( )1 1

2 2 2 2

2 2 2

2 3

2

1 4

11 1 1 1

1 12 3 2 2 3 3 3 2 2 3 2 3

13 2 3 2 2 3 2 2

)

2sin

cot

cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( ) ( )

cot Re( ) cot Im( ) ( ) Re( ) (

n

n n

n n n n

n n n

A A

n P A

ne Q e c Q B

ne Q ne Q e c e c Q e c e c Q B B

ne Q ne Q e c e c Q e c

µ µ

µ µ µ µ

µ µ µ

φ

φ

φ φ

φ φ

− −

+ −

+ − +

+ − − + + + − +

+ − − − +2

13 3 2 3

2

1 4

) Im( ) ( )

2sin

n

n

e c Q i B B

n Q B

µ

φ

− + −

+ −

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( )1 2 2 1 2 21 2 3 2 3 1 2 3 2 3Re( )( ) Im( ) ( ) Re( )( ) Im( ) ( )n n n n n n

nw P A P A A P i A A Q B Q B B Q i B Bµ µ µ µ µ µφ = + + + − + + + + −

( )

( )( )

1

2 2 2 2 2 2

1

2 2 2

1 1

3 2 3 3 2 3

21

1 1 4

1 1

3

1sin1 1 1 1Re( ) Im( ) ( ) Re( ) Im( ) ( )

sin sin sin sin

cot 2 12 2

1sin

1 1Re( ) Im( )sin sin

nn

n n n n

n n

n

n n

u ne P A

ne P ne P A A ne P ne P i A A

n nP n n P A

ne Q B

ne Q ne Q

θ µ

µ µ µ µ

µ

µ µ

φφ

φ φ φ φ

φ

φ

φ φ

= −

− + + + − −

+ − + − +

− +

( )( )

2 2 22 3 3 2 3

21

1 1 4

1 1( ) Re( ) Im( ) ( )sin sin

cot 2 12 2

n n

n n

B B ne Q ne Q i B B

n nQ n n Q B

µ µφ φ

φ −

+ + − −

+ − + − +

In deriving the above relation we used the recursive relations:

1

1

cot ( 1)( )

cot ( 1)( )

i

i i

i

i i

nn n

i i

nn n

i i

dPn P n n P

ddQ

n Q n n Qd

µµ µ

µµ µ

φ µ µφ

φ µ µφ

= − + − − +

= − + − − +

(22)

Using matrix formulation, the displacement functions can be expressed as follows:

( )( )( )

[ ]( )( )( )

[ ][ ] ,,,

n

n

n

U uW T w T R CU u

φ φ

θ θ

φ θ φφ θ φφ θ φ

= = (23)

The vector C is given by the expression:

( ) ( ) 1 2 3 2 3 4 1 2 3 2 3 4TC A A A i A A A B B B i B B B= + − + − (24)

The elements of matrix [ ]R are given in the Appendix.

In the finite element method, the vector C is eliminated in favor of displacements at elements nodes. At

each finite element node, the three displacements (axial, transversal and circumferential) and the rotation

are applied. The displacement of node i are defined by the vector:

(21)

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Tii i in

i n n ndwu w udxφ θδ

= (25)

The element in Fig. 3 with two nodal lines (i and j) and eight degrees of freedom will have the following

nodal displacement vector:

[ ]

Ti ji i i i j j jn n

n n n n n nj

dw dwu w u u w u A Cd dφ θ φ θ

δδ φ φ

= = (26)

With

( )( )

1 1 2 2 2

2 2 2 1 1

2 2 2

1 1 11 1 2 3 2 3

1 1 13 2 2 3 1 1

1 12 3

cot cot Re( ) Re( ) Im( ) ( )

cot Im( ) Re( ) Im( ) ( ) cot

cot Re( ) Re( ) Im( ) (

n n n n nn

n n n n n

n n n

dw n P c P A n P c P c P A Ad

n P c P c P i A A n Q c Q B

n Q c Q c Q

µ µ µ µ µ

µ µ µ µ µ

µ µ µ

φ φφ

φ φ

φ

− − −

− − −

− −

= − + + − + − +

+ − + + − + − + + − + −

2 2 2

2 3

1 13 2 2 3

)

cot Im( ) Re( ) Im( ) ( )n n n

B B

n Q c Q c Q i B Bµ µ µφ − −

+

+ − + + − (27)

The terms of matrix [ ]A are obtained from the values of matrix [ ]R and ndwdx

are given in the appendix.

Now, pre-multiplying by [ ] 1A − equation (26) one obtains the matrix of the constant Ci as a function of

the degree of freedom:

[ ] 1 i

jC A

δδ

− =

(28)

Finally, one substitutes the vector C into equation and obtains the displacement functions as follows:

( )( )( )

[ ][ ][ ] [ ]1,,,

i i

j j

UW T R A NU

φ

θ

φ θδ δ

φ θδ δ

φ θ

= =

(29)

The strain vector ε can be determined from the displacement functions , ,U U Wφ θ and the deformation

–displacement as:

[ ] [ ][ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]10 00 0

i i

j j

T TQ C Q A B

T Tδ δ

εδ δ

− = = =

(30)

Where matrix [ ]Q is given in the appendix.

This relation can be used to find the stress vector, equation (4), in terms of the nodal degrees of freedom

vector:

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[ ][ ] i

jP B

δσ

δ

=

(31)

Based on the finite element formulation, the local stiffness and mass matrices are:

[ ] [ ] [ ][ ]

[ ] [ ] [ ]

T

locA

T

locA

k B P B dA

m h N N dAρ

=

=

∫∫

∫∫ (32)

Where ρ the density and h is the thickness of shell.

The surface element of the shell wall is 2 sindA R d dφ φ θ= . After integrating overθ , the preceding

equations become

[ ] [ ] [ ][ ] [ ]1 2 1 1 1sinj

i

T TT

lock A R Q P Q d A A G A

φ

φ

π φ φ− − − −

= =

[ ] [ ] [ ] [ ]1 2 1 1 1sinj

i

T TT

locm h A R R R d A h A S A

φ

φ

ρ π φ φ ρ− − − −

= =

∫ (33)

In the global system the element stiffness and mass matrices are

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

1 1

1 1

TT

TT

k LG A G A LG

m h LG A S A LGρ

− −

− −

=

= (34)

Where

[ ]

sin cos 0 0 0 0 0 0cos sin 0 0 0 0 0 0

0 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 sin cos 0 00 0 0 0 cos sin 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

i i

i i

j j

j j

LG

φ φφ φ

φ φφ φ

− = − (35)

From these equations, one can assemble the mass and stiffness matrix for each element to obtain the mass

and stiffness matrices for the whole shell: [ ]M and [ ]K . Each elementary matrix is 8x8, therefore the

final dimensions of [ ]M and [ ]K will be 4(N+1) where N is the number of elements of the shell.

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13

φ0

3. Numerical results

In order to test the efficiency and the accuracy of our model, we used the theory developed in this

paper to calculate the natural frequencies and modes of uniform thin elastic spherical shell, which were

both non-shallow(φ0 <30) and shallow, of various dimensions and under different boundary conditions.

These cases have already been investigated by other authors using others methods. For purposes of

comparisons among the natural frequencies obtained, it is eminently practical at this stage to introduce

the non-dimensional natural frequency:

12

REρω Ω =

(36)

Where:

ω is the natural angular frequency.

R is the radius of the reference surface.

ρ is the density.

E is the modulus of elasticity.

Fig.4. Definition of angle φ0

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14

3.1 Case 1: clamped spherical shell with φ0 =10°

Narassihan and Alwar [9] investigated the case of an axisymmetric clamped spherical shell. The

analysis is based on the application of the Chebyshev-Galerkin spectral method for the evaluation of free

vibration frequencies and mode shapes. Sai Ram and Sreedhar Babu [12] analyzed the same case with the

classical finite element method using 80 elements. Each element is an eight noded degenerated

isoparametric shell element with nine degrees of freedom at each node. With our model and using 6 finite

elements, the natural were computed, the results are shown in table 1.

Mode Present Sai Ram and Sreedhar babu[12] Narassihan and Alwar [9]

1 1.4861 1.4577 1.4588

2 2.2498 2.2931 2.2999

3 4.4779 4.5773 4.5461

Table 1: Normalized natural frequencies for 10° clamped spherical shell with 200Rh

=

3.2 Case 2: clamped spherical shell with φ0 =30°

This case was investigated analytically by Kalnins [1] using classical theory and transverse

vibration theory. With our theory, we used 8 finite elements to study the spherical shell with the results

shown in table 2. The frequencies we obtained with our model are very comparable to Kalinin’s values.

The maxim displacement values were:

( )max

max

3.61WUφ

= ( )

max

max

1.54WUθ

=

It was observed that at lowest natural frequency, motion of the spherical shell is mainly dominated by its

radial component.

Mode Present theory Kalnins [1]

1 1.169 1.168

2 2.224 2.589

3 3.303 3.230

4 4.200 4.288

5 4.923 4.683

Table 2: Normalized natural frequencies for 30° clamped spherical shell with 20Rh

=

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15

3.3 Case 3: spherical shell φ0 =60° under the three boundary conditions: clamped, simply upported

and free

The free axisymmetric vibration of the spherical shell in this case was studied by Kalnins [2],

Cohen [3], Navaratna [4], Webster [5], Greene et al [7], Tessler and Spiridigliozzi [8], Gautham and

Ganesan [10] and Buchanan and Rich [13]. In the present investigation, the shell was investigated with

10 elements; the results are given respectively for clamped, simply supported and free shells in tables 3, 4

and 5.

The natural modes corresponding to the lowest shell natural frequencies under the two boundary

conditions are illustrated in figures 5 and 6. They reveal that at the lowest natural frequency, spherical

shell motion is radial.

It is easy to see that all displacementsUφ , W and Uθ are all zero at the top (φ = 0 ) of the spherical shell.

Mode Kalnins[2] Navaratna [4] Webster

[5]

Tessler and

Spiridigliozzi

[8]

Gautham

and Ganesan

[ 10]

Buchanan

and Rich

[13]

Present

theory

1 1.006 1.008 1.007 1.000 1.001 1.001 1.031

2 1.391 1.395 1.391 1.368 1.373 1.370 1.496

3 - 1.702 1.700 1.673 1.678 1.675 1.760

4 - 2.126 2.095 - - 2.094 2.089

5 2.375 2.387 2.386 2.260 - 2.256 2.276

6 3.486 3.506 3.851 3.213 - 3.209 3.311

7 3.991 3.996 4.062 3.965 - 3.964 3.775

8 - 4.159 4.151 - - 4.060 4.073

9 4.947 5.001 5.962 4.442 - 4.427 4.826

10 - 6.037 6.208 5.773 - 5.740 5.777

Table 3: Normalized natural frequencies for 60° clamped spherical shell with 20Rh

=

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16

Table 4: Normalized natural frequencies for 60° simply supported spherical shell with 20Rh

=

Fig.5. Displacements versus φ coordinate for clamped spherical shell φ0 =60

Mode Kalnins [2]Navaratna

[4]

Greene et

al [7]

Cohen

[3]

Gautham

and

Ganesan

[ 10]

Buchanan

and Rich

[13]

Present

Theory

1 0.962 0.963 0.974 0.959 - 0.956 0.981

2 1.334 1.338 1.338 1.325 1.315 1.308 1.412

3 - 1.653 1.652 1.646 1.639 1.612 1.646

4 2.128 2.131 2.162 - - 2.044 2.038

5 - 2.141 - - - 2.059 2.115

6 3.176 3.185 - - - 2.965 2.934

7 3.988 3.933 - - - 3.837 3.871

8 - 4.159 - - - 4.000 4.017

9 4.575 4.601 - - - 4.148 4.138

10 - 6.031 - - - 5.608 5.773

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17

Table 5: Normalized natural frequencies for 60° free spherical shell with 20Rh

=

Fig.5. Displacements versus φ coordinate for simply supported spherical shell φ0 =60

Mode Kalnins [2] Navaratna [4] Webster [5] Present theory

1 0.931 0.932 0.931 0.938

2 1.088 1.094 1.089 1.062

3 1.533 1.544 1.535 1.426

4 2.348 2.363 2.360 2.425

5 2.544 2.548 2.551 2.725

6 - 2.982 2.985 2.944

7 3.497 3.519 4.023 4.264

8 - 4.971 4.950 4.973

9 4.951 4.980 5.548 5.793

10 5.230 5.543 6.224 6.605

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18

3.4 Case 4: Spherical shell with φ0 =90°

Kraus [6] investigated the case of simply supported spherical shell using a general theory which

included the effects of transverse shear stress and rotational inertia. For cases both with and without these

effects, he determined the natural frequencies for the shell motion that was independent of θ for

circumferential mode number n = 0. Tessler and Spiridigliozzi [8], Gautham and Ganesan [11] analyzed

the case of clamped hemispherical shell. Ventsel et al. [14] studied the case of simply supported

spherical shell using the boundary elements method for various circumferential mode

numbers ( )n n n= = =0 1 2, , . With our model and using 12 finite elements, the natural frequencies were

computed for clamped and simply supported shells. The results are shown respectively in table 7 and

table 8. The maximum displacements values are:

( )max0.3381Uφ = max 0.2317W = ( )max

0.0854Uθ =

The result is that at the lowest natural frequency, the motion of spherical shell is predominately by the

axial displacement.

Mode Tessler and

Spiridigliozzi [8]

Gautham and

Ganesan [11]

Present

theory

1 0.8481 0.8439 0.8327

2 1.2328 1.2317 1.1919

3 1.5902 1.5808 1.5041

4 1.9435 1.9267 1.9161

Table 7 : Normalized natural frequencies for 90° clamped spherical shell with 10R

h=

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19

Table 8: Normalized natural frequencies for 90° simply supported spherical shell 4. Conclusion

The purpose of the investigation described in this paper is to determine the natural frequencies

and shape modes of free vibrations of spherical shell. The modal is based on hybrid approach combining

the classical finite element method and the classical shell theory. This theoretical approach is much more

precise than usual finite element methods because the displacement functions are derived from exact

solutions of equilibrium equations for spherical shells. The mass and stiffness matrices are determined by

numerical integration.

The results obtained for spherical shells with different angles and different boundary conditions

are compared with results available in the literature. Very good agreement was found. This approach

resulted in a very precise element that leads to fast convergence and less numerical difficulties from the

computational point of view. Because of its use of classical shell theory for the displacement functions,

the presented method may easily be adapted to take fluid-structure effects into account. A paper under

preparation on the effect fluid on vibrations of shells confirms this approach . For the same reason, we

can obtain the high as well as low frequencies with very good accuracy.

Mode Kraus [6]

10Rh

=

Kraus [6]

50Rh

=

Ventsel et

al.[14]

200Rh

=

Present

theory

50Rh

=

1 0.8060 0.7548 0.7441 0.7579

2 1.2054 0.9432 0.9281 0.9034

3 1.6179 1.0152 0.9693 0.9499

4 1.9051 1.1082 - 1.1089

5 2.7205 1.2523 - 1.2759

6 2.9301 1.4576 - 1.4723

7 4.0274 1.6558 - 1.6237

8 5.5142 1.7636 - 1.7634

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20

Appendix

11 411 2

12 422

214 44

2 2

( , , ) cot( )

1 1cot( ) cot( ) cot( )sin( ) sin( )

1

U UP PL U U W W W

R R

U UP PU W U W

R R

UP P WR R R

φ φφ θ

θ θφ φ

φ

φφ φ φ

φ φ φφ φ θ φ θ

φ φ φ

∂ ∂ ∂ = + + + + ∂ ∂ ∂ ∂ ∂ ∂ + + + + + + + ∂ ∂ ∂

∂ ∂ ∂ + + − ∂ ∂ ∂

2

2

2 215 45

2 2 2 2 2

51212

cot( )

1 1 1 1 1cot( ) cot( ) cot( ) cot( ) cot( )sin( ) sin( )sin ( ) sin ( )

U W

P P U UW W W WU UR R R

UPPW

R R

φ

θ θφ φ

φ

φφ φ

φ φ φ φ φφ φ θ φ φ θ φφ θ φ θ

φ

∂ ∂+ − ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + − − + + − − ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ − + + ∂

52222

254242 2

225 55

2 2 2

33 632

cot( )

1 cot( ) cot( )sin( )

1 cot( )

1 1 1cot( ) cot( ) cot( )sin( ) sin ( )

1

P UPU W

R RUPP W

R R R

P P U W WUR R R

P PR R

θφ

φ

θφ

φ

φ φφ θ

φφ φ

φ φ φφ θ φφ θ

∂ − + + + ∂ ∂ ∂ − + − ∂ ∂

∂ ∂ ∂ − + + − − ∂ ∂∂

+ +

236 66

2

51212 2

52222

1 cot( )sin( ) sin( )

1 1 1 cot( ) 2cot( ) 2sin( ) sin( ) sin( ) sin( )

1( , , )sin( )

UUU

UP P U W WUR R R

UPPL U U W W

R R

PPR R

ϕθθ

ϕθθ

φφ θ

φφ θ φ φ θ

φφφ θ φ φ θ φ θ φ φ θ

φ θ φ

∂ ∂∂+ − ∂ ∂ ∂

∂ ∂∂ ∂ ∂ + + + − + − ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ = + + ∂ ∂

+ +

254242 2

225 55

2 2 2

33 632

1 1 cot( )sin( ) sin( )

1 1sin( )

1 1 1 1cot( ) cot( )sin( ) sin( ) sin ( )

1

UU W

UPP WR R R

P P U W WUR R R

P P UR R

θφ

φ

θφ

θ

φφ θ φ θ

φ θ φ φ

φ φφ θ φ θ φφ θ

φ φ

∂ ∂ + + ∂ ∂ ∂ ∂ ∂ + + − ∂ ∂ ∂

∂ ∂ ∂ ∂ + + + − − ∂ ∂ ∂∂

∂∂ + + + ∂ ∂ 2

36 662

1cot( ) 2 cot( ) cot( )sin( ) sin( )

1 1 cot( ) 2 1 cot( )cot( ) 2 2 cot( ) 2sin( ) sin( ) sin( ) sin( ) si

U UUU U

U UP P U UW WU UR R R

ϕ ϕθθ θ

ϕ ϕθ θθ θ

φ φ φφ θ φ φ θ

φ φφ φφ φ φ θ φ θ φ φ θ φ φ θ

∂ ∂ ∂− + + − ∂ ∂ ∂

∂ ∂ ∂ ∂∂ ∂ ∂ + + + − + − + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

22 cot ( )n( ) sin( )

W W t φφ θ φ φ θ

∂ ∂− ∂ ∂ ∂

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21

11 213

12 22

214 242 2 2

215 252 2 2 2

41 5

( , , )

1 cot( )sin( )

1 1cot( ) cot( )sin( ) sin ( )

UP PL U U W W

R R

UP PU W

R RUP P W

R R

P P U W WUR R

P P

φφ θ

θφ

φ

θφ

φ

φφ θ

φ φ

φ φφ θ φφ θ

∂ = − + + ∂ ∂ − + + + ∂

∂ ∂ − + − ∂ ∂ ∂ ∂ ∂ − + + − − ∂ ∂∂

−+ 1

2

41 512

42 522

cot( )

1 sin( )sin( )

1 1cot( ) cot( ) cot( )sin( ) sin( )

U UW W

R

U UP W P W

R

P P U UU W U W

R

φ φ

φ φ

θ θφ φ

φφ φ φ

φφ φ φ θ φφ

φ φ φφ φ θ φ θ

∂ ∂ ∂+ − + ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂+ + + + ∂ ∂ ∂ ∂ ∂

− ∂ ∂ ∂+ + + − + + ∂ ∂ ∂

42 522

2 244 54

3 2 2

443

1 1 1sin( ) cot( ) cot( )sin( ) sin( )sin( )

cot( )

1 sin( )sin( )

U UP U W P U W

R

U UP P W WR

UP

R

θ θφ φ

φ φ

φ φ φφ φ φ θ θ φ θφ

φφ φ φφ φ

φφ φφ

∂ ∂ ∂ ∂ ∂

+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ − ∂ ∂ ∂

+ − − − ∂ ∂ ∂∂ ∂

∂∂ ∂+

∂ ∂

2 2

542 2

2 245 55

3 2 2 2 2

3

1 1 1 1cot( ) cot( ) cot( ) cot( ) cot( )sin( ) sin( )sin ( ) sin ( )

1sin( )

UW WP

P P U UW W W WU UR

PR

φ φ

θ θφ φ

φ θ φφ φ

φ φ φ φ φφ φ θ φ φ θ φφ θ φ θ

φ

∂ ∂ ∂ ∂− + − ∂ ∂ ∂∂ ∂

− ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − − − + − − ∂ ∂ ∂ ∂ ∂∂ ∂

+

( )

2 2

45 552 2 2 2

263

2

1 1 1 1sin( ) cot( ) cot( ) cot( ) cot( )sin( ) sin( )sin ( ) sin ( )

1 cot( ) 3cotsin( )sin( )

U UW W W WU P U

UP UU

R

θ θφ φ

ϕθθ

φ φ φ φ φφ φ φ θ φ θ φ θ φφ θ φ θ

φ φφ θ φ φ θφ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ − − + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂

∂ ∂∂ ∂+ + − + ∂ ∂ ∂ ∂ ∂

( )2 2 2

663

1 cot( )sin( )

cot( ) cot( )1 2 1 2cot( ) 2 3cot cot( ) 2sin( ) sin( ) sin( ) sin( ) sin( ) sin( )sin( )

UUU

U UP U UW W W WU UR

ϕθθ

ϕ ϕθ θθ θ

φθ φ φ θ

φ φφ φ φφ θ φ φ θ φ θ φ φ θ φ φ θ φ θ φ φ θφ

∂ ∂+ − ∂ ∂

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂+ + − + − + + − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

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22

( )1 1

11 1 1(1,1) cot n nR e n P e c Pµ µφ −= − +

2 2 2 2

1 12 3 2 2 3 3 3 2 2 3(1, 2) cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( )n n n nR ne P ne P e c e c P e c e c Pµ µ µ µφ φ − −= − − + + + −

2 2 2 2

1 13 2 3 2 2 3 2 2 3 3(1,3) cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( )n n n nR ne P ne P e c e c P e c e c Pµ µ µ µφ φ − −= − − − + +

2

1(1, 4)2sin

nnR Pφ

= −

( )1 1

11 1 1(1,5) cot n nR e n Q e c Qµ µφ −= − +

2 2 2 2

1 12 3 2 2 3 3 3 2 2 3(1,6) cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( )n n n nR ne Q ne Q e c e c Q e c e c Qµ µ µ µφ φ − −= − − + + + −

2 2 2 2

1 13 2 3 2 2 3 2 2 3 3(1,7) cot Re( ) cot Im( ) ( ) Re( ) ( ) Im( )n n n nR ne Q ne Q e c e c Q e c e c Qµ µ µ µφ φ − −= − − − + +

2

1(1,8)2sin

nnR Qφ

= −

1(2,1) nR Pµ=

2(2, 2) Re( )nR Pµ=

( )2

2,3 Im( )nR Pµ=

( )2,4 0R =

1(2,5) nR Qµ=

2(2,6) Re( )nR Qµ=

( )2

2,7 Im( )nR Qµ=

( )2,8 0R =

( )11

13,1sin

nR ne Pµφ= −

2 2 231 1(3,2) Re( ) Im( )

sin sinn nR ne P ne Pµ µφ φ

= − −

2 2 231 1(3,3) Re( ) Im( )

sin sinn nR ne P ne Pµ µφ φ

= −

( )( )2

11 1(3, 4) cot 2 1

2 2n nn nR P n n Pφ −= − + − +

( )11

13,5sin

nR ne Qµφ= −

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23

2 2 231 1(3,6) Re( ) Im( )

sin sinn nR ne Q ne Qµ µφ φ

= − −

2 2 231 1(3,7) Re( ) Im( )

sin sinn nR ne Q ne Qµ µφ φ

= −

( )( )2

11 1(3,8) cot 2 1

2 2n nn nR Q n n Qφ −= − + − +

(1, ) (1, )A j R j= , (2, ) (2, )A j R j= , (3, ) ( )ndwA j jdφ

= , (4, ) (3, )A j R j= with iφ φ= (5, ) (1, )A j R j= ,

(6, ) (2, )A j R j= ; (7, ) ( )ndwA j jdφ

= , (8, ) (3, )A j R j= with jφ φ=

j=1,…,8

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24

( )

( )

( ) ( )

1 1

2

2

2

2 111 1 1 12

22 2 3 3 2 2

23 2 2 3 3 2

12 2 3 3 3 2 2 3

1 1(1,1) ( cot ) 1 cotsin

1 1(1,2) ( cot ) 1 Re( )sin

1 1( cot ) Im( )sin

1 1cot Re( ) cot

n n

n

n

n

eQ e c ne n P c PR r

Q e c e c ne n PR

e c e c ne n PR

e c e c P e c e cR R

µ µ

µ

µ

µ

φ φφ

φφ

φφ

φ φ

= + + + −

= + + + +

+ − + +

− + − −

( )

( )

( ) ( )

2

2

2

2 2

1

23 2 2 3 3 2

22 2 3 3 2 2

1 13 2 2 3 2 2 3 3

2 2

1

Im( )

1 1(1,3) ( cot ) Re( )sin

1 1( cot ) 1 Im( )sin

1 1cot Re( ) cot Im( )

1(1,4) ( 1) cot ( 2)(2 sin 2

n

n

n

n n

n

P

Q e c e c ne n PR

e c e c ne n PR

e c e c P e c e c PR R

n nQ n P nR R

µ

µ

µ

µ µ

φφ

φφ

φ φ

φφ

− −

= − − + +

+ + + + +

+ − − +

= + − −

( )

( )

( )

1 1

2

2

2

11

2 111 1 1 12

22 2 3 3 2 2

23 2 2 3 3 2

12 2 3 3

11)sin

1 1(1,5) ( cot ) 1 cotsin

1 1(1,6) ( cot ) 1 Re( )sin

1 1( cot ) Im( )sin

1 1cot Re( )

n

n n

n

n

n

n P

eQ e c ne n Q c QR r

Q e c e c ne n QR

e c e c ne n QR

e c e c QR

µ µ

µ

µ

µ

φ

φ φφ

φφ

φφ

φ

+

= + + + −

= + + + +

+ − + +

− + − ( )

( )

( )

( ) ( )

2

2

2

2 2

13 2 2 3

23 2 2 3 3 2

22 2 3 3 2 2

1 13 2 2 3 2 2 3 3

2

cot Im( )

1 1(1,7) ( cot ) Re( )sin

1 1( cot ) 1 Im( )sin

1 1cot Re( ) cot Im( )

1(1,8) ( 1) cot2 sin

n

n

n

n n

e c e c QR

Q e c e c ne n QR

e c e c ne n QR

e c e c Q e c e c QR R

nQ nR

µ

µ

µ

µ µ

φ

φφ

φφ

φ φ

φφ

− −

= − − + +

+ + + + +

+ − − +

= +2

11 1

1( 2)( 1)2 sin

n nnQ n n QR φ

−− − +

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25

1 1

2 2

2 2

2 111 12

2 232 2 2

1 12 2 3 3 3 2 2 3

32

1 1(2,1) 1 ( cot ) cotsin

1 1 1(2,2) 1 ( cot ) Re( ) ( cot ) Im( )sin sin

1 1( )cot Re( ) ( ) cot Im( )

1(2,3) (sin

n n

n n

n n

eQ ne n P c PR r

neQ ne n P n PR R

e c e c P e c e c PR R

neQ nR

µ µ

µ µ

µ µ

φ φφ

φ φφ φ

φ φ

− −

= − + +

= − + − +

+ + + −

=

( ) ( )

2 2

2 2

1

2 22 2

1 13 2 2 3 2 2 3 3

2 21

1 1

21 2

1 1cot ) Re( ) 1 ( cot ) Im( )sin

1 1( )cot Re( ) ( ) cot Im( )

1 1(2,4) 1 cot ( 2) 12 sin 2 sin

1 1(2,5) 1 ( cot )sin

n n

n n

n n

n

P ne n PR

e c e c P e c e c PR R

n nQ n P n n PR R

eQ ne n QR

µ µ

µ µ

µ

φ φφ φ

φ φ

φφ φ

φφ

− −

+ + − +

− − + +

= − + + − +

= − + +

1

2 2

2 2

2

111

2 232 2 2

1 12 2 3 3 3 2 2 3

2 2322 2

cot

1 1 1(2,6) 1 ( cot ) Re( ) ( cot ) Im( )sin sin

1 1( )cot Re( ) ( ) cot Im( )

1 1 1(2,7) ( cot ) Re( ) 1 ( cot )sin sin

n

n n

n n

n

c Qr

neQ ne n Q n QR R

e c e c Q e c e c QR R

neQ n Q ne nR R

µ

µ µ

µ µ

µ

φ

φ φφ φ

φ φ

φ φφ φ

− −

= − + − +

+ + + −

= + + − +

( ) ( )

2

2 2

1 13 2 2 3 2 2 3 3

2 21

1 1

Im( )

1 1( )cot Re( ) ( ) cot Im( )

1 1(2,8) 1 cot ( 2) 12 sin 2 sin

n

n n

n n

Q

e c e c Q e c e c QR R

n nQ n Q n n QR R

µ

µ µφ φ

φφ φ

− −

− − + +

= − + + − +

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26

1 1

2 2

2 2

11 1 1

2 3

1 12 2 3 3 3 2 2 3

3

2 1 2 1(3,1) ( 1) cotsin sin

2 1 2 1(3,2) ( 1) cot Re( ) ( 1) cot Im( )sin sin

2 1 2 1( ) Re( ) ( ) Im( )sin sin

2 1(3,3) ( 1) cot Resin

n n

n n

n n

n nQ e n P e c PR Rn nQ e n P e n P

R Rn ne c e c P e c e c P

R RnQ e n

R

µ µ

µ µ

µ µ

φφ φ

φ φφ φ

φ φ

φφ

− −

= + −

= + + +

− + − −

= − +

( ) ( ) ( )( )

2 2

2 2

1

2

1 13 2 2 3 2 2 3 3

2 11 12

1 1

2 1( ) ( 1) cot Im( )sin

2 1 2 1( ) Re( ) ( ) Im( )sin sin

1(3,4) 1 2 cot 2 1 cot2 sin2 1 2 1(3,5) ( 1) cot

sin sin

n n

n n

n n

n

nP e n PR

n ne c e c P e c e c PR R

n nQ n n n P n n PR Rn nQ e n Q c Q

R R

µ µ

µ µ

µ µ

φφ

φ φ

φ φφ

φφ φ

− −

+ +

+ − − +

= + − + + − − +

= + −1

2 2

2 2

2 2

1

2 3

1 12 2 3 3 3 2 2 3

3 2

3 2

2 1 2 1(3,6) ( 1) cot Re( ) ( 1) cot Im( )sin sin

2 1 2 1( ) Re( ) ( ) Im( )sin sin

2 1 2 1(3,7) ( 1) cot Re( ) ( 1) cot Im( )sin sin

2 (

n

n n

n n

n n

n nQ e n Q e n QR R

n ne c e c Q e c e c QR Rn nQ e n Q e n Q

R Rn e c e

R

µ µ

µ µ

µ µ

φ φφ φ

φ φ

φ φφ φ

− −

= + + +

− + − −

= − + + +

+ −

( ) ( ) ( )( )

2 2

1 12 3 2 2 3 3

2 11 12

1 2 1) Re( ) ( ) Im( )sin sin

1(3,8) 1 2 cot 2 1 cot2 sin

n n

n n

nc Q e c e c QR

n nQ n n n Q n n QR R

µ µφ φ

φ φφ

− −

− +

= + − + + − − +

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27

( ) ( )

( ) ( ) ( )

( ) ( )

( )

1 1

2

2

2 111 1 1 12 2 2

22 2 3 3 22 2

23 2 2 3 32 2

2 2 3 32

11 1(4,1) 1 1 cot cotsin

1 1(4,2) 1 1 cot Resin

1 11 cot Imsin

1 1 cot

n n

n

n

eQ e c n e n P c PR R

Q e c e c n e n PR

e c e c ne n PR

e c e cR

µ µ

µ

µ

φ φφ

φφ

φφ

− −= − + − + −

= − + + − +

+ − − + +

− − + ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2

2

2

2

1 13 2 2 32

23 2 2 3 32 2

22 2 3 3 22 2

13 2 2 3 2 2 3 32 2

1Re 1 cot Im

1 1(4,3) 1 cot Resin

1 11 1 cot Imsin

1 11 cot Re 1 c

n n

n

n

n

P e c e c PR

Q e c e c ne n PR

e c e c n e n PR

e c e c P e c e cR R

µ µ

µ

µ

µ

φ φ

φφ

φφ

φ

− −

− − −

= − − − + +

+ − + + − +

+ − − − − + ( )

( )

( ) ( )

( ) ( ) ( )

2

1 1

2

1

2 21

1 12 2

2 111 1 1 12 2 2

22 2 3 3 22 2

3 22

ot Im

1 1(4,4) 1 cot ( 2)( 1)2 sin 2 sin

11 1(4,5) 1 1 cot cotsin

1 1(4,6) 1 1 cot Resin

1

n

n n

n n

n

P

n nQ n P n n PR R

eQ e c n e n Q c QR R

Q e c e c n e n QR

e cR

µ

µ µ

µ

φ

φφ φ

φ φφ

φφ

= + − − +

−= − + − + −

= − + + − +

+ ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

2

2 2

2

22 3 3 2

1 12 2 3 3 3 2 2 32 2

23 2 2 3 32 2

22 2 3 3 22 2

11 cot Imsin

1 11 cot Re 1 cot Im

1 1(4,7) 1 cot Resin

1 11 1 cotsin

n

n n

n

e c ne n Q

e c e c Q e c e c QR R

Q e c e c ne n QR

e c e c n e nR

µ

µ µ

µ

φφ

φ φ

φφ

φφ

− −

− − + +

− − + − − −

= − − − + +

+ − + + − +

( )

( ) ( ) ( ) ( )

( )

2

2 2

1 13 2 2 3 2 2 3 32 2

2 21

1 12 2

Im

1 11 cot Re 1 cot Im

1 1(4,8) 1 cot ( 2)( 1)2 sin 2 sin

n

n n

n n

Q

e c e c Q e c e c QR R

n nQ n Q n n QR R

µ

µ µφ φ

φφ φ

− −

+ − − − − +

= + − − +

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28

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

1 1

2 2

2 2

12 11 12 2 2

2 2322 2 2 2

1 12 2 3 3 3 2 2 32 2

32

11(5,1) 1 cot cotsin

1 1(5,2) 1 cot Re cot Imsin sin

1 11 cot Re 1 cot Im

(5,3)

n n

n n

n n

enQ e n P c PR R

nenQ e n P n PR R

e c e c P e c e c PR RneQR

µ µ

µ µ

µ µ

φ φφ

φ φφ φ

φ φ

− −

− = − + +

= − + − +

+ − + + − −

= ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )( )

( )

2 2

2 2

2 222 2 2

1 13 2 2 3 2 2 3 32 2

11 12 2

212 2

1 1cot Re 1 cot Imsin sin

1 11 cot Re 1 cot Im

1 1(5,4) 1 cot 2 12 sin 2 sin

1(5,5) 1 cotsin

n n

n n

n n

nn P e n PR

e c e c P e c e c PR R

n nQ n P n n PR R

nQ e nR

µ µ

µ µ

φ φφ φ

φ φ

φφ φ

φφ

− −

+ + − +

− − − + − +

= − + + − +

= − +

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

1 1

2 2

2 2

2

1 112

2 2322 2 2 2

1 12 2 3 3 3 2 2 32 2

232 2 2

1cot

1 1(5,6) 1 cot Re cot Imsin sin

1 11 cot Re 1 cot Im

1(5,7) cot Re 1sin

n n

n n

n n

n

eQ c Q

R

nenQ e n Q n QR R

e c e c Q e c e c QR Rne nQ n Q eR R

µ µ

µ µ

µ µ

µ

φ

φ φφ φ

φ φ

φφ

− −

−+

= − + − +

+ − + + − −

= + + −

( ) ( )

( ) ( ) ( ) ( )

( ) ( )( )

2

2 2

22 2

1 13 2 2 3 2 2 3 32 2

11 12 2

1cot Imsin

1 11 cot Re 1 cot Im

1 1(5,8) 1 cot 2 12 sin 2 sin

n

n n

n n

n Q

e c e c Q e c e c QR R

n nQ n Q n n QR R

µ

µ µ

φφ

φ φ

φφ φ

− −

+

− − − + − +

= − + + − +

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29

( )( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

1 1

2 2

2 2

11 1 12 2

2 32 2

1 12 2 3 3 3 2 2 32 2

32

2 1 2 1(6,1) 1 1 cot 1sin sin

2 ( 1) 1 2 ( 1) 1(6,2) 1 cot Re cot Imsin sin

2 1 2 11 Re 1 Imsin sin

2 ( 1)(6,3)

n n

n n

n n

n nQ n e P e c PR R

n n n nQ e P e PR R

n ne c e c P e c e c PR R

n nQ eR

µ µ

µ µ

µ µ

φφ φ

φ φφ φ

φ φ

− −

= + − + −

+ += − +

− − + − − −

+= − ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

( )( )

2 2

2 2

22

1 13 2 2 3 2 2 3 32 2

2 11 12 2 2

12

1 2 ( 1) 1cot Re 1 cot Imsin sin

2 1 2 11 Re 1 Imsin sin

1(6,4) 1 2 cot 2 1 cot2 sin2 1(6,5) 1 1 co

sin

n n

n n

n n

n nP e PR

n ne c e c P e c e c PR R

n nQ n n n P n n PR RnQ n e

R

µ µ

µ µ

φ φφ φ

φ φ

φ φφ

φ

− −

++ −

+ − − − − +

= + − + + − − +

= + − ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

1 1

2 2

2 2

2

11 12

2 32 2

1 12 2 3 3 3 2 2 32 2

32 2

2 1t 1sin

2 ( 1) 1 2 ( 1) 1(6,6) 1 cot Re cot Imsin sin

2 1 2 11 Re 1 Imsin sin

2 ( 1) 1 2 ( 1)(6,7) cot Resin

n n

n n

n n

n

nQ e c QR

n n n nQ e Q e QR R

n ne c e c Q e c e c QR R

n n n nQ e QR R

µ µ

µ µ

µ µ

µ

φφ

φ φφ φ

φ φ

φφ

− −

+ −

+ += − +

− − + − − −

+ += − + ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

2

2 2

2

1 13 2 2 3 2 2 3 32 2

2 11 12 2 2

11 cot Imsin

2 1 2 11 Re 1 Imsin sin

1(6,8) 1 2 cot 2 1 cot2 sin

n

n n

n n

e Q

n ne c e c Q e c e c QR R

n nQ n n n Q n n QR R

µ

µ µ

φφ

φ φ

φ φφ

− −

+ − − − − +

= + − + + − − +

In deriving the above relation we used the recursive relations:

( )( ) ( )( )

( )( ) ( )( )

22 1

2 2

22 1

2 2

11 cot cot 1sin

11 cot cot 1sin

nn n

nn n

d Pn n n n P n n P

d

d Qn n n n Q n n Q

d

µµ µ

µµ µ

µ µ φ φ µ µφ φ

µ µ φ φ µ µφ φ

= − − + + + − − − +

= − − + + + − − − +

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30

References

[1] Kalnins A., Free non-symmetric vibrations of shallow spherical shells, Journal of the Acoustical Society of America 33 (1961) 1102-1107.

[2] Kalnins A., Effect of bending on vibration of spherical shell, Journal of the Acoustical Society of America 36 (1964) 74-81.

[3] Cohen G.A., Computer analysis of asymmetric free vibrations of ring stiffened orthotropic shells of revolution, AIAA Journal 3 (1965) 2305-2312.

[4] Navaratna D. R., Natural Vibration of Deep Spherical Shells, AIAA Journal 4 (1966) 2056-2058.

[5] Webster J.J., Free vibrations of shells of revolution using ring finite elements, International Journal of Mechanical Sciences 9(1967) 559-570.

[6] Kraus H., Thin elastic shells, John Wiley and Sons, New York, 1967.

[7] Greene B.E., Jones R.E., Mc Lay R.W. and Strome D.R., Dynamic analysis of shells using doubly curved finite elements, Proceedings of The Second Conference on Matrix methods in Structural Mechanics, TR-68-150(1968) 185-212

[8] Tessler A., Spirichigliozzi L., Resolving membrane and shear locking phenomena in curved deformable axisymmetric shell element, International Journal for Numerical Methods in Engineering, 26(1988) 1071-1080.

[9] Narasimhan M.C., Alwar R.S., Free vibration analysis of laminated orthotropic spherical shell, Journal of Sound and Vibration, 54(1992) 515-529.

[10] Gautham B. P., Ganesan N., Free vibration analysis of thick spherical shells, Computers and Structures Journal, 45(1992) 307-313.

[11] Gautham B. P., Ganesan N., Free vibration characteristics of isotropic and laminated orthotropic shell caps, Journal of Sound and Vibration, 204(1997) 17-40.

[12] Sai Ram K.S., Sreedhar Babu T., Free vibration of composite spherical shell cap with and without a cutout, Computers and Structures Journal, 80(2002) 1749-1756.

[13] Buchanan G.R., Rich B.S., Effect of boundary conditions on free vibrations of thick isotropic spherical shells, Journal of Vibration and Control 8(2002) 389-403.

[14] Ventsel E.S, Naumenko V., Strelnikova E., Yeseleva E., Free vibrations of shells of revolution filled

with fluid, Journal of Engineering Analysis with Boundary Elements 34(2010) 856-862

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