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
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
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/
2
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
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)
4
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.
Uθ
W
Uφ
2X
θ
1X
X3
Fig.1. Geometry of spherical shell
dθ
θ
Mθ Mθφ
Mφ
Mφθ
Qθ
Nφ
Nθφ Nθ
Qφ
Nφθ
φ dφ
Fig.2. Stress resultants and stress couples
5
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:
6
( )( )( )
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)
7
iUφ
i
dWdφ
iW jφ
iφ
iUθ θ
Fig. 3. Spherical frustum element
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
4
4 (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:
9
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
− + −
+ −
10
( )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)
11
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:
12
[ ][ ] 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.
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
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
=
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
=
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
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
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=
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
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
∂ ∂− ∂ ∂ ∂
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
∂ ∂+ − ∂ ∂
∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂+ + − + − + + − + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
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= −
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
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
−− − +
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
− −
−
− − + +
= − + + − +
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
− −
−
− +
= + − + + − − +
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
− −
−
+ − − − − +
= + − − +
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
− −
−
+
− − − + − +
= − + + − +
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
−
−
= − − + + + − − − +
= − − + + + − − − +
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
Top Related