While several numerical approaches exist for the vibration analysis of thin shells, there is a lack of analytical approaches to address this problem. This is due to complications that arise from coupling between the midsurface and normal coordinates in the transverse differential equation of motion (TDEM) of the shell. In this research, an Uncoupling Theorem for solving the TDEM of doubly curved, thin shells with equivalent radii is introduced. The use of the uncoupling theorem leads to the development of an uncoupled transverse differential of motion for the shells under consideration. Solution of the uncoupled spatial equation results in a general expression for the eigenfrequencies of these shells. The theorem is applied to four shell geometries, and numerical examples are used to demonstrate the influence of material and geometric parameters on the eigenfrequencies of these shells.

References

References
1.
Liew
,
K. M.
,
Lim
,
C. W.
, and
Kitipornchai
,
S.
,
1997
, “
Vibration of Shallow Shells: A Review With Bibliography
,”
ASME Appl. Mech. Rev.
,
50
(
8
), pp.
431
444
.
2.
Qatu
,
M. S.
,
2002
, “
Recent Research Advances in the Dynamic Behavior of Shells: 1989–2000—Part 2: Homogeneous Shells
,”
ASME Appl. Mech. Rev.
,
55
(
5
), pp.
415
434
.
3.
Narita
,
Y.
, and
Leissa
,
A.
,
1986
, “
Vibrations of Completely Free Shallow Shells of Curvilinear Planform
,”
ASME J. Appl. Mech.
,
53
(
3
), pp.
647
651
.
4.
Tornabene
,
F.
, and
Erasmo
,
V.
,
2008
, “
2-D Solution for Free Vibrations of Parabolic Shells Using Generalized Differential Quadrature Method
,”
Eur. J. Mech. A: Solids
,
27
(
6
), pp.
1001
1025
.
5.
Leissa
,
A. W.
,
1973
,
Vibration of Shells
, Ohio State University, Columbus, OH.
6.
Leissa
,
A. W.
,
1969
,
Vibration of Plates
,
Ohio State University
,
Columbus, OH
.
7.
Ventsel
,
E.
, and
Krauthammer
,
T.
,
2001
,
Thin Plates and Shells: Theory, Analysis, and Applications
,
Marcel Dekker
,
New York
.
8.
Love
,
A. E. H.
,
1888
, “
The Small Free Vibrations and Deformation of a Thin Elastic Shell
,”
Philos. Trans. R. Soc. London. A
,
179
, pp.
491
546
.
9.
Soedel
,
W.
,
2005
,
Vibration of Shells and Plates
,
Taylor and Francis
,
Oxfordshire, UK
.
10.
Wilkinson
,
J. P.
, and
Kalnins
,
A.
,
1965
, “
On Nonsymmetric Dynamic Problems of Elastic Spherical Shells
,”
ASME J. Appl. Mech.
,
32
(
3
), pp.
525
532
.
11.
Wilkinson
,
J. P.
,
1965
, “
Natural Frequencies of Closed Spherical Shells
,”
J. Acoust. Soc. Am.
,
38
(
2
), pp.
367
368
.
12.
Naghdi
,
P. M.
, and
Kalnins
,
A.
,
1962
, “
On Vibrations of Elastic Spherical Shells
,”
ASME J. Appl. Mech.
,
29
(
1
), pp.
65
72
.
13.
Kalnins
,
A.
,
1964
, “
Effect of Bending on Vibrations of Spherical Shells
,”
J. Acoust. Soc. Am.
,
36
(
1
), pp.
74
81
.
14.
Ross
,
E. W.
,
1965
, “
Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells
,”
ASME J. Appl. Mech.
,
32
(
3
), pp.
553
561
.
15.
Niordson
,
F. I.
,
1984
, “
Free Vibrations of Thin Elastic Spherical Shells
,”
Int. J. Solids Struct.
,
20
(
7
), pp.
667
687
.
16.
Plaut
,
R. H.
, and
Johnson
,
L. W.
,
1984
, “
Optimal Forms of Shallow Shells With Circular Boundary
,”
ASME J. Appl. Mech.
,
51
(
3
), pp.
526
530
.
17.
Liew
,
K. M.
,
Peng
,
L. X.
, and
Ng
,
T. Y.
,
2002
, “
Three-Dimensional Vibration Analysis of Spherical Shell Panels Subjected to Different Boundary Conditions
,”
Int. J. Mech. Sci.
,
44
(
10
), pp.
2103
2117
.
18.
Irie
,
T.
,
Yamrada
,
G.
, and
Muramoto
,
Y.
,
1985
, “
Free Vibration of a Point-Supported Spherical Shell
,”
ASME J. Appl. Mech.
,
52
(
4
), pp.
890
896
.
19.
Jiang
,
S.
,
Yang
,
T.
,
Li
,
W. L.
, and
Du
,
J.
,
2013
, “
Vibration Analysis of Doubly Curved Shallow Shells With Elastic Edge Restraints
,”
ASME J. Vib. Acoust.
,
135
(
3
), p.
034502
.
20.
Shi
,
P.
,
Kapania
,
R. K.
, and
Dong
,
C. Y.
,
2015
, “
Free Vibration of Curvilinearly Stiffened Shallow Shells
,”
ASME J. Vib. Acoust.
,
137
(
3
), p.
031006
.
21.
Chakravorty
,
D.
, and
Bandyopadhyay
,
J. N.
,
1995
, “
On the Free Vibration of Shallow Shells
,”
J. Sound Vib.
,
185
(
4
), pp.
673
684
.
22.
Singh
,
A. V.
,
1985
, “
Asymmetric Modes and Associated Eigenvalues for Spherical Shells
,”
ASME J. Pressure Vessel Technol.
,
107
(
1
), pp.
77
82
.
23.
Singh
,
A. V.
,
1991
, “
On Vibrations of Shells of Revolution Using Bezier Polynomials
,”
ASME J. Pressure Vessel Technol.
,
113
(
4
), pp.
579
584
.
24.
Petyt
,
M.
,
2010
,
Introduction to Finite Element Vibration Analysis
,
Cambridge University Press
,
New York
.
25.
Choi
,
S. T.
, and
Chou
,
Y. T.
,
2003
, “
Vibration Analysis of Non-Circular of Curved Panels by the Differential Quadrature Method
,”
J. Sound Vib.
,
259
(
3
), pp.
525
539
.
26.
Tornabene
,
F.
,
Brischetto
,
S.
,
Fantuzzi
,
N.
, and
Viola
,
E.
,
2015
, “
Numerical and Exact Models for Free Vibration Analysis of Cylindrical and Spherical Shell Panels
,”
Composites, Part B
,
81
, pp.
231
250
.
27.
Artioli
,
E.
, and
Viola
,
E.
,
2006
, “
Free Vibration Analysis of Spherical Caps Using a GDQ Numerical Solution
,”
ASME J. Pressure Vessel Technol.
,
128
(
3
), pp.
370
378
.
28.
Bryan
,
A.
,
2017
, “
Free Vibration of Thin Shallow Elliptical Shells
,”
ASME J. Vib. Acoust.
,
140
(
1
), p.
011004
.
29.
Bryan
,
A.
,
2017
, “
Free Vibration of Thin Spherical Shells
,”
ASME J. Vib. Acoust.
,
139
(
6
), p.
061020
.
30.
Potter
,
M. C.
, and
Goldberg
,
J.
,
1995
,
Mathematical Methods
,
Great Lakes Press
,
Okemos, MI
.
31.
Kreyszig
,
E.
,
1995
,
Advanced Engineering Mathematics
,
Wiley
,
New York
.
32.
Gutiérrez-Vega
,
J. C.
,
2000
, “Formal Analysis of the Propagation of Invariant Optical Fields in Elliptic Coordinates,” Ph.D. thesis, INAOE, Monterrey, Mexico.
33.
United States, National Bureau of Standards
,
1951
,
Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors
,
Columbia University Press
,
New York
.
34.
Young
,
P.
,
2009
, “Helmholtz's and Laplace's Equations in Spherical Polar Coordinates: Spherical Harmonics and Spherical Bessel Functions,”
Physics 116C Lecture Notes
, University of California, Santa Cruz, CA.http://physics.ucsc.edu/~peter/116C/helm_sp.pdf
35.
Anon
,
N. D.
, 2006, “Spherical Harmonics,”
Physics 221A Lecture Notes
, University of California Berkeley, Berkeley, CA.http://sandman.berkeley.edu/221A/sphericalharmonics.pdf
You do not currently have access to this content.