Abstract
This paper focuses on free vibration of hemi-ellipsoidal shells with the consideration of the bending rigidity and nonlinear terms in strain energy. The appropriate form of the energy functional is formulated based on the principle of virtual work and the fundamental form of surfaces. Natural frequencies and their corresponding mode shapes are determined using the modified direct iteration method. The obtained results, which show a close agreement with previous research, are compared with those obtained based on the membrane theory. The effect of the support condition, thickness, size ratio, and volume constraint condition on frequency parameters and mode shapes is demonstrated. With the bending rigidity, shell thickness has a significant impact on the frequency, especially in higher vibration modes and in shells with a considerable thickness but the frequency parameter converges to that determined by using the membrane theory while the reference radius-to-thickness ratio is increasing. In addition, accounting for the bending rigidity solves the issue of determining natural frequencies and mode shapes of the shells using the membrane theory without the volume constraint condition. The obtained results also indicate that the free vibration analysis with bending is essential for the hemi-ellipsoidal shell with a base radius-to-thickness ratio of less than 100, which gives over 2.84% difference compared with that of the shell derived by membrane theory, and this allows engineers to perform the analysis in more applications.
References
1.
Krivoshapko
, S. N.
, 2007
, “Research on General and Axisymmetric Ellipsoidal Shells Used as Domes, Pressure Vessels, and Tanks
,” ASME Appl. Mech. Rev.
, 60
(6
), pp. 336
–355
. 2.
Jha
, A. K.
, Inman
, D. J.
, and Plaut
, R. H.
, 2002
, “Free Vibration Analysis of an Inflated Toroidal Shell
,” ASME J. Vib. Acoust.
, 124
(3
), pp. 387
–396
. 3.
Li
, K.
, Zheng
, J.
, Liu
, S.
, Ge
, H.
, Sun
, G.
, Zhang
, Z.
, Gu
, C.
, and Xu
, P.
, 2019
, “Buckling Behavior of Large-Scale Thin-Walled Ellipsoidal Head Under Internal Pressure
,” Thin Walled Struct.
, 141
, pp. 260
–274
. 4.
Karamanos
, S. A.
, Patkas
, L. A.
, and Platyrrachos
, M. A.
, 2006
, “Sloshing Effects on the Seismic Design of Horizontal-Cylindrical and Spherical Industrial Vessels
,” ASME J. Pressure Vessel Technol.
, 128
(3
), pp. 328
–340
. 5.
Shariati
, S. K.
, and Mogadas
, S. M.
, 2011
, “Vibration Analysis of Submerged Submarine Pressure Hull
,” ASME J. Vib. Acoust.
, 133
(1
), p. 011013
. 6.
Yazdi
, M. S.
, Rostami
, S. A. L.
, and Kolahdooz
, A.
, 2016
, “Optimization of Geometrical Parameters in a Specific Composite Lattice Structure Using Neural Networks and ABC Algorithm
,” J. Mech. Sci. Technol.
, 30
(4
), pp. 1763
–1771
. 7.
Altenbach
, H.
, and Eremeyev
, V.
, 2016
, “Thin-Walled Structural Elements: Classification, Classical and Advanced Theories, New Applications,” Shell-Like Structures
, H.
Altenbach
, and V.
Eremeyev
, eds., Springer
, Cham
, pp. 1
–62
.8.
Tangbanjongkij
, C.
, Chucheepsakul
, S.
, and Jiammeepreecha
, W.
, 2019
, “Large Displacement Analysis of Ellipsoidal Pressure Vessel Heads Using the Fundamental of Differential Geometry
,” Int. J. Press. Vessels Pip.
, 172
, pp. 337
–347
. 9.
Tangbanjongkij
, C.
, Chucheepsakul
, S.
, and Jiammeepreecha
, W.
, 2020
, “Analytical and Numerical Analyses for a Variety of Submerged Hemi-Ellipsoidal Shells
,” ASCE J. Eng. Mech.
, 146
(7
), p. 04020066
. 10.
Jiammeepreecha
, W.
, and Chucheepsakul
, S.
, 2017
, “Nonlinear Free Vibration Analysis of a Toroidal Pressure Vessel Under Constrained Volume Condition
,” Int. J. Struct. Stab. Dyn.
, 19
(10
), p. 19501189
. 11.
Lamb
, H. M. A.
, 1882
, “On the Vibrations of a Spherical Shell
,” Proc. London Math. Soc.
, S1-14
(1
), pp. 50
–56
. 12.
Love
, A. E. H.
, 1888
, “The Small Free Vibrations and Deformations of a Thin Elastic Shell
,” Philos. Trans. R. Soc., A
, 179
, pp. 491
–546
. 13.
Kalnins
, A.
, 1964
, “Effect of Bending on Vibrations of Spherical Shells
,” J. Acoust. Soc. Am.
, 36
(1
), pp. 74
–81
. 14.
Kalnins
, A.
, and Biricikoglu
, V.
, 1972
, “Theory of Vibration of Initially Stressed Shells
,” J. Acoust. Soc. Am.
, 51
(5B
), pp. 1697
–1704
. 15.
Jiammeepreecha
, W.
, and Chucheepsakul
, S.
, 2017
, “Nonlinear Axisymmetric Free Vibration Analysis of Liquid-Filled Spherical Shell With Volume Constraint
,” ASME J. Vib. Acoust.
, 139
(5
), p. 051016
. 16.
Ross
, E. W.
, Jr., 1965
, “Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells
,” ASME J. Appl. Mech.
, 32
(3
), pp. 553
–561
. 17.
Ross
, Jr. E. W.
, and Matthews
, W. T.
, 1967
, “Frequencies and Mode Shapes for Axisymmetric Vibration of Shells
,” ASME J. Appl. Mech.
, 34
(1
), pp. 73
–80
. 18.
Kunieda
, H.
, 1984
, “Flexural Axisymmetric Free Vibrations of a Spherical Dome: Exact Results and Approximate Solutions
,” J. Sound Vib.
, 92
(1
), pp. 1
–10
. 19.
Singh
, A. V.
, and Mirza
, S.
, 1985
, “Asymmetric Modes and Associated Eigenvalues for Spherical Shells
,” ASME J. Pressure Vessel Technol.
, 107
(1
), pp. 77
–82
. 20.
Alijani
, F.
, and Amabili
, M.
, 2014
, “Non-linear Vibrations of Shells: A Literature Review From 2003 to 2013
,” Int. J. Non Linear Mech.
, 58
, pp. 233
–257
. 21.
Meish
, V. F.
, 2005
, “Numerical Solution of Dynamic Problems for Reinforced Ellipsoidal Shells Under Nonstationary Loads
,” Int. Appl. Mech.
, 41
(4
), pp. 386
–391
. 22.
Meish
, V. F.
, and Maiborodina
, N. V.
, 2008
, “Nonaxisymmetric Vibrations of Ellipsoidal Shells Under Nonstationary Distributed Loads
,” Int. Appl. Mech.
, 44
(9
), pp. 1015
–1024
. 23.
Ventsel
, E. S.
, Naumenko
, V.
, Strelnikova
, E.
, and Yeseleva
, E.
, 2010
, “Free Vibrations of Shells of Revolution Filled With a Fluid
,” Eng. Anal. Boundary Elem.
, 34
(10
), pp. 856
–862
. 24.
Chernobryvko
, M. V.
, and Avramov
, K. V.
, 2016
, “Natural Vibrations of Parabolic Shells
,” J. Math. Sci.
, 217
(2
), pp. 229
–238
. 25.
Eslaminejad
, A.
, Ziejewski
, M.
, and Karami
, G.
, 2019
, “Vibrational Properties of a Hemispherical Shell With Its Inner Fluid Pressure: An Inverse Method for Noninvasive Intracranial Pressure Monitoring
,” ASME J. Vib. Acoust.
, 141
(4
), p. 041002
. 26.
Eslaminejad
, A.
, Ziejewski
, M.
, and Karami
, G.
, 2019
, “An Experimental–Numerical Modal Analysis for the Study of Shell-Fluid Interactions in a Clamped Hemispherical Shell
,” Appl. Acoust.
, 152
(5
), pp. 110
–117
. 27.
Breslavsky
, I. D.
, and Amabili
, M.
, 2018
, “Nonlinear Vibrations of a Circular Cylindrical Shell With Multiple Internal Resonances Under Multi-Harmonic Excitation
,” Nonlinear Dyn.
, 93
(1
), pp. 53
–62
. 28.
Amabili
, M.
, Balasubramanian
, P.
, and Ferrari
, G.
, 2016
, “Travelling Wave and Non-Stationary Response in Nonlinear Vibrations of Water-Filled Circular Cylindrical Shells: Experiments and Simulations
,” J. Sound Vib.
, 381
, pp. 220
–245
. 29.
Zippo
, A.
, Barbieri
, M.
, Iarriccio
, G.
, and Pellicano
, F.
, 2020
, “Nonlinear Vibrations of Circular Cylindrical Shells With Thermal Effects: an Experimental Study
,” Nonlinear Dyn.
, 99
(1
), pp. 373
–391
. 30.
Amabili
, M.
, and Balasubramanian
, P.
, 2020
, “Nonlinear Vibrations of Truncated Conical Shells Considering Multiple Internal Resonances
,” Nonlinear Dyn.
, 100
(1
), pp. 77
–93
. 31.
Chorfi
, S. M.
, and Houmat
, A.
, 2010
, “Non-Linear Free Vibration of a Functionally Graded Doubly-Curved Shallow Shell of Elliptical Plan-Form
,” Compos. Struct.
, 92
(10
), pp. 2573
–2581
. 32.
Bryan
, A.
, 2018
, “Free Vibration of Thin Shallow Elliptical Shells
,” ASME J. Vib. Acoust.
, 140
(1
), p. 011004
. 33.
Bryan
, A.
, 2018
, “Free Vibration of Doubly Curved Thin Shells
,” ASME J. Vib. Acoust.
, 140
(3
), p. 031003
. 34.
Shim
, H. J.
, and Kang
, J.-H.
, 2004
, “Free Vibrations of Solid and Hollow Hemi-Ellipsoids of Revolution From a Three-Dimensional Theory
,” Int. J. Eng. Sci.
, 42
(17–18
), pp. 1793
–1815
. 35.
Kang
, J.-H.
, 2007
, “Field Equations, Equations of Motion, and Energy Functionals for Thick Shells of Revolution With Arbitrary Curvature and Variable Thickness From a Three-Dimensional Theory
,” Acta Mech.
, 188
(1
), pp. 21
–37
. 36.
Ko
, S. M.
, and Kang
, J.-H.
, 2017
, “Free Vibration Analysis of Shallow and Deep Ellipsoidal Shells Having Variable Thickness With and Without a Top Opening
,” Acta Mech.
, 288
(12
), pp. 4391
–4409
. 37.
Chen
, J.-H.
, and Huang
, T.
, 1985
, “Appropriate Forms in Nonlinear Analysis
,” ASCE J. Eng. Mech.
, 110
(10
), pp. 1215
–1226
. 38.
Cook
, R. D.
, Malkus
, D. S.
, Plesha
, M. E.
, and Witt
, R. J.
, 2002
, Concepts and Applications of Finite Element Analysis
, Wiley
, New York
.39.
Belytschko
, T.
, Liu
, W. K.
, Moran
, B.
, and Elkhodary
, K. I.
, 2014
, Nonlinear Finite Elements for Continua and Structures
, Wiley
, WS
.40.
Langhaar
, H. L.
, 1962
, Energy Methods in Applied Mechanics
, Wiley
, New York
.41.
Sohn
, K.-A.
, Juttler
, B.
, Kim
, M.-S.
, and Wang
, W.
, 2002
, “Computing Distance Between Surfaces Using Line Geometry
,” Proceedings of 10th Pacific Conference on Computer Graphics and Applications
, Beijing, China
, Oct. 9–11
, IEEE
, pp. 236
–245
.42.
Flugge
, W.
, 1960
, Stresses in Shells
, Springer-Verlag
, Berlin
.43.
Langhaar
, H. L.
, 1964
, “Foundations of Practical Shell Analysis,” Lecture Note
, University of Illinois at Urbana-Champaign
, Champaign, IL
.44.
Dym
, C. L.
, 1990
, Introduction to the Theory of Shells
, Taylor and Francis
, New York
.45.
Radenković
, G.
, Borković
, A.
, and Marussig
, B.
, 2021
, “Nonlinear Static Isogeometric Analysis of Arbitrarily Curved Kirchhoff-Love Shells
,” Int. J. Mech. Sci.
, 192
, p. 106143
. 46.
Rajasekaran
, S.
, and Murray
, D. W.
, 1973
, “Incremental Finite Element Matrices
,” J. Struct. Div.
, 99
(12
), pp. 2423
–2438
. 47.
Prathap
, G.
, and Varadan
, T. K.
, 1978
, “The Large Amplitude Vibration of Hinged Beams
,” Comput. Struct.
, 9
(2
), pp. 219
–222
. 48.
Jiang
, S.
, Yang
, T.
, Li
, W. L.
, and Du
, J.
, 2013
, “Vibration Analysis of Doubly Curved Shallows With Elastic Edge Restraints
,” ASME J. Vib. Acoust.
, 139
(5
), p. 051016
. 49.
Kim
, J. G.
, 1998
, “A Higher-Order Harmonic Element for Shells or Revolution Based on the Modified Mixed Formulation
,” Ph.D. dissertation
, Seoul National University
, Seoul
.50.
Artioli
, E.
, and Viola
, E.
, 2006
, “Free Vibration Analysis of Spherical Shells Using G.D.Q. Numerical Solution
,” ASME J. Pressure Vessel Technol.
, 128
(3
), pp. 370
–378
. 51.
Lee
, J.
, 2009
, “Free Vibration Analysis of Spherical Caps by the Pseudospectral Method
,” J. Mech. Sci. Technol.
, 23
(1
), pp. 221
–228
. 52.
Pulngern
, T.
, Halling
, M. W.
, Chucheepsakul
, S.
, and Poovarodom
, N.
, 2006
, “On the Free Vibrations of Variable-Arc-Length Beams: Analytical and Experimental
,” ASCE J. Struct. Eng.
, 132
(5
), pp. 772
–780
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