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Journal of Engineering Mathematics

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01-04-2014

Eigenoscillations of a thin-walled azimuthally closed, axially open shell of revolution

Authors: I. Gavrilyuk, M. Hermann, V. Trotsenko, Yu. Trotsenko, A. Timokha

Published in: Journal of Engineering Mathematics | Issue 1/2014

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Abstract

This paper generalizes earlier authors’ results on the analytical approximation of the singularly perturbed boundary problem describing the eigenoscillations of a thin-walled axisymmetric shell. The asymptotic behavior of the eigenmodes at the clamped ends is studied, and a set of trial functions capturing this behavior is constructed to be used in the Ritz method. Illustrative numerical examples demonstrate a fast convergence so that the eigenmodes are accurately approximated in a uniform metric together with their second-, third-, and fourth-order derivatives. The numerical results are validated by comparing them with an asymptotic eigensolution and computations done by the ANSYS codes based on the finite-element method.

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Gavrilyuk I, Hermann M, Trotsenko V, Trotsenko Yu, Timokha A (2010) Axisymmetric oscillations of a cupola-shaped shell. J Eng Math 65:165–178CrossRefMathSciNet

Leissa AW (1993) Vibration of shells. Acoustical Society of America, New York

Anderson GL (1970) On Gegenbauer transforms and forced torsional vibrations of thin spherical shells. J Sound Vib 12:265–275ADSCrossRefMATH

Göller B (1980) Dynamic deformations of thin spherical shells based on analytical solutions. J Sound Vib 73:585–596ADSCrossRefMATH

Mukherjee K, Chakraborty SK (1985) Exact solution for larger amplitude free and forced oscillation of a thin spherical shell. J Sound Vib 100:339–342ADSCrossRef

Al-Jumaily AM, Najim FM (1997) An approximation to the vibrations of oblate spheroidal shells. J Sound Vib 207:561–574ADSCrossRef

Reissner E (1941) A new derivation of the equations for the deformation of elastic shells. Am J Math 62(1):177–184CrossRefMathSciNet

Mushtari KhM, Galimov KZ (1957) Non-linear theory of thin elastic shells. NASA Technical Report, NASA-TT-F-62, Washington, DC

Mushtari KhM, Sachenkov AV (1958) Stability of cylindrical and conical shells of circular cross section, with simultaneous action of axial compression and external normal pressure. NACA Technical Memorandum, NACA TM 1433, Washington, DC

10.

Reissner E, Wan FYM (1969) On the equations of linear shallow shell theory. Stud Appl Math 48:133–145MATH

11.

Koiter WT (1966) Summary of equations for modified, simplest possible accurate linear theory of thin circular cylindrical shells. Rep 442, Lab Tech Mech, TH, Delft

12.

Koiter WT (1966) On the nonlinear theory of thin elastic shells. I-Introductory sections. II: Basic shell equations. III: Simplified shell equations. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series B 69(1):1–54

13.

Green AE, Naghdi PM (1968) The linear elastic cosserat surface and shell theory. Int J Solids Struct 4:585–592CrossRefMATH

14.

Ilgamov MA (1969) Oscillations of elastic shells containing liquid and gas. Nauka, Moscow (in Russian)

15.

Libai A, Simmonds JG (1998) Nonlinear theory of elastic shells. Cambridge University Press, Cambridge, UKCrossRefMATH

16.

Ross EW Jr (1966) Asymptotic analysis of the axisymmetric vibration of shells. J Appl Mech 33:553–561CrossRef

17.

Nau RW, Simmonds JG (1973) Calculation of the low natural frequencies of clamped cylindrical shells by asymptotic methods. Int J Solids Struct 9:591–605CrossRefMATH

18.

Vlasov VZ (1949) General theory of shells and its application in technology. Gostechizdat, Moscow (in Russian)

19.

Gavrilyuk I, Hermann M, Trotsenko Yu, Timokha A (2010) Eigenoscillations of three- and two-element flexible systems. Int J Solids Struct 47:1857–1870CrossRefMATH

20.

Wasow W (2002) Asymptotic expansions for ordinary differential equations. Dover, New York

21.

Verhulst F (2005) Methods and applications of singular perturbations: boundary layers and multiple timescale dynamics. Springer Science+Business Media, New YorkCrossRef

22.

O’Malley RE Jr (1991) Singular perturbation methods for ordinary differential equations. Springer Science, New YorkCrossRefMATH

23.

Goldenveizer AL, Lidski VB, Tovstik PE (1979) Free vibrations of thin elastic shells. Nauka, Moscow (in Russian)

24.

Alumyan NA (1960) On fundamental system of integrals of equation on small axisymmetric steady-state oscillations of an elastic conical shell of revolution. Izv AN Est SSR, Ser Technol Phys Math Sci 10(1):3–15 (in Russian)

25.

Lubimov VM, Pshenichnov GI (1976) The perturbation method in shallow shell theory. Mech Solids 11(4):120–124

26.

Tovstik PE (1975) Low-frequency oscillations of a convex shell of revolution. Izv AN SSSR, Mech Solid Bodies 6:110–116 (in Russian)

27.

Vishik MI, Lusternik LA (1957) Regular degeneracy and boundary layer for linear differential equations with a small parameter. Uspekhi Math Nauk 12(5):3–122MATH

28.

Marchuk GI (1982) Methods of numerical mathematics, 2nd edn. Springer, New YorkCrossRefMATH

29.

Doolan EP, Miller JJH, Schilders WHA (1980) Uniform numerical methods for problems with initial and boundary layers. Boole Press, DublinMATH

Title: Eigenoscillations of a thin-walled azimuthally closed, axially open shell of revolution
Authors: I. Gavrilyuk
M. Hermann
V. Trotsenko
Yu. Trotsenko
A. Timokha
Publication date: 01-04-2014
Publisher: Springer Netherlands
Published in: Journal of Engineering Mathematics / Issue 1/2014
Print ISSN: 0022-0833
Electronic ISSN: 1573-2703
DOI: https://doi.org/10.1007/s10665-013-9626-9

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