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Vibration Analysis of Thick Hyperboloidal Shells of Revolution from a Three-Dimensional Analysis  

심현주 (중앙대학교 공과대학 건축공학과)
강재훈 (중앙대학교 공과대학 건축공학과)
Publication Information
Journal of the Computational Structural Engineering Institute of Korea / v.16, no.4, 2003 , pp. 419-429 More about this Journal
Abstract
A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of thick, hyperboloidal shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u/sub r/, u/sub θ/, u/sub z/ in the radial, circumferential, and axial directions, respectively, we taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential(strain) and kinetic energies of the hyperboloidal shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the hyperboloidal shells of revolution. Numerical results are tabulated for eighteen configurations of completely free hyperboloidal shells of revolution having two different shell thickness ratios, three variant axis ratios, and three types of shell height ratios. Poisson's ratio (ν) is fixed at 0.3. Comparisons we made among the frequencies for these hyperboloidal shells and ones which ate cylindrical or nearly cylindrical( small meridional curvature. ) The method is applicable to thin hyperboloidal shells, as well as thick and very thick ones.
Keywords
three-dimensional analysis; hyperboloidal shell of revolution; thick shell; vibration; Ritz method;
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1 Krivoshapko, S. N., 'Static, Vibration, and Buckling Analyses and Applications to One- Sheet Hyperboloidal Shells of Revolution', Applied Mechanics Reviews, Vol.55, No.3, 2002, pp.241-270   DOI   ScienceOn
2 Stoneking, J. E., 'Free Vibrations of Shells of Revolution with Variable Thickness', Nuclear Engineering and Design, Vol.24, 1973, pp.314-321   DOI   ScienceOn
3 Gould, P. L. and Basu, P. K., 'Dynamische Untersuchung von aturzugkhltrmen', Konstr. Ingenieurbau. Ber., Vol.31, 1978, pp.82-83
4 Delpak, R. and Hague, W. M., 'An Experimental and Theoretical Investigation of the Frequencies and Mode Shapes of Axisymmetric Shell Models', Journal of Sound and Vibration, Vol.72, 1980, pp.235-249   DOI   ScienceOn
5 Kustovskiy, V. D., 'A Finite Element Solution of Free Vibration Problems for Thin Elastic Shells of Revolution Bounded by Parallels(in Russian)', Nadezhnost sudov. mashin, Nikolaev: NKI, 1983, pp.54-65
6 Sokolnikoff, I. S., Mathematical Theory of Elasticity, Second Edition, McGraw-Hill Book Co., New York, 1956
7 Ritz, W., 'Über eine neue Methode zur Lsung gewisser Variationsprobleme der mathematischen Physik', Journal fr die Reine und Angewandte Mathematik, Vol.135, 1909, pp.1-61
8 Sen Subir K. and Gould Phillip L., 'Free Vibration of Shells of Revolution Using FEM', L. Eng. Mech. Div. Proc. ASCE, Vol.100, No.2, 1974, pp.283-303
9 Tupikin, A. I., 'A Free Vibration Investigation of a Cooling Tower Shell with the Help of a Finite Element Method(in Russian)', Tr. TsNIISKa, Vol.43, 1975, pp.28-46
10 Leissa, A. W., Vibration of Shells, U.S. Government Printing Office, Washington D.C., Reprinted by The Acoustical Society of America, 1973; 1993
11 Nelson, R. L and Thomas, D. L., 'Free Vibration Analysis of Cooling Towers with Column Supports', Journal of Sound and Vibration, Vol.57, No.1, 1978, pp.149-153   DOI   ScienceOn
12 Belikov, G. I. and Tarasov, A. A., 'Free Vibration Optimization of Geometric Parameters of Cooling Tower Shells(in Russian)', Stroit. mech. i raschet sooruzheniy, Vol.4, 1982, pp.12-16
13 Wenda, L. and Shi-qiao, G., 'The Effect of Local Geometric Imperfections of Rational Shell on Its Natural Frequencies and Modes', Appl. Math. and Mech., Vol.8, No.11, 1987, pp.1013-1018   DOI
14 Nelson, R., 'Stresses in Shell structures', Journal of Sound and Vibration, Vol.79, No.3, 1981, pp.397-414   DOI   ScienceOn
15 Neal, B. G., 'Natural Frequencies of Cooling Towers', J. Strain Analysis, Vol.2, No.2, 1967, pp. 127-133   DOI
16 Andronov, V. I., Boltuhov, A. A., and Povolotskiy, B. N., 'Natural Vibrations of Cooling Tower Shells(in Russian)', Stroit. meh. i raschet soor., Vol.4, 1966, pp.16-19
17 Kolkunov, N. V. and Zveryaev, E. M., 'Cooling Tower Shell Dynamics(in Russian)', Teoriya i eksper. issled. prostranst. konstruktsiy. Primenenie obolochek v ing. soor.: Intern. Congress, Moscow, Vol.3, 1985, pp.146-155
18 Lin, A. N, Mozaffarian, H., and Helpingstine, M., 'Measured Dynamic Response of Hyperbolic Shell', Proc. 3rd US Nat. Conf. Earthquake Eng., Charleston, S.C., Vol.2.-El Cerrito, Calif., 1986, pp.1407-1418
19 Kantorovich, L. V. and Krylov, V. I., Approximate Methods in Higher Analysis, Noordhoff, Gronigen, The Netherlands, pp.266-268, 1958
20 McGee, O. G. and Leissa, A. W., 'Three- Dimensional Free Vibrations of Thick Skewed Cantilever Plates', Journal of Sound and Vibration, Vol.144, pp.305～322; Errata 149, 1991, pp.539-542   DOI   ScienceOn
21 Kosawada, T., Suzuki, K., and Takahashi, S., 'Axisymmetric Free Vibrations of Shells of Revolution Having General Meridional Curvature(in Japanes)', Trans. JSME. C52, No. 473, 1986, pp.209-215
22 Yangang, Z. and Jinren, J., 'Dynamic Analysis of Hyperbolic Cooling Towers(in Chinese), Eartquake Eng. and Eng. Vibr., Vol.14, No.1, 1994, pp.61-71
23 Carter, R. L., Robinson, A. R., and Schnobrich, W. C., 'Free and Forced Vibrations of Hyperboloidal Shells of Revolution', Civil Engineering Studies, Structural Research, Univ. of Illinois, Vol.334, 1968
24 Lyahina, L. I., 'Natural Frequencies and Modes of a Hyperboloidal Cooling Tower Shell with Local Axisymmetrical Imperfections(in Russian)', MKE raschotah zhelezob. plastin i obol., Frunze: FPI, 1979, pp.118-126
25 Anon, Report of the Committee of Inquiry into Collapse of Cooling Towers at Ferrybridge, Monday 1 Nov., Central Electricity Generating Board, London, England, 1965
26 Abramek, W., 'Drgania wlasne hiperboloidalnej chlodni kominowej(in Polish)', Inz. i bud., Vol.46, No.9, 1989