Abstract
An asymptotic process for evaluating the frequencies of free axisymmetric vibrations of transversely isotropic hollow cylinders is proposed. This process is developed in detail for a cylinder with hinge-supported ends and free lateral surfaces. The approaches which make it possible to construct algorithms for identifying their natural frequencies within the given interval are tested on model problems. The results from the Kirchhoff-Love and Ambartsumyan theories are compared with those from the 3D elasticity theory. In the first term of an asymptotic expansion, two frequencies coinciding with those obtained using the applied shell theory are found and a countable set of frequencies absent in this theory is determined.
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REFERENCES
A. L. Gol'denveizer, V. B. Lidskii, and P. E. Tovstik, Free Vibrations of Thin Elastic Shells [in Russian], Nauka, Moscow (1997).
O. D. Oniashvili, Some Dynamic Problems of the Theory of Shells [in Russian], Izd. Akad. Nauk SSSR, Moscow (1957).
V. S. Gontkevich, Natural Vibrations of Plates and Shells [in Russian], Naukova Dumka, Kiev (1964).
I. A. Birger and Ya. G. Panovko, Strength, Stability, and Vibrations. Handbook. Vol. 3 [in Russian], Mashinostroenie, Moscow (1968).
J. D. Achenbah, “An asymptotic method to analyze the vibrations of an elastic layer,” Trans. ASME. Ser. E, J. Appl. Mech., 36, No. 1 (1969).
O. K. Aksentyan, “Determination of the frequency of natural vibrations of round plates,” Prikl. Matem. Mekh., 40, Iss. I, 112-119 (1976).
V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kiev (1981).
Yu. A. Ustinov, “On some features of the asymptotic method as applied to studying vibrations of thin inhomogeneous elastic plates,” in: Proc. I All-Union School on the Theory and Numerical Methods for Calculating Shells and Plates, Tbilisi (1975), pp. 395-403.
M. F. Mekhtiev, “Free vibrations of an isotropic hollow cylinder,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 5, 83-88 (1985).
M. F. Mekhtiev, “Free vibrations of a closed thin hollow sphere,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 6, 159-164 (1986).
S. G. Lekhnitskii, Elasticity Theory of Anisotropic Bodies [in Russian], Nauka, Moscow (1977).
A. S. Kosmodamianskii and V. A. Shaldyrvan, Thick Multiply Connected Plates [in Russian], Naukova Dumka, Kiev (1978).
V. B. Lidskii and V. A. Sadovnichii, “Asymptotic formulas for the roots of one class of integer functions,” Matem. Sbornik, 75, No. 4, 558-566 (1968).
U. K. Nigul', “On the roots of the Lamb equation for deformations of a plate antisymmetric about the midsurface,” Izv. Akad. Nauk Estonsk. SSR, No. 3, 284-293 (1963).
S. A. Ambartsumyan, General Theory of Anisotropic Shells [in Russian], Nauka, Moscow (1974).
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Mekhtiev, M.F., Fomina, N.I. Free Vibrations of Transversely Isotropic Hollow Cylinders. Mechanics of Composite Materials 38, 55–68 (2002). https://doi.org/10.1023/A:1014060907719
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DOI: https://doi.org/10.1023/A:1014060907719