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IssuesArchive of Issues2002-6pp.123-133

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Yu. E. Senitskii, "Dynamics of inhomogeneous non-shallow spherical shells," Mech. Solids. 37 (6), 123-133 (2002)
Year 2002 Volume 37 Number 6 Pages 123-133
Title Dynamics of inhomogeneous non-shallow spherical shells
Author(s) Yu. E. Senitskii (Samara)
Abstract On the basis of a refined theory taking into account shear strains and rotation inertia of cross-sections [1], an axially-symmetric initial-boundary value problem is formulated for an elastically fixed inhomogeneous spherical shell. Variation of physical-mechanical material characteristics with respect to the thickness of the structure is caused by radiation, thermal, and chemical fields acting on one of its surfaces and called induced inhomogeneity factors [2]. In the calculation scheme, this variation is taken into account by the introduction of two arbitrary dimensionless positive function, f1(z) and f2(z), of the thickness coordinate z.

A new exact (in the framework of the adopted model) solution is obtained in the form of finite integral transforms [3, 4] by the method of expansion with respect to vector-valued eigenfunctions. An essential point is that the corresponding spectral problem and inversion formulas take into account multiple eigenvalues (internal resonances) [5-7] which affect the dynamic response of non-shallow shells. While solving the problem, we also introduce internal friction forces in accordance with the refined frequency-independent Voigt model [8, 9]. The problem formulated in such terms has never been studied before.

Fairly complete studies of integrability in inhomogeneous elasticity problems exist only for problems with a static loading [10]. Much less attention has been given to dynamical problems for continuously inhomogeneous shells. In fact, closed form solutions (in the framework of the formulated mathematical models) of dynamical problems have been constructed only for shallow spherical shells [11, 12]. These results were obtained without taking into account viscoelastic resistance and internal resonances typical of such shells. As shown in [5, 6] by the analysis of the equations of motion, resonance interaction of flexural and tangential shapes of vibrations in non-shallow shells is not an exceptional phenomenon and must be taken into account when finding the frequency spectrum of such shells. However, this question about the dynamical reaction remains open even for homogeneous shells.
References
1.  E. I. Grigolyuk and I. T. Selezov, Nonclassical Theories of Vibration of Beams, Plates, and Shells. Advances in Science and Technology. Mechanics of Deformable Solids [in Russian], VINITI, Moscow, 1973.
2.  V. V. Petrov, I. G. Ovchinnikov, and Yu. M. Shikhov, Design of Structural Elements in Corrosive Environment [in Russian], Izd-vo Saratovsk. Gos. Un-ta, Saratov, 1987.
3.  Yu. E. Senitskii, "A multi-component generalized finite integral transform and its applications to nonstationary problems in mechanics," Izv. Vuzov. Matematika, No. 4, pp. 57-63, 1991.
4.  Yu. E. Senitskii, "Convergence and uniqueness of representations defined by the formula of multi-component generalized finite integral transform," Izv. Vuzov. Matematika, No. 9, pp. 53-56, 1991.
5.  A. L. Goldenveiser, V. B. Lidskii, and P. E. Tovstik, Free Vibrations of Thin Elastic Shells [in Russian], Nauka, Moscow, 1979.
6.  E. A. Dain, S. A. Lukovenko, and N. V. Khar'kova, On the Problem of Internal Resonances in the Theory of Vibration of Thin Shells. Preprint No. 97 [in Russian], IPM RAN, Moscow, 1977.
7.  A. D. Lizarev, and N. B. Rostanina, Vibrations in Metal-polymeric and Homogeneous Spherical Shells [in Russian], Nauka i Tekhn., Minsk, 1984.
8.  A. I. Tseitlin, "On linear models of frequency-independent internal friction," Izv. AN SSSR. MTT [Mechanics of Solids], No. 3, pp. 18-28, 1978.
9.  A. I. Tseitlin and A. A. Kusainov, Methods for Taking into Account Internal Friction in Dynamic Problems for Structures [in Russian], Nauka, Kaz. SSR, Amla-Ata, 1987.
10.  V. A. Lomakin, Theory of Elasticity for Inhomogeneous Bodies [in Russian], Izd-vo MGU, Moscow, 1976.
11.  Yu. E. Senitskii, "An axially symmetric dynamical problem for an inhomogeneous shallow spherical shell with finite shear stiffness," Prikl. Mekhanika, Vol. 30, No. 9, pp. 50-57, 1994.
12.  Yu. E. Senitskii, "On the integrability of the dynamical initial-boundary value problem for an inhomogeneous shallow spherical shell," Vestnik Samarsk. Un-ta, No. 2(8), pp. 106-121, 1998.
13.  V. L. Berdichevskii, Variational Principles in Continuum Mechanics [in Russian], Nauka, Moscow, 1983.
14.  A. Kalnins, "Effect of bending on vibration of spherical shell," J. Acoust. Soc. America, Vol. 36, No. 1, pp. 74-83, 1964.
15.  Yu. E. Senitskii, "On some identities used for the solution of boundary value problems by the method of finite integral transforms," Diff. Uravneniya, Vol. 19, No. 9, pp. 1636-1638, 1983.
16.  Yu. E. Senitskii and S. A. Lychev, "On the determination of the norm of finite integral transform kernels and their applications," Izv. Vuzov. Matematika, No. 8, pp. 60-69, 1999.
Received 3 May 2000
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