Dynamics of a console cylindrical panel
A.V. Ivanilova*, D.V. Tarlakovskiia,b**
aMoscow Aviation Institute (National Research University), Moscow, 125993 Russia
bResearch Institute of Mechanics, Moscow State University, Moscow, 119192 Russia
E-mail: *kru0303@mail.ru, **tdvhome@mail.ru
Received December 26, 2017
Abstract
A one-dimensional stationary problem of the dynamics of a console circular panel with applied normal pressure has been discussed. The boundary value problem has been considered for various shells, such as the Kirchhoff–Love shell and the Donnel–Mushtari shell with extensible and non-extensible middle plane. The possibility of finding an analytical solution for the above-said shells has been considered as well.
The solution of the problem has been sought in the form of using integral expressions with Green's functions as their core. Green's functions would be the boundary value problem's solution with pressure in a form of Dirac delta function. An analytical algorithm has been developed to calculate Green's functions value for any fluctuation frequency. This algorithm is based on the numerical (in case of the Kirchhoff–Love shell and the Donnel–Mushtari shell with extensible middle plane) and precise (in any other cases) solution of the characteristic equation and also on developing a common solution in a matrix form for the boundary value problem. Examples of the calculations of dynamics of a console circular panel with applied concentrated force (Green's function) and normal pressure have been given.
Comparison of various shells with the applied concentrated and distributed loads has been performed.
Keywords: console circular panel, Kirchhoff–Love shell, Donnel–Mushtari shell, stationary boundary value problem, eigenfunction, Green's function
Acknowledgments. The study was supported by the Russian Science Foundation, project no. 14-49-00091.
References
1. Strutt J.W. Teoriya zvuka [Theory of Sound]. Moscow, Gostehteorizdat, 1955. (In Russian)
2. Lyamshev L.M. Otrazhenie zvuka tonkimi plastinkami i obolochkami v zhidkosti [Reflection of Sound by Thin Plates and Shells in Liquid]. Moscow, Izd. Akad. Nauk SSSR, 1955. 73 p. (In Russian)
3. Gurovich Yu.A. On the sound insulation of a rectangular plate at low frequencies. Akust. Zh., 1978, vol. 24, no. 4, pp. 508–515. (In Russian)
4. Igumnov L.A., Lokteva N.A., Paimushin V.N., Tarlakovskii D.V. Soundproof properties of a one-dimensional three-layer plate. J. Math. Sci., 2014, vol. 203, no. 1, pp. 104–113. doi: 10.1007/s10958-014-2093-7.
5. Paimushin V.N., Tarlakovskii D.V., Gazizullin R.K., Lukashevich A. Investigation of different versions of formulation of the problem of soundproofing of rectangular plates surrounded with acoustic media. J. Math. Sci., 2017, vol. 220, no. 1, pp. 59–81. doi: 10.1007/s10958-016-3168-4.
6. Lokteva N.A., Serdyuk D.O., Tarlakovskii D.V. Influence of incident wave shape on sound-insulating properties of a rectangular plate with complex structure. Tr. MAI, 2015, no. 82. Available at: http://trudymai.ru/published.php?ID=58602. (In Russian)
7. Lokteva N.A., Serdyuk D.O., Tarlakovskii D.V. Investigation of soundproof properties of a three-layer plate. Izv. Vyssh. Uchebn. Zaved., Mashinostr., 2016, no. 1, pp. 167–171. (In Russian)
8. Lokteva N.A., Paimushin V.N., Serdyuk D.O., Tarlakovskii D.V. The interaction between the plane wave and the plate with limited height in soil. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 64–74. (In Russian)
9. Vlasov V.Z. Obshchaya teoriya obolochek [General Theory of Shells]. Moscow, Gostekhizdat, 1949. 784 p. (In Russian)
10. Novozhilov V.V., Chernykh K.F., Mikhailovsky E.I. Lineinaya teoriya tonkikh obolochek [Linear Theory of Thin Shells]. Leningrad, Politekhnika, 1991. 656 p. (In Russian)
11. Timoshenko S.P., Voinovsky-Krieger S. Plastinki i obolochki [Plates and Shells]. Moscow, Nauka, 1966. 635 p. (In Russian)
12. Ogibalov P.M., Koltunov M.A. Obolochki i plastiny [Shells and Plates]. Moscow, Izd. Mosk. Univ., 1969. 695 p. (In Russian)
13. Volmir A.S. Obolochki v potoke zhidkosti i gaza (zadachi aerouprugosti) [Shells in Fluid and Gas Flow (Aeroelasticity Problems)]. Moscow, Nauka, 1976. 416 p. (In Russian)
14. Galimov K.Z., Paimushin V.N., Teregulov I.G. Osnovaniya nelineinoi teorii obolochek [Foundations of the Nonlinear Theory of Shells]. Kazan, Fen, 1996. 216 p. (In Russian)
15. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnykh sredakh [Waves in Continuous Media]. Moscow, Fizmatlit, 2004. 472 p. (In Russian)
16. Sbornik zadach po soprotivleniyu materialov s teoriei i primerami [Collection of Problems on Resistance of Materials with Theory and Examples]. Gorshkov A.G., Tarlakovskii D.V. (Eds.). Moscow, FIZMATLIT, 2003. 632 p. (In Russian)
17. Tarlakovskii D.V., Fedotenkov G.V. Obshchie sootnosheniya i variatsionnye printsipy matematicheskoi teorii uprugosti [General Relations and Variational Principles of the Mathematical Theory of Elasticity]. Moscow, Izd. MAI-PRINT, 2009. 112 p. (In Russian)
For citation: Ivanilov A.V., Tarlakovskii D.V. Dynamics of a console cylindrical panel. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 3, pp. 528–543. (In Russian)
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