Equations for the parametric resonance of a three-layered cylindrical shell with composite layers stiffened with ribs and containing a hollow isotropic cylinder are obtained. The shell is loaded by axial forces and an external pressure varying harmonically in time. The influence of the cylinder is modeled as an elastic foundation whose modulus of subgrade reaction was determined from equations of 3D elasticity theory. Their solution is sought in the form of a trigonometric series in the axial coordinate. The infinite system of inhomogeneous differential equations of Mathieu–Hill type obtained is solved using a trigonometric series in time. Using a numerical example, the main regions of instability were obtained for the first time, and plots for the critical frequencies on the channel radius, the elastic modulus of cylinder material, and the number and height of ribs are found. The mathematical model proposed extends the range of relevant scientific and applied problems in studying three-layered stiffened shells.
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References
V. N. Bakulin, “Investigation of the influence of the cutout dimensions on the stress-strain state of three-layer shells with load-bearing layers of composite materials,” J. Phys. Conference Ser. Mater. Sci. Eng., 714, 012002 (2020).
A. M. Lipanov, S. A. Karskanov, S. L. Cheernyshev, and I. I. Lipatov, “Theoretical investigation of conditions of the origination of high-speed buffeting,” Bulletin of Udmurt University, Mathematics, Mechanics, Computer Sciences, 29, No. 3, 382-395 (2019) [in Russian].
A. S. Vol’mir, Shells in Flow of Liquid and Gas. Aeroelasticity Problems, Fizmatlit, Moscow (1976) [in Russian].
V. N. Bakulin, “An efficient model for layer-by-layer analysis of three-layered irregular cylindrical shells of revolution,” Doklady Akademii Nauk, 478, No. 2, 148-152 (2018).
V. N. Bakulin, “Block-layer-by-layer approach for analyzing the stress-strain state of three-layered irregular cylindrical shells of revolution,” Prikladnaya Matematika I Mechanika, 85, No. 3, 383-395 (2021).
V. N. Paimushin, “Theory of moderately large deflections of sandwich shells having a transversely soft core and reinforced along their contour,” Mech. Compos. Mater., 53, No. 1, 1-16 (2017).
V. N. Bakulin, I. F. Obraztsov, and V. A. Potopakhin, Dynamic Problems of Nonlinear Theory of Multilayered Shells: Action of Intense Thermal Power Loads of Concentrated Energy Flows [in Russian], Fizmatlit, Moscow (1998).
E. I. Grigolyuk and P. P. Chulkov, Stability and Vibrations of Three-layered Shells [in Russian], Mashinostroenie, Moscow (1973) .
E. I. Starovoitov and D. and V. Leonenko, “Resonance vibrations of a three-layered cylindrical shell with elastic core,” Mech. Compos. Mater. and Struct., 22, No. 1, 60-68 (2016).
V. N. Bakulin, E. N.Volkov, and A. Ya. Nedbai, “Dynamic stability of a cylindrical shell reinforced by longitudinal ribs and a hollow cylinder under the action of axial forces,” J. Eng. Phys. Thermophys., 89, No. 3, 747-753 (2016).
V. N. Bakulin and A. Ya Nedbai, “Dynamic stability of a cylindrical shell stiffened with longitidunal ribs of of piecewise constant thickness under action of axial load,” Doklady Akademii Nauk, Physics, Tekhn. Nauki, 495, 43-49 (2020).
A. A. Mailybaev and A. P. Seiranyan, Multiparameter Stability Problems. Theory and Applications in Mechanics [in Russian], Fizmatlit, Moscow (2009).
Jin-Yih Kao, Chun-Sheng Chen, and Wei-Ren Chen, “Parametric vibration response of foam-filled sandwich plates under periodic loads,” Mech. Compos. Mater., 48, No. 5, 525-538 (2012).
P. M. Ogibalov and M. A. Koltunov, Shells and Plates [in Russian], GITTL, Moscow, (1956).
S. A. Vol’mir, Nonlinear Dynamics of Plates and Shells [in Russian], Nauka, Moscow (1972).
S. D. Algazin and I. A. Kijko, Flutter of Plates and Shells [in Russian], Nauka, Moscow (2006).
V. N. Bakulin, E. N. Volkov, and A. Ya. Nedbai, “Flutter of a layered cylindrical shell stiffened with annular ribs and loaded with axial forces,” Doklady Akademii Nauk, 463, No. 4, 414-417 (2015).
V. N. Bakulin, M. A. Konopel’chev, and A. Ya. Nedbai, “Flutter of a laminated cantilever cylindrical shell with a ringstiffened edge,” Russ. Aeronautika, 61, No. 4, 517-523 (2018).
V. G. Moskvin, “Stability of circular cylindrical shell made from linear viscoelastic material in a supersonic gas flow,” Proc. 8th All-Union Conf. on Theory of Shells and Plates, Nauka, Moscow, (1962), 527-531.
S. A. Bochkarev and S. V. Lekomtsev, “ Study of panel flutter of circular cylindrical shells made of functional-gradient material,” PNRPU Mechanikha Bulet, No. 1, 57-75 (2014).
V. N. Bakulin, E. N. Volkov, and A. I. Simonov, “Dynamic stability of a cylindrical shell under alternating axial external pressure,” Russian Aeronautika, 60, No. 4, 508-513 (2017).
V. N. Bakulin, V. N. Bokov, and A. Ya. Nedbai, “Aeroelastic stability of a cylindrical composite shell at a bilateral flow,” 53, No. 6, 801-808 (2018).
Yu. S. Solomonov, V. P. Georgievskii, A. Ya. Nedbai, and E. N. Volkov, “Dynamic stability of layered cylindrical shell stiffened with circular ribs and cylinder under external pressure,” Mech. Compos. Mater. and Struct., 19, No.4, 61-623 (2013).
V. V. Bagdasaryan, O. V. Kuznetsov, and I. S. Malyutin, ‘On parametric resonance of cylindrical shell stiffened with longitudinal ribs,” Vopr. Mat. Fiz. Vibration Teor. Issue 3, 89-96 (1975).
V. N. Bakulin, E. V. Danilkin, and A. Ya. Nedbai, “Dynamic stability of a cylindrical shell stiffened with a cylinder and longitudinal diaphragms at external pressure,” J. Eng. Phys. Thermophys., 91, No. 2, 537–543 (2018).
Yu. S. Solomonov, V. P. Georgievskii, A. Ya. Nedbai, and V. A. Andryushin, Applied Problems of Mechanics of Composite Cylindrical Sheels [in Russian], Fizmatgiz, Moscow (2014).
I. S. Malyutin, “Stability of three-layered orthotropic cylindrical shells directly stiffened by ribs,” Applied Mech., 15, No. 7, 20-26 (1979).
V. V. Bolotin, Dynamic Stability of Elastic Systems [in Russian], GITTL, Moscow (1956).
V. N. Bakulin and A. Ya. Nedbai, “Dynamic stability of composite cylindrical shell of linear-variable thickness under the action pulsed external pressure,” J. Eng. Phys. Thermophys, 94, No. 2, 525-533 (2021).
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To the blessed memory of the Corresponding Member of the Science Academy of USSR Vsevolod Ivanovich Feodosyev in the year of his 105th birthday (05.05.1916—09.24.1991)
Translated from Mekhanika Kompozitnykh Materialov, Vol. 57, No. 5, pp. 887-900, September-October, 2021. Russian DOI: 10.22364/mkm.57.5.06.
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Bakulin, V.N., Boitsova, D.A. & Nedbai, A.Y. Parametric Resonance of a Three-Layered Cylindrical Composite Rib-Stiffened Shell*. Mech Compos Mater 57, 623–634 (2021). https://doi.org/10.1007/s11029-021-09984-9
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DOI: https://doi.org/10.1007/s11029-021-09984-9