PERIODIC, ALMOST PERIODIC AND CHAOTIC FORCED OSCILLATIONS OF A SHALLOW CANTILEVER SHELL WITH GEOMETRICALLY NON-LINEAR DEFORMATION

Authors

S. Ye. Malyshev, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova St., Kharkiv, 61002, Ukraine), e-mail: malsea@ukr.net

K. V. Avramov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharsky St., Kharkiv, 61046, Ukraine), e-mail: kvavramov@gmail.com, ORCID: 0000-0002-8740-693X

V. N. Konkin, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova St., Kharkiv, 61002, Ukraine)

Abstract

A nonlinear dynamical system with a finite number of degrees of freedom is obtained. It describes the forced oscillations of a shallow shell during its geometrically nonlinear deformation. In order to derive this dynamic system, a method of specified forms is used. In the region of the first main resonance, the Neimark-Sacker bifurcations are investigated. As a result of these bifurcations, almost periodic oscillations arise, which are then transformed into chaotic ones. The properties of these oscillations are investigated.

Keywords: nonlinear periodic oscillations of a shallow shell, stability of oscillations, almost periodic oscillations, chaotic oscillations

References

1. Amabili, M. & Paidoussis, M. P. (2003). Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid structure interaction. Appl. Mech. Reviews, vol. 56, iss. 4, pp. 349–381. https://doi.org/10.1115/1.1565084
2. Alijani, F. & Amabili, M. (2014). Non-linear vibrations of shells: A literature review from 2003 to 2013. International Journal of Non-Linear Mechanics, vol. 58, no. 1, pp. 233–257. https://doi.org/10.1016/j.ijnonlinmec.2013.09.012
3. Avramov, K. V. & Mikhlin, Yu. V. (2010). Nelinejnaja dinamika uprugih sistem. V 2-h t. T. 1. Modeli, metody, javlenija. Moscow: NIC «Reguljarnaja i haoticheskaja dinamika», In-t komp’juter. issled., 704 p.
4. Amabili, M. (2008). Nonlinear vibrations and stability of shells and plates. Cambridge: Cambridge Press. https://doi.org/10.1017/CBO9780511619694
5. Awrejcewicz, J., Kurpa, L., & Osetrov, A. (2011). Investigation of the stress-strain state of the laminated shallow shells by R-functions method combined with spline-approximation. ZAMM. J. Appl. Mathematics and Mechanics, vol. 91, iss. 6, pp. 458–467. https://doi.org/10.1002/zamm.201000164
6. Avramov, K. V. & Breslavsky, D. (2013). Vibrations of shallow shells rectangular in the horizontal projection with two freely supported opposite edges. Mechanics of Solids, vol. 48, iss. 2, pp. 186–193. https://doi.org/10.3103/S0025654413020106
7. Avramov, K. V., Papazov, S. V., & Breslavsky, D. (2017). Dynamic instability of shallow shells in three-dimensional incompressible inviscid potential flow. Journal of Sound and Vibration, vol. 394, pp. 593–611. https://doi.org/10.1016/j.jsv.2017.01.048

Received 19 July 2017

Published 30 September 2017