 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12855 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4811 |
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| B.D. Annin, A.Yu. Vlasov, Yu.V. Zakharov, and K.G. Okhotkin, "Study of Static and Dynamic Stability of Flexible Rods in a Geometrically Nonlinear Statement," Mech. Solids. 52 (4), 353-363 (2017) |
| Year |
2017 |
Volume |
52 |
Number |
4 |
Pages |
353-363 |
| DOI |
10.3103/S002565441704001X |
| Title |
Study of Static and Dynamic Stability of Flexible Rods in a Geometrically Nonlinear Statement |
| Author(s) |
B.D. Annin (Lavrent'ev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, pr. akad. Lavrentyeva 15, Novosibirsk, 630090 Russia)
A.Yu. Vlasov (Siberian State Aerospace University, pr. im. gazety "Krasnoyarskiy rabochiy" 31, Krasnoyarsk, 660014 Russia)
Yu.V. Zakharov (Siberian State Aerospace University, pr. im. gazety "Krasnoyarskiy rabochiy" 31, Krasnoyarsk, 660014 Russia)
K.G. Okhotkin (Academician M.F. Reshetnev Information Satellite Systems, ul. Lenina 52, Zheleznogorsk, Krasnoyarsk region, 662972 Russia; Federal Research Center "Krasnoyarsk Scientific Center of the Siberian Branch of the Russian Academy of Sciences," ul. Akademgorodok 50, Krasnoyarsk, 660036 Russia, okg2000@mail.ru) |
| Abstract |
We study static and dynamic stability problems for a thin flexible rod subjected to axial compression with the geometric nonlinearity explicitly taken into account. In the case of static action of a force, the critical load and the bending shapes of the rod were determined by Euler. Lavrent'ev and Ishlinsky discovered that, in the case of rod dynamic loading significantly greater than the Euler static critical load, there arise buckling modes with a large number of waves in the longitudinal direction. Lavrent'ev and Ishlinsky referred to the first loading threshold discovered by Euler as the static threshold, and the subsequent ones were called dynamic thresholds; they can be attained under impact loading if the pulse growth time is less than the system relaxation time. Later, the buckling mechanism in this case and the arising parametric resonance were studied in detail by Academician Morozov and his colleagues.
In this paper, we complete and develop the approach to studying dynamic rod systems suggested by Morozov; in particular, we construct exact and approximate analytic solutions by using a system of special functions generalizing the Jacobi elliptic functions. We obtain approximate analytic solutions of the nonlinear dynamic problem of flexible rod deformation under longitudinal loading with regard to the boundary conditions and show that the analytic solution of static rod system stability problems in a geometrically nonlinear statement permits exactly determining all possible shapes of the bent rod and the complete system of buckling thresholds. The study of approximate analytic solutions of dynamic problems of nonlinear vibrations of rod systems loaded by lumped forces after buckling in the deformed state allows one to determine the vibration frequencies and then the parametric resonance thresholds. |
| Keywords |
flexible rod, geometric nonlinearity, static stability, dynamic stability |
| References |
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| 17. | N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik,
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|
| Received |
03 April 2017 |
| Link to Fulltext |
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