 | | 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: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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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 |
| 1. | E. P. Popov,
Nonlinear Static Problems for Thin Beams
(OGIZ, Leningrad-Moscow, 1948)
[in Russian]. |
| 2. | E. P. Popov,
Theory and Calculation of Flexible Elastic Rods
(Nauka, Moscow, 1986)
[in Russian]. |
| 3. | S. V. Levyakov, "States of Equilibrium and Secondary Buckling of a
Straight Rod Loaded by an Axial Force," Zh. Prikl. Mekh. Tekhn.
Fiz 42 (2), 153-159 (2001) [J. Appl. Mech. Tech. Phys.
(Engl. Transl.) 42 (2), 321-327 (2001)]. |
| 4. | S. P. Timoshenko,
Stability of Rods, Plates, and Shells
(Nauka, Moscow, 1971)
[in Russian]. |
| 5. | A. S. Vol'mir,
Stability of Deformable Systems
(Nauka, Moscow, 1967)
[in Russian]. |
| 6. | V. I. Arnold,
Mathematical Methods of Classical Mechanics
(Nauka, Moscow, 1989)
[in Russian]. |
| 7. | Ya. G. Panovko and I. I. Gubanova,
Stability and Vibrations of Elastic Systems
(Nauka, Moscow, 1987)
[in Russian]. |
| 8. | M. A. Lavrent'ev and A. Yu. Ishlinsky, "Dynamic Buckling Modes of
Elastic Systems," Dokl. Akad. Nauk SSSR 64 (6), 779-782
(1949). |
| 9. | L. Collatz,
Eigenvalue Problems with Engineering Applications
(Akad. Verlag, Leipzig, 1963; Fizmatgiz, Moscow, 1968). |
| 10. | L. I. Shkutin,
"Numerical Analysis of the Branched Forms of Bending for a Rod,"
Zh. Prikl. Mekh. Tekhn. Fiz.
42 (2), 141-147 (2001)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
42 (2), 310-315 (2001)]. |
| 11. | Yu. V. Zakharov and K. G. Okhotkin,
"Nonlinear Bending of Thin Elastic Rods,"
Zh. Prikl. Mekh. Tekhn. Fiz.
43 (5), 124-131 (2002)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.)
43 (5), 739-744 (2002)]. |
| 12. | Yu. V. Zakharov and K. G. Okhotkin, "Elliptic Functions and
Problems of Bending of Thin Rods and Plates," Vestnik KrasGU,
No. 3, 44-52 (2004). |
| 13. | A. D. Polyanin, V. F. Zaitsev, and A. I. Zhurov,
Methods for Solving Nonlinear Equations of Mathematical Physics and Mechanics
(Fizmatlit, Moscow, 2005)
[in Russian]. |
| 14. | H. Bateman and A. Erdélyi,
Higher Transcendental Functions, Vol. 3
(McGraw Hill, New York, 1955; Nauka, Moscow, 1967). |
| 15. | E. L. Ince, "The Periodic Lamé Functions," Proc. Roy. Soc. Edinburgh 40, 47-63 (1940). |
| 16. | A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and T. P. Tovstik,
"Buckling Problem for a Rod Longitudinally Compressed by
a Force Smaller Than the Euler Critical Force,"
Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 28-39 (2016)
[Mech. Solids (Engl. Transl.)
51 (3), 263-272 (2016)]. |
| 17. | N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik,
"Ishlinsky-Lavrent'ev Problem. Development of the idea," in
Proc. XI All-Russia Meeting in Fundamental Problems of
Theoretical and Appllied Mechanics, Kazan, August 20-24. 2015
(KFU, Kazan, 2015), pp. 2636-2638
[in Russian]. |
| 18. | N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik,
"The Ishlinsky-Lavrent'ev Problem at the Initial Stage of
Motion," Dokl. Ross. Akad. Nauk 463 (5), 543-546 (2015)
[Dokl. Phys. (Engl. Transl.) 60 (8), 368-371 (2015)]. |
| 19. | N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik,
"Initial Stage of Motion in the Lavrent'ev-Ishlinsky Problem on
Longitudinal Shock on a Rod," Dokl. Ross. Akad. Nauk 465 (3), 302-306 (2015) [Dokl. Phys. (Engl. Transl.) 60 (11),
519-523 (2015)]. |
| 20. | N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, "Stability of a
Rod in Long-Term Axial Compression," Probl. Prochn. Plastichn.
77 (1), 40-48 (2015). |
|
| Received |
03 April 2017 |
| Link to Fulltext |
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