 | | 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: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3212 |
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| << Previous article | Volume 48, Issue 5 / 2013 | Next article >> |
| A.K. Belyaev, D.N. Il'in, and N.F. Morozov, "Dynamic Approach to the Ishlinsky-Lavrent'ev Problem," Mech. Solids. 48 (5), 504-508 (2013) |
| Year |
2013 |
Volume |
48 |
Number |
5 |
Pages |
504-508 |
| DOI |
10.3103/S002565441305004X |
| Title |
Dynamic Approach to the Ishlinsky-Lavrent'ev Problem |
| Author(s) |
A.K. Belyaev (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr-t 61, St. Petersburg, 199178 Russia, vice.ipme@gmail.ru)
D.N. Il'in (St.-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia, dimakrechet@mail.ru)
N.F. Morozov (St.-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia, morozov@nm1016.spb.edu) |
| Abstract |
The dynamic stability problem for a hinged rod in the case of a stepwise axial load is considered. The method of expansions in series in natural vibration modes is systematically applied to the longitudinal as well as bending vibrations. The longitudinal vibrations result in the occurrence of longitudinal periodic forces which, in turn, cause unstable bending vibrations. An application of the Galerkin method leads to a system of ordinary differential equations with periodic coefficients, which can be reduced to Mathieu type equations. The instability domains, whose shape depends on the spectral properties of longitudinal and bending vibrations, the damping value, and the longitudinal force, are constructed. An example of nonstandard location of the instability domains is given; namely, the rod parameters are chosen so that the twelfth transverse vibration mode caused by the first longitudinal mode turns out to be unstable. The expression for the minimum value of a stepwise load leading to instability of this transverse vibration mode is obtained. |
| Keywords |
dynamic transverse stability of the rod, longitudinal and bending vibrations, Mathieu equation |
| References |
| 1. | M. A. Lavrent'ev and A. Yu. Ishlinskii, "Dynamic Shapes of
Buckling of Elastic Systems," Dokl. Akad. Nauk SSSR 64
(6), 779-782 (1949). |
| 2. | A. Yu. Ishlinskii,
"An Example of Bifurcation Resulting in the Appearance of Unstable Forms
of Stationary Motion,"
Dokl. Akad. Nauk SSSR
117 (1), 47-49 (1957). |
| 3. | A. Yu. Ishlinskii, S. V. Malashenko, and M. E. Temchenko,
"Branching of the Stable Dynamical Equilibrium Positions of a Mechanical System,"
Izv. Akad. Nauk SSSR. OTN. Mekh. Mashinostr.,
No. 8, 53-61 (1958). |
| 4. | N. F. Morozov and P. E. Tovstik,
"Dynamics of a Rod on Longitudinal Impact,"
Vestnik S.-Peterburg Univ. Ser. I. Mat. Mekh. Astr.
No. 2, 105-111 (2009). |
| 5. | V. V. Bolotin,
The Dynamical Stability of Elastic Systems
(Gostekhizdat, Moscow, 1956; Holden-Day, California, 1964). |
| 6. | A. S. Vol'mir,
Stability of Deformable Systems
(Nauka, Moscow, 1967)
[in Russian]. |
| 7. | V. A. Palmov,
Vibrations of Elasto-Plastic Bodies
(Nauka, Moscow, 1976; Springer, Berlin, 1998). |
|
| Received |
06 March 2013 |
| Link to Fulltext |
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