 | | 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: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4810 |
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| A.P. Markeev, "On the Stability of Nonlinear Vibrations of Coupled Pendulums," Mech. Solids. 48 (4), 370-379 (2013) |
| Year |
2013 |
Volume |
48 |
Number |
4 |
Pages |
370-379 |
| DOI |
10.3103/S0025654413040031 |
| Title |
On the Stability of Nonlinear Vibrations of Coupled Pendulums |
| Author(s) |
A.P. Markeev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, markeev@ipmnet.ru) |
| Abstract |
The motion of two identical pendulums connected by a linear elastic spring is studied. The pendulums move in a fixed vertical plane in a homogeneous gravity field. The nonlinear problem of orbital stability of such a periodic motion of the pendulums is considered under the assumption that they vibrate in the same direction with the same amplitude. (This is one of the two possible types of nonlinear normal vibrations.) An analytic investigation is performed in the cases of small vibration amplitude or small rigidity of the spring. In a special case where the spring rigidity and the vibration amplitude are arbitrary, the study is carried out numerically. Arbitrary linear and nonlinear vibrations in the case of small rigidity (the case of sympathetic pendulums) were studied earlier [1, 2]. |
| Keywords |
pendulum, nonlinear vibrations, resonance, stability |
| References |
| 1. | A. Sommerfeld,
Mechanics
(Izd-vo Inostr. Lit., Moscow, 1947)
[in Russian]. |
| 2. | A. P. Markeev, "Nonlinear Vibrations of Sympathetic Pendulums,"
Nelin. Din. 6 (3), 605-621 (2010). |
| 3. | A. M. Zhuravskii,
Reference Book in Elliptic Functions
(AN SSSR, Moscow-Leningrad, 1941)
[in Russian]. |
| 4. | P. F. Byrd and M. D. Friedman,
Handbook of Elliptic Integral for Engineers and Physicists
(Springer, Berlin-Gottingen-Heidelberg, 1954). |
| 5. | A. P. Markeev,
Theoretical Mechanics
(NITs "Regular and Chaotic Mechanics", Moscow-Izhevsk, 2007)
[in Russian]. |
| 6. | A. P. Markeev,
"An Algorithm for Normalizing Hamiltonian Systems in the Problem of the Orbital Stability of Periodic Motions,"
Prikl. Mat. Mekh.
66 (6), 929-938 (2002)
[J. Appl. Math. Mech. (Engl. Transl.)
66 (6), 889-896 (2002)]. |
| 7. | V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics
(Editorial URSS, Moscow, 2002)
[in Russian]. |
| 8. | A. P. Markeev,
Libration Points in Celestial Mechanics and Space Dynamics
(Nauka, Moscow, 1978)
[in Russian]. |
| 9. | E. T. Whittaker,
Analytical Dynamics
(Gostekhizdat, Moscow-Leningrad, 1937)
[in Russian]. |
| 10. | I. G. Malkin,
Theory of Stability of Motion
(Nauka, Moscow, 1966)
[in Russian]. |
| 11. | V. A. Yakubovich and V. M. Starzhinskii,
Linear Differential Equations with Periodic Coefficients and Their Applications
(Nauka, Moscow, 1972)
[in Russian]. |
| 12. | A. P. Markeev,
"Stability of Equilibrium States of Hamiltonian Systems: a Method of Investigation,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 6, 3-12 (2004)
[Mech. Solids (Engl. Transl.)
39 (6), 1-8 (2004)]. |
| 13. | A. P. Markeev,
Linear Hamiltonian Systems and Several Problems of Satellite Motion Stability w.r.t. Center of Mass
(IKI, NITs "Regular and Chaotic Mechanics", Moscow-Izhevsk, 2009)
[in Russian]. |
| 14. | V. Ph. Zhuravlev and D. M. Klimov,
Applied Methods in the Theory of Vibrations
(Nauka, Moscow, 1988)
[in Russian]. |
|
| Received |
15 February 2013 |
| Link to Fulltext |
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