 | | 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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| O.V. Kholostova, "On Stability of Relative Equilibria of a Double Pendulum with Vibrating Suspension Point," Mech. Solids. 46 (4), 508-518 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
4 |
Pages |
508-518 |
| DOI |
10.3103/S0025654411040029 |
| Title |
On Stability of Relative Equilibria of a Double Pendulum with Vibrating Suspension Point |
| Author(s) |
O.V. Kholostova (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993 Russia, kholostova_o@mail.ru) |
| Abstract |
We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical.
The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1-3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6-9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting. |
| Keywords |
double pendulum, relative equilibrium, Mathieu equation, stability resonance |
| References |
| 1. | M. J. O. Strutt,
Lame'sche-Mathieusche-und verwandte Funktionen in Physik und Technik
(Springer, Berlin, 1932; Gostekhizdat, Kharkov-Kiev, 1935). |
| 2. | N. W. Mac-Lachlan,
Theory and Applications of Mathieu Functions
(Clarendon Press, Oxford, 1947; Izd-vo Inostr. Lit., Moscow, 1953). |
| 3. | J. J. Stoker,
Nonlinear Vibrations in Mechanical and Electrical Systems
(Wiley, New York-London, 1950; Izd-vo Inostr. Lit., Moscow, 1953). |
| 4. | A. P. Markeyev,
"The Behaviour of a Non-Linear Hamiltonian System with one Degree of Freedom at the Boundary
of a Parametric Resonance Domain,"
Prikl. Mat. Mekh.
59 (4), 569-580 (1995)
[J. Appl. Math. Mech. (Engl. Transl.)
59 (4), 541-551 (1995)]. |
| 5. | B. S. Bardin and A. P. Markeyev,
"The Stability of the Equilibrium of a Pendulum for Vertical Oscillations of the Point of Suspension,"
Prikl. Mat. Mekh.
59 (6), 922-929 (1995)
[J. Appl. Math. Mech. (Engl. Transl.)
59 (6), 879-886 (1995)]. |
| 6. | A. Stephenson,
"On a New Type of Dynamical Stability,"
Mem. Proc. Manch. Lit. Phil. Soc.
52 (2, 8), 1-10 (1908). |
| 7. | A. Erdélyi,
"Über die Kleinen Schwingungen eines Pendels mit Oszillierendem Aufhängepunkt,"
ZAMM
14 (4), 235-247 (1934). |
| 8. | P. L. Kapitsa,
"Pendulum with Vibrating Suspension,"
Uspekhi Fiz. Nauk
44 (1), 7-20 (1951). |
| 9. | P. L. Kapitsa,
"Dynamical Stability of a Pendulum with Vibrating Suspension Point,"
Zh. Éksp. Teor. Fiz.
21 (5), 588-597 (1951)
[Soviet Phys. JETP (Engl. Transl.)]. |
| 10. | A. Stephenson,
"On Induced Stability,"
Phil. Mag. Ser. 7
17, 765-766 (1909). |
| 11. | T. G. Strizhak,
Methods for Studying 'Pendulum'-Type Dynamical Systems
(Nauka, Alma-Ata, 1981)
[in Russian]. |
| 12. | O. V. Kholostova,
"On the Motions of a Double Pendulum with Vibrating Suspension Point,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 2, 25-40 (2009)
[Mech. Solids (Engl. Transl.)
44 (2), 184-197 (2009)]. |
| 13. | A. P. Markeyev,
"A Constructive Algorithm for the Normalization of a Periodic Hamiltonian,"
Prikl. Mat. Mekh.
69 (3), 355-371 (2005)
[J. Appl. Math. Mech. (Engl. Transl.)
69 (3), 323-337 (2005)]. |
| 14. | V. A. Yakubovich and V. M. Starzhinskii,
Parametric Resonance in Linear Systems
(Nauka, Moscow, 1987)
[in Russian]. |
| 15. | A. P. Markeev,
Libration Points in Celestial Mechanics and Space Dynamics
(Nauka, Moscow, 1978)
[in Russian]. |
| 16. | A. M. Lyapunov,
"On Stability of Motion in a Special Case of the Three-Body Problem,"
in Collection of Works, Vol. 1
(Izd-vo AN SSSR, Moscow-Leningrad, 1954),
pp. 327-401
[in Russian]. |
| 17. | V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics
(Springer-Verlag, New York, 1988; Editorial URSS, Moscow, 2002). |
| 18. | J. Moser,
"New Aspects of the Theory of Stability of Namiltonian Systems,"
Comm. Pure Appl. Math.
11 (1), 81-114 (1958). |
| 19. | J. Glimm,
"Formal Stability of Namiltonian Systems,"
Comm. Pure Appl. Math.
17 (4), 509-526 (1964). |
| 20. | A. P. Markeev,
"Stability of Planar Rotations of a Satellite in a Circular Orbit,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 63-85 (2006)
[Mech. Solids. (Engl. Transl.)
41 (4), 46-63 (2006)]. |
|
| Received |
23 April 2009 |
| Link to Fulltext |
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