 | | 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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| A.P. Markeev, "To the Problem of Accuracy for the Pendulum Clock on a Vibrating Base," Mech. Solids. 53 (5), 573-583 (2018) |
| Year |
2018 |
Volume |
53 |
Number |
5 |
Pages |
573-583 |
| DOI |
10.3103/S0025654418080113 |
| Title |
To the Problem of Accuracy for the Pendulum Clock on a Vibrating Base |
| Author(s) |
A.P. Markeev (Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia; Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia; Moscow Institute of Physics and Technology (State University), 9 Institutskiy per., Dolgoprudny, 141700 Russia, anat-markeev@mail.ru) |
| Abstract |
The problem of the influence of periodic vibrations of the suspension point of a physical pendulum on its nonlinear oscillations in the vicinity of a stable vertical equilibrium position is investigated. The vibrations are assumed to be periodic and occur in the plane of the pendulum motion.
The approximate, time-independent equations of motion are obtained. For the case of pendulum oscillations in the vicinity of the vertical equilibrium position that is allowed by approximate equations, the action - angle variables have been introduced and it has been shown that most of the trajectories of the approximate system are also preserved in the complete system. In the plane of two dimensionless parameters: the oscillation amplitudes of the pendulum and the parameter characterizing the difference in vibration intensity values of the suspension point in the horizontal and vertical directions there are areas in which the pendulum clock is fast and areas in which it is slow. |
| Keywords |
pendulum, vibration, stability |
| References |
| 1. | P.L. Kapitza,
"Dynamic stability of the pendulum with vibrating suspension point,"
Zh. Eksp. Teor. Fiz.
51 (21), 588-597 (1951). |
| 2. | P.L. Kapitza,
"Pendulum with Vibrating Suspension,"
Usp. Fiz. Nauk
44 (1), 7-20 (1951). |
| 3. | T.G. Strizhak,
Methods for Studying "Pendulum"-Type Dynamical Systems
(Nauka, Alma-Ata, 1981) [in Russian]. |
| 4. | I.I. Blekhman,
Vibrational Mechanics
(Nauka, Moscow, 1994). |
| 5. | O.V. Kholostova,
Problem of Dynamics of Solids with Vibrating Suspension
(Ins-t Comp. Issl., Moscow-Izhevsk, 2016) [in Russian]. |
| 6. | V.I. Yudovich,
"Vibrodynamics and Vibrogeometry of Mechanical Systems with Constraints,"
Usp. Mekh.
4 (3), 26-158 (2006). |
| 7. | A.P. Markeev,
"The Equations of the Approximate Theory of the Motion of a Rigid Body with a Vibrating Suspension Point,"
Prikl. Mat. Mekh.
75 (2), 193-203 (2011)
[J. Appl. Math. Mech. (Engl. Transl.)
75 (2), 132-139 (2011)]. |
| 8. | V.I. Arnol'd, V.V. Kozlov, and A.I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics
(Editorial URSS, Moscow, 2002) [in Russian] |
| 9. | V.F. Zhuravlev and D.M. Klimov,
Applied Methods in the Theory of Oscillations
(Nauka, Moscow, 1988) [in Russian]. |
| 10. | G.E.O. Giacaglia,
Perturbation Methods in Non-Linear Systems
(Springer, New York, 1972; Nauka, Moscow, 1979). |
| 11. | I.G. Malkin,
Theory of Motion Stability
(Nauka, Moscow, 1966) [in Russian]. |
| 12. | A.A. Andronov, A.A. Vitt, and S.E. Khaikin,
Theory of Oscillation
(Fizmatgiz, Moscow, 1959) [in Russian]. |
| 13. | V.N. Rubanovskii and V.A. Samsonov,
Stability of Steady Motions in Examples and Problems
(Nauka, Moscow, 1988) [in Russian]. |
| 14. | S.V. Bolotin, A.V. Karapetyan, E.I. Kugushev, and D.V. Treschev,
Theoretical Mechanics
(Izd-vo Akademiya, Moscow, 2010) [in Russian]. |
| 15. | N.G. Chetaev,
Stability of Motion. Works in Analytical Mechanics
(Izdat. AN SSSR, Moscow-Leningrad, 1962) [in Russian]. |
| 16. | A.M. Zhuravsky,
Handbook of Elliptic Functions
(Izdat. AN SSSR, Moscow-Leningrad, 1941) [in Russian]. |
| 17. | I.S. Gradstein and I.M. Ryzhik,
Tables of Integrals, Sums, Series and Products
(Fizmatgiz, Moscow, 1962) [in Russian]. |
| 18. | P.F. Byrd and M.D. Friedman,
Handbook of Elliptic Integral for Engineers and Physicists
(Springer, Berlin; Gottingen; Heidelberg, 1954). |
| 19. | A.P. Markeev,
Theoretical Mechanics
(NIZ "Regulyarnaya i Haoticheskaya Dinamika," Moscow-Izhevsk, 2007) [in Russian]. |
| 20. | I.G. Malkin,
Several Problems of Theory of Nonlinear Vibrations
(Nauka, Moscow, 1956) [in Russian]. |
| 21. | J. Mozer,
"Lectures on Hamiltonian Systems,"
Mem. AMS.,
No. 81, 1-60 (1968). |
| 22. | A.I. Neishtadt,
"Estimates in the Kolmogorov Theorem on Conservation of Conditionally Periodic Motions,"
Prikl. Mat. Mekh.
45 (6), 766-772 (1981)
[J. Appl. Math. Mech.
45 (6), 766-772 (1981)] |
|
| Received |
05 April 2017 |
| Link to Fulltext |
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