 | | 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
|
| In English (Mech. Solids): | | 4760 |
|
| << Previous article | Volume 51, Issue 6 / 2016 | Next article >> |
| M.A. Novikov, "Methods for Obtaining Sufficient Conditions for the Stability of Autonomous Conservative Systems," Mech. Solids. 51 (6), 643-653 (2016) |
| Year |
2016 |
Volume |
51 |
Number |
6 |
Pages |
643-653 |
| DOI |
10.3103/S0025654416060030 |
| Title |
Methods for Obtaining Sufficient Conditions for the Stability of Autonomous Conservative Systems |
| Author(s) |
M.A. Novikov (Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, ul. Lermontova 134, Irkutsk, 664033 Russia, nma@icc.ru) |
| Abstract |
A computational method for obtaining sufficient conditions for the stability of the stationary solution of autonomous conservative systems is proposed in the paper. This method is adapted to linear autonomous gyroscopic systems with three degrees of freedom. It is based on the positive definiteness of a parametric quadratic form composed of the gyroscopic force matrices and the potential function. The control parameters for the stability of the zero solution of the gyroscopic system are the entries of the gyroscopic force matrix. The algorithm of the computational method includes estimating one gyroscopic force parameter in the equation constructed from a necessary stability condition.
A special example is used to demonstrate the application of this algorithm. Comparison is performed with some well-known methods for obtaining sufficient conditions on the basis of an incomplete set of first integrals of motion. It is shown that the positive definiteness of the modified potential energy may result in stable as well as unstable motions. |
| Keywords |
autonomous conservative system, characteristic equation, stability of motion, positive (or negative) definite quadratic form, gyroscopic system |
| References |
| 1. | A. M. Lyapunov,
"General Problem of Stability of Motion,"
in Collective Works
(Izdat. AN SSSR, Moscow-Leningrad, 1956)
Vol. 2, pp. 7-263
[in Russian]. |
| 2. | N. G. Chetaev,
Stability of Motion. Works in Analytical Mechanics
(Izdat. AN SSSR, Moscow-Leningrad, 1962)
[in Russian]. |
| 3. | A. M. Letov,
Stability of Nonlinear Controlled Systems
(Fizmatgiz, Moscow, 1962)
[in Russian]. |
| 4. | A. M. Letov,
Mathematical Theory of Control Processes
(Nauka, Moscow, 1981)
[in Russian]. |
| 5. | B. P. Demidovich,
Lectures on Mathematical Theory of Stability
(Nauka, Moscow, 1967)
[in Russian]. |
| 6. | G. V. Kamenkov,
Stability of Motion, Oscillations, Aerodynamics, Vol. 1
(Nauka, Moscow, 1971)
[in Russian]. |
| 7. | G. V. Kamenkov,
Stability and Oscillations of Nonlinear Systems, Vol. 2
(Nauka, Moscow, 1972)
[in Russian]. |
| 8. | I. G. Malkin,
Theory of Stability of Motion
(Nauka, Moscow, 1966)
[in Russian]. |
| 9. | D. R. Merkin,
Introduction to the Theory of Stability of Motion
(Nauka, Moscow, 1971)
[in Russian]. |
| 10. | P. A. Kuz'min,
"Quadratic Integrals of Linear Mechanical Systems,"
Prikl. Mat. Mekh.
24 (3), 575-577 (1960)
[J. Appl. Math. Mech. (Engl. Transl.)
24 (3), 856-860 (1960)]. |
| 11. | L. A. Burlakova and V. D. Kregov,
"Routh-Lyapunov Theorem in Systems with Linear Integrals,"
in Direct Method in Stability and Its Applications
(Nauka, Novosibirsk, 1981),
pp. 151-165
[in Russian]. |
| 12. | I. M. Lakhadanov,
"On Stabilization of Potential Systems,"
Prikl. Mat. Mekh.
39 (1), 53-58 (1975)
[J. Appl. Math. Mech. (Engl. Transl.)
39 (1), 45-50 (1975)]. |
| 13. | I. M. Lakhadanov,
"On Quadratic Integrals of Linear Autonomous Systems,"
Prikl. Mat. Mekh.
42 (3), 555-557 (1978)
[J. Appl. Math. Mech. (Engl. Transl.)
42 (3), 579-581 (1978)]. |
| 14. | N. N. Polyakov, S. A. Zegzhda, M. P. Yushkov,
Theoretical Mechanics
(Vysshaya Shkola, Moscow, 2000)
[in Russian]. |
| 15. | P. Appel,
Theoretical Mechanics, Vol. 2: Dynamics of Systems. Analytical Mechanics
(Fizmatgiz, Moscow, 1960)
[in Russian]. |
| 16. | L. A. Pars,
A Treatise on Analytic Mechanics
(Wiley, New York, 1965; Nauka, Moscow, 1971). |
| 17. | E. T. Whittaker,
Analytical Dynamics
(Udmurdskii Universitet, Izhevsk, 1999)
[in Russian]. |
| 18. | F. R. Gantmakher,
Theory of Matrices
(Nauka, Moscow, 1992)
[in Russian]. |
| 19. | E. A. Barbashin,
Lyapunov Functions
(Nauka, Moscow, 1970)
[in Russian]. |
| 20. | M. A. Novikov,
"On Necessary Conditions for Stability of a Gyroscopic System,"
Sovr. Tekhnol. Sist. Anal. Model.,
No. 3, 80-86 (2013). |
| 21. | M. A. Novikov,
"On Sufficient Conditions for Stability of a Linear Gyroscopic System,"
Sovr. Tekhnol. Sist. Anal. Model.,
No. 4, 23-33 (2013). |
| 22. | I. S. Berezin and N. P. Zhidkov,
Computational Methods
(GIFML, Moscow, 1962)
[in Russian]. |
| 23. | R. M. Bulatovich,
"The Stability of Linear Potential Gyroscopic Systems when the Potential Energy Has a Maximum,"
Prikl. Mat. Mekh.
61 (3), 385-389 (1997)
[J. Appl. Math. Mech. (Engl. Transl.)
61 (3), 371-375 (1997)]. |
| 24. | C. J. Lanczos,
The Variational Principles of Mechanics
(Univ. Toronto, Toronto, 1949; Mir, Moscow, 1965). |
| 25. | V. V. Kozlov,
"Gyroscopic Stabilization and Parametric Resonance,"
Prikl. Mat. Mekh.
64 (5), 739-745 (2001)
[J. Appl. Math. Mech. (Engl. Transl.)
64 (5), 715-721 (2001)]. |
| 26. | T. V. Sal'nikova,
"The Stability of Linear Potential Gyroscopic Systems,"
Prikl. Mat. Mekh.
70 (1), 35-39 (2006)
[J. Appl. Math. Mech. (Engl. Transl.)
70 (1), 32-35 (2006)]. |
| 27. | B. L. Van der Waerden,
Contemporary Algebra, Vol. 2
(ONTI NKTP SSSR, Moscow-Leningrad, 1937)
[in Russian]. |
|
| Received |
16 April 2014 |
| Link to Fulltext |
|
| << Previous article | Volume 51, Issue 6 / 2016 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English