 | | 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
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| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8071
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| In English (Mech. Solids): | | 4811 |
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| S. A. Agafonov, "On the stability of a circular system subjected to nonlinear dissipative forces," Mech. Solids. 44 (3), 366-371 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
3 |
Pages |
366-371 |
| DOI |
10.3103/S0025654409030054 |
| Title |
On the stability of a circular system subjected to nonlinear dissipative forces |
| Author(s) |
S. A. Agafonov (Bauman Moscow State Technical University, 2-ya Baumanskaya 5, Moscow, 105005 Russia) |
| Abstract |
A circular system is a mechanical system subjected to
potential forces and positional nonconservative forces (circular forces). The latter linearly depend on the coordinates and are characterized by a skew-symmetric matrix. The influence of linear dissipative forces on the stability of a circular system is ambiguous: on the one hand, they can stabilize a stable circular system (making it asymptotically stable); on the other hand, they
can destabilize it [1-4]. The action of linear dissipative forces on a circular system results in the so-called
destabilization paradox: the stability threshold decreases
by a finite value.
A detailed survey of this phenomenon can be found in [5]. The destabilization effect is also preserved under the action of nonlinear dissipative forces. The influence of these forces on the stability of the Ziegler pendulum with a tracking force was studied in [6]. It was shown that the critical value of the tracking force decreases by a finite value. A similar effect was discovered in the analysis of a continual system in [7].
In the present paper, we study how nonlinear dissipative forces affect the stability of the equilibrium of a circular mechanical system with two degrees of freedom. The stability problem is solved without any references to specific mechanical systems. The results are used to analyze the stability of a gimbal gyro with allowance
for dry friction in the rotor bearings. |
| References |
| 1. | S.A. Agafonov,
"To the Stability of Nonconservative Systems,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 1, 47-51 (1986)
[Mech. Solids (Engl. Transl.)
21 (1), 46-50 (1986)]. |
| 2. | D.R. Merkin,
Introduction to Theory of Stability of Motion
(Nauka, Moscow, 1971) [in Russian]. |
| 3. | O.N. Kirillov and A.P. Seyranian,
"Stabilization and Destabilization of a Circulatory System
by Small Velocity-Dependent Forces,"
J. Sound Vibr. 283 (3-5), 781-800 (2005). |
| 4. | O.N. Kirillov,
"A Theory of the Destabilization Paradox in Non-Conservative Systems,"
Acta Mech. 174 (3-4), 145-166 (2005). |
| 5. | A.P. Seyranian,
"Destabilization Paradox in Nonconservative Systems,"
Uspekhi Mekh. 13 (2), 89-124 (1990). |
| 6. | P. Hagedorn,
"On the Destabilizing Effect of Non-Linear Damping in Nonconservative Systems with Follower Forces,"
Int. J. Nonlin. Mech. 5 (2), 341-358 (1970). |
| 7. | S.A. Agafonov and D.V. Georgievskii,
"Dynamic Stability of a Beam That has Nonlinear Internal Viscosity and is Subjected to a Follower Force,"
Dokl. Ross. Akad. Nauk 396 (3), 339-342 (2004)
[Dokl. Phys. (Engl. Transl.) 49 (5), 332-335 (2004)]. |
| 8. | G.V. Kamenkov,
Selected Works, Vol. 1: \Motion Stability. Oscillations. Aerodynamics
(Nauka, Moscow, 1971) [in Russian]. |
| 9. | L.G. Khazin and E.E. Shnol,
Stability of Critical Equilibrium States
(Center of Biological Studies, Pushchino, 1985) [in Russian]. |
| 10. | V.Ph. Zhuravlev,
"Nutational Self-Oscillation of a Free Gyroscope,"
Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 13-16 (1992)
[Mech. Solids (Engl. Transl.) 27 (6), 11-14 (1992)]. |
|
| Received |
07 July 2005 |
| Link to Fulltext |
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