Abstract
The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the most complicated one in the problem of stability analysis of mechanical systems with a nonstationary parameter at the vector of dissipative forces. The lemma of the estimate from below for the norm of the restoring torque in the neighborhood of the stabilized motion of a rigid body and two theorems on asymptotic stability of the stabilized motion of a body are proven. It is shown that the sufficient conditions of asymptotic stability found in the theorems are close to the necessary ones. The results of numerical simulation illustrating the conclusions obtained in this study are presented.
Similar content being viewed by others
References
V. A. Sarychev, Problems of Artificial Satellites Orientation (VINITI, Moscow, 1978), in Ser.: Advances in Science and Technology. Space Research, Vol. 11 [in Russian].
N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method (Springer-Verlag, New York, 1977).
L. Hatvani, “On partial asymptotic stability and instability. III. Energy-like Ljapunov functions,” Acta Sci. Math. 49, 157–167 (1985).
H. R. Srirangarajan and P. J. Banait, “Analysis of Duffing’s oscillator equation with time-dependent parameters,” J. Sound Vib. 233, 435–440 (2000). doi 10.1006/jsvi.1999.2819
L. Hatvani, “The effect of damping on the stability properties of equilibria of non-autonomous systems,” J. Appl. Math. Mech. 65, 707–713 (2001).
Y. Shen, S. Yang, and X. Liu, “Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method,” Int. J. Mech. Sci. 48, 1256–1263 (2006). doi 10.1016/j.ijmecsci.2006.06.003
J. Sun, Q.-G. Wang, and Q.-C. Zhong, “A less conservative stability test for second-order linear time-varying vector differential equations,” Int. J. Control 80, 523–526 (2007).
M. Onitsuka, “Uniform asymptotic stability for damped linear oscillators with variable parameters,” Appl. Math. Comput. 218, 1436–1442 (2011). doi 10.1016/j.amc.2011.06.025
A. Yu. Aleksandrov, “The stability of the equilibrium positions of non-linear non-autonomous mechanical systems,” J. Appl. Math. Mech. 71, 324–338 (2007).
A. Yu. Aleksandrov and A. A. Kosov, “Asymptotic stability of equilibrium positions of mechanical systems with a nonstationary leading parameter,” J. Comput. Syst. Sci. Int. 47, 332–345 (2008). doi 10.1134/S1064230708030027
A. Yu. Aleksandrov and A. V. Platonov, “On the preservation of asymptotic stability of mechanical systems under the evolution of dissipative forces rezulting in their disappearance,” Sist. Upr. Inf. Tekhnol. 50 (4), 4–7 (2012).
Encyclopedia of Mathematics, Ed. by I. M. Vinogradov and M. Hazewinkel (Sov. Entsiklopediya, Moscow, 1982; Kluwer, Dordrecht, 1989), Vol.3.
V. I. Zubov, Stability of Motion (Vysshaya Shkola, Moscow, 1973) [in Russian].
V. I. Zubov, Dynamics of Controlled Systems (Vysshaya Shkola, Moscow, 1982) [in Russian].
E. Ya. Smirnov, Some Problems of the Mathematical Control Theory (Leningr. Gos. Univ., Leningrad, 1981) [in Russian].
K. A. Antipov and A. A. Tikhonov, “Parametric control in the problem of spacecraft stabilization in the geomagnetic field,” Autom. Remote Control 68, 1333–1344 (2007). doi 10.1134/S000511790708005X
A. Yu. Aleksandrov and A. A. Tikhonov, “Electrodynamic stabilization of earth-orbiting satellites in equatorial orbits,” Cosmic Res. 50, 313–318 (2012). doi 10.1134/S001095251203001X
K. A. Antipov and A. A. Tikhonov, “Electrodynamic control for spacecraft attitude stability in the geomagnetic field,” Cosmic Res. 52, 472–480 (2014). doi 10.1134/S001095251406001X
A. Yu. Aleksandrov, K. A. Antipov, A. V. Platonov, and A. A. Tikhonov, “Electrodynamic attitude stabilization of a satellite in the Konig frame,” Nonlinear Dyn. 82, 1493–1505 (2015). doi doi 10.1007/s11071-015-2256-1
A. Yu. Aleksandrov, K. A. Antipov, A. V. Platonov, and A. A. Tikhonov, “Electrodynamic stabilization of artificial earth satellites in the König coordinate system,” J. Comput. Syst. Sci. Int. 55, 296–309 (2016). doi 10.1134/S1064230716010020
D. R. Merkin, Introduction to the Theory of Stability (Nauka, Moscow, 1987; Springer-Verlag, New York, 1997).
Vibrations in Engineering. Handbook, Vol. 2: Oscillations of Nonlinear Mechanical Systems, Ed. by I. I. Blekhman (Mashinostroenie, Moscow, 1979) [in Russian].
E. I. Rivin, Passive Vibration Isolation (Am. Soc. Mech. Eng., New York, 2003).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.Yu. Aleksandrov, A.A. Tikhonov, 2017, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2017, Vol. 62, No. 4, pp. 633–643.
About this article
Cite this article
Aleksandrov, A.Y., Tikhonov, A.A. Attitude stabilization of a rigid body in conditions of decreasing dissipation. Vestnik St.Petersb. Univ.Math. 50, 384–391 (2017). https://doi.org/10.3103/S1063454117040021
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1063454117040021