Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8569Research ArticleStabilization of Redundantly Constrained Dynamic SystemMukharlyamovR GDepartment of Theoretical Physics and Mechanicsrobgar@mail.ruDeressaChernet TugeDepartment of Theoretical Physics and Mechanicschernettuge@gmail.comPeoples’ Friendship University of Russia150120151607208092016Copyright © 2015,2015This article addresses the issue of constraint stabilization in a dynamic system. The well known Lagrange’s equation of motion of second order is used for modelling the dynamics of a mechanical systems considered in this paper. It is known that Baumgarte’s method of constraint stabilization does not avoid the problem of singularity of mass matrices that may result from redundancy of constraints and as a result it fails to run simulations near and at singularity points. A generalized Baumgarte’s method of constraint stabilization is developed and the stability of the developed method is ascertained by Lyapunov’s direct method. The developed method avoids using the same correction parameters for all constraints under discussion. The usual Baumgarte’s method, which uses the same correction parameters, becomes a particular case of the one developed in this article. Moreover, a modified Lagrange’s equation is constructed in a way that explains all the details of its derivation. The modified Lagrange’s equation improves Lagrange’s equation of motion in such a way that, it addresses the issue of redundant constraints and singular mass matrices. As it is the case in Baumgarte’s method, the usual Lagrange’s equation is a particular case of the improved method developed in this paper. Besides, a numerical example is provided in order to demonstrate the effectiveness of the methods developed. Finally, the carried out simulations show asymptotic stability of the trajectories and run without problem at singularity points.stabilitygeneralized Baumgarte’s methodmodified Lagrange’s equationsingular mass matricesredundant constraintsLyapunov’s direct methodстабилизацияобобщённый метод Баумгартамодифицированные уравнения Лагранжасингулярная массовая матрицаизбыточные ограниченияпрямой метод Ляпунова[Мухарлямов Р.Г. Уравнения движения механических систем. - Москва: РУДН, 2001.][Мухарлямов Р.Г. Численное моделирование в задачах механики // Вестник РУДН. Сер. «Прикладная математика и информатика». - 1995. - № 1.][Mukharlyamov R.G. On the Equations of Kinematics and Dynamics of Constrained Mechanical Systems // Multibody System Dynamics. - 2001. - No 6. - Pp. 17-28.][Baumgarte J. Stabilization of constraints and integrals of motion in dynamical systems // Computer Methods in Applied Mechanics and Engineering. - 1972. - No 1. - Pp. 1-16.][Bayo E., Lendesma R. Augmented Lagrangian and Mass-Orthogonalprojection Methods for Constrained Multibody Dynamics // Applied Mathematics and Mechanics. - 1996. - No 9. - Pp. 113-130.][de Jal´on G., Bayo E. Kinematic and Dynamic Simulation of Multi-body Systems: the Real-Time Challenge. - Berlin: Springer, 1994.][Chernet T.D. Constructing Dynamic Equations of Constrained Mechanical Systems // Bulletin of PFUR. Series Mathematics, Information Science, Physics. - 2013. - No 3. - Pp. 92-104.][Mukharlyamov R. G. Stabilization of the Motions of Mechanical Systems in Prescribed Phase-Space Manifolds // Applied Mathematics and Mechanics. - 2006. - Vol. 70. - Pp. 210-222.][Blajer W. Advanced Design of Mechanical Systems: From Analysis to Optimization. - New York: Springer Wien, 2009.]