Теоретическая механика UDC 531.3 Solving Differential Equations of Motion for Constrained Mechanical Systems R. G. Mukharlyamov∗, A. W. Beshaw† ∗ Department of Theoretical Mechanics Peoples’ Friendship University of Russia Miklukho-Maklaya, 6, Moscow, Russia, 117198 † Department of Mathematics Bahr Dar University Ethiopia, Bahr Dar This paper presents an investigation of modeling and solving system of differential equations in the study of mechanical systems with holonomic constraints. <...> A method is developed for constracting equation of motion for mechanical system with constraints. <...> A technique is developed how to approximate the solution of the problem that is obtained from modeling of kinematic constraint equation which is stable. <...> A perturbation analysis shows that velocity stabilization is the most efficient projection with regard to improvement of the numerical integration. <...> How frequently the numerical solution of the ordinary differential equation should be stabilized is discussed. <...> A new approach is applied for constructing and stabilyzing Runge-Kutta numerical methods. <...> The Runge-Kutta numerical methods are reformulated in a new approach. <...> Key words and phrases: numerical integration, kinematic constraint, stable solution, Taylor expansion, row decomposition. 1. <...> In the present paper, to estimate the deviations from the constraint equations, we introduce the equations of constraints and the equations of constraint perturbations and define the notions of stable and asymptotically stable constraints. 82 Bulletin of PFUR. <...> Modeling of Kinematic Mechanical System equations: Kinematic state of mechanical system can be described by ordinary differential y˙ = v(y, t), where y(t0) = y0, y ∈ Rn. <...> In the application of this method one should have in mind the possibility of integration error accumulation for equation (4), which in the course of time, leads to the destruction of the constraint equation (2). <...> By numerical integration, we mean to compute, from y0 (the initial condition), each successive result y1, y2, y3, . . . that satisfy equation they describe are so large, that a purely mathematical analysis is not possible. <...> Setting a = Kf and using the right side of (10) construct the equation yn+1 = yn +τ ˙yn. <...> For a differential equation the following assertion holds [2]: Theorem 2. <...> Runge-Kutta Method Runge-Kutta methods introduce values between tn and tn+1 <...>