Vietnam Journal of Mechanics, NCST of Vietnam Vol. 25, 2003, No 2 (170 - 185) . NUMERICAL SOLUTION FOR CONSISTENT INITIAL CONDITIONS OF CONSTRAINED MECHANICAL SYSTEMS DINH VAN PHONG Department of Applied Mechanics, Hanoi University of Technology Abstract. The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique. 1. Introduction The system of equations of motion of a mechanical system leads to the problem of solving the system of ordinary differential equations (ODE) or the mixed sys- tem of differential-algebraic equations (DAE). If the set of independent generalised coordinates is used, the system of equations of motion has the form of ordinary differential equation. In contrast, if the set of dependent generalised coordinates is used, the system of equations of motion will have the form of differential-algebraic equations. Due to the generality and complexity of these systems of equations nu- merical integration is usually applied in order to get the solution. Various schemes for ordinary differential equations and differential-algebraic equations can be used for this purpose. Here in this article we will focus our attention on the problem of the initial values of the system of differential-algebraic equations, since unlike the ordinary differential equations initial conditions for differential-algebraic equations can not be chosen arbitrarily: they have to meet the constraints equation and their derivatives. This is one major difference between differential-algebraic equations and ordinary differential equations. It can be seen as one source of difficulties in numerically solving differential-algebraic equations. The problem of consistency of initial conditions is a central part for all types of differential-algebraic equations which must be addressed as part of the problem solution, see e.g. [2), [9), [18), [19), [21] etc. In recent years many mathematical researchers were interested in the general problem of consistency of initial condition for first order ordinary differential equations which can be described in general by equation: f(y, y', x) = 0 (1.1) 170