tailieunhanh - A form of equations of motion of a mechanical system in quasi-coordinates

In the present paper the form of equations of motion is written in quasi-coordinates. These equations are solved with respect to quasi-accelerations, which allow to define the motion of a holonomic and nonholonomic systems by a closed set of algebraic - differential equations. The reaction forces of constraints imposed on the system under consideration are calculated by means of a simple algorithm. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 21, 1999, No 1 (45 -56) A FORM OF EQUATIONS OF MOTION OF A MECHANICAL SYSTEM IN QUASI-COORDINATES Do SANH Hanoi University of Technology, Vietnam ABSTRACT. In [3, 4, 5] the form of equations of motion in holonomic coordinates has constructed. The equations obtained give an effective tool for investigating complicated systems. In the present paper the form of equations of motion is written in quasi-coordinates. These equations are solved with respect to quasi-accelerations, which allow to define the motion of a holonomic and nonholonomic systems by a closed set of algebraic - differential equations. The reaction forces of constraints imposed on the system under consideration are calculated by means of a simple algorithm. For illustrating the eft'ectivness of this form of equations an example is considered. 1. Introduction As known [1, 6, 7), in some cases the expression of kinetic energy of a mechanical system is written conveniently in quasi-velocities, for example, in the case of bodies moving about a fixed point. For nonholonomic systems the constraints are often written in quasi-velocities, for example, a rigid body rolls without sliding on a plane. In such a case, we can apply the Lagrange's equations with multipliers or the equations in quasi-coordinates. However, as shown in [6] these methods are very complicated. In connection with this, for the method of Lagrange's multipliers it is necessary to eliminate undeterminate multipliers, but for the Lagrange's equations in quasi-coordinates we have to calculate complicated indices. These problems can be avoided by means of generalizing the equations obtained in [3, 5]. 2. Equations of motion of a holonomic mechanical system in quasicoordinates Let us consider a holonomic mechanical system of n degrees of freedom. Denote by qi the Lagrangian coordinates and by Q,-generalized forces (i = 1, n). 45 Kinetic energy of the system under consideration is of the .