tailieunhanh - A form of equation of motion for chaplyghin

In the present paper, a form of equations of motion for the Chaplyghin's systems is introduced. The scheme for writing these equations is very simple, because they are established by means of only the matrix of inertia of Lagrangian function and are written in the matrix form. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 22, 2000, No 1 (54 - 64) A FORM OF EQUATION OF MOTION FOR CHAPLYGHIN'S SYSTEMS . ' Do SANH Hanoi University of Technology 1. Introduction The Chaplyghin's systems play an important role in engineering. There exist some methods for establishing these systems. Lt is necessary to pay attention to the Chaplyghin's equations which are written by means of the Lagrangian function in independent variables. By this, the mentioned equations are a closed set of second order differential equations. The application of the Lagrangian function for writing equations of motion is an advantageous point of the Chaplyghin's method. However, the algorithm of establishing these equations is complicated enough, especially for the large systems. In the present paper, a form of equations of motion for the Chaplyghin's systems is introduced. The scheme for writing these equations is very simple, because they are established by means of only the matrix of inertia of Lagrangian function and are written in the matrix form. It is necessary to emphasize that the equations obtained are very convenient for automatically programming on computers. 2 . . The form of equations of motion for Chaplyghin's systems Let us consider a mechanical system with n holonomic coordinates qi (i = 1, 2, . , n). Assume that the matrix of inertia of the system is denoted by A which is an n x n positive define symmetric matrix. The kinetic energy of the system is of the form: T- !·TA. 2q q, () where q is an n x 1 matrix of generalized velocities. The letter located on the high right corner denotes the transposition. The generalized force corresponding to the coordinate qi is denoted by Qi 54 (i = 1, 2, . , n). Then x 1 matrix of generalized forces is denoted by Q, that is: () Suppose that the system under consideration is subjected to constraints of the form: k Qr= L brjQj, r = 1, s; k = n - s. () j=l Let us consider the case when .