TAILIEUCHUNG - An algorithm for deriving equations of motion of constrained mechanical system

The article deals with the form of equations of motion of mechanical system with constraints. For holonomic systems the number of differential equation is equal to the degrees of freedom, without regard to the number of chosen coordinates. The possibilities of computer processing (symbolical and numerical) are shown. Two simple examples demonstrate the described technique. | VietRam Journal of Mechanics, NCST of Vietnam Vol. ~1, 1999, No 1 (36 - 44) AN ALGORITHM FOR DERIVING EQUATIONS OF MOTION OF CONSTRAINED MECHANICAL SYSTEM DINH VAN PHONG Hanoi University of Technology, Vietnam ABSTRACT. The article deals with the form of equations of motion of mechanical system with constraints. For holonomic systems the number of differential equation is equal to the degrees of freedom, without regard to the number of chosen coordinates. The possibilities of computer processing (symbolical and numerical) are shown. Two simple examples demonstrate the described technique. 1. Introduction There are a lot of techniques for building equations of motion of mechanical systems. The conventional approaches could be divided into two groups, see .[7], [8]. In the first one, a minimal set of Lagrangian variables, equal to a degree of freedom, is chosen, in order to define the system configuration. The number of differential equations is minimal and equal to a degree of freedom. The drawback of this technique is complexity of equations of motion and even the computer processing {. by recursive algorithm) is time consuming. The second group of methods uses a larger number of coordinates in combination with constraints. The form of system of equations is simple, permitting computer generations. However~ a final mixed system of differential-algebraic equations is large, including not only Lagrangian coordinates, but also so-called Lagrange multipliers. The total equations consist of differential equations which number is equal to number of chosen coordinates, and equations of constraints. In the present paper we will show that it is possible to derive the equations of motion with only a minimum of differential equations. Moreover there exists the possibility of calculation of reaction forces. 2. The form of equations of motion Let us consider the dynamical system with m degrees of freedom. For this system we choose n Lagrangian coordinates qi, i = 1,

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