TAILIEUCHUNG - About the gibbs-appel equations for multibody systems

In this paper a matrix form of Gibbs-Appel function is recommended for multibody dynamics formulations. The form proposed in this paper seems to be more clear and suitable for automatic generation of dynamical equations of motion. The advantages followed from the formulation proppsed are illustrated through an example. | Vietnam Journal of Mechanics, VAST, Vol. 28, No. 4 (2006) , pp. 225 - 229 ABOUT THE GIBBS-APPEL EQUATIONS FOR MULTIBODY SYSTEMS NGUYEN VAN KHANG Department of Applied Mechanics, Hanoi University of Technology Abstract. In this paper a matrix form of Gibbs-Appel function is recommended for multi- body dynamics formulations. The form""·proposed in this paper seems to be more clear and suitable for automatic generation of dynamical equations of motion. The advantages followed from the formulation prop_psed are illustrated through an example. 1. INTRODUCTION Gibb-Appel equations were introduced by in 1879 [1] and were subsequently studied and formalized by Appel in 1900 [2] . The Gibbs-Appel equations for the nonholonomic syst em constituted of n particles have been shown in a lot of specialist books [3,4]. In this paper we consider a system S constituted of p rigid bodies. The Gibbs-Appel function is defined as [5,6] G = ~ ja 2 dm () (S) and the equations of motion are obtained by fJG - = 8 'Tri (i = 1, ., J) rri , () where rri is t he quasi-active force, 'Tri is the quasi-coordinate, f is the number of degrees of freedom of t he system. . The subj ect of this paper is the proposition of a useful matrix form of Gibbs-Appel function of multibody system. The formulation is aimed at more automatic and clear generation of t he dynamical equations of motion. 2. THE MATRIX FORMULATION OF GIBBS-APPEL FUNCTION As shown in Fig. 1, which depicts a rigid body in the three-dimensional space, the global position of an arbitrary point on the body can be written as r= f'A +u () By differentiating Eq. () with respect to time one obtains the velocity of an arbitrary point on the rigid body () v= VA +w x a. The time derivative of Eq. (2 .2) gives acceleration relation as a:= a:A +a x a+ w x (w x ii) () 226 Nguyen Van Khang 1 - 2dm -a 2 ti Fig. 1. Rigid body in the space By substituting Eq. (2 .3) into Eq. (), one .

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