NUMERICAL MODELLING IN DYNAMICS OF HINGED AND TETHERED SYSTEMS I. Stroe 1 , P. Pârvu 2 1 Mechanics Department, “Politehnica” University of Bucharest 2 Aerospace Sciences Department, “Politehnica” University of Bucharest 313 Spl. Indepentei, Bucharest, ROMANIA E-mail: ion.stroe@gmail.com, parvu@aero.pub.ro ABSTRACT Some problems of dynamics for tethered and hinged (very short tether) bodies are presented in this paper. General problem of kinematics of systems is presented in the first part of the paper. For bodies in gravitational field the motion with respect movable references is important to be known (2). Equations of relative motion of the rigid body are used (3) . Motions of rigid bodies with articulation joints and of tethered bodies are analyzed. In the case of rigid body motion equation of mass center are completed with motion equation of rotation with respect to mass center. The two kinds of equations (of mass center and of rotation with respect to mass center) can be parted only in particular cases (4) . When the motion of a system of bodies which compose a large orbital station is described with reference frames having origin in the center of attractive body (Earth) the problem of integration of motion equations presents some difficulties (5) , because some coordinates (like vector radii) have very great values, and others (like distances between bodies) have very small values. Some difficulties can be avoided if relative motion of the system is studied with respect to a reference frame with known motion. Relative motion study isn’t imposing by integration considerations; this is imposing by practical aspects. The models and the elaborated method allow solving a large number of problems of bodies’ systems dynamics in gravitational field. The most general approach for the solution of the dynamic response of systems is the direct numerical integration of the dynamic equations. Many different numerical techniques have previously been presented; however, all approaches can fundamentally be classified as either explicit or implicit integration methods. All explicit methods are conditionally stable with respect to the size of the time step. Implicit methods can be conditionally or unconditionally stable; however, larger time steps may be used. There exist a large number of accurate, higher-order, multi-step methods that have been developed for the numerical solution of differential equations. These multistep methods assume that the solution is a smooth function in which the higher derivatives are continuous. The exact solution of many nonlinear systems requires that the accelerations are not smooth functions. Based on a significant amount of experience (1) , it is the conclusion of the authors that only single-step, implicit, unconditional stable methods must be used for the step-by-step analysis of practical systems. KINEMATICS OF SYSTEMS OF RIGID BODIES Let us consider two bodies, (i) and (j), having constrained motions through a coupling mechanism, which is made precise by points , (Fig. 1). The motion of the body (i), with respect to the inertial reference frame is determined by the position vector of mass center and by matrix , which gives the attitude of trihedral, jointed with (i) body, with respect to the 0 0 0 0 z y x O reference frame. In the same way are defined position vector and matrix for the body (j). Each body, (i) or (j), has 6 degrees-of-freedom, when it is a free body. The number of degrees-of-freedom is reduced by the number of constrains which are imposed by the coupling mechanism (5) . If the general motion of bodies (i) and (j) with respect the inertial reference frame are known, then the relative motion of the body (i) with respect to body (j) can be determined by the vector 1 and by the matrix , which gives the attitude of (i) body with respect to the (j) body, . 2