Dynamic modeling of multi-body system based on Gauss's principle

Based on the Gauss's principle of least constraint, the dynamic modeling of a multi-body system connected by an elastic cable with varied lengths and large deformation in gravitational field of the earth was proposed in this paper. The practical background of the topic is the release process of a tethered satellite. The Kirchhoff's method was applied to transform the deformation of the elastic cable to rotation of rigid cross section along the centerline of the cable. Since the local small deformation of the cable can be accumulated limitlessly along the arc-coordinate, the Kirchhoff's model is suitable to describe the super-large deformation of elastic rod. In present paper the Gauss's constraint function of the system of rigid-flexible bodies in gravitational field of the earth was derived, and the geometric constraint conditions concerning relative position of bodies in space were considered using the Lagrange's multipliers. Therefore the dynamical model of the system was established in the form of conditional extremum problem. Applying the approach of Gauss's principle the real motion of the system can be obtained by the variation method directly through seeking the minimal value of constraint function without differential equations. The unified form of the model does not changed for different topologic constructions of the system, and it is unnecessary to distinct the tree system or system with loops. In the case of multi-body system with automatic control, the dynamic analysis can be combined with the optimization for different technique objectives.