Dynamics of multibody systems in space environment; Lagrangian vs. Eulerian approach

摘要

The paper describes the motion of a multibody in space environment: by space environment we mean space-varying gravity, gradient forces, control forces, if any. (1) In the Eulerian approach, the motion of each individual member is described through kinematic parameters: (a) position of its CM with respect to the inertial frame; (b) rotation of the members with respect to the inertial frame; amplitude of the elastic modes (free-free). The said parameters are of different order of magnitudes, and therefore an adequate separation of them is highly desirable. Therefore, individual positions are replaced by overall position of the system (of the order of Earth's radius), and by the motion of each bar relative to it (of the order of members dimension), and for modes amplitudes modal equations are used. It should be noted, however, that the above-described motion parameters are redundant, and we must introduce: (a) reactions between members, (b) equations of compatibility of the same number of reactions. In summary, (ⅰ) the set of unknowns is: motion parameters, reactions, control forces; (ⅱ) the equations are equilibrium, compatibility, control. Control is introduced by prescribing the motion of some members, produced by control moments of forces. By simple matrix algebra, it is reduced to a system with motion parameters (overall + local) only. (2) In the Lagrangian approach, motion parameters are selected which are already consistent with compatibility conditions. In this case, as customarily, the expression of kinetic, potential, elastic energy is written, and the application of Lagrangian techniques provides directly the solving system. No reactions and compatibility equations appear here, however; for control purpose, prescribed motion law must again be introduced. Comparison of the two approaches shows perfect agreement (as one should have expected), since they are both exact models referring to the same physical system. In general, however, the Eulerian approach lends itself to a better understanding of physical facts, in particular, of the entity of the reactions and of the corresponding structural stresses.