Motion Equations of Rigid Bodies Systems with Closed Kinematic Chains in Hamiltonian Variables

Abstract

Constructing methods of mathematical models for mechanical systems with closed kinematic loops are considered. All connections are holonomic. At the first stage, the mechanical system is reduced to a system with a tree structure by certain vertices (bodies) splitting in the original graph. The motion equations are constructed in matrix form with respect to Hamiltonian variables: generalized coordinates and impulses.  At the second stage, additional constraints are taken into account in the motion equations using Lagrange multipliers at the level of kinematic relations. These connections lead to closure of kinematic chains. The motion equations are formed according to recurrent formulas using a minimum set of primary information about the structure, geometric, kinematic, mass-inertial and power characteristics of a mechanical system. Iterative and finite algorithmsfor solving the obtained equations with respect to the integration variables are constructed. The wholeprocess of primary information preparation is demonstrated in the formation of a mathematical model in the proposed form on the example of one mechanical system with two closed cycles. The simulation results show that the algorithms described in the article provide a numerical solution that is more resistant to computational errors than methods based on the classical approach, in which the motions equation are closed by doubly differentiated equations of additional constraints.