Computerized generation of motion equations using variational graph-theoretic methods

Severe tolerances on mechanical components have created increasingly stringent demands on the quality of new mechanical designs. The mathematical models used to simulate the various types of mechanical systems these days need to incorporate an optimization algorithm capable of minimizing the number of matrix multiplications when deriving symbolically the equations of motion. The method is based on a simplistic topological approach which is incorporated into an efficient variational graph-theoretic process used to solve these non-linear problems. The system is represented by a linear graph, in which nodes represent reference frames on rigid bodies, and edges represent components that connect these frames. By selecting a proper spanning tree for the graph, the analyst can choose the set of coordinates appearing in the final system of equation. The procedure casts, simultaneously, the Lagrange’s equations of motion and the kinematic constraints into a symmetrical format which yields a symbolic solution. The algorithm serves as a basis for a computer program which generates the equations of motion in symbolic form, and provides the time varying response of the system. The effectiveness of this approach is demonstrated in the analysis of a spatial four-bar mechanism and an articulated semi-trailer vehicle.