Numerical Integration of the Equations of Motion

It was shown in Chapter 5 how the application of the laws of dynamics to constrained multibody systems leads to a set of differential algebraic equations (DAE). These can be transformed to second order ordinary differential equations (ODE) by proper differentiation of the kinematic constraint equations, by use of an independent set of coordinates, or by penalty formulations. A stable and accurate integration of both DAE and ODE is of great importance for the solution of the equations of motion. Although analytical solutions may be found for some simple cases, the number and complexity of the equations resulting from the majority of multibody systems requires numerical solutions. Because the theory of ordinary differential equations has been known for a long time, the stability, convergence, and accuracy of many methods have been studied in great detail. This has led to a wide use of these methods as compared to the differential algebraic equations, not so thoroughly known at this stage. As a consequence, many of the computer programs currently available for the computer-aided analysis and design of multibody systems rely on well-established methods for the solution of ODE.