Simulation of mechanisms - Solving equations using software

Here we are in the downstream modelling phase, which consists in numerically integrating the mathematical models previously obtained (see files ). There are numerous integration methods: Euler, Runge-Kutta, Adams-Bashforth, Adams-Moulton, Backward Differentiation Formula, Gear, to name but a few. Certainly the most crucial problem associated with mechanism simulation is that of numerically solving systems of algebraic-differential equations (ADEs). In this case, simulation can be undertaken at the cost of a "mathematical reformulation" of the problem. Techniques such as coordinate partitioning, the projection method, Baumgarte stabilization or the penalty method can be used. In addition, the transformation of certain numerical methods (from their explicit to their implicit form, not to be confused with the explicit and implicit forms given to the model in ), can also be used to simulate algebraic-differential systems (Runge-Kutta Implicit -IRK- method, for example).

Throughout this section, we assume that the mechanism has no hyperstatic unknowns or, in other words, that the C matrix for the first-order binding equations is of full rank. Furthermore, n represents the number of generalized coordinates and L the number of binding equations, as in previous sections.