Mechanism simulation - Topology, geometry, kinematics

Software such as ADAMS, DADS, MECHANICA, COMPAM, SYMPACK, etc. are no doubt well known. The various dossiers entitled "Simulation des mécanismes" (Mechanism simulation) to be published in Techniques de l'Ingénieur are designed to explain the theory behind them.

These are software programs that provide information on the motion and actions between solids in a complex system. Before the existence of today's powerful computing resources, putting a problem with even 5 or more degrees of freedom into equations was extremely tedious work. Solving it, even for very simple cases, proved even more difficult, if not impossible, and very often we were content to consider it in the very restrictive case of small movements around a known state of motion.

With the development of information technology, these questions can now be addressed. But it is useful to be aware of the pitfalls that can arise when using software of the type mentioned above, which can vary from one program to another depending on the principles they implement. That's the purpose of these files. On the other hand, very often an industrial company wishes to have complete control over its simulation tool, and builds its own for a very specific category of applications. Engineers will find most of the elements useful for this kind of work. All these elements are part of a scientific specialty known as "multibody systems theory". The best-known authors in this field are Haug, Nikravesh, Wittenburg, Schielen, Shabana...

There are two main categories of mechanism simulation software, depending on the type of coordinates used to describe the motion.

The former, often the oldest, but also, for this reason in particular, the most widespread, describe the position of each undeformable solid relative to the reference (or Galilean) frame by, in general, six coordinates, and each link gives rise to a number of link equations or constraints equal to six minus the degree of freedom of the link envisaged. In this way, all the equations of dynamics for each solid are in the same form. This systematic aspect, well-suited to programming, was the reason for the use of this description in the early days of multibody theory. On the other hand, knowledge of the equations of dynamics in literal form is of no interest. ADAMS and DADS fall into this category. So we can't expect them to provide equations that might help understanding.

The latter define the solids in relation to those that precede them in the kinematic chain, by a number of magnitudes (abscissae or angles) equal to the degree of freedom of the link that joins them together. This approach is more familiar to the mechanic of General Mechanics. As we shall see, the number of connection equations...