Modeling and simulation of a laser deflecting system

In this report a model of Differential and Algebraic Equations (DAE) for the dynamics of a rotating laser deflecting system, which is a multibody system, is derived. Classical numerical integration software isn't however able to solve these equations, so the possiblity to transform the model into a set of Ordinary Differential Equations (ODE) is investigated. Elimination of the algebraic equations leads to an unmanageable model or yield a stiff numerical problem. This results in a very time consuming simulation. Therefore techniques to solve DAE system directly are examined. First some characteristics of DAE's like their index and structure are explained. It appears that multibody systems always result in an index three DAE. Then the possibilities classical (ODE based) techniques like Backward Differential and Runge-Kutta methods are discussed. Also special techniques for index three DAE's are mentioned: index reduction, Baumgarte stabilization, projection method, stabilizing Lagrange multipliers and state space transformation. The backward difference based code DASSL is used to simulate the rotating system. The results are accurate and obtained much faster than by the above mentioned simulation technique. This makes it possible to compare the behaviour of the nonlinear system to that of the linearized system that is also derived in this report. Only for very small angles the linear model is a good approximation. For large angles there are considerable differences. Furthermore the sensors of the deflecting system are discussed. It appears that these sensors can give rise to unaccuracy of the 3-D position measurement system in which the deflecting system is implemented.