Index Reduction and Regularisation Methods for Multibody Systems

This paper presents regularisation methods for di_erential{algebraic equations of mechanical systems. These systems can be described via systems of di_erential and algebraic equations that usually have di_erential index three. Such systems are typically gained by the use of multibody simulation tools, where the system is composed of connected rigid and exible bodies and di_erent joints and the underlying equations are derived automatically by the software. The considered methods in this paper are applicable to di_erential{algebraic equations of index three and are divided into three basic approaches: index reduction with di_erentiation, stabilisation by projection and methods based on state space transformation. The methods using di_erentiation are the substitution of the constraint equations by derivatives, the Baumgarte- Method and the Pantelides-Algorithm with the use of Dummy Derivatives. Furthermore two methods using projection are considered: the orthogonal projection method and the symmetric projection method. The next approach uses a local coordinate transformation to reduce the index. Lastly the Gear-Gupta-Leimkuhler formulation is considered. At the end the advantages and disadvantages of all these methods are discussed and a basic outline on the functionality and the requirements for the implementation of each method is given.