Implicit reduced involutive forms and their application to engineering multibody systems

Authors:
Wenqin Zhou;David J. Jeffrey;Gregory J. Reid;Chad Schmitke;John McPhee
Affiliations:
Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada;Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada;Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada;Systems Design Engineering, University of Waterloo, Waterloo, Ontario, Canada;Systems Design Engineering, University of Waterloo, Waterloo, Ontario, Canada
Venue:
IWMM'04/GIAE'04 Proceedings of the 6th international conference on Computer Algebra and Geometric Algebra with Applications
Year:
2004

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Abstract

The RifSimp package in Maple transforms a set of differential equations to Reduced Involutive Form. This paper describes the application of RifSimp to challenging real-world problems found in engineering design and modelling. RifSimp was applied to sets of equations arising in the dynamical studies of multibody systems. The equations were generated by the Maple package Dynaflex, which takes as input a graph-like description of a multibody mechanical system and generates a set of differential equations with algebraic constraints. Application of the standard RifSimp procedure to such Differential Algebraic Equations can require large amounts of computer memory and time, and can fail to finish its computations on larger problems. We discuss the origin of these difficulties and propose an Implicit Reduced Involutive Form to assist in alleviating such problems. This form is related to RifSimp form by the symbolic inversion of a matrix. For many applications such as numerically integrating the multibody dynamical equations, the extra cost of symbolically inverting the matrix to obtain explicit RifSimp form can be impractical while Implicit Reduced Involutive Form is sufficient. An approach to alleviating expression swell involving a hybrid analytic polynomial computation is discussed. This can avoid the excessive expression swell due to the usual method of transforming the entire input analytic differential system to polynomial form, by only applying this method in intermediate computations when it is required.