Nonlinear Dynamics 18: 357–382, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. A Fully Symbolic Model of Multibody Systems Containing Flexible Plates A. EL OUATOUATI, P. FISETTE ∗ , and D. A. JOHNSON Department of Mechanical Engineering, University of Louvain-la-Neuve, 1348 Louvain-la-Neuve, Belgium (Received: 5 February 1998; accepted: 29 September 1998) Abstract. The modelling of flexible elements in mechanical systems has been investigated via several methods issuing from both the field of multi-body dynamics and the area of structural mechanics and vibration theory. As regards the multibody approach, recursive formulations in relative coordinates are quite suitable and efficient for a large variety of applications. Such a formalism is developed here for a general multibody system containing flexible plates and in such a way that its full symbolic generation is possible within the ROBOTRAN program [1]. Keywords: Multibody dynamics, flexible plate, symbolic implementation. 1. Introduction The modelling of flexible plates in mechanical systems has been investigated in the liter- ature via several methods issuing from both the field of multibody dynamics and the area of structural mechanics and vibration theory (see, for instance, [2]). A multibody approach using a recursive formalism in relative coordinates is adopted here. While leading to a highly non-linear system with a dense mass matrix, relative coordinates allow the setting up of the minimal set of equations of motion for open-loop systems. Such a formalism is developed in this contribution for a general multibody system containing flexible plates and in such a way that its fully symbolic generation is possible within the ROBOTRAN symbolic program [1]. Both for flexible beams and for flexible plates, previous contributions [2, 15] have already proposed rigorous formalisms in which the flexible elements undergo large overall motion: in the present contribution, which follows a previous work related to the case of flexible beams [3], we also aim at modelling flexible elements undergoing arbitrary motions, but a pure multibody approach using relative coordinates is taken here: besides the theoretical aspects of the model, its implementation in a fully symbolic form is discussed and has been introduced in the multibody program ROBOTRAN [1], initially developed for rigid multibody systems. Using the same philosophy as in [3] and [6], the way the equations of motion are formulated clearly distinguishes between the joint equations and the deformation equations of the var- ious bodies. The recursive computation approach, which has proved to be most appropriate when considering relative joint variables, is all the more advantageous in the present context that specific symbolic procedures in ROBOTRAN are able to optimize the computation of any recursive scheme by removing the superfluous equations and by vectorizing the scheme. Moreover, the program has been completed with new modules of symbolic derivation which are able to compute in a purely recursive manner the tangent matrices of the system with ∗ Chercheur qualifi´ e FNRS.