On rigid components and joint constraints in nonlinear dynamics of ¯exible multibody systems employing 3D geometrically exact beam model Adnan Ibrahimbegovi  c a, * , Sa  õd Mamouri b a ENS Cachan/LMT, 61, av. du president Wilson, 94235 Cachan cedex, France b Department of GSM, Division MNM, Laboratory of G2MS, Compi  egne University of Technology, UTC, BP-529, 60206 Compi  egne, France Received 30 March 1999 Abstract In this work, we discuss the ®nite element implementation of the internal constraints in a three-dimensional (3D) geometrically exact beam model which can be formulated as holonomic constraint relationships. Model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component. Appropriate modi®- cations of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time-integration schemes for nonlinear dynamics. Consequently, only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved, thus avoiding the use of the extra variables (the Lagrange multipliers) and the pertinent diculties in integrating the system of dierential±algebraic equations. A number of numerical simulations considering dynamic analysis of multibody systems with rigid±¯exible components and joint constraints are presented in order to illustrate ver- satility of the proposed procedure. Ó 2000 Elsevier Science S.A. All rights reserved. Keywords: Flexible multibody; Dynamics; Constraints; Beam 1. Introduction Starting with the seminal works of Simo and Vu-Quoc [23,24] a number of publications have addressed the issues pertaining to statics and dynamics of geometrically exact beam model for ®nite rotations (e.g., see [3,5,6,12,15,18,20,25], among others). The key ingredients explaining the success of this model are its ability to provide ®nite strain measures which are valid for any size of displacements and rotations, as well as the simplicity in accounting for inertial effects with a chosen inertial frame. In this respect, the geometrically exact beam model represents a signi®cant departure from the previously proposed formulations for statics and dynamics of beams, such as the natural approach (e.g., [1]) or ¯oating frame approach (e.g., see [8]), which rely crucially on small strain assumption in order to decouple rigid body modes from deformation modes. The geometrically exact model is advantageous to these previous approaches when it comes to dynamics, since it eliminates a rather cumbersome Coriolis acceleration term and simpli®es the treatment of inertial effects. Despite this disadvantage, the ¯oating frame approach (yet referred to as Kane's approach of gener- alized speeds, e.g., see [21]) is still largely used in analysis of multibody systems, or mechanisms, where several components are joined together and where some of them can be considered as rigid (e.g., see a www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 188 (2000) 805±831 * Corresponding author. 0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 3 6 3 - 1