INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 42, 1071–1090 (1998) ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS WITH SPATIAL BEAMS USING MIXED VARIATIONAL PRINCIPLES a B. M. QUADRELLI 1; *;† AND S. N. ATLURI 2; ‡ 1 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, U.S.A. 2 University of California, Los Angeles, CA 90095, U.S.A. ABSTRACT A general nite element formulation is presented for dynamic analysis of spatial elastic beams, for small strains, in a multi-body conguration. The tangent maps associated to the nite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form. A corotational method and a mixed variational method are used to compute the tangent stiness matrix. The tangent constraint matrices are obtained using consistent linearization of the constraint equations. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include: dynamic analysis of exible beam structures and multi-exible body systems with open and closed kinematic loops. A comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used. ? 1998 John Wiley & Sons, Ltd. KEY WORDS: exible multibody dynamics; nite rotation; beam nite elements; variational principles 1. INTRODUCTION This paper deals with the modelling of the spatial dynamic behaviour of homogeneous, isotropic and linear elastic one-dimensional deformable bodies, such as beams or rods, undergoing arbitrarily large rotations and translations, and small strains. The beam may be connected to other beams by means of kinematic constraints to form an open or closed-loop multi-exible body mechanical system. A previous paper, 1 of which the present one is a sequel, has investigated two dierent methods to compute the tangent stiness of the elastic beam. In particular, the elastic tangent stiness matrix and residual vector for a space beam were derived with two dierent approaches: a primal, Total Lagrangean (TL), corotational approach, and a mixed, Updated Lagrangean (UL) approach. The idea of that derivation was to formally adopt a variational setting to derive the eld equations of motion of constrained exible bodies. Previously, a consistent variational approach for multi-exible body dynamics with beams has only been presented by Cardona and Geradin. 2 In their work, they used the Principle of Virtual Work and methods of non-linear structural * Correspondence to: B. M. Quadrelli, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, U.S.A. E-mail: marco@grover.jpl.nasa.gov † Sta Engineer ‡ Director, Center for Aerospace Research and Education a This paper is presented to Prof. Franz Ziegler of the Technical University of Vienna, on the occasion of his 65th birthday CCC 0029–5981/98/061071–20$17.50 Received 10 December 1996 ? 1998 John Wiley & Sons, Ltd. Revised 6 February 1998