Constrained dynamics of geometrically exact beams P. Betsch, P. Steinmann Abstract Geometrically exact beams are regarded from the outset as constrained mechanical systems. This viewpoint facilitates the discretization in space and time of the underlying continuous beam formulation without using rotational variables. The present semi-discrete beam equations assume the form of differential–algebraic equa- tions which are discretized in time. The resulting energy– momentum scheme satisfies the algebraic constraint equations on both configuration and momentum level. Keywords Nonlinear finite elements, Constrained mechanical systems, Structural dynamics, Multibody systems, Energy–momentum methods 1 Introduction The present work deals with the dynamics of nonlinear beams in three-dimensional space. Modern finite element formulations for beams are inherently related to the ap- proximation of finite rotations in space and time, see, for example, Crisfield [11, Chapter 17]. However, finite elements based on the space interpolation of rotational variables may be afflicted with problems such as non- objective and path-dependent solutions, see Crisfield and Jelenic ´ [12, 20]. Moreover the use of rotational variables may further burden the discretization in time, see Jelenic ´ and Crisfield [19] for an investigation of alternative time- stepping procedures. The present work is based on the so-called ‘geometri- cally exact beam theory’ which has been at the heart of many previously developed beam finite elements, see, for example, [10, 12, 15–22, 25–28]. All of these works make use of rotational variables in the discretization process. In contrast to that, the present discretization approach does not rest on rotational variables. This is achieved by regarding nonlinear beams from the outset as constrained mechanical systems. In particular, we extend the con- strained semi-discrete beam formulation in [8, Section 3.1] to the dynamic range. An outline of the remaining part of the paper is as fol- lows. Section 2 contains a short summary of the computa- tional treatment of constrained mechanical systems needed subsequently. Section 3 provides a first illustration of the present approach within the context of rigid body dynam- ics. Then in Sect. 4 the present approach is extended to the semi-discrete beam formulation. Numerical examples are given in Sect. 5 and conclusions are drawn in Sect. 6. 2 Numerical treatment of constrained mechanical systems In this section we consider finite-dimensional mechanical systems subject to holonomic constraints. For the present purposes it suffices to restrict our attention to autonomous systems. Within the Lagrangian framework of classical mechanics, the motion of a constrained mechanical system can be described by d dt oL o _ q   oL oq ¼ X m b¼1 k b rU b ðqÞ ð1Þ The configuration of the mechanical system is character- ized by the vector of coordinates qðtÞ2 R n . The coordi- nates have to satisfy m holonomic constraint conditions of the form U b ðqÞ¼ 0; b ¼ 1; ... ; m ð2Þ These constraints restrict possible motions to a ðn  mÞ- dimensional constraint manifold Q¼fqðtÞ2 R n jU b ðqÞ ¼ 0; b ¼ 1; ... ; mg. Accordingly, the constraint manifold Q is regarded as embedded in a real, n-dimensional vector space. The constraint functions U 1 ; ... ; U m : R n ! R are assumed to be irreducible, i.e. the gradients rU b ðqÞ are linearly independent for q 2Q. The constraints are enforced by means of b ¼ 1; ... ; m Lagrange multipliers k b ðtÞ2 R. They account for the forces of constraint, given by the expression on the right-hand side of (1). The La- grangian L : R n  R n ! R is assumed to be of the form Lðq; _ qÞ¼ 1 2 _ q  M _ q  V ðqÞ ð3Þ where V : R n ! R is a potential energy function and M is a n  n mass matrix. For the standard case of a regular Lagrangian, o 2 L=o _ qo _ q ¼ M is non-singular. In view of our numerical treatment of the differential- algebraic equations (DAEs) in (1) and (2) the corresponding Hamiltonian form is more appropriate. The passage to the Hamiltonian formulation rests on a change of the variables ðq; _ qÞ to the phase space variables ðq; pÞ, where pðtÞ2 R n is the conjugate momentum defined by Computational Mechanics 31 (2003) 49–59 Ó Springer-Verlag 2003 DOI 10.1007/s00466-002-0392-1 49 P. Betsch (&), P. Steinmann Chair of Applied Mechanics, Department of Mechanical Engineering, University of Kaiserslautern, Germany e-mail: pbetsch@rhrk.uni-kl.de Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.