Constrained dynamics of geometrically exact beams

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 equations which are discretized in time. The resulting energymomentum scheme satisfies the algebraic constraint equations on both configuration and momentum level. IntroductionThe present work deals with the dynamics of nonlinear beams in three-dimensional space. Modern finite element formulations for beams are inherently related to the approximation 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 nonobjective and path-dependent solutions, see Crisfield and Jelenić [12,20]. Moreover the use of rotational variables may further burden the discretization in time, see Jelenić and Crisfield [19] for an investigation of alternative timestepping procedures.The present work is based on the so-called 'geometrically exact beam theory' which has been at the heart of many previously developed beam finite elements, see, for example, [10,12,[15][16][17][18][19][20][21][22][25][26][27][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 constrained semi-discrete beam formulation in [8, Section 3.1] to the dynamic range.An outline of the remaining part of the paper is as follows. Section 2 contains a short summary of the computational treatment of constrained mechanical systems needed subsequently. Section 3 provides a first illustration of the present approach within the context of rigid body dynamics. 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. Numerical treatment of constrained mechanical systemsIn 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 byThe configuration of the mechanical system is characterized by the vector of coordinates qðtÞ 2 R n . The coordinates have to satisfy m holonomic constraint conditions of the formThese constraints restrict possible motions to a ðn À mÞ-dimensional constraint manifold Q ¼ fqðtÞ 2 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 irreduc...

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Flexible Multibody Dynamics Finite Element Formulation Applied to Structural Progressive Collapse Analysis

“…in Mamouri (2000 and, Ibrahimbegović et al (2003), Romero (2008), Ghosh andRoy (2009), Betsch andSteinmann (2003), Jelenić and Crisfield (2001) and Romero and Armero (2002). Most of those studies state that the Newmark time integration procedure cannot be used in nonlinear dynamics as a whole, however, finite rotation based methods are not the unique strategy to solve nonlinear dynamics.…”