On the dynamics in space of rods undergoing large motions — A geometrically exact approach

Simo, J. C.; Vu‐Quoc, L.

doi:10.1016/0045-7825(88)90073-4

Cited by 567 publications

(395 citation statements)

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“…In the field of MBS geometric integration, special attention is devoted to structure preserving methods that exploit rich geometric structure of rigid body rotational dynamics (see [7,16,25,58,67,73,84,85,87,89,90,97,104,135,146,147,152] and references cited therein). To this end, rigid body rotational dynamics is studied most conveniently as Lie-Poisson system that is defined on so * (3) (the dual space of so (3)).…”

Section: Geometric Integration Of Mbs Models In Absolute Coordinatesmentioning

confidence: 99%

Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

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Section: Geometric Integration Of Mbs Models In Absolute Coordinatesmentioning

confidence: 99%

Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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“…For the constitutive material behaviour, we choose a simple linear viscoelastic one that is called 'viscoelastic of complexity one' in [1,2,3]. The elastic parameters can be straightforwardly deduced from material and geometric ones [28,39]. Concerning the damping model, we note that it is macroscopic and phenomenological, it comprises not only pure material damping, but also miscellaneous damping mechanisms.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the dynamical analysis of fully nonlinear beams and rods in 3D is even today a challenging problem, both from the viewpoint of modeling and from the viewpoint of the efficient numerical solution of the resulting model equations [14,21,23,36,38,39]. In the present paper, we combine an objective/frame-indifferent geometrically exact space discretisation of Kirchhoff and Cosserat rods with standard methods for the time integration of the equations of motion for constrained mechanical systems [4,17,22].…”

Section: Introductionmentioning

confidence: 99%

“…We distinguish three basic types of classical rod models. In hierarchical descending order, these are the Cosserat, the extensible Kirchhoff and the inextensible Kirchhoff model [1,2,3,9,13,14,21,23,24,25,26,27,28,32,38,39]. Table 1 presents a short overview, including numerical problems in time integration to be discussed in this article.…”

Section: Introductionmentioning

confidence: 99%

“…(But we will see that this is actually not a severe drawback.) Our starting point for the continuous rod models is the exposition in [3,38,39]. For the constitutive material behaviour, we choose a simple linear viscoelastic one that is called 'viscoelastic of complexity one' in [1,2,3].…”

Section: Introductionmentioning

confidence: 99%

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Numerical aspects in the dynamic simulation of geometrically exact rods

Lang

Arnold

2012

Applied Numerical Mathematics

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Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrisecl by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests

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