The Vectorial Parameterization of Motion

Bauchau, Olivier A.; Choi, Jou-Young

doi:10.1115/detc2003/vib-48304

Cited by 43 publications

(89 citation statements)

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“…(iv) The number of unknowns is reduced from eight for (p, v) to seven for (p, ν). This is minimal on velocity level, but still singularity free -compared to any three-dimensional -or 'vectorial' [6] -parametrisation of SO (3), which necessarily must have singularities.…”

Section: Numerical Problems In Time Integrationmentioning

confidence: 99%

“…The first two curvatures correspond to vectorial parametrisations with the sine resp. tangent generator family in [6]. The third curvature choice can be considered as the limit case for κ → ∞ [32].…”

Section: Discrete Geometrically Exact Rodsmentioning

confidence: 99%

“…The choice of a discrete curvature measure is by no means unique. We propose several choices from literature [6,7,10,24,27,32,40], which can be expressed in terms of the material unit axis u n and the material angle ϕ n of difference rotation. Typical examples are…”

Section: Discrete Geometrically Exact Rodsmentioning

confidence: 99%

“…Of course, it is well known [6] -and most probably due to Euler. However, we give a very short and compact proof for the reader's convenience.…”

mentioning

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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Section: Numerical Problems In Time Integrationmentioning

confidence: 99%

Section: Discrete Geometrically Exact Rodsmentioning

confidence: 99%

Section: Discrete Geometrically Exact Rodsmentioning

confidence: 99%

“…Of course, it is well known [6] -and most probably due to Euler. However, we give a very short and compact proof for the reader's convenience.…”

mentioning

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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“…Their temporal discrete form can be derived and reduced according to the discrete null space method. This procedure has the advantage of circumventing the difficulties associated with rotational parameters [34] and it can be generalized easily to the modelling of geometrically exact beams and shells and to multibody systems consisting of theses structures as developed in [35][36][37].…”

Section: Introductionmentioning

confidence: 99%

Discrete mechanics and optimal control for constrained systems

Leyendecker

Ober‐Blöbaum

Marsden

et al. 2010

Optim. Control Appl. Meth.

130

105

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SUMMARYThe equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange-d'Alembert principle. This paper derives a structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem.

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On the Modelling of “Fork” Supports Using geometrically Exact Thin‐walled Beam Finite Elements

Gonçalves¹,

Ritto‐Corrêa²

2022

ce papers

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This paper discusses issues related to the modelling of so-called "fork" supports using geometrically exact 3D thin-walled beam finite elements whose rotations are parametrized using the rotation vector. This study is motivated by the fact that isolated members with "fork" supports are commonly employed in numerical and experimental studies, for the development of design rules for steel members. These supports are defined in the small displacement range, restraining the cross-section in-plane displacements and the torsional rotation of the beam (along the beam initial axis). This definition is usually extended to the large displacement case for beam finite element models, whereas in experimental tests and shell finite element models, a two-axis gimbaltype support is used instead. This paper thus aims at (i) showing how the two-axis gimbal support may be modelled using beam elements which are based on a rotation vector parametrization of finite rotations and (ii) highlighting the differences between the two support types. Numerical examples are presented to illustrate the differences, which are quite significant for moderate-tolarge displacements, but do not seem to affect the collapse load of standard steel members.

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The Vectorial Parameterization of Motion

Cited by 43 publications

References 0 publications

Numerical aspects in the dynamic simulation of geometrically exact rods

Numerical aspects in the dynamic simulation of geometrically exact rods

Discrete mechanics and optimal control for constrained systems

On the Modelling of “Fork” Supports Using geometrically Exact Thin‐walled Beam Finite Elements

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