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A discrete mechanics approach to Cosserat rod theory – Part 1: static equilibria

  • A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchho's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the conguration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null{space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates rst integrals of the equilibrium equations to Lie group actions on the conguration bundle, so{called symmetries. The symmetries relevant for rod mechanics are frame{indierence, isotropy and uniformity. We show that a completely analogous and self{contained theory of discrete rods can be formulated in which the arc{length is a discrete variable ab initio. In this formulation, the potential energy density is dened directly on pairs of points along the arc{length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identies exact rst integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application. Copyright c 2010 John Wiley & Sons, Ltd.

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Author:P. Jung, S. Leyendecker, J. Linn, M. Ortiz
URN (permanent link):urn:nbn:de:hbz:386-kluedo-16485
Serie (Series number):Berichte des Fraunhofer-Instituts für Techno- und Wirtschaftsmathematik (ITWM Report) (183)
Document Type:Report
Language of publication:English
Year of Completion:2010
Year of Publication:2010
Publishing Institute:Fraunhofer-Institut für Techno- und Wirtschaftsmathematik
Date of the Publication (Server):2010/07/21
Tag:Lagrangian mechanics; Noether’s theorem; Special Cosserat rods; discrete mechanics; frameindifference
Faculties / Organisational entities:Fraunhofer (ITWM)
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011