Mixed Eulerian–Lagrangian description in the statics of a flexible belt with tension and bending hanging on two pulleys

Abstract

Studying the mechanics of thin, axially moving strings, beams or plates (e.g.: belt drives, cable cars, . . . ) at mixed Eulerian-Lagrangian description, which features the transformation of material coordinates to spatial ones, is more appropriate than the classical material (Lagrangian) one, see [1,2]. Aiming at testing a newly proposed non-material finite element formulation we study the statics of a looped belt as a rod hanging in contact with two pulleys. Numerical experiments demonstrate rapid mesh convergence as well as correspondence to semi-analytical results and conventional Lagrangian finite element solutions.This is an open access article under the terms of the Creative Commons Attribution-Non-Commercial-NoDerivs Licence 4.0, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Coulomb dry friction contact in a non‐material shell finite element model for axially moving endless steel belts

“…This variational equation is accompanied by contact constraints for both normal and tangential contact, which may be written in the form of so called Kuhn-Tucker conditions [4]. It is worth noting that the augmented treatment of normal contact diverges in the given case due to Dirac-type jumps of the transverse force in the first contact points, see [2]. In order to improve accuracy, we further adopt an augmented Lagrangian strategy for tangential contact via introduction of an approximate Lagrange multiplier λ ⊥ , see [4].…”

“…Hence, we refer to [1,2] for details on the mixed E.L. kinematic description of structural finite elements. Hence, we refer to [1,2] for details on the mixed E.L. kinematic description of structural finite elements.…”

“…The article primarily focuses on how to properly account for the Coulomb dry friction contact between the belt and the rigid drums. Hence, we refer to [1,2] for details on the mixed E.L. kinematic description of structural finite elements. The main consequence of the E.L. approach is that the finite element nodal points cannot take part in the axial motion of the belt, but may only move in planes perpendicular to the circumferential coordinate σ.…”

“…In order to improve accuracy, we further adopt an augmented Lagrangian strategy for tangential contact via introduction of an approximate Lagrange multiplier λ ⊥ , see [4]. It is worth noting that the augmented treatment of normal contact diverges in the given case due to Dirac-type jumps of the transverse force in the first contact points, see [2]. Formulas (2) demonstrate how the contact contributions to the above variational principle are processed at every integration point.…”