Eulerian description of non-stationary motion of an idealized belt-pulley system with dry friction

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

“…Besides ν, which denotes the transverse deflection, we have chosen the material displacements as the second major unknown. Usings rather than s for discretization is beneficial, as one avoids trouble with fulfilling the looping condition, as the material coordinate s jumps by L S where the two endpoints of the belt are connected, see [3]. The finite element approximations read…”

“…Seeking validation for a novel mixed Eulerian-Lagrangian (E.-L.) finite element scheme, we check its correspondence to semi-analytical results as well as classical Lagrangian type finite element calculations. In contrast to the latter type the mixed finite elements allow for locally refined meshes even if the belt is moving along its contour [3]. The initial stretch factor λ = L S /L G defines the material length of the belt in its straight reference configuration L S by its ratio to the geometric length L G = 2 R π + 2 H of the contour r G .…”

Flexible belt hanging on two pulleys: Contact problem at non-material kinematic description

“…Referring the interested reader to a detailed discussion of the voluminous literature on axially moving continua and Eulerian (spatial) kinematic description in structural mechanics to Chen (2005); Vetyukov et al (2016Vetyukov et al ( , 2017a; Vetyukov (2018), here we point out the intrinsic drawbacks of conventional finite element schemes with Lagrangian (material) kinematics when applied to the considered sort of problems as for example done by Dufva et al (2007). Letting the finite element mesh move between the qualitatively different domains, namely the free spans and the zones of sticking or sliding contact between the belt and the pulleys, one inevitably experiences numerically induced oscillations in the solution, see also the benchmark example in (Oborin et al, 2018). Moreover, material description implies regular re-meshing if a finer discretization is desired near the contact domains of a long belt drive, thus impacting the efficiency and accuracy of the analysis.…”

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

“…Hence, we refer to [1,2] for details on the mixed E.L. kinematic description of structural finite elements. Furthermore, less integration points leave or enter the contact zones during time integration, which prevents spurious oscillations that typically arise in a pure Lagrangian framework due to material transport [3]. This is particularly beneficial for the contact treatment as it allows for refined finite element meshes in the contact regions, see Fig.…”

“…2. Furthermore, less integration points leave or enter the contact zones during time integration, which prevents spurious oscillations that typically arise in a pure Lagrangian framework due to material transport [3]. Also, the contact state at a given point σ in space varies slowly for subsequent time increments, which promotes convergence.…”