We propose a non-material finite element scheme for modelling large deformations of a closed flexible rod supported by two rigid pulleys in the field of gravity. The mixed Eulerian-Lagrangian kinematic description of circumferential and transverse displacements is beneficial for simulations of moving belt drives. The necessary C 1 inter-element continuity in a compound coordinate system with Cartesian and polar domains requires a nonlinear finite element approximation. The theoretically predicted singular reaction force distribution prevents us from using the technique of Lagrange multipliers for normal contact. A novel semi-analytical solution of the static problem based on the integration of the equations of the nonlinear theory of rods in the free spans as well as in the segments of contact with pulleys is presented for the sake of validation. We demonstrate the mutual convergence of simulation results for a benchmark problem and additionally justify them by comparison against conventional Lagrangian finite element solutions.
We seek the steady state motion of a slack two-pulley belt drive with the belt modelled as an elastic, shear-deformable rod. Dynamic effects and gravity induce significant transverse deflections due to the low pre-tension. In analogy to belt-creep theory, it is assumed that each contact region between the belt and one of the pulleys consists of a single sticking and a single sliding zone. Based on the governing equations of the rod theory, we for the first time derive the corresponding boundary value problem and integrate it numerically. Furthermore, a novel mixed Eulerian-Lagrangian finite element scheme is developed that iteratively seeks the steady state solution. Finite element solutions are validated against semi-analytic results obtained by numerical integration of the boundary value problem. Parameter studies are conducted to examine solution dependence on the stiffness coefficients and the belt pre-tension.
This comprehensive review primarily concerns axially moving flexible structures in problems involving distributed structure-to-solid contact. The distinguishing features of axially moving structures are presented in terms of prevalent studies regarding models with simplified support conditions. Subsequent sections focus on the particular difficulties of treating contact problems with classical structural theories, on the appropriate non-material kinematic description for travelling structures, on the proper formulation of established mechanical principles for open systems and on the category of Arbitrary Lagrangian–Eulerian (ALE) approaches, which are frequently applied for the development of application-oriented finite element schemes. Novel analytical and numerical transient solutions for the benchmark problem of an axially moving beam, which is travelling across a rough surface between two misaligned joints, are presented to illustrate particular challenges as well as to highlight perspectives for future research activities. There are 177 references cited in this paper.
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