Mixed kinematic modelling and simulation of slack belt drives using structural theories of rods and shells

The cumulative thesis is concerned with the development of numerical simulation models for slack belt drives. For this sake, classic Lagrangian finite elements are inefficient, as they require uniform meshes or frequent re-meshing, and also prone to produce spurious numeric oscillations because of the ever recurring motion of nodal points across the contact zone boundaries. The mixed Eulerian–Lagrangian (MEL) kinematic description overcomes these deficiencies by means of a transformation that replaces the axial material coordinate with a spatial one, which is aligned with the primary direction of axial motion. In continuation of previous research, we apply the MEL approach for the first time to the benchmark problem of a two-pulley belt drive. The introduction of a problem-specific compound coordinate system with a looped spatial coordinate allows to decouple the gross axial motion of material particles in circumferential direction from the transverse deflections of the belt. Finite elements based on this novel description reside at fixed points of the looped Eulerian coordinate, while material is transported through the mesh in axial direction. We consider planar string and rod models of the belt and verify the finite element simulations against semi-analytic reference solutions, as obtained through numerical integration of the corresponding boundary value problems. A spatial parametrisation of the primary fields facilitates the deduction of the governing system of ordinary differential equations, which is later restated in a form accessible to standard purpose solvers. We seek slack static configurations of the belt in frictionless contact with the pulleys, simulate the quasistatic transient run up or compute the steady-state motion. In regard to the latter, a novel iterative concept is developed that avoids the time consuming simulation of the transient motion and enables a direct solution of the stationary problem in the finite element framework. The assumed Coulomb dry friction law amounts to a belt creep theory type of solution in the contact domains. In general, the iterative augmented Lagrangian multiplier method is the preferred strategy to enforce the contact constraints in the finite element schemes, but it fails in certain cases owing to the particularities of the contact response of the employed structural theories; then, the inherently simpler penalty regularisation method is used instead. In the framework of the industrial cooperation project LaLaBand, funded by the Austrian Research Promotion Agency, grant number 861493, we develop a shell finite element model in MEL kinematic description for the simulation of the phenomenon of lateral run-off in a slack steel belt drive with account for the geometric imperfections of the belt as well as the tilting motion of the steering drum. Coulomb dry friction is replaced with an elastic contact model for simplicity and increased robustness of the scheme. The construction of consistent finite element approximations in the MEL formulation for the shell requires the introduction of various extended shape functions to satisfy the C1 continuity condition. The successful validation of the shell finite element model against a series of physical experiments concludes the application oriented part of the research.