# Axially Moving Structures

## We are always on the move!

Various technical devices and engineering processes feature “axially moving structures”, that is: thin-walled parts, which move across a given domain of interest in a particular direction. Typical examples are: Belt and chain drives, conveyor belts or a moving strip of metal in a rolling mill. During operation the structure may stray from the desired path (usually the steady motion in axial direction) owing to certain mechanical phenomena, like: self-excited vibrations of a transmission belt, geometric imperfections of a steel conveyor belt or out-of-plane buckling of a metal strip during hot rolling. Such disturbances may seriously impact the quality of an industrial product, interrupt a continuous process or even result in a catastrophic failure of the axially moving structure.

In view of these technological challenges, the Research Group Mechanics of Solids focuses on mathematical modelling, simulation and model-based control design for axially moving structures. The research in this field builds upon our expertise in nonlinear mechanics of thin-walled structures, numerical simulation and control design as well as our extensive practical experience.

## Mathematical modelling

Neither the Lagrangian (or material) nor the Eulerian (or spatial) kinematic description are ideal when it comes to the mathematical modelling of axially moving structures. The material volume, which is residing in the considered spatial domain, is changing in time, and it is thus inefficient to follow mechanical processes in material particles. On the other hand, a truly spatial perspective is not well suited to capture the actual deformation of the structure contained in the control volume. The mixed Eulerian-Lagrangian description, proposed by the Research Group Mechanics of Solids, resolves this dilemma very effectively. Our team frequently applies this hybrid modelling strategy in fundamental and application-oriented research projects on axially moving structures.

## Examples and Publications

The vast amount of literature on the mechanics of planar belt drives provides a knowledge-base for reference as well as an opportunity to compare novel approaches against. In particular, the more recent contributions from the Research Group Mechanics of Solids on this topic focus on: transient dynamics, belt-to-pulley (structure-to-solid) contact interaction as well as finite element modelling in the mixed Eulerian-Lagrangian framework; see for example:

• Evgenii Oborin et al.; “Eulerian description of non-stationary motion of an idealized beltpulley system with dry friction”; International Journal of Solids and Structures 147 (2018); https://doi.org/10.1016/j.ijsolstr.2018.04.007
• Yury Vetyukov; “Non-material finite element modelling of large vibrations of axially moving strings and beams”; Journal of Sound and Vibration 414 (2018); https://doi.org/10.1016/j.jsv.2017.11.010
• Jakob Scheidl et al.; “Steady Motion of a Slack Belt Drive: Dynamics of a Beam in Frictional Contact With Rotating Pulleys”; ASME J. Appl. Mech. 87-12 (2020); https://doi.org/10.1115/1.4048317

Shell- and rod finite element models to both accurately and efficiently simulate the undesirable phenomenon of lateral run-off in a slack steel belt drive were developed in the framework of an industrial research cooperation project, see:

• Jakob Scheidl et al.; “Mixed Eulerian–Lagrangian shell model for lateral run-off in a steel belt drive and its experimental validation”; International Journal of Mechanical Sciences 204 (2021); https://doi.org/10.1016/j.ijmecsci.2021.106572
• Christian Schmidrathner et al.; “Non-material finite element rod model for the lateral run-off in a two-pulley belt drive”; Z. Angew. Math. Mech. 102-1 (2022); https://doi.org/10.1002/zamm.202100135

Industrial processes of continuous metal forming are another key topic of our research on axially moving structures; see for example our publications concerning metal rolling and sheet metal roll forming:

• Yury Vetyukov et al.; “Nonlinear model of an axially moving plate in a mixed Eulerian–Lagrangian framework”; Acta. Mech. 227 (2016); https://doi.org/10.1007/s00707-016-1651-0
• Yury Vetyukov et al.; “Mixed Eulerian–Lagrangian description in materials processing: deformation of a metal sheet in a rolling mill”; Int. J. Numer. Meth. Eng. 109-10 (2016);  https://doi.org/10.1002/nme.5314
• Emin Kocbay et al.; “Stress resultant plasticity for plate bending in the context of roll forming of sheet metal”; Int. J. Numer. Meth. Eng. 122-18 (2021); https://doi.org/10.1002/nme.6760

## Contact

TU Wien

Institute of Mechanics and Mechatronics

Research Unit of Mechanics of Solids

Getreidemarkt 9 / E325-02 / 5th floor

1060 Vienna

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