Practical relevance, technical difficulties in maintaining the desired regime of motion, non-trivial and even sometimes counter-intuitive behavior are coupled with challenges, intrinsic for the theoretical investigation of axially moving structures, see Refs. [1,2,3] for one-dimensional strings and beams, see also [4,5] for plates. The mathematical models traditionally feature a spatial (or Eulerian) description with unknown displacements, forces, moments, etc. considered as functions of a fixed coordinate in the axial direction with the boundary conditions applied at given points. On the other hand, the basic equations of structural mechanics are available in the material (Lagrangian) form, when the mechanical fields are observed in material points [6].
The focus of the present talk lies in transforming the structural mechanics formulations to the spatial form using the principles of analytical mechanics and preserving the geometrically exact kinematic description. Problem specific non-material variational equations for finite deformations and vibrations of axially moving strings, beams and plates result in numerical schemes with material particles moving across the finite element mesh [5,7,8]. Further, we address a shell model and treat the challenging problem of an endless steel belt, moving between two rotating drums. Being used in various industrial applications (laminate production, continuous casting of polymer films, etc.), such belts tend to run off the drums in the lateral direction. The model-based control design requires an efficient and reliable mathematical model of the process. The dry friction contact, geometric imperfections and essentially nonlinear three-dimensional behavior make the latter problem particularly challenging for numerical analysis [9].
[1] L.-Q. Chen, "Analysis and Control of Transverse Vibrations of Axially Moving Strings", ASME Applied Mechanics Reviews, vol. 58, pp. 91-116, 2005.
[2] J. A. Wickert, "Nonlinear vibration of a traveling tensioned beam", International Journal of Non-Linear Mechanics, vol. 27, no. 3, pp. 503-517, 1992.
[3] V. Eliseev and Y. Vetyukov, "Effects of deformation in the dynamics of belt drive", Acta Mechanica, vol. 223, pp. 1657-1667, 2012.
[4] M. H. Ghayesh, M. Amabili, and Paїdoussis, "Nonlinear dynamics of axially moving plates", Journal of Sound and Vibration, vol. 332, no. 2, pp. 391-406, Jan. 2013.
[5] Y. Vetyukov, P. G. Gruber, and M. Krommer, "Nonlinear model of an axially moving plate in a mixed Eulerian-Largangian framework", Acta Mechanica, vol. 227, pp. 2831-2842, 2016.
[6] Yu. Vetyukov, Nonlinear Mechanics of Thin-Walled Structures, p. 272 (Springer, Wien, 2014).
[7] Y. Vetyukov, P. G. Gruber, M. Krommer, J. Gerstmayr, I. Gafur, and G. Winter, "Mixed Eulerian-Lagrangian description in materials processing: deformation of a metal sheet in a rolling mill", International Journal for Numerical Methods in Engineering, vol. 109, pp. 1371-1390, 2017.
[8] Y. Vetyukov, "Non-material finite element modelling of large vibrations of axially moving strings and beams", Journal of Sound and Vibration, vol. 414, pp. 299-317, 2018.
[9] J. Scheidl and Y. Vetyukov, "Coulomb dry friction contact in a non-material shell finite element model for axially moving endless steel belts", PAMM, vol. 19, no. 1, p. e201900260, 2019.
