Keywords: axially moving beam, absolute nodal coordinate formulation, Bernoulli-Euler beam, frictional contact.

The present paper deals with the simulation of an axially moving strip using beam finite elements with a constant superimposed axial velocity. The finite element is developed in the framework of the absolute nodal coordinate formulation introduced by Shabana [1], which is suitable for large deformation elements. It is based on a third order Bernoulli-Euler element by Gerstmayr and Irschik [2], where nodal positions and slope vectors are used as degrees of freedom.

In contrast to conventional, straightforward computations involving axially transported strips, the considered beam finite elements do not move along the strip axis during the simulation, but keep a fixed undeformed configuration in time. To account for the inertia forces stemming from the transport of mass through the beam element, according terms are added to the standard equations of motion. These terms are derived from an extended form of Lagrange's equations, which was introduced by Irschik and Holl [3]. The proposed Lagrange-Eulerian approach leads to a structurally different, but equivalent formulation for a Bernoulli-Euler beam as given by the fundamental works of Vu-Quoc and Li [4] or Behdinan and co-workers [5]. With this technique, stationary solutions in the pre-critical velocity range can be computed by a quasi-static computation, which is much less time-consuming than a corresponding dynamic simulation. To include dynamic effects such as vibrations or flutter of the strip, a time-dependent scheme is used. Additionally, a contact formulation for modelling the frictional contact between an axially moving strip and rotating rolls is presented. In contact computations, the fact that the elements keep a fixed reference configuration is a major benefit. The definition of possible contact surfaces is alleviated greatly, as they remain fixed throughout the computation, and can easily be resolved by a finer finite-element discretisation of the strip.

1

A.A. Shabana, "Definition of the slopes and the finite element absolute nodal coordinate formulation", Multibody System Dynamics, 1(3), 339-248, 1997. doi:10.1023/A:1009740800463

2

J. Gerstmayr, H. Irschik, "On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach", Journal of Sound and Vibration, 318, 461-487, 2008. doi:10.1016/j.jsv.2008.04.019

3

H. Irschik, H. Holl, "The equations of Lagrange written for a non-material volume", Acta Mechanica, 153, 231-248, 2002. doi:10.1007/BF01177454

4

L. Vu-Quoc, S. Li, "Dynamics of sliding geometrically-exact beams: large angle maneuver and parametric resonance", Comput. Methods Appl. Mech. Engrg., 120, 65-118, 1995. doi:10.1016/0045-7825(94)00051-N

5

K. Behdinan, M.C. Stylianou, B. Tabarrok, "Dynamics of flexible sliding beams - non-linear analysis part I: formulation", Journal of Sound and Vibration, 208(4), 517-539, 1997. doi:10.1006/jsvi.1997.1167

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