A non-material shell finite element model is developed and applied to the example problem of a slack steel belt moving on two rotating drums. For the first time in the open literature we demonstrate an approach for predicting the time evolution of the lateral run-off velocity of the belt in response to its geometric imperfection and angular drum misalignment. We adopt a novel Eulerian–Lagrangian kinematic description featuring a mixed parametrisation of the configurational space with a Eulerian circumferential coordinate and two Lagrangian coordinates for the transverse and lateral deflections. A nonlinear finite element approximation provides the necessary $C^1$ inter-element continuity in this compound coordinate system. Using the model of elastic tangential contact, we account for the convective term in the local increments of the relative displacement between the contacting surfaces during the time integration. A thorough convergence study with respect to the mesh and time step sizes justifies the approach. Together with the successful validation against the results of a series of physical experiments, this makes the present contribution an important step towards a model-based controller design.

开发了一种非材料壳有限元模型，并将其应用于在两个旋转滚筒上运动的松弛钢带的示例问题。我们首次在公开文献中展示了一种方法，用于预测带的横向径流速度的时间演变，以响应其几何缺陷和角鼓未对准。我们采用了一种新颖的欧拉-拉格朗日运动学描述，其特征是配置空间的混合参数化具有欧拉周向坐标和横向和横向偏转的两个拉格朗日坐标。非线性有限元近似提供了必要的$C^1$该复合坐标系中的元素间连续性。使用弹性切向接触模型，我们考虑了时间积分期间接触表面之间相对位移的局部增量中的对流项。关于网格和时间步长的彻底收敛研究证明了该方法的合理性。连同针对一系列物理实验结果的成功验证，这使得目前的贡献成为迈向基于模型的控制器设计的重要一步。