Flexural–torsional modeling and equilibria of three-dimensional axially moving beams

The equilibrium equations for the coupled flexural–flexural–torsional–extensional mechanics of an axially moving beam with initial tension are studied. The form of the governing equations is discussed, and numerical solutions are presented for two types of boundary misalignment that both result in out-of-plane flexural–torsional equilibria. Solutions to the governing equations are obtained while retaining all non-linear terms. With all non-linear terms, the governing equations cannot be expressed in first-order form and appear to be missing boundary conditions, thus complicating the numerical solution. The results are compared to an existing reduced model, thereby confirming previously employed simplifying assumptions. Finally, an alternative form of the equations of motion is derived that are simpler and consists of only polynomial non-linearities. The new equations are shown to capture the behavior studied with only a small reduction in accuracy compared to other models.