An Extending Dynamic Elastica: Impact With a Surface

The study of the dynamic transport of paper, film, and tape is of increasing importance as process speeds increase. In earlier machines, transport velocities were slow enough that inertia loads did not appreciably affect the deformation of the material. In many applications, material is pushed from a channel or clamp and can be modeled as a cantilevered beam or plate until it reaches another guide or barrier. In this paper, we are concerned with modeling the behavior of such a cantilevered sheet after it strikes a guide. The guide may be any two-dimensional curve. We are able to determine leading edge velocities and reaction forces necessary to keep the sheet from penetrating the guide. Friction is included in the model. The sheet is allowed to exit the channel at an arbitrary angle and may have a nonconstant transport velocity. Inertial loads become noticeable at speeds greater than one copy per second and are necessary for the resolution of impact loads.