Mechanics of lateral positioning of a translating tape due to tilted rollers: Theory and experiments

Abstract

Abstract A mechanics based model to describe the lateral positioning of a thin, tensioned, translating tape over a tilted roller is introduced, based on the assumption that the transport velocity of the tape should match the surface velocity of the roller when there is sufficient traction. It is shown that this condition requires the slope of the neutral axis of the tape and the slope of the centerline of the tilted roller to be the same over the wrapped segment. An extension of this model is discussed including the possibility of circumferential and lateral sliding, depending on the velocity difference between the tape and the roller. The new model is incorporated into a generalized model of a tape path that consists of numerous rollers as well as the appropriate boundary conditions for the take-up and supply reel dynamics. The nonlinear equation of motion is solved numerically, and the steady state solution is found by an implicit time stepping algorithm. An experimental setup with one tilting roller, two or three nearly ideally oriented rollers and two reels is used for verification of the model. The effects of roller tilt angle, tape wrap angle, and the lengths of the free-tape spans upstream and downstream of the tilted roller on the steady state lateral tape position are investigated experimentally and by simulations. The experiments show that the circumferential position of the wrap on the upstream side of a tilted roller has the strongest effect on pushing the tape in the lateral direction. The total wrap angle around the roller has a smaller effect. It was also shown that the tape segments upstream and downstream of the tilted roller interact, and the combined effect results in a different overall lateral tape response in steady state.