Vibration and guiding of moving media with edge weave imperfections

This paper examines the steady-state-forced vibration of a moving medium that is guided by a partial elastic foundation, and where geometric imperfections on the mediumʹs edge act as an excitation source. Such a system is of technical interest in the areas of web handling and magnetic tape transport where externally pressurized air bearing guides are sometimes used to control lateral position. The axially moving strip is modeled here as a string that is guided by elastic foundation segments, and that is subjected to traveling wave excitation as the edgeʹs imperfection interacts with the foundation. The equation of motion for this “moving medium and moving load” system incorporates a skew-symmetric Coriolis acceleration component that arises from convection. The governing equation is cast in state-space form, with one symmetric and one skew-symmetric operator, as is characteristic of gyroscopic systems. Through modal analysis, the forced response of the system is obtained to the complex harmonic excitation associated with the interaction between the edgeʹs weave pattern and the guides. Parameter studies are presented in the transport speed, foundation stiffness, guide placement, guide width, and imperfection wavelength. Of potential technological application, for a given wavelength of the edgeʹs imperfection, it is possible to reduce the mediumʹs vibration at a certain location by judiciously selecting the locations and spans of the foundation segments.