Interaction between two web spans because of a misaligned downstream roller

Abstract

Fourth order differential equations for two web spans are solved, with the required eight boundary conditions reflecting conditions at the upstream end of the first span, the downstream end of the second span (at the misaligned roller), and at the roller between the two spans, where slippage occurs. Required relationships between the moment and lateral force at the slipping roller are derived from analysis of distributed frictional vectors across the web, along with the auxiliary variable Ks which is determined by the curvature and slope of the web at the intermediate roller. The solution of three equations with three unknowns is accomplished by iteration, then the three parameters are substituted into the solutions of the differential equations.

The patch of contact between the web and the intermediate roller is assumed to be rigid, as supported by many observations and photographs. The distributed loading on the intermediate roller is assumed to be uniform, but the effects of a nonuniform loading as imposed by a web wrapping the roller are examined and found to be little different from the effects of a uniform loading.

Besides identifying the conditions which lead to failure of a guide, interaction of spans is established as a potentially major source of erratic lateral behavior of a web, including a newly identified ''jump" phenomenon. Another result includes identification of conditions which would cause a slack edge.

Because of the potentially severe problems of lateral behavior when interaction occurs, simple methods for prevention of interaction are recommended for installation of steering guides. Also, tolerances for parallelism of stationary rollers for avoidance of interaction are recommended.

The results of the analysis compared favorably with existing experimental data for the case of the web wrapping the slipping roller, even when the steering was great enough to cause a slack edge.