Ke models: Theoretical dynamic subsystems of longitudinal web strain

Abstract

Efficient design and optimization of many production processes often require models which predict transient and steady state web strains. To date, much attention has been given to modeling web strains much less than unity. Considerably less attention however, has been given to modeling strains of relatively lower modulus materials. A particular related challenge often involves selection of dancers vs. load cells as feedback devices in tension control systems.

This paper explores derivations of theoretical "Ke" models as primitive functions of roller motions. At a fundamental level, simple linear and nonlinear differential equations exist for each strain component or "subsystem" independent of others. Combinations can determine total strains in web spans including those at inputs and outputs of dancer rollers and within festoons. Validity is retained at any value of strain including zero and negative values (compression). The author demonstrates that mathematical equations of high web strains instead of becoming unwieldy, can be applied with accuracy and with a large degree of natural elegance. Applied classical control theory allows users a natural intuition when interpreting results which are primarily outputs of computer simulations.

The "free web span" has been extensively studied within the web handling community and is again examined here as a 1st section of web under any dynamic strain feeding into a 2nd section of web between two driven rollers. A free web span Ke based model is compared to a first order approximate model of the same physical system while applying step changes to roller velocities. Both models are compared as final values of strain approach extremely high values toward infinity. Using Ke models, all strain-time trajectories in the free web span as a result of step changes to roller velocities are shown to be sections of an S-shaped curve designated "The Universal Strain Time Curve". The output of the first order approximate model, when plotted on the Universal Strain Time Curve (USTC), reveals that the first order approximate model may often be applied with acceptable results for strains from 0 through 25%. Finally, an example model of a tension control system with load cell feedback demonstrates how consecutively higher order subsystems may be included as elements of a Ke Subsystem Library.

A practical and intuitive method of modeling web strains of any value has been developed here and may be applied by scientists and engineers having a basic knowledge of classical control system theory. With relatively accurate input data, effects on strain resulting from various roller inertias, web span lengths, dancers vs. load cells, and many other design decisions can be simulated. For both high and low modulus materials, Ke models provide a high degree of accuracy when simulating web strains during process design and optimization. This research is applicable to a broad spectrum of webs from thin plastics to paper, textiles, flat metals, wires, films, belts, foils, strips, threads, fabrics, and composites which are manufactured in rolling processes. The academic derivation process which has been applied also reinforces a useful framework to solve similar scientific problems.