Resonant Frequencies in Web Process Lines Due to Idle Rollers and Spans

This paper presents the results of a study on the resonant frequencies of a system of idle rollers and web spans. A simple linear model that describes the behavior of tension in a web span is considered. It is shown experimentally on a web platform that the simple model is capable of predicting resonant frequencies in an idler system consisting of idle rollers and spans between two driven rollers. This model is used to investigate the behavior of the minimum resonant frequency as a function of the number of idle rollers and span lengths. Analytical solutions for computing the minimum resonant frequency for the one-idler system (one idle roller and two spans) and the two-idler system (two idle rollers and three spans) are given. Discussions on how to maximize the minimum resonant frequency for two-, three-, and four-idler systems based on numerical analysis are also given. An analytical approximation of the minimum resonant frequency for equal span lengths and any number of idle rollers is derived. This approximation is a lower bound on the minimum resonant frequency and tends to converge to it, with the increase in the number of idle rollers. This is useful in getting a quick and reliable estimate of the minimum resonant frequency in festoons/accumulators which contain many idle rollers and web spans of equal lengths. The results of this study are expected to be useful to the web machine designer and developers of web tension control systems.