New Spatially Dependent Models for Web Lateral Dynamics and Model-Based Designs for Control of Web Lateral Position and Slope

Abstract

The control of web lateral behaviour is important to obtain quality products and enable new technologies in R2R manufacturing. In this work, regulation of web lateral position and slope at any point within the span is investigated. For this, a new model for the web lateral behaviour rep­resented by spatially dependent lateral transfer functions was obtained, by including the entry rule directly in the boundary conditions and then applying a one dimensional (1-D) temporal Laplace transform to both the governing equations and boundary conditions. The proposed approach over­comes one of the key limitations of the existing methods which provide web lateral position only on the rollers. Determination of slope, moments, and shear forces within the span are also readily obtained by the proposed method, which was not the case in prior models. The approach also significantly simplifies the consideration of shear (relevant for short spans), in addition to bending, which has been found to be difficult to handle in the past model development studies. The new governing equations provide mechanisms to analyze web lateral behavior within spans, study prop­agation of lateral disturbances, and aid in the development of closed loop lateral control systems in emerging applications that require precise lateral positioning of the web. As an example of their application, common guide configurations are analyzed using the new model, and their influence in the span is presented. The determination of moments and shear may be useful for the research to study and predict wrinkles and other web failures.

Using the proposed model several controllers were developed and analyzed, and it is concluded that controlling the web lateral position and slope requires controlling the rotation and translation of the guide roller independently. Further, it is concluded that web slope is required as feedback. Currently available commercial web guides have only one control action that simultaneously con­trols the translation and rotation through a kinematic chain, thus incapable of web slope control. Since there are no available sensors to measure web slope directly, observers to estimate not only the web slope, but also the disturbances were developed. For this, it was assumed that only edge sensors (that measure web lateral position) are available; as results of this analysis conditions for observability, controllability, and determination of the minimum number of edge sensors and their location are also presented. The edge sensor location is based in a new approach that search for the minimum norm of the observer gains along the span. It is shown that less sensors than the estab­lished by previous studies are required for the slope estimation. Finally, a model based controller to obtain web lateral position and slope regulation that actively rejects the lateral disturbances in the system is developed. Results from numerical simulations of representative situations to support the developments and discussions are provided. An experimental platform and a device capable to displace and rotate a roller independently was developed. This new guide system was employed to validate the proposed spatially dependent lateral transfer functions, the effectiveness of the ob­servers, and verify that adding the the web slope as feedback in the controllers improves the web lateral behaviour.

This study is particularly relevant for high-precision lateral regulation within the span that may be required for emerging R2R applications in flexible and hybrid electronics, such as nanoimprint­ing, printing, and deposition processes.