Web tension behavior in the presence of eccentric rollers: Modeling and validation

Since rotating machinery is used to transport the web on rollers, it is common to observe periodic oscillations in measured signals such as web tension and web transport speed. These periodic oscillations are more prevalent in the presence of non-ideal machine elements such as eccentric rollers and out-of-round material rolls. One of the main objectives in transport of webs is to maintain tension at a prescribed value. Tension regulation affects almost all key aspects of web transport including printing, registration, wrinkle formation, winding, etc. Therefore, models of web tension and web transport velocity in the presence of non-ideal rollers which can accurately predict measured behavior will be beneficial to the analysis of web transport under various dynamic conditions and in the design of suitable control systems.The focus of this paper is on modeling the effect of eccentric rollers on web tension. The governing equations for web velocity on an eccentric roller and web tension in spans adjacent to the eccentric roller are presented and discussed. To solve these governing equations, one requires the knowledge of the entry and exit point of the web on the eccentric roller as it rotates and the length of web spans adjacent to the eccentric roller; a method in obtaining this information is described. To corroborate the models and the developed approach, data from experiments on a web platform are compared with model simulations and results are presented and discussed.Mechanical and Aerospace Engineerin

Modeling and identification of the source of oscillations in web tension

Although there has been much work in dynamic modeling of different web handling elements and web longitudinal behavior, efforts to systematically validate models by experimentation on a web platform are non-existent. Existing literature has extensively used dynamic models for numerical analysis and/or design of control systems without adequate experimental validation of the models. One of the well-known modeling techniques for creating a model for the entire web line is based on the concept of primitive elements. In this approach every primitive element of the web line is modeled separately using first principles approach, and then the entire web line model is obtained by appropriately combining the primitive element models. The goal of this paper is to present results from recent investigations on validation of key primitive element models. Model refinement and modifications are also considered when sufficient level of agreement between model and experimental data was not obtained.Since the dynamic model for web tension in a span is nonlinear, designing experiments for web line model validation is a formidable task. There are a few known model validation techniques for nonlinear systems but these do not provide any clear procedures that can be applied to the web line. Therefore, the approach taken in this study was to consider test cases of experimentation that mimic typical web line operations in the industry such as acceleration/deceleration of the line and running the line at a constant speed. A number of test cases for model simulations and experimentation were considered. A representative sample of the results is shown and discussed.Data from the simulations of existing models did not contain the oscillations found in measured tension signals. This study also considered refinement and modifications of dynamic models that would lead to better agreement between the model and experimental data. It was found that span length variations introduced by out-of-round and/or eccentric rollers are the direct cause of oscillations in the tension signal. A refined model for web tension that includes span length variations is given. Comparison of the data from simulations of the refined model and experimental data shows a high level of agreement. These results are shown and discussed.Mechanical and Aerospace Engineerin

Computation of span length variations due to out-of-round material rolls

It is well known that non-ideal elements such as out-of-round/eccentric material rolls affect web tension. However, the mechanism through which these non-ideal components induce tension oscillations was not clear previously. In a companion paper (Modeling and Identification of the Source of Oscillations in Web Tension) it is shown that an out-of-round/eccentric material roll produces length variations in the web span adjacent to the roll. These length variations are the main reason for oscillations in the tension signal; this was experimentally verified in the companion paper. In order to reproduce these tension oscillations in model simulations, it was necessary to include span length variations in the tension dynamics models. Given a generic profile for the out-of-round unwind roll, determination of the length of the adjacent span as a function of time as material is released from the roll is a formidable task. The focus of this paper is on finding a relationship between the shape of the out-of-round material roll and length of the span adjacent to it.The simplest case to analyze is the length variations due to an eccentric roller. Considering the geometry of the problem, it is possible to find an expression in closed form that gives the length of the web span as function of the angular displacement of the roller. The expression for the rate of change of span length as a function of angular velocity is obtained by direct differentiation of the closed form expression. Finding closed form expressions for length of the span adjacent to an out-of-round material roll even for simple cases, such as an elliptical roll, is not trivial.As a starting point, an elliptical material roll is taken into consideration. To find the length of the web span between the material roll and the idle roller it is necessary to find the line tangent to both of them. An analytical approach to the problem did not provide any insights into finding a closed form expression for span length as a function of angular displacement of the material roll. To overcome this problem a convex optimization problem is formulated and an efficient numerical approach is developed to obtain the common tangent to the material roll and the first idle roller. Once the common tangent is obtained, span length and rate of change of span length can be found numerically as well. The algorithm and related pertinent discussions are given. Incorporation of this algorithm into web line model simulation software will enable better correlation of model and experimental tension data.Mechanical and Aerospace Engineerin

Modeling of laminated webs

A dynamic model of the longitudinal behavior of a laminated web is developed. A single web model that takes into account both thermal and hygral strains is developed first from first principles; the model assumes heat transfer in the region of wrap and free web span and moisture diffusion in the free web span. A classical one-dimensional heat equation is considered in the transverse direction to determine the heat transfer in the region of wrap. In the free web section, a lumped capacitance model is used to investigate heat transfer from the web surface. Moisture diffusion from the web surface is assumed to follow Fickian diffusion, which is used to determine hygral strain in the web. Mechanical and physical properties of a laminated web consisting of two isotropic webs of different material are stated using the rule-of-mixtures. The developed single web model and the laminate properties are used to derive a dynamic model for a laminated web span immediately downstream of the laminator rolls.Mechanical and Aerospace Engineerin

New governing equation for web tension by employing a Neo-Hookean material model

In this work, we derive a governing equation for web tension in a span by employing a Neo-Hookean material model that is applicable for transport of web materials under both small and large strains. This governing equation may be employed to study the evolution of tension within a span as well as propagation of tension variations from span to span as the web is transported in the machine. First, we find the stretch in a web span and relate it to web tension via a Neo-Hookean material model; the Neo-Hookean model is linear for small strain and nonlinear otherwise. Second, we conduct a dimensional analysis by defining several key coefficients that aid in grouping the machine and web material parameters separately in order to obtain a compact system of governing equations; this representation may be utilized to efficiently study the impact of web and roller properties on transport behavior

Lateral control of a web using estimated velocity feedback

The focus of this paper is on lateral control of a web using estimated motor velocity feedback. A reduced state velocity observer is designed to estimate the motor velocity based on the measured lateral position of the web and the motor input. Estimated velocity is used for inner-loop motor velocity feedback instead of measured velocity from a tachometer. Two approaches are investigated in the design of the reduced state velocity observer; the first is based on the motor dynamics and the web lateral dynamics and the second is based on the motor dynamics and the static gain of the web lateral dynamics. The second approach results in a simple low-order velocity observer when compared to the first approach.The proposed designs are experimentally investigated on a Fife remotely pivoted steering guide. The performance of the lateral control system with estimated motor velocity feedback is compared with the tachometer feedback and results are discussed. Representative experimental data from the two approaches indicated above is presented. Experimental results on the example considered shows that the observer can successfully replace the tachometer to close the inner velocity loop in lateral control systems.Mechanical and Aerospace Engineerin