Roll diameter configuration definitions:
Fig. 1 Winding Applications Roll Configuration
In winding applications the diameter at which Line acceleration ends is defined as RACC, and occurs near core. The diameter at which Line deceleration starts is defined as RDEC., and occurs near full roll.
Fig. 2 Unwinding Applications Roll Configuration
In unwinding applications the diameter at which Line acceleration ends is defined as RACC, and occurs near full roll. The diameter at which Line deceleration starts is defined as RDEC., and occurs near core.
The time domain equations that describe winding and unwinding of rolls are recursive and given below.
Where, assuming a perfect transmission system, and no clutch or roll winding slippage, the motor and roll rotational speeds are defined as:
The author has derived a simple set of exact analytic equations that solve the above recursive equations and provide the control engineer an equation set which can be used for calculating approximate winding and unwinding times and estimating the roll diameter at any point in time during the build-up/wind-down of the roll. This is accomplished without the need to solve (1) and (2) with a numerical integration based equation solver.
The paper is divided into four sections. The first section is composed of this introduction and the paper nomenclature, the second section defines the equation set, the third section provides some useful examples. Observations and conclusions are presented in the fourth section
The set of equations presented in this paper provide a framework for scheduling winding and unwinding operations in strip/web processing lines. The potential application of these equations is broad in scope. They can be used to optimize many aspects of winding, including providing guidance in selecting motors. It is left up to the reader to identify specific applications in which they may be applied, based on a given set of operational requirements. The author has employed these and other equations found in , , , ,  in a well received proprietary optimization program that configures recipes for winding and unwinding operations, while specifying the least expensive motor for the given application, and the specified design constraints.
The equations presented in this paper are not complete in so far as the effects of applying an "s-curve" to the speed reference are not considered. Also, the effects of air-entrainment, and radial pressure are ignored. Given that the metals industries do not usually employ significant "s-curved" speed references, and both air-entrainment and radial pressure phenomena are only slightly significant in foil applications, the equations presented here-in can be deployed in those applications with little or no error in the results.
One final caveat is that the accuracy of the equations is linearly dependant on the accuracy of the material thickness measurement. For example, if the thickness measurement is incorrect by 1 part in 100, the calculated radius will also be incorrect by the same amount at any instant in time.