Regulating the lateral position of the web material in a roll to roll machine is critical for a wide variety of converting applications. Variations in lateral disturbances can result in substandard finished product, or in extreme cases may result in material breakage which leads to machine downtime. The lateral variations are primarily caused due to roller misalignment, web splices, web thickness variations, roller thickness variations, poor quality input roll, etc. Even with a perfectly aligned machine with well machined rollers, poor quality raw material (with thickness variations or with wound-in edge variations) will result in lateral variations. It is necessary to control these variations actively using a closed-loop feedback control system. In order to design a high performance closed-loop feedback control system it is essential to understand the lateral dynamics of the web on the rollers.
Lateral Dynamics of the Web on Rollers
The lateral dynamics of the web between two rollers is modeled based on Bernoulli beam theory developed by Dr. John Shelton. The lateral displacement is mainly caused due to roller misalignment, web splices, web thickness variations, roller thickness variations, etc. The lateral dynamic model developed in article titled Optimal Web Guiding considers the effect of roller misalignment, roller displacement and web displacement on the lateral dynamics. And the boundary conditions are introduced to extend the model for multiple spans.
The dynamics model is derived by introducing a few assumptions. Some of these assumptions may not be satisfied in a real industrial conditions, which makes the case for active control of lateral position using web guides.
- Web is elastic
- Lateral dynamics is due to bending and effect of shear is neglected
- Traction is maintained between the web and the roller
- Web thickness variation is negligible
- Rollers are perfectly round
Simulation results with the following transport conditions are shown in this white paper.
Effect of Web Displacement between Two Fixed Rollers
The dynamics of the web between two fixed rollers (that are aligned properly) is mainly influenced by the lateral displacement seen at the upstream roller. The lateral disturbances can be caused by various reasons, such as, web splices, web thickness variations, roller misalignment, etc., and in this white paper we see the propagation of disturbances in a span between two fixed rollers. Frequency domain Bode plots are used to illustrate the transport of disturbances in a span between two fixed rollers. Figure 1 shows the fixed span under consideration with the input being the lateral misalignment at the upstream roller and the output being the lateral displacement at the end of the span measured at the downstream roller.
The frequency response of this span is shown in Figure 2. The top plot is the magnitude plot and the bottom plot shows the phase plot. The transfer function that describes the relation between the web position at an upstream roller (input) and the web displacement at the downstream roller (output) within a single span is available in the Optimal Web Guiding paper. As seen from this plot the bandwidth of the system (lateral dynamics) is just over 10 Hz. Note that the system is a non-minimum phase system because of the right half plane zeros in the transfer function.
As the parameters move away from the nominal values the bandwidth of the lateral dynamics is changed. Ideally one would desire to have a low bandwidth system so that the web absorbs the high frequency disturbance from being propagated to downstream spans. Hence it is necessary to understand the effect of various web parameters that increase or decrease the bandwidth of disturbance propagation.
Change in web tension and modulus of the web, within the range listed in the table above, have no significant change in the frequency response 3 . However, as the transport velocity changes there is a significant variation in the bandwidth of the system as seen in Figure 3 (4 Hz to 18 Hz as the speed changes from 5 m/s to 20 m/s).
Lateral Dynamics on an Offset-Pivot Guide
In order to actively correct lateral disturbances, the dynamics of the web on the guide (an offset-pivot guide or a displacement guide as seen in Figure 4 is typically used for guiding applications) must be well understood. Depending on the magnitude and frequency of the disturbance the guide mechanism and the sensor should have the required bandwidth.
One factor that needs to be considered is the magnitude of understeering or oversteering of the web based on the guide installation (position of the pivot axis). For the nominal conditions listed in the previous section, a guide installation can result in understeering of about 33% to oversteering of about 50%. Typically offset-pivot guides are installed for understeering or neutral steering configuration (the location of the pivot point dictates the steering condition, where x1 is the distance from the the downstream guide roller as shown in Figure 5). And ideally it is desired to have neutral steering where the offset-pivot guide is a true displacement guide.
Figure 6 shows the Bode plot for an understeering configuration (x1 = 350 mm and L = 250 mm) for various transport speeds with the nominal values for the rest of the parameters. The plot represents the magnitude and phase response of a system with a transfer function whose input is the guide displacement and the output is the lateral position of the web on the guide roller.
For an understeering configuration the ratio of web displacement to the guide displacement is less than one and it reaches one as the transport speed increases. Likewise the ratio of web displacement to guide displacement is more than one for oversteering configuration. The understeering and oversteering configuration is achieved by the guide mechanism design. For displacement guides these conditions cannot be achieved by guide installation.
In summary the lateral dynamics of the web is significantly influenced by the speed of transport and the web span parameter (a function of Young’s modulus of the web, span length and the web tension in the span). As the speed increases or as the span length reduces more and more disturbances are propagated within the web guiding system. At lower speed or with longer spans these lateral disturbances are absorbed by the web. With a well designed displacement guide it is possible to obtain a pure displacement action (neutral steering) with the same bandwidth for lateral correction as the actuator bandwidth. If need be an over steering option can also be designed for some applications.