# Profile Design of the Grooved Die and Rolling Force Prediction in the Cold Pilger Rolling Process
## 2 Rolling Force Prediction in the Cold Pilger Rolling Process
### Main Process Variables in Rolling Force Prediction  
The cold pilger rolling method employs a pair of rolls to reduce the outer diameter of the tube while proceeding back and forth in the working zone. The rolls are also employed to control the inner diameter of the tube using the mandrel. At this stage, it is important to select the main process variables to predict the rolling force [15; 16]. The main process variables are listed in Table 2.  
The feed amount (\(m\)) of the tube is an important process variable that affects the material production speed and the rolling force during the pilger rolling process. Figure 6 shows the tube feed in this process. To calculate the feed amount of the tube, the number of times (\(u\) = 10-12 times) that a point contacts the roll on the conical surface of the deforming tube and turn angles (\(\theta\) = 30-36\({}^{\circ}\)) are used, as shown in Figure 7. Otherwise, the surface status of the tube is unclear. In addition, the tube is set to rotate at least ten times in the working zone. The principal deformation zone in the cold pilger rolling is the conical shape part of the tube with a diameter that increases during rolling. At the end of each stroke, the mandrel is moved forward by a constant feed amount. The tube feed amount, which is the first process variable, was set to 2 to 6 mm, which is the practically applicable range based on the roller specifications and field conditions.  
The second process variable is the reduction area (\(R.A.\)). The \(R.A.\) is calculated by Equation (1) using the difference of the cross-sectional area of the material before and after the pilgering.  
\[R.A.=\frac{A_{o1}-A_{o2}}{A_{o1}}\qquad[\%] \tag{1}\]  
Here, \(A_{o1}\) and \(A_{o2}\) are the material cross-sectional areas before and after the pilgering. In the pilgering process, the \(R.A.\) has a significant impact on the pilger die shape, forming load, and final product quality. In general, if the \(R.A.\) is large at the forming end with respect to the center of the working zone, it may cause mandrel breakage or deterioration of the dimensional accuracy of the pilgered product. Therefore, the appropriate setting of the \(R.A.\) is a critical task.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Value** \\ \hline Feed amount (\(m\)) & 3 mm \\ Turn angle (\(\theta\)) & 30\({}^{\circ}\) \\ Reduction area (\(R.A.\)) & 70\% \\ Roll radius (\(R\)) & 303 mm \\ Working zone (\(L\)) & 380 mm \\ Principal deformation zone (\(PDZ\)) & 190 mm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input parameters for calculating rolling force.  
Figure 5: Schematic description of cold pilger rolling process.  
In this study, the die groove was designed to situate most of the \(R.A.\) in the principal deformation zone (\(PDZ\)), as shown in Figure 8. In the \(PDZ\), the \(R.A.\) range was set to 50-80% with consideration of the pilger rolling characteristics [1]. The third process variable is the roll-die radius \(R\) and die-groove angle \(\varphi\), as shown in Figure 9. Both variables are highly dependent on the die groove shape, and the roll radius is determined by the roller specifications.  
Figure 8: Principal deformation zone of cold pilger rolling process.  
Figure 6: Tapered cross-section between rolls and mandrel during cold pilger rolling process.  
Figure 7: Cross-section view of the rolling tube and tools at a given position on the z-axis.  
Figure 9: Process variables of cold pilger rolling process.