# 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
### Rolling Force Prediction According to the Die Groove Shape  
To date, theoretical analyses of the pilgering process have not been comprehensively conducted. Obtaining information on the involved rolling forces is important for setting stress levels, necking and fracturing of materials, and establishing optimal process conditions. Although the finite element method has been widely used for analysis in this area, it is not easily applied in actual fields owing to the difficulty of constructing the finite element model and the long computation time it requires. In this study, we, therefore, propose a rolling force prediction model for the cold pilger rolling process using the Geleji equation [17]. The main variables of the predictive model are the product dimensions of the initial tube and the final product, the shape and dimension of the roll and mandrel, the pipe feed amount, and the pipe turn angle. The proposed method is relatively simple and enables easy identification of the influence of the main variables on the rolling forces. To simplify the problem of rolling force prediction, the following assumptions are made:  
**Assumption 1**. : _The elastic deformation of the material is ignored. Large deformation occurs during the forming process, and the proportion of the elastic deformation is insignificant in this process; thus, it can be ignored._  
**Assumption 2**. : _In cold pilger rolling, when the roll moves forward and then backward, the tube is fed in a forward direction by a certain feed amount and is simultaneously turned by approximately 30\({}^{\circ}\). The feed rate and rotation speed of the roll are independent of the tube deformation._  
**Assumption 3**. : _As shown in Figure 10, the file-up amount caused by rolling during the cold pilger rolling process is assumed to be \(d_{ox}=d_{o(x+ld)}\)._  
In this study, the Geleji equation was applied to predict the forming load during the cold pilger rolling process. The steps of the calculation procedure are outlined below. **1**. Enter the dimensions, working zone distance (mm), and the tube feed amount (mm) of the initial tube and final product as initial values: outer diameter of the initial tube (\(d_{01}\)), inner diameter(\(d_{11}\)), wall thickness(\(\delta_{1}\)); outer diameter of the pilger rolled product (\(d_{02}\)), inner diameter(\(d_{2}\)), wall thickness(\(\delta_{2}\)); instantaneous outer diameter of the pilger rolling product(\(d_{ox}\)), wall thickness(\(\delta_{x}\)); working zone(\(L\)); tube feed amount (\(m\)); tube outer diameter angle (\(\varphi\)), inner diameter angle (\(\psi\)); tube turn angle (\(\theta\)). **2**. Change the tube inner diameter/outer diameter angles during the pilger rolling process. \[\varphi\approx\frac{d_{o1}-d_{o2}}{2L} \tag{2}\]  
\[\psi=\frac{d_{i1}-d_{i2}}{2L} \tag{3}\]  
**3**. Calculate the feed amount, \(m\). Assume the number of times a point is in contact with the roll on the conical surface of the deforming tube (\(u=10\)-\(12\)). \[m=\frac{l_{1}}{u} \tag{4}\]  
Figure 10: Roll pressure on contact surface during cold pilger rolling process. In this case, \(l_{1}\) represents the minimum straight length of the initial tube required for the whole one revolution of the tube and is set as 40 mm consideration of the on-site roller specification. **4. ** Calculate the rolling force by using the surface pressure and contact area (Figure 10). \[\text{F}=\rho_{m}\left(\frac{d_{ox}+d_{o(x+l_{d})}}{2}\right)\cdot\frac{\pi}{2} \cdot l_{d} \tag{5}\]  
Here, \(\rho_{m}\) is the mean rolling surface pressure, \(l_{d}\) denotes the contact length, and \(d_{ox}\) and \(d_{o(x+l_{d})}\) represents the outer diameter of the tube before and after the contact point with the roll during the pilger rolling.