# Establishment and Numerical Analysis of Rolling Force Model Based on Dynamic Roll Gap
## 4 Numerical Analysis of the Rolling Force Model  
The accuracy of the newly developed rolling force model can be verified by conducting numerical simulations and field tests. The numerical calculation accuracy can be determined based on comparing the model predictions and the actual results. For instance, a field test was conducted on producing thin strip steel in a large cold rolling enterprise, as depicted in Figure 2. The industrial parameters related to material processing, such as strip width, inlet and outlet thickness, front and rear tension, reduction amount, and velocity, are provided in Tables 1 and 2. The simulation test is conducted using the data presented in Tables 1 and 2. By utilizing the numerical calculation process depicted in Figure 3, the results obtained from the new rolling force model are compared with the calculation results of the original rolling force model. In the rolling force measurement, due to the influence of various factors, to characterize the rationality of the measurement results, uncertainty factors such as instrument resolution, environmental conditions, and measurement methods are considered [21], and the data statistics are shown in Table 3. The comparison data, as shown in Table 4, are sorted out for analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rolling Pass** & **Measured Rolling** & **New Rolling Force** & **Original Rolling** \\
**NO** & **Force (kN)** & **(kN)** & **Force (kN)** \\ \hline
1 & 9370 \(\pm\) 0.00018 & 8976.25 & 8019.84 \\
2 & 10320 \(\pm\) 0.00018 & 9643.87 & 9024.54 \\
3 & 9900 \(\pm\) 0.00018 & 9708.86 & 9492.02 \\
4 & 8980 \(\pm\) 0.00018 & 8876.64 & 8388.03 \\
5 & 9500 \(\pm\) 0.00018 & 8304.13 & 8110.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical calculation results of each rolling pass. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rolling Pass** & **Inlet** & **Outlet** & **Post Tension** & **Front** & **Relative** & **Speed** & **Friction** \\
**Pass** & **Thickness** & **Thickness** & **\(\tau_{o}\) (MPa)** & **Tension** & **Amount of** & **“_u_ (m/s)** & **“_Coefficient_**\(\mu\)** \\
**NO** & **\(h_{i}\) (mm)** & **\(h_{o}\) (mm)** & **\(h_{o}\) (mm)** & **\(\tau_{i}\) (MPa)** & **\(\tau_{i}\) (MPa)** & **\(\tau_{i}\) (m/s)** & **“_u_ (m/s)** & **“_u_ \\ \hline
1 & 2.775 & 2.198 & 110 & 46 & 20.8 & 9.5 & 0.05 \\
2 & 2.198 & 1.790 & 140 & 110 & 18.6 & 12.3 & 0.05 \\
3 & 1.790 & 1.382 & 150 & 140 & 22.8 & 16.1 & 0.05 \\
4 & 1.382 & 1.056 & 180 & 150 & 23.6 & 20.4 & 0.04 \\
5 & 1.056 & 0.870 & 31 & 180 & 17.6 & 20.9 & 0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the rolling production site data of each rolling pass. Figure 3: Numerical calculation process. The numerical results indicate that the rolling force model proposed in this paper exhibits good consistency with the actual measurements, with an error range of approximately 8%. Compared to the original numerical calculation model that only considers elastic-plastic deformation, the proposed model aligns more closely with the actual rolling force observed in industrial production. These findings provide evidence of the established model's high prediction accuracy and ability to represent the cold rolling forming process accurately. The intermediate pass plays a crucial role in the stability of cold tandem rolling [22]. In this study, the focus is on intermediate passes. The relationship between the friction coefficient and the rolling force is depicted in Figure 4. It can be observed that the rolling force increases with an increase in the friction coefficient. However, beyond a friction coefficient of 0.2, the rate of increase in rolling force tends to decrease. While a smaller friction coefficient leads to energy savings, it also results in poor rolled surface quality due to the fluid dynamic state at the working interface of the roll and the rolled piece, as well as the lack of constraints on grain deformation on the rolled surface.