# Establishment and Numerical Analysis of Rolling Force Model Based on Dynamic Roll Gap
## 3 Consider the Dynamic Roll Gap Cold Rolling Process Model
### Decoupling of Roll-flattening Radius and Rolling Force Model  
Based on the rolling force model and the roll-flattening radius calculation model, a coupling relationship between them can be observed. Although the results are typically obtained through simultaneous iterative calculations, the editing of the calculation program tends to be complex, and non-convergence may occur. Therefore, decoupling the two models can yield an explicit formula for the rolling force model.  
Equations (12) and (6) are substituted into Equation (5) and converted into the form of a roll-flattening radius, and replaced by the following equation:  
\[\mathrm{a}_{1}=\{\overline{K}-[\alpha\tau_{i}+(1-\alpha)\tau_{o}]\}B\sqrt{ \Delta h}\]  
\[\mathrm{a}_{2}=1.08-1.02\varepsilon\]  
\[\mathrm{a}_{3}=\frac{1.79\varepsilon\mu\sqrt{1-\varepsilon}}{\sqrt{h_{o}}}\]  
\[\mathrm{a}_{4}=\frac{2}{3}\sqrt{\frac{1-\
u^{2}}{E}\overline{K}\frac{h_{o}}{ \Delta h}}\{\overline{K}-[\alpha\tau_{i}+(1-\alpha)\tau_{o}]\}B\sqrt{\Delta h}\]  
where \(\mathrm{a}_{1},\mathrm{a}_{2},\mathrm{a}_{3},\mathrm{a}_{4}\) represent parameters independent of the roll-flattening radius, and the rolling force model is determined by the following equation:  
\[P=\mathrm{a}_{1}\sqrt{r}(\mathrm{a}_{2}+\mathrm{a}_{3}\sqrt{r^{\prime}})+ \mathrm{a}_{4}\sqrt{r^{\prime}} \tag{13}\]  
The roll flattening radius of Equation (8) is sorted out in a parametric way:  
\[\mathrm{b}_{1}=r\]  
\[\mathrm{b}_{2}=\frac{16(1-\
u^{2})r}{\pi EB\Delta h^{\prime}}\]  
It can be known that \(\mathrm{b}_{1}\) and \(\mathrm{b}_{2}\) are parameters independent of the rolling force, and the roll-flattening radius is determined by the following equation:  
\[rt=\mathrm{b}_{1}+\mathrm{b}_{2}P \tag{14}\]  
Through transformation Equation (14), obtain  
\[P=\frac{r^{\prime}-\mathrm{b}_{1}}{\mathrm{b}_{2}} \tag{15}\]  
Substituting Equation (15) into Equation (13), a quadratic equation with one variable about \(\sqrt{r^{\prime}}\) can be obtained as follows:  
\[(\mathrm{a}_{1}\mathrm{a}_{3}\mathrm{b}_{2}-1)(\sqrt{r^{\prime}})^{2}+( \mathrm{a}_{1}\mathrm{a}_{2}\mathrm{b}_{2}+\mathrm{a}_{4}\mathrm{b}_{2}) \sqrt{r^{\prime}}+\mathrm{b}_{1}=0\]If \((a_{1}a_{3}b_{2}-1)<0\), the equation has a positive solution, and the following equation determines the roll-flattening radius after decoupling:  
\[\sqrt{r}=\frac{-(a_{1}a_{2}b_{2}+a_{4}b_{2})-\sqrt{(a_{1}a_{2}b_{2}+a_{4}b_{2})^{ 2}-4(a_{1}a_{3}b_{2}-1)}}{2(a_{1}a_{3}b_{2}-1)} \tag{16}\]  
In conclusion, the rolling force can be separated via plugging Equation (16) into Equation (13), and you can obtain the rolling force display formula shown below:  
\[P=a_{1}\times\frac{-(a_{1}a_{2}b_{2}+a_{4}b_{2})-\sqrt{(a_{1}a_{2}b_{2}+a_{4}b _{2})^{2}-4(a_{1}a_{3}b_{2}-1)}}{2(a_{1}a_{3}b_{2}-1)}\]  
\[\times[(a_{2}+a_{3}\times\frac{-(a_{1}a_{2}b_{2}+a_{4}b_{2})-\sqrt{(a_{1}a_{2} b_{2}+a_{4}b_{2})^{2}-4(a_{1}a_{3}b_{2}-1)}}{2(a_{1}a_{3}b_{2}-1)})]\]  
\[+a_{4}\times\frac{-(a_{1}a_{2}b_{2}+a_{4}b_{2})-\sqrt{(a_{1}a_{2}b_{2}+a_{4}b _{2})^{2}-4(a_{1}a_{3}b_{2}-1)}}{2(a_{1}a_{3}b_{2}-1)}\]