## 2 Tension Modeling and Its Analysis  
### Tension Modeling  
A demonstration of tension application during the rolling process is shown in Figure 2. A is the exit clamp center. B is the steady rolling zone center. C is the entry clamp center. \(l_{2}\) is the transient distance between B and C. \(x_{\text{a}}\), \(x_{\text{b}}\), and \(x_{\text{c}}\) are, respectively, the displacements of A, B, and C. \(T_{\text{h}}\) is the forward tension. \(T_{\text{H}}\) is the backward tension. \(h\) is the exit thickness. \(H\) is the entry thickness.  
It can be concluded from the geometric relationship that the strain of the strip under the increment of forward tension is  
\[\varepsilon=\frac{x_{\text{a}}-x_{\text{b}}}{l_{\text{0h}}+x_{\text{b}}} \tag{1}\]  
where \(l_{\text{0h}}\) is the equivalent original length of the workpiece on the exit side.  
According to [PERSON]'s law, when the material undergoes elastic deformation, the stress and the strain satisfy the equation  
\[T_{\text{h}}-T_{\text{0h}}=E\varepsilon A_{\text{h}} \tag{2}\]  
Figure 2: Demonstration of tension loading.  
where \(E\) is the elastic modulus of the material; \(T_{\rm 0h}\) is the initial forward tension before rolling; \(A_{\rm h}\) is the cross-sectional area of the strip on the exit side.  
According to the Delhi Princeton formula, forward slip can be expressed as  
\[S_{\rm h}=\frac{\gamma^{2}}{h}R \tag{3}\]  
where \(\gamma\) is a neutral angle; \(R\) is the roller radius.  
The neutral angle is obtained according to the force balance condition of the rolling zone [14]  
\[\gamma=\frac{\alpha}{2}(1-\frac{\alpha}{2f})+\frac{T_{\rm h}-T_{\rm H}}{4 \overline{p}fwR} \tag{4}\]  
where \(f\) is the friction coefficient; \(w\) is the width of the strip; \(\overline{p}\) is the average rolling pressure; \(\alpha=\cos^{-1}(\frac{R-\Delta h}{R})\); \(\Delta h\) is the pass reduction.  
By substituting Equation (4) into Equation (3), the following equation is obtained:  
\[S_{\rm h}=\frac{\left[\frac{\alpha}{2}(1-\frac{\alpha}{2f})+\frac{T_{\rm h}-T_ {\rm H}}{4\overline{p}fwR}\right]^{2}}{h}R \tag{5}\]  
According to the definition of the forward slip, the exit speed is  
\[v_{\rm b}=v(1+S_{\rm h}) \tag{6}\]  
where \(v\) is the rolling speed of the mill.  
Integrating both sides of Equation (6), \(x_{\rm b}\) can be obtained as  
\[x_{\rm b}=\int_{0}^{t}v(1+S_{\rm h})\ \mathrm{d}t \tag{7}\]  
By substituting Equations (1) and (7) into Equation (2), the following equation is obtained:  
\[T_{\rm h}=\frac{A_{\rm h}E}{l_{\rm 0h}+\int_{0}^{t}v(1+S_{\rm h})\ \mathrm{d}t}[x_{\rm a}-\int_{0}^{t}v(1+S_{\rm h})\ \mathrm{d}t]+T_{\rm 0h} \tag{8}\]  
Combining Equation (8) with the principle of metal second flow being equal, the back slip model is obtained. The backward tension differential equation can then be obtained using the same method mentioned above.  
\[T_{\rm H}=\frac{A_{\rm H}E}{l_{\rm 0H}-\frac{h}{H}\int_{0}^{t}v(1+S_{\rm h}) \ \mathrm{d}t}[\frac{h}{H}\int_{0}^{t}v(1+S_{\rm h})\ \mathrm{d}t-x_{\rm c}]+T_{\rm 0H} \tag{9}\]  
where \(l_{\rm 0H}\) is the equivalent original length of the workpiece on the entry side; \(T_{\rm 0H}\) is the initial backward tension before rolling. From the above modeling process, the block diagram of the tension model can be derived, as shown in Figure 3.  
Figure 3: Block diagram representation of the tension model.