# Research on the Transportation of Low-Strength Composites Sheets
## 4 The Optimal Sag  
If the sheet between rolls is too long or too short, it will have large tensions inside. To determine the relationship between the length of the sheet between rolls, the maximum tension in the sheet,  
Figure 6: The shape of the moving thin-sheet materials between rolls when position of rolls varies.  
and the position and size of rolls, define the length between rolls \(L\) as the length of the sheet between \(A\) and \(D\) as shown in Figure 4. For a given \(\theta_{1}\), \(L\) can be expressed as:  
\[L\equiv \overbrace{AB}^{}+\widehat{BC}+\widehat{CD}\] \[= R_{1}(\theta_{1}-\theta_{s})+\int_{x_{B}}^{x_{C}}\sqrt{1+\dot{y}^ {2}}\mathrm{d}x+R_{2}(\theta_{2}+\theta_{s})\] \[= \frac{C_{1}}{\rho btg}(\tan\theta_{1}+\tan\theta_{2})+R_{1}( \theta_{1}-\theta_{s})+R_{2}(\theta_{2}+\theta_{s}) \tag{18}\]  
Equation (11) indicates that the maximum tension exists at either point \(B\) or point \(C\) (the higher one). So the maximum stress in the sheet is:  
\[\sigma_{max} = \frac{T_{max}}{bt}=\frac{\max\{T_{B},T_{C}\}}{bt} \tag{19}\] \[= 2\rho v^{2}-\frac{\lambda}{bt}+\rho g\max\{y_{B},y_{C}\}\] \[= 2\rho v^{2}-\frac{\lambda}{bt}+\rho g\max\{D_{h}+R_{1}\cos\theta _{1},R_{2}\cos\theta_{2}\}\]  
Define the dimensionless forms of \(L\) and \(\sigma_{max}\) as:  
\[L\equiv\frac{L}{R_{1}}=\tilde{C_{1}}(\tan\theta_{1}+\tan\theta_{2})+\theta_{ 1}+\tilde{r}\theta_{2}+(\tilde{r}-1)\theta_{s} \tag{20}\]  
\[\tilde{\sigma}_{max}\equiv\frac{\sigma_{max}-\rho v^{2}}{R_{1}\rho g}=\max\{ \tilde{D_{h}}+\cos\theta_{1},\tilde{r}\cos\theta_{2}\}-\tilde{C_{3}} \tag{21}\]  
It should be pointed out that for some specific parameters \(R_{1},R_{2},D_{l},D_{h}\) (when \((R_{1}-R_{2}+D_{h})D_{h}<0\)), the maximum tension point may change between \(B\) and \(C\) as the length changes. For example, when \(R_{1}=0.2,R_{2}=1.0,D_{l}=3.0,D_{h}=0.4\), as shown in Figure 7, point \(C\) has greater tension when the sheet is tensioned and point \(B\) posesses greater tension when the sheet is relaxed.  
The relationship between \(\tilde{\sigma}_{max}\) and \(L\) is calculated with Equations (20) and (21) when \(\tilde{D_{l}}\) varied and \(\tilde{D_{h}}=0\) is shown in Figure 8. Figure 8 (A), (B), (C), (D) for \(\tilde{r}=0.2,1.0,1.5,1.8\) respectively.  
Figure 7: Maximum tension point may shift between \(B\) and \(C\).  
The plot of \(\bar{\sigma}_{max}\) as a function of \(\tilde{L}\) is shown in Figure 9, when \(\tilde{D_{h}}\) varied and \(\tilde{D_{l}}=2.5\). Figure 9 (A), (B), (C), (D) for \(\tilde{r}=0.2,1.0,1.5,1.8\) respectively.  
Figures 8 and 9 show that \(\bar{\sigma}_{max}\) increases fast when \(\tilde{L}\) is close to the length of \(AD\), which means that the sheet is tensioned. It is also shown that the slope of \(\bar{\sigma}_{max}-\tilde{L}\) curve coverages to 0.5 when \(\tilde{L}\) increases. It has been shown that for a set of specific position and size parameters \(R_{1},R_{2},D_{l},D_{h}\), there is one optimum length \(\tilde{L}\) which minimizes the maximum stress inside the sheet. Examples of this optimum pose are shown in Figure 10.  
\[\tilde{L}=\arg\min_{L}\sigma_{max} \tag{22}\]In practical engineering, \(\hat{L}\) can be estimated from the length of \(AD\). Define \(L_{0}=AD\). Figure 11 shows \(\hat{L}/L_{0}\) when \(\tilde{D_{l}}\in[1.5,5]\), \(\tilde{D_{h}}\in[-10,10]\). Figure 11 (A), (B), (C), (D) for \(\tilde{r}=0.2,1.0,1.5,1.8\) respectively.  
The approximate value of \(\hat{L}\) is:  
\[\hat{L}\approx 1.25L_{0}=1.25\sqrt{D_{h}^{2}+4D_{l}^{2}+(R_{1}-R_{2})^{2}} \tag{23}\]  
The estimation method of the optimum sheet sag shown in Equation (23) is a fast way to know how the sheet should be located to lower the maximum stress in the sheet.