## 3 Experiment and Result Analysis
### Binary Fitting of Twisting Tension to Spindle Angular Speed and Yarn Fineness
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & & \(\mathbf{Fitting}\) \\ \hline
1 & \(P_{d1}=1.52\times 10^{7}\cdot e^{m_{1}}-1.47\times 10^{-4}T^{2}+1.26T+211.48\), \(m_{1}=1-e^{\frac{888\cdot 10^{8}-m_{1}}{83\cdot 10^{8}-m_{2}}}\) & 0.66834 \\
2 & \(P_{d2}=-0.00446\omega^{2}+8.79241\omega-1.82192\times 10^{-4}T^{2}+1.14944T+4.60594 \times 10^{-4}\omega T-4315.98\) & 0.80733 \\
3 & \(P_{d3}=-8158.42947e^{m_{2}}+4721.93\), \(m_{2}=\frac{\omega}{1588\cdot 912}-\frac{T}{3060\cdot 676}\) & 0.81977 \\
4 & \(P_{d4}=1.91396\omega+0.95618T-1286.85\) & 0.82736 \\
5 & \(P_{d5}=0.00656\omega^{0.96149}\times 10^{8}\omega T^{0.8231}\) & 0.83979 \\
6 & \(P_{d6}=\frac{-1.038\times 10^{6}\omega+167228.127}{-451.902\omega-0.10452\omega^{2 }+254.797}+1.01817\omega^{2}+0.18961\omega T+1\) \\
7 & \(P_{d8}=-8.37\times 10^{-10}\omega+4.19\times 10^{-6}\omega^{-4}-0.00813\omega^{3} +7.656\omega^{2}-3525.9\omega\) & 0.98081 \\
8 & \(P_{d8}=-0.00407\omega^{2}-7.15227\times 10^{-4}T^{2}+8.92584\omega+3.19332T-5816.53\) & 0.98601 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Binary fitting equations of balloon yarn tension of the direct-twisting machine. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Twisting Configuration** & **Experimental Item** &
\begin{tabular}{c} **Spindle Angular** \\ **Velocity (rad/s)** \\ \end{tabular} & **Yarn Fineness (dtex)** & **Twisting Tension (cN)** \\ \hline
1 & 837.76 & 930 & 1193.0 \\
2 & 890.12 & 930 & 1208.0 \\
3 & 942.48 & 930 & 1355.0 \\
4 & 785.40 & 1100 & 1361.0 \\
5 & 837.76 & 1100 & 1427.4 \\
6 & 890.12 & 1100 & 1526.6 \\
7 & 942.48 & 1100 & 1628.8 \\
8 & 733.04 & 1440 & 1344.8 \\ twisting tray/yam storage & 9 & 785.40 & 1440 & 1417.6 \\ tray/spindle can: & 10 & 837.76 & 1440 & 1520.6 \\
220 mm/120 mm/266 mm & 11 & 890.12 & 1440 & 1630.2 \\
12 & 942.48 & 1440 & 1789.4 \\
13 & 733.04 & 1670 & 1948.6 \\
14 & 785.40 & 1670 & 2036.2 \\
15 & 837.76 & 1670 & 2145.8 \\
16 & 890.12 & 1670 & 2247.8 \\
17 & 733.04 & 2200 & 2007.4 \\
18 & 785.40 & 2200 & 2268.8 \\
19 & 837.76 & 2200 & 2415.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measurement values of twisting tension under different yarn fineness and spindle angular velocity. sample ordinates, and \(y^{\prime}\) is the fitting of \(x_{i}\) value, the coefficient of determination is shown in Equation (5) [19; 20]. \[R^{2}=1-\frac{\sum\limits_{i=1}^{N}{(y_{i}-y^{\prime})}^{2}}{\sum\limits_{i=1}^{N }{(y_{i}-\overline{y})}^{2}} \tag{5}\]  
The coefficient of determination \(R^{2}\) value range is [0, 1]. The closer \(R^{2}\) is to 1, the better the fit of the regression line to the observed values; Conversely, the closer \(R^{2}\) is to 0, the worse the fit of the regression line to the observed values [21]. It can be seen from the Table 4 that the coefficient of determination \(R^{2}\) of the fitting equation of No. 1 was the smallest, which was 0.66834, indicating that the empirical equation of No. 1 had a poor fitting degree. The fitting equation of No. 8 had the largest value of determination coefficient \(R^{2}\), which was 0.98601, indicating that the fitting degree was relatively good. By comparing the fitting equations, the fitting equation with the largest coefficient of determination \(R^{2}\) was selected as the best fitting model. Therefore, the fitting equation of the relationship between balloon yarn tension, spindle angular velocity, and yarn fineness is shown in Equation (6), where the coefficient of determination \(R^{2}\) = 0.98601. The fitting surface obtained according to the fitting equation is shown in Figure 8. The unit of \(P_{d}\) is cN. \[P_{d}=-0.00407\omega^{2}-7.15227\times 10^{-4}T^{2}+8.92584\omega+3.19332T-5816.53 \tag{6}\]