## 3 Results
### Development of a Method for Repairing Main Pipelines
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pipeline Position Survey Method**} & \multicolumn{3}{c}{**Pipeline Axis Bending Polynomial Coefficients**} \\ \cline{2-5}  & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{e}\) \\ \hline Laser scanning & \(-6.211\times 10^{-5}\) & \(3.023\times 10^{-3}\) & \(-1.318\times 10^{-2}\) & \(-5.539\times 10^{-3}\) \\ Photographing cross markers & \(-6.111\times 10^{-5}\) & \(3.084\times 10^{-3}\) & \(-1.347\times 10^{-2}\) & \(-5.541\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Polynomial coefficients for the curved axis of the pipeline in the case of its laser scanning and photographing of markers. Figure 18: The curve of the deflection of the pipeline obtained by the method of laser scanning and photographing cross markers. The calculated values of the reaction forces in the hydraulic cylinders of the devices and the centering forces obtained from the results of photographing the cross markers glued along the side generatrix of the pipeline and its laser scanning differ by less than 5%, which indicates the reliability of the laser scanning data when assessing the position of the pipeline in order to calculate the studied parameters, as well as the correctness of the above calculation algorithms. Determining the Convergence of the Results of Calculations Based on the Proposed Mathematical Model with the Results of Finite Element Modeling of the Process of Cutting the Main Pipeline  
In this section, we will create a finite element model of the process of cutting the main pipeline in the ANSYS software. To do this, we build a three-dimensional model of the main pipeline with an outer diameter \(D=1.02\) m, an inner diameter \(d=0.996\) m and a length \(l=40\) m. Using the transient structural dynamic processes modeling module, we implement a bend in the pipeline, after which we create supports in the cross sections where the device grips are installed. Next, we implement the cutting of the pipeline by removing elements of the finite element mesh in the place of cutting with a width of 1 cm. After removing the mesh elements, the ends of the pipeline are displaced. At this moment, we measure the reaction forces arising in the created supports, which will be equal to the reaction forces arising in the hydraulic cylinders of the devices when fixing the position of the ends of the pipeline in the process of cutting it. The supports in this case act as an analogue of the hydraulic cylinders of the fixing and centering devices. Next, we apply the centering forces calculated by the proposed mathematical model to the pipeline, after which we measure the height position of the ends of the pipeline. We carry out the bending of the pipeline to an elastic bending radius equal to \(1500D\), which is localized in a section with a coordinate of 25 m, in which the pipeline is cut. Centering of the ends of the pipeline is carried out at the height of the right end \(h=-0.1604\) m.  
The graph of the modeled pipeline deflection is shown in Figure 19. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Value Obtained by** & **Value Obtained by** & **Deviations of the** \\  & **Laser Scanning** & **Photographing Cross Markers** & **Values, \%** \\ \hline \(X_{1\text{\tiny C}}\), N & 125.576 & 129.152 & 2.769 \\ \(X_{2\text{\tiny C}}\), N & \(-948.374\) & \(-971.726\) & 2.403 \\ \(P_{1\text{\tiny N}}\) & 927.2 & 927.437 & 0.026 \\ \(P_{2\text{\tiny N}}\) & \(-348.757\) & \(-348.85\) & 0.027 \\ \(M_{0}\), N-m & 452.215 & 452.336 & 0.027 \\ \(Q_{0}\), N & \(-522.731\) & \(-522.875\) & 0.028 \\ \(\sigma_{max}\), MPa & 88.231 & 88.208 & 0.026 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of the calculation of the reaction forces arising in the hydraulic cylinders and the forces required to center the end of the pipeline used during the laboratory research. Figure 19: The graph of the modeled pipeline deflection.