# Study on the Position Deviation between the Iron Roughneck's Spin-Rollers and the Drilling Tool
## 2 Materials and Methods
### Analysis of Position Deviation between the Spin-Roller and the Drilling Tool
#### 2.2.2 Theoretical Analysis  
Figure 2 is the analysis diagram of the deviation after the spin-rollers hold the drill pipe tightly. The circles drawn with solid lines in the figure represent the positions of the drill pipe and the roller when the axis is not offset, and the dashed circles represent the location of the when the axis is offset. If the eccentricity of the drill pipe and the roller is too large, the drill pipe and the iron roughneck will vibrate greatly during screwing, resulting in poor contact between the roller and the drill pipe, affecting the stability of screwing and possibly damaging the thread of the drill tool. The eccentricity relationship between driving rollers and drill pipe is analyzed as follows:  
As shown in Figure 2, take the axis position of the follow-up roller as the fixed reference. Assuming that all three rollers are in contact with the drilling tool, the deviation of the drilling tool in the \(x\) direction is \(\delta\). Under the condition of no offset, the axial distance \(L_{1}\) between the follow-up roller and the drill pipe is:  
\[L_{1}=R_{\text{a}}+R_{\text{b}} \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tubular Connection** & **Rated Pressure of** & **Spin Speed** & **Spin Torque** \\
**od Range** & **Hydraulic System** & **Spin Speed** & **Spin Torque** \\ \hline
89–311 mm & 25 MPa & 100 rpm & 2373 Nm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Technical parameters of the spinner mechanism. After the deviation occurs, set the included angle between the connecting line between the follow-up roller and the drill pipe axis and the \(y\) axis as \(\theta\), and then it is evident from the figure:  
\[\sin\theta=\frac{\delta}{R_{\text{a}}+R_{\text{b}}} \tag{2}\]  
So \(\theta\) can be expressed as follows:  
\[\theta=\arcsin\frac{\delta}{R_{\text{a}}+R_{\text{b}}} \tag{3}\]  
After the offset, the drill pipe and the follow-up roller are still tangential to the outer surface, so the distance between the two axis lines along the \(y\)-axis direction is:  
\[L_{1y}=L_{1}\cos\theta=(R_{\text{a}}+R_{\text{b}})cos\theta \tag{4}\]  
The deviation of the drill pipe in the \(y\)-axis direction \(\delta_{1y}\) is:  
\[\delta_{1y}=\left|L_{1}-L_{1y}\right|=\left|(R_{\text{a}}+R_{\text{b}})(1-\cos \theta)\right| \tag{5}\]  
assuming the relative positions of the components of the spinner mechanism remain unchanged. When there is no offset, the center angle between the axis of the follow-up roller and the axis of the driving roller relative to the axis of the drill pipe is \(\beta\). The included angle between the axis connecting line of the driving roller, the follow-up roller, and the \(y\)-axis is \(\alpha\). The axial distance between the follow-up roller and the driving roller is \(L_{2}\). The diameters of the driving roller and the follow-up roller are equal, so the axial distance between the follow-up roller and the drill pipe is also \(L_{1}\). According to the sine theorem, \(L_{2}\) and \(L_{1}\) meet the following relationship:  
\[\frac{L_{2}}{\sin\beta}=\frac{L_{1}}{\sin\alpha} \tag{6}\]  
where \(\alpha\) and \(\beta\) meet the following relationship:  
\[\alpha=\frac{\pi-\beta}{2} \tag{7}\]  
Therefore, the expression of \(L_{2}\) can be obtained as follows:  
\[L_{2}=\sin\beta\frac{L_{1}}{\sin\alpha}=\sin\beta\frac{R_{\text{a}}+R_{\text{ b}}}{\sin\left(\frac{\pi-\beta}{2}\right)} \tag{8}\]  
Figure 2: Diagram of deviation analysis.