## 2 Geometric Modeling of 6 \(\times\) K31 WS + FC Wire Rope
### Parametric Equation of the Wire Centroid Line  
The precise 3D geometry of the wire rope can be obtained by sweeping the cross-section of the wire along the centroid line of each wire constituting the wire rope. Thus, the parametric equations of the wire rope need to be acquired in advance. The global coordinate system \(O_{1}\)-\(XYZ\) and the relative coordinate system \(O_{2}-\overrightarrow{n}\overrightarrow{b}\overrightarrow{t}\) or the Frenet frame, are established to obtain the parametric equations of the wire centroid line. As shown in Figure 2, the origin \(O_{1}\) is located at the center of the bottom of the rope core, the origin \(O_{2}\) is located at the center of the outer wire section, which is the endpoint of the centroid line of a single-helix wire, \(\overrightarrow{n}\), \(\overrightarrow{b}\), \(\overrightarrow{t}\) are the unit principal normal vector, the unit slave normal vector, and the unit tangent vector at this point, respectively. The position of point \(A\) on the centroid line of the single-helix wire is represented by vector \(\overrightarrow{R}\) in \(O_{1}\)-\(XYZ\), the position of point \(B\) on the centroid line of the double-helix line is represented by vector \(\overrightarrow{Q}\), the vector formed by the connection between point \(B\) and the origin \(O_{1}\) is represented by \(\overrightarrow{P}\), and \(\theta\) is the angle between vector \(\overrightarrow{R}\) and the \(X\) axis in the \(XO_{1}Y\) plane. The parameter equation of the centroid line of a single-helix wire is \(R(\theta)=[r_{\rm r}\)cos\(\theta_{\rm r}\), \(r_{\rm r}\)sin\(\theta_{\rm r}\), \(r_{\rm r}\)\(\theta_{\rm r}\)tan\(\beta_{\rm r}\)]T, where \(r_{\rm r}\), \(\theta_{\rm r}\), and \(\beta_{\rm r}\) are the helix radius, position angle, and helix angle of the wire, respectively. Then, the unit vectors of the Frenet frame can be obtained as:  
\[\left\{ \begin{array}{l}\overrightarrow{t}=\frac{R^{\prime}(\theta)}{R^{ \prime\prime}(\theta)}=\left[\begin{array}{c}-\sin\theta_{\rm r}\sin\beta_{ \rm r}\\ \cos\theta_{\rm r}\cos\beta_{\rm r}\\ \sin\beta_{\rm r}\end{array}\right]\\ \\ \overrightarrow{b}=\frac{R^{\prime}(\theta)\times R^{\prime\prime}(\theta)}{|R ^{\prime}(\theta)\times R^{\prime\prime}(\theta)|}=\left[\begin{array}{c} \sin\theta_{\rm r}\sin\beta_{\rm r}\\ -\cos\theta_{\rm r}\sin\beta_{\rm r}\\ \cos\beta_{\rm r}\end{array}\right]\\ \\ \overrightarrow{n}=\overrightarrow{b}\times\overrightarrow{t}=\left[\begin{array} []{c}-\cos\theta_{\rm r}\\ -\sin\theta_{\rm r}\\ 0\end{array}\right]\\ \end{array} \right. \tag{1}\]  
The transfer matrix from the global coordinate system to the Frenet frame is represented by \(T\):  
\[T=\left\{ \begin{array}{ccc}\overrightarrow{n}&\overrightarrow{b}& \overrightarrow{t}\end{array} \right\}^{T}=\left[ \begin{array}{ccc}-\cos \theta_{\rm r}&-\sin\theta_{\rm r}&0\\ \sin\theta_{\rm r}&-\cos\theta_{\rm r}\sin\beta_{\rm r}&\cos\beta_{\rm r}\\ -\sin\theta_{\rm r}\cos\beta_{\rm r}&\cos\theta_{\rm r}\cos\beta_{\rm r}&\sin \beta_{\rm r}\end{array} \right] \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Serial Number** & **Mark** & **Number of Wires** & **Radius (mm)** \\ \hline Rope core & I & \(R_{1}\) & 1 & 7.4 \\ Center wire & II & \(R_{2}\) & 1 & 0.85 \\ Wire in inner layer & II & \(R_{3}\) & 6 & 0.85 \\ Wire in middle & IV & \(R_{4}\) & 6 & 0.75 \\ layer & V & \(R_{5}\) & 6 & 0.95 \\ Wire in outer layer & VI & \(R_{6}\) & 12 & 1.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Radius of the rope core and wires in a 6 \(\times\) K31 WS + FC wire rope. Any point on the centroid line of a double-helix wire is represented in the Frenet frame as \([r_{si}\cos\theta_{si},r_{si}\sin\theta_{si},0]^{T}\), where \(r_{si}\) and \(\theta_{si}\) represent the helix radius and position angle of each kind of double-helix wire, respectively, and it can be represented in \(O_{1}\)-\(XYZ\) as:  
\[ \begin{bmatrix}x_{si}\\ y_{si}\\ z_{si}\end{bmatrix}  =