## 2 Theoretical Derivation of PIC Model
### Contact Pressure Caused by Cable Bending  
The third significant source of contact pressure between wires is generated due to the bending of the cables. The cable ends are anchored at the saddle points, and during the vibration process, the cables will experience a certain degree of bending. As individual elements, parallel wires primarily experience tensile stress, and the contact pressure between wires from different layers mainly originates from the vertical radial force of the tensile stress in the wires. The magnitude of the radial force is related to the tensile force in the wires. As Figure 9 shows, at the contact positions of wires from different layers, the vertical pressure generated by the axial tensile force in the wires is given by:  
\[p_{v}=\frac{2T\sin(d\varphi/2)}{dl}=\frac{2T\sin(d\varphi/2)}{\rho d\varphi}= \frac{T}{\rho} \tag{10}\]  
The pressure transmitted from the upper-layer wires to the lower-layer is given by:  
\[p_{w}=\frac{p_{v}}{2\cos\phi}=\frac{p_{v}}{\sqrt{3}} \tag{11}\]  
where \(p_{v}\) is the vertical component of the wire axial tensile force, \(p_{w}\) is the \(p_{v}\) component force transmitted from the upper-layer wires to the lower-layer, \(T\) is the tensile force of wires, \(\rho\) is the curvature radius of the wire, and \(d\varphi\) and \(dl\) are the angle of bending and the length of the wire. This type of wire transmission has an accumulative nature; the contact pressure from the upper-layer wires is transmitted to the lower-layer and superposed wires. The transmission pattern conforms to the two-dimensional closely arranged contact force transmission model under concentrated force. Research indicates that the location of maximum stress does not always occur at the centerline but progressively expands layer by layer outward. It symmetrically distributes over two layers along the centerline, exhibiting a distinct bimodal phenomenon [36; 37]. As Figure 10 shows, a single wire affects the wires in the conical region below it during the bending of cable, while such an effect decreases as one moves towards the inner layers. Figure 9: Vertical force distribution in different layers, (**a**) the vertical force caused by cable bending. (**b**) distribution of vertical force on the two wires. The PIC model can be used to compute the extent of the influence exerted by the curvature of the wire on other wires in the cable by employing a force transmission model. This calculation can be expressed in terms of the transmission ratio of force and fluctuation momentum of force. Assuming there exists the following relationship between the transmission ratio of force and the fluctuation momentum of force [36]:  
\[ \begin{split} f_{(i-1,j+1)}&=\frac{1}{2}+\omega_{(i, j)}\\ f_{(i-1,j-1)}&=\frac{1}{2}-\omega_{(i,j)}\end{split}  \tag{12}\]  
where \(\omega_{(i,j)}\) is the fluctuation momentum of force for the wire \((i,j)\), \(f_{(i-1,j-1)}\) is the transmission ratio of force from wire \((i,j)\) to wire \((i-1,j-1)\), and \(f_{(i-1,j+1)}\) is the transmission ratio of force from wire \((i,j)\) to wire \((i-1,j+1)\). It can be observed that the force transmission ratio is composed of two parts: one part is the average distribution quantity \(1/2\), and the other part is the force fluctuation momentum \(\omega_{(i,j)}\). Therefore, the compression exerted by the bending of an individual wire on the lower-layer wires simplifies the contact pressure between different layers of wires as follows:  
\[F_{(i,j)}=F_{(i+1,j-1)}\cdot\left(\frac{1}{2}+\omega_{(i+1,j-1)}\right)+F_{(i +1,j+1)}\cdot\left(\frac{1}{2}-\omega_{(i+1,j+1)}\right) \tag{13}\]  
The fluctuation momentum of force will influence the distribution of contact pressure between different layers of cables, thereby affecting the difference in contact pressure between layers. In practical applications, it should be considered in conjunction with values for parameters such as the number of cable layers and wire radius.