# Dynamic Simulation of Underground Cable Laying for Digital Three-Dimensional Transmission Lines
## 3 Construction of Cable Laying System  
### Modeling of Cables and Traction Ropes  
Taking full account of the stiffness and physical characteristics of cables and traction ropes, we use the shaft sleeve force Bushing [10] method to establish cables and traction rope models to maximize the actual actions, such as stretching, shearing, twisting, and bending, of the cables and traction ropes themselves during cable laying construction [11]. The specific steps are as follows: first, the micro-element method is used to divide the cable and the traction rope into a number of micro-segment cylinders with different radii according to equal lengths; then, the shaft sleeve force Bushing is used to impart a mutual force to the two micro-segment cylinders; and finally, the macro command of the ADAMS virtual prototype is used to traverse all the micro-segment cylinders and connect them to  
Figure 1: Target device model. (**a**) Cable tray model. (**b**) Pulley model. (**c**) Cable conveyor model.  
Figure 2: Cable tunnel environment model.  
establish a cable and traction rope model. Figure 3 shows the force model of the micro-section cylinder of the cable and traction rope.  
The bushing force Bushing connection method adds a flexible connection force to the two by defining six components of force and moment between two micro-sections of cylinders. The calculation formula for the bushing force is [12]:  
\[\left[\begin{array}{c}F_{z}\\ F_{y}\\ F_{z}\\ T_{x}\\ T_{y}\\ T_{x}\end{array}\right]=\left[\begin{array}{ccccc}K_{11}&0&0&0&0&0\\ 0&K_{22}&0&0&0&0\\ 0&0&K_{33}&0&0&0\\ 0&0&0&K_{44}&0&0\\ 0&0&0&K_{55}&0\\ 0&0&0&0&K_{66}\\ \end{array}\right]-\left[\begin{array}{c}C_{11}&0&0&0&0&0\\ 0&C_{22}&0&0&0&0\\ 0&0&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\\ \end{array}\right]\left[\begin{array}{c}v_{x}\\ v_{y}\\ v_{x}\\ v_{y}\\ w_{z}\end{array}\right]+\left[\begin{array}{c}F_{x0}\\ F_{y0}\\ F_{z0}\\ T_{x0}\\ T_{z0}\\ T_{z0}\end{array}\right] \tag{1}\]  
In Equation (1), \(R_{i}\) and \(\theta_{i}\) are, respectively, the relative linear displacement and relative angular displacement of the I-Marker coordinate system on the first micro-segment cylinder relative to the J-Marker coordinate system on the second micro-segment cylinder; \(V_{i}\) and \(W_{i}\) are, respectively, the relative linear velocity and relative angular velocity of the I-Marker coordinate system on the first micro-segment cylinder relative to the J-Marker coordinate system on the second micro-segment cylinder; \(F_{i0}\) and \(T_{i0}\) are, respectively, the initial force and moment of the model; \(K\) and \(C\) are, respectively, the stiffness coefficient and damping coefficient of the model.  
The calculation formula for the rigidity coefficient matrix (tension, shear, torsion, and bending) of the shaft sleeve force Bushing is:  
\[\begin{array}{l}K_{11}=\frac{EA}{L}\\ K_{22}=K_{33}=\frac{EA}{L}\\ K_{44}=\frac{G\pi d^{4}}{32L}\\ K_{55}=K_{66}=\frac{EI}{L}=\frac{E\pi d^{4}}{64L}\end{array} \tag{2}\]  
where \(E\) is the elastic modulus of the cable and the traction rope; \(G\) is the shear modulus of the cable and the traction rope; \(A\) is the cross-sectional area of the cable and the traction rope; \(L\) is the length of the small cylindrical micro-section of the cable and the traction rope; \(I\) and \(d\) are the moment of inertia and diameter of the cable and the traction rope, respectively. The Poisson's ratio of cables and traction ropes is 0.3.  
A cable model of 64/110 kV-YJQ03 \(-\)800 mm\({}^{2}\) was selected and simplified based on the research issues. Because the tensile damping factor has little impact on the operating stability of the entire system, no adjustment was made. The torsional damping factor has a great influence on the change of motion and was selected between 1 and 10 according to the actual needs of the site. Because there were too many parts in the model, the ADAMS/View command language was used to model cables and traction ropes.  
Figure 3: Cylindrical force model of cable and traction rope.