# Dynamic Modeling and Simulation of a Cyclic Towing System for Underwater Vehicles
## 2 Model Development
### External Load Modeling  
During the operation of the cyclic towing system, the system is primarily subjected to external forces, such as tension and fluid resistance. Tension plays a crucial role in the system, as the tensioning device, by applying force to the cable, prevents nonlinear slackening of the long cable, thereby enhancing system stability. In this study, the tension is set as a constant and is added to the generalized force matrix.  
During full underwater operation tests in an experimental water tank, the underwater vehicle is subjected to fluid resistance. To this end, the fluid resistance of the underwater vehicle in the water tank was calculated and analyzed using the computational fluid dynamics software Fluent (v.5.0.4). Additionally, simulation software was used to measure the fluid resistance and drag coefficient of the underwater vehicle under different operating conditions. Using SolidWorks software (2024), a 3D model was created based on the geometric dimensions of the underwater vehicle, disregarding minor features during the modeling process. The front and rear of the underwater vehicle were modeled as semi-ellipsoidal, with the middle section modeled as a cylinder measuring 1 m in length and 0.4 m in diameter. For the fluid computation domain, the upstream boundary was set as a velocity inlet, the downstream boundary was set as a free outlet, and the surrounding boundaries and the computational model were treated as solid walls. Mesh generation employed inflation and localized meshing. The inflation boundary was set to the vehicle's outer surface, with a maximum of 15 layers, a growth rate of 1.2, and a transition ratio of 0.25. In the localized meshing, the element size for the outer surface of the fluid domain was set to 0.2 m, while the element size for the main area of interest, the vehicle's outer surface, was set to 0.005 m. This resulted in 965,892 nodes and a total of 2,568,133 mesh elements. The turbulence model used was the standard k-epsilon model with standard wall functions, and the coupled algorithm was applied to solve the pressure-velocity coupling equations, with 5000 iterations. The fluid resistance experienced by the underwater vehicle increases with its speed; therefore, the fluid resistance was calculated for the vehicle at speeds of 1 m/s, 2 m/s, 3 m/s, and 4 m/s. Using Fluent, the drag on the underwater vehicle in the fluid domain under different speed conditions can be obtained, with the drag coefficient calculated according to the fluid resistance formula [16]:  
\[C_{d}=\frac{f_{s}}{0.5\rho v^{2}A} \tag{6}\]  
The variables in Equation (6) are defined as: \(v\)--velocity of the vehicle, \(f_{s}\)--hydrodynamic resistance, \(\rho\)--water density, \(A\)--projected area of the underwater vehicle relative to the water flow direction, and \(C_{d}\)--drag coefficient.  
After conducting four simulation runs, the dimensionless drag coefficients were averaged, resulting in an average drag coefficient of 0.248 for an underwater vehicle in the direction of travel, as shown in Table 1. Therefore, the water resistance can be calculated using the following formula:  
\[f_{s}=\frac{1}{2}C_{d}\rho Av^{2}=15.4v^{2} \tag{7}\]