# Dynamic Modeling and Simulation of a Cyclic Towing System for Underwater Vehicles
## 4 Analysis of Key Parameter Influences  
### Different Input Torques  
In Case Study 1, by applying different torques to the towing wheel, setting the tension force on the cable to 3500 N, keeping other parameters unchanged, and using the parameter settings from Table 2, the underwater vehicle's constant speed phase achieves the target speeds of 1 m/s, 2 m/s, 3 m/s, and 4 m/s. Figure 6 presents the displacement and speed of the underwater vehicle, the tension and deformation of the equally divided sections of the cable between the towing wheel and the underwater vehicle, and the relationship between the input torque of the towing wheel and the speed of the underwater vehicle.  
Figure 5: Validation of the dynamic model for the cyclic towing system.  
As shown in Figure 6e, when the maximum input torques reach 504 N-m, 1026 N-m, 1538 N-m, and 2078 N-m, the underwater vehicle's constant speed phases correspond to 1 m/s, 2 m/s, 3 m/s, and 4 m/s, respectively. The displacement of the underwater vehicle is 3.973 m, 8.005 m, 11.889 m, and 15.88 m, respectively. As shown in Figure 6b, during the 2-4 s constant speed phase, the speed fluctuations of the underwater vehicle are within the ranges of 0.991 m/s to 1.01 m/s, 1.996 m/s to 2.051 m/s, 2.96 m/s to 3.06 m/s, and 3.952 m/s to 4.126 m/s, respectively. When the towing wheel is controlled according to the ideal input curve, although the underwater vehicle can move with the towing device, the flexibility and elasticity of the cable cause varying degrees of speed fluctuations during the constant speed phase, leading to unstable motion. In fact, as the input torque increases, the operating speed of the underwater vehicle becomes faster, and the speed fluctuations become more pronounced.  
In the cyclic towing system, long-distance, flexible cables are key components, as their deformation becomes a critical focus in system operations due to their relaxation nonlinearities. Traditional studies on cables often assume that the cable is a rigid body. For instance, in commercial software such as ADAMS, cables are typically discretized as small spheres connected by generalized forces. However, actual flexible cables are composed of multiple intertwined strands, and when subjected to tension, they exhibit complex viscoelastic mechanical behaviors. To better approximate real-world conditions, this paper adopts the Kelvin-Voigt model as the mechanical model for the cable. This model consists of a spring and a damper, allowing for simultaneous consideration of the cable's stiffness and damping characteristics, thus more accurately simulating the elastic and viscous behavior of the cable during stretching. According to Figure 6c,d, the maximum tension in the cable occurs during the acceleration phase, with values of 6341 N, 7362 N, 8377 N, and 9424 N, respectively. At these points, the maximum deformation of the cable is 0.284 m, 0.344 m, 0.404 m, and 0.465 m, corresponding to maximum elongation rates of 0.73%, 0.89%, 1.04%, and 1.2%, respectively. During the constant speed phase, the average tension in the cable is 4698 N, 5001 N, 5304 N, and 5596 N, respectively, while the corresponding average deformation of the cable is 0.188 m, 0.206 m, 0.224 m, and 0.241 m. During the  
Figure 6: Dynamic curves for the underwater vehicle and cable under different input torques.  
acceleration phase, as the load increases, the tension in the cable significantly rises, which in turn enhances elastic stress, leading to more pronounced tensile deformation of the cable. During the constant speed phase, although the tension decreases, it still gradually increases with the load. The higher tension causes greater deformation, making the experimental underwater vehicle more prone to speed fluctuations. In the deceleration phase, as the system's kinetic energy gradually decreases, the tension in the cable also reduces, leading to a corresponding decrease in the extent of its deformation.