# Dynamic Modeling and Simulation of a Cyclic Towing System for Underwater Vehicles
## 2 Model Development  
### System Dynamics Modeling  
When establishing the model, considering that the depth of the system remains constant during the experiment and that the underwater vehicle moves at a fixed depth underwater, the longitudinal state vector of the system is not taken into account. Additionally, the complex accompanying components and structures of the cyclic towing system are neglected, simplifying the dynamic model of the cyclic towing system. Underwater vehicles are considered as a mass block, and after simplification and analysis, a system with multiple state variables is obtained. The system mainly consists of the towing wheel, guiding wheel, cable circuit, underwater vehicle, and tensioning device. In the cyclic towing system, all components are primarily connected in series through a cable, which plays a crucial role. During the operation of the underwater vehicle, the length of the cable continuously changes over time, mainly affecting the mass and stiffness of the cable. Therefore, the cable is discretized into a spring-damper system [15]. The dynamic model of the cyclic towing system in this paper is based on the following three assumptions:  
1. The lateral elastic deformation of the cable does not reach its elastic limit and follows [PERSON]'s law;
2. The density and cross-sectional area of each segment of the cable are identical, ensuring continuity;
3. The longitudinal displacement of the cable is neglected; only the effect of the lateral deformation of the cable on the system is considered. The parameters in Figure 2 are defined as follows: \(m_{1},~{}m_{2},~{}m_{3}\)--equivalent mass of the towing wheel and tensioning device, equivalent mass of the guiding wheel and tensioning device, and the mass of the underwater vehicle; \(m_{a1},~{}k_{a1},~{}c_{a1},~{}k_{a2},~{}c_{a2}\)--equivalent mass, stiffness, and damping of each equally divided segment of the cable between the towing wheel and the underwater vehicle; \(m_{a2},~{}k_{a3},~{}c_{a3},~{}k_{a4},~{}c_{a4}\)--equivalent mass, stiffness, and damping of each equally divided segment of the cable between the guiding wheel and the underwater vehicle; \(m_{a3},~{}k_{a5},~{}c_{a5},~{}k_{a6},~{}c_{a6}\)--equivalent mass, stiffness, and damping of each equally divided segment of the cable between the guiding wheel and the towing wheel; \(k_{0},~{}c_{0},~{}k_{7},~{}c_{7}\)--stiffness and damping of the tensioning device; \(j_{1},~{}r_{1},~{}\theta_{a}\)--rotational inertia of the towing wheel, radius of the towing wheel, and the angular displacement of the towing wheel; \(j_{2},~{}r_{2},~{}\theta_{b}\)--rotational inertia of the guiding wheel, radius of the guiding wheel, and the angular displacement of the guiding wheel. The structural force analysis of the cyclic towing system is shown in Figure 3. In the model, all linear displacements to the right are considered positive, and all angular displacements in the counterclockwise direction are considered positive. The dynamic equations of the cyclic towing system are established based on the Lagrange equation:  
\[\frac{d}{dt}(\frac{\partial T}{\partial\dot{x}_{i}})-\frac{\partial T}{\partial x _{i}}+\frac{\partial U}{\partial x_{i}}+\frac{\partial D}{\partial\dot{x}_{i}}= F_{i} \tag{1}\]  
The variables in Equation (1) are defined as: \(T\)--system kinetic energy, \(U\)--system potential energy, \(D\)--system dissipation function, \(x_{i}\)--system generalized coordinates, \(\dot{x}_{i}\)--system generalized velocities, and \(F_{i}\)--system generalized forces. The system's kinetic energy \(T\) is given by:  
\[ \begin{array}{l}T=\frac{1}{2}m_{1}\dot{x}_{1}^{2}+\frac{1}{2}m_{2}\dot{x}_{2 }^{2}+\frac{1}{2}m_{3}\dot{x}_{3}^{2}+\frac{1}{2}m_{a1}\dot{x}_{a1}^{2}\\ +\frac{1}{2}m_{a2}\dot{x}_{a2}^{2}+\frac{1}{2}m_{a3}\dot{x}_{a3}^{2}+\frac{1} {2}f_{1}\dot{\theta}_{a}^{2}+\frac{1}{2}f_{2}\dot{\theta}_{b}^{2}\end{array}  \tag{2}\]  
The system's potential energy \(U\) is given by:  
\[