# Motion Response of the Submersible Underwater Towed System as Cable Length Changes
## 1 Introduction
[PERSON] [17] performed simulations and discovered that both the speed at which the cable is towed and the speed at which the cable is released have an impact on the shape of the cable. However, the ultimate shape of the cable is primarily determined by the speed at which it is towed. [PERSON] [18] developed a simplified model of towed cable motion by solving a linear elastic model that disregards bending stiffness and a dynamic multibody dynamics model that simulates the stowage of the towed cable with finite segments. [PERSON] [19] designed an underwater towed system with a variable cable length that took into account the impacts of axial loads, shear, bending, and torsion. [PERSON] [20] established a dynamic model of variable cable length during ship circular rotational motion and investigated the towed cable structure and stresses at different speeds and depths of deployment. The majority of the literature has examined the distinct impacts of the maneuvering motion or the deployment and retraction of the cable in isolation. For instance, [PERSON] [8] and [PERSON] [9] primarily concentrated on the variable-speed linear motion, whereas [PERSON] [15] and [PERSON] [16] examined the cable retraction and deployment process but did not thoroughly research the relationship between the two. The intricate scenario involving the simultaneous coordination of maneuvering motion and cable deployment is nearing practical implementation; however, it remains an area with limited research. For instance, [PERSON] [10] and [PERSON] [14] discussed several instances that were coupled, but the level of detail and scope were insufficient. This paper utilizes the lumped mass method as a theoretical framework to describe the hydrodynamic properties of the tow cable, as depicted in Figure 1. In this method, the towed cable is discretized as a series of elastic, frictionless light springs, modeling only the axial and torsional properties of the cable. The inter-spring segments can be conceptualized as comprising two coaxial telescopic rods that are linked by axial and torsional spring dampers. The bending properties of the cable are depicted by rotational spring dampers positioned at both ends of the cable segment, connecting the segment to the node. The cable does not have to be axisymmetric since different values of bending stiffness can be specified for the two orthogonal bending planes. The nodes of the system concentrate the mass of the towed cable, fluid drag, added mass, gravity, and buoyancy. The nodes of each half-segment are combined and allocated to the nodes at the end of the segment, while the kinematic relationship between the nodes is utilized as a condition for continuity. [PERSON]'s second law states that the equations governing the motion of the cable being towed can be determined and solved using numerical methods in an inertial coordinate system. This detailed view consists of three nodes and two line segments. The towed cable model structure is composed of three types of spring dampers. The first type is the axial spring damper, which is employed to simulate the axial stiffness and damping in the cable. It applies equal and opposite effective tension to the nodes at the ends of the line segment. The second type is the bending spring damper, which is employed to simulate the bending properties of the cable. The third type is a torsion spring damper, which is employed to simulate the torsional stiffness and damping of the cable. It exerts equal and opposite torque moments on the nodes located at both ends of the segment. The lumped mass method used in this paper considers the towed cable as a series of mass points, which allows for a better simulation of the physical behavior of actual towed cables. The model established by this method is a discrete model. Physical properties such as mass and stiffness of the towed cable vary discontinuously over the length, and the discrete model can capture these variations more accurately. Continuous models may require solving composite partial differential equations.