# Motion Response of the Submersible Underwater Towed System as Cable Length Changes
## 1 Introduction
Discrete models simplify the mathematical treatment and numerical calculations by splitting the cable into multiple simple mass units. Discrete models are typically faster to compute and are suitable for  
Figure 1: Actual cable and lumped mass method models. real-time simulation and control applications, which is especially important in analyzing the dynamic response of towed systems. [PERSON] [21] initially proposed the lumped mass method in a study of the kinematic response of an underwater anchor chain. In this method, the relationship between the nodes is used as a continuity condition, and the added mass is taken into consideration to divide the time differential vector and formulate a set of equations of motion for the microelements of the towed cable. [PERSON] [22] examined the behavior of the faired cable during maneuvering by employing the lumped mass technique. The simulation results were then compared to the findings obtained from actual sea trials to confirm the accuracy of the predicted towed system using the lumped mass method. [PERSON] [23] described the process of anchor chain calculation based on the lumped mass method and compared the differences between different types of anchor chains, which further verified the reliability of the lumped mass method in calculating slender cables. [PERSON] [24] created a comprehensive three-dimensional kinematic model of a towed system using the lumped mass method and examined its kinematic traits. The cable is considered to be an elastic cable within the linear elasticity range, and there is a potential for high-frequency vibration on the towed cable. [PERSON] [25] examined the impact of tension, bending moment, seabed contact, and torque on towed cables in force analysis. He expanded on the lumped mass method and studied the motion characteristics of cables like anchor chains, risers, and towed cables in different situations through numerical simulations. The lumped mass method is characterized by a distinct physical interpretation, a straightforward methodology, and broad applicability and scalability. It has been widely adopted by engineers and technicians since its inception. Most studies have focused on single maneuvering motions or maneuvering motions with fixed cable lengths. Some studies have focused on the effects of specific parameters, but there have been fewer systematic analyses of the combined effects of multiple parameters. In this paper, we combine the maneuvering motion and the dynamic adjustment of towed cable length, which is closer to the complexity in real operation. And the effects of different motion and structural parameters on motion response, such as buoyancy-to-gravity ratio, acceleration time, and rotational speed of circular motion, are analyzed to find the relationship between parameters and motion response. A theoretical model of a towed cable with a non-constant length for underwater towed systems is developed using the lumped mass approach in this paper. The simple maneuvering motion and the composite maneuvering motion are simulated by the cable dynamics computational simulation tool Orcaflex, and the laws of the influence of different cable length change speeds on the motion response during the maneuvering motion of the submersible are found. This paper uses cable dynamics computational simulation tools to simulate the operation of retracting and releasing towed cables during accelerated motion and circular rotational maneuvers. The goal is to simplify the equations of motion under composite maneuvering actions and obtain the laws of motion response of the towed cable length changes during the submersible's movements. The speed of cable retraction and deployment directly influences the magnitude of shock effects experienced by the towed cable. The rate of change in cable length has the greatest impact on the depth change of the towed body when the cable deployment maneuver is executed while the submersible is accelerating.