## 1 Introduction  
As the demand for exploration in deepwater areas escalates, for reasons such as oil and gas exploration and the deployment of specialized detection equipment, the marine industry faces increasing challenges in deploying large and heavy structures in deep-sea environments [1]. In deep-sea deployment operations, the accurate evaluation of the dynamic characteristics of large floating structures, the sinking positions of seabed equipment, and the tension of steel cables is crucial for ensuring the safety and functionality of work barges [2; 3; 4; 5]. To effectively evaluate these key factors, establishing a time-domain dynamic coupling system with nonlinear characteristics is essential. However, accurately simulating the deployment process of coupled systems in deep-sea environments continues to be a significant challenge. Given the unique environmental conditions, the accuracy of scale model experiments has yet to be validated; moreover, the cost of at-sea trials remains prohibitively high. Consequently, there is a common reliance on numerical analysis methods for preliminary simulations and predictions of the deployment process. Furthermore, selecting an appropriate installation method is critically important for impacting operational processes. Various deepwater installation methods have emerged in marine engineering, including vertical installation [6; 7], HCSS theory [8], pulley system theory [9], Y theory [10], and the Pendulum Installation Method (PIM). As an emerging method, the Pendulum Installation Method (PIM) addresses limitations in deepwater and ultra-deepwater areas, such as the \"resonance zone\" challenge, commonly known as \"snatch loading\" [11], thereby mitigating cyclic tension changes in cables caused by sudden loads and minimizing safety risks to equipment and personnel. Furthermore, the PIM does not necessitate precise control of the dynamic positioning systems of two vessels, thus simplifying the requirements for vessel control systems and facilitating the precise positioning of the seabed equipment deployment point. This approach meets the requirements for low-cost and non-vessel-specific operations. The deepwater deployment using the PIM is carried out through the coordinated operation of two vessels. The process is illustrated in Figure 1. Upon reaching the installation site, one end of the cable is connected to the large floating structure, while the other end is attached to the work vessel. The transport vessel lifts the floating structure using a crane, moves to the designated swinging point, and subsequently releases it. The structure utilizes its own weight to swing down near the seabed. The Pendulum Installation Method is specially designed for the installation of heavy structures in ultra-deepwater environments and is applicable in water depths of up to 3000 m. Petrobras successfully adopted this method for the first time through numerical analysis, scaled model tests, and full-scale offshore trials, installing heavy objects weighing 280 and 200 tons at depths of 1845 m and 1900 m, respectively [12; 13]. [PERSON] et al. [14] employed FLUENT software for object modeling and conducted time-domain coupled analysis between ships, cables, and objects using the hydrodynamic calculation software SESAM. They reduced the operating cable to a spring model and observed that the variation in internal tension of the cable during the pendulum installation process was relatively stable, with no sudden loads identified that could lead to catastrophic failure. [PERSON] et al. [15], utilizing MOSES software, conducted nonlinear coupled numerical simulations of the pendulum installation theory, taking into full account the interactions between the vessel, mooring system, hoisting cable, and production facilities. [PERSON] et al.