## 1 Introduction  
The quality of shafting alignment directly affects the invisibility, reliability, and safety of a submarine. A poor shafting alignment state will cause violent vibration, abnormal wear of the bearings, and even shaft deformation and fracture in severe cases [1]. With the requirements for the acoustic stealth of modern submarine continuously increasing, the design scheme of elastically supported submarine propulsion shafting has been proposed under this background. Various elastic vibration isolators, such as the large floating raft, the air spring, and the airbag isolator, are utilized for elastically supported submarine propulsion shafting to attenuate the vibration energy transmitted from the shafting to the hull, thus achieving the goal of vibration and noise reduction. In recent years, advancements in elastic support technology have catalyzed the development of novel vibration isolation isolators. [PERSON] et al. [2] developed an intelligent floating raft system incorporating automatic pressure-adjusting air springs. Integrating a fuzzy logic-based self-tuning PID controller with the magneto-rheological damper, [PERSON] et al. [3] established a semi-active vibration isolator used for marine diesel engine systems. [PERSON] et al. [4] engineered a digital twin-driven air spring vibration isolation system for marine shafting featuring real-time alignment control. However, elastic vibration isolators will cause elastic deformation at the bearing supports of a shafting, which will change the deflection of the shafting and affect its alignment state. Therefore, it is necessary to precisely analyze the alignment characteristics of elastically supported submarine propulsion shafting and subsequently achieve alignment optimization based on the analytical results. With the emergence and continuous development of the design scheme of elastically supported shafting, many scholars have begun to study the shafting alignment calculation under elastic boundary conditions, in order to obtain more scientific and accurate results. [PERSON] et al. [5] developed finite element code for marine shafting alignment with the elasticity of the bearing supports considered, and analyzed the alignment state of a heavy shafting under rigid and elastic boundary conditions. Taking the ship hull deformation into account, [PERSON] [6] calculated the alignment state of a marine shafting, and believed that the stiffness and damping characteristics of the boundary conditions should be considered. Furthermore, he developed software for shafting alignment calculations under elastic boundary conditions using the finite element method [7]. [PERSON] [8] proposed the concept of elastic alignment for the propulsion shafting of ultra-large ships, and established an elastic alignment calculation model that incorporates elastic boundary conditions, including elastic hull deformation, as well as oil-film stiffness and damping of each bearing. [PERSON] et al. [9] presented the design scheme of a ship shaft line on elastic supports with a coefficient of rigidity, and analyzed the influence of elastic properties of deadwood bearings on shaft line operability. [PERSON] et al. employed [10] various aft stern tube bearing support models, and comparatively analyzed the alignment characteristics of a ship shafting with a large span of propeller shaft under different elastic support conditions. The elastic deformation generated under elastic support conditions is an important dynamic factor affecting shafting alignment. [PERSON] [11] analyzed the displacement of a propulsion shaft line and a crankshaft axis caused by thermal deformation. Using the finite element method and the jack-up method, [PERSON] et al. [12] analyzed the influence of hull deformation caused by draught change on the alignment of the shafting for a large ship, and believed that hull deformation is one of the key factors contributing to the bearings' elastic displacements. Based on comprehensive calculations and analyses, [PERSON] et al.