## 1 Introduction  
Modern advanced ship propulsion systems have put forward more stringent requirements for acoustic stealth [1]. In the process of ship propulsion, the thrust bearing is a crucial component for transmitting propulsive force to the hull, and its vibrations can lead to resonance throughout the entire hull structure. Consequently, there has been a significant increase in the development of vibration-damping thrust bearings, both domestically and internationally. Currently, the primary methods for reducing vibration and noise include rubber vibration damping [2], hydraulic vibration damping [3], and polymer material vibration damping [4], among others. These techniques have achieved some success, particularly in minimizing vibrations in the middle and high-frequency ranges. However, the effectiveness of vibration damping in the middle and low-frequency bands has not yet met expectations. As a result, there is an urgent need to explore new technologies to enhance the acoustic stealth of thrust bearings, particularly in improving vibration and noise reduction capabilities in the middle and low-frequency ranges. Permanent magnet thrust bearings offer a natural advantage in vibration and noise reduction. They utilize the magnetic force generated by permanent magnets to achieve contactless levitation of the rotor, enabling the rotor and stator to remain mutually separated without the need for lubrication or friction during operation. This unique feature makes permanent magnet thrust bearings especially suitable for use in contactless transmission systems, such as spacecraft [5], flywheel energy storage systems [6], and high-precision machine tools [7]. Due to their contactless operation, permanent magnet thrust bearings provide key advantages, including frictionless motion, low noise, and high-speed capabilities. Given the excellent performance of permanent magnet bearings, many researchers have focused on various aspects, such as bearing structure design, bearing capacity, stiffness, optimization, and experimental methods. In flywheel energy storage systems, permanent magnet bearings have shown promising application prospects and have attracted considerable attention. One study [8] proposed a numerical method for the structural design of a new type of permanent magnet bearing for flywheel energy storage systems, enabling rapid determination of the bearing's force, stiffness, and damping characteristics, which was experimentally validated. Another study [6] introduced a system utilizing both superconducting magnetic bearings and permanent magnet bearings, where superconducting magnetic bearings suppress the vibration of the flywheel rotor while permanent magnet bearings passively control the rotor's position, demonstrating excellent control effectiveness. To improve the bearing capacity of permanent magnet bearings, another approach [9] combined permanent magnet bearings with fluid bearing technology. This approach explores the possibility of achieving both low starting torque in permanent magnet bearings and high bearing capacity in fluid bearings under a single-bearing arrangement. Experimental results have shown an improvement in bearing capacity. In centrifuge rotating bearings, which are subjected to extreme working conditions such as high rotational speeds, traditional contact bearings are prone to metal fatigue, cracks, and other failures that compromise their operational lifespan. To address this, one study [10] proposed a centrifuge-specific permanent magnetic bearing, leveraging the non-contact nature of permanent magnetic bearings to achieve high rotational speeds and extend bearing life. The bearing capacity and stiffness of permanent magnet bearings are crucial mechanical characteristics. To investigate how structural parameters--such as thickness, width, number of layers, cross-sectional area, and air gap--affect the mechanical properties of radially magnetized permanent magnet bearings, researchers [11] conducted in-depth studies, providing valuable insights into structural design.