## 1 Introduction  
An oil-lubricated bearing carries the risk of lubricating oil leakage, which could pollute seawater quality and harm marine ecosystems. Therefore, researchers are turning their attention to a cleaner and more environmentally friendly water-lubricated rubber bearing (WLRB) [1, 2, 3]. A WLRB uses water as a lubricant, which is more environmentally friendly and has better heat dissipation performance than oil. However, at normal temperature and pressure, the viscosity of water is much lower than that of oil, and the smaller viscosity makes the water film bearing capacity of a WLRB much smaller than that of an oil-lubricated bearing [4, 5, 6]. Submarines often operate at low speed and with a heavy load, making it difficult to form a water film in the bearing that supports dynamic lubrication, resulting in the bearing having a mixed lubrication or even a dry friction state [7, 8, 9]. Rubber, often used as a bearing liner, can generate frictional vibration noise under these conditions, which affects the navigation of warships [10, 11, 12]. Therefore, the frictional noise of the WLRB during operation needs to be reduced. At present, there are several theories to explain the generation of friction noise, which are the stick-slip theory [13, 14], mode-coupling theory [15], self-locking-sliding theory [16], and friction-negative slope of the relative sliding velocity curve theory [17]. In the field of noise in WLRBs, researchers also conducted a lot of scientific research work [18, 19, 20, 21]. As early as the 1990s, [PERSON] [22] proposed the nonlinear variation of the friction coefficient, laying the foundation for the theory of the friction-negative slope of the relative sliding velocity curve by later generations. [PERSON] [23] studied the stick-slip behavior of polymer materials under water lubrication and low-speed conditions on a friction testing machine and found that the speed affects deformation and, thus, stick-slip. [PERSON] [24] used a disk-slider to study the stick-slip vibration characteristics and found that the change in the disk speed has a significant impact on the vibration. Therefore, when studying the vibration characteristics, it is necessary to consider the time-varying velocity. [PERSON] [25] studied the mechanism and process of the friction-induced vibration and torsional vibration coupling and visually displayed the coupling process of the friction-induced vibration and torsional vibration in the tailshaft system. When the engine is in the starting state, the WLRB is in both a low-speed state and a speed time-varying state. Based on the above research status, the vibration is affected by multiple factors at this time, so it is necessary to analyze the vibration situation during startup. The traditional way to study the frictional vibration behavior of the WLRB is through experiments [26; 27; 28]. With the development of finite element software, many researchers began to use finite element software to analyze the frictional vibration behavior. Scholars initially studied the braking noise of friction brake discs. For example, [PERSON] [29] conducted a complex modal analysis of disc brakes using ABAQUS and found that reducing the friction coefficient and increasing the stiffness of the friction disc can reduce the vibration frequency of the friction noise. [PERSON] [30] used complex mode analysis and transient dynamics in ABAQUS to study the vibration behavior between metal sliding blocks and found that the vibration frequency is very sensitive to the sliding speed and contact pressure. [PERSON] also showed the time-varying contact pressure between the sliding blocks, indicating that the jumping of the sliding blocks is the main source of noise, which revealed a new mechanism of the noise source and laid the foundation for future research. [PERSON] [31] used finite element software to analyze the resonance frequency of a four-ball wear testing machine and analyzed the correlation between the lubricant and friction coefficient.