## 1 Introduction  
Single-screw compressors can offer reliable and efficient compression services in several industrial fields. In recent years, water-lubricated single-screw compressors (WSSCs)have increasingly prevailed in the fields of food, pharmaceuticals, petrochemical industry, and new energy owing to their advantages, such as their energy consumption, their oil-free nature in discharging air, lower manufacturing cost, and relatively lower maintenance cost compared to other types of compressors [1, 2, 3].The WSSC structure is shown in Figure 1; the WSSC utilizes one screw rotor to mesh with two star wheels which are located on separated shafts mounted at right angles to the main rotor axis. As the screw rotor rotates, spaces form between the screw groove, the star wheel and the casing, and then gas from suction port flows into fill this space, which increases until the particular screw groove moves out of mesh with the star wheel. Continued rotation brings the screw groove full of gas into mesh with the next star wheel on the other side of the rotor; the trapped volume decreases, and the gas pressure increases as the star wheel sweeps along the screw groove. The final part of the process is the discharge phase, where all the gas is swept out of the screw groove by the star wheel. With the proposal and implementation of a multi-column envelope meshing pair profile [4], breakthroughs have been achieved on the rapid wear issue of star wheel tooth surfaces in WSSCs. On the basis of the adaption of a new meshing profile, WSSCs with large displacement and a high compression ratio have been developed in numerous Chinese companies. However, rapid wear and scuff often occur in their star wheel's thrust bearings due to the increase in and fluctuation of forces, which has become a main barrier to the development of WSSCs. At present, two types of axis thrust bearings are used in WSSCs. The first type is rolling bearings lubricated with grease, which should be tightly sealed and relatively difficult for frictional heat to dissipate. This condition consequently increases the compressor manufacturing cost and operational failure of the rolling bearings [5; 6]. Moreover, a rolling bearing appears to encounter numerous failures when radium and axis pulsating forces act simultaneously. The other type is water-lubricated thrust bearings composed of a high hardness shaft and graphite sleeve, in which no hydraulic film is formed. Axial forces acting on the star wheel shaft make solid materials that are contacted directly, resulting in the graphite shaft sleeve wearing out fast. Water lubricated bearings are apt for use in WSSCs owing to the small meshing forces and minimal friction between their star wheel and screw tooth. In WSSCs, the air-water mixture is discharged from the discharge port at the end of the compression process and enters the gas-water separator thereafter for separation. After cooling in a cooler, lubricating water is injected into the compression chamber. Given that pressure in the gas-water separator is the same as the discharge pressure, the pressure difference between the exhaust pressure and suction pressure is often the injection pressure difference before and after the nozzle. WSSCs can automatically supply high-pressure water that can be injected into the hydrostatic thrust bearings of compressors to balance the axial forces and reduce the contact wear between two surfaces. With the continuous development of the application of water-lubricated bearings in machinery, research has mainly focused on their reliability and bearing performance. [PERSON] et al. analyzed the influence of the guide bearing load of large vertical pumps on the structure, design, manufacturing construction and installation errors of pump unit and assembly [7]. They established the probabilistic load theory of guide bearing for the first time and studied the modeling method and characteristics of bearing probability load. [PERSON] et al.