Numerical and experimental study on air thrust foil bearing performance with improved structural model based on real-time taper inlet height

： Air thrust foil bearings are key bearings of micro turbo-machinery (such as micro gas turbines, turbo blowers, and air compressors). The bearing load capacity is affected by many factors, and the taper inlet height of bearing structure is closely related to the load capacity. In many previous literature the taper inlet height, as a constant value, was used to calculate film thickness distribution. However, the reality is that the foil will be squeezed by the pressure generated between runner disk and top foil, which makes taper inlet height change during iteration. Therefore, the actual bearing taper inlet height should be chosen properly instead of the constant taper inlet height when iterating. In this paper, an improved computational model of film thickness for adjusting the taper inlet height in real-time is proposed. The relationship between the maximum bearing load capacity and taper inlet height at different rotor speeds of two models is obtained through numerical simulation. It is found that the optimal taper inlet height of the new model is larger than that of the old model. Three types of bearings with different taper inlet height (20μm, 70μm, 114μm) had been tested and the maximum load capacity at different rotor speeds had been obtained. Finally, test data and the simulation results of the two models are compared. It is found that the simulation results of the two models are quite different when the taper inlet height is near the optimal taper inlet height, and the new model is more agree well with the test data.


Introduction
The air foil thrust bearing is a kind of bearing that can be applied to a micro-turbo machine, which balances the axial force generated by the pressure difference between the compressor and the turbo-machine [1][2][3][4]. Compared with traditional rolling bearings or sliding bearings, air foil thrust bearings have many advantages, such as: no pollution to the environment due to using air as a lubricant; high reliability because it doesn't need a separate oil supply device; high speed and wide temperature range on operating, etc. [5][6][7]. With the continuous development of technology, the demand for thrust bearings is increasing. Therefore, it is critical to obtain the characteristics of the thrust bearing more accurately.
The development of air foil journal bearings has been relatively mature [8][9][10]. Later, researchers began to devote themselves to the research of air thrust foil bearings [11][12][13][14][15][16][17][18]. In 1983, Heshmat H. and Walowit J. A. established the foundation model of the bump foil equivalent to spring structure, and gave a calculation model [6]. From then on, the theoretical research on the air foil thrust bearing began. Later, Roger Ku [19] studied the influence of bearing design parameters on its dynamic stiffness and damping characteristics. In 1999, Iordanoff [20] established a stiffness calculation model that considers the friction of bump structures. The finite difference method is used to solve the gas Reynolds equation, and the analysis of the structure was simplified for faster calculation. In the same year, Heshmat and Xu [21] used the finite element method and the finite difference method to analyze the bearing characteristics. The pressure distribution and film thickness distribution were simulated and provides a research method for air foil bearing.
In some literatures, the models considering foil deformation had been also proposed [5,[22][23][24][25][26]. In 2008, Park D. J. and Kim C. H. etc. studied the influence on air thrust bearing characteristics with tilts and slip flow. At the same time, the influence of different taper inlet heights on bearing load, friction torque, stiffness and damping coefficients are simulated. However, this paper only contains theoretical analysis and lacks relevant test data [24]. Lee and Kim etc. developed a model that predicted deflection with variable axial load. They used deflection of the elastic foundation in the air film height equation. Three types of bearings with different outer radii had been were tested to verify the theoretical model. Nevertheless, this paper didn't research the influence of the taper inlet height on bearing characteristics [25]. In 2015, Andres and Ryu etc. had present the physical analysis and numerical model prediction of the static and dynamic forced performance of air thrust foil bearing. Similarly, the foil deformation had been taken into account in the model. However, this paper doesn't focus on the taper inlet height [26]. In 2017, Kim and Park etc. studied the influence of bearing geometric parameters (taper angle, taper inlet height) on the maximum bearing load capacity. They had found the existence of an optimal taper inlet height when the minimum film thickness is constant or the rotor speed is constant. However, they didn't consider the real-time variation of taper inlet height [22].
The air thrust foil bearing will be damaged if the bearings load capacity can't be predicted accurately. Therefore, it is very important to improve the accuracy of theoretical model in predicting bearing load capacity. In many literatures, the taper inlet height, as a constant value, was used to calculate film thickness distribution. However, the reality is that the foil will be squeezed by the pressure, which makes taper inlet height change during iteration. In order to predict bearing load capacity more accurately, an improved calculation model that can adjust the taper inlet height in real-time has been proposed. Finally, the superiority of the proposed model is proved by test.

Theoretical model
As in the previous literatures, for this air thrust foil bearings, assuming that the viscosity of the air is constant. An isothermal air model has been used and the pressure has been considered in the gas film according to the ideal gas law Pv RT  = . [6] Therefore, the steady film flow between the runner disk and the thrust bearing surface can be written as the following dimensionless Reynolds equation: Where: 2 0 The structure design of thrust bearing foil is shown in Fig.1. In addition, the taper inlet height HT of bearing can be changed by adjusting the height of shim. The film thickness between the runner disk and the thrust bearing can be expressed by the following formula: Where C is the film gap between runner disk and the thrust bearing trailing edge, ( , ) gr is the taper height and ( , ) wr is the foil deformation.
A dimensionless film thickness formula is obtained: Where h is the dimensionless film thickness, ( , )  In the previous literature, the calculation of the taper height is based on the following formula [5]:

HT Hbump
In fact, the foil will deform because of the pressure acting on the thrust bearing, which causes directly the taper inlet height HT to change ceaselessly. The schematic diagram is shown in Fig.2. Therefore, the taper inlet height after the change (the taper inlet height after the change is referred to as the actual taper inlet height actual H ) should be used when iterating, which is closer to the real physical model. The actual taper inlet height is equal the difference between initial taper inlet height and foil deformation. There: The model 2 variables describing is as shown in Fig. 3. Therefore, this paper uses the following formula to calculate the distribution of film thickness: The corresponding dimensionless boundary conditions are: The characteristics of the thrust bearing mainly include the bearing load capacity and friction torque, which can be expressed by the following formula [27]:

Fig. 3 Code flow chart
The Reynolds equation and the film thickness equation are solved by the finite element method and the finite difference method, and then the pressure distribution and film thickness distribution under steady state conditions will be obtained. The bearing load capacity and friction torque can be calculated by the pressure distribution and film distribution. Write a code to calculate the bearing characteristics, the whole process are shown in Fig. 3. In the iteration process, HT should be replaced by Hactual before calculating the film thickness distribution. Table 1 lists the parameters of air thrust foil bearing for simulation calculation. The thrust bearing outer diameter RO used in the simulation is 72mm, the inner diameter RI is 18mm, and the thickness of the top foil Ttop is 0.2mm. The total stiffness of all bump foils K is 11.58MN/m. The predicted maximum load capacity versus taper inlet height for different rotor speed had been simulated. In this paper, it is considered that the bearing load capacity is the maximum load capacity when the minimum film thickness Hm is 4μm.  Fig.4 shows the predicted maximum bearing load capacity versus taper inlet height for different rotor speed under old model. The prediction results show that there is an optimal taper inlet height to maximize load capacity. Moreover, the optimal taper inlet height increases with the increasing rotor speed. The predicted maximum load capacity increases sharply with the increasing taper inlet height from 0 to the optimal taper inlet height. The predicted maximum load capacity decreases slowly with the increasing taper inlet height beyond the optimal taper inlet height. In exactly the same way, Fig.5 shows the predicted maximum bearing load capacity versus taper inlet height for different rotor speed under new model. Overall, the trend of the new model is consistent with the old model.   Fig.4 and Fig.5. It shows the comparison of simulation results between old model and new model. It should be noted that the optimal taper inlet height of new model is larger than old model (The whole curve shifted to the right) at the same rotor speed. The higher the rotor speed, the larger the maximum load capacity. This will lead to a larger amount of foil deformation, so the difference between the two models is more obvious at high speed. In particular, the predicted bearing load capacity of the two models is quite different before the optimal taper inlet height. The gap between the predicted load capacities by the two models gradually decrease and tend to be consistent when taper inlet height over the optimal value. Therefore, more attention should be paid to this difference when the optimal taper inlet height is chosen as the geometric parameter of bearing.       7 shows the pressure distribution of a single pad for taper inlet height HT=20μm and rotor speed ω=28krpm under different models. Overall, the peak pressure simulated by new model is smaller than old model. This result also agrees with Fig.6. In addition, Fig.8 shows the foil deflection distribution of a single pad for taper inlet height HT=20μm and rotor speed ω=28krpm under different models. The foil deformation is relatively small because the predicted maximum load capacity of the new model is relatively low. Fig.9 shows the film thickness distribution of a single pad for taper inlet height HT=20μm and rotor speed ω=28krpm under different models. The film thickness of new model is smaller than old model.

Experiment and data analysis 4.1 Test rig introduction
The physical diagram of the thrust bearing and test rig used in this paper is shown in the Fig.10, including the motor drive unit, the thrust bearing unit, the piston-loading unit and the relevant data acquiring unit. The shaft is coupled to the high-speed motor via a coupling and the speed of motor can be adjusted arbitrarily by an inverter. The piston-loading unit includes piston shaft, cylinder and other connecting parts. The axial load is supplied and changed by adjust the piston-loading unit and load cell is used to response the size of load capacity. Otherwise, the force sensor (not shown in physical diagram) and the friction torque rod are used to measure the friction torque. Finally, the electrical signals collected by these sensors are collected into the computer through the relevant data acquiring unit.
The plane run-out of run disk is 2μm in the experiment. The first-order critical speed of shaft is 32,000rpm and the fundamental frequency vibration of the rotor is 6μm at 25,000 rpm, which meets the requirements of the experiment in principle.

Data analysis and discussion
In this paper, three types of bearings with different taper inlet heights were tested at different rotor speeds. The relationship between friction torque and load capacity had been obtained by continually loading the tested bearing. Fig.11 is photos of the air thrust foil bearing with taper inlet height HT =20μm before and after test. It can be seen that the wear of wear-resistant coating mainly occurs in the flat region. Table2 lists the air thrust foil bearing parameters for test.

Conclusion
In this paper, a computational model for adjusting the taper inlet height of the bearing in real-time has been proposed. The improved calculation model can adjust the taper inlet height in real-time according to the foil deflection. In the iteration process (code), initial taper inlet height HT had been replaced by actual taper inlet height Hactual before calculating the film thickness distribution.
The predicted maximum load capacity versus taper inlet height for different rotor speed had been simulated. The prediction results show that there is an optimal taper inlet height to maximize load capacity, and the optimal taper inlet height of new model is larger than old model at the same rotor speed. Moreover, the predicted bearing load capacity of the two models is quite different before the optimal taper inlet height.
Three types of air thrust foil bearing with different taper inlet height had been tested. The maximum bearing load capacity of different types of bearings at different rotor speeds had been obtained. The test results had been compared with the simulation results. It can be found that the new model performs better than the old model when the taper inlet height HT= 20μm(less than the optimal taper inlet height). Moreover, the higher the rotor speed, the better the performance of the new model will be. The predicted results of the two models are almost the same when the taper inlet height far beyond the optimal value. Therefore, more attention should be focused on the characteristics of real-time variable taper inlet height when the taper inlet height of the bearing is no more than the optimum value.

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.