Tribo-dynamics modeling and analysis of key friction pairs in scroll compressor with floating fixed scroll design

The fixed scroll of the scroll compressor is usually immovable, and the deformation of the scrolls may lead to the operation stuck, which may be avoided by designing the fixed scroll as floating one. However, the introduction of floating fixed scroll will inevitably cause the variation of the dynamic characteristics, so it is necessary to establish a simulation model to explore the influence of floating fixed scroll. In this study, a tribo-dynamics model considering floating fixed scroll is established, which couples the key friction pairs composed of fixed scroll, orbiting scroll and frame to reveal the influence of floating scroll on tribological and dynamics performance. The variation of the lubricant viscosity and boundary pressure as well as the deformation of components are included in the model to consider the real working conditions. The results show that compared with the fixed scheme, the floating scheme provides a compensation space for the components deformation, which is conducive to the operation stability; The floating scheme can not only ensure the sealing of the compression chamber, but also reduce the friction work by 53% compared with the fixed clearance of 4µm.


Introduction
Scroll compressor is widely used in the air conditioning industry due to its advantages of low noise, low vibration and high efficiency (Du et al., 2022;K. Kim et al., 2021). The main components of the scroll compressor to realize the compression function include the fixed scroll (FS), the orbiting scroll (OS), the Oldham's ring (OR), the frame and the crankshaft, as shown in Figure 1.
The OS is driven by the crankshaft and compresses the refrigerant through an orbital motion in the FS. The keys on the upper surface of the OR are assembled on the OS keyway, and the keys on the lower surface are assembled on the frame keyway. Therefore, the OS orbits with a constant radius instead of rotating motion (Qiang, 2010). The FS and OS mesh into a crescent compression chamber, and the FS scroll surface is in contact with the OS tooth root surface to seal the compression chamber. Similarly, if the tooth height of OS is equal to that of FS, it also has sealing effect. The ring surface of the frame contacts with the bottom surface of the OS to form a thrust surface.
In order to avoid operation jamming caused by deformation of components, the FS of some types of compressors are designed to float axially in a small range.
CONTACT Xianghui Meng xhmeng@sjtu.edu.cn The introduction of floating FS will inevitably affect the performance of the compressor. However, there is no theoretical research on floating FS in the existing literature, so it is necessary to establish a simulation model to reveal the effect mechanism of floating FS on tribology and dynamics. In order to achieve this goal, the key friction pairs of scroll compressor should be analyzed first. Typically, three groups of key friction pairs are composed of the FS, the OS and the frame, and the thrust friction pair plays a supporting role, while the scroll axial friction pairs (including the FS scroll surface vs. the OS tooth root surface, the OS scroll surface vs. the FS tooth root surface) play a sealing role. As shown in Figure 1, these three groups of friction pairs are coupled with each other, so the motion and force of each component can be accurately simulated only when the coupling effect of components is considered in the model. Some classic researches provide reference for the simulation of the thrust surface of the scroll compressor. For example, Lee et al. (2009) simulated various operating conditions of the scroll compressor through the thrust bearing test rig, explored the influence of composite lubricating oil on the friction characteristics of the thrust bearing. Based on the boundary adaptive coordinate system, Kim et al. (2010) explored the effect of the thrust surface groove on the dynamic characteristics of the OS. Ishii et al. (2014) proposed a new lubrication model of the frame thrust bearing and found that elastic deformation can form a lubrication wedge to increase the oil film pressure. Although these works have some value, they do not consider the interaction of the FS, OS and thrust surface. In recent years, the tribo-dynamics studies on of the axial scroll friction pairs with high citation rate are represented by the following work. Ahn et al. (2016) studied the scroll thrust bearing of high side shell type scroll compressor and considered the effect of ambient pressure on oil film pressure, but the internal scroll of this compressor does not form lubrication clearance. Wang et al. (2022) used interpolation to calculate the pressure distribution of the scroll without establishing a detailed lubrication calculation model. Kim et al. (2020) established the quasi-dynamics model of the scroll compressor, including the oil film pressure on the scroll surfaces, but their model did not consider the inertia forces and moments. For the high-speed scroll compressor, the influence of the inertia part on the overall dynamic performance cannot be ignored. It can be seen that these studies have greatly simplified the lubrication and dynamics modeling of the scroll friction pairs, and a more detailed model needs to be established.
The radial clearance between scrolls affects the compression efficiency, so some scholars have established the simulation model of scroll from the perspective of leakage. For example, Sun et al. (2022) established the lubrication model of scroll radial clearance, and discussed the influence of structural parameters on the tangential leakage. Liu et al. (2022) analyzed the influence of scroll radial clearance on the time average performance of scroll compressor. Zheng et al. (2022) found that arranging continuous sealing grooves on the FS sidewall could control the tangential leakage. Since the radial forces between scrolls are smaller than the radial gas loads, the influence of the radial forces on the tribo-dynamics can be ignored. In addition, the temperature and boundary pressure of the scrolls change along its shape, and the friction pairs deform under the action of forces, which have important effects on the tribo-dynamics. Therefore, it is necessary to consider these factors in the simulation model to ensure accuracy, and some commercial software can efficiently solve these problems. For example, Zeinalzadeh and Pakatchian (2021) used FLUENT to simulate the gas pressure acting on the transonic rotor of the compressor, and proposed a scheme to improve the efficiency of the compressor. Zhang et al. (2022) used FLUENT to calculate the gas pressure of the gas micro bearing of the compressor, and analyzed the influence of the flow field on the bearing capacity. Ren et al. (2022) also used FLUENT to calculate the flow field of the compressor impeller, and explored the influence of the eccentric effect of the impeller on the operating stability. These studies provide valuable reference for the application of commercial software.
To sum up, the existing works mainly focus on the single friction pair of the scroll compressor and lack the research on the floating FS, and ignore the necessary factors in the lubrication and dynamic model. Therefore, this study establishes a tribo-dynamics model of the FS-OS-Frame system with floating FS, the model considers the variations of viscosity, boundary pressure and friction pair deformation. The effect of the floating FS on the dynamic characteristics and friction loss is discussed in detail. It provides theoretical support for the optimal design of scroll compressor.

Structure and coordinate system
During the operation of scroll compressor, the OS moves in x and y directions driven by the crankshaft. Due to the clearance between the components, the OS has the freedom of axial (z direction) movement and overturning around the x-axis and y-axis. The FS is considered to have only axial movement. Figure 2 shows the motion and dimension symbols of the compression chamber components. The reference plane for the z direction displacement (e OS,z ) of the OS is the frame thrust surface, and the reference plane for the displacement (e FS,z ) of the FS is the upper disk surface of the OS at the initial position. It should be noted that this study also investigates the tribo-dynamics of the fixed design of the FS. The fixed design means that the FS is completely stationary and has a fixed clearance C FO with the frame, as shown in Figure 2.
The overturning angles of the OS around the x-axis and y-axis are ϕ OS,x and ϕ OS,y , and the rotation center  is the mass center. The dimensions of components and the starting angle of the polar coordinate system are indicated in Figure 2. Where, l FS and l OS are the scroll tooth heights of the FS and OS, l OT is the thickness of the OS disc, r in and r out are the inner radius and outer radius of the thrust surface respectively. It should be noted that l FS of this type of compressor is 5μm longer than l OS , so there will be an assembly clearance C as of 5μm on the OS scroll surface. The clearance between the friction pairs can be obtained according to the attitude of the FS and the OS, which can be used to calculate the oil film pressure and contact pressure. The structural parameters and material parameters of each friction pair are listed in Tables 1 and 2 respectively, and these symbols are distinguished according to the subscripts.

Mathematical model
In this section, the tribo-dynamics model of scroll compressor including FS, OS and frame is introduced. The dynamics model is established based on Newton Euler equation, and the mixed lubrication model is established in polar coordinate system.

Dynamics model
The forces acting on the FS and the OS are shown in Figure 3. The forces on the FS are simplified because only  its z direction movement is considered. In Figure 3(a), the back of the FS is affected by the back gas force F gas,back and limiting force F limit . For this type of compressor, when e FS,z reaches maximum floating displacement d limit , it is constrained by F limit . Axial gas force in compression chamber is F gas,z , the value of the force acting on the FS and the OS is equal, but the direction is opposite. The force F FS generated by the FS scroll surface acts on the FS and the OS with equal magnitude and opposite direction, and so does the force F OS generated by the OS scroll surface. The forces on the FS also include its own gravity G FS and inertia force F FS,zi . Through the above analysis, the force balance equation of the FS is established as follows: FS,z ,ë FS,z , are the acceleration in z direction of the FS, respectively.
In Figure 3(b), in addition to the above forces, the OS is also subjected to the gas forces F gas,x and F gas,y , the friction forces F fFS,x and F fFS,y derived from F FS , and the friction forces F fOS,x and F fOS,y derived from F OS in x and y directions. The support force F thrust on the thrust surface and the friction forces F fthrust,x and F fthrust,y in x and y directions. The components of the driving force of the crankshaft in the x and y directions are F drive,x and F drive,y respectively. The OS is also affected by gravity G OS and the inertial forces F OS,xi , F OS,yi and F OS,zi in x, y and z directions. The force balance equations of the OS are as follows: The forces exerted on the OS that do not pass through the center of mass will generate overturning moments. F gas,x and F gas,y generate moments M gas,y and M gas,x around the y-axis and x-axis. The moments generated by F OS and F FS are M OS,x , M OS,y , M FS,x and M FS,y , and the moments generated by F fFS,x , F fFS,y , F fOS,x and F fOS,y are M FSf ,y , M FSf ,x , M OSf ,y and M OSf ,x , respectively. The moments generated by F thrust are M TH,x and M TH,y , and the moments generated by F fthrust,x and F fthrust,y are M THf ,y and M THf ,x . The moment balance equations are as follows: where M OS,xi and M OS,yi are the inertia moments of the OS rotating around the x-axis and y-axis, are the angular acceleration of the OS around the x-axis and y-axis, respectively. Combining Equations (1) ∼ (6) to solveë FS,z ,ë OS,z , ϕ OS,x andφ OS,y , the dynamics matrix is as follows: ⎡ These forces generated by friction pairs include oil film force and contact force, that is, The subscripts oil and c represent the oil film force and contact force, and FS, OS and TH represent the FS scroll surface, the OS scroll surface and the frame thrust surface respectively.

Lubrication model
The average Reynolds equation is used to calculate the oil film pressure, and the GT model is used to calculate the contact pressure. The average Reynolds equation expression in polar coordinate system is adopted (Beschorner et al., 2009;Bhushan, 2013): where p is the average lubricant pressure, h is the oil film thickness, η is the viscosity of lubricating oil. φ θ and φ r are the pressure flow factors, φ s is the shear flow factor (Patir & Cheng, 1978, 1979, and φ c is the contact factor (Wu & Zheng, 1989). θ and r are the circumferential angle and radius of the computational node, v θ and v r are the tangential and radial velocities of the OS in polar coordinates. The OS moves with a constant orbiting radius under the drive of the crankshaft, as shown in Figure 4. At any time, the line velocity component of each point on the OS are the same, so v θ and v r can be calculated (Pietrowicz et al., 2002;Rak & Pietrowicz, 2020). Taking point P as an example, the velocities can be expressed as: where R e and α are the orbiting radius and angle, ω OS is the crankshaft speed, ω OS = 120π/s.  According to the viscosity temperature property of lubricating oil (AE1294) and the proportion of refrigerant (R410A), the corresponding viscosity can be obtained. The temperature of scroll surfaces is calculated by ANSYS 19.2. Through the grid independence analysis, the optimal number of meshes is determined, as shown in Table 3. Since the temperature at the same position on the scroll changes little at different times, the temperature can be regarded as changing only with position and not with time (Y. Liu et al., 2012). The time domain average viscosities η FS and η OS of scroll surfaces are shown in Figure 5, and the viscosity η TH at each point on the thrust surface is constant 0.001Pa · s.
The Reynolds boundary conditions are used to solve the Equation (8), where p b is the boundary pressure, Γ are the boundary lines of the computational domain, Ω is the cavitation domain of the oil film rupture. The boundary lines of the frame thrust surface are r in and r out , and the suction pressure (1.088 MPa) is taken as its boundary pressure p b,TH . The boundary lines of the scroll surfaces are the inner and outer involutes, and the boundary pressures of FS scroll surface (p b,FS ) and OS scroll surface (p b,OS ) are calculated by FLUENT 19.2, the optimal number of meshes used is shown in Table 3. Since p b,FS and p b,OS change continuously along the involute but change little with time, the time domain averages of the boundary pressures are used to calculate the oil film pressure on the scroll surface, as shown in Figure 6.  The oil film thicknesses of three groups of friction pairs are expressed as, h OS = e FS,z − e OS,z + r os sin θ · ϕ OS,x − r os cos θ · ϕ OS,y wherein, h FS and h OS represent the oil film thickness of the scroll surface of the FS and the OS, h TH represent the oil film thickness of thrust surface. r fs , r os , r th respectively represent the radius of computational nodes of the FS, OS and frame. D FS , D OS and D TH are the composite deformations of the FS, OS and frame in z direction respectively, which are calculated by ANSYS 19.2, the optimal number of meshes used is shown in Table 3. In order to ensure the accuracy of the lubrication calculation, the grid independence test is carried out under the initial dynamics condition (see Table 5), and the test results are listed in Table 4 (relative error is acceptable within 0.1%).
According to the grid independence test results in Table 4, the number of nodes (θ × r) in the lubrication zone of the thrust surface and the scroll surface is 101 × 201 and 101 × 251 respectively.

Contact model
When the clearances between the friction pairs are small enough, the asperities of the rough surface will contact, resulting in the contact force p c . The most commonly used method for calculating contact pressure on rough surfaces is the statistical model proposed by Greenwood and Tripp, namely the GT model (Greenwood & Tripp, 1967). The calculation formula of the GT model is shown in Equation (14), which can effectively solve the contact pressure of rough surface with acceptable accuracy (Hu et al., 1994). where , E 1 , E 2 and ν 1 , ν 2 are the elastic modulus and Poisson ratios of the contact surfaces, respectively. ξ is the equivalent areal asperity density, β is the equivalent curvature radius of asperity, σ is the equivalent standard deviation of asperity height. F 5/2 (h * ) is the probability density function of the surface roughness with Gaussian distributed asperities. Dimensionless clearance h * = h/σ .
For the convenience of numerical calculation, the following fitting formula is used to express F 5/2 (h * ): The oil film forces and contact forces can be obtained by integrating the oil film pressure and contact pressure in the calculation domain, and the corresponding moments can be obtained by further considering the force arms. It should be noted that, in the case of dry friction, contact damping should also be considered . But the friction pairs in this study have both contact and lubrication, so the contact damping can be ignored.

Numerical solution method
Based on the finite volume method, the numerical solution equation of (8) in polar coordinates is established. As shown in Figure 7, to meet the mass conservation of the flow in the calculation unit, the total flow entering the cell is equal to the increased flow in the cell, which can be expressed as: (q 1 − q 2 ) · r + (q 3 − q 4 ) · θ = q 5 · θ · r (16) where the flows over the cell surfaces are Substitute Equations (17 ∼ 21) into Equation (16) to obtain The relaxation iteration method is adopted for solution:p wherep n i,j represents the nth iteration result,p n−1 i,j represents the n-1st iteration result, p n i,j represents the pressure calculated according to Equation (24) based on the last iteration; ω is the relaxation factor. When the load of two iterations meets the relative accuracy ε 1 = 10 −6 , the calculation is considered to be convergent, that is, The modified extended backward differentiation formulae (MEBDF) proposed by Cash (Cash, 2000) is adopted to solve the ordinary differential equations (ODEs) in Equation (7). To solve these ODEs, the first step is to express the four second-order variables as firstorder ODEs (Fang et al., 2021): e 0 OS,z ,φ 0 OS,x ,φ 0 OS,y ) T , and then the initial value problem in the first-order ODEs is obtained as follows: where y 1 ∼ y 4 can be obtained by y 5 ∼ y 8 , y 5 ∼ y 8 are the moving or rotating accelerations of the FS and the OS, which are determined by the resultant forces and moments in Equation (7). Through iterative calculation of simulation program, the optimal initial values in y 0 are determined, which are conducive to rapid convergence of dynamics calculation, as shown in Table 5 Figure 8.
Step 1: Set calculation parameters (calculation nodes, scroll compressor structure and material parameters) and quadratic function profile parameters.
Step 2: Calculate the deformations of the friction pairs and the lubricant temperature by ANSYS 19.2, and then obtain the viscosity from the temperature and lubricant properties; Calculate the boundary pressure through FLUENT19.2.
Step 3: Initialize the dynamics parameter y, including the initial attitude (e, ϕ) and speed (ė,φ) of the FS and OS.
Step 4: Calculate the oil film pressure p oil and contact pressure p c of two groups of friction pairs. Then, obtain the forces and moments in B 11 ∼ B 41 .
Step 5: Solve the dynamics with the MEBDF algorithm, and obtain the dynamics variables y t+ t of the FS and OS in the next step. Step 6: Determine whether the dynamics results meet the periodic convergence. If the relative error is less than the allowable error ε (0.001), the calculation is completed. Otherwise, take y clycle end as the new initial value y 0 , and repeat the calculation from θ = 0 until the period converges.

Results and discussions
Before the tribo-dynamics simulation of scroll compressor, the simulation method was verified. Based on the model in this study, and using the scroll compressor simulation parameters and conditions in the work of Kim et al. (2020), a simulation calculation was carried out. The detailed simulation conditions can be found in Kim's literature. Figure 9 shows the variation of the OS attitude angles (ϕ OS,x and ϕ OS,y ) in one cycle. Since the detailed data of viscosity, boundary pressure and deformation in Kim's study cannot be obtained, the corresponding data of this study is used for verification calculation, so there are some slight differences between the results of the two works, but the results are generally consistent, which proves that the simulation method in the current study is reliable.
Based on the tribo-dynamics model established in the previous sections, the simulation of scroll compressor is carried out under the conditions of floating FS and fixed FS respectively, and the fixed clearance C FO includes three dimensions: 4, 5 and 6 μm. The simulation results are discussed in this section, in which the object discussed in Section 4.1 is the frame thrust surface, and the objects discussed in Section 4.2 are the scroll surfaces of FS and OS. The effects of the floating scheme of FS on the friction loss are discussed in Section 4.3.
The cooperative enterprise provides the gas loads in a cycle, as shown in Figure 10. It should be noted that the directions of F gas,z and F gas,back are downward, so they are both negative values. When the orbiting angle turns to about 270°, the absolute value of F gas,back begins to drop sharply, which is due to the compressor running to the exhaust stage.
The displacements of the FS and OS are shown in Figure 11. When the FS is designed to be movable, the displacement e FS,z ranges between 4.47 µm and 5.56 µm. When the orbiting angle is near 270°, the FS begins to float to a large extent, and finally reaches the maximum value of 5.56 µm at orbiting angle is 342°. The reason for the floating of the FS can be seen from the z direction gas forces. At most times, F gas,back is greater than F gas,z , and the FS is pressed downward, and when F gas,back is less than F gas,z , the FS will obviously float. However, the time period when F gas,back is less than F gas,z is very short, and the FS will not float continuously. This figure also shows the instantaneous velocity de FS,z /dt of the floating FS, which will affect the oil film squeezing effect between the scroll surfaces, and then affect the oil film force.
The results of the OS movement and overturning in Figure 11 show that if the fixed clearance C FO increases, the e OS,z of one cycle will also increase. When the FS is free, the value of e OS,z is slightly less than that when C FO = 5µm, which means that when C FO = 4µm, the OS is subject to additional constraints, while when C FO = 6µm, the constraints are not enough. The attitude of the OS can represent the forces and moments acting on it, so it is necessary to analyze its overturning angles. When C FO = 4µm, the overturning angle ϕ OS,x in one cycle is significantly different from that when C FO = 5μm and 6μm, which indicates that the small clearance limits the motion attitude of the OS. ϕ OS,x and ϕ OS,y of the floating FS are close to the scheme of C FO = 5µm, indicating that the OS is subject to similar constraints in these two schemes. The kinematics results of FS and OS can provide reference for the design of axial clearance in the compression chamber.

Dynamic analysis of thrust surface
The variations of oil film force F oil,TH and contact force F c,TH on the thrust surface in one cycle are shown in Figure 12. When orbiting angle is 61°, the maximum F oil,TH of C FO =4µm is 29076 N, while the maximum values under other C FO are less than 10000 N. It is worth noting that the floating FS will make the change trend of F oil,TH different from that of the fixed FS, that is, the F oil,TH increases first and begins to decrease when the orbiting angle is about 270°. It can be seen from Figure 11 that when the orbiting angle about 270°, the F gas,back starts to decrease, and the OS moves up with the FS, resulting in this phenomenon. The variations of contact forces F c,TH also show the law that the larger C FO is, the smaller the force is. When orbiting angle is 61°, the contact forces of C FO = 4µm, 5µm and 6µm are 1755N, 376N and 251N respectively. The contact force of the floating FS scheme is slightly higher than that of C FO = 5µm. The variations of the forces on the thrust surface in one cycle show that small clearance makes the forces greatly increased, which is more likely to cause wear. The floating FS has less influence on the contact force and more significant influence on the oil film force, which can be attributed to that the OS moves with the FS, and the oil film on the thrust surface is continuously squeezed or stretched.
The minimum oil film thickness H min,TH of the thrust friction pair can further explain the variations of forces. As shown in Figure 13, the H min,TH increases with the increase of C FO , when the FS is designed to be floating, the result of H min,TH is close to that of C FO = 5μm. When the orbiting angle is 61°, there is a valley value of H min,TH due to the serious deformation of the thrust friction pair (D TH = −1.5 ∼ 0.4µm). Figure 14 shows the cloud diagrams of the oil film pressure distribution on the thrust surface of orbiting angle is 61°. In the case of C FO = 4µm, H min,TH is very small, and D TH causes the peak value of the oil film pressure of 271 MPa. In the case of C FO = 5μm and 6μm, the peak values are 38.1 and 30.3 MPa, respectively. When the FS is designed to be movable, the peak value of oil film pressure is 38 MPa. Figure 15 shows the contact pressure distribution at orbiting angle is 61°, which is similar to the oil film pressure distribution -the deformation of the friction pair causes the peak of the contact pressure. When C FO = 4µm, 5µm and 6µm, the peak values of contact pressure are 12.3 , 3.35 and 2.42 MPa respectively. When the FS is designed to be movable, the peak value of contact pressure is 3.59 MPa.

Dynamic analysis of scroll surface
The clearance and forces of the scroll friction pair are analyzed in this section. It should be noted that the dynamics laws of the OS scroll surface vs. the FS tooth root surface are similar to those of the FS scroll surface vs. the OS tooth root surface, so only the results of the latter are shown here.
The oil film forces F oil,FS and contact force F c,FS exerted on the OS by the FS scroll are shown in Figure 16, and oil film forces are relative values of the ambient pressure. According to the coordinate system established in this study, when the oil film forces are negative, it means that the oil film forces on the OS are greater than the ambient pressure, and vice versa. Figure 16 shows that under the fixed FS scheme, the effect of C FO on the oil film forces is not significant, and the oil film forces relative to the ambient pressure (C FO = 5, 6µm)  are around 0. When C FO = 4µm, the variation of oil film force is slightly obvious, which is because the smaller the clearance is, the more significant the effect of deformation on the hydrodynamic pressure effect is. In the scheme of floating FS, F oil,FS is obviously different from other schemes and can be analyzed in two stages. When the orbiting angle is between 270°∼ 315°, the FS will obviously float and the oil film will be stretched, resulting in F oil,FS less than the ambient pressure; When the crankshaft angle is between 110°∼ 270°, F gas,back makes the FS move downward and the oil film is squeezed, so F oil,FS is greater than that under the fixed scheme.
According to the directions of the coordinate system, the values of F c,FS are negative. In the case of C FO = 4µm, when the expansion deformation of the scroll surfaces is serious (orbiting angle = 61°), the contact force can reach 23492 N. With the increase of C FO , this phenomenon will be reduced. When the orbiting angle is between 110°∼ 300°, F c,FS of floating FS is obviously greater than that under C FO = 5µm. Although this may increase the friction loss, it is beneficial to the sealing of the air chamber. In the scheme with C FO = 6µm, F c,FS in a cycle is always less than 1N, which indicates that excessive increase of C FO will result in no axial contact between FS and OS in one cycle. Therefore, the sealing performance of the compression chamber is reduced.
The minimum oil film thickness of the FS scroll surface is H min,FS as shown in Figure 17. When the orbiting  still has sharp points, but when the FS can float, H min,FS changes smoothly. This means that the floating FS can avoid sudden decrease of clearance, thus maintaining stability. Figure 17 also shows the composite deformation D FS of the FS scroll surface when orbiting angle = 61°. The peak value of D FS reaches 2.8μm, which explains why H min,FS has a sharp point at this moment. Figure 18 shows the distribution of oil film pressure p oil,FS of the FS scroll surface when the orbiting angle is 61°. The peak value of p oil,FS appears in the innermost region of the scroll due to the high boundary pressure in this region (see Figure 6). In the scheme of C FO = 4µm, p oil,FS at the outermost circle of the scroll is less than the boundary pressure, which is due to the deformation of the FS scroll surface, and the phenomenon is no longer obvious when C FO is increased. In addition, because the scroll surface is narrow, p oil,FS in most areas is close to the boundary pressure. In the floating FS scheme, p oil,FS is less than the boundary pressure, which is attributed to that FS is floating upward at this moment and the oil film is stretched, as shown in Figure 11. Figure 19 shows the distribution of contact pressure p c,FS of FS scroll surface when the orbiting angle is 61°. When C FO = 4µm, due to the deformation of the scroll surface, the peak value of p c,FS is 83.4 MPa, which is much Figure 19. Contact pressure distribution on the FS scroll surface at orbiting angle is 61°.
higher than other schemes. In the schemes of C FO = 5µm and floating FS, the peak pressures are 14.8 and 10.9 MPa respectively, while the peak pressure of C FO = 6µm is only 0.9 MPa. From the above comparison, the floating FS scheme can effectively avoid the sudden increase of p c,FS , to ensure the stability of the compressor. Although increasing C FO can also reduce the contact pressure, it is detrimental to the seal. Figure 20 shows the friction loss P f in one cycle, where (a), (b) and (c) are the friction loss of the FS scroll surface P f ,FS , the OS scroll surface P f ,OS and the thrust surface P f ,TH respectively, and (d) is the friction work of three groups of friction pairs under the scheme of fixed FS and floating FS. Equation (29) is the expression of P f (Li et al., 2021;R. Liu et al., 2020):

Friction loss
where v is the relative sliding speed of the friction pair, φ f and φ fs are the shear stress factors, and their detailed descriptions can refer to the work of Patir and Cheng (Patir & Cheng, 1979). It can be seen that P f ,FS is the dominant friction loss and P f ,OS is the smallest. The smaller the C FO , the greater the friction loss, especially when C FO = 4µm, almost all P f ,FS values are above 2000 W in one cycle. The friction loss of the floating FS scheme increases in the gas compression stage and decreases significantly in the exhaust stage, which is due to the variation of the friction pair clearances. It can be seen from Figure 20(d) that when C FO increases from 4µm to 6µm, the friction work decreases by 69%, however, in the case the excessive friction pair clearance will weaken the compression chamber seal. For example, when C FO = 6µm, F c,FS is always less than 1 N, which means that there is almost no contact between friction pairs. The friction work of C FO = 5µm and floating scheme is reduced by 58% and 53% respectively compared with C FO = 4µm. However, considering the stability and sealing between scrolls, the comprehensive performance of floating FS is superior to the scheme of C FO = 5µm.

Conclusion
In this study, a tribo-dynamics model of scroll compressor with floating fixed scroll is independently developed based on the Fortran language, and the influences of the floating fixed scroll on the compressor performance are analyzed. The primary conclusions are summarized as follows: (1) To accurately investigate the influence of the floating fixed scroll on the tribological and dynamics performance of the scroll compressor, it is necessary to consider the coupling effect of the fixed scroll, the orbiting scroll and the frame as well as the actual operating environment, and establish a tribo-dynamics coupling model of the key friction pairs. (2) In the fixed scheme, it is difficult to consider both the sealing performance and friction loss of the compressor. However, in the floating scheme, the scroll can adjust its position flexibly under the action of axial gas forces to maintain the sealing in the compression stage and reduce the friction loss in the exhaust stage. In addition, the floating fixed scroll can provide compensation space for deformation, which is conducive to the stability of compressor operation.
(3) Compared with the scheme of 4μm fixed clearance, the friction work of fixed clearance of 5μm, 6μm and floating fixed scroll is reduced by 58%, 69% and 53% respectively. Considering the comprehensive performance, the floating fixed scroll is the optimal scheme.
The current research has revealed the influence mechanism of floating fixed scroll on the tribology and dynamics of key friction pairs, which can provide a reference for the optimal design of scroll compressor. However, due to the limitation of experimental conditions, the simulation model has not been verified. At present, the cooperative enterprise is conducting real machine experiments according to the conclusions of this study. In the future, the model will be further optimized according to the experimental results to make it more accurate and reliable.