Squeeze Film Effect and Its Influence on the Lubrication Characteristics of Slipper Oil Film under Pressure Pulse

. Tis study aims at the lubricant flm between the slipper and swashplate of an axial piston pump. Based on a single piston/slipper assembly model, a mathematical model for the squeeze flm efect is developed according to the pressure-fow transfer process as well as the dynamics of the piston/slipper assembly. Te squeeze flm efect on the slipper lubricant flm has been predicted by nonlinear numerical calculations and computational fuid dynamic (CFD). Te slipper lubricant flm thickness and leakage fow, sealing land pressure distribution, and load carrying capacity are evaluated as important parameters to indicate the performance of slipper oil flm. Te infuence of piston/slipper assembly geometric parameters, oil compressibility, and oil viscosity on the squeeze flm efect under pressure pulse has been studied, and the efect of squeeze flm speed on the support force of slipper lubricant flm has been given out.


Introduction
In the process of diversifed and extreme applications of axial piston pumps, the performance requirements of the key friction pairs are becoming higher and higher, and the slipper-swashplate pair is one of the key friction pair of axial piston pumps, which is most prone to wear and thus afects the reliability of the piston pumps. During the operation of an axial piston pump, a micrometer-thick oil flm exists between the slipper and swashplate, which plays the role of lubrication and support and is the key to protect the friction pair to avoid wear. When its load carrying capacity is not enough to resist the external load force, the slipper friction pair will be in the extreme working condition of boundary lubrication or even dry friction, and the surface of the slipper or swashplate will wear quickly, causing the pump to fail to operate properly. Te oil flm lubrication characteristics of the slipper directly afect the working performance of the piston pump. Te study on the oil flm of the slipper and the squeeze flm efect on the performance has an important guiding signifcance for the structural design of the axial piston pump.
Many theoretical analyses and experimental research studies on the fuid lubrication characteristics between the slipper and swashplate of axial piston pump have been continuously explored in recent years. Canbulut [1] used experimental measurements to investigate the impact of surface roughness, damping orifce, pressure in the piston chamber, and rotation speed on the leakage fow of the slipper-swashplate pair under the hydrostatic support, but these experiments did not show the characteristics of the lubricant flm between slipper and swashplate at the moment of the pressure pulse in the piston chamber. Harris et al. [2] developed a numerical model to predict the lift and tilt behavior of the slipper-swashplate pair under a constant temperature feld. Tey investigated and verifed by experiments that the load and overturning characteristics of the slipper may lead to contact between slipper and swashplate. However, the squeezing flm efect was not considered. Deeken [3][4][5] analyzed the oil flm characteristics of the slipper pair in a hydraulic pump by virtual prototype simulation and verifed the simulation results through experiments. Manring et al. [6] studied the infuence of different ball and socket structures of slipper on the oil flm characteristics of slipper pairs through experiments. Borghi et al. [7] established a static model of slipper to predict the critical speed of excessive increase of the oil flm thickness between the slipper and swashplate and paid attention to the infuence of spring pressure on the maximum rotation speed of the pump. In this model, the pressure change of the piston chamber and the viscous friction of the piston pair were considered. Hooke et al. [8,9] have studied the efects of the oil conditions and slipper surface conditions on the performance of slipper and measured the oil flm thickness in a short time during the operation of the piston pump. Tey have conceived that in the slipper structure designed according to the hydrostatic bearing method, the compression force of the slipper is not only provided by the hydrostatic supporting force but also balanced by the dynamic pressure efect of the oil flm between the slipper and the swashplate. And they compared the bearing capacity of the slipper pair under diferent damping orifces through experiments and verifed the balance performance of the slipper pair under diferent static pressure efect designs. Te Ivantysynova and Huang [10] team coupled the dynamic characteristics of the slipper with the multiphysical feld and obtained more accurate results than with the single physical feld. Te CASPAR software program was compiled to study the hydrodynamic, dynamic, and temperature characteristics, and the viscosity, temperature, and pressure of the oil and the compressibility of the oil were considered. Schenk [11] established a new model of slipper pair, used this to solve the dynamic pressure feld and thickness feld of slipper pair during operation, and verifed the working characteristics of the oil flm of slipper pair by displacement sensor. Zhang et al. [12,13] studied the spin motion of axial piston pump slippers through experiments; they used sensors to measure the spin speed of the slipper while the pump was operating, they also used sensors and other means to measure the thickness of the slipper oil flm at multiple points and they obtained some experimental sample data. A theoretical study has been studied the efect of surface roughness on the hydrodynamic lubrication of squeeze flm between a sphere and a fat plate by Naduvinamani et al. [14]. It is found that azimuthal roughness could increase the load carrying capacity and squeeze flm time compared to smooth condition. Qing-rui and Hou [15] have investigated the squeeze flm efect on hydro-viscous drive speed regulation and obtained an approximate 30% increase in oil flm loading capacity for the squeeze flm efect. Tey also found that the squeeze flm efect is very slight in the early stages of the startup process, but the squeeze flm efect becomes more and more remarkable as the oil flm thickness decreases in the later stages. Te negative ofset surface roughness pattern was found to increase the load carrying capacity and oil flm squeeze time through numerical analysis by Bujurke et al. [16]. Walicka [17] has obtained the pressure distribution of the Schulman fuid squeeze oil flm efect on porous surfaces by a continuous approximation method. Haidak and Wang [18] realized the damage and fatigue analysis of the slipper/ swashplate interface by predicting the solid deformation, strain, and wear of the solid for a given material. Ye et al. [19,20] studied the bearing capacity of the surface-textured slipper, introduced the elastohydrodynamic lubrication model, analyzed the bearing capacity of the oil flm of the slipper pair under diferent speeds and load pressures and considered the comprehensive efects of inclination angle, area density, depth diameter ratio, and other factors. In addition, they also established the thermal elastohydrodynamic lubrication model of the slipper oil flm for the piston pump and discussed the deformation of the slipper and the distribution of slipper oil flm thickness, pressure, and temperature. Te efects of working conditions and structural parameters of slippers, such as oil flm thickness, pressure, temperature and leakage fow, on the lubrication performance of TEHD are studied, and the experiments show that this research has certain guiding signifcance for the structural optimization of slippers. Based on the equation of motion, generalized Reynolds equation, energy equation, and Fourier heat equation, Hashemi et al. [21] established the multibody dynamics calculation model of a swashplate axial piston pump and discussed the infuence of the holding device on slipper friction and temperature.
Te squeeze flm phenomenon exists between the slipper and swashplate of an axial piston pump, as these two lubricating surfaces are near each other at a high speed during the pump operation. Tis study focuses on the squeeze flm efect at the sliding interface between the slipper and swashplate. To further predict the squeeze efect on the lubricant flm, a numerical model will be developed. Te numerical model of the squeeze flm efect considers the multiphysics and multiscale, and it can predict the squeeze flm efect under the pressure pulse. Figure 1 shows the axial piston pump and its component parts.

Mathematical Model
In this section, the mathematical model of the squeeze flm efect on slipper lubricant flm will be derived from the dynamics of the piston/slipper assembly and the pressurefow transfer of fuid. We can predict the force and motion trend of the piston/slipper assembly based on the dynamics model under the squeeze flm efect. At the same time, we can predict the pressure distribution and fow transfer on the slipper oil flm under the squeeze flm efect according to the fuid lubrication model. In order to simplify the model, only the hydrostatic nature of the fuid is considered. Figure 2 gives the external forces acting on the piston/slipper assembly. Te assembly is balanced by compression force and support force. Te compression force contains F p , F k , F a , F f , where F p is the hydraulic force caused by oil pressure from the displacement chamber, F k is the spring force, F a is the inertia force due to the movement of the piston/slipper assembly, F f is the friction force caused by friction reaction of the cylinder bore. In this paper, the support force generated by the lubricant flm between slipper and swashplate is divided into hydrostatic support force F N1 and the squeeze support force F N2 . Tese support forces play a role of the lifting the piston/slipper assembly.

Dynamics Model of the Piston/Slipper Assembly.
In the process of modeling squeeze flm efect, only the oil pressure from the bottom of the piston is considered, as the other forces only afect the value of the compression force. Tey have no infuence on the analysis of the squeeze flm efect. Tus, the force balance equation of the piston/ slipper assembly parallel to direction of the shaft axis can be written as follows: where the hydraulic force can be given by 2.2. Hydrostatic Support Force. Due to diferential pressure at the inlet and outlet of the slipper sealing land, a hydrostatic support force exists on the sealing land between the slipper and swashplate. Te pressure drop causes oil to fow from the slipper pocket to the pump chamber, creating a pressure distribution on the sealing land. Figure 3 shows the schematic diagram of the slipper pair. Te support forces resulting from pressure distribution on the sealing land of the slipper [22] can be described as follows: According to the abovementioned equation, the hydrostatic support force on the sealing land of the slipper is infuenced by the geometric parameters of the slipper, independent of the flm thickness. Figure 2, during operation of the hydraulic pump, a lubrication oil flm of thickness h is formed between the slipper and swashplate. Te piston continuously suctions and discharges the pressure at the bottom of the piston constantly switches. Ten the oil flm of the slipper is squeezed and thinned at a speed v. Meanwhile, there is a certain amount of fow to be excluded along the circle to the piston chamber. Ten the squeeze flm efect produces a pressure feld, which means the squeeze flm efect produces a support force.

Support Force from the Squeeze Film Efect. As shown in
Te microelementary ring at radius of r is used as an object to study.
Under the efect of pressure drop, the fow through this microelement ring band is presented as follows: Similarly, the pressure distribution [22] along the radius is as follows:  Figure 1: Axial piston pump and its component parts. Shock and Vibration Te support force formed by the squeeze flm efect can be expressed as follows: Te dynamics of piston/slipper assembly under the squeeze flm efect is modeled as follows:

Fluid Lubrication Model of Slipper Pads under the Squeeze
Film Efect. As shown in Figure 4, oil from the piston chamber fows through the piston orifce into the slipper socket, then through the slipper orifce into the slipper pocket, and fnally through the gap between the slipper and swashplate into the pump chamber. A layer of lubricating oil flm is formed between the slipper and swashplate. Te pressure-fow transfer process in the piston/slipper assembly is analyzed, and then a fuid lubrication theoretical model of the slipper lubricant flm under squeeze flm efect is instituted.
Te following assumptions were made: (1) Neglecting the efects of surface tolerances, slipper and swashplate defects, and surface structure on oil flm properties (2) Disregarding the dynamic pressure efect on the oil flm of the slipper, it is considered that the bottom surface of the slipper is always parallel to the plane of the swashplate (3) Leakage at the slipper socket is not considered (4) Assume that oil density does not change with temperature.
Oil from the piston chamber passes through the piston orifce to the slipper socket and creates a pressure drop. Te fow equation through the piston orifce [22] is as follows: Oil enters the slipper pocket through the slipper orifce, and the fow equation through the slipper orifce [22] can be given by where q 1 is equals to q 2 ; thus, Oil from the slipper pocket fows through the gap between slipper and swashplate to the pump chamber, and q 3 can be written as follows: Te pressure pulse causes a change in force at the bottom of the piston, the force which causes the piston/slipper assembly move towards the swashplate, and then the slipper lubricant flm is squeezed. Te flm thickness decreases due to the squeeze flm efect on the slipper. For the expression of equation, we need to make another assumptions, assuming that the initial oil flm thickness is h 0 , the change of oil flm thickness is x, so that, the oil flm thickness can be expressed as where the pump chamber pressure is 0, and then equation (11) becomes Te fow rate over the sealing land of the slipper can be given by where q 4 is the fow produced by the squeeze flm efect, q 4 can be given by According to the compressibility equation of the slipper pair fuid,  Shock and Vibration q 1 , q 3 , and q 4 can be integrated in equation (14), and then the fuid lubrication model of slipper under the squeeze flm efect is as follows: Equations (7) and (17) presented above are the global relationships that may be used to predict the dynamic behavior of the squeeze flm efect on the slipper/swashplate pad system. Equation (7) represents the dynamics model of the piston/slipper assembly, while equation (17) expresses the pressure-fow transfer process of slipper oil flm under the squeeze flm efect. According to equation (17), we can see that the force conditions and motion trends have an impact on the pressure-fow transfer process, while according to equation (7), we can also get that the pressure distribution and fow transfer afect the dynamics model of the piston/slipper assembly. Only when the two models are combined with each other can we better predict the lubrication characteristics of slipper oil flms under the squeeze flm efect.

Nonlinear Model of the Squeeze Film Effect on Simulink
During the operation of the hydraulic piston pump, the pressure at the bottom of the piston is constantly transformed between high and low pressure due to the continuous rotation of the cylinder. When the pressure at the bottom of the piston changes from the low pressure zone to the high pressure zone, the pressure change in the piston chamber is equivalent to creating a pressure pulse at the bottom of the piston. Terefore, the pressure pulse can most directly refect the lubrication characteristics of the slipper oil flm under excitation conditions. Tus, in the Simulink model, we used the pressure pulse function to research the squeeze flm efect, and also in the subsequent study of the fuid feld properties, we applied the squeeze flm efect under the action of the pressure pulse.
Simulink is used to model the theoretical model of the squeeze flm process and analyze the efect of parameter variation on the squeeze flm process of the slipper pair. And the Simulink model is shown in Figure 5. Te pressure pulse at the bottom of the piston is simulated by using the step function, and 28 MPa is taken.
According to the model, the squeeze flm efect with diferent geometric parameters and diferent oil viscosities will be studied. Before that, the initial values of the parameters were obtained, and they are listed in Table 1. Te diferent geometric parameters and oil viscosity are selected as a single variable to calculate the model. In order to reveal the properties of lubricant flm, the slipper pocket pressure, lubricant flm thickness, and leakage will be used to analyze the performance of lubricant flm.
From Figure 6(a) and Table 2, under the pressure pulse of 28 MPa, as the r 2 (mm) increases (from 0.6 to 1.2), the slipper pocket pressure increases (from 27.81 MPa to 27.98 MPa), but it takes less time for the slipper pocket pressure to build up (from 6 × 10 −5 s to 5 × 10 −6 s). Tese can reveal that as the radius of the piston orifce increases, the damping efect weakens and the response time for pressure buildup continues to decrease. According to Figure 6(b) and Table 2, the time for the slipper pocket pressure to build up increase (from 3 × 10 −6 s to 8 × 10 −6 s) when the r 3 (mm) increases (from 0.2 to 0.5) and the change of the slipper pocket pressure is no longer signifcant. For Figure 6(c) and Table 2, when the r 5 (mm) increases (from 12 to 13.5), the slipper pocket pressure decreases (from 27.99 MPa to 27.98 MPa). Although this impact on the pressure value is slight, the efect on the time needed for pressure buildup is remarkable. And the time decreases (from 5 × 10 −4 s to 2 × 10 −5 s). Meanwhile, r 5 can represent the width of the slipper sealing land. Te increase in the width of slipper sealing land leads to an increase in the time required of slipper pocket pressure. Figure 6(d) and Table 2 show that (a) under the pressure pulse of 28 MPa, the slipper pocket pressure increases (from 27.97 MPa to 27.975 MPa) when the h 1 (mm) increases (from 0.5 to 2); (b) the time for slipper pocket pressure to build up increases (from 1 × 10 −5 s to 2 × 10 −5 s). In fact, h 1 can refect the volume of the slipper pocket. Tis numerical model of the squeeze flm efect considers the oil compressibility, so we checked the infuence of oil compressibility on the pressure buildup of the slipper pocket under the squeeze flm efect using the parameter h 1 . Figure 6(e) shows the efect of oil viscosity; it has the same trend as Figure 6(c). Figures 7(a), 7(b), and 7(d) and Table 2 show that (a) the variation of r 2 (mm), r 3 (mm), and h 1 (mm) has almost no infuence on the oil flm thickness under the squeeze flm efect; (b) the lubricant flm thickness increases when r 2 increases; (c) the lubricant flm thickness decreases while r 3 or h 1 increases. According to Figures 7(c) and 7(e) and Table 1, (a) with the increase of r 5 (mm) (from 12 to 13.5), the lubricant flm thickness increases (from 2.125 to 6.203); (b) the lubricant flm thickness increases (from 2.360 to 4.663) when μ (Pa·s) increases (from 0.01 to 0.04); and (c) increasing either r 5 or μ weakens the squeeze flm efect on the lubricant flm.
With reference to Figure 8(a) and Table 2, it is observed that at the moment of the pressure pulse, the leakage fow rate of the slipper-swashplate interface decreases (from 5.021 L/min to 1.834 L/min) when the r 2 (mm) increases Shock and Vibration 5 (from 0.6 to 1.2). However, in the end, the leakage fow rate will become the lowest when r 2 is the smallest. From Figure 8(b) and Table 2, there is a gradual decline in the leakage fow rate due to the decreases of r 3 . Te radius of the piston orifce and radius of the slipper orifce have opposite efects on the slipper leakage fow rate. For Figures 8(c) and 8(e) and Table 2, the width of the sealing land and the oil viscosity have the same tendency to afect the slipper leakage fow rate. All those trends in leakage are similar to Figure 8(a). As shown in Figure 8(d) and Table 2, the height of the slipper pocket and the radius of the slipper orifce have the same tendency to afect the leakage of the slipper pair. From Figures 6 to 8, in the process of squeezing the lubricant flm, both geometric parameters and working condition parameters afect the pressure buildup in the slipper pocket. Under the action of the pressure pulse, the leakage fow rate and oil flm thickness are mainly afected by the establishment of slipper pocket pressure. Te establishment of the slipper pocket pressure determines the support force buildup between the slipper and swashplate. When the time required to establish the support force becomes longer, the lubricant flm will be quickly squeezed and thinned because of the squeeze flm efect. At the same time, the leakage fow rate caused by the squeeze flm efect will increase. After the slipper pocket pressure is built up, squeeze flm speed becomes slower, and then the leakage fow rate is related to the lubricant flm thickness at the next time. Te smaller the flm thickness is, the smaller the leakage is. Although the squeeze flm efect leads to slipper leakage fow rate, which ultimately causes power loss, it shows that the establishment of the lubricating oil flm between the slipper and swashplate ensures the proper action of the piston/slipper assembly under pressure pulse from the piston chamber at the moment.

Fluid Simulation of the Squeeze Film Effect
In this section, the Integrated Computer Engineering and Manufacturing Code for Computational Fluid Dynamics (ICEMCFD) is used to verify the squeeze flm efect of the slipper pair. All the simulations are conducted using the commercial software package ANSYS 19.0, in which the ICEMCFD analysis module is used to calculate the squeeze process of the lubricant flm of the slipper. A 3D CAD model of the lubricant flm is created by the SolidWorks software, and then the model is imported into the ANSYS ICEMCFD. Te model contains geometric parameters such as the inner and outer radius of the slipper sealing land and the assumed initial thickness of the slipper pads lubricant flm. Also, the structure includes two surfaces of the lubricant flm between the slipper and swashplate. Te fnite element model of     Figure 13 illustrates the support forces for the piston/ slipper assembly generated by the squeeze flm efect. Te higher squeeze speed corresponds to a higher support force, and the result is due to factor that the flm thickness becomes lower due to the squeeze flm efect. And the hydrostatic support force remains unchanged because the squeeze flm efect has no efect on it.

Conclusion
In this paper, the squeeze flm efect on the slipper pair is introduced. Te mathematical model for the squeeze flm efect is derived, in which varying structural parameters of the slipper/piston assembly and oil viscosity have an impact on the lubricant flm's performance. Te calculations and simulations on the lubricant flm between the slipper and the swashplate with diferent parameters are achieved and different pressure distribution and support forces on the sealing land with varying squeeze velocity are obtained. Te reasons for these results are analyzed. And from the mathematical modeling and simulation analysis, we can get    F N2 : Squeeze support force (N) θ: Swashplate angle (°) m: Piston/slipper assembly mass (kg) a: Piston/slipper assembly acceleration (m/s 2 ) r 1 : Piston radius (mm) r 2 : Piston orifce radius (mm) r 3 : Slipper orifce radius (mm) r 4 : Slipper sealing land inner radius (mm) r 5 : Slipper sealing land outer radius (mm) r: Radius of any point on slipper sealing land (mm) h: Slipper oil flm thickness (mm) μ: Oil viscosity (Pa · s) l: Piston orifce length (mm) C d : Flow rate coefcient ρ: Oil density (kg/m 3 ) E: Volume oil elastic modulus (MPa) V 1 : Slipper pocket volume (m 3 ) h 1 : Slipper pocket height (mm) h 0 : Initial oil flm thickness (mm).

Data Availability
Te data used to support the fnding of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.