Coupled evolution of piston asperity and cylinder bore contour of piston/cylinder pair in axial piston pump

Fei LYU, Junhui ZHANG,*, Shoujun ZHAO, Kun LI, Bing XU,Weidi HUANG, Hogong XU, Xiohen HUANG

a State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, China

b Beijing Institute of Precision Mechatronics and Controls, Beijing 100076, China

c Qing’an Group Co., Ltd., Xi’an 710077, China

KEYWORDS Asperity;Contour;Coupled evolution;Piston/cylinder pair;Wear process

Abstract The wear condition of the piston/cylinder pair is crucial to the performance and reliability of the axial piston pump.The hard piston surface, the soft cylinder bore surface, and the interface oil film affects each other during the wear process.Specifically,in the mixed lubrication region,the geometry of the hard piston surface asperity directly affects the wear of soft cylinder bore surface, while the asperities may deform or even degrade when penetrating and sliding against the cylinder bore.So far, there is no suitable method to simulate their coupled evolution.This paper proposed a wear process simulation model considering the real-time interaction between the elasto-plastic deformation of the piston surface asperity, the wear contour of the cylinder bore,and the lubrication condition of the interface.An offline library of the elasto-plastic constitutive behavior of the asperity based on the finite element method (FEM) is established as a part of the simulation model to precisely analyze the deformation and degradation of the asperity and quickly invoke them in the numerical wear process simulation.The simulation and experimental results show that the piston asperity and the cylinder bore contour converge to a steady state after running-in for about 0.5 h.The distribution of the simulated asperity degradation and wear depth is also verified by the experiment.

1.Introduction

Higher power density is not only the superiority but also the trend of the hydraulic systems.Enhancing the nominal working pressure is an effective way to improve power density.1–4In the past few decades, the system pressure of the aircraft has been enhanced from 21 MPa to 35 MPa.However, higher pressure leads to the challenge of the reliability of key friction pairs in axial piston pumps, which play the role of ‘‘heart”in hydraulic systems.5–8As a positive displacement pump, the axial piston pump sucks and discharges hydraulic medium by changing the volumes of piston chambers.The piston/cylinder pair is the key friction pair to ensure the lubrication and seal of the piston chamber.So the reliability of the piston/cylinder pair is significant for axial piston pumps.9The force state of the piston is shown in Fig.1.The hydraulic medium with high pressure produces a force FpOPon the bottom of the piston.The supporting force from the slipper FsSPis applied on the ball joint of the piston to balance the axial forces.However, the slipper slides on the inclined swashplate,so the piston is subjected to a large lateral force Ftwhich consists of the lateral component of the supporting force FsSPand the friction between the slipper and the swashplate FfWS.Excessive lateral force makes the piston tilt in the cylinder bore.

The force state of the piston/cylinder pair is severe,and the risk of wear failure is high due to the large time-varying lateral force.The wear of the piston/cylinder pair is gradual and accumulative,and the failure occurs when the wear accumulates to a certain extent.10Therefore,clarifying the wear process is significant for prolonging the service life of piston/cylinder pair by bearing surface design and even achieving predictive maintenance of axial piston pump.

In the piston/cylinder pair, the piston surface is generally harder than the cylinder bore surface, so the wear mainly occurs on the cylinder bore surface.As shown in Fig.1, due to the tilting piston, the contact condition of the piston/cylinder interface is distributed.In the region where the metallic contact occurs,like condition Fig.1(b),the scratches,grooves,and ripples may be generated on the cylinder bore surface by the hard asperity of the piston surface, and then, the abrasive wear occurs.11,12In the region where the metallic contact is severe, like condition Fig.1(a), adhesive wear may occur on the cylinder bore surface.13–16Therefore, the wear mechanism is determined by the local lubrication conditions.The surface contour gradually changes by the distributed wear.Simultaneously, as the geometric boundary, the wear contour of the cylinder bore affects the lubrication condition.Considering the coupling between the wear contour and the load-bearing and lubrication parameters (LBLPs), the wear prediction methods of the friction pairs in the axial piston pump were proposed,17–19and the accumulative wear process of the softer surface was predicted.

Fig.1 External force state of piston and distributed contact condition of interface.

However, the evolution of the piston/cylinder pair consists of not only the variation in the wear contour of the cylinder bore but also the degradation of the characteristics of the piston surface.Although the wear does not occur on the hard surface, the asperity, which ‘cuts’the cylinder bore surface as the‘‘tool”, will be deformed by the cutting force.20,21The deformation of the asperity could be plastic when the force state is severe.22Based on the research of several scholars,23–25Akbarzadeh and Khonsari gave the criterion for judging whether the asperity is in elastic contact,elasto-plastic contact,or plastic contact.26However, the contact condition of the asperity was simplified to Hertz contact, and the influence of the cutting force on the asperity deformation was not considered.Ghatrehsamani et al.supplemented the effects of the cutting force.The asperity was regraded to a cantilever beam,and the cutting force was applied on the end of the beam.The deformation of the beam was calculated based on the beam theory.27However, the actual shape of the asperity of the piston is oblate,which is not suitable to be analyzed by beam theory.It is noticed that the above methods are analytical.The contact condition and the shape of the asperity continuously change during the degradation process, and the analytical method is suitable to deal with the variations.Therefore,although the simplifications in these analytical methods make the deformation model of the asperity different from reality,the analytical method is still widely used to analyze the degradation process of the asperity.

It is well known that the numerical method is more accurate for the analysis of the complicated cutting deformation of the asperity.24,28,29But the difficulty of frequently changing model parameters and the large computational cost make the numerical method does not suitable for dealing with the continuous variation of the contact condition and the asperity shape during the degradation process.Furthermore, the degradation of the piston surface asperity and the wear of the cylinder bore affect each other during the operation, and the coupling between them is absent in the existing research.

To address these issues,a simulation method,which considers the coupling between the degradation of the piston surface asperity,the LBLPs,and wear contour of the cylinder bore,is proposed in this paper.An offline asperity deformation library is obtained by the finite element method (FEM).The plastic deformation of the asperity is invoked to simulate the unrecoverable degradation of the asperity.The total deformation of the asperity is invoked to the numerical calculation of the lubrication condition and the distributed calculation of the cylinder bore wear.The characteristics of the piston surface asperity, the cylinder bore contour and the LBLPs can be exchanged in real-time to simulate the coupled evolution of the piston/cylinder pair.The rest of this paper is outlined as below:In Section 2,the wear mechanism of the piston/cylinder pair, the offline piston surface asperity deformation library,and the simulation method for the coupled evolution of the piston/cylinder pair are introduced.In Section 3, the simulation results are analyzed, and they are verified by experiments in Section 4.The conclusions and outlooks are given in Section 5.

2.Model establishment

2.1.Distributed wear mechanism of cylinder bore

As is clear from Fig.1, the lateral force applied to the piston causes the piston to tilt.There are two main reaction forces to balance the external forces applied on the piston: oil film supporting force and metallic contacting force.The supporting force of the oil film is generated by hydrodynamic effect and squeezing effect of the oil film between the clearance of the piston/cylinder pair.These two effects are described by the Reynolds Equation30,31:

where p is the hydrodynamic pressure, h is the oil film thickness,μ is the viscosity of the hydraulic medium,RCis the cylinder bore radius, vPzis the piston axial sliding speed of the piston, vPθis the circumferential sliding speed of the piston which is obtained by the coupling dynamic model of the piston and the slipper.32,33It can be seen from the right side of Eq.(1)that the thickness of the oil film thickness varies with position and time.On the whole interface of the piston/cylinder pair,the solid contact conditions are different everywhere.

When the external force is too large to be supported by the oil film, the oil film may break in the regions where the reaction force should be large, and then the metallic contact force takes the responsibility.In this condition, the adhesive wear occurs and the Archard wear model can be used to calculate the adhesive wear volume Vwa34:

where psis the solid contact stress,Asis the solid contact area,σsis the yield strength of the cylinder bore, kais the adhesive wear coefficient which is determined by experiments,35and s is the sliding distance which is calculated by the piston sliding speeds as Eq.(3):

where t is the simulation time.Another wear mechanism occurring in the interface of the piston/cylinder pair is the sliding wear caused by the hard asperity of the piston,and it is the focus of this paper.The asperity of the piston surface may contact the cylinder bore surface in the regions where the oil film is thin.Because that the piston surface is much harder than the cylinder bore surface,the asperity of the piston could penetrate and‘cut’the cylinder bore when sliding.21According to previous experimental research,the initial asperity of the piston can be simplified as a cone.36The wear volume can be calculated by the product of the sliding distance and the penetrating section area of the asperity.The schematic diagram of the sliding wear is shown in Fig.2.In the figure,hAis the asperity height,hpis the penetrated depth, and dAis the base diameter of the asperity.

The wear volume per unit sliding distance equals the section area of the penetrating asperity, as expressed in Eq.(4)30:

Fig.2 Schematic diagram of cylinder bore wear caused by piston surface asperity.

where Vwsis the sliding wear volume.

According to the three lubrication conditions in Fig.1 and their corresponding wear mechanisms, the calculation method of wear volume of the cylinder bore V can be summarized as Eq.(5):

2.2.Deformation of the asperity

Fig.3 Schematic diagram of deformation of piston surface asperity.

It can be seen from Eq.(4)that the asperity height hAis critical to the sliding wear volume.As shown in Fig.3(a), when the wear occurs, the resistance FCPAloaded on the asperity could deflect the asperity.22After deflection, the section area of the penetrating asperity is shown in Fig.3(b).Assumed that the penetrating section area is still triangle, and it is calculated by Eq.(6):

where ΔhAtis the total deformation of the asperity height.Therefore, the asperity height after deformation is significant to the wear mechanism of the piston/cylinder bore.The deformation of the asperity could be elasto-plastic.The elastic deformation only affects the transient wear volume, and then the effect disappears.The plastic deformation, which permanently changes the asperity height on the piston surface, continuously affects the wear volume during the wear process.Therefore,both the total deformation and the plastic deformation of the asperity height need to be obtained.According to strain–displacement relationship, the total deformation of the asperity height ΔhAtcan be obtained by integrating the strain component εzzalong the radial direction of the piston, as shown in Eq.(7):

In Eq.(7), the strain component εzzcan be replaced by the plastic strain component, and then the plastic deformation ΔhApcan be obtained in the same way.To separate the plastic deformation from the total deformation, the plastic strain in the asperity needs to be obtained.

Fig.4 Establishment of FEM model of piston surface asperity and cylinder bore.

The FEM is used to calculate the total strain and the plastic strain in the penetrating and sliding asperity.Firstly,the geometry characteristics of the asperity are needed.The piston surface has been measured by a surface roughness profiler(SRP),and the shape of the asperity without filtering was output.As Fig.4(a) shows, several positions were zoomed in.The results show that the asperity height is about 0.15 μm and the base diameter of the asperity about 5.5 μm.Based on the measured geometry parameters, the FEM model is established as shown in Fig.4(b).To clearly display the shape of the asperity, the figure is scaled in the Xf-direction by a factor of 25%.Table 1 shows the mechanical properties of the piston material and the cylinder bore material.

The speed and displacement in the Xf-direction are applied to the asperity to simulate the relative sliding of the asperity and the cylinder bore, and the displacement in the Zfdirection is applied to the asperity to generate the different oil film thicknesses.The speed in the Xf-direction is defined as a constant of 1 m/s according to the average speed of the piston.17The displacements in the Xf-direction and Zfdirection are defined as shown in Fig.5(a).So that the deformations under different oil film thicknesses can be simulated based on this FEM model.The simulation results of the plastic strain component in Zf-direction are shown in Fig.5(b).

The total deformation of the asperity height can be directly obtained by tracking the node of the asperity peak, and it can be expressed by Eq.(8):

The total and the plastic deformations of the asperity at the different positions can be obtained by substituting the corresponding oil film thickness into Eq.(10) and Eq.(11).If the oil film thickness decreases,then total and plastic deformations of the asperity height increase according to the curves in Fig.6.If the oil film thickness increases,further deformation does notoccur.However, the asperity height has been changed due to the former plastic deformation, which means that in the next wear process, the initial asperity height is no longer 0.15 μm.Therefore, in addition to the oil film thickness, the initial asperity height is also a variable that needs to be considered to obtain the degradation process of the asperity.The FEM models with different initial asperity heights are established,and the simulation results are shown in Fig.7.

Table 1 Mechanical properties of piston material and cylinder bore material.

Fig.5 Displacements definition and simulation results.

Fig.6 Deformations of asperity height under different oil film thicknesses.

According to Fig.6 and Fig.7,the asperity height deformations under different oil film thicknesses and different initial asperity heights are fitted into Fig.8.Simultaneously, Eq.(10) and Eq.(11) can be expanded into Eq.(12) and Eq.(13):

The degradation of the piston surface asperity can be clarified.There exists a time interval Δt between two calculation steps of asperity height.If the plastic deformation of the current time step is greater than that of the previous time step,the asperity height at this time hAtdecreases,and the degradation value is given by Eq.(13).Otherwise, the asperity height does not change.The degradation mechanism of the asperity height can be expressed by Eq.(14):

2.3.Interaction between the asperity, the wear contour and the LBLPs

According to Eq.(15), the LBLPs of the piston/cylinder bore,like the oil film thickness h and the solid contact stress ps,directly affect the wear characteristics of piston and cylinder bore.In turn, the wear also affects the LBLPs.The influence of the asperity height of the piston surface on LBLPs is firstly analyzed.The LBLPs are derived by the piston force equilibrium equations which are based on the Reynolds Equation of the piston/cylinder pair.Patir and Cheng proposed an average flow model considering the effect of roughness, and the Reynolds Equation was revised as Eq.(16)37,38:

where φθis the pressure flow factors that revise the pressure flow rates under the rough surface in the circumferential direction, φzis that in the axial direction, φsθand φszare the shear flow factors which represent the additional flow due to the sliding asperities.For convenience,the roughness is assumed to be isotropic.Therefore, the flow factors in the axial and circumferential directions are equal, which can be approximately expressed as Eq.(17) and Eq.(18)37:

Fig.7 Deformation vs oil film thickness under different initial asperity heights.

Fig.8 Asperity height deformations under different oil film thicknesses and different initial asperity heights.

where σ is the standard deviation of the roughness function of the piston.In this paper,the roughness asperities of the piston are simplified to cones, and the relation between the asperity parameters and the standard deviation σ can be given by Eq.(19):

The Reynolds equation needs to be solved by numerical method which decomposes the Reynolds Equation into several linearized equations.To guarantee the convergence, the solving domain is discretized into na× ncpressure nodes and(2na+1)×(2nc-1)thickness nodes as shown in Fig.9,where nais the pressure node number along the axial direction,and ncis the pressure node number along the circumferential direction.

Fig.9 Schematic diagram of discretized oil film.

The finite difference method (FDM) is used to express the partial derivatives of p(i, j) in the θ-direction and the zdirection:

Replace the partial derivative in Eq.(21)with the difference quotient shown in Eq.(20).Then the relationship between the p(i, j) and the pressure of neighboring nodes can be expressed as Eq.(23):

Above all, the pressure of each node can be expressed by linear Eq.(23).The two boundaries of the z-direction are given by the piston chamber pressure and the case pressure.39The cyclic tridiagonal matrix algorithm (CTDMA) can be used to iteratively solve the pressure of each point starting from the boundaries.40

It can be seen from Eq.(24) that the pressure coefficients and the source term are affected by the flow factors.The flow factor of each node can be given by the corresponding asperity height hA.This is the effect of piston surface roughness on the LBLPs.

The cylinder bore contour effects the distributed oil film thickness h(i, j), and it has been analyzed in the former research as shown in Eq.(25)35:

where w(i, j) is the wear depth of the (i, j) node, e1and e2are the eccentricities in the horizontal direction of the two ends of the piston, respectively, e3and e4are the eccentricities in the vertical direction of the two ends of the piston, respectively,DCand DPare the diameters of the cylinder bore and the piston,respectively,Lois the length of the oil film,lhais the axial interval between the oil film thickness nodes.

Fig.10 Schematic diagram of wear process simulation.

The mechanism of the coupling between the LBLPs and the wear characteristics of the piston/cylinder pair is elaborated by the above analysis, and then the coupled evolution of the piston/cylinder pair can be clarified.The main steps of the simulation are described below:

? Import the initial cylinder bore contour and the initial piston roughness characteristics into the simulation model.

? Calculate the LBLPs, and the effects of the cylinder bore contour and the piston roughness are considered.

? Import the calculated LBLPs into the wear calculation as boundary conditions.Calculate the wear depth of the cylinder bore and the deformation of the asperity height at each node.

? Subtract the current wear depth of cylinder bore and degradation of asperity height from the cylinder bore contour and the asperity height at the previous time step (for the first time step,they are initial cylinder bore contour and initial asperity height).The updated cylinder bore contour and asperity height are obtained in this step.

? Replace the corresponding parameters in the LBLPs calculation and wear calculation with the updated cylinder bore contour and asperity height.Then repeat step 2 and start a new calculation cycle.Simultaneously, the current cylinder bore contour and asperity height are output and stored.The output of each calculation cycle constitutes the evolution of the piston/cylinder pair.

? While the simulation time t exceeds the set value te,the simulation ends.

The schematic diagram of the wear process simulation is illustrated in Fig.10.

3.Simulation analysis

The evolution of the piston/cylinder pair consists of the degradation of piston surface roughness and the wear of cylinder bore, and the evolution can be analyzed based on the simula-tion model.It should be noted that rise of the piston/cylinder pair temperature is not considered in the simulation model.The temperature is set to be 55°C at which the piston/cylinder pair reaches thermal equilibrium.41Some of the simulation parameters are shown in Table 2.

Table 2 Simulation parameters of piston/cylinder pair evolution model.

In order to clarify the general variation in the piston roughness and the cylinder bore contour during the wear process,the variables that can reflect the general characteristics of the piston roughness and the cylinder bore contour need to be defined.The variable δAreflects the total degradation of the piston surface asperity height,and Vwis the total wear volume of the cylinder bore.They are given by Eq.(26) and Eq.(27):

The simulation results of δAand Vware plotted in Fig.11.The piston roughness and the cylinder bore contour respectively converge to a steady state after a specific running-in period.The running-in period of the piston roughness is 0.47 h.At this time, the cylinder bore contour is not steady yet, and the wear rate of the cylinder bore keeps at a low degree until the wear contour reaches the steady state after 0.69 h.According to the convergence process of the piston roughness and the cylinder bore contour,the evolution of the piston/cylinder pair can be divided into the following stages:

Fig.11 General evolution of piston roughness and cylinder bore contour.

Stage 1: Both the piston roughness and the cylinder bore contour change.

Stage 2:The piston roughness reaches the steady state,and the cylinder bore contour continues to change with a smaller wear rate.

Stage 3: The piston roughness and the cylinder bore contour both reach the steady state.

After dividing the evolution of the piston/cylinder pair into three stages,the distributed piston roughness and cylinder bore contour, which illustrate the wear characteristics of the piston/cylinder pair during evolution in detail, can be analyzed.The distribution of the wear depth hw(l, θ) in the three stages is illustrated to analyze the evolution of the cylinder bore contour.As shown in Fig.12, the cylinder bore surface is expanded to an unwrapped plane for intuitively displaying the locations of the worn regions.The two ends of the cylinder bore in a specific circumferential range are severely worn.To clearly compare the discrepancy between the contours in the three stages, the evolutions of the circumferential profile of the cylinder bore at the two ends are displayed in Fig.13.In stage 1, the wear regions and the wear depths expand rapidly.According to Fig.11,the wear rate of the cylinder bore is small in stage 2.As a result, the variation of the cylinder bore contour from stage 2 to stage 3 is not obvious.In stage 2,the cylinder bore profile near the swash plate side reaches the steady state.Although the cylinder bore is slightly worn near the valveplate side, the wear regions do not expand.

The distribution of the piston roughness is drawn in Fig.14.Since the piston roughness degrades only in stage 1,the roughness distribution in stage 3 is not shown.It should be noted that the distribution along the circumferential direction is averaged due to the piston spinning, so the roughness distributes only along the axial direction.Fig.14 shows that the roughness mainly degrades at the bottom and the head of the piston.After running-in stage, the degradation lengths of the piston roughness definitively range from 5 mm to 28 mm and 70 mm to 85 mm.

The relative position of the bearing interface and the piston surface can be used to analyze the range of the asperity degradation.As shown in Fig.15, x1and x2are the coordinates of the valveplate-side end of the interface and the position of the swashplate-side end of the interface, respectively.During one rotation, x1and x2vary from 4.34 mm to 28.19 mm and 69.34 mm to 85.00 mm respectively, which almost completely coincides with the degradation lengths of the piston surface roughness.Therefore, the load-bearing and lubrication conditions at the two ends of the bearing interface are severe,and play an important role in the wear of the piston/cylinder pair.

The wear of the cylinder bore and the degradation of the piston surface asperity are calculated based on the LBLPs,so the evolution of LBLPs can explain the variation in the piston and the cylinder bore.The average hydrodynamic pressure Spand the average solid contact stress Ssare used to evaluate the LBLPs, and they are calculated by Eq.(28) and Eq.(29):

Fig.12 Entire cylinder bore contour in different stages.

Fig.13 End cylinder bore profiles in different stages.

Fig.14 Distributions of piston roughness in different wear stages.

As shown in Fig.16, in stage 1, the average hydrodynamic pressure increases, and the average solid contact stress decreases.The variations in the piston/cylinder pair enhance the hydrodynamic effect of the oil film and decrease the metallic contact in the running-in stage.

Fig.15 End positions of piston/cylinder interface.

4.Experimental validation

4.1.Design of experiment

The experiment was carried out to validate the simulated asperity degradation process and the wear process of the cylinder bore.The structural parameters and the operating conditions of the tested axial piston pump were consistent with the simulation model.The hydraulic medium used in the test is ISO 46 hydraulic oil, and the main properties are shown in Table 3.

Fig.16 Evolution of LBLPs.

Table 3 Main properties of hydraulic medium.

Before the test,the piston surface was measured by the surface roughness profiler (SRP), and the cylinder bore contour was measured by the coordinate measuring machine (CMM).Then the axial piston pump was assembled to operate on the test bench.In order to demonstrate the evolution of the piston/cylinder pair, the wear characteristics of the piston and the cylinder bore were measured several times during the operation of the axial piston pump.The intervals for dissembling the pump and measuring the wear characteristics were set at:0.1, 5, 2, 3 h, and 4 h.The flow chart of the experiment is shown in Fig.17.

4.2.Validation of asperity degradation process

Fig.18 Validation of degradation of the piston asperity.

On the piston surface,the roughness value was measured every 1 mm along the axial direction.The measured roughness in the entire length of the piston surface after different test times is shown in Fig.18(a), and the picture of the piston which has operated for 9.6 h is shown in Fig.18(b).Consistent with the simulation results, the measurement results show that the piston surface roughness degrades at the lengths near the piston bottom and the piston head.And the picture shows that the two lengths of surface are polished.

In order to quantitatively evaluate the degradation of the piston surface, the piston is divided into two sections which respectively contain the two degraded lengths, and the degraded values in the two sections are defined as Eq.(30)and Eq.(31):

In order to exclude the effect of the absolute measurement error while comparing the simulation and experimental results,the simulation and experimental results were separately normalized.The relationship between the degraded value in these two sections and the operating time is shown in Fig.19.

Fig.17 Flow chart of experiment.

Fig.19 Quantified comparison between simulation and experimental results of asperity degradation.

The roughness in the two lengths both degrades until operating for about 0.6 h.When achieving the steady state,the simulation result shows that the asperity degradation in Section 1 is higher than that in Section 2 by 14.9%.The experimental result slightly fluctuates in the steady state.So, the average of the values at 2.6 h, 5.6 h and 9.6 h is used to evaluate the asperity degradation in the steady state.The experimental results show that the asperity degradation in Section 1 is higher than that in Section 2 by 13.7%.The degradation processes of the piston roughness obtained by the simulation method show high consistency with the experimental results.

4.3.Validation of the cylinder bore wear process

The cylinder bore surface was scanned by the CMM probe,and the coordinates of the surface along the scanned trajectories were output.The unwrapped cylinder bore contours after different test times can be drawn as shown in Fig.20(a).The corresponding simulation results are shown in Fig.20(b).

The experimental results and the simulation results both show that the wear condition is severe at the two sides of the cylinder bore,and the severely worn regions range from about 180° to 360° along the circumferential direction.To quantitively evaluate the wear distribution of the cylinder bore, Vw1and Vw2are defined to represent the total wear volume of the valveplate side and the swashplate side of the cylinder bore,respectively.c

Fig.20 Validation of wear of cylinder bore contour.

Fig.21 Quantified comparison between simulation and experimental results of cylinder bore wear.

In order to exclude the effect of the absolute measurement error while comparing the simulation and experimental results,the simulation and experimental results were separately normalized.The wear volumes Vw1and Vw2derived from experimental results and simulation results are displayed in Fig.21.

The two sides of the cylinder bore were both worn until operating for 0.6 h.When achieving the steady state, the simulation result shows that the wear volume of the valveplate side of the cylinder bore is higher than that of the swashplate side by 24.6%.The experimental result slightly fluctuates in the steady state due to the CMM error.So, the average of the wear volumes at 2.6 h, 5.6 h and 9.6 h is used to evaluate the wear in the steady state.The experimental results show that the wear volume of the valveplate side of the cylinder is higher than that of the swashplate side by 23.0%.The experimental results show that the simulation model has a high accuracy in predicting the wear process and the wear distribution.

5.Conclusions and outlooks

In this paper,the real-time interaction mechanism between the piston asperity degradation, the cylinder bore wear and the load-bearing and lubrication condition during the wear process is elaborated.The coupled evolution of the piston asperity, the cylinder bore contour, and the LBLPs is described below:

(1) The wear rate of cylinder bore is large in 0.47 h, and then the cylinder bore contour continues to change with a small wear rate until being steady at 0.69 h.During the running-in period, the cylinder bore contour gradually becomes partially chamfered, and the wear volume near the valveplate side is 24.6% higher than that near the swashplate side.

(2) The piston surface asperity converges to a steady state after a running-in period lasting about 0.47 h.The asperity degradation lengths of the piston surface are consistent with the variation ranges of the relative positions of the two ends of the bearing interface and the piston surface.

(3) In the running-in periods of the piston and the cylinder bore, the average hydrodynamic pressure of the oil film increases, and the average solid contact stress between the piston and the cylinder bore decreases.It means that the load-bearing and lubrication condition of the piston/cylinder pair is gradually improved until the piston and the cylinder bore reach the steady state.

It can be seen from the coupled evolution of the piston/-cylinder pair that the running-in process can gradually eliminate the harsh contact condition.So, the running-in process plays a vital role in reducing the wear and enhancing the service life the of the piston/cylinder pair.How to load the running in process of the piston/cylinder pair may affect the steady wear performance and has become a noteworthy problem.42,43The simulation method proposed in this paper can be used to analyze not only the performance in the running-in process, but also the steady performance under different running-in load.By this way, the running-in load can be optimized to obtain the best steady wear performance.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research was financially supported by the National Key Research and Development Program of China (No.2018YFB2001101), the National Outstanding Youth Science Foundation of China(No.51922093),and the National Natural Science Foundation of China (No.51890882).