Performance optimization for high speed axial piston pump considering cylinder block tilt

Hogong XU, Junhui ZHANG,*, Shoujun ZHAO, Junyun CHEN,Weidi HUANG, Xiohen HUANG, Fei LYU, Qi SU, Bing XU

a State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310058, China

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

c Jiangsu Jinling Intelligent Manufacturing Research Institute Co, Ltd, Nanjing 210006, China

KEYWORDS

Abstract Increasing the rotating speed is considered as an efficient approach to upgrade the power-to-weight ratio in an axial piston pump,but penalized by more leakage and more severe wear resulting from the adverse cylinder block tilt.Previous studies mainly focused on the bearing characteristic of the valve plate/cylinder block pair, but the spline coupling also plays a key role in the undesired cylinder block tilt, which has been little studied.A theoretical model for the rotating assembly is presented to investigate the effect of the spline coupling length on the cylinder block tilt and the performance of the valve plate/cylinder block pair.A typical high-speed axial piston pump with the displacement of 5.2 mL/r at 10000 r/min was studied by simulation and experiment.It shows that the optimal spline coupling length is one value increased by 2 mm from the original,bringing a remarkable leakage reduction under the high-speed condition by decreasing the cylinder block tilting angle.The experiment result matches well with the simulation.The influences of the spline coupling on the cylinder block tilt and the leakage were demonstrated.

1.Introduction

Due to the benefits of high reliability, high power-to-weight ratio, and high efficiency, the high-speed axial piston pump(HSAPP) is commonly employed in aerospace applications,such as the flight control system in the airplane,the thrust vector control system in the rocket,and so on.1–3The schematic of HSAPP is illustrated in Fig.1.The spline coupling,which consists of the spline on the shaft and the hub on the cylinder block, transmits the driven torque from the shaft to the cylinder block,causing the cylinder block to slide on the valve plate surface.Meanwhile, the piston-slipper assemblies reciprocate in the cylinder block bores due to the inclined swash plate.The slippers slide along the swash plate surface.And these three kinds of moving parts are so-called friction pairs:the piston/cylinder block pair, the valve plate/cylinder block pair,and the slipper/swash plate pair.The three friction pairs are the major source of the leakage,and the wear of the pump also occurs in these friction pairs.Ideally, thin oil film exists in the friction pairs, which prevents severe wear at the cost of small leakage.

Fig.1 Inner structure of HSAPP.

Sensitivity to weight in aerospace applications puts forward the growing demand for a higher power-to-weight ratio,so the rotating speed is continuously increasing for the HSAPP.However, high rotating speed brings several problems, such as pressure pulsation,cavitation, heat problem, and the tilting of the rotating assembly.4The tilting of the rotating assembly changes the ideal shape of the oil film in the friction pairs,especially the tilt of the cylinder block will lead to the wedge-shaped oil film in the valve plate/cylinder block pair,which is the largest scale friction pair in the axial piston pump.The tilting angle of the cylinder block is sharply enlarged at high rotating speed, especially under low pressure,5and it brings dramatic drain leakage in the pump and decreases the volumetric efficiency, even failing to meet the outlet pressure requirement.Moreover, the high rotating speed also increases the PV value in the lubricating interfaces, resulting in severe wear of the pump.

The bearing characteristic of the valve plate/cylinder block pair is affected by the cylinder block tilt, which has been observed experimentally,6–9and the cylinder block tilt has also been studied using a variety of theoretical models and simulation models.10–15These studies mainly focused on industrial pumps, which always operate at low rotating speeds below 3000 r/min.However, the inertia effect of the piston-slipper assemblies is significant at high speed,and it results in an obvious increase of the tilting moment.The tilting moment increases the cylinder block tilting angle, which is the primary cause of the dramatic leakage at high rotating speed.

Optimizing the bearing characteristic of the valve plate/-cylinder block pair could weaken the cylinder block tilt to reduce leakage.And several efforts have been conducted in the past decades to optimize the bearing characteristic of the valve plate/cylinder block pair.The first type of effort is the geometry optimization of the valve plate.Shang and Ivantysynova found that the energy dissipation increases with the sealing land widths at high pressure due to the improved bearing characteristic of the oil film.16Geffroy et al.added the additional pockets on the cylinder block kidney ports and the valve plate kidney ports to increase the local bearing characteristic, which decreased the cylinder block tilting angle and the metal-to-metal contact.17,18Achten et al.proposed an annular pressure pocket in the sealing land of the valve plate to eliminate substantial wear effectively.19The second type of effort is the surface texture.Murrenhoff et al.conducted the studies on the textured valve plate and analyzed the influence of the dimple geometry parameters on the oil film bearing characteristic theoretically and experimentally.20Wang et al.experimentally studied the overall efficiency in an axial piston motor in which the valve plate is textured, and the effects of the dimple size and the dimple number are investigated.21Deng et al.pointed out that the maximum bearing characteristic can be obtained when the dimple depth equals the oil film thickness.22Chen et al.found that the textured surface improves the pump efficiency by reducing the wear and the cylinder block tilting angle.23Furthermore, they optimized the dimple shape on the valve plate by a multi-objective optimization model.24The third type of effort is the waved surface in the friction pair.Baker and Ivantysynova have investigated the effect of the micro-structured cylinder block, and the results showed that the oil film thickness increases obviously when adopting the waved surface valve plate.25Zecchi et al.further analyzed the effect of the waved valve plate considering the additional solid body heat transfer in the friction pair.12Chacon then took the effect of the offset angle of the wave on the valve plate into account and optimized the parameters of the wave to minimize the power loss.26Shin and Kim found that the cylinder block tilt would even take a turn for the worse under the improper surface design27.

The above efforts merely focused on improving the bearing characteristic of the valve plate/cylinder block pair, but the spline coupling also affects the movement of the cylinder block.Hence,few researchers have put their eyes on the effects of the shaft and the spline coupling.Zhang et al.analyzed the cylinder block considering the shaft deformation,but the cylinder block was assumed to connect with the shaft rigidly in this study.28Chao et al.presented the spline design criterion according to the mechanism analysis of the cylinder block and provided the location limitations of the spline on the cylinder block.29This criterion presented the minimum length of the spline,but the lengths of the splines in the pumps were significantly longer than the theoretical minimum lengths.As for the HSAPP, the moment due to the inertia effect increases exponentially at high rotating speed.Although the hydrodynamic effect causes the bearing characteristic of oil film to rise with increased rotational speed, it is only effective until fullfilm lubrication is reached in the valve plate/cylinder block pair.5As a result, the spline coupling would play a key role in bearing the tilting moment in the HSAPP,especially at high speed.

Improving the bearing characteristic of the spline coupling,which arises from the misalignment of the spline coupling,seems to compensate for the bearing capacity limitation in the valve plate/cylinder block pair to enhance the pump performance.Hence, this study presents a dynamic model of the rotating assembly, coupled with the spline coupling meshing model,to investigate the influence of the spline coupling length on the cylinder block tilt and optimize the spline coupling length to improve the performance of the valve plate/cylinder block pair.The remaining paper is structured as follows:Section 2 presents the dynamic model; Section 3 presents and discusses the theoretical findings; Section 4 shows the experimental validation; and Section 5 gives the conclusion.

2.Model setup

The comprehensive dynamic model of the rotating assembly consists of four parts, as shown in Fig.2.The major component of the dynamic model is the rotor dynamic model, which is established according to the rotating assembly geometry.The other four parts are described as generalized forces applied on the cylinder block, respectively, including the interactions within the spline coupling, the valve plate/cylinder block pair,the piston/cylinder block pair, and the central spring.The established model is shown as follows:

where Mris the mass matrix, Cris the damping matrix, Gris the gyroscopic matrix, and Kris the stiffness matrix of the rotating assembly, respectively.qris the generalized coordinates vector.Ω is the rotating speed of the pump.Fvp, Fsc,Fsp, Fcsare the excitation from the valve plate/cylinder block pair, the spline coupling, the piston/cylinder block pairs, and the central spring, respectively.It needs to be noticed that the axial movement of the cylinder block is small enough to affect the compression of the central spring, so the excitation from the central spring Fcsis constant, which is not described in detail.

2.1.Rotor dynamic model

The rotor dynamic model is established through the finite element method.The cylinder block and the shaft are modeled by the Timoshenko beam,so the elasticity and inertia can be considered.As shown in Fig.2, the shaft consists of 20 beam elements and 21 nodes,and the cylinder block spline has 12 beam elements and 13 nodes.Each cylinder block node takes all the degrees of freedom (DOFs) into account, while the axial displacements of the shaft nodes are ignored since the axial displacement of the shaft is limited by the preloaded angular contact ball bearing.The total DOFs of the rotor dynamic model is 183.The characteristics of the bearings are simplified as the linear spring-damper models, and the stiffness and damping coefficients are coupled to the stiffness matrix and damping matrix of the rotor dynamic model, respectively.

Fig.2 Illustration of dynamic model of rotating assembly.

2.2.Interaction of spline coupling

The shaft supports the cylinder block through an involute spline coupling whose bearing characteristic depends on the meshing status.The radial force due to the inclined swash plate brings about the bending deformation of the spline coupling and the driving torque results in the torsional deformation.As a result,the spline coupling deviates from the ideal meshing status, which is illustrated in Fig.3.

The bending deformation results in the misalignment of the spline coupling axes and only the parallel misalignment is taken into account in each spline coupling element, which can be obtained according to the movements of nodes from the rotor dynamic model.osiand ocirepresent the center of the ith shaft spline element and the center of the ith cylinder block spline element, respectively.The misalignment and the azimuth angle for the ith spline coupling element can be obtained by the movements of osiand ocias follows

where xsi, ysiare the radial displacements of the shaft spline element, xci, yciare the radial displacements of the cylinder block spline element, and the subscript i represents the ith spline coupling element.The torsional deformation has an effect on the meshing status along the z-axis.And the relative torsion angle of the ith spline coupling element can be obtained as

Fig.3 Misalignment of spline coupling.

where ?si, ?ciare the shaft torsional angle and cylinder block torsional angle on the ith spline coupling element,respectively.The meshing status considering the deformations can be described by the clearance distribution of the teeth pair on the spline coupling element as follows

where δijis the clearance of the jth spline tooth pair on the ith spline coupling element, Δ is the backlash of the spline coupling, γ is the pressure angle of the spline coupling, φ is the rotating angle of the shaft,n is the teeth number,rsis the pitch diameter of the spline.The three terms of Eq.(5) on the right side represent the clearance due to the design and the fabrication,the bending deformation, and the torsional deformation,respectively.When the clearance δijbecomes negative, it indicates that the corresponding tooth pair is engaged, and the meshing force of the jth spline tooth pair on the ith spline coupling element Fijcan be expressed as

where Kspis the contact stiffness of the spline tooth element,and L is the axial length of the spline coupling element,which is calculated according to the previous method.30Then the meshing forces of the teeth pairs provide the radial forces and axial torques,which are the components of the generalized force vector Fscdue to the spline coupling.The radial force along the x-axis Fsxi, the radial force along the y-axis Fsyi,and the torque around the z-axis Tszion the ith shaft spline element can be expressed as

The forces and torques applied on the cylinder block spline elements are the reaction of that exerted on the shaft nodes.

2.3.Interaction in the valve plate/cylinder block pair

The cylinder block is supported by the valve plate/cylinder block pair along the z-axis, and the bearing capacity mainly results from the oil film and the asperity contact in the valve plate/cylinder block pair.In our previous work, an analytical bearing model has been derived considering the trade-off between the simulation accuracy and the time consumption5.

As shown in Fig.4, the oil film consists of two areas: the sealing lands and the main kidney land.The pressure field in the main kidney land area is related to the piston chamber pressure and the kidney ports pressure of the valve plate,which varies with the shaft rotating angle periodically.Hence,the bearing force and moment of the oil film in the main kidney land area can be calculated by numerical method at special rotating angles, and the bearing force and moment at an arbitrary angle can be obtained by interpolation.

As for the sealing lands, the pressure distribution is described by the one-dimensional Reynolds equation since the widths of the sealing lands are much smaller than the radius.

where r is the sealing lands radius,h is the oil film thickness and is affected by the tilt angle of the cylinder block α,θ is the angle coordinate.The analytical bearing force and moment of the oil film in the sealing land areas are calculated by integrating the pressure field of the oil film.Although the analytical model is expressed in a sophisticated form,the simulation time consumption is much smaller than the numerical model which involves nested iterative loops31.

In addition, the asperity contact is considered through the Greenwood –Tripp model.

where the K′, and E′are the elastic factor and the composite elastic modulus, respectively.F5/2(h/σ) is the form function.

Finally, the total excitation forces from the oil film can be obtained by summing the above three parts.

2.4.Interaction in the piston/cylinder block pair

Fig.5 Excitation forces applied on a single piston-slipper assembly.

The interaction in the piston/cylinder block pair is simplified as excitation force, which has also been analyzed in our previous work.5In this study, the excitation from the piston/cylinder block pairs is considered to be applied on the end of the spline coupling near the valve plate,as shown in Fig.5.According to the mechanical analysis of the piston-slipper assembly, the bearing force Fpcand moment Mpcapplied on the slipperpiston assembly from the cylinder block can be obtained.It is noticed that the force due to the piston chamber pressure Fp,the centrifugal force Fcf, the inertia force Fif, and the reaction from the swash plate Fsware considered.The friction in the slipper/swash plate pair Fsfis also taken into account, while the friction in the piston/cylinder block pair is ignored since it is quite smaller than the pushing force from the piston chamber pressure.Finally,the excitation forces applied on the cylinder block from the slipper-piston assemblies are the sum of the reactions of Fpcand Mpc.

2.5.Workflow

The flow chart of the established dynamic model is shown in Fig.6.The mass matrix Mr,the damping matrix Cr, the gyroscopic matrix Gr, and the stiffness matrix Krare calculated by the rotor dynamic model according to the rotating assembly geometry.And then,the initial excitation on the cylinder block can be obtained according to the working condition and the initial status of the rotating assembly.The dynamic characteristics of the rotating assembly can be obtained by solving the dynamic model: Eq.(1) is rearranged as the state space equation in Fig.6 and then solved by the variable step Runge Kutta algorithm to obtain the movements of the rotating assembly in the next time step.The excitation on the cylinder block,as well as the state space equation,is updated by the calculated movements.Then solve the updated state space equation to obtain the result in the next time step until the time step exceeds the preset time(0.1 s).

Fig.6 Flow chart of calculation.

The spline coupling length is one of the most important geometry parameters of the rotating assembly, and changing the spline coupling length will lead to the variation of the matrices of the rotordynamic model.And the interaction between the spline coupling also will be affected.On the other hand,the location of the forces from the piston-slipper assemblies is also changed when changing the spline coupling length,according to Fig.5.And these changes lead to the variation of the simulation results, which can illustrate the effect of the spline coupling length.The movements of the rotating assembly under different spline coupling lengths can be calculated according to the workflow in Fig.6, respectively.Finally, the results are saved and post-processed to obtain the tilt behavior of the rotating assembly and the performance of the valve plate/cylinder block pair, which can help to find the optimal spline coupling length.

3.Analysis and discussion

As shown in Fig.7,the spline coupling length can be described by the distances from the spline coupling to the cylinder block bottom face.29And the distance from the spline coupling end near the swash plate to the cylinder block bottom face Lssdetermines the axial size of the cylinder block, which affects the size of the whole pump.As a result, the Lssis close to the theoretical limitation obtained by analyzing the application point of the spline reaction,which aims to improve the powerto-weight ratio.28The distance from the spline coupling end near the valve plate to the cylinder block bottom face Lsvis much smaller than its theoretical limitation, and it means the length of the spline coupling is longer than the theoretical result, which is considered to ensure the torque transfer from the shaft to the cylinder block.Hence, this study focuses on the change of Lsvwhen investigating the influence on the cylinder block tilt.

The deviation from the initial spline coupling length ΔL is adopted to describe different spline coupling lengths, and the positive ΔL represents a longer spline coupling and vice versa.A case study of an HSAPP with nine piston-slipper assemblies is conducted in this study, and the main parameters of the HSAPP are shown in Table 1.

Fig.7 Description of spline coupling length.

Table 1 Main parameters of HSAPP.

The rotating assembly tilt at the maximum rotating speed with the initial spline coupling length is investigated firstly.It should be noted that the maximum outlet pressure is set at 18 MPa since a pressure controller is adopted in the HSAPP,and the displacement will be decreased when the outlet pressure reaches around 19 MPa.And the minimum outlet pressure is set at 3 MPa because the pressure drop of the system under the maximum flow rate exceeds 2 MPa.As shown in Fig.8(a), the tilting angle of the cylinder block bottom face decreases gradually when the outlet pressure increases.Furthermore, the rotating assembly tilt under low pressure(3 MPa) and high pressure(18 MPa) is illustrated in Fig.8(b)and 8(c), respectively, where the displacements and deformations along the x-axis and y-axis are scaled 500:1.

The shaft deformation mainly appears along the y-axis,and the deformation of the shaft is enlarged obviously under high outlet pressure.Besides, the cylinder block also shows slight deformation along the y-axis affected by the deformed shaft through the spline coupling, especially under high outlet pressure.The relative deflection between the shaft spline and the cylinder block spline can be found, which is the so-called misalignment of the spline coupling,and it results in the ununiform meshing of the spline teeth, which provides the bearing moment to prevent the cylinder block from tilting.It is noted that the shaft spline shows the opposite tilting angle to that of the cylinder block bottom face along the y-axis, and it means that the enlarged shaft deformation could improve the bearing characteristic of the spline coupling, which helps to decrease the cylinder block tilting angle.This is the reason that the tilting angle of the cylinder block bottom face decreases gradually with increasing pressure.

The spline coupling length may have different effects on the cylinder block tilt under different outlet pressure due to the elastic deformations of the shaft and the cylinder block.In order to present the effect of the spline coupling length clearly,the shaft tilt and the cylinder block tilt are presented by their axes,as shown in Fig.9.It is noted that the tilt near the spline coupling is paid attention to since changing the spline coupling length mainly affects its meshing status.The colored circle dots mark the position of the splines with different lengths on the axes.According to Fig.9(a), the spline coupling length has a more evident effect on the cylinder block tilt than that of the shaft under low pressure(3 MPa).With the increasing spline coupling length, the tilting angles of the whole cylinder block decrease obviously, while the shaft tilting angles are less affected.The deformations of the cylinder block and the shaft are slight, which barely affects the misalignment of the spline coupling.Hence, the misalignment of the spline coupling is reduced obviously,and it means that the longer spline coupling can improve the bearing characteristic of the spline coupling under low pressure.The tilting angle of the cylinder block bottom face shows the same trend as that of the cylinder block spline.Under high pressure(18 MPa),the cylinder block tilting angle along the x-axis is less affected by the spline coupling length, but the misalignment of the spline coupling is still reduced since the tilting angle of the shaft along the x-axis is significantly increased with the increase of the spline coupling length, as shown in Fig.9(b).In contrast, the shaft deforms obviously along the y-axis, and the longer spline coupling has a stronger influence on the cylinder block deformation.It can be found that the meshing status of the spline coupling along the y-axis can be divided into two parts by the maximum displacement position of the shaft spline,as shown in Fig.9(b).The misalignment of the spline coupling on the right side increases with the increasing spline coupling length, while the misalignment on the left side decreases.As a result, the tilting angle of the cylinder block bottom face may increase with a longer spline coupling, and it means that a longer spline coupling would weaken the bearing characteristic of the spline coupling under high pressure.

The meshing status of the spline coupling is analyzed to investigate the effect of the spline coupling length on the bearing capacity.The spline teeth are numbered in Fig.10(a), and the clearance distributions of the spline coupling with different lengths are shown in Fig.10(b) and 10(c).The negative clearance indicates the meshing position of the spline pair, and the smaller clearance means a larger deformation of the meshing tooth pair.The misalignment of the spline coupling leads to variations of the clearance in both axial and circumferential directions.Under 3 MPa, the maximum meshing deformation is located at the spline tooth pair 11 at the swash plate side,and it decreases gradually towards the valve plate side.At the valve plate side, tooth pair 11 is dismissed, and the maximum meshing deformation occurs in tooth pair 3, which is located at the opposite side of tooth pair 11.Under 18 MPa,the meshing deformation is larger than that under 3 MPa since the torque transmitted by the spline coupling is increased.The maximum meshing deformation at the swash plate side is also located at the spline tooth pair 11, while the maximum meshing deformation at the valve plate side is located at the spline tooth pair 8.And the circumferential clearance distribution of the spline coupling at the swash plate side is quietly larger than at the valve plate side since the misalignment is larger at the swash plate side.Increasing the spline coupling length can decrease the maximum meshing deformation and increase the meshing area under 3 MPa since the misalignment between the spline coupling is reduced,as shown in Fig.9(a).But under 18 MPa, the maximum meshing deformation at the swash plate side is barely affected by the spline coupling length.On the contrary,the maximum meshing deformation at the swash plate side increases slightly with a longer spline coupling since the misalignment of the spline coupling at the valve plate side is enlarged, as shown in Fig.9(b).

Fig.8 Rotating assembly tilt at high rotating speed.

The changes of the meshing status finally affect the bearing capacity of the shaft to the cylinder block.And the bearing force distribution of the spline coupling under different spline coupling lengths is shown in Fig.11.Increasing the spline coupling length enlarges the distribution area of the bearing force,so the maximum radial force can be reduced while the total radial force and the moment still can be provided to support the cylinder block.And it is the reason that the smaller misalignment between the spline coupling can be found with a longer spline coupling under 3 MPa.However, the radial force from the cylinder block causes the deformation of the shaft along the y-axis.Especially the reduction of the shaft deformation is significantly enlarged under 18 MPa, and its effect on the cylinder block tilt can not be neglected.The enlarged distribution area and the reduced distribution force with a longer spline coupling results in the reduction of the shaft deformation, as shown in Fig.9(b).It can be found that maximum shaft deformation occurs at the area of the spline coupling near the valve plate,and the maximum reduction also occurs at the same place.And it means that the shaft spline rotates slightly counterclockwise around the + x-axis, so the cylinder block spline also rotates slightly, and it will increase the tilting angle of the cylinder block bottom face.

The cylinder block tilt further affects the wedged oil film in the valve plate/cylinder block pair, and the thickness distribution can be described by the wedge angle and the minimum thickness of the oil film.The effects of the spline coupling length on the wedge angle and the minimum thickness of the oil film are shown in Fig.12.The spline coupling length interval is 1 mm while the outlet pressure interval is 3 MPa,so total of 60 different combinations of the pressure and spline coupling length are investigated.The spline coupling length shows a greater effect on the wedge angle and the minimum oil film thickness under low outlet pressure.With the increasing spline coupling length,the wedge angle of the oil film decreases gradually under low outlet pressure, while the wedge angle decreases first and then increases under high outlet pressure.Besides, the minimum oil film thickness decreases when the spline coupling length increases under all the outlet pressure.The changes in the wedged oil film affects the valve plate/-cylinder block pair performance, such as leakage and wear.The leakage can be calculated by

where μ is the dynamic viscosity of the oil.The wear is positively correlated with the PV value, which is related to the asperity contact in the valve plate/cylinder block pair as

Fig.9 Effects of spline coupling length on rotating assembly tilt.

The leakage and the PV value with different spline coupling lengths are shown in Fig.13.According to Fig.13(a),when the outlet pressure increases, the leakage with the initial spline coupling increases first and then decreases, and the leakage under low outlet pressure is obviously larger than that under high outlet pressure.The leakage under low outlet pressure can be reduced by a longer spline coupling since the wedge angle and the oil film thickness decrease, and the reduction is more significant under lower outlet pressure.The leakage under high outlet pressure decreases first and then increases with the increasing the spline coupling length which is also similar to the tendency of the wedge angle of the oil film, and the higher the outlet pressure is, the more obvious the leakage increases.On the contrary,the PV value increases with the outlet pressure with the initial spline coupling, and the maximum PV value is near 30 MPa?m/s under 18 MPa.The PV value increases with the increasing spline coupling under low outlet pressure and increases first and then decreases under high outlet pressure at the same time,as shown in Fig.13(b).To investigate the effects of the spline coupling length on the leakage and the PV value more clearly, the changes of the leakage and the PV value relative to that with initial spline coupling are illustrated in Fig.13(c) and 13(d).According to Fig.13(c), with the increasing spline coupling length, the leakage reduction under low outlet pressure increases sharply at first,and then gradually tends to a constant value.And the leakage under high outlet pressure will be enlarged obviously when the spline coupling increases by over 4 mm.As a result, it can be found that increasing the spline coupling length by no more than 4 mm can decrease the leakage under low outlet pressure significantly, while the leakage under high outlet pressure is affected slightly.On the other hand, a longer spline coupling may increase the PV value, especially the PV value under around 6 MPa, as shown in Fig.13(d).When increasing the PV value within 3 MPa.m/s(10%of the maximum PV value)is acceptable,the spline coupling length should be increased by no more than 2 mm, and the leakage under 3 MPa can be reduced by up to 56.79%with a 2 mm longer spline coupling.It is noted that only the effect of the spline coupling length is investigated in this study.The PV value can be further limited by optimizing the geometry of the valve plate/cylinder block pair, and the wear resistance of the sliding surface also can be improved by the cermet coating and heat treatment32,33.

4.Experimental validation

Fig.10 Meshing status of spline coupling.

The cylinder block tilt and the performance of the HSAPP are measured to validate the effectiveness of the optimized spline coupling length.The test rig has been built before, as shown in Fig.14.The cylinder block tilt is measured by four eddy current sensors, and the rotating speed is measured by the tachometer synchronously.And the sensors measure the distances to the cylinder block cylindrical surface.The test rig could regulate the rotating speed and the outlet pressure at 500–16000 r/min and 0–28 MPa, respectively.The inlet pressure is boosted by an auxiliary pump and is maintained at around 0.6 MPa.The operating conditions and the performances of the pump can be monitored, such as the leakage temperature, the leakage flow, the outlet flow, and the input torque from the servo motor.

And the details of how to obtain the cylinder block tilt from the measured signals have been described in our previous work.5The real cylinder block displacement is obtained by eliminating the run-out signals from the measured signals of the eddy current sensors.According to Fig.15(a), the obvious lag can be found between the measured signal and the run-out signal,since the tachometer can not obtain the real-time rotating angle of the shaft.And then the lag can be eliminated by the signal alignment method to extract the real displacement of the cylinder block.Finally, the cylinder block tilting angle is calculated by

where x1, x2are the radial displacements along the x-axis of the cylinder block, and y1, y2are the radial displacements along the y-axis, z1, z2are the axial distances from the two groups of sensors to the spline coupling, as shown in Fig.15(b).

Two cylinder blocks with different spline lengths are manufactured,one of which has the same geometry as the original cylinder block to minimize the effect of wear of the original cylinder block, and the other is the optimized cylinder block with a 2 mm longer spline while other geometry parameters remain the same.The test pump replaces the original cylinder block with the newly manufactured cylinder blocks successively, and then the performances of the test pump under different outlet pressure(3 MPa,6 MPa,12 MPa,and 18 MPa)at the maximum rotating speed(10000 r/min) are measured.

Fig.11 Distribution of radial force within spline coupling.

Fig.12 Effects of spline coupling length on wedged oil film characteristics.

The measured tilting angles and the simulated tilting angles of the cylinder block under different pressure are compared in Fig.16, where the simulated angles and the measured tilting angles are normalized separately to weaken the absolute measurement error.The cylinder block tilting angles under different outlet pressure can be reduced by the optimized cylinder block,and the reduction of the tilting angle is more significant under a lower outlet pressure.The consistency between the simulated angles and the measured angles could confirm that the longer spline coupling helps to decrease the cylinder block tilting angle effectively.

Fig.13 Effects of spline coupling length on performance of valve plate/cylinder block pair.

Fig.14 Test rig for HSAPP.

Fig.15 Measurement of cylinder block tilting angle.

Fig.16 Cylinder block tilting angle.

The optimized spline coupling reduces the pump leakage under different outlet pressure, as shown in Fig.17, especially the leakage under low outlet pressure.The leakage under 3 MPa can be reduced from 3.12 L/min to 2.36 L/min by 24.48%.Although the drain leakage comes from all three friction pairs, it still can prove the effectiveness of the optimized spline coupling on improving the performance of the valve plate/cylinder block pair.The leakage reduction could not only improve the volumetric efficiency of the pump but also prevent the potential that the outlet pressure can not meet the requirement due to the huge leakage.

5.Conclusion

Fig.17 Drain leakage of whole pump.

In the paper, the spline coupling length is optimized to reduce the leakage at the high rotating speed in an HSAPP.The optimized spline coupling length is obtained by analyzing the pump performance under different outlet pressure based on a dynamic model of the rotating assembly.And the experimental test validates the effectiveness.The following conclusions can be taken from the simulation and experimental results:

1) The rotating assembly tilt at high rotating speed can be obtained by the established model.The misalignment of the spline coupling results in the ununiform meshing of the spline teeth, which provides the bearing moment.The deformations of the cylinder block and the shaft contribute to the misalignment, especially the enlarged deformation of the shaft under high outlet pressure improves the bearing characteristic of spline coupling, which helps to decrease the cylinder block tilt angle.

2) The spline coupling length has a different influence on the cylinder block tilt under different outlet pressure at high rotating speed.The deformations of the cylinder block and the shaft are slight under the low outlet pressure,and a longer spline coupling could decrease the cylinder block tilting angle by decreasing the misalignment of the spline coupling.But under the high outlet pressure, the shaft deformation is enlarged, and it affects the cylinder block deformation significantly,especially with a longer spline coupling.Hence,increasing the spline coupling length enlarges the cylinder block tilting angle under high outlet pressure.

3)The cylinder block tilt further affects the wedged oil film performance in the valve plate/cylinder block pair.The leakage under low outlet pressure can be reduced by a longer spline coupling since the wedge angle and the minimum thickness of the oil film decrease.However, the leakage under high out pressure may be enlarged obviously with a much longer spline coupling, and the PV value under low outlet pressure also increases obviously when the spline coupling length increases,which limits the spline coupling length.

In future work,the collaborative optimization of the spline coupling and the valve plate/cylinder block pair will be conducted to find out the optimal parameters of the rotating assembly, which could further improve the cylinder block tilt and the pump 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

This study was co-supported by the National Key R&D Program of China(No.2019YFB2005101),National Outstanding Youth Science Foundation of China (No.51922093), the National Natural Science Foundation of China (No.52105075),the National Natural Science Foundation of China(No.51890882), the Natural Science Foundation of Zhejiang Province (No.LQ21E050022).