Multi-objective optimization of surface texture for the slipper/swash plate interface in EHA pumps

Abstract

Well-designed surface textures can improve the tribological properties and the efficiency of the electro-hydrostatic actuator (EHA) pump under high-speed and high-pressure conditions. This study proposes a multi-objective optimization model to obtain the arbitrarily surface textures design of the slipper/swash plate interface for improving the mechanical and volumetric efficiency of the EHA pump. The model is composed of the lubrication film model, the component dynamic model considering the spinning motion, and the multi-objective optimization model. In this way, the arbitrary-shaped surface texture with the best comprehensive effect in the EHA pump is achieved and its positive effects in the EHA pump prototype are verified. Experimental results show a reduction in wear and an improvement in mechanical and volumetric efficiency by 1.4% and 0.8%, respectively, with the textured swash plate compared with the untextured one.

Key words： electro-hydrostatic actuator; axial piston pump; slipper/swash plate interface; multi-objective optimization; surface texture

1 Introduction

The high-speed axial piston pumps are widely used as the main power source in the electro-hydrostatic actuators (EHAs) due to the advantages of compact structure, high power density, and high reliability. The EHA pump has a broad operating range and needs to start up instantly and change shaft rotating direction frequently. Therefore, full fluid film lubrication may be difficult to be established, and severe abrasion of sliding parts will appear, especially in the slipper/swash plate interface. The slipper/swash plate interface is one of the three major tribological contacts of the axial piston pump, and it plays the function of sealing and bearing [1]. Its friction torque and leakage flow determine the mechanical and volumetric efficiencies of the EHA pump [2]. However, the dynamic behaviors of slippers are extremely complex. The external forces such as piston–slipper ball joint friction force and unbalanced centrifugal inertia force generate micro motions between the slipper and the swash plate, such as tilting motion [3], squeezing motion [4], and spinning motion [5]. Thus, the geometry variation in lubricant film thickness will result in uneven hydrodynamic pressure distribution, and mixed friction areas with severe lubrication conditions will occur [6]. Notably, oxidation and blackening wear traces even appear on the surfaces of components, which greatly affect the efficiency and the lifetime of the EHA pump [7].

The laser surface texturing (LST) technique, which is considered a feasible option for changing the surface topography, can reduce friction and wear of various industrial applications [8]. The enhanced performances are dependent on the abilities of the micro dimples to induce additional hydrodynamic pressure, store lubricants, lower shear gradient, and trap wear debris [9]. Thus, the LST technique is applied to improve the tribological properties of tribosystems. However, the geometric parameters of surface textures are sensitive to the type of contacts and working conditions, and an improperly designed texture pattern will result in detrimental effects on the tribological performance [10].

In the published theoretical studies, most of the studies on surface texture optimization mainly focus on the thrust bearing [11–16] and the mechanical seal [17–19]. However, the minimum gap height between two specimens is constant and does not depend on the oil film pressure and the external load, while the lubricant film thickness and boundary conditions of the lubricating interfaces in the axial piston pump are time-varying and unevenly distributed, which is profoundly different from thrust bearings and mechanical seals. To overcome this difficulty and apply the optimal LST to the friction pairs in axial piston pumps, many researchers, including the authors’ team, have made efforts [8,20–23]. A previous study by the authors’ team established the hydrodynamic lubrication model and proposed an optimization model for the multi-scale textured cylinder/valve plate interface [23]. These studies provide the basis for the LST optimization model for the slipper/swash plate interface. However, the following two problems remain to be addressed.

First, the piston–slipper assemblies have multiple degrees of freedom, and their boundary velocities stem from the linear reciprocating and rotating macro-motions and the spinning micro-motions, whose kinematic and hydrodynamic models are different from the cylinder/valve plate interface [24]. Tang et al. [25] and Spencer [26] measured the oil film thickness, pressure, and temperature distributions of the slipper/swash plate interface. However, spinning motion was always ignored or replaced by a constant value while establishing the lubrication model. The experiment results obtained by Zhang et al. [27] confirmed the presence of the spinning motion and showed that the spinning speed approximately varied with shaft speed. Ransegnola et al. [28] introduced a mutual interaction model considering the friction at the ball-socket interface, which also concluded that the spinning speed was highly dependent on the operation conditions. Thus, a spinning motion prediction model is strongly needed to obtain a relatively reliable and accurate velocity boundary of the slipper/swash plate interface in this study.

Some of the effects of the LST on the axial piston pump are contradictory to each other. Murrenhoff et al. [29,30] first applied surface texturing techniques to the friction pairs of the axial piston pump. The results indicate that the surface texture plays the most significant role in improving the carrying capacity of the oil film under the condition of high speed and pressure, which creates a thicker oil film thickness and a higher leakage at the same time. In recent years, the research on surface texture for the slipper/swash plate interface shows that the optimization goals are mainly to analyze the load-carrying capacity of friction pair interfaces [31] and the leakage [32] independently. Therefore, a multi-objective optimization model is needed for the LST to deal with the contradictory effects and comprehensively enhance the performance of the pump.

In this study, an optimization model for the multi-scale textured slipper/swash plate interface in the EHA pump is proposed. A slipper dynamic model is established considering the spinning speed to avoid inaccurate calculation of the velocity boundary caused by the slipper spinning motion. Instead of focusing on mechanical losses or volumetric losses independently, the oil film load-bearing capacity, viscous friction torque, tilting angle, and leakage are all regarded as optimization objectives. The optimal Pareto solution sets of the surface textured dimple shapes of the slipper/swash plate interface are obtained. The EHA pump prototype equipped with the textured or untextured swash plate is tested on the high-speed pump test rig. The experimental results demonstrate the positive effects of the optimal surface texture on wear reduction and efficiency improvement in EHA pumps.

2 Multi-objective optimization model of the slipper/swash plate interface

2.1 Dynamics

To avoid the long-term mixed friction caused by the serious tilting behavior of the slipper, or the solid contact caused by the difficulty of forming the lubricant film which will finally result in the wear of the component, the dynamics of the piston–slipper assembly is analyzed.

The global coordinate system O‒XYZ, the additional coordinate system O_sw‒X_swY_swZ_sw, and the additional coordinate system o_s‒x_sy_sz_s of the slipper are defined as shown in Fig.1. The piston–slipper assembly is subjected to F_N, which is generated by the piston chamber pressure F_p, the reciprocating inertia force F_a, and the axial friction force F_f between the cylinder and the piston:

Fig.1 Axial piston pump kinematics: (a) cylinder and piston–slipper assembly, (b) slipper, and (c) spinning motion of piston and slipper. ODC: outer dead center, IDC: inner dead center.

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(1)F_{\mathrm{N}}=\left(F_{\mathrm{p}}+F_{\mathrm{a}}+F_{\mathrm{f}}\right)/\phantom{\left(F_{\mathrm{p}}+F_{\mathrm{a}}+F_{\mathrm{f}}\right)\cos {\beta }_{\mathrm{s}}}\cos {\beta }_{\mathrm{s}},

(2)F_{\mathrm{N}y}=F_{\mathrm{N}}\sin {\beta }_{\mathrm{s}},

(3)F_{\mathrm{N}\mathit{\text{z}}}=F_{\mathrm{N}}\cos {\beta }_{\mathrm{s}},

where β_s is the swash plate angle, and F_Ny and F_Nz are side forces produced by the swash plate in y- and z-axis, respectively.

The load-bearing capacity F_oils of the slipper/swash plate interface needs to be balanced with F_N and the spring-back force F_r which serves to hold the slippers against the retainer. The total torques around the x_s- and the y_s-axis include T_oilsx and T_oilsy produced by the unevenly distributed pressure fields, T_τsx and T_τsy caused by viscous friction of oil film, friction torque caused by friction force F_fps of the spherical joint, and torque produced by centrifugal force F_cs acting on the centroid of slippers. The viscous friction of the oil film on the slipper land also produces the torque T_τsz, which affects the spinning motion of the slipper as well:

(4)T_{\tau \mathrm{s}\mathit{\text{z}}}+T_{\mathrm{f}\mathrm{p}\mathrm{s}}=I_{\mathit{\text{z}}\mathrm{s}}\frac{\mathrm{d}{\omega }_{\mathit{\text{z}}\mathrm{s}}}{\mathrm{d}t},

(5)T_{\mathrm{f}\mathrm{p}\mathrm{s}}=F_{\mathrm{f}\mathrm{p}\mathrm{s}}\cdot \left(r_{\mathrm{j}}\cos {\beta }_{\mathrm{s}}\right),

where I_{\mathit{\text{z}}\mathrm{s}} is the inertia moment of the slipper about the z_s-axis, {\omega }_{\mathit{\text{z}}\mathrm{s}} is the spinning speed of the slipper, t is time, T_fps is the friction torque of the slipper caused by the spherical joint, and r_j is the radius of the spherical joint.

The force and moment equilibrium equations of slippers are given in Eq. (6):

(6)\{\begin{array}{c}{F}_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}}+F_{\mathrm{N}}+F_{\mathrm{r}}=0,\\ T_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}x}+T_{\tau \mathrm{s}x}+F_{\mathrm{f}\mathrm{p}\mathrm{s}}\sin {\phi }_{\mathrm{s}}\left(r_{\mathrm{j}}\sin {\beta }_{\mathrm{s}}\right)+F_{\mathrm{c}\mathrm{s}}{l}_{\mathrm{c}\mathrm{s}}=0,\\ T_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}y}+T_{\tau \mathrm{s}y}+F_{\mathrm{f}\mathrm{p}\mathrm{s}}\cos {\phi }_{\mathrm{s}}\left(r_{\mathrm{j}}\sin {\beta }_{\mathrm{s}}\right)=0,\end{array}

with

(7)F_{\mathrm{r}}=F_{\mathrm{s}\mathrm{p}}/\phantom{F_{\mathrm{s}\mathrm{p}}\left(n_{\mathrm{p}}\cdot \cos {\beta }_{\mathrm{s}}\right)}\left(n_{\mathrm{p}}\cdot \cos {\beta }_{\mathrm{s}}\right),

where φ_s is the rotation angle of o_s in the O_sw‒X_swY_swZ_sw system, l_cs is the length between the center of the spherical joint and the centroid of the slipper, F_sp is spring force in the cylinder center, and n_p is the number of the piston.

2.2 Pressure governing equations

The fluid domain of the lubricant film of the slipper/swash plate interface is ring-shaped. As shown in Fig.2, the coordinate system o_so‒x_soy_soz_so is defined for the fluid domain, and it is obtained by translating the o_s‒x_sy_sz_s system from the origin o_s to the origin o_so. In the fine-meshed region determined by the radial coordinate R_ds and R_de, and the azimuth coordinate θ_ds and θ_de, four identical micro-dimples are formed, which are arranged in two rows along the radial direction and two columns along the circumferential direction. As shown in Fig.3, the optimization model uses four dimples rather than a single one to consider interactions between adjacent dimples. The parameters of arbitrary shapes are a set of radial coordinates R_op, which are distributed at equal angular intervals in the direction of 0°–360°. As shown in Fig.3 [23], too many decision variables R_op will result in a spiny divergent shape which is difficult to process in the application. Therefore, the number of R_op is set as 12 [23] in this model.

Fig.2 Oil film domain of the slipper/swash plate interface.

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Fig.3 Arbitrary dimple shapes characterized by R_op and four identical micro-dimples [23].

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The gap height h of any point P (r, θ) on the slipper/swash plate interface can be derived from the functions h₁, h₂, and h₃, which are the heights of points A, B, and C determined by the time-dependent external load, as shown in Eq. (8):

(8)\begin{array}{rl}h=& \frac{1}{\sqrt{3}{R}_h}\left(h_2-h_3\right)r\sin \theta +\frac{1}{3R_h}\left(2h_1-h_2-h_3\right)r\cos \theta \\ & +\frac{1}{3}\left(h_1+h_2+h_3\right),\end{array}

where R_h is the oil film radius, and h₁, h₂, and h₃ are the gap heights of points A, B, and C, respectively.

The macro-motion of the slipper forms an elliptical trajectory on the X_sw‒Y_sw plane. The relationship between the origin o_s and the shaft rotation angle φ in the coordinate system O_sw‒X_swY_swZ_sw is given as Eq. (9):

(9)\{\begin{array}{c}{R}_{\mathrm{o}\mathrm{s}}=R_{\mathrm{d}}\sqrt{1+{\tan }^2{\beta }_{\mathrm{s}}{\cos }^2\phi },\\ {\phi }_{\mathrm{s}}=\arctan \left(\cos {\beta }_{\mathrm{s}}\tan \phi \right),\end{array}

with

(10)\phi =\omega t,

where R_os is the length between o_s and the Z_sw-axis, R_d is the distribution radius of slippers, and \omega is rotation speed.

Considering the spinning speed \omega_zs of the slipper, the radial and circumferential velocities of any point P (r, θ) can be expressed as Eq. (11). A more detailed explanation of the slipper spinning motion is included in Chao et al. [1]:

(11)\{\begin{array}{c}{v}_r={\omega }_{\mathrm{s}}{R}_{\mathrm{s}}\sin \left(\theta -\sigma \right),\\ v_{\theta }={\omega }_{\mathrm{s}}{R}_{\mathrm{s}}\cos \left(\theta -\sigma \right)+{\omega }_{z\mathrm{s}}r,\end{array}

with

(12){\omega }_{\mathrm{s}}=\frac{\mathrm{d}{\phi }_{\mathrm{s}}}{\mathrm{d}t},

(13)R_{\mathrm{s}}=\sqrt{R_{\mathrm{o}\mathrm{s}}^2+r^2+2R_{\mathrm{o}\mathrm{s}}r\cos \theta },

(14)\sigma =\arcsin \left(r\sin \theta /\phantom{r\sin \theta R_{\mathrm{s}}}{R}_{\mathrm{s}}\right),

where v_r and v_θ are radial and circumferential velocities at any position (r, θ) of the slipper bottom, respectively, {\omega }_{\mathrm{s}} is the speed of o_s about the Z_sw, R_s is the length between point P (r, θ) and the Z_sw-axis, and σ is the angle between P‒O_sw and o_s‒O_sw.

The leakage flow rate of the slipper/swash plate interface is given by Eq. (15):

(15)Q_{\mathrm{s}\mathrm{s}}={\int }_0^{\text{2π}}{{\int }_0^h\left(v_{\mathrm{o}r}r\right)|}_{r\in \left[R_{\mathrm{i}\mathrm{n}},R_{\mathrm{o}\mathrm{u}\mathrm{t}}\right]}\mathrm{d}\mathit{\text{z}}\mathrm{d}\theta ,

with

(16)v_{\mathrm{o}r}=\frac{1}{2\eta }\frac{\mathrm{\partial }p}{\mathrm{\partial }r}\left({\mathit{\text{z}}}^2-h\mathit{\text{z}}\right)+v_r\frac{\mathit{\text{z}}}{h},

where Q_ss is leakage, v_or is the radial velocity of the oil, R_in and R_out are the inner and outer radii of the slipper land, respectively, η is the oil dynamic viscosity, and p is the oil film pressure.

The oil film between the slipper/swash plate interface is assumed to be incompressible and laminar. At the same time, the velocity of the fluid attached to the wall is considered same as that of the wall without considering the wall slip phenomenon [14,33]. The Reynolds equation [34] in polar coordinates is derived as shown in Eq. (17):

(17)\begin{array}{rl}& \frac{1}{\eta r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r{h}^3\frac{\mathrm{\partial }p}{\mathrm{\partial }r}\right)+\frac{1}{\eta r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(h^3\frac{\mathrm{\partial }p}{\mathrm{\partial }\theta }\right)\\ =& 6\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r{v}_{r}h\right)+6\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(v_{\theta }h\right)+12\frac{\mathrm{\partial }h}{\mathrm{\partial }t}+\frac{3\rho }{10\eta r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(v_{\theta }^2{h}^3\right),\end{array}

where ρ is oil density.

The load-bearing capacity F_oils, viscous friction torque about the z_s-axis T_τsz, and viscous friction torque about the Z_sw-axis T_τszg can be solved by the Reynolds equation with the pressure and velocity boundary conditions:

(18)F_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}}={\int }_0^{\text{2π}}{\int }_{R_{\mathrm{i}\mathrm{n}}}^{R_{\mathrm{o}\mathrm{u}\mathrm{t}}}pr\mathrm{d}r\mathrm{d}\theta ,

(19)T_{\tau \mathrm{s}\mathit{\text{z}}}={\int }_0^{\text{2π}}{\int }_{R_{\mathrm{i}\mathrm{n}}}^{R_{\mathrm{o}\mathrm{u}\mathrm{t}}}{{\tau }_{\mathrm{o}\theta }|}_{\mathit{\text{z}}=h}\cdot r^2\mathrm{d}r\mathrm{d}\theta ,

(20)\begin{array}{rl}{T}_{\tau \mathrm{s}\mathit{\text{z}}\mathrm{g}}=& {\int }_0^{\text{2π}}{\int }_{R_{\mathrm{i}\mathrm{n}}}^{R_{\mathrm{o}\mathrm{u}\mathrm{t}}}\left[{{\tau }_{\mathrm{o}\theta }|}_{\mathit{\text{z}}=h}\cdot \cos \left(\theta -\sigma \right)+{{\tau }_{\mathrm{o}r}|}_{\mathit{\text{z}}=h}\cdot \sin \left(\theta -\sigma \right)\right]\\ & \cdot R_{\mathrm{s}}r\mathrm{d}r\mathrm{d}\theta ,\end{array}

with

(21){{\tau }_{\mathrm{o}\theta }|}_{\mathit{\text{z}}=h}=\frac{h}{2r}\frac{\mathrm{\partial }p}{\mathrm{\partial }\theta }+\eta \frac{v_{\theta }}{h},

(22){{\tau }_{\mathrm{o}r}|}_{\mathit{\text{z}}=h}=\frac{h}{2}\frac{\mathrm{\partial }p}{\mathrm{\partial }r}+\eta \frac{v_r}{h}-\frac{\rho h}{4r}{v}_{\theta }^2,

where τ_oθ and τ_or are circumferential and radial viscous shear stresses, respectively.

In the simulation model, the main dimensions, working conditions, and oil parameters of the EHA pump are listed in Tab.1.

Tab.1 EHA pump parameters and operating conditions considered in this study

Parameter	Value
Distribution radius of slippers, Rd	0.02 m
Piston radius, rp	0.005 m
Inlet pressure, pin	5 × 105 Pa
Outlet pressure, pout	28 × 106 Pa
Case drain pressure, p0	1×105 Pa
Atmospheric saturation pressure	3 × 104 Pa
Shaft speed, n	10000 r/min
Swash plate angle, βs	0.13 rad (7.5°)
Temperature	60 °C
Oil density at 60 °C	837.4 kg/m3
Oil kinematic viscosity at 60 °C	10.3 mm2/s

2.3 Simulation model

The flowchart of the multi-objective optimization model of the slipper/swash plate interface is shown in Fig.4. The model is mainly composed of three parts: the oil film characteristic module, the force and torque balance module, and the multi-objective optimization module based on the non-dominated sorting genetic algorithm II (NSGA-II), which is confirmed to be an effective method to solve the multi-objective optimization problem [23].

Fig.4 Flowchart of the multi-objective optimizing procedure for the slipper/swashplate interface. IDC: inner dead center, ODC: outer dead center.

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Simultaneously, the model needs to consider not only the tilting motion but also the spinning motion of the piston and the slipper. Thus, the oil film characteristic module needs to be executed constantly to obtain pocket pressure p_ops which is regarded as one of the current pressure boundary conditions to satisfy the flow continuity equation. Then, p_ops is constantly called by the force and torque balance module. Finally, when the slipper reaches the force and torque balance, the spinning motion driving or hindering torque can be substituted into the spinning motion module. The module also solves the spinning angular acceleration at the current moment t and the spinning velocity at the next moment (t + Δt) to obtain new velocity boundary conditions. Regarding one entire shaft revolution as a cycle, the model needs to be executed for at least eight cycles under any working conditions to acquire periodic variation of lubricant film characteristics and micro-motion characteristics of rotating components.

During the simulation process, if the surface textures produced by each individual in a population are all evaluated for eight cycles, then the simulation time will be prolonged thousands of times. Thus, the multi-objective optimization module is set to start the additional fine-meshed region optimization where the slipper is most likely to produce wear due to the excessive tilting angle. The changing surface topography only varies the load capacity of the textured fine-meshed region but not the entire slipper/swash plate interface. As a result, the difference between local load-bearing capacity at the textured fine-meshed region F_{oils{textured}} and untextured region F_{oils{untextured}}, namely ΔF_oils, is selected as a surrogate objective of F_oils as shown in Eq. (23):

(23)\mathrm{\Delta }{F}_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}}=F_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}\left\{\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}\right\}}-F_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{s}\left\{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}\right\}}.

According to Eqs. (18) and (19), the mechanical efficiency η_m is expressed as Eq. (24) under the simulation condition:

(24){\eta }_{\mathrm{m}}=\frac{\left(p_{\mathrm{o}\mathrm{u}\mathrm{t}}-p_{\mathrm{i}\mathrm{n}}\right)V_{\mathrm{s}}}{2\text{π}\left[T_{\mathrm{s}\mathrm{s}}+\left(T_{\tau \mathrm{s}\mathit{\text{z}}}+0.2\right)\cos {\beta }_{\mathrm{s}}\sqrt{1-{\cos }^2\phi {\sin }^2{\beta }_{\mathrm{s}}}\right]},

where V_s is the displacement of the pump, and T_ss is the output torque of the shaft in addition to viscous friction torque of the slipper/swash plate interface. η_m increases with the decrease in T_τsz. As a result, T_τsz can be considered the optimization objective.

Based on the authors’ previous attempts at different depths during the study, the recommended value of dimple depth in this simulation is 5 μm. Under the fixed dimple depth condition, the objective of the simulation model is to find the optimal radial coordinate array R_op of the dimple shape. This way maximizes the load-bearing capacity ΔF_oils and minimizes the tilting angle α_s, the viscous friction torque T_τsz, and the leakage Q_ss of the slipper/swash plate interface.

The NSGA-II parameters utilized in the multi-objective optimization model are listed in Tab.2.

Tab.2 Parameters of multi-objective optimization model based on NSGA-II

Parameter	Value
Dimple depth	5 μm
Number of Rop elements	12
Population size	60
Pareto fraction	0.35
Crossover fraction	0.8
Migration fraction	0.2
Lower bounds	≥ 0.1 mm
Upper bounds	≤ 0.2 mm
Objectives	1. Maximization ΔFoils
	2. Minimization Tτsz
	3. Minimization αs
	4. Minimization Qss

3 Simulation results and discussion

The simulated slipper tilting angle α_s and the tilting azimuth angle of the untextured slipper/swash plate interface θ_αs are shown in Fig.5. In the last three revolutions, the range of the maximum tilting angle varies from 0.00564° to 0.00578°, which is located at the azimuth angle of 175.5°–179.1° and the shaft rotation angle of 308°–314°. The slipper tends to form the maximum tilting angle on the low-pressure side near the outer dead center (ODC). Based on the simulation data, the additional fine-meshed region where the slipper has a maximum tilting angle is then located. The specific parameters are listed in Tab.3.

Tab.3 Fine-meshed region positional parameter and starting shaft rotation angle φ_op

Parameter	Value
θds	3 rad (171.90°)
θde	3.19 rad (182.70°)
Rds	5×10−3 m
Rde	6.05×10−3 m
φop	36.83 rad (2110°)

Fig.5 Periodic changes of the slipper tilting angle and the tilting azimuth angle.

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The rank of individuals is realized by comparing it with the objective function at the next shaft rotation angle (φ_op + \omegaΔt). The Pareto non-dominated solution set can be obtained after the algorithm searches exceed the stall generations and the average change in the spread reaches the pre-set tolerance. After the multi-objective optimization model reaches the termination condition, the Pareto optimal solution set is generated. The dimple shapes represented by the optimal solution are shown in Fig.6.

Fig.6 Dimple shape represented by the optimal solution on the Pareto front.

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The Pareto front of the slipper/swash plate interface is depicted in Fig.7. The dimple shapes of the 9th and 10th results are similar to the arc-sided square, which produce the maximum lubricant film capacity of load carrying and minimum tilting angle. Meanwhile, the dimple shapes of the 5th and 21st results are approximated as water-drop shapes, which minimize the leakage flow. The dimple shape of the 1st result is circular, which decreases the viscous friction torque. The dimple shape parameters R_op which conclude the representative optimization results are listed in Tab.4. When the radial size accuracy is only 1 μm, the dimple shapes between the 9th and 10th results and the 5th and 21st results are considered identical. In addition, the overall approximate water-drop-shaped dimple is nearly similar to a circle because of its small sharp angle.

Tab.4 Optimal R_op with corresponding dimple shapes

Dimple shape	Rop/mm	Shape
1	[0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100]	Circle
5	[0.100, 0.108, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.103, 0.100, 0.100]	Similar to water drop
9	[0.185, 0.200, 0.200, 0.142, 0.179, 0.175, 0.130, 0.156, 0.200, 0.148, 0.193, 0.173]	Approximate arc-sided square
10	[0.185, 0.200, 0.200, 0.142, 0.179, 0.175, 0.130, 0.156, 0.200, 0.148, 0.193, 0.173]	Approximate arc-sided square
21	[0.100, 0.108, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.100, 0.103, 0.100, 0.100]	Similar to water drop

Fig.7 Pareto front with the optimal results.

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The pressure and gap height distributions produced by the optimal dimple shapes of the slipper/swash plate interface are shown in Fig.8. The dynamic pressure caused by different dimple shapes has obvious differences. The pressure distribution curves along the cylinder rotation direction produced by the approximate water-drop-shaped dimples (No. 5) and the approximate arc-sided square-shaped dimple (No. 9) are shown in Fig.9. The positions where the maximum dynamic pressures are generated and the positions at which the atmospheric saturation pressures are reached are different. The maximum dynamic pressure produced by the approximate arc-sided square-shaped dimple (No. 9) is about 5 MPa higher than that produced by the approximate water-drop-shaped dimples (No. 5). This condition allows the approximate arc-sided square-shaped dimple (No. 9) to generate maximum load-bearing capacity.

Fig.8 Pressure distribution and gap height distribution of the textured region with Nos. 1, 5, and 9 dimple shapes.

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Fig.9 Pressure distribution curves along the circumferential direction of the slipper land with Nos. 5 and 9 dimple shapes.

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The load-bearing capacity of the untextured region is 1.653 N. The rate of changes in four optimizing targets (r_perf) is calculated as

(25)r_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}=\left(V_{\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}-V_{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}\right)/\phantom{\left(V_{\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}-V_{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}\right)V_{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}}{V}_{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}},

where V is replaced by F_oils, α_s, T_τsz, and Q_ss when evaluating the load bearing capacity, tilt angle, friction torque, and leakage, respectively.

As shown in Fig.10, the surface texture plays a positive role in improving the capacity of load carrying, but it weakly reduces the viscous friction torque, tilting angle, and leakage. Although approximate arc-sided square-shaped dimple (No. 9) has little effect on reducing viscous friction torque and leakage, it increases the load-bearing capacity of the slipper/swash plate interface by 123.77% compared with the untextured one, whose optimization effect is significantly greater than that of approximate water-drop-shaped dimple (No. 5).

Fig.10 Rate of changes in four optimizing targets with the optimal results.

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4 Experiment verification

4.1 Experimental scheme

The real effects of the surface texture scheme in the EHA pump are evaluated on a test rig that is used in the previous work, as shown in Fig.11, to verify the optimal solution of arbitrary-shaped surface textures for the slipper/swash plate interface discussed in Section 3. The test rig can provide a rotational speed of up to 16000 r/min and can measure variables such as inlet, outlet, and leakage pressure, outlet and leakage flow, shaft speed, and shaft torque.

Fig.11 Test rig used for evaluating the performance of EHA pumps. EHA: electro-hydrostatic actuator.

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According to the simulation results of optimal dimple shapes in Section 3, the approximate arc-sided square-shaped dimple (No. 9) generates the maximum load-bearing capacity and the minimum tilting angle. Therefore, the wear marks of the approximate arc-sided square-shaped (No. 9) textured swash plate are compared with those of the untextured one. The texture pattern is fabricated on the swash plate using a Q-switched Nd:YVO4 laser machine with a wavelength of 355 nm, a frequency of 30 kHz, and a maximum output power of 5 W [35]. Slippers are prone to tilt toward the splined shaft. Thus, the surface texture is arranged in the negative y_so-axis region and the total number of dimples is 1224 (radial 3 rows × circumferential 192 rows + radial 3 rows × circumferential 216 rows). By transforming the slipper bottom coordinate from the o_s‒x_sy_sz_s coordinate system to the O_sw‒X_swY_swZ_sw coordinate system, the surface topography parameters of the static swash plate surface can be converted to the slipper land surface which is revolving around. The textured swash plate and dimples on the surface are shown in Fig.12.

Fig.12 Textured swash plate and dimples on the surface.

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The tested components and their abbreviations are the textured swash plate (TSP) and the untextured swash plate (UTSP). Before the test, the components are cleaned in an ultrasonic acetone bath to remove the contaminants. The assembly of the swash plate with the texture design is shown in Fig.13. After the test, the macroscopic views of the swash plate are observed by a confocal laser scanning microscope (CLSM).

Fig.13 Assembly of the swash plate with texture design.

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For better observations of the surface wear marks of the components, a relatively severe operating condition is chosen. When the EHA pump pressure reaches 28 MPa, the tilting angle of the swash plate will be reduced to 1.2°, and the pump only needs to run for a very short time. Thus, the prototype is operated for 20 min at the maximum rotational speed of 10000 r/min and maximum pressure condition of 28 MPa. The tested EHA pump prototype will be equipped with TSP or UTSP.

4.2 Wear of the swash plate surface

The macroscopic views of the UTSP and TSP interface are shown in Fig.14. The black boxes correspond to the position where the slipper causes the largest tilting angle. Compared with the wear marks in the black box of UTSP, the wear marks of TSP at the same place are not obvious. The characteristics discussed above can be confirmed by the CLSM scanned image of the UTSP and the TSP as shown in Fig.15. Deep plowing marks of 1–2 μm generated by wear debris can be found on the UTSP surface, which are more obvious than that of the TSP. This phenomenon indicates that the slippers sliding against the UTSP have a greater degree of tilting behavior, which results in serious abrasive wear. By contrast, the surface texture of the TSP weakens the tilting behavior of the slipper and enhances the load-bearing capacity at the position where the slipper causes the largest tilting angle. The ability of the dimples to trap wear debris is also proven.

Fig.14 Photos of the tested TSP and UTSP marked with rectangular frames. IDC: inner dead center, TSP: textured swash plate, UTSP: untextured swash plate, ODC: outer dead center.

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Fig.15 CLSM scanned images of the TSP and the UTSP after testing at the position where the slipper causes the largest tilting angle. TSP: textured swash plate, UTSP: untextured swash plate, CLSM: confocal laser scanning microscope.

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4.3 Efficiencies of EHA pump prototype

The performance data of the EHA pump prototype equipped with TSP and UTSP are shown in Tab.5. The mechanical efficiency η_m and volumetric efficiency η_v are calculated according to Eqs. (26) and (27), respectively:

Tab.5 Efficiencies and output torques of EHA prototype at 10000 r/min, 28 MPa

EHA pump prototype equipped with	Mechanical efficiency, ηm/%	Volumetric efficiency, ηv/%	Output torque, Tsh/(N·m)
TSP	87.1	81.1	18.6
UTSP	85.7	80.3	18.9

(26){\eta }_{\mathrm{m}}=\frac{\left(p_{\mathrm{o}\mathrm{u}\mathrm{t}}-p_{\mathrm{i}\mathrm{n}}\right)V_{\mathrm{g}}}{2\text{π}{T}_{\mathrm{s}\mathrm{h}}}\times 100\mathrm{\%},

(27){\eta }_{\mathrm{v}}=\frac{q_{\mathrm{o}\mathrm{u}\mathrm{t}}}{V_{\mathrm{g}}n}\times 100\mathrm{\%},

where q_out is the outlet flow and T_sh is the output torque of the shaft of the prototype.

Tab.5 shows that the TSP improves the mechanical efficiency by 1.4%. The direct reason for the improvement of mechanical efficiency is that the TSP reduces the T_sh of the prototype by 0.3 N·m. However, the improvement effect of the TSP on the volumetric efficiency of the EHA pump prototype is not obvious, and it is less than 1%.

5 Conclusions

From all the simulation and experiment results performed, the following conclusions can be drawn:

A multi-objective optimization model for the multi-scale textured slipper/swash plate interface based on the NSGA-II is proposed. The optimal Pareto solution sets of the surface textured dimple shapes are obtained. In the simulation, the approximate arc-sided square-shaped dimple (No. 9) reduces the friction torque, tilting angle, and leakage, and it increases the load-bearing capacity of the slipper/swash plate interface by 123.77%. In the experiment, the No. 9 dimple improves the mechanical and volumetric efficiencies of the EHA pump prototype by 1.4% and 0.8%, respectively.

The effects of thermoelastic deformation of the solid parts will be considered in the future work to refine the model. The three modules of the optimization model are also required to further improve the computational efficiency and shorten the simulation time.

6 Nomenclature

Abbreviations
CLSM	Confocal laser scanning microscope
EHA	Electro-hydrostatic actuator
LST	Laser surface texturing
IDC	Inner dead center
ODC	Outer dead center
TSP	Textured swash plate
UTSP	Untextured swash plate
Variables
Fa	Reciprocating inertia force
Fcs	Slipper centrifugal force
Ff	Axial friction force between the cylinder and the piston
Ffps	Friction force of the slipper caused by the spherical joint
FN	Normal force produced by the swash plate of the piston‒slipper assembly
FNy, FNz	Side force produced by the swash plate in y- and z-axis, respectively
Foils	Load-bearing capacity force of the slipper/swash plate interface
Foils{textured}, Foils{untextured}	Local load-bearing capacity at the textured fine-meshed and untextured regions, respectively
Fp	Piston chamber pressure
Fr	Spring-back force
Fsp	Spring force in the cylinder center
ΔFoils	Difference of local load-bearing capacity at textured region and untextured region
h	Gap height
Izs	Inertia moment of the slipper about the zs-axis
lcs	Length between the center of the spherical joint and the centroid of the slipper
n	Shaft speed
np	Number of the piston
p	Oil film pressure
p0	Case drain pressure
pin, pout	Inlet and outlet pressures, respectively
pops	Pocket pressure
qout	Outlet flow
Qss	Leakage
r	Radius
rj	Radius of the spherical joint
rp	Piston radius
rperf	Rate of changes in four optimizing targets, i.e., Foils, αs, Tτsz, and Qss
Rd	Distribution radius of slippers
Rds, Rde	Radial coordinate of the fine-meshed region
Rh	Oil film radius
Rin, Rout	Inner and outer radii of the slipper land, respectively
Ros	Length between os and the Zsw-axis
Rs	Length between any position (r, θ) and the Zsw-axis
Rop	Array of the radial coordinates defining the arbitrary dimple shape
t	Time
Tfps	Friction torque of the slipper caused by the spherical joint
Toilsx, Toilsy	Torques produced by the unevenly distributed pressure fields of the oil film about the xs- and ys-axis, respectively
Tsh	Output torque of the shaft of the prototype
Tss	Output torque of the shaft in addition to viscous friction torque of the slipper/swash plate interface
Tτsx, Tτsy, Tτsz, Tτszg	Viscous friction torques about the xs-, ys-, zs-, and Zsw-axis, respectively
vor	Radial velocity of the oil
vr, vθ	Radial and circumferential velocities at any position (r, θ) of the slipper bottom, respectively
Vs	Displacement of the pump
αs	Tilting angle of the slipper
βs	Swash plate angle
η	Oil dynamic viscosity
ηm	Mechanical efficiency
ηv	Volumetric efficiency
θ	Azimuth angle
θαs	Tilting azimuth angle of the slipper
θds, θde	Azimuth coordinate of the fine-meshed region
ρ	Oil density
σ	Angle between any position (r, θ)‒Osw and os‒Osw
φ	Shaft rotation angle
φop	Starting shaft rotation angle of the optimization process
φs	Rotation angle of os in the Osw‒XswYswZsw system
\omega	Rotation speed
{\omega }_{\mathrm{s}}	Rotation speed of os about the Zsw-axis
{\omega }_{\mathit{\text{z}}\mathrm{s}}	Spinning speed of the slipper
τoθ, τor	Circumferential and radial viscous shear stresses, respectively

Acknowledgements

The research was financially supported by the National Key R&D Program of China (Grant No. 2018YFB2001101), the National Outstanding Youth Science Foundation of China (Grant No. 51922093), the National Science Foundation for Young Scientists of China (Grant No. 51905473), the Major Science and Technology Projects in Ningbo, China (Grant No. 2019B10054), and the Open Foundation of the State Key Laboratory of Mechanical Transmissions, China (Grant No. SKLMT-ZDKFKT-202001).

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