J. Cent. South Univ. (2021) 28: 2375-2393

DOI: https://doi.org/10.1007/s11771-021-4776-9

Lubrication characteristics of external return spherical hinge pair of axial piston pump or motor under combined action of inclination and offset distance

WANG Lei(王雷)^{1, 2}, DENG Hai-shun(邓海顺)^{1, 3}, GUO Yong-chun(郭永存)^{1, 3},WANG Chuan-li(王传礼)^{1, 2}, HU Cong(胡聪)¹

1. School of Mechanical Engineering, Anhui University of Science and Technology, Huainan 232001, China;

2. Anhui Key Laboratory of Mine Intelligent Equipment and Technology, Anhui University of Science andTechnology, Huainan 232001, China;

3. State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mines,Anhui University of Science and Technology, Huainan 232001, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: External return mechanism is a mechanical structure applied to axial piston pumps. To study its lubrication characteristics, the Reynolds equation applied to an external return spherical hinge pair was deduced based on the vector equation of relative-motion velocity of the external return spherical hinge pair under the influence of external swash plate inclination and offset distance. The results show that the total friction, axial leakage flow, and maximum value of the maximum oil-film pressure increase with increasing pump-shaft speed and decrease with increasing offset distance in one working cycle when the external-swash-plate inclination is constant. However, the varying offset distance has little effect on the axial leakage flow. The maximum value of the maximum oil-film pressure decreases with increasing external-swash-plate inclination and the total leakage flow increases with increasing external-swash-plate inclination in one working cycle when the offset distance is constant. It can be seen that the abovementioned parameters are important factors that affect the lubrication characteristics of external return spherical hinge pairs. Therefore, the complex effects of different coupling parameters should be comprehensively considered in the design of the external return mechanism.

Key words: axial piston motor or pump; external return mechanism; external swash plate inclination; offset distance; lubrication performance

Cite this article as: WANG Lei, DENG Hai-shun, GUO Yong-chun, WANG Chuan-li, HU Cong. Lubrication characteristics of external return spherical hinge pair of axial piston pump or motor under combined action of inclination and offset distance [J]. Journal of Central South University, 2021, 28(8): 2375-2393. DOI: https://doi.org/10.1007/ s11771-021-4776-9.

1 Introduction

Return mechanisms are devices that connect the three friction pairs of a plunger pump or motor [1-4] (namely the plunger pair [5, 6], slipper pair [7-9], and port pair [10, 11]). The return mechanism preloads the slipper and port pairs and forms a static-pressure support between them to ensure that they are in a state of hydrodynamic lubrication. It also provides the return force of the plunger to guarantee a volume change in the cavity between the plunger and cylinder block. The return mechanism plays an important role in the operational performance of the three friction pairs, and its importance has attracted the attention of researchers around the world.

External return mechanisms [12] are the major components of dual-drive axial piston motors or pumps (Figure 1). They mainly comprise external spherical hinges and external retainer plates. Because its structure is similar to that of a spherical plain bearing, the research results and methods pertaining to spherical plain bearings can be used for reference. XIANG et al [13] optimized the structure of GEZ101ES for a radial spherical plain bearing using a finite-element model. By keeping the assembly dimension unchanged, they obtained the optimal value of the ball diameter under static loading through a numerical experiment. LU et al [14] analyzed the linear velocity and pressure distribution in the contact area of the friction pair under three modes of swivel and tilt motion. They also combined the swing and tilt motions of the self-lubricating radial spherical plain bearing, deduced the wear-life model suitable for such a bearing, and obtained the relationship between the bearing-life ratio and pendulum angle. XUE et al [15] used the classical Archard adhesion-wear theory to establish a three-dimensional finite-element model of self-lubricating spherical plain bearings under the condition of swinging wear. They found that the maximum contact pressure and the maximum wear depth appeared in the central-contact region and that the maximum contact pressure decreased with increasing number of swings. Using the Fang model for non-conformal and conformal contact between spherical surfaces, FANG et al [16] proposed an accurate theoretical solution for calculating the conformal-contact-pressure distribution in spherical plain bearings and investigated the relationships between structural and contact-mechanical parameters. They also discussed the influence of the free-edge effect on the distribution of radial and axial contact pressure. QIU et al [17] investigated the influence of rare-earth treatment on the friction-wear and bonding performance of Kevlar and PTFE-fiber-blended fabric spherical bearings. TANG et al [18] studied how surface-structure and roughness parameters influence the friction performance of a spherical plain bearing, as well as the effects of dimple depth and density on the friction coefficient. HE et al [19] deduced the Reynolds equation for the lubricating grease of a spherical fixed-set bearing in a roller bit in spherical coordinates and analyzed the influence of eccentricity and bearing gap on the bearing capacity of an oil film. TANG et al [20] deduced the Reynolds equation for joint loading and studied the influence of the power-law index, inclination of the inner ring, and swing velocity on the lubrication performance.

Figure 1 Structure of dual-drive axial piston pump/motor:

Although there are certain similarities between external return spherical hinge pairs and spherical plain bearings, these differ in terms of motion characteristics and structural design, meaning that the results of research on joint bearings cannot be directly applied to external return spherical hinge pairs. In the case of motion characteristics, the relative-motion trajectory between the external retainer plate and external spherical hinge has an approximate “raindrop” shape [21] and there are complex spatial-motion states in a working cycle (the main axis rotates through 360°). In the present study, to reduce the difficulty associated with the processing and installation of parts in the structural design of the external-return mechanism, a certain offset distance is included for the external-retainer plate relative to the center of the external spherical hinge. In addition, to achieve a different displacement, a dual-drive axial piston motor also is operated under different swash-plate inclinations. In the working cycle of the external return mechanism, a convergence gap is formed between the external retainer plate and spherical hinge under the combined action of the offset distance and external-swash-plate inclination. Due to the difference in the relative speed between the two friction surfaces of the external return spherical hinge pair, the lubricant can enter into the convergence system and act as a hydrodynamic lubricator.

Different offset distances and external swash plate inclinations greatly affect the spatial-motion state of the external-return mechanism. XU et al [22, 23] established mathematical and virtual-prototype models of the relative motion of the retainer plate and spherical hinge and obtained the relative-motion traces of these components. LIU et al [24] established the dynamic law for the lubricating oil in a slipper pair of axial-piston pumps and discussed the influence of different restraining devices on the slipper’s dynamic characteristics. DENG et al [25] investigated the influence of the structural parameters of the external-return mechanism on the friction power of the contact areas between the external retainer plate and external spherical hinge. WANG et al [26] studied the film-lubrication mechanism of the return mechanism using its kinematic characteristics and the modified Jakobsson-Floberg-Olsson (JFO)-cavitation algorithm.

The lubrication performance of the external return spherical hinge pair has been studied only for small external swash plate inclinations, and the offset distance has not been considered. Therefore, it is necessary to study its lubrication performance under the combined action of different offset distances and external swash plate inclinations to further improve the motion characteristics of the external return mechanism and the lubrication theory of external retainer plate-external spherical hinge friction pairs.

2 Equations

2.1 Kinematic equation of external return mechanism

The external return mechanism is shown in Figure 1. The external spherical hinge is affixed to the cylindrical block through the juncture, and the external retainer plate is driven by the external row slipper. Because the external swash plate’s vertical center line is inclined with respect to the center line of the pump shaft, the external retainer plate rotates elliptically relative to this line.

To analyze the kinematic characteristics of the external return mechanism, principal and subordinate coordinates similar to Ref. [27] were established. As shown in Figure 2, the principal coordinate system O-xyz is consolidated on the external spherical hinge; the origin is coincident with the inner contour of this hinge and the O-x axis is taken as the axial direction of the pump shaft and is used to describe the hinge’s rotation. The subordinate coordinate O₁-x₁y₁z₁ is consolidated on the external retainer plate, and the O₁-x₁ axis is taken as the axial direction of this plate and is used to describe its rotary motion. The axis O₁-y₁ is the radial direction of the external retainer plate; β is the external swash plate inclination; b_f is the offset distance; R is the radius of the external retainer plate, and R_M is the radius of the external spherical hinge. The point Q is the intersection between the O₁-y₁ axis at the center of the external retainer plate and surface of this plate. B is the width of the external retainer plate, B₁ is the distance between Q and the plate’s right end face, and B₂ is the distance between Q and the plate’s left end face.

                            (1)

B=B₁+B₂                                                  (2)

Figure 2 Principal and subordinate coordinates of external return mechanism

To further analyze the influence of the external return spherical hinge pair’s mode of motion on the lubrication characteristics of the external return mechanism, a coordinate diagram describing this motion is established, as shown in Figure 3. ω_b is the angular velocity of the external spherical hinge’s rotation; ω_r is the instantaneous angular velocity of the external retainer plate; and ω is the angular velocity of the pump/motor shaft. To analyze the lubrication characteristics of the external return spherical hinge pair, the spherical-coordinate system shown in Figure 4 is established. The coordinate system’s origin is the center of the ball at the inner contour of the external spherical hinge, with the O-x axis being the central line of the pump shaft and the O-y axis being the radial direction of the external spherical hinge.

Figure 3 Motion position and coordinate system

Figure 4 Circumferential force on microscopic fluid element

Unless otherwise specified, the x, y, and z unit vectors of the O-xyz coordinate systems in Figures 2-4 are represented by i, j, and k, respectively. The unit vectors of the three orthogonal axes of the spherical-coordinate system r, θ, and φ are e_r, e_θ and e_φ, respectively. Intercept an arbitrarily small fluid element m in the lubrication film and its circumferential force is shown in Figure 4.

Because the relative motion of the external return mechanism significantly influences the lubrication characteristics, the lubrication problem is studied according to the vector equation of the relative speed of the external return spherical hinge pair in Ref. [27].

        (3)

where k=(R²cos²αcos²β-Rb_fcosαsin(2β)+b_f²sin²β+R²sin²α)/

The abovementioned relative-velocity vector equation was obtained under the assumption that the external retainer plate and external spherical hinge are point-contact models, it describes the relative velocity of the contact point Q between them. To study the influence of the velocity distribution of all contact points of the pair on its lubrication characteristics, the velocity distribution of the entire contact surface must be determined. Therefore, the value of β is changed to include all contact points of the external return spherical hinge pair,i.e., The relative-velocity distribution of this pair is obtained as shown in Figure 5. The velocity-distribution diagram shown in Figure 5 is consistent with the offset distance b_f=0.005 m in Ref. [27].

As shown in Figure 4, the Reynolds equation is in spherical coordinates, making it necessary to convert the vector equation of velocity from Cartesian coordinates to spherical ones. This transformation has the form:

      (4)

From Eqs. (3) and (4), the vector equation of relative velocity in the spherical-coordinate system can be written as:

                          (5)

where .

The vector expression of the angular velocity ω_r in the coordinate system o-xyz is:

            (6)

where β₁ is the angle between O-x₁ and O-x axis. When β₁=β, ω_r is the angular velocity of the point Q.

Figure 5 Velocity distribution of external-return/spherical-hinge pair:

The expression for the radius vector R_m of a contact point m₁ on the external return spherical hinge pair in the coordinate system o-xyz is [21]:

     (7)

From Eqs. (4), (6), and (7), the vector equation of the absolute velocity of contact point m₁ in the spherical-coordinate system can be obtained:

        (8)

The vector equation of the angular velocity ω_b in the o-xyz coordinate system can be written as:

                        (9)

Because the external spherical hinge rotates with the pump/motor shaft, the vector expression for the reference velocity of point m₁ in the spherical-coordinate system is as follows:

(10)

Because the composite-motion form of the external return spherical hinge pair significantly influences the lubrication characteristics, the velocity boundary conditions should be considered when solving the Reynolds equation. ζ is the distance from the outer surface of the external retainer plate to an arbitrarily small fluid element m of the oil film. When ζ=0, the velocity vector of m is caused by the external retainer plate; when ζ=h, this vector is caused by the external spherical hinge.

According to Eqs. (8) and (10), the velocity-boundary conditions of the external return spherical hinge pair can be obtained as follows:

When ζ=0:

      (11)

When ζ=h:

      (12)

where u_r, u_θ and u_φ are the velocity components in the directions r, θ and φ, respectively, of the spherical-coordinate system.

2.2 Boundary conditions

As the external return spherical hinge pair can be used as a special spherical plain bearing, we can adopt Reynolds boundary conditions that are more in line with the actual situation. These boundary conditions are presented here.

2.2.1 Circumferential boundary conditions of external return spherical hinge pair

The starting point of the oil film is:

.

Its ending point is:

.

The continuous place of the oil film is:

;

where p₀ is the environmental pressure.

2.2.2 Axial-boundary conditions of external return spherical hinge pair

Figure 6 presents the axial-boundary conditions of external return spherical hinge pair, where:

                        (13)

where

To describe the influence of the velocity distribution of all contact areas along the width B of the external return mechanism on the oil film’s lubrication characteristics, the variation range of θ is taken as [θ₁, θ₂], where:

                    (14)

                    (15)

where

When θ=θ₁, U_r describes the relative velocity of any point on the circle Q₁S₁ on the right boundary in Figure 6(b); when θ=π/2+θ_N+β, U_r describes the relative velocity of any point on the circle QS on the right boundary in Figure 6(b); and when θ=θ₂, U_r describes the relative velocity of any point on the circle Q₂S₂ on the left boundary in Figure 6(b). The variation of the axial-boundary conditions of the external return spherical hinge pair through the angle β₁, [β-b₁, β+b₂] not only considers the trajectory of all contact points on the width B of the external retainer plate in the Reynolds equation of the pair but also solves the boundary conditions for the existence of the external-swash-plate inclination β. By introducing the angle θ_N, the factor of the offset distance b_f is taken into account in the lubrication analysis of a spherical hinge pair.

Figure 6 Axial-boundary conditions of external return spherical hinge pair:

2.3 Reynolds equation

The following basic assumptions [28] are made: the volume and inertia forces of the lubricant are ignored, the grease-lubrication film does not slide on the solid interface, the thickness h of the grease-lubrication film is very small and the pressure change along the film thickness in the direction r of Figure 4 is ignored, and the lubricant is an isothermal fluid with constant viscosity along the direction of lubrication-film thickness. Let r be the distance from the center of the hinge to any point m in the contact-oil film between the hinge and retainer plate and ζ be the distance from the outer surface of the external retainer plate to the small fluid element m. Then, r=R+ζ, r≈R, /r=/ζ.

The force on an arbitrarily small fluid element m intercepted in the lubrication film is shown in Figure 4.

From the force balance of small fluid element:

                           (16)

where τ_θ and τ_φ are respectively the shear-stress components along the θ and φ directions in the spherical-coordinate system.

According to the properties of Newtonian fluid:

                           (17)

Substituting Eq. (17) into Eq. (16) and using the basic assumptions, we obtain:

                      (18)

Let us integrate Eq. (18) twice in the direction of ζ and use the velocity boundary condition of Eq. (11). The expressions for u_φ and u_θ can be written as follows:

          (19)

(20)

where C₂=

By combining Eqs. (17)-(20), the shear-stress component can be obtained as:

                        (21)

where

According to the basic assumptions, we know that η_θ=η_φ=η; thus, the equation of continuity is as follows:

                 (22)

Integrating Eqs. (19) and (20) on ζ and substituting them into Eq. (22), we obtain:

 (23)

As can be seen from the Eq. (5),.

Equation (23) is the Reynolds equation corresponding to the combined action of the offset distance and inclination of the external return spherical hinge pair. It considers the rotation of the external spherical hinge as well as the oblique reciprocating motion of the external retainer plate. As such, it is a Reynolds equation for complex motion.

2.4 Oil film thickness

During the operation of the external return mechanism, the combined action of the radial and axial loads and lubrication-film pressure causes the external retainer plate to generate eccentricity relative to the external spherical hinge. Radial eccentricity is generated under the action of the radial load and lubrication-film pressure and axial eccentricity is generated under the action of the axial load and lubrication-film pressure. The following film-thickness equation was presented in Ref. [29]:

          (24)

where c is the radius clearance of the external return spherical hinge pair, c=R_M-R,

2.5 Dimensional normalization

Dimensionless parameters:

                  (25)

From Eqs. (24) and (25), the dimensionless Reynolds equation for the external return mechanism can be written as:

               (26)

where

Substituting the dimensionless parameter of Eq. (25) into Eq. (20), the dimensionless expression of can be obtained:

 (27)

When this parameter is substituted into Eq. (21), the dimensionless expressions of and are:

(28)

 (29)

where =0, and are the components of fluid-friction-shear stress on the surface of the external retainer plate.

From Eqs. (24) and (25), the dimensionless form of the oil-film-thickness equation can be written as:

            (30)

2.6 Loading capacity of oil film

                    (31)

2.7 Friction, friction coefficient, and oil-film-friction power

The friction of an oil film is given by:

                   (32)

When ζ=0, is the fluid-friction-shear stress on the surface of the external retainer plate.

The friction coefficient of the oil film is:

                             (33)

And the dimensionless oil-film-friction power is given by:

                 (34)

where ΔV_i,j is the relative velocity of any point in the oil-film-lubrication area in Eq. (5); P_i,j is the dimensionless pressure at any point in the area of oil-film lubrication; and μ_i,j is the friction coefficient of the area of oil-film lubrication.

2.8 Dimensionless axial leakage flow

           (35)

3 Method of analysis

Our method of analysis included the following steps:

1) Enter the calculated parameters, such as the structural parameters of the external return mechanism, pump-shaft speed, external swash plate inclination, working pressure, and dynamic viscosity of the lubricating oil.

2) Solve the Reynolds equation by combining the finite-difference method and over-relaxation-iteration method. Divide angle θ (axial) into 72 nodes and angle φ (circular) into 360 nodes and calculate the oil-film pressure.

3) Apply Eqs. (5), (27), (28) and (29) to determine the relative velocity vector, component of axial velocity, and component of shear stress in the spherical coordinate system.

4) Apply Eqs. (31)-(35) to determine the loading capacity, friction, friction coefficient, friction power, and axial leakage flow, respectively.

4 Results and analysis

Taking the external return mechanism of a dual-drive axial piston pump/motor as our research subject, we analyzed the influence of the relative velocity of the external return spherical hinge pair on the pair’s lubrication characteristics under different offset distances and external swash plate inclinations. Table 1 shows the main parameters of the external-return mechanism of the dual-drive axial-piston pump/motor designed in this study. We selected the main parameters that affect the working conditions of the external-return mechanism as n=1500 r/min (low calibration speed), n=2500 r/min (medium speed), n=3500 r/min (high speed), b_f=0.00269 m (low calibration-offset distance), b_f=0.00300 m (medium offset distance), b_f=0.00319 m (high offset distance), external swash plate inclination β=10°(low calibration inclination), β=14°(medium inclination), and β=18 (high inclination). The lubrication characteristics of the external return mechanism were studied under different working conditions.

Table 1 Main parameters

We employed these parameters in our calculation. The friction and lubrication characteristics of the external return spherical hinge pair are discussed in terms of the maximum oil-film pressure, axial leakage flow, and oil-film-friction power.

Figure 7 presents the pressure distribution of the pair’s oil film at offset distances b_f of 0.00241, 0.00269, 0.00300, and 0.00319 m. Upon increasing the value of b_f, the pressure of the oil film decreases accordingly. However, the maximum value or range of oil-film pressures at different offset distances appear at about the 54th node of the axial angle. It can be seen from Section 3 that the axial angle is equally divided into 72 nodes. Therefore, as shown in Figure 6(b), the range of the maximum pressure of the oil film appears near the point Q.

4.1 Influence of offset distance and pump-shaft speed on lubrication characteristics of external return spherical hinge pair

Figures 8-10 show the variation in the lubrication-performance parameters of the external return spherical hinge pair in one working cycle (pump shaft rotates 360°) at offset distances b_f of 0.00269, 0.00300, and 0.00319 m when n=1500, 2500, and 3500 r/min, respectively. Tables 2-4 show the corresponding total friction power, total axial-leakage flow and the maximum value of the maximum oil-film pressure of external return spherical hinge pair in one working cycle. As shown in Figure 8, for a constant pump-shaft-rotation-angle position, the maximum oil-film pressure of the external return mechanism increases with decreasing offset distance under different pump-shaft speeds. At a low pump-shaft speed (1500 r/min), the maximum value of the maximum oil-film pressure for external return spherical hinge pair gradually transitions from a pump-shaft-rotation angle of 90° to 180° (bottom dead point) with decreasing offset distance. At the medium and high pump-shaft speeds (2500 and 3500 r/min, respectively) under varying offset distances, the maximum oil-film pressure has a double peak in one working cycle. The maximum value is located near the bottom dead point and the minimum value near a pump-shaft angle of 90°. The change in the offset distance has little effect on the variation laws for the maximum oil-film pressure of a friction pair.

Figure 7 Pressure distribution of oil film:

Figure 8 Maximum pressure of oil film under different offset distances, b_f:

Figure 9 Variation rate of pressure under different conditions

Figure 10 Axial-leakage flow under different offset distances:

Table 2 Maximum dimensionless value of oil-film pressure

Table 3 Total axial-leakage flow of external-return/spherical-hinge pair in one cycle

Table 4 Total friction power of external return spherical hinge pair in one working cycle

At low and medium offset distances (0.00269 and 0.00300 m, respectively), when the pump-shaft speed changes, the variation laws for the maximum oil-film pressure in the working cycle remain essentially the same. However, at a medium-offset distance (0.00300 m) with low (1500 r/min) or medium (2500 r/min) pump-shaft speeds, the variation laws for the maximum oil-film pressure exhibit a certain difference around a pump-shaft-rotation angle of 90°. At a high offset distance (0.00319 m), when the pump-shaft speed changes, the variation laws for the maximum oil-film pressure in the working cycle exhibit obvious differences at different times. As shown in Table 2 and Figure 9, the maximum value of the maximum oil-film pressure for external return mechanism increases with increasing pump-shaft speed in a working cycle. The higher the pump-shaft speed, the higher the maximum value of the maximum oil-film pressure. In contrast, the maximum value of the maximum oil-film pressure for external return mechanism increases with decreasing offset distance in a working cycle. The smaller the offset distance, the higher the growth rate of the maximum value of pressure.

As shown in Figure 10, at different pump-shaft speeds, the variation laws of the axial-leakage-flow rate are essentially the same under different offset distances. The maximum value of the axial-leakage-flow rate appears around a pump-shaft-rotation angle of 90°, because this position corresponds to the greatest relative velocity along the x-axis and the swing and impact of the external retainer plate are severe. The zero value of the axial-leakage flow appears at the top and bottom dead points of the external return mechanism. After the bottom dead point, the axial-leakage flow becomes negative, indicating that the direction of leakage turned in the opposite direction, achieving a certain sealing function. From Table 3 and Figure 11, we find that, at different pump-shaft speeds, the trends of increasing axial-leakage flow and total axial-leakage flow of the external-return mechanism with increasing offset distance are not obvious in a working cycle. However, at different offset distances, these trends become obvious within one working cycle as pump-shaft speed increases, but the rate of increase of the total axial leakage flow remains constant.

Figure 11 Variation rate of total axial leakage flow under different pump-shaft speeds

As shown in Figure 12, a change in the offset distance has little effect on the variation laws of the maximum friction power, and a double-peak phenomenon appears in one working cycle. Under different offset distances, the peak-value of the oil-film friction power appears at the same pump-shaft-rotation angle and the two peaks are close to each other. The maximum friction power of the oil film at pump-shaft-rotation angles of 0°(top dead point) and 180°(bottom dead point) is zero. This quantity is also zero around a pump-shaft-rotation angle of 72°. However, this slightly changes with variation in the offset distance. After the bottom dead point, the maximum friction power of the oil film increases more quickly. From Table 4 and Figure 12, we see that, at the same pump-shaft speed, total friction power and peak value of the maximum friction power in one working cycle of the external return mechanism decreases with increase of the offset distance. At the same offset distance, the change in the pump-shaft speed has little effect on the variation laws for the maximum oil-film friction power in one working cycle. The total friction power and peak value of the maximum friction power in one working cycle of the external return mechanism show an obvious increase with increasing pump-shaft speed. Figure 13 show that the increase rate of total frictional power increases with the increasing of pump-shaft speed, but it increases with the decreasing of offset distance.

Figure 12 Maximum friction power under different offset distances, b_f:

Figure 13 Variation rate of total friction power under different conditions

4.2 Influence of offset distance and external-swash-plate inclination on lubrication characteristics of the external return spherical hinge pair

Figures 14 and 15 respectively present the variation of lubrication-performance parameters of the external return spherical hinge pair during one working cycle under different external swash plate inclinations (β=10°, 14° and 18°) and different offset distances (0.00269, 0.00300 and 0.00319 m). Table 5 shows the total axial-leakage flow of the pair during one working cycle.

As shown in Figure 14, under different offset distances, the maximum value of the maximum oil-film pressure of the friction pair during one working cycle decreases with the increase of external-swash-plate inclination. For the external return mechanism at low offset distances (0.00269 m), the variation laws of maximum film pressure are consistent under all swash-plate inclinations. The maximum value of the maximum oil-film pressure appears near the bottom dead point, and the maximum oil-film pressure gradually increases near the pump-shaft-rotation angle of 90°. There are two peaks in one working cycle, and the maximum oil-film pressure between these peaks (22°-186°) decreases with increasing external-swash-plate inclination. For the external return mechanism at a medium offset distance (0.00300 m), the variation laws for the maximum film pressure are also consistent under different external swash plate inclinations. The maximum value of the maximum oil-film pressure appears near the bottom dead point. There are three peaks in one working cycle, and the maximum oil-film pressure between the two peaks (22°-186°) decreases with increasing external-swash-plate inclination. For the external return mechanism at medium and high inclinations, the change of the external-swash-plate inclination has little effect on the variation laws of maximum oil-film pressure over one working cycle at high offset distances. The maximum value of the maximum oil-film pressure appears near a pump-shaft-rotation angle of 90°, and there are two peaks in one working cycle. However, at low external-swash-plate inclination (10°), the maximum oil-film pressure only has a single peak in each working cycle. At a pump-shaft-rotation angle of 13°-176°, the maximum oil-film pressure decreases with the increase of the external-swash-plate inclination, and at 176°-212°, the maximum pressure increases with inclination. Under a constant external-swash-plate inclination, the maximum oil-film pressure of the external retainer plate-external spherical hinge friction pair varies greatly with the increase of the offset distances over one working cycle, and exist the variation rule of double peak, three peak and single peak.

Figure 14 The maximum oil-film pressure under different external swash plate inclinations:

Figure 15 Axial-leakage flow under different external swash plate inclinations, β:

Table 5 Total axial-leakage flow of external return spherical hinge pair during one working cycle

As shown in Figure 15, for the external return mechanism at different offset distances, the variation laws of axial-leakage flow are basically the same over one working cycle, regardless of the external-swash-plate inclination. The maximum value of axial-leakage flow generally appears near a pump-shaft-rotation angle of 90°, at which the swing and impact of the external retainer plate are severe. With increasing external-swash-plate inclination, the maximum value of the axial-leakage flow will increase accordingly. At the same time, the swing and impact of the external retainer plate will be intensified. However, zero values of axial-leakage flow always appear at the top and bottom dead points of the external return mechanism. Between these dead points, the axial-leakage flow increases along with the external-swash-plate inclination at the same pump-shaft-rotation angle. After the bottom dead point, the axial-leakage flow becomes negative, indicating that the leakage direction reverses. The absolute value of the axial-leakage-flow rate increases with increasing external-swash-plate inclination at the same pump-shaft-rotation angle, enhancing the sealing effect. However, as shown in Table 5 and Figure 16, the total axial-leakage flow of the external return mechanism increases along with the external-swash-plate inclination per working cycle under different offset distances. However, the increase of the external swash plate inclinations makes the growth rate of the total axial leakage flow decrease. The total axial-leakage flow of the external return mechanism increases slowly with the increase in the offset distance within one working cycle under various external swash plate inclinations.

Figure 16 Variation rate of total axial leakage under different conditions

5 Experiment and discussion

As can be seen from Figure 7, for external-swash-plate inclination β=14°, the pressure distribution of the oil film changes greatly with offset distance b_f. According to Eq. (24), when the offset distance is set to 0.00241, 0.00269, 0.00300, or 0.00319 m, the radial clearance is 0.00004, 0.00005, 0.00006, or 0.00007 m, respectively. At external-swash-plate inclination β=14°, offset distances of 0.00241, 0.00269, 0.00300 and 0.00319 m have the maximum oil-film pressures that are always around the 55th node in the axial direction. Thus, it can be seen that the axial angle is equally divided into 72 nodes. Therefore, according to Figure 6(b), the maximum-pressure range of the oil film appears near point Q, which is the intersection point between the O₁-y₁ axis at the center of the external retainer plate and the surface of that plate.

In the axial direction of the external-retainer plate, the range of the maximum oil-film pressure always appears near point Q, and the corresponding friction and wear at that point is more obvious [30]. To further verify the effect of friction and wear conditions on the external return mechanism, we test the external return mechanism using a self-developed testing device [27]. The test rig is shown in Figure 17. The test rig mainly includes the test axial piston pump, drive motor, and hydraulic pump station. The test pump was driven by the motor using a coupling. Working conditions and main parameters are shown in Table 1. The axial piston pump operates at the speed of 1500 r/min and working pressure of 31.5 MPa. The external retainer plate is made of 42CrMo alloy steel, while the external spherical hinge is made of steel 45 and plated with brass on its inner surface. The number of plungers in internal and external row is set at 10 each row. The external-swash-plate inclination is 14° and the offset distance of the external retainer plate is 0.00269 m. After testing, the experimental sample was taken out for measurement, as shown in Figure 18.

Figure 17 Axial piston pump test rig

In the friction and wear experiment, friction marks were found on the surface of the external retainer plate due to the external spherical hinge, as shown in Figure 18(b). The obvious friction marks appear near point Q, which is consistent with the simulation-analysis results.

According to the analysis of Figures 7, 8 and 14, the oil-film area has axial nodes 0-72 and circumferential nodes 0-212. Compared with the lubrication area of the joint bearing, the offset distance causes the oil-film-convergence wedge area of the external return spherical hinge pair to expand, forming a larger dynamic pressure effect interval. This is conducive to the fluid lubrication of the external return spherical hinge pair. At the same time, increasing the external-swash-plate inclination has a similar effect. Thus, unlike spherical plain bearings, the external return spherical hinge pair can improve its lubrication characteristics by adjusting the offset distance and the external-swash-plate inclination. It also has a larger combination form and optimization space in structural design than a joint bearing has.

Figure 18 Friction and wear experiment of external return spherical hinge pair:

According to the analysis of Figures 12(a)-(c), under different offset distances, pump-shaft speeds, and calibrated external swash plate inclinations, the maximum friction power of the oil film of the external return mechanism near the top and bottom dead points and pump-shaft rotation angles of 2° and 72° is 0. However, the power produced by the friction force is converted into the rotational and vibrational kinetic energy and heat of the external return spherical hinge pair, thereby indicating that little kinetic energy or heat is generated at these four corners. The maximum values of the oil-film friction power appear near pump-shaft-rotation angles of 42°, 144°, and 212°, where a lot of heat and kinetic energy are generated. However, the heat generated during high-speed operation of the external return spherical hinge pair will reduce the oil’s viscosity and the loading capacity of the spherical hinge pair, thereby further aggravating the friction and wear on the surface of the external retainer plate and the external spherical hinge. Therefore, we should pay special attention to the wear condition of these areas when using the external return spherical hinge pair to judge its working condition.

6 Conclusions

In this study, the Reynolds equation for the friction pair comprising the external retainer plate and external spherical hinge was deduced. We discussed the friction and lubrication characteristics of the external return mechanism under different working conditions (external-swash-plate inclination, offset distance, and pump-shaft speed) and obtained the following conclusions.

1) Under most working conditions of the external return mechanism, the maximum value of the maximum oil-film pressure appears near the bottom dead point of the external return spherical hinge pair. However, for the external return mechanism at high offset distance and low pump-shaft speed, the maximum value of the maximum oil-film pressure appears near a pump-shaft-rotation angle of 90° under different external swash plate inclinations. The maximum value of the maximum oil-film pressure for the external return mechanism increases with increasing the pump-shaft speed, increases with decreasing the offset distance in a working cycle, and decreases with increasing the external swash plate inclinations.

2) For the external return mechanism at low pump-shaft speed, when the external swash plate inclination is constant, the variation law of the maximum oil film pressure of the external return spherical hinge pair exhibits large differences with increasing offset distance in a working cycle. For the external return mechanism at medium and high pump-shaft speed, the curve of the maximum oil film pressure exhibits two peaks under different offset distances.

3) When the pump-shaft speed and external-swash-plate inclination are constant, the increasing trends of axial-leakage flow and total axial-leakage flow with increase of offset distance are not obvious. When the offset distance is constant, the total axial-leakage flow increases along with pump-shaft speed and external-swash-plate inclination. Between the top and bottom dead points, the axial-leakage flow increases along with the external-swash-plate inclination. After the bottom dead point, the sealing effect enhances with the increase of the external-swash-plate inclination.

4) Under different pump-shaft speeds, the variation laws of maximum friction power of the oil film remain basically the same for a given offset distance over one working cycle. Double peaks also occur over one working cycle. When the pump-shaft speed is constant, total friction power and peak value of the maximum friction power in one working cycle of the external return mechanism decreases with increase of the offset distance. When the offset distance is constant, total friction power and the peak value of the maximum friction power increase noticeably, along with the pump-shaft speed in one working cycle.

Nomenclature

β

External swash plate inclination (°)

β₁

The angle between O-x₁ and the O-x axis (°)

b_f

Offset distance (m)

R

Radius of external retainer plate (m)

R_M

Radius of external spherical hinge (m)

B

Width of external retainer plate (m)

B₁

The distance from the point Q to the right end face of the external retainer plate (m)

B₂

The distance from the point Q to the left end face of the external retainer plate (m)

ω_b

Angular velocity of the external spherical hinge (rad/s)

ω_r

Instantaneous angular velocity of the external return plate (rad/s)

ω

Angular velocity of the pump/motor shaft (rad/s)

m

Intercept an arbitrary small fluid element

m₁

A contact point between external retainer plate and external spherical hinge

u_r

The velocity components in the direction of r (m/s)

u_θ

The velocity components in the direction of θ (m/s)

u_φ

The velocity components in the direction of φ(m/s)

r

The distance from any point m in the contact oil film between the external retainer plate and external spherical hinge to the center of the external spherical hinge (m)

ζ

The distance from the outer surface of external retainer plate to the small fluid element m (m)

τ_θ

Fluid shear stress component in the θ direction (MPa)

τ_φ

Fluid shear stress component in the φ direction (MPa)

c

Radius clearance (m)

ε

Eccentricity

h

Film thickness (m)

η

Lubricating oil viscosity (Pa·s)

Loading capacity of oil film

Friction of oil film

μ

Friction coefficient of oil film

P^φ

Dimensionless oil film friction power

Dimensionless axial leakage flow

p₀

Environment pressure (MPa)

P

Dimensionless pressure of oil film

p

Pressure of oil film (MPa)

n

Pump-shaft speed (r/min)

ε_x

Axial eccentricity

ε_y

Radial eccentricity

beat

Relaxation factor

ρ

The constant fluid (oil) density (kg·m^-3)

Contributors

WANG Lei and DENG Hai-shun provided the concept and edited the draft of manuscript. WANG Lei and HU Cong conducted the literature review and wrote the first draft of the manuscript. WANG Lei, GUO Yong-chun and WANG Chuan-li edited the draft of manuscript.

Conflict of interest

WANG Lei, DENG Hai-shun, GUO Yong-chun, WANG Chuan-li, and HU Cong declare that they have no conflict of interest.

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(Edited by ZHENG Yu-tong)

中文导读

柱塞泵/马达外回程球铰副倾角偏距共同作用下的润滑特性

摘要：外回程机构是应用于轴向柱塞泵的机械结构，为研究其润滑特性，基于外斜盘倾角和偏距影响下的外回程球铰副相对运动速度矢量方程，推导出适用于外回程球铰副的Reynolds方程，对比分析了不同工况下外回程球铰副的摩擦润滑特性。结果表明：当外斜盘倾角一定时，外回程机构一个工作循环中最大油膜压力的最大值和总摩擦功率均随着转速的增加而增加、随着偏距的增加而减少；而偏距的变化对轴向泄露流量的影响甚微。当偏距一定时，一个工作循环中最大油膜压力的最大值随着外斜盘倾角的增加而减少，总轴向泄露流量随着偏距的增加而增加。由此可知，以上参数是影响外回程球铰副润滑特性的重要因素。因此在外回程机构设计时，需综合考虑不同参数耦合作用下复杂影响。

关键词：轴向柱塞泵或马达；外回程机构；外斜盘倾角；偏距；润滑特性

Foundation item: Project(GXXT-2019-048) supported by the University Synergy Innovation Program of Anhui Province, China; Project(51575002) supported by the National Natural Science Foundation of China; Project(gxbjZD11) supported by the Top-Notch Talent Program of University (Profession) in Anhui Province, China

Received date: 2020-07-07; Accepted date: 2021-02-04

Corresponding author: DENG Hai-shun, PhD, Professor; Tel: +86-554-6668934; E-mail: dhs1998@163.com; ORCID: https://orcid.org/ 0000-0001-5979-7243