Hydrostatic Bearing Characteristics Investigation of a Spherical Piston Pair with an Annular Orifice Damper in Spherical Pump

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Article Hydrostatic Bearing Characteristics Investigation of a Spherical Piston Pair with an Annular Oriﬁce Damper in Spherical Pump

Dong Guan 1,2,* , Zhuxin Zhou 1 and Chun Zhang 1

1 College of Mechanical Engineering, Yangzhou University, Yangzhou 225127, China; [email protected] (Z.Z.); [email protected] (C.Z.) 2 Changchai Company Limited, Changzhou 213002, China * Correspondence: [email protected]; Tel.: +86-176-2586-3985

Abstract: The spherical pump is a totally new hydraulic concept, with spherical piston and hydrostatic bearing, in order to eliminate the direct contact between the piston and cylinder cover. In this paper, the governing Reynolds equation under spherical coordinates has been solved and the hydrostatic bearing characteristics are systematically investigated. The operating sensitivities of the proposed spherical hydrostatic bearing, with respect to the piston radius, film beginning angle, film ending angle, film thickness, and temperature, are studied. The load carrying capacity, pressure drop coefficient, stiffness variation of the lubricating films, leakage properties, and leakage flow rates are comprehensively discussed. The related findings provide a fundamental basis for designing the high-efficient spherical pump under multiple operating conditions. Besides, these related results and mechanisms can also be utilized to design and improve other kinds of annular orifice damper spherical hydraulic bearing systems. Keywords: spherical pump; hydrostatic bearing; annular orifice damper; bearing properties;

Citation: Guan, D.; Zhou, Z.; leakage analysis Zhang, C. Hydrostatic Bearing Characteristics Investigation of a Spherical Piston Pair with an Annular Oriﬁce Damper in Spherical Pump. 1. Introduction Coatings 2021, 11, 1007. https:// The hydrostatic bearing consists of a ﬂuid layer between two surfaces, whose purpose doi.org/10.3390/coatings11081007 is to avoid direct surface-to-surface contact [1–6]. Generally, hydrostatic bearings are utilized in heavy equipment, to carry large loads between two surfaces that are moving Academic Editor: Giuseppe Carbone relative to each other at a low speed [4]. The spherical pump is a totally new hydraulic concept and consists of a pump with a Received: 15 July 2021 spherical piston. As demonstrated in the previous work [7–14], the rotation speed of the Accepted: 20 August 2021 Published: 23 August 2021 spherical piston is only half the rotational speed of its shaft. It is, therefore, characterized by its relatively low rotation speed. Additionally, the spherical piston has a large load

Publisher’s Note: MDPI stays neutral carrying capacity, because of the pump’s spatial structure and special working mechanism. with regard to jurisdictional claims in Hence, it is proper to utilize a hydrostatic bearing to support the spherical piston and to published maps and institutional affil- mitigate direct contact between the spherical piston and cylinder block. iations. Extensive studies have been conducted to investigate hydrostatic bearing performance [4,5,15]. The operating sensitivity of hydrostatic bearings, with respect to pressure- induced deformation, were studied by Manring [4]. The latter have described the pressure distributions, flow rates, and load carrying capacities of the bearings, by using lubrication equations for low Reynolds’ number flow. Expressions were developed to consider defor- Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. mations of both concave and convex shapes, and the impact of both shapes were compared. This article is an open access article However, the bearing they studied is a flat thrust bearing, rather than a spherical bearing. distributed under the terms and In another paper, Nie [5] has proposed a hydrostatic slipper bearing, with an annular orifice conditions of the Creative Commons damper in a water hydraulic axial piston motor, and the reaction forces, including friction Attribution (CC BY) license (https:// forces, centrifugal forces, and the piston dynamics, are considered. They established a creativecommons.org/licenses/by/ characteristic equation for hydrostatic slipper bearings and the influences of different struc- 4.0/). tural parameters, such as clearance, supporting length, and damping length, are included

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in their studies. However, they did not consider the leakage and flow rate properties of the hydrostatic bearing. Pang et al. [15] studied the hydrostatic lubrication of the slipper, in a high-pressure plunger pump, by considering the pressure-viscosity effects of the lubricating surface. These authors also investigated the dynamic stiffness properties of the oil film. However, they did not cover the effect of temperature on hydrostatic bearing performance. In addition, the slipper is also supported by a flat hydrostatic bearing. Yamaguchi [16–21] conducted abundant research on hydrostatic bearings. They established a mixed lubrication model [16,17,19], which consists of the Greenwood–Williamson model [22] for non-lubricated rough surfaces in contact, as well as the Patir and Cheng mean flow model [23,24]. The effects of surface roughness, eccentric, loading pressure and rotation speed on the bearing’s friction, as well as flow rate and power losses, were studied theoretically in [16,17]. A test rig was established to investigate the mixed lubrication characteristics of hydrostatic bearings [19]. The experimental results show good agreement with the proposed theoretical model. The properties of disk-type hydrostatic thrust bearings supporting concentric loads, simulating the major bearing/seal parts of water hydraulic motors and pumps, are studied in [20]. Both theoretical and experimental approaches are utilized, to investigate the relationships between film thickness, flow rate leakage, working pressure, and load carrying capacity. Later, a theoretical model was used to investigate bearings that are loaded eccentrically. The power loss properties, stiffness, and load carrying capacities of the disk-type hydrostatic thrust bearings are analyzed theoretically in [21]. Another related research was conducted by Hooke et al. [25–28]. However, all their studies cover disk-type hydrostatic thrust bearings, rather than spherical bearings. Static and dynamic analyses on spherical bearings were investigated by Goeka and Booker, using finite element methods [29]. Xu and Jiang [30,31] have proposed a self- compensation hydrostatic spherical hinge, which can provide a large load carrying capacity. The effects of centripetal force and surface roughness on the bearing performance of spherical fitted bearings were studied by Yacout [32]. However, all these publications are generally focused on traditional hydrostatic bearings. It should be noted that the structure and working mechanism are different for the hydrostatic bearings that are utilized in spherical pumps. In this study, the operating sensitivity of the proposed spherical hydrostatic bearing, with respect to the piston radius R, film beginning angle θ1, film ending angle θ2, film thickness h, and temperature t, are also studied. By using the continuity equation of incompressible, lubricating fluid, related expressions are derived for describing the load carrying capacity, pressure drop coefficient, stiffness variation of the lubricating film, leakage properties, the effect of temperature on leakage, and the flow rate of the leakage.

2. Hydrostatic Bearing Model for Spherical Cylinder–Piston Pair 2.1. Theoretical Analysis As shown in Figure1, any point can be expressed as M(x, y, z) in Cartesian coordinates as follows:  =  xM r sin θcosϕ yM = r sin θ sin ϕ (1)  zM = r cos θ CoatingsCoatings2021 2021, ,11 11,, 1007 x FOR PEER REVIEW 33 ofof 1819

FigureFigure 1.1. CoordinateCoordinate systems.systems. and also in the spherical coordinates (r, θ, ϕ). The latter are related to Cartesian coordinates and also in the spherical coordinates (r, θ, φ). The latter are related to Cartesian coordi- (x, y, z) by the following: nates (x, y, z) by the following: p  r = x2 + y2 + z2   θ = arccos √ z x2+y2+z2 (2)  222 rxyz=++y  ϕ = arctan x  z where r represents the distance (radius)θ = arccos from a point M to the origin O, θ indicates the(2) polar angle from the positive z-axis, with 0 ≤ θ ≤xπ222,++ andyzϕ is the azimuthal angle in the x–y x ≤ ϕ ≤ π plane from the -axis, with 0  2 . y Therefore, we can express theϕ Navier-Stokes= arctan equations for the incompressible lubricating ﬂuid with uniform viscosity in a sphericalx coordinates system, (r, θ, ϕ), as follows:  where r represents the distanceu2+u2 (radius) from a point M to the origin O, θ indicates  Dur − θ ϕ = − ∂p + + the polar angle from ρ theDt positiver z-axis, with∂r 0f r≤ θ ≤ π, and φ is the azimuthal angle in the  x–y plane from the x-axis, with 0 ≤ φ ≤ 2π.  h ∂u 2u cot θ ∂uϕ i  ∇2u − 2ur − 2 θ − θ − 2 ( ) Therefore, we µcan expressr r 2the Navier-Stor2 ∂θ kesr2 equationsr2 ∂ϕ for the 3aincompressible lubri-  sin θ cating fluid with uniform viscosity in a spherical coordinates system, (r, θ, φ), as follows:  u2 cot θ  Duθ uθ ur ϕ 1 ∂p  ρ Dt + r − r = − r + fθ+ 22+ ∂θ (3)  Duuuθϕ ∂ p ρ r −=−++   h f u ∂uϕ i  ∇2u + 2 ∂urr − θ − 2 cot θ ( ) Dt rµ θ ∂ rr2 ∂θ 2 2 r2 θ ∂ϕ 3b   r sin θ sin  h i   Duϕ uϕur uθ uϕ cot θ ∂1 ∂p 22u ρ ∂uu+ 2cot+ θ = 2− uϕ + fϕ+ μ ∇−2 r −Dt θθ −r r −r sin θ ∂ϕ − ur 22 2 2 (3 1)  rr h ∂∂θθϕ r∂u ruϕsin ∂u i  µ ∇2u + 2 r − + 2 cot θ θ (3c)  ϕ r2 sin θ ∂ϕ r2 sin2 θ r2 sin θ ∂ϕ 2 θ  Duθθ u u uϕ cot 1 ∂p whereρ ρ and+−p representr the density =−++ and pressuref of the lubricating ﬂuid, u , u and u  ∂θ θ r θ ϕ are the velocitiesDt r in the three r coordinate r directions, and µ indicates the dynamic viscosity.  (3) The body force components are denoted by∂fr, fθ and fϕ, and the Laplacian operator is  22cot∂u uθ θ uϕ as follows:μ ∇+2u r − −(3 − 2)  θ 2222 2 rr∂∂2θθθϕ1 ∂sin2 ∂ r sin1 ∂ ∂ 1 ∂  ∇ = (r ) + (sinθ ) + (4) r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂2 ϕ  Du u u u u cotθ ∂ ρ ϕϕ++r θϕ =−1 p + + and the operator D/Dt is denoted by the following: fϕ  Dt r rrsinθϕ∂  D ∂ ∂ uθ ∂ uϕ ∂ ∂u = uϕ+ ur + θ ∂+u (5) μ ∇+2 22cotrDt −∂t +∂r r ∂θ θr sin θ ∂ϕ − uϕ 2222 (3 3)  rrrsinθϕ∂∂ sin θ sin θϕ 2.2. Modeling of the Piston–Cylinder Hydrostatic Bearing

Inwhere this section,ρ and p therepresent model the of the density piston–cylinder and pressure bearing of the pair lubricating is established fluid, basedur, uθ and on theuφ are theory the mentionedvelocities in in the Section three 2.1 coordinate. directions, and μ indicates the dynamic viscosity.Figure The 2body depicts force the components piston and are cylinder, denoted with by the fr, f dashedθ and fφ, line and representing the Laplacian the operator inner surfaceis as follows: of the cylinder body. The lubricating system consists of high-pressure lubricating ﬂuid, Pr, being injected into the piston’s top gap, at which time the bearing ﬁlm

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∂∂ ∂ ∂ ∂2 2211 1 ∇=()r + (sin)θ + (4) 22∂∂θθ ∂ ∂ θ 222 θ ∂ϕ rr r rsin r sin and the operator D/Dt is denoted by the following:

D ∂∂u ∂uϕ ∂ =+u +θ + ∂∂r ∂θθ ∂ϕ (5) Dt t r r r sin

2.2. Modeling of the Piston–Cylinder Hydrostatic Bearing In this section, the model of the piston–cylinder bearing pair is established based on the theory mentioned in Section 2.1. Coatings 2021, 11, 1007 Figure 2 depicts the piston and cylinder, with the dashed line representing the inner4 of 18 surface of the cylinder body. The lubricating system consists of high-pressure lubricating fluid, Pr, being injected into the piston’s top gap, at which time the bearing film is established on the piston’s surface and cylinder body, which is an annulus structure. This an- nulusis established structure on is thealso piston’s defined surface as the revolute and cylinder joint, body, which which is the isjoint an annulusof the piston structure. and cylinderThis annulus body. structure Therefore, is alsoin spherical deﬁned coordinates, as the revolute (r, joint,θ, φ), whichthe continuity is the joint equation of the pistonof the lubricatingand cylinder fluid body is as (Video follows S1). [33]: Therefore, in spherical coordinates, (r, θ, ϕ), the continuity equation of the lubricating ﬂuid is as follows [33]: ∂ 11∂∂ 1uϕ 2 ++=()θ 1 ∂ ()rur 1 ∂ uθ sin1 ∂uϕ 0 (6) rr2 ∂∂r2u + rsinθθ(u sin θ) + r sin θ ∂= ϕ0 (6) r2 ∂r r r sin θ ∂θ θ r sin θ ∂ϕ

FigureFigure 2. 2. TheThe configuration configuration of of spherical spherical bear bearinging between between piston and cylinder. As the fluid flows uniformly from the piston pin, it can be concluded that the velocity As the fluid flows uniformly from the piston pin, it can be concluded that the velocity uϕ is 0, which is perpendicular to the inlet direction. Furthermore, because the lubricating ufilmφ is thickness0, which is is perpendicular at the micrometer to the scale, inlet we direction. can assume Furthermore, that the flow because in the ther direction lubricating also film thickness is at the micrometer scale, we can assume that the flow in the r direction vanishes and ur = 0. Therefore, Equation (7) can be transformed to the following: also vanishes and ur = 0. Therefore, Equation (7) can be transformed to the following: ∂u ∂uθ = − cot θu (7) ∂θ θ =−cotθθu ∂θ θ (7) Supposing the lubricating fluid is steadily flowing, the acceleration of the three differ- ∂u ent directionsSupposing are the∂ ulubricatingr = 0, ∂uθ = fluid0 and is steadilyϕ = 0, flowing, respectively. the acceleration The eliminating of the lubricating three dif- ∂t ∂θ ∂ϕ ∂ u2∂+u2 ∂u uϕ fluid inertia yields θ ϕr ==00. Additionally,θ = 0 ∂ ==00and u = 0, since the piston is r∂ ∂θ ∂ϕ∂ϕ ϕ ferentsymmetrical directions along are the ϕtdirection., Therefore, and Equation (3), simplifiesrespectively. tothe The following: eliminating

 p ∂ + µ 2 ∂uθ + 2uθ cot θ = 0 (8a)  ∂r r2 ∂θ r2   h 2 2 i ρuθ ∂uθ 1 ∂p 2 ∂uθ ∂ uθ cot θ ∂uθ 1 ∂ uθ uθ (8) r = − r + µ r + 2 + 2 + 2 2 − 2 2 (8b)  ∂θ ∂θ ∂r ∂r r ∂θ r ∂θ r sin θ   ∂p ∂ϕ = 0 (8c)

Substituting Equation (7) into Equation (8a), we arrive at the following:

∂p = 0 (9) ∂r indicating that the lubricating oil pressure, p, only varies with θ. Therefore, the pressure distribution along the θ direction can be obtained based on the ﬂow theory between the parallel surfaces and Equation (8b).

dp 2 ∂u ∂2u = µr θ + θ (10) dθ r ∂r ∂r2 Coatings 2021, 11, 1007 5 of 18

The velocity uθ can be obtained by integrating r into the following:

r dp a u = − + b (11) θ 2µ dθ r

where a and b are two different integral constants. Substituting the boundary conditions as follows: uθ|r=R = 0 (12) and the following: uθ|r=R+h = 0 (13) into Equation (11), the velocity distribution at different θ can be expressed as follows: r2 − r(2R + h) + R(R + h) dp u = (14) θ 2µr dθ

where h is the clearance between the piston surface and the internal surface of the cylinder body. Consequently, the lubricating ﬂuid ﬂow can be obtained from the following:

R R+h Q = R 2πr sin θuθdr (15) πh3 sin θ dp = − 6µ dθ

Substituting h = ecosθ into Equation (15), and evaluating the integral by using the separation of the variables approach, yields the following:

Z p Z θ 6µQ = dp 3 3 dθ (16) ps θ1 πe cos θ sin θ

where the eccentricity e indicates the distance between the piston center and the center of the cylinder body. The symbol ps denotes the inlet lubricating fluid pressure of the spherical piston, and θ1 represents the initial angle of the lubricating film. Therefore, the pressure distribution of the lubricating fluid can be expressed as follows:

3µQ tan θ = − + 2 − 2 p ps 3 2 ln tan θ tan θ1 (17) πe tan θ1

As shown in Figure2, the fluid pressure p = 0 when θ = θ2, where θ2 is the lubricating film ending angle. By substituting this boundary condition into Equation (17), the leaking lubricating fluid at the piston surface can be obtained as follows:

3 psπe Q = h i (18) 3µ 2 ln tan θ2 + tan2 θ − tan2 θ tan θ1 2 1

The substitution of Equation (18) into Equation (17) yields a pressure distribution as follows: " 2 ln tan θ + tan2 θ − tan2 θ # tan θ1 1 p = ps 1 − (19) 2 ln tan θ2 + tan2 θ − tan2 θ tan θ1 2 1

When eliminating the bearing force between the clearance of d2 – d1, the bearing force of the spherical piston can be obtained as follows:

F = R θ2 (2πR sin θ) × (p cos θ) × Rdθ θ1 h i 2 2 2 2 2 tan θ2 (20) πR ps cos θ1(tan θ2−tan θ1)−2 sin θ1 ln = tan θ1 tan θ2 2 2 2 ln +tan θ2−tan θ1 tan θ1 Coatings 2021, 11, 1007 6 of 18

and Equation (20) can be simpliﬁed as follows:

F = psSe (21)

h i 2 2 2 2 2 tan θ2 πR cos θ1(tan θ2−tan θ1)−2 sin θ1 ln where S = tan θ1 represents the spherical piston effective e tan θ2 2 2 2 ln +tan θ2−tan θ1 tan θ1 bearing area. We can observe that it only varies with the lubricating angles θ1 and θ2. Generally, the initial angle of the lubricating ﬁlm, θ1, is related to the piston pin diameter, d1. The bigger the piston pin, the larger the beginning angle, which can be expressed as follows: d θ = arcsin 1 (22) 1 2R

The final angle of the lubricating film, θ2, is predominantly related to the pump working capability, i.e., the pump displacement. 3. Bearing Properties Analysis In this section, the properties of the hydrostatic pair, under different structural parameters, will be analyzed, based on the above established theory. Firstly, the effective bearing area, Se is affected by both the angles θ1 and θ2, and the piston bearing capacity is subsequently dependent on Se. The initial angle θ1 is specified by the piston pin diameter d1. Therefore, both d1 and θ2 may impact the hydrostatic bearing ◦ working capacity directly. Generally, θ2 is less than 90 , due to the fact that the lubricating film thickness, h, is equal to ecosθ and h > 0. 3.1. Variation in Hydrostatic Pressure The effects of the piston radius and the piston pin diameter on the bearing pressure are studied in this section. The variation tendency of the hydrostatic pressure with the piston radius are sketched ◦ in Figure3. The bearing force is 5000 N and the lubricating film final angle θ2 is set to 60 , as this is the most utilized angle for spherical pistons. From Figure3, we can observe that when the angle θ2 and the bearing force are constants, the hydrostatic pressure of the bearing pair, ps, decreases with the piston radius. The larger the piston radius, the smaller the needed hydrostatic pressure is. This is due to the fact that the bearing area increases with the piston radius. However, the hydrostatic pressure changes significantly differently, with d1/R reaching a minimum value when d1/R is equal to 0.5. To study this phenomenon further, Figure4 is presented. The plot horizontal axis is the ratio between the piston pin diameter and the piston radius. The final angle ranges from ◦ ◦ ◦ 60 to 80 , with an interval of 5 . Generally, a larger angle θ2 requires a lower hydrostatic pressure when the bearing force is fixed. The lowest hydrostatic pressure occurs when θ2 ◦ ◦ is 80 , while the highest hydrostatic pressure occurs when θ2 is 60 , as demonstrated in Figure4. However, for a specific angle θ2, multiple variation tendencies appeared. When ◦ the angle is 80 , the hydrostatic pressure increases almost linearly with d1/R. In other words, if we do not consider the effect of structural strength, a smaller d1/R is desired for the best piston lubricating conditions. As the angle θ2 gradually decreases, this circumstance ◦ ◦ ◦ ◦ changes. When θ2 is 75 , 70 , 65 , and 60 , the lowest hydrostatic pressure occurs when d1/R is 0.38, 0.44, 0.48, and 0.5, respectively. A minimum hydrostatic pressure occurs when d1/R is between 0.2 and 1. Furthermore, the smallest hydrostatic pressure occurs at a ◦ ◦ higher d1/R ratio, when the angle θ2 decreases from 80 to 60 . Coatings 2021, 11, x FOR PEER REVIEW 7 of 19

The final angle of the lubricating film, θ2, is predominantly related to the pump working capability, i.e., the pump displacement.

3. Bearing Properties Analysis In this section, the properties of the hydrostatic pair, under different structural parameters, will be analyzed, based on the above established theory. Firstly, the effective bearing area, Se is affected by both the angles θ1 and θ2, and the piston bearing capacity is subsequently dependent on Se. The initial angle θ1 is specified by the piston pin diameter d1. Therefore, both d1 and θ2 may impact the hydrostatic bearing working capacity directly. Generally, θ2 is less than 90°, due to the fact that the lubricating film thickness, h, is equal to ecosθ and h > 0.

3.1. Variation in Hydrostatic Pressure The effects of the piston radius and the piston pin diameter on the bearing pressure are studied in this section. The variation tendency of the hydrostatic pressure with the piston radius are sketched in Figure 3. The bearing force is 5000 N and the lubricating film final angle θ2 is set to 60°, as this is the most utilized angle for spherical pistons. From Figure 3, we can observe that when the angle θ2 and the bearing force are constants, the hydrostatic pressure of the bearing pair, ps, decreases with the piston radius. The larger the piston radius, the smaller the needed hydrostatic pressure is. This is due to

Coatings 2021, 11, 1007 the fact that the bearing area increases with the piston radius. However, the hydrostatic7 of 18 pressure changes significantly differently, with d1/R reaching a minimum value when d1/R is equal to 0.5.

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In other words, if we do not consider the effect of structural strength, a smaller d1/R is desired for the best piston lubricating conditions. As the angle θ2 gradually decreases, this circumstance changes. When θ2 is 75°, 70°, 65°, and 60°, the lowest hydrostatic pressure occurs when d1/R is 0.38, 0.44, 0.48, and 0.5, respectively. A minimum hydrostatic pressure occurs when d1/R is between 0.2 and 1. Furthermore, the smallest hydrostatic pressure FigureFigure 3. 3. EffectsEffects of of piston piston radius radius on on hydrostatic hydrostatic pressure. occurs at a higher d1/R ratio, when the angle θ2 decreases from 80° to 60°. To study this phenomenon further, Figure 4 is presented. The plot horizontal axis is the ratio between the piston pin diameter and the piston radius. The final angle ranges from 60° to 80°, with an interval of 5°. Generally, a larger angle θ2 requires a lower hydrostatic pressure when the bearing force is fixed. The lowest hydrostatic pressure occurs when θ2 is 80°, while the highest hydrostatic pressure occurs when θ2 is 60°, as demonstrated in Figure 4. However, for a specific angle θ2, multiple variation tendencies appeared. When the angle is 80°, the hydrostatic pressure increases almost linearly with d1/R.

Figure 4. Effects of d1/R on hydrostatic pressure (θ2 = 60°,◦ F = 5000 N). Figure 4. Effects of d1/R on hydrostatic pressure (θ2 = 60 , F = 5000 N).

Therefore, thethe optimumoptimumd d11/RR ratioratio that that corresponds corresponds to to the the lowest lowest hydrostatic hydrostatic pressure can be found in FigureFigure2 2,, for for each each speciﬁc specific angle angle θθ22.. ThisThis isis particularlyparticularly helpfulhelpful toto knowknow when designingdesigning newnew sphericalspherical pumps.pumps. 3.2. Variation in Bearing Force 3.2. Variation in Bearing Force The bearing force change with the ending angles is illustrated in Figure5. The piston The bearing force change with the ending angles is illustrated in Figure 5. The piston radius is 20 mm, with a hydrostatic pressure of the lubricating ﬁlm of 15 MPa. radius is 20 mm, with a hydrostatic pressure of the lubricating film of 15 MPa.

Figure 5. Effects of final angle θ2 on bearing force (R = 20 mm, Ps = 15 MPa).

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Figure 4. Effects of d1/R on hydrostatic pressure (θ2 = 60°, F = 5000 N).

Therefore, the optimum d1/R ratio that corresponds to the lowest hydrostatic pressure can be found in Figure 2, for each specific angle θ2. This is particularly helpful to know when designing new spherical pumps.

3.2. Variation in Bearing Force Coatings 2021, 11, 1007 8 of 18 The bearing force change with the ending angles is illustrated in Figure 5. The piston radius is 20 mm, with a hydrostatic pressure of the lubricating film of 15 MPa.

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2 FigureFigure 5.5. EffectsEffects ofof finalfinal angleangle θθ2 onon bearingbearing forceforce ((RR == 2020 mm,mm, PsPs == 15 MPa). From Figure 5, we can observe that the bearing force increases at different growth From Figure5, we can observe that the bearing force increases at different growth rates rates with the piston final angle. Take d1/R = 0.6 as an example, its bearing force grows with the piston final angle. Take d /R = 0.6 as an example, its bearing force grows almost almost linearly when the final angle1 is between 60° and 85°. However, the growth rate linearly when the final angle is between 60◦ and 85◦. However, the growth rate decreases decreases when θ2 is higher◦ than 85°. Furthermore, the d1/R ratio will also affect the piston when θ2 is higher than 85 . Furthermore, the d1/R ratio will also affect the piston bearing bearing force. A smaller ratio, d1/R , leads to a higher increment of the bearing force, when force. A smaller ratio, d1/R, leads to a higher increment of the bearing force, when θ2 varies θ2 varies◦ from ◦60° to 90°. This leads to the conclusion that the smaller the d1/R is, the higher from 60 to 90 . This leads to the conclusion that the smaller the d1/R is, the higher the bearingthe bearing force force is, under is, under the samethe same conditions. conditions.

3.3. Variation in Pressure Drop CoefficientCoefficient An equivalentequivalent fluidfluid bridgebridge ofof the the hydrostatic hydrostatic bearing bearing system system is is illustrated illustrated in in Figure Figure6. Lubricating6. Lubricating oils oils are are injected injected from from the the clearance clearance between between the the piston piston pin pin andand thethe cylindercylinder cover. The fluidfluid resistanceresistance ofof thethe clearanceclearance isis denoteddenoted byby RRcc,, whichwhich isis aa constant,constant, andand RRgg represents thethe adjustableadjustable fluidfluid resistanceresistance ofof thethe hydrostatichydrostatic bearingbearing pair.pair.

Figure 6. Equivalent ﬂuidfluid bridge.

Based on the continuity theorem,theorem, wewe obtainobtain thethe following:following:

pspppc =s = c (23) Rg Rc (23) Rg Rc

where pc = pr − ps represents the pressure drops at the clearance of the piston pin and the where pc = pr − ps represents the pressure drops at the clearance of the piston pin and the cylinder cover. The pressure drop of the lubricating pair is indicated by ps. Therefore, the cylinder cover. The pressure drop of the lubricating pair is indicated by ps. Therefore, the pressure drop ratio of the piston and the whole system can be expressed as follows: pressure drop ratio of the piston and the whole system can be expressed as follows: pspRRs α α= ===ss (24) pr Rc ++Rs (24) pRRrcs

This is also called the pressure drop coefficient. Rc is the fluid resistance between the clearance of the piston pin and the cylinder cover. It is expressed as follows: 12μμll 96 R == c π dh3 π ()− 3 (25) 1 c dd12 d 1

where d1 and d2 are the diameters of the piston pin and its mating hole, respectively. The

axial length of the revolute joint is l, and the clearance between the piston pin and the matching hole of the cylinder cover is denoted by (d2 − d1)/2. The fluid resistance between the spherical surface of the piston and cylinder cover is denoted by Rs, which is expressed as follows: θ μθ22−+ θ tan 2 3tantan2ln21 tanθ (26) R = 1 s π 3 h Substituting Equations (25) and (26) into Equation (24), we obtain the following:

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This is also called the pressure drop coefﬁcient. Rc is the ﬂuid resistance between the clearance of the piston pin and the cylinder cover. It is expressed as follows:

12µl 96µl = = Rc 3 3 (25) πd1hc πd1(d2 − d1) Coatings 2021, 11, x FOR PEER REVIEW 10 of 19

where d1 and d2 are the diameters of the piston pin and its mating hole, respectively. The axial length of the revolute joint is l, and the clearance between the piston pin and the matching hole of the cylinder cover is denoted by (d2 − d1)/2. The ﬂuid resistance between the spherical surface ofR the piston and cylinder cover is denoted by Rs, which is expressed α ==s 1 as follows: h μπ3 i RR+ 12 2lh2 tan θ2 cs +×3µ tan θ2 − tan θ1 + 2 ln 1 3 tan θ1 (27) Rs = π dh tanθ (26) 1 c 3tantan2lnμθπh3 22−+ θ 2 21tanθ Substituting Equations (25) and (26) into Equation (24), we obtain the following:1

12Rl s 1 α == = = 1 (27) If we set KBC 3 and K BS 12µl πh3 , then Equa- Rc + Rs 1 + × h i dh1 c πd h3 2 2 tan θ2tanθ 1 c 3µ22tan θ2−tan θ1+2 ln 2 3tanθθ−+ tan 2lntan θ1 21 θ tan 1 12l 1 If we set K = and K = h i , then Equation (27) can be tion (27) can be BCexpressedd h3 as follows:BS 2 2 tan θ2 1 c 3 tan θ2−tan θ1+2 ln tan θ1 expressed as follows: 1 α = 1 (28) = + 3 α 1 KKhBC BS3 (28) 1 + KBCKBSh

The KKBCBC andand KKBSBS areare structural structural parameters parameters of of the the revolute revolute joint and bearing pair, andand both relate toto theirtheir ownown structurestructure andand geometrygeometry properties.properties. EquationEquation (28)(28) isis alsoalso calledcalled aa characteristic equation, which reﬂectsreflects thethe bearingbearing characteristicscharacteristics ofof thethe entireentire hydrostatichydrostatic bearing system. The effectseffects of ofK BCKBCand andK BSKBSon on the the bearing bearing characteristics characteristics areillustrated are illustrated in Figures in Figures7 and8 .7 Theand piston8. The pin,piston diameter pin, diameterd1, is 5 d mm1, is and5 mm its and length, its length,l, is 5 mm l, is as5 mm well. as well.

Figure 7. Effect of hcc on bearing characteristics.

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(a) Effect of θ1 (b) Effect of θ 2

Figure 8. Effect of KBS on bearing characteristics. Figure 8. Effect of KBS on bearing characteristics.

In Figure7, the clearance of the revolute joint, hc, varies from 0.02 mm to 0.1 mm, In Figure 7, the clearance of the revolute joint, hc, varies from◦ 0.02 mm◦ to 0.1 mm, with an interval of 0.02 mm. The structural angles θ1 and θ2 are 15 and 75 , respectively. Fromwith an Figure interval7, we of can 0.02 observe mm. The that structural the pressure angles drop θ1 and decreases θ2 are 15° with and various 75°, respectively. speeds, whenFrom theFigure lubricating 7, we can film observe thickness that increases. the pressure When drop the decreases lubricating with film various thickness speeds, is smallerwhen the than lubricating 0.02 mm, film or thicker thickness than increase 0.12 mm,s. When its variation the lubricating has little film impact thickness on the is smaller than 0.02 mm, or thicker than 0.12 mm, its variation has little impact on the pres- pressure drop coefficient. When hc is equal to 0.02 mm, the pressure drop coefficient is moresure sensitivedrop coefficient. to the lubricating When hc is film equal thickness, to 0.02 especiallymm, the pressure when the drop thickness coefficient varies isfrom more 0.02sensitive mmto to 0.08the lubricating mm. This meansfilm thickness, that the especially bearing film when stiffness the thickness is higher varies in this from range. 0.02 Whenmm toh c0.08increases mm. This from means 0.02 that to 0.1, the thebearing pressure film stiffness drop graph is higher becomes in this more range. and When more hc flat,increases and the from sensitivity 0.02 to 0.1, of the the pressure pressure drop drop coefficient graph becomes on the more lubricating and more film flat, thickness and the decreasessensitivity gradually. of the pressure This means drop that coefficient the lubricating on the film lubricating stiffness decreasesfilm thickness as hc increases.decreases Obviously,gradually. aThis higher means bearing that stiffnessthe lubricating can be film obtained stiffness when decreases using a smalleras hc increases. clearance Obvi-hc, butously, the lubricatinga higher bearing film thickness stiffness mustcan be locate obtained within when an appropriate using a smaller range. clearance hc, but the lubricatingThe effects offilmKBS thicknesson the bearing must locate characteristics within an areappropriate demonstrated range. in Figure8. The effects ofTheθ1 andeffectsθ2 areof K illustratedBS on the bearing in Figure characteristics8a,b, respectively. are Wedemonstrated can observe in that, Figure similarly 8. The toeffects Figure of7 ,θ when1 and θ the2 are lubricating illustrated film in Figure thickness 8a,b, is re smallerspectively. than We 0.02 can or thickerobserve than that, 0.12, simi- thelarly pressure to Figure drop 7, coefficientwhen the lubricating decrease very film slowly. thickness is smaller than 0.02 or thicker than ◦ ◦ ◦ ◦ ◦ 0.12,In the Figure pressure8a, θ 2dropis 60 coefficientand θ1 has decrease 5 , 15 andvery 25 slowly.values. When θ1 increases from 5 to ◦ 25 , theIn pressureFigure 8a, drop θ2 is coefficient 60° and θ cure1 has moves 5°, 15° left, and meaning 25° values. that Whenα becomes θ1 increases more sensitive from 5° to to ◦ h25°,and the higher pressure lubricating drop coefficient film stiffness cure values moves are left, realizable. meaning In that Figure α becomes8b, θ1 is more equal sensitive to 10 , ◦ ◦ whileto h andθ2 increases higher lubricating from 60 to film 80 stiffness. The pressure values drop are realizable. coefficient In curve Figure moves 8b, θ to1 is the equal right to when10°, whileθ2 increases, θ2 increases which from is totally 60° to different80°. The pressu comparedre drop to θ coefficient1. Obviously, curve the moves lubricating to the filmright stiffness when θ decreases2 increases, with whichθ2. Furthermore,is totally different when compared comparing to Figure θ1. Obviously,8a,b, wefind the lubricat- that θ2 hasing a film larger stiffness effect ondecreases the pressure with dropθ2. Furthermore, coefficient than whenθ1. comparing Figure 8a,b, we find that θ2 has a larger effect on the pressure drop coefficient than θ1. 3.4. Lubricating Film Stiffness Analysis 3.4. Lubricating Film film Stiffness stiffness Analysis is defined as follows:

Lubricating film stiffness is defined as follows:dF J = − (29) dhdF J =− (29) where F denotes the bearing force, which is defineddh by Equation (21), and h indicates the lubricating film thickness. Therefore, the lubricating film stiffness represents the bearing where F denotes the bearing force, which is defined by Equation (21), and h indicates the force change rate with different film thicknesses. Substituting Equations (21), (24) and (28) lubricating film thickness. Therefore, the lubricating film stiffness represents the bearing into Equation (29), leads to the following: force change rate with different film thicknesses. Substituting Equations (21), (24) and (28) into Equation (29), leads to the following:3K K p S h2 J = BC BS r e (30) 3 2 (1 + KBCKBSh )

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3KKpSh2 = BC BS r e J 2 (30) + 3 ()1 KKhBC BS Coatings 2021, 11, 1007 11 of 18 From Equation (30), we can observe that the stiffness of the lubricating film is not only affected by the structural parameters KBC and KBS, but it is also affected by the hydrostatic pressure pr, effective bearing area Se and thickness of lubricating film h. The extremes From Equation (30), we can observe that the stiffness of the lubricating film is not only of the film thickness, h, can be obtained when ∂∂=Jh/0 and the maximum lubricating affected by the structural parameters KBC and KBS, but it is also affected by the hydrostatic film thickness is as follows: pressure pr, effective bearing area Se and thickness of lubricating film h. The extremes of the film thickness, h, can be obtained when ∂J/∂h = 0 −and1 the maximum lubricating film = ()3 (31) thickness is as follows: hkkmax 2 BC BS − 1 hmax = (2kBCkBS) 3 (31) Substituting Equation (31) into Equations (28) and (30), we can obtain the maximum pressureSubstituting drop coefficient Equation α (31)= 2/3 into and Equations the maximum (28) and bearing (30), stiffness we can obtain as follows: the maximum pressure drop coefficient α = 2/3 and the maximum bearing stiffness as follows: 42/3 2 JpSKK= 2/3 ()3 (32) max 4 r e BC BS2 J = 3 p S (K K ) 3 (32) max 3 r e BC BS Then, a dimensionless stiffness coefficient, δ, can be determined. The ratio between δ EquationsThen, (30) a dimensionless and (32) is as stiffness follows: coefficient, , can be determined. The ratio between Equations (30) and (32) is as follows: 2/3 J 9 ()KK h2 δ == BC BS2/3 2 J 9 (KBCKBS) h (33) δ = = 2/3 3 2 (33) Jmax 2/34 ()1+ KKh3 2 Jmax 4 (1 + KBCBCKBS BSh )

TheThe effectseffects ofof KKBCBC and KBS onon the the dimensionless dimensionless stiffness stiffness coefficients coefﬁcients of the lubricating ﬁlmfilm areare illustratedillustrated inin FiguresFigures9 9 and and 10 10.. Both Both the the diameter diameter and and the the length length of of the the piston piston pin pin areare 55 mm.mm.

c FigureFigure 9.9. Effect of hhc on dimensionless stiffness coefﬁcient.coefficient.

From Figures9 and 10, we can observe that the dimensionless stiffness coefficient increases with the lubricating film thickness, h, firstly, and arrives at a maximum value of one. Then, it decreases from this maximum value, with different variation tendencies. Figure9 demonstrates that a smaller piston pin clearance makes the maximum stiffness occur at a smaller film thickness. As demonstrated in Figure 10a, when the angle θ2 is a constant, a larger θ1 will make the stiffness curve move left. Conversely, when θ1 is a constant and θ2 is increased, the curves of the dimensionless stiffness coefficients will move to the right. This means that the maximum stiffness occurs when the lubricating film is thicker. Coatings 2021, 11, x FOR PEER REVIEW 13 of 19

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(a) Effect of θ (b) Effect of θ 1 2

FigureFigure 10. 10.Effect Effect of ofK KBSBS onon dimensionlessdimensionless stiffnessstiffnesscoefﬁcient. coefficient.

TheFrom structural Figures parameters9 and 10, weKBC canand observeKBS will that increase the dimensionless when increases stiffnessθ1, decreases coefficienthc andincreasesθ2. Therefore, with the lubricating it can be concluded film thickness, that whenh, firstly,KBC andand arrivesKBS increase, at a maximum the maximum value of valueone. Then, of these it decreases curves will from move this to maximum the left, which value, makes with thedifferent film thickness variation of tendencies. the maximum Fig- stiffnessure 9 demonstrates smaller, and, that finally, a smaller impacts piston the pin hydrostatic clearance bearing makes the performance. maximum stiffness occur at a smaller film thickness. As demonstrated in Figure 10a, when the angle θ2 is a constant, 3.5. Leakage Analysis a larger θ1 will make the stiffness curve move left. Conversely, when θ1 is a constant and θ2 is Aincreased, proper fit the clearance curves shouldof the dimensionless exist between sphericalstiffness coefficients pistons and will cylinder move bodies. to the right. Con- sequently,This means leakage that the will maximum inevitably stiffness exist through occurs when the clearance. the lubricating In this film section, is thicker. the leakage properties of lubricating surfaces will be investigated and discussed. The structural parameters KBC and KBS will increase when increases θ1, decreases hc The leakage characteristics of the lubricating surface are illustrated by Equation (19), and θ2. Therefore, it can be concluded that when KBC and KBS increase, the maximum value whichof these demonstrates curves will thatmove leakage to the is left, directly whic proportionalh makes the tofilm lubricating thickness pressure of the maximumps, eccen- 3 µ tricitystiffnesse , smaller, and inversely and, finally, proportional impacts to the the hydrostatic dynamic viscosity bearing coefficientperformance.. Furthermore, it is also related to the structural parameters θ1 and θ2. 3.5. LeakageThe results Analysis shown in Figure 11 are used to investigate the effect of eccentricity e on leakage. Here, the dynamic viscosity is selected as 0.01 Pa·s, and θ and θ are 15◦ and 75◦, A proper fit clearance should exist between spherical pistons1 and2 cylinder bodies. respectively. Three different lubricating pressures are selected at 5 MPa, 10 MPa, and 15 MPa.Consequently, From Figure leakage 11, we will can inevitably observe exist that through leakage increasesthe clearance. sharply In this with section, eccentricity the leak-e. Furthermore,age properties the of higher lubricating the lubricating surfaces will pressure, be investigated the larger and the leakage.discussed. When eccentricity is smallerThe leakage than 0.08 characteristics mm, the changes of the in leakagelubricating are slight.surface When are illustratede exceeds by 0.08, Equation the leakage (19), changewhich demonstrates rate grows rapidly. that leakage Therefore, is directly a proper proportional lubricating clearanceto lubricating is essential pressure at differentps, eccen- 3 workingtricity e , conditions. and inversely proportional to the dynamic viscosity coefficient μ. Furthermore, it is also related to the structural parameters θ1 and θ2. The results shown in Figure 11 are used to investigate the effect of eccentricity e on leakage. Here, the dynamic viscosity is selected as 0.01 Pa·s, and θ1 and θ2 are 15° and 75°, respectively. Three different lubricating pressures are selected at 5 MPa, 10 MPa, and 15 MPa. From Figure 11, we can observe that leakage increases sharply with eccentricity e. Furthermore, the higher the lubricating pressure, the larger the leakage. When eccentricity is smaller than 0.08 mm, the changes in leakage are slight. When e exceeds 0.08, the leakage change rate grows rapidly. Therefore, a proper lubricating clearance is essential at different working conditions.

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FigureFigure 11. 11. Effect Effect of of eccentricity eccentricity e e on on leakage. leakage.

TheThe effectseffects of ofof lubricating lubricatinglubricating pressure pressurepressure on onon leakage leakleakageage performance performanceperformance are areare illustrated illustratedillustrated in Figure inin FigureFigure 12. 12.The12. The dynamicThe dynamicdynamic viscosity viscosityviscosity coefﬁcient coefficientcoefficientµ and μ eccentricityμ andand eccentricityeccentricitye are 0.01 ee areare Pa · 0.01s0.01 and Pa·sPa·s 0.05 and mm,and 0.050.05 respectively. mm,mm, re-respectively.Fromspectively. Figure From From 12, we Figure Figure can observe 12, 12, we we thatcan can observe theobserve leakage th thatat increases the the leakage leakage linearly increases increases with lubricatinglinearly linearly with with pressure. lubri- lubri- catingHowever,cating pressure. pressure. the different However, However, structural the the di differentfferent parameters structural structuralθ1 and parameters parametersθ2 will have θ θ11 and and multiple θ θ22 will will effects have have multiple multiple on their effectsleakageeffects on on characteristics. their their leakage leakage characteristics. characteristics.

FigureFigure 12.12. Effect Effect ofof lubricatinglubricating pressurepressure on on leakage. leakage.

◦ ◦ ◦ TheThe effects effectseffects of of θ θ11 are areare demonstrated demonstrated demonstrated in in in Figure Figure Figure 12a, 12a, 12 a,where where where θ θ22 θis is2 60° is60° 60 and andand θ θ11 is isθ 15°, 5°,is 15°, 15°, 5 , and 15and, ◦ ◦ ◦ 25°.and25°. 25WhenWhen. When θθ11 increases,increases,θ1 increases, fromfrom from 5°5° to 5to to25°,25°, 25 thethe, the correcorre correspondingspondingsponding leakageleakage leakage increasesincreases increases gradually.gradually. Obviously,Obviously, aa smallersmaller θ θ111 is isis helpful helpfulhelpful to toto prevent preventprevent leakage leakageleakage of ofof the thethe lubricating lubricatinglubricating pair. pair.pair. The TheThe impacts impactsimpacts ◦ ◦ ◦ ◦ ofof θ θθ22 are are illustrated illustrated in in in Figure Figure Figure 12b, 12b, 12b, when when when θ θ11 θis is1 15°, is15°, 15 while while, while θ θ22 is isθ 260°, 60°,is 6070° 70° ,and and 70 80°, and80°, respectively. respectively. 80 , respec- Distincttively.Distinct Distinct fromfrom thethe from θθ11 dependence,thedependence,θ1 dependence, thethe increaseincrease the increase inin θθ22 willwill in θ 2 prohibitprohibitwill prohibit leakageleakage leakage effectively.effectively. effectively. Fur-Fur- thermore,Furthermore,thermore, the the theeffects effects effects of of θ θ of22 on onθ2 leakage onleakage leakage are are arestronger stronger stronger than than than θ θ11. . θ1. Therefore,Therefore, an anan effective effectiveeffective way wayway to toto prohibit prohibitprohibit leakage leakageleakage from fromfrom lubricating lubricatinglubricating pairs pairspairs is toisis todecreaseto decreasedecreaseθ1 θandθ11 and and increase increase increaseθ2 ,θ withθ22, ,with with a larger a a larger largerθ2 beingθ θ22 being being more more more helpful helpful helpful than than than a smaller a a smaller smallerθ1. θ θ11. . 3.6. Effect of Temperature on Leakage Properties 3.6.3.6. Effect Effect of of Temperature Temperature on on Leakage Leakage Properties Properties Lubricating medium temperatures will inevitably increase when the spherical pump LubricatingLubricating medium medium temperatures temperatures will will inevitably inevitably increase increase when when the the spherical spherical pump pump is running. On the one hand, the friction of the lubricating pair generates part of the heat isis running. running. On On the the one one hand, hand, the the friction friction of of the the lubricating lubricating pair pair generates generates part part of of the the heat heat

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flow. Leakage of the lubricating medium will also cause temperature increases, and a rise flow.in temperature Leakage of will the further lubricating decrease medium the lubricant will also causeviscosity. temperature increases, and a rise in temperatureThe viscosity will of further the lubricating decrease themedium lubricant is its viscosity. sensitivity to temperature. Both theo- reticalThe and viscosity experimental of the lubricatingstudies are conducted medium is to its investigate sensitivity their to temperature. relation [34,35]. Both In theo- this reticalstudy, andthe utilized experimental viscosity–temperature studies are conducted equation to investigate is expresse theird as relationfollows [[36]:34,35 ]. In this study, the utilized viscosity–temperature equationβ ()− is expressed as follows [36]: μμ= tt0 0e (34) β(t−t0) µ = µ0e (34) where μ0 represents the initial viscosity, β denotes the viscosity–temperature coefficient, and t0 is the initial temperature. where µ0 represents the initial viscosity, β denotes the viscosity–temperature coefficient, Substituting Equation (34) into Equation (18), we can obtain the relationship between and t0 is the initial temperature. temperatureSubstituting and Equationleakage, as (34) follows: into Equation (18), we can obtain the relationship between temperature and leakage, as follows: peπ 3 Q = s θ 3 β ()tt− tanpsπe 22 (35) Q = 32lntantanμ e 0 h 2 +−θθi (35) 021β(t−t ) tan θ2 2 2 3µ0e 0 2 ln tanθ+ tan θ2 − tan θ1 tan θ1 1 TheThe leakageleakage characteristicscharacteristics under different temperatures are illustrated in Figure 1313.. ◦ ◦ −−11 TheThe initialinitial temperaturetemperaturet 0t0is is20 20 °C;C;viscosity-temperature viscosity-temperature coefficient coefficientβ βis is 1/20 1/20 C°C ; the lu- bricatingbricating pressurepressurep pss isis 1515 MPa;MPa; andand thethe eccentricityeccentricity isis 0.050.05 mm.mm. TheThe structuralstructural parametersparameters ◦ ◦ θθ11 andand θθ22are are 1515° andand 6060°,, respectively.respectively. FromFrom FigureFigure 13 13,, we we can can observe observe that that the the leakage leakage curves curves of the of lubricatingthe lubricating pair increasepair in- nonlinearlycrease nonlinearly with temperature. with temperature. Leakage Leakage increases increases sharply sharply with with temperature, temperature, when when the temperaturesthe temperatures are below are below 80 ◦C. 80 At °C. temperatures At temperat higherures higher than 80 than◦C, both80 °C, the both leakages the leakages curves arecurves at approximately are at approximately 0.12 L/min. 0.12 L/min. This is This because is because the viscosity the viscosity of the lubricationof the lubrication decreases de- tocreases almost to 0almost Pa·s when 0 Pa·s the when temperature the temperature is higher is than higher 80 ◦ C.than The 80 initial °C. The dynamic initial viscositydynamic alsoviscosity has analso impact has an on impact the leakage on the performance, leakage performance, namely, the namely, higher the the higher initial the dynamic initial viscosity,dynamic theviscosity, larger the larger amount the of amount leakage. of leakage.

FigureFigure 13.13. EffectEffect ofof temperaturetemperature tt onon leakage.leakage.

3.7.3.7. Flow Rate CharacteristicsCharacteristics ofof LeakageLeakage MediumMedium TransparentTransparent leakageleakage ofof thethe bearingbearing ﬁlmfilm cancan bebe expressedexpressed as as follows: follows: Q ===V S = V 2ππθR sin θh (36) QVSf rfrf r fr Vfr fr 2sin R h (36)

where Vfr is the flow rate of a specific surface, which is perpendicular to the piston axis, and the specific surface is constituted by the clearance between the piston and the cylinder cover. The area of the specific surface is denoted by Sfr, which is equal to 2πRsinθh.

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where Vfr is the flow rate of a specific surface, which is perpendicular to the piston axis, Coatings 2021, 11, 1007 and the specific surface is constituted by the clearance between the piston and the cylinder15 of 18 cover. The area of the specific surface is denoted by Sfr, which is equal to 2πRsinθh. Substituting Equation (18) and h = ecosθ into Equation (36), the flow rate of the sur- facesSubstituting with different Equation θ angles (18) can and beh obtained= ecosθ into from Equation the following: (36), the ﬂow rate of the surfaces with different θ angles can be obtained from theep2 following: V = s fr e2 pθ tan s2 22 (37) Vf r =3μR sin 2θθθh 2ln+− tan tani (37) tan θ2 2 212 3µR sin 2θ 2 ln θ+ tan θ2 − tan θ1 tantanθ1 1

The flowflow ratesrates ofof thethe leakage leakage media media are are displayed displayed in in Figure Figure 14 .14. The The structural structural angles anglesθ1 ◦ ◦ andθ1 andθ2 areθ2 are 15 15°and and 75 ,75°, respectively, respectively, the pistonthe piston radius radius is 20 is mm, 20 mm, and theand dynamic the dynamic viscosity vis- µcosityand eccentricityµ and eccentricitye are 0.01 e are Pa· 0.01s and Pa·s 0.05 and mm, 0.05 respectively. mm, respectively. The pressure The drop,pressureps, isdrop, 8 MPa, ps, 9is MPa,8 MPa, and 9 MPa, 10 MPa. and The 10 MPa. effects The of angleeffectsθ ofon angle the flow θ on rate the areflow similar rate are to asimilar quadratic to a curve.quad- Theratic lowestcurve. The flow lowest rate occurs flow rate when occursθ is 45when◦, and θ is the 45°, highest and the flow highest rate flow takes rate place takes when placeθ ◦ ◦ ◦ iswhen 15 andθ is 7515° forand this 75° specificfor this structure.specific structure. The flow The rate flow decreases rate decreases when θ

Figure 14. Flow rate of leakage medium.

4. Conclusions The hydrostatic bearing bearing characteristics characteristics of of a aspherical spherical piston piston pair pair are are studied studied theoret- theo- reticallyically in this in this paper. paper. A piston–cylinder A piston–cylinder hydrostatic hydrostatic bearing bearing model model is first is firstestablished. established. The Thesmallest smallest hydrostatic hydrostatic pressure pressure is needed is needed when when the the ratio ratio of of the the piston piston pin-to-piston pin-to-piston radius equals 0.5.0.5. The The pressure pressure drop drop coefficient coefficient decreases decreases with lubricatingwith lubricating film thickness. film thickness. A smaller A revolutesmaller revolute joint clearance joint clearance and ending and angle ending can angle make can the make pressure the pressure drop coefficient drop coefficient decrease moredecrease sharply, more whilesharply, the while film beginning the film beginni angle hasng angle an inverse has an effect inverse on theeffect pressure on the droppres- coefficient.sure drop coefficient. The appropriate The appropriate clearance ranges clearance from 0.04ranges to 0.12 from mm 0.04 and to it 0.12 is beneficial mm and to it im- is proving the performance of the spherical bearing. The maximum film stiffness occurs when beneficial to improving the performance of the spherical bearing. The maximum film stiff- the pressure drop coefficient α is 2/3. A smaller revolute joint clearance and film ending ness occurs when the pressure drop coefficient α is 2/3. A smaller revolute joint clearance angle can make the maximum stiffness occur when the lubricating oil film is thinner. Both a and film ending angle can make the maximum stiffness occur when the lubricating oil smaller beginning angle and a larger ending angle can prohibit leakage in the lubricating film is thinner. Both a smaller beginning angle and a larger ending angle can prohibit media. Leakage increases sharply with temperature, when the temperature is below 80 ◦C, leakage in the lubricating media. Leakage increases sharply with temperature, when the and it is around 0.12 L/min when the operating temperature is higher than 80 ◦C. temperature is below 80 °C, and it is around 0.12 L/min when the operating temperature Though this study is conducted to investigate the hydrostatic bearing characteristics is higher than 80 °C. of the spherical pump, these related results and mechanisms can also be utilized to design and improve other kinds of annular oriﬁce damper spherical hydraulic bearing systems.

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Supplementary Materials: The following are available online at https://www.mdpi.com/article/10 .3390/coatings11081007/s1, Video S1: Motion of the spherical pump. Author Contributions: Conceptualization, D.G. and C.Z.; methodology, D.G. and Z.Z.; software, Z.Z.; validation, Z.Z. and C.Z.; formal analysis, Z.Z. and C.Z. investigation, D.G. and Z.Z.; resources, D.G.; data curation, Z.Z.; writing—original draft preparation, Z.Z.; writing—review and editing, D.G.; visualization, Z.Z.; supervision, D.G.; project administration, D.G.; funding acquisition, D.G. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Natural Science Foundation of China, Grant No.52005433, Jiangsu Province Natural Science Foundation, Grant No.BK20180933, Natural Science Foundation of Jiangsu Higher Institutions, Grant No.19KJB460028 and 20KJB460001, Special Cooperation Foun- dation for Yangzhou & YZU, Grant No.2020182, the Qing Lan and High-end Talent Project from Yangzhou University, Jiangsu Association for Science and Technology “Young Talent Support Project”. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

a, b integral constants d1 piston pin diameter d2 cylinder hole diameter e eccentricity fr, fθ, fϕ body force in three coordinate directions F bearing force h clearance between the piston and the internal surface of the cylinder body hc (d2 − d1)/2 revolute joint clearance hmax maximum lubricating film thickness J lubricating film stiffness Jmax maximum lubricating film stiffness KBC, KBS structural parameters l axial length of revolute joint Q lubricating fluid flow p pressure of lubrication fluid Pc Pr − Ps revolute joint pressure drop Pr injected lubricating fluid pressure ps lubricating pair pressure drop R piston radius Rc revolute joint fluid resistance Rs bearing pair fluid resistance Se effective bearing area S f r 2πRh sin θ specific surface area t temperature ur, uθ, uϕ fluid velocities in three coordinate directions Vf r specific surface flow rate α Ps/Pr hydrostatic bearing pressure drop ratio δ J/Jmax dimensionless stiffness coefficient ρ density of lubrication fluid θ1 film (piston) beginning angle θ2 film (piston) ending angle µ dynamic viscosity coefficient

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