coatings Article
Effect of Surface Topography and Structural Parameters on the Lubrication Performance of a Water-Lubricated Bearing: Theoretical and Experimental Study Zhongliang Xie 1 , Zhushi Rao 2 and Huanling Liu 1, * 1 2
*
School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China;
[email protected] Laboratory of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200240, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-029-88203115
Received: 14 October 2018; Accepted: 10 December 2018; Published: 2 January 2019
Abstract: This study explored the influence of the surface topography of a bushing on the lubrication performance of a water-lubricated bearing. Bushing deformations were considered in the mathematical model. Theoretical calculations and experiments were performed. The test data corresponded well with the simulation. The main stiffness and cross stiffness coefficients were measured and compared with the theoretical values, and the empirical formula of friction coefficient was fitted based on the test data. Keywords: water-lubricated bearing; surface topography; dynamic characteristics; empirical formula of friction coefficient; lubrication performance
1. Introduction In recent decades, the rapid development of the ship-building industry has witnessed the large and extensive application of water-lubricated bearings, due to their unparalleled advantages over other bearings. Many problems have occurred in this application, which, in turn, has promoted the study of lubrication mechanism and lubrication performance of these types of bearings. A growing interest has been given to investigating the lubrication performance of water-lubricated plain journal bearings, especially the influence of the surface topography of bearing coatings. Researchers make great efforts to improve the lubrication performance [1–3] of the bearing, such as the exploitation of surface textures [4–6], the optimum design of bearing structures [7–9], and the introduction of longitudinal grooves. Many studies [10–12] focused on the effect of surface topography [13] on the bearing performance [10,14–20]. For example, Tala-Ighil et al. [21] investigated the modeling of journal bearing characteristics. They found that the lubrication performance of the textured bearing improved significantly with appropriate surface texture geometry and texture distribution. Brito et al. [22] explored the effect of grooves in single and twin axial groove journal bearings under varying load directions. Their results also showed that the friction coefficient deceased compared with the smooth surface bearing. The characteristics of textured journal bearings with consideration of thermal effects were analyzed by Tala-Ighil et al. [23] using the finite difference method (FDM). In other references [24–29], scholars have investigated lubrication performance from different perspectives. Dadouche et al. [30] investigated the operational performance of textured journal bearings lubricated with contaminated fluid. Special attention was focused on the load-carrying capacity, friction, and wear under different contamination levels in the lubricant. Results indicated the effectiveness of textures Coatings 2019, 9, 23; doi:10.3390/coatings9010023
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in capturing contaminant particles and reducing the possibility of failure. Lu et al. [31] performed experiments on the friction characteristic of journal bearings with dimpled bushings manufactured using machining and chemical etching techniques. Their results indicated that a bushing with etched dimples over the entire circumference offered a better frictional performance than a bushing with dimples etched on half of its circumference. Tala-Ighil et al. [32] presented the influence of a textured area on the lubrication performance of hydrodynamic journal bearings. Analysis indicated that a textured area on a bearing bushing could effectively increase the load-carrying capacity and increase the minimum film thickness in the main load-carrying area, while decreasing the friction coefficient at the same time. Cristea et al. [33] studied lubrication performance (transient pressure and temperature field measurements) of journal bearings with circumferential grooves, where the operating modes were lightly loaded from startup to steady-state thermal stabilization. Fluid film pressure, temperature field, friction torque, and lubricant side leakage were detected simultaneously through experimental methods. Their results indicated that film rupture starts from cavitation downstream and the minimum film thickness. This occurred because of the existence of the circumferential grooves and the surface topography. Xie et al. [34–36] explored the lubrication states, lubrication performance with consideration of the bushing macro deformation using theoretical simulation and experimental verification. Their results indicated that the surface roughness had a significant influence on the lubrication state transition and the lubrication performance. A summary of the coupled factors on the lubrication states was also given. However, even though researchers have carried out much work, investigation of the lubrication performance of plain journal bearings is rather insufficient. Only a few studies address the significance of surface topography and bushing deformation on lubrication performance, particularly the experimental verification of the bearings. In view of the above problems, this study focused on the influence of surface topography on the lubrication performance of water-lubricated bearings. The research sheds light on the lubrication mechanism of the bearing and has a certain significance for guiding the design of such bearings. 2. Theoretical Model 2.1. Mathematic Model Figure 1 shows the practical plain journal bearing, with the surface topography. The bearing is completely submerged in the lubricant. It carries the vertical load. Surface topography effect is considered in the analysis. For the convenience of simulation, the bushing and journal surfaces are equivalent to the bushing with combined surface roughness, while the journal is absolutely smooth (as shown in Figure 1a). Figure 1b presents the closer view of the surface roughness. In the Cartesian coordinates, the modified Reynolds Equation with consideration of surface topography effect is as follows: ∂ ρh3 ∂p ∂ ρh3 ∂p ∂(φc ρhT ) ∂(ρφs ) ∂(φc ρhT ) φx + φz = 6U + 6Uσs + 12 ∂x µ ∂x ∂z µ ∂z ∂x ∂x ∂t
(1)
Surface topography is considered using these parameters: φx , φz are the pressure flow factors, φs is the shear flow factor, and φc is the contact factor. For the Equation above, the pressure distribution of the fluid field is governed by the structure and operating parameters, as well as the film thickness. The modified formula of the film thickness is as follows: h = h0 + δh + δ1 + δ2
(2)
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Figure Figure 1. 1. Schematic Schematic representation representation of ofthe thefilm filmthickness thicknessin incontact contactwith withrough roughsurfaces. surfaces.
δh is the macroscopic elastic deformation of the bearing bushing. The macroscopic elastic δh is the macroscopic elastic deformation of the bearing bushing. The macroscopic elastic deformation of the bushing surface due to the normal hydrodynamic effect is calculated using the deformation of the bushing surface due to the normal hydrodynamic effect is calculated using the following Boussinesq Formula: following Boussinesq Formula: 2 x p(ξ, ζ) q δh = dξdζ (3) p ( ξ, ζ ) 02 πE δh = Ω ( x − ξ)2 + (y − ζ)2dξdζ (3) 2 2 πE '
( x − ξ) + ( y − ζ )
Ω
If a groove is considered in the model, the film thickness should be modified as follows: If a groove is considered in the model, the film thickness should be modified as follows: ( h0 + δh + δ1 + δ2 , 0 ≤ θ ≤ θs , θf ≤ θ ≤ 2π h= h0 + δh + δ + δ + δ , θ θ ≤θfθ< h0 + δh +1 δ1 +2 δ2 , groove 0θ≤s < s , θf ≤ θ ≤ 2 π
(4)
h= (4) h0 + δh + δ1 + δ2 + δgroove , θs < θf < θ where θs , θf are the start angle and the end angle of the groove, respectively δgroove is the height of the fluid film in the groove. the end of the groove, respectively δgroove is the height of the where θs, θf are the start angle and δgroove = dangle 0 − C (1 + cos(θ − π/2)) fluid film in the groove. The surface plastic macro deformation can also be added into the calculation if needed. However, δgroove d0 − C 1 +impact, cos ( θ −and π / 2corrosion ) for composite materials due to their wear= resistance, properties, the calculation of plastic macro deformation is rather cumbersome. The equilibrium Equation for the hydrodynamic → The surface plastic macro deformation can also be added into the calculation if needed. force, contact force, and external load P are calculated by: However, for composite materials due to their wear resistance, impact, and corrosion properties, the calculation of plastic macro deformation cumbersome. The equilibrium Equation for the → is→rather → = 0 P + F fluid + W asp (5) hydrodynamic force, contact force, and external load P are calculated by:
(
→
where hydrodynamic force F fluid = lubrication (ML) model. Contact force where hydrodynamic force F fluid = Formula of friction coefficient:
)
2π R1 R film= pressure are determined by the mixed (5) P +pdθdz F fluid and + W asp 0
0 0 → W 1 2asp π
=
R1 2π R
p asp dθdz.
0 0 pdθdz and film pressure are determined by the mixed
0 0
2π + α f = α1 f1asp f total 2 fluid lubrication (ML) model. Contact force W = dθdz . asp = 1, αp,asp α1 + α α 2 1 2 ∈ [0, 1]
(6)
0 0
of friction whereFormula f asp is the frictioncoefficient: coefficient due to micro-asperities contacts effect and f fluid is the friction coefficient due to “viscous effect” (i.e., the shearing stress of the fluid molecules).
f total = α1 f asp + α 2 f fluid
α1 + α 2 = 1, α1 , α 2 ∈ [ 0,1]
(6)
where fasp is the friction coefficient due to micro-asperities contacts effect and ffluid is the friction coefficient due to “viscous effect” (i.e., the shearing stress of the fluid molecules).
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2.2. Boundary Conditions For water-lubricated bearings, the commonly used boundary conditions include the Reynolds, Jakobsson-Floberg–Olsson (JFO), and Sommerfeld boundary conditions. For different boundary conditions, the calculated lubrication performance differs. For JFO boundary conditions [20], the calculated pressure distribution is limited to the positive pressure region, and both the cavitation upper and lower boundaries are based on the mass conservation equation. In the cavitation region, the pressure is equal to the cavitation pressure Pcav , whereas in the rupture region (x0 ,z0 ), Px0 ,z0 = Pcav ,
∂P ∂P = =0 ∂x x= x0 ∂z z=z0
(7)
For Reynolds boundary conditions, the nature rupture boundary condition, which considers the water film to be continuous, and the end point of the film are a nature rupture phenomenon. The film will fracture automatically after striking the minimum film thickness. Generally speaking, Reynolds boundary conditions are closer to the practical engineering operations: (
z = 0, P = Pa , z = z2 , dP dz = 0 0 < z < z2 , P = P(z), z2 < z < 2π, P = Pa ,
dP dz
=0
(8)
In this study, the above two boundary conditions are used to investigate the influence on lubrication performance. For the bearing-rotor system, when the rotor is disturbed, the support force of the film will change correspondingly. If the disturbance is small, the force can be expanded through the Taylor series near the equilibrium point, then: Fx = Fx0 + Fy = Fy0 +
∂Fx ∂x ∆x ∂Fy ∂x ∆x
+ +
∂Fx ∂y ∆y + ∂Fy ∂y ∆y +
∂Fx . . ∆x ∂x ∂Fy . .
∂x
+
∆x +
∂Fx . . ∆y + . . . ∂y ∂Fy .
∆y + . . .
.
∂y
(9)
where Fx , Fy are the components of the film forces and Fx0 , Fy0 are the components of the film forces on the equilibrium position. When the disturbance is small, the high order components of the second order and above are ignored, whereas the first order components are retained: (
.
.
∆Fx = Fx − Fx0 = k xx ∆x + k xy ∆y + c xx ∆ x + c xy ∆y . . ∆Fy = Fy − Fy0 = k yx ∆x + k yy ∆y + cyx ∆ x + cyy ∆y
(10)
For the dynamic coefficients of the film:
k xx = c xx =
∂Fx ∂x 0 , k xy ∂Fx . ,c ∂ x 0 xy
k xx k yx
!
k xy k yy
!
c xx cyx
!
=
= =
Z 1 Z 2π 0
=
0
Z 1 Z 2π 0
=
∂Fx ∂y 0 , k yx ∂Fx . ,c ∂y 0 yx
0
Z 1 Z 2π 0
0
= =
∂Fy ∂x 0 , k yy ∂Fy . ,c ∂ x 0 yy
!
px
sin x − cos x
!
py
sin x − cos x
!
px
sin x − cos x
= =
∂Fy ∂y 0 ∂Fy . ∂y 0
(11)
dxdz
(12)
dxdz
(13)
dxdz
(14)
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c xy cyy
!
=
Z 1 Z 2π 0
0
py
sin x − cos x
! dxdz
(15)
3. Experimental Section 3.1. Test Apparatus Figure 2 presents the experimental apparatus, and Figure 3 illustrates the simple line drawing of the system. For this study, a multi-function bearing-rotor coupled system was used. Different types of bearings could be tested, for example, plain journal bearings, rolling element bearings, and thrust bearings. The shaft was supported by two hydrostatic bearings between the coupling and the test bearing. These two hydrostatic bearings stiffened the rotor system and decreased the deformation of the shaft when vertical load was applied on the tested bearing. They decreased the influence of overhung load and improved the accuracy of the system. The lubricant tank contained the lubricant (for this test, it was water), and the test bearing was completely immersed into the lubricant to guarantee full film lubrication. For the lubricant, water at room temperature (20 ◦ C) was used. Then, the bearing was fully submerged into the water tank; thermal effects were negligible during the experiment. In the case of water, the viscosity is almost independent of the temperature, which means that the viscosity remains constant. Four displacement sensors were installed in the housing along the circumferential direction. The test bearing was subjected to external load through a hydraulic cylinder system. Maximum velocity of the shaft was 6000 rpm. Specific pressure for the bearing was 0–2 MPa, which could be adjusted according to the practical operating mode. The test bearing was subjected to external load through the vertical load unit. Figure 4a presents the load unit on the bearing. Two force sensors and four displacement sensors were installed on the apparatus. Force sensor #1 measured the external load in the vertical direction, and Force sensor #2 measured the tangential force in the circumferential direction. Four displacement sensors were uniformly distributed along the circumferential direction and measured the rotor vibration and the film thickness. The bearing was completely immersed into the lubricant in order to guarantee full-film lubrication. gives a closer view of the external load unit. Coatings 2018, Figure 8, x FOR 4b PEER REVIEW 6 of 20
Figure2.2.Physical Physicallayout layoutof ofthe thebearing-rotor bearing-rotortest testrig rigsystem. system. Figure
Figure 3. Line drawing of the experiment system: 1-motor; 2-damper; 3-coupling; 4-shaft; 5-
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Figure2.2.Physical Physicallayout layoutof ofthe thebearing-rotor bearing-rotortest testrig rigsystem. system. Figure
Figure drawing of the system: 1-motor; 2-damper; 3-coupling; 4-shaft; 5-hydrostatic Figure3.3.3.Line Line drawing ofexperiment the experiment experiment system: 1-motor; 2-damper; 3-coupling; 4-shaft; 55Figure Line drawing of the system: 1-motor; 2-damper; 3-coupling; 4-shaft; bearing; 6-tested bearing; 7-shaft sleeve; 8-hydraulic cylinder; 9-leading bar; 10-force sensor; hydrostatic bearing; 6-tested bearing; 7-shaft sleeve; 8-hydraulic cylinder; 9-leading bar; 10-force hydrostatic bearing; 6-tested bearing; 7-shaft sleeve; 8-hydraulic cylinder; 9-leading bar; 10-force 11-eddy–current sensor; 12-water tank; 13-water; 14-base.14-base. sensor;11-eddy–current 11-eddy–current sensor;12-water 12-water tank;13-water; 13-water; 14-base. sensor; sensor; tank;
(a) (a)
(b) (b)
Figure4.4.Closer Closerview viewof ofthe theload loadunit. unit.(a) (a)load loadunit; unit;(b) (b)closer closerview viewof ofthe theload loadunit. unit. Figure
3.2. Experimental Procedure 3.2.Experimental ExperimentalProcedure Procedure 3.2. In In order order to to measure measure the the film filmthickness, thickness, four four displacement displacement sensors sensors were were mounted mounted along along the the In order to measure the film thickness, four displacement sensors were mounted along the circumferential direction (as seen in Figure 5a,b). If the exact value of the radial clearance circumferential direction can be seen in Figure 5a,b). If the exact value of the radial clearance is circumferential direction (as can be seen in Figure 5a,b). If the exact value of the radial clearance is CCis 00, , C , then assume the initial value of the radial clearance to be C. With given parameters, the film then assume the initial value of the radial clearance to be C. With the parameters, the film 0 then assume the initial value of the radial clearance to be C. With the given parameters, the film thickness distribution and the pressure distribution can be calculated theoretically. Furthermore, thicknessdistribution distributionhhhand andthe thepressure pressuredistribution distributionpppcan canbe becalculated calculatedtheoretically. theoretically.Furthermore, Furthermore, thickness the distances between the bushing and shaft can also be obtained: d10 , d20 , d30 , d40 . At the same time, the distances can also be measured: d1 , d2 , d3 , d4 . If |d10 − d1 | ≤ 0.5µm, |d20 − d2 | ≤ 0.5µm, |d30 − d3 | ≤ 0.5µm, |d40 − d4 | ≤ 0.5µm, the radial clearance of the bearing is C. Otherwise, one can modify the value of the clearance and restart the loop until it is convergent. The convergent criterion was 0.5 µm. Then, after the minimum value of the film thickness distribution is found, the minimum film thickness hmin is obtained.
the distances between the bushing and shaft can also be obtained: d10, d20, d30, d40. At the same time, the distances can also be measured: d1, d2, d3, d4. If d10 − d1 ≤ 0.5μm, d 20 − d 2 ≤ 0.5μm，d30 − d3 ≤ 0.5μm, d 40 − d 4 ≤ 0.5μm , the radial clearance of the bearing is C. Otherwise, one can modify the value of the clearance and restart the loop until it is convergent. Coatings 2019, 9,The 23 convergent criterion was 0.5 μm. Then, after the minimum value of the film thickness 7 of 20 distribution is found, the minimum film thickness hmin is obtained.
Figure of measuring pointspoints for the bearing. displacement sensors alongsensors the circumferential; Figure 5.5.Layout Layout of measuring for the (a) bearing. (a) displacement along the (b) relative positions of the displacement sensors. circumferential; (b) relative positions of the displacement sensors.
3.3. Test Bearing 3.3. Test Bearing Geometry parameters of the test bearings can be found in Table 1. The measured dry friction Geometry parameters of the test bearings can be found in Table 1. The measured dry friction coefficient between steel and bushing is 0.13. Therefore, in order to explore the effect of surface coefficient between steel and bushing is 0.13. Therefore, in order to explore the effect of surface topography on lubrication performance of the bearing, several sets of test bearings with different topography on lubrication performance of the bearing, several sets of test bearings with different surface roughness Sa were processed (as can be seen in Table 2). surface roughness Sa were processed (as can be seen in Table 2). Table 1. Geometry of the bearing. Table 1. Geometry of the bearing. Description Symbol Value Dimension
Description Width Width Diameter Diameter L/D ratio L/D ratio Radial clearance Radial clearance Clearance ratio Clearance ratio Velocity Velocity External load External load
Symbol L L D D L/D L/D C C Ψ Ψ V V F F
Value Dimension 80 mm 80 mm 62 mm 62 mm 1.30 1.30 −− 0.03–0.07 mm 0.03–0.07 mm 0.096h–2.25h 0.096‰–2.25‰ −− 0.001–10 m/s 0.001–10 m/s 80–6000 NN 80–6000
In the experiment, the following parameters were measured: F is the total external load exerted Inbearing, the experiment, the following parameters were F is the total external on the which includes the bearing gravity, Tf ismeasured: the tangential force, and Tf × R isload the exerted friction on the bearing, which includes thethe bearing gravity, is the tangential andbearing Tf × R gravity. is the friction torque. Other parameters include following: D Tisf the diameter andforce, G is the torque. the following: D is the diameter and G is the bearing gravity. TheOther forceparameters applied on include the bearing is as follows: The force applied on the bearing is as follows:
Fr = F − G
Fr = F − G
(16) (16)
The coefficient coefficient of of friction friction of of the the bearing bearing is is as as follows: follows: The
× RR Tf × 22×× T f f== F ×f D Fr r× D
(17) (17)
bearing, R R == 205 mm, D = 62 mm, G == 150 For the test bearing, 150 N, N, the the expression expression of the friction coefficient is as follows: 205Tf f = (18) 31Fr The measuring accuracy of the eddy–current sensor was 0.1 µm, whereas the sampling frequency was 1000 Hz. The nonlinear error ≤ ±0.1%, and the response frequency was 10 kHz. The measuring accuracy of the force sensor was 0.1 N. During the test, the data acquisition system should balance to ensure the equilibrium position of the shaft. The test data were recorded about 180 s under each working mode. Before the data acquisition, the test bearing should run for several minutes. The friction
(18)
31Fr
The measuring accuracy of the eddy–current sensor was 0.1 μm, whereas the sampling frequency was 1000 Hz. The nonlinear error ≤ ±0.1%, and the response frequency was 10 kHz. The measuring accuracy of the force sensor was 0.1 N. During the test, the data acquisition system should Coatings 2019, 9, 23 8 of 20 balance to ensure the equilibrium position of the shaft. The test data were recorded about 180 s under each working mode. Before the data acquisition, the test bearing should run for several minutes. The frictionfluctuate values fluctuate periodically withamplitude. a small amplitude. A moredescription detailed description values periodically with timewith withtime a small A more detailed of the test of the test bearing caninbethe found in the[34,35]. reference [34,35]. bearing can be found reference 4. Results and Discussion 4.1. Measurement of 4.1. Measurement of Surface Surface Topography Topography Figure of the Table 22 gives Figure 6a–d 6a–d shows shows the the 3D 3D morphology morphology distribution distribution of the four four test test bearings. bearings. Table gives the the characteristic parameters of the surface topography for the four tested samples. S is the arithmetic characteristic parameters of the surface topography for the four tested samples. Saa is the arithmetic mean height; V mp peak material volume, which represents the mean height; height;SSqq isisthe theroot rootmean meansquare square height; Vmpisisthe the peak material volume, which represents part that will be worn out in the test; and V is the pit void volume. vv the part that will be worn out in the test; and Vvv is the pit void volume. One One thing thing to to note note for for scientific scientific validity validity is is that that more more than than one one sample sample extract extract per per surface surface should should be investigated. In the experiment, dozens of samples were extracted for one surface. Characteristic be investigated. In the experiment, dozens of samples were extracted for one surface. Characteristic surface Mathematic algorithms algorithms exist within the the Universal Universal surface parameters parameters were were measured measured for for the the samples. samples. Mathematic exist within Profilometer that can automatically deduce the characteristic surface parameters for the surface. Profilometer that can automatically deduce the characteristic surface parameters for the surface. In In this study, we only present one 3D morphology distribution contour for each test surface. this study, we only present one 3D morphology distribution contour for each test surface.
(a)
(b)
(c)
(d)
Figure 6. 3D morphology distribution of the four test samples (the height scale is not the same in all subfigures). (a) sample 1#; (b) sample 2#; (c) sample 3#; (d) sample 4#.
From variation of of the the real real surface surface topography topography for for the the four four tested tested samples. samples. From Table Table 22 we we can can see see the the variation For mean height and the For samples samples #1, #1, #2, #2, #3, #3, and and #4, #4, the the arithmetic arithmetic mean height and the root root mean mean square square height height increase increase correspondingly with the surface roughness. After the experiment, characteristic parameters were significantly reduced, which shows the improvement of the surface topography. Typically, in the start-up and shut-down stage of the bearing-rotor coupled system, the water-lubricated plain journal bearing is at low speed and heavy load state and the film is thin (film thickness may be less than 10 µm in most cases). In some extreme operating conditions, the film thickness may be just a few microns. Water film thickness has almost the same order of magnitude as the bushing interface topography (machining accuracy of composite material bushing >1.6 µm), thus the film thickness ratio is very small. Direct contacts of micro-asperities take place under certain conditions.
significantly reduced, which shows the improvement of the surface topography. Typically, in the start-up and shut-down stage of the bearing-rotor coupled system, the waterlubricated plain journal bearing is at low speed and heavy load state and the film is thin (film thickness may be less than 10 μm in most cases). In some extreme operating conditions, the film thickness Coatings 2019,may 9, 23 be just a few microns. Water film thickness has almost the same order of magnitude 9 of 20 as the bushing interface topography (machining accuracy of composite material bushing >1.6 μm), thus the film thickness ratio is very small. Direct contacts of micro-asperities take place under certain These undoubtedly affect the fluid Therefore, we willwe investigate further and conditions. These undoubtedly affecthydrodynamic. the fluid hydrodynamic. Therefore, will investigate further experimentally verify the lubrication state transitions of the bearing. and experimentally verify the lubrication state transitions of the bearing. Table Table2.2.Characteristic Characteristicparameters parametersofofthe thesurface surfacetopography topographyfor forthe thefour fourtested testedsamples. samples.SSa :a:arithmetic arithmetic mean : peak material volume; V vv: pit : pitvoid voidvolume. volume. root mean mean square square height; height; VVmp mp: peak material volume; Vvv meanheight; height;SSqq:: root Before/After Parameter Unit Parameter Before/After Experiment Experiment Before Sa Before µm After Sa After µm Before Before µm Sq Sq After After µm Before Before µm3 /µm2 VmpVmp 3 2 After After µm /µm Before µm3 /µm2 Before V After µm3 /µm2 Vvv vv After
Unit S1 S1 S2 S3 S2 S3 S4 S4 μm 1.439 (±0.0288) 2.046 (±0.0409) 2.107 (±0.0421) 3.134 (±0.0627) 1.439 (±0.0288) 2.046 (±0.0409) 2.107 (± 0.0421) 3.134 2.238 (±0.0627) μm 1.222 (±0.0244) 1.914 (±0.0383) 1.853 (±0.0371) (±0.0448) 1.222 (±0.0244) 1.914 (±0.0383) 1.853 (±0.0371) 2.238 (±0.0448) μm 1.995 (±0.0399) 2.656 (±0.0531) 2.821 (±0.0564) 4.028 (±0.0842) 1.995 (±0.0399) 2.656 (±0.0531) 2.821 (±0.0564) 4.028 (±0.0842) μm 1.668 (±0.0334) 2.216 (±0.0452) 2.029 (±0.0406) (±0.0574) 1.668 (±0.0334) 2.216 (±0.0452) 2.029 (± 0.0406) 2.870 2.870 (±0.0574) 0.183 (±0.0037) 0.197 (±0.0039) 0.230 (± 0.0046) 0.323 0.323 (±0.0065) μm³/μm² 0.183 (±0.0037) 0.197 (±0.0039) 0.230 (±0.0046) (±0.0065) 0.141 (±0.0028) 0.158 (±0.0032) 0.208 (± 0.0041) 0.305 0.305 (±0.0061) μm³/μm² 0.141 (±0.0028) 0.158 (±0.0032) 0.208 (±0.0041) (±0.0061) 0.171 (±0.0034) 0.206 (±0.0041) 0.254 (±0.0051 0.259 (±0.0052) μm³/μm² 0.171 (±0.0034) 0.206 (±0.0041) 0.254 (±0.0051 0.259 (±0.0052) 0.165 (±0.0033) 0.162 (±0.0032) 0.211 (±0.0042) 0.213 (±0.0043) μm³/μm² 0.165 (±0.0033) 0.162 (±0.0032) 0.211 (±0.0042) 0.213 (±0.0043)
4.2. 4.2.Verification Verificationofofthe theModel Model The and reliability reliability ofofthe themodel modeland and algorithm were prerequisites for study. the study. The accuracy accuracy and algorithm were prerequisites for the The The accuracy of the model was verified by comparing the results with reference [20]. Figure 7 presents accuracy of the model was verified by comparing the results with reference [20]. Figure 7 presents the theschematic schematicdiagram diagramof ofthe therelative relativeposition positionof ofSections Sections11and and22on onthe thetest testbearing. bearing.Sections Sections11and and22 are the two separate sections on the bearing in the axial direction. Section 1 is located at the 1/5 are the two separate sections on the bearing in the axial direction. Section 1 is located at the 1/5LLof of the bearing, whereas Section 2 is located at the 3/5 L of the bearing. Figure 8 shows the comparison the bearing, whereas Section 2 is located at the 3/5 L of the bearing. Figure 8 shows the comparison of ofpressure pressuredistribution distribution between between this this research research and and the the reference, reference, for for Sections Sections 11 and and 22(as (asshown shownin in Figure circumferential direction along thethe liquid filmfilm at the sections of the Figure7). 7).Pressure Pressuredata dataininthe the circumferential direction along liquid at two the two sections of bearing were extracted and compared with the theoretical analysis and experimental results in the the bearing were extracted and compared with the theoretical analysis and experimental results in reference, as shown in Figure 8. 8. the reference, as shown in Figure
Figure Figure7.7.Schematic Schematicdiagram diagramof ofSections Sections11and and2.2.
From FromFigure Figure8,8,we wecan cansee seethat thatthis thisstudy’s study’ssimulation simulationcorresponds correspondswell wellwith withthe theliterature literatureunder under two boundary conditions, the Reynolds boundary condition and the Jakobsson-Floberg-Olsson two boundary conditions, the Reynolds boundary condition and the Jakobsson-Floberg-Olsson (JFO) (JFO) boundary boundary condition, condition, which which proves proves the the accuracy accuracyof ofthe themodel modeland andalgorithm. algorithm. Pressure Pressure under under the the Reynolds boundary conditions shows better correspondence with the reference [20]. The following calculation is based on this model under the Reynolds boundary conditions.
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Reynolds Coatings 2019,boundary 9, 23
conditions shows better correspondence with the reference [20]. The following 10 of 20 calculation is based on this model under the Reynolds boundary conditions.
(a)
(b)
Figure8.8.Comparison Comparisonof ofthe thepressure pressuredistribution distributionbetween betweenthis thisresearch researchand andreference reference[20]. [20].(a) (a)Section Section1;1; Figure (b)Section Section2.2. (b)
4.3. 4.3.Effect EffectofofElastic ElasticDeformation Deformationofofthe theBushing Bushing Figure Figure 99 shows shows the the effect effect ofof eccentricity eccentricity ratio ratio epsilon epsilon on on the the pressure pressure distribution distribution in in the the circumferential direction. As the eccentricity ratio epsilon increases, the maximum value of pressure circumferential direction. As the eccentricity ratio epsilon increases, the maximum value of pressure increases, shows a strong nonlinearity. At theAt same mainthe load-carrying area decreases. increases,which which shows a strong nonlinearity. thetime, samethetime, main load-carrying area Figure 10 shows the effect of bushing thickness on the pressure distribution in the circumferential decreases. direction. the thicker the bushing, the smaller film maximum pressure. FigureWe 10 observe shows that the effect of bushing thickness on the the pressure distribution in the The main load-carrying area also increases with the bushing thickness increase. Correspondingly, circumferential direction. We observe that the thicker the bushing, the smaller the film maximum the load-carrying capacity and friction coefficient can be the perspective pressure. The main load-carrying area alsodecrease. increasesThis with theexplained bushing from thickness increase. ofCorrespondingly, energy, that is, if the bushingcapacity is absolutely rigid without elasticdecrease. deformation, load thebearing load-carrying and friction coefficient This the canexternal be explained will be balanced by the film thickness. However, under the same conditions, the bushing deformation from the perspective of energy, that is, if the bearing bushing is absolutely rigid without elastic will absorb partthe of the energy andwill share externalby load with the However, water film. under The thicker the deformation, external load bethe balanced thetogether film thickness. the same bushing, the more energy the bushing will absorb thereby reducing the maximum pressure of the film. conditions, the bushing deformation will absorb part of the energy and share the external load This is also the bushing’s mechanism of cushion and shock absorption. together with the water film. The thicker the bushing, the more energy the bushing will absorb Figure 11 shows the effect of elastic deformation on is the dimensionless thereby reducing the maximum pressure of the film. This also the bushing’sload-carrying mechanism ofcapacity cushion and the friction force. The rigid model represents the model which does not consider the deformation and shock absorption. of theFigure bushing, whereas elastic model deformation considers theondeformation. Load-carrying capacity and 11 shows thethe effect of elastic the dimensionless load-carrying capacity friction bothforce. increase eccentricity ratio. capacity with elastic and theforce friction The with rigidthe model represents theLoad-carrying model which does not consider thedeformation deformation isofsmaller than that of the rigid model, whereas friction force is higher than that of the rigid model. the bushing, whereas the elastic model considers the deformation. Load-carrying capacity and The difference the rigidwith and elastic models gradually with the eccentricity friction force between both increase the eccentricity ratio. increases Load-carrying capacity withratio. elastic Figure 12isshows thethan effectthat of elastic the dynamic coefficients. Withthan the increase in deformation smaller of the deformation rigid model,on whereas friction force is higher that of the the eccentricity ratio, the dynamic coefficients K , K , K , K , C , C , C , C , increase for the rigid xx and xy yx yy models xx xy gradually yx yy rigid model. The difference between the rigid elastic increases with the and elastic models. eccentricity ratio. Dynamic coefficients with elastic deformation are smaller than that of the rigid model. The existence elastic decreases the rigidity of thecoefficients. bearing, which beneficial Figure 12 showsof the effectdeformation of elastic deformation on the dynamic Withisthe increase for the stability of the bearing-rotor system. Elastic deformation decreases the maximum pressure and in the eccentricity ratio, the dynamic coefficients Kxx, Kxy, Kyx, Kyy, Cxx, Cxy, Cyx, Cyy, increase for the rigid dynamic characteristics. The difference between the rigid and elastic are models gradually increases and elastic models. Dynamic coefficients with elastic deformation smaller than that of the with rigid the eccentricity ratio. model. The existence of elastic deformation decreases the rigidity of the bearing, which is beneficial
for the stability of the bearing-rotor system. Elastic deformation decreases the maximum pressure and dynamic characteristics. The difference between the rigid and elastic models gradually increases with the eccentricity ratio.
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Figure 9. Effect of eccentricity ratio on the pressure distribution. Figure 9. 9. Effect of of eccentricity ratio ratio on the the pressuredistribution. distribution. Figure Figure 9. Effect Effect of eccentricity eccentricity ratio on on thepressure pressure distribution.
Figure 10. 10. Effect Effect of of bushing bushing thickness thickness on onthe thepressure pressuredistribution. distribution. Figure Figure 10. Effect of bushing thickness on the pressure distribution. Figure 10. Effect of bushing thickness on the pressure distribution.
(a) (a) (a)
(b) (b) (b)
Figure 11. elastic deformation on the performance: (a) load-carrying capacity; (b) friction Figure 11. Effect Effectofof elastic deformation onlubrication the lubrication performance: (a) load-carrying capacity; Figure 11. Effect of elastic deformation on the lubrication performance: (a) load-carrying capacity; (b) friction Figure 11. Effect of elastic deformation on the lubrication performance: (a) load-carrying capacity; (b) friction coefficient. (b) friction coefficient. coefficient. coefficient.
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(b)
(a)deformation on the dynamic coefficients: (a) stiffness (b)coefficient; (b) damping Figure 12. Effect of elastic coefficient. Figure 12. 12. Effect of elastic deformation on the dynamic dynamic coefficients: coefficients: (a) stiffness coefficient; (b) damping damping Figure coefficient. coefficient.
4.4. Minimum Film Thickness 4.4. Film 4.4. Minimum Minimum Film Thickness Thickness The minimum film thickness is the major judging criterion for the lubrication regime’s transition minimum film thickness major criterion for the and The the lubrication performance. Specific pressure, velocity viscosity regime’s are the transition dominant The minimum film thicknessis is the the major judging judging criterion forand the lubrication lubrication regime’s transition and the lubrication performance. Specific pressure, velocity and viscosity are the dominant influencing influencing factors on the minimum film thickness. It is ofvelocity vital significance to examine the minimum and the lubrication performance. Specific pressure, and viscosity are the dominant factors on thefactors minimum film thickness. It isthickness. of vital significance examine thetominimum thickness film thickness and its factors. influencing on influencing the minimum film It is of vitaltosignificance examine film the minimum and its influencing factors. Figure 13and presents the measured film thickness its influencing factors.minimum film thickness as a function of velocity under Figure 1313presents thethe measured minimum film thickness as aspecific function of velocity different different specific pressures formeasured the test bearing. Under thethickness same the minimum film Figure presents minimum film as a pressure, function of under velocity under specific pressures for the test bearing. Under the same specific pressure, the minimum film thickness thickness increases with the velocity. For different specific pressures, the changing rule is different. different specific pressures for the test bearing. Under the same specific pressure, the minimum film increases the velocity. different pressures, the changing is different. the When thewith bearing iswith subjected to a light load (specific pressure of 0.125 MPa), therule minimum film thickness increases theFor velocity. Forspecific different specific pressures, therule changing isWhen different. bearing is rises subjected ainlight load (specific pressure ofmoderately 0.125 MPa),of the minimum thickness thickness sharply the low rises in the medium velocity range,rises and When the bearing istosubjected tovelocity a light range, load (specific pressure 0.125 MPa), film the minimum film sharply in the low velocity range, rises moderately in the medium velocity range, and remains the same remains the same in the high velocity range, which illustrates that the contribution of velocity to thickness rises sharply in the low velocity range, rises moderately in the medium velocity range, and in the high velocity range, which illustrates that the contribution of velocity to minimum film thickness minimum film thickness greater than range, that ofwhich specificillustrates pressure.that When bearing is subjected a remains the same in the is high velocity thethe contribution of velocitytoto is greater than that of specific pressure. When the bearing is subjected to a medium load (specific medium load (specific pressure of 0.156, 0.313 and 0.625 MPa), the minimum film thickness increases minimum film thickness is greater than that of specific pressure. When the bearing is subjected to a pressure of 0.156, 0.313 and 0.625velocity MPa), minimum film thickness increases almost linearlyincreases over the almost linearly over the entire range. In the caseMPa), of a high load condition (specific pressure medium load (specific pressure of 0.156,the 0.313 and 0.625 the minimum film thickness entire velocity range. In the case of a high load condition (specific pressure of 0.938 MPa), the minimum of 0.938linearly MPa), the minimum film thickness increase theload velocity over the entirepressure velocity almost over the entire velocity range. In theslowly case ofwith a high condition (specific film thickness increase slowly with the velocity over the entire velocity range, which indicates that the range, which indicates that the contribution of specific pressure to minimum film thickness is greater of 0.938 MPa), the minimum film thickness increase slowly with the velocity over the entire velocity contribution of specific pressure to minimum film thickness is greater than that of velocity. than that of velocity. range, which indicates that the contribution of specific pressure to minimum film thickness is greater than that of velocity.
Figure Figure 13. 13. Minimum Minimum film film thickness thickness under under different different specific specific pressures. pressures.
Figure 13. Minimum film thickness under different specific pressures.
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4.5. 4.5.Stiffness StiffnessCoefficients Coefficients 4.5. Stiffness Coefficients The of Thestiffness stiffness ofthe thebearing bearingin inthe thevertical verticaldirection directionKKyyyyisisequal equalto tothe theratio ratioof ofthe theload loadincrement increment ΔΔFFyyThe in to The isis as in the the yy direction direction to the the displacement displacement increment in the the yto y direction. direction. The expression as Δyyisin stiffness of the bearing in the verticalincrement direction KΔ equal the ratio of theexpression load increment yy ∆F follows: y in the y direction to the displacement increment ∆y in the y direction. The expression is as follows: follows:
ΔΔFFy F −−FFFy1 FFy 2 − ∆F KKyy= = y y= == y2y 2 y1y1 Kyy yy = ∆y yy22− ΔΔyy −−yyy111
(19) (19) (19)
Figure 14 shows the relationship between the external load and the relative displacement the Figure load and the relative displacement inin the y Figure14 14shows showsthe therelationship relationshipbetween betweenthe theexternal external load and the relative displacement in the 3 N, 3the 3 yydirection. When the load is in the range 1.50–4.50 × 10 N, the linearity of the curve is better. The direction. When the load is in the range 1.50–4.50 × 10 linearity of the curve is better. The slope direction. When the load is in the range 1.50–4.50 × 10 N, the linearity of the curve is better. The 7N − 771N slope curve the coefficient, ××m 10 m its of the of curve the stiffness coefficient, Kyy = 2.1KKyy × , and theoretical value value is Kyy is =is slope ofthe therepresents curverepresents represents thestiffness stiffness coefficient, yy=10 =2.1 2.1 10 N m−1−1,its ,and and itstheoretical theoretical value 7 − 1 77N m KKyyyy==×2.86 aarelative error of as in 2.86 10 ××N10 m a relative error of 26.57%, as shown in Table 3. 3.3. 2.86 10 N with m−1−1with with relative error of26.57%, 26.57%, asshown shown inTable Table
Figure 14. the vertical direction. Figure14. 14.Curve Curveof ofthe theexternal externalload loadand andthe therelative relativedisplacement displacementininthe thevertical verticaldirection. direction. Figure
Similarly, Similarly, the cross stiffness coefficient isisequal equal to the ratio of the load increment ∆F Similarly,the thecross crossstiffness stiffnesscoefficient coefficientKKyx Kyxyxis equalto tothe theratio ratioof ofthe theload loadincrement increment∆F ∆Fyyyin inthe theyy direction direction to the displacement increment ∆x in the direction. The expression isisas as follows: directionto tothe thedisplacement displacementincrement increment∆x ∆xin inthe thexxxdirection. direction.The Theexpression expressionis asfollows: follows:
ΔΔFFy F −−FFFy1y1 FFy 2 − ∆F KKyx= = y y= == y2y 2 y1 Kyx yx = ∆x xx22− ΔΔxx −−xxx111
(20) (20) (20)
For Forthe thesame sameconditions conditionsas asin inFigure Figure14, 14,Figure Figure15 15presents presentsthe therelationship relationshipbetween betweenthe theexternal external load the relative displacement direction. curve represents the stiffness loadand andthe therelative relativedisplacement displacementin inthe thexxxdirection. direction.The Theslope slopeof ofthe thecurve curverepresents representsthe thestiffness stiffness 1 , and its theoretical value is K = 3.29 × 6N −1 awith 666 −1 −1,− coefficient, ==2.72 2.72 10 10 m a relative coefficient, KKyx yx N error coefficient,K yx= 2.72×× ×10 10 NNm mm ,and andits itstheoretical theoreticalvalue valueisisKKyxyx=yx =3.29 3.29××10 1066N Nm m−1−1with with arelative relative error error of 21.00%, as shown in Table of as in 3.3. 3. of21.00%, 21.00%, asshown shown inTable Table
Figure horizontal direction. Figure15. 15.Curve Curveof ofthe theexternal externalload loadand andthe therelative relativedisplacement displacementininthe thehorizontal horizontaldirection. direction.
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Table 3. Dynamic coefficient of the bearing. Coatings 2018, 8, x FOR PEER REVIEW
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Theoretical (N/m)
Relative Error
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7 bearing. 26.57% Table coefficient the 2.863.×Dynamic 107 2.10 ×of10
Kyy Kyx
4.6. Empirical Formula of
Test (N/m)
Item Kyy Kyx Friction
3.29 × 106
Theoretical (N/m) 2.86 × 107 3.29 × 106 Coefficient
2.72 × 106
Test (N/m) 2.10 × 107 2.72 × 106
21.00%
Relative Error 26.57% 21.00%
Figure 16 shows the of experimental verification of the friction coefficient for two test bearings with 4.6. Empirical Formula Friction Coefficient different surface roughness. Specifically, σ1 and σ2 represent bearings #1 and #2, respectively. Test data Figure 16 shows the experimental verification of the friction coefficient for two test bearings with correspond well with the simulation, for the high velocity region, and the relative error is very small, different surface roughness. Specifically, σ1 and σ2 represent bearings #1 and #2, respectively. Test especially for bearingwell #2.with the simulation, for the high velocity region, and the relative error is very data correspond For the practical small, especially forengineering bearing #2. bearings, it is generally difficult to accurately measure the film thickness.For Friction coefficient is the commonly used parameter, whichtocan be measured exactly [37–40]. the practical engineering bearings, it is generally difficult accurately measure the film thickness. Friction coefficient is the commonly used which canand be measured exactly [37–40]. Figure 17a,b shows the relationship between theparameter, friction coefficient the velocity under different shows the= relationship theψfriction coefficient and the velocity under clearance Figure ratios, 17a,b namely, (a) ψ 0.5h, (b) ψ between = 1.0h, (c) = 1.5h, and (d) ψ = 2.0h. From the figures different clearance ratios, namely, (a) ψ = 0.5‰, (b) ψ = 1.0‰, (c) ψ = 1.5‰, and (d) ψ = 2.0‰. From we observe that as the velocity increases, the friction coefficient decreases sharply in the low speed the figures we observe that as the velocity increases, the friction range, and decreases moderately in the medium and high speedcoefficient range. decreases sharply in the low speed range, and decreases moderately in the medium and high speed range. Figure 18a,b shows the relationship between the friction coefficient and the specific pressure Figure 18a,b shows the relationship between the friction coefficient and the specific pressure under different clearance ratios, namely, (a) ψ = 0.5h, (b) ψ = 1.0h, (c) ψ = 1.5h, and (d) ψ= 2.0h. under different clearance ratios, namely, (a) ψ = 0.5‰, (b) ψ = 1.0‰, (c) ψ = 1.5‰, and (d) ψ= 2.0‰. FromFrom the figures we observe thatthat thethe friction coefficient with the theincrease increaseinin the the figures we observe friction coefficientdecreases decreasesmoderately moderately with specific pressure over the entire speed range [41]. the specific pressure over the entire speed range [41].
Figure 16. Verification of friction coefficient(σ (σ11 represents represents bearing σ2 σrepresents bearing #2). Figure 16. Verification of friction coefficient bearing#1,#1, 2 represents bearing #2).
(a)
(b)
(a)
(b) Figure 17. Cont.
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(a)
(b)
(c)
(d)
Figure 17.and the velocity for different clearance ratios ψ: Figure 17. Relationship between the friction coefficient (a) ψ = 0.5h, (b) ψ = 1.0h, (c) ψ = 1.5h, and (d) ψ = 2.0h.
(a)
(b)
1 (c)
(d)
Figure 18 and the specific pressure for different clearance Figure 18. Relationship between the friction coefficient ratios ψ: (a) ψ = 0.5h, (b) ψ = 1.0h, (c) ψ = 1.5h, and (d) ψ = 2.0h.
Generally speaking, it is of great concern to estimate the friction coefficient and evaluate the lubrication performance. The main influencing factors affecting the bearing’s performance include the specific pressure, the clearance ratio, the velocity, and the width to diameter ratio. The friction coefficient is closely related to these parameters. Based on the tested data, the relationship between the friction coefficient and the above factors was obtained through fitting. The effects of the bearing’s structural parameters and operating conditions on tribological properties were examined. The empirical formula developed with velocity, specific pressure, and clearance ratio is as follows: f (v, p, ψ) = α1 · e(−α2 v+α3 ) · pα4 · ψα5 + α6
(21)
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f ( v, p, ψ ) = α1 ⋅ e(
− α 2 v + α3 )
⋅ p α 4 ⋅ ψ α5 + α 6
(21)
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where p is the specific pressure, which is equal to the external load divided by the equivalent loaded F area, p = total , with the unit N m−2; A is the equivalent loaded area, which is equal to the diameter A where p is the specific pressure, which is equal to the external load divided by the equivalent loaded multiplied L, with the unit m2; ψ is the clearance ratio, which is equal to the radial Ftotal by the width, A = D × − area, p = A , with the unit N m 2 ; A is the equivalent loaded area, which is equal to the diameter c clearance divided by A the=eccentricity, ; α1 to are andratio, through fitting experimental multiplied by the width, D × L, with ψ= the unit m2α;6ψ is coefficients the clearance which is equal to the radial ce clearance divided by the eccentricity, ψ = ; α to α are coefficients and through fitting experimental 1 4 gives 6 the estimated values of the coefficients α1 to α6. e data, estimated values can be obtained. Table data, The estimated be1 to obtained. 4 gives the estimated values thefound coefficients range ofvalues valuescan for α α6 and theTable recommended values for α1 to α6 areof also in Table α 4.1 to α6 .
The range of values for α1 to α6 and the recommended values for α1 to α6 are also found in Table 4. Table 4. Estimated values of the coefficients.
Table 4. Estimated values of the coefficients. Item
Item α 1 α1 α2 α3 α4 α5 α6
α2 α3 α4 α5 α6
Range of the Value
Range of the Value 0.02468–0.1206 0.02468–0.1206 5.5180–7.8470 5.5180–7.8470 0.1018–0.2226 0.1018–0.2226 0.1035–0.4069 0.1035–0.4069 0.3372–0.6800 0.3372–0.6800 0.02758–0.02997 0.02758–0.02997
Recommended Value
Recommended Value 0.0967 0.0967 6.6460 6.6460 0.1531
0.1531 0.2745 0.4439 0.4439 0.0288 0.0288 0.2745
Thus, the empirical formula of the friction coefficient obtained through fitting the measured data
Thus, the empirical formula of the friction coefficient obtained through fitting the measured data is as follows: is as follows: +v 0.1531 0.2745 −6.646 + 0.1531) ) f 1 (v, p,f1ψ( )v,= 0.0967 · e(−⋅6.646v ψ0.4439 + 0.0288 p, ψ e( ⋅ ·pp0.2745 ⋅ ψ ·0.4439 + 0.0288 (22) (22) ) = 0.0967 Theoretical calculations, experimental results and the fitting data are plotted in Figure 19, where in Theoretical calculations, experimental results and the fitting data are plotted in Figure 19, where (a) the clearance ratio is 1.0hand in (b) the clearance ratio is 1.5h. Fitting curves correspond well in (a) the clearance ratio is 1.0‰ and in (b) the clearance ratio is 1.5‰. Fitting curves correspond well with the experimental data, while some deviation occurs compared to the theoretical calculations. with the experimental data, while some deviation occurs compared to the theoretical calculations. Nevertheless, the empirical formula isisbeneficial optimumdesign design structures, rationale Nevertheless, the empirical formula beneficial for for the the optimum of of structures, the the rationale for selecting the parameters (e.g., the clearance ratio, the specific pressure) and the prediction for selecting the parameters (e.g., the clearance ratio, the specific pressure) and the prediction of of tribological characteristics. tribological characteristics.
(a)
(b)
Figure Fitting curveofofthe thefriction friction coefficient: coefficient: (a) =1‰, (b)(b) ψ=1.5‰. Figure 19. 19. Fitting curve (a)ψψ=1h, ψ=1.5h.
The empirical formula usedtotopredict predict the the friction friction coefficient ofof thethe bearing. Figure 20 20 The empirical formula cancan bebe used coefficient bearing. Figure shows the comparison of the empirical formula values and the experimental data. For P 1 and P2 the shows the comparison of the empirical formula values and the experimental data. For P1 and P2 the specific pressure is 0.15 and 0.20 MPa, respectively, and the clearance ratio is 1.0h. Predicted values using the empirical formula correspond well with experimental data in the medium and high velocity region, while deviations occur in the low velocity region. Figure 21 shows the three-dimensional distribution of the friction coefficient with the velocity and the clearance ratio. From Figure 21 it is clear that the optimum range of the clearance ratio is approximately from 0.8hto 2.0h. This is of important significance for guiding the parameter optimization and structure design of such bearings.
usingwhile the empirical formula with experimental data in the medium and high velocity region, deviations occurcorrespond in the lowwell velocity region. region, while deviations occur in the low velocity region. Figure 21 shows the three-dimensional distribution of the friction coefficient with the velocity 21 shows three-dimensional of optimum the frictionrange coefficient the velocity and theFigure clearance ratio.the From Figure 21 it is distribution clear that the of thewith clearance ratio is and the clearance ratio. From Figure 21 it is clear that the optimum range of the clearance ratio is approximately from 0.8‰ to 2.0‰. This is of important significance for guiding the parameter approximately from 0.8‰ to 2.0‰. This is of important significance for guiding the parameter Coatings 2019, 9, 23 17 of 20 optimization and structure design of such bearings. optimization and structure design of such bearings.
Figure20. 20.Fitting Fittingcurve curveof of the the friction friction coefficient ψ Figure coefficient 1h). coefficient((ψ (ψ===1‰). 1‰).
Figure 21. Fitting curve of the friction coefficient. Figure Figure 21. 21. Fitting curve of of the the friction friction coefficient. coefficient.
5. Conclusions 5. 5. Conclusions Conclusions This study has a certain significance for guiding the future investigation of the lubrication state This has for guiding future the state transition of water-lubricated plain journal An formula wasof using test This study study has aa certain certain significance significance forbearings. guiding the the empirical future investigation investigation ofproposed the lubrication lubrication state transition of water-lubricated plain journal bearings. An empirical formula was proposed using data fitting. Predicted values corresponded well with the experiment, and they will be beneficial for transition of water-lubricated plain journal bearings. An empirical formula was proposed using test test data fitting. Predicted values corresponded well with the experiment, and they willconclusions be beneficial forbethe the optimum structure design of the bearing. From the analysis, the following can data fitting. Predicted values corresponded well with the experiment, and they will be beneficial for optimum structure designdesign of the of bearing. From the analysis, the following conclusions can be made: themade: optimum structure the bearing. From the analysis, the following conclusions can be • The The existenceofofbushing bushingdecreases decreasesthe the dimensionless dimensionless pressure. •made: existence pressure.With Withthe theincrease increaseininthe thebushing bushing thickness, the dimensionless pressure decreases correspondingly; thickness, the dimensionless pressure decreases correspondingly; • The existence of bushing decreases the dimensionless pressure. With the increase in the bushing With the increase in the eccentricity ratio, the dimensionless load-carrying capacity and the • • thickness, With the increase in the eccentricity ratio, the dimensionless load-carrying capacity and the friction the dimensionless pressure decreases correspondingly; friction force increase. The existence of bushing deformation (elastic model) decreases the loadforce increase. The in existence of bushing deformation (elastic model) decreases the load-carrying • With the increase the eccentricity ratio, the dimensionless load-carrying capacity and the carrying capacity but increases the friction force; capacity but increases the friction force; friction force increase. The existence of bushing deformation (elastic model) decreases the load• carrying With thecapacity increasebut in the eccentricity ratio, the dimensionless stiffness and damping coefficients increases the friction force; increase. The existence of the bushing deformation (elastic model) decreases the dynamic characteristic coefficients; • Under the same specific pressure, with the increase in the speed, the minimum film thickness increases. Under the same speed, with the increase in the specific pressure, the minimum film thickness decreases. Specific pressure and velocity are the dominant influencing factors on the measured minimum film thickness;
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The empirical formula of friction coefficient with velocity, specific pressure, and clearance ratio is obtained based on experimental data. The empirical formula is beneficial for the optimum design of structures and the prediction of tribological characteristics.
These conclusions are useful for the structure design, analysis, and optimization of journal bearings. For future research, we will continue to improve the test rig and the measurement accuracy. The mathematical model will also need to consider more comprehensive influencing factors. Author Contributions: Conceptualization, Z.X. and Z.R.; Data curation, Z.X.; Formal analysis, Z.X.; Funding acquisition, Z.X. and Z.R.; Investigation, Z.X. and H.L.; Methodology, Z.X.; Project Administration, Z.X. and Z.R.; Resources, Z.X.; Software, Z.X.; Supervision, Z.X.; Validation, Z.X.; Visualization, Z.X.; Writing–Original Draft, Z.X.; Writing–Review and Editing, Z.X., Z.R., and H.L. Funding: This research was funded by the National Natural Science Foundation of China (No. 1167020010). Acknowledgments: The authors thank Chunxiao Jiao and Xiuli Zhang in Shanghai Jiao Tong University for their contributions to the experiments. Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature C Rb , Rj e ε L, D p hmin δh h h0
radial clearance = Rb − Rj bearing and journal radii eccentricity eccentricity ratio, e/c width and diameter of bearing hydrodynamic pressure minimum film thickness macroscopic elastic deformation of the bushing real film thickness nominal film thickness
P ω ρ µ Tf k xx , k xy , k yx , k yy Ob , O j θ σ1 , σ2 δ1 , δ2 σ λ T E φs φc φ x , φz θs , θ f δgroove ψ G c xx , c xy , cyx , cyy
total external load
→
angular velocity = 2πN lubricant density lubricant viscosity tangential force coefficient of stiffness bearing, journal left angular coordinate RMS surface roughness of two surfaces roughness height of two surfaces combined surface roughness film thickness ratio bushing thickness combined elastic modulus shear flow factor contact factor pressure flow factors start angle and end angle of the groove height of fluid film in the groove clearance ratio bearing gravity coefficient of damping
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6. 7. 8. 9. 10. 11.
12. 13. 14. 15. 16.
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