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Experimental research on the interface lubrication regimes transition of water lubricated bearing
Mechanical Systems and Signal Processing 136 (2020) 106522
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Experimental research on the interface lubrication regimes transition of water lubricated bearing Zhongliang Xie, Huanling Liu ⇑ School of Electro-Mechanical Engineering, Xidian University, No. 2 Taibai South Road, Xi’an 710071, China
a r t i c l e
i n f o
Article history: Received 8 March 2019 Received in revised form 14 July 2019 Accepted 10 November 2019
Keywords: Water lubricated bearing Surface topography effects Lubrication regimes transition Lift-off speed Friction coefficient Tribological characteristics
a b s t r a c t This paper aims to experimentally investigate the influences of surface topography and operating conditions on lubrication regimes transition of water lubricated bearing in submarine. Simulations are also performed under the same settings and operating conditions. The calculated and experimental results agree well. Effects of surface roughness, external loads, radial clearances, the acceleration and deceleration process on the lubrication regimes transition are further investigated. Characteristic parameters of the lubrication regimes transition are summarized. Reasonable explanations are given to illustrate the results as well as the relative errors. It has certain guiding significance for further step investigation on influences of surface topography on lubrication regimes transition of such bearings. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The acoustic stealth performance is an important indicator to evaluate the operational capability of underwater submarines. Reducing the radiated noise and the harmonic target strength can not only improve the acoustic stealth of naval underwater submarine, but also can effectively improve the capability of the first enemy attack. With the improvement of the equipment’s low-noise performance and the hydrodynamic performance of the underwater submarines, the effects of the stern structure coupled vibration radiated noise on the acoustic stealth performance have become increasingly prominent. One of the main sources of the coupled vibration is the water lubricated plain journal stern tube bearing, which plays crucial roles in the ship’s propulsion system (as can be seen in Fig. 1). The mechanical properties of the bearing have significant influences on the dynamic response of the rotor system and the acoustic stealth performances. Recent years have witnessed the broad application of water lubricated bearings, with their unique advantages over the conventional bearings. Problems occur in the practical application and have drawn the research interests of many researchers. The former scholars have done researches in the lubrication mechanism [1–3], lubrication performances [4–6], optimum design [7–9] as well as the dynamic characteristics [10] of the bearings. Researchers [11–15] have also paid attentions to the modeling and simulating roughness effects on lubrication regimes. Among them, only several literatures [16–21] focus on the lubrication regimes transition. For instances, classical Reynolds equation was modified by Patir in 1978 [17,18] to accommodate the case of fluid flow associated with rough surfaces. Sahlin [19,20] and Hsu [21,22] investigate the lubrication performances of mixed lubrication with consideration of measured surface topography with modified mixed lubrication model. ⇑ Corresponding author. E-mail address: [email protected] (H. Liu). https://doi.org/10.1016/j.ymssp.2019.106522 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.
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Nomenclature c Rb ; Rj e r1 ; r2 L; D p hmin dh h h !0 P
x q l G w Ob ; Oj h
e
d1 ; d2
r k dgroov e E /s /c /x ; /z hs ; hf T Tf
radial clearance = Rb Rj bearing and journal radii eccentricity RMS surface roughness of two surfaces width and diameter of bearing hydrodynamic pressure the minimum film thickness macroscopic elastic deformation of bushing real film thickness nominal film thickness total external load angular velocity = 2pN lubricant density lubricant viscosity bearing gravity clearance ratio bearing, journal center angular coordinate eccentricity ratio, e=c roughness height of two surfaces combined surface roughness film thickness ratio fluid depth in the groove combined elastic modulus shear flow factor contact factor pressure flow factors the start angle and the end angle of the groove the bushing thickness tangential force
He [23] investigates the marine stern tube bearing with consideration of shaft deformation and cavitation, as well as the lubrication regimes transition from mixed lubrication (ML) to hydrodynamic lubrication (HL) transition. The modified model incorporates surface topography effects and covers all the three lubrication regimes. He also calculates the friction coefficient and establishes the relationship between friction and lubrication regimes transition. Critical speed that transition from mixed lubrication to hydrodynamic lubrication are explored. Lorentz [24] studies the influences of surface roughness on lubrication performances and lubrication regimes. Hu [25] and Zhu [26] develop deterministic method with consideration of surface topography effects to simulate the three lubrication regimes, the boundary lubrication regime (BL), the mixed lubrication regime (ML) and the hydrodynamic regime (HL). They also study the lubrication regimes transition with experimental results under different operating conditions. Lubrication regimes transition has significant effects on the operating stability and lifetime of the bearing-rotor coupled system. For the actual bearing and rotor surface, undoubtedly, lubrication regime on the interface are extremely complicated
Fig. 1. Schematic diagram of submarine stern structure with water lubricated bearing [1].
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and always in dynamic fluctuation states under different operating conditions. Lubrication regimes transition occurs in the process. Xie [27–31] determines the calculation of friction coefficient through theoretical method, experimentally verifies the simulation results. Stribeck curve is an important tool to analyze the tribological characteristic. Based on this, he has made attempts to establish the relationship between the friction coefficient and the lubrication regimes transition (BL to ML, ML to HL). Kraker [32] simulates the friction coefficient and surface-contact pressure through mixed-EHL lubrication model. He also investigates the lubrication regimes transition. Isaksson [33] studies the effects of surface threedimensional topography on the tribological characteristics with three different lubrication regime (BL, ML and HL). However, researches on the effects of surface topography and the lubrication regimes transition are relatively insufficient. Little literature is available. Only a few of them have addressed the importance of effects of surface topography [34–36] on lubrication regimes transition for such bearings. Fig. 2(a) and (b) show schematic diagram for practical water lubricated bearings with surface topography effects. From Fig. 2 we can see, surface topography effects are important factors for the lubrication regime analysis, especially in the thin film thickness region. This paper experimentally investigates the influences of surface topography and operating conditions on the lubrication regimes transition. In order to observe the micro interface topography vividly, surface roughness parameters are measured with high-resolution electron microscope. A set of specially designed experimental devices for the bearing are exploited. Simulations are also done under the same settings and operating conditions. Calculated friction coefficients are compared with tested results. The calculated and experimental results agree rather well. Effects of surface roughness, external loads and radial clearances on the lubrication regimes transition are further investigated. Characteristic parameters (lift-off speed & the critical friction coefficient) of the lubrication regimes transition are summarized. Reasonable explanations are given to illustrate the results as well as the relative error. The symbols are summarized in the Nomenclature section and discussed in details in the following section. 2. Theoretical model Fig. 3(a) presents the film thickness in rough surfaces contact, Fig. 3(b) presents the enlarged drawings of the film thickness in the load carrying area. For water lubricated bearing, the modified Reynolds equation [28,29]:
! ! @ qh3 @p @ qh3 @p @ ð/c qhT Þ @ ðq/s Þ @ ð/c qhT Þ þ ¼ 6U /x /z þ 6U rs þ 12 @x @z @x @x @t l @x l @z
ð1Þ
The modified fluid film thickness is obtained through:
h ¼ h0 þ dh þ d1 þ d2
Fig. 2. Schematic diagram for practical water lubricated bearings with surface topography [23].
ð2Þ
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Fig. 3. Film thickness in rough surfaces contact (enlarged drawings of film thickness).
dh is the macroscopic elastic deformation of the bushing. Macroscopic elastic deformation due to the normal hydrodynamic effect is calculated by Boussinesq formula:
dh ¼
2 pE0
ZZ X
pðn; fÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dndf ðx nÞ2 þ ðy fÞ2
ð3Þ
If lubrication groove is considered (as shown in Fig. 4), then the film thickness should be modified:
h¼
h0 þ dh þ d1 þ d2 ; h0 þ dh þ d1 þ d2 þ dgroov e ;
0 6 h 6 hs ; hf 6 h 6 2p hs < hf < h
ð4Þ
where dgroov e is the fluid depth in the groove. dgroov e ¼ d0 C ð1 þ cosðh p=2ÞÞ.
! The equilibrium equation for the hydrodynamic force, contact force and external load P are calculated by:
! ! ! P þ F fluid þ W asp ¼ 0
ð5Þ
R 1 R 2p R 1 R 2p ! ! where the hydrodynamic force F fluid ¼ 0 0 pdhdz, the contact force W asp ¼ 0 0 pasp dhdz, and the film pressure is determined by Eq. (1). A general formula of friction coefficient in water lubricated bearing is:
f total ¼ a1 f asp þ a2 f fluid
ð6Þ
a1 þ a2 ¼ 1; a1 ; a1 2 ½0; 1
Fig. 4. Lubrication groove of the bearing [31].
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where f asp is the friction coefficient due to micro-asperities contacts effect. f fluid is the friction coefficient due to viscous effect, i.e. the shearing stress of the fluid molecules. Fig. 5 shows the relationship between the friction coefficient and the micro-asperities contacts effect and the fluid viscous effect. In the low velocity region (V < V 1 , f fluid ! 0, f total f asp ), the micro-asperities contacts effect dominates the total friction coefficient; while in the high velocity region (V > V 2 , f asp ! 0, f total f fluid ), the fluid viscous effect dominates the total friction coefficient. For the actual bearing, we can obtain the Stribeck curve and determine the lubrication regimes transition point P1 and P2. Because for different lubrication regimes, the dynamic models for the bearing differs. Therefore, the lubrication regime significantly influences the dynamic characteristics of the whole bearing-rotor system. There actually exists turning points for the lubrication regimes transitions in the operating process. In the research process, lubrication regimes transition point from ML to HL is at the vicinity of the minimum friction coefficient point. It has been verified by references [28,29]. 3. Experiment introduction 3.1. Experimental equipment Fig. 6 gives physical map of the test rig. It is a multi-function bearing rotor system test equipment. It can test the water lubricated bearing, rolling element bearing as well as the thrust bearing according to the experimental design. Its main functions are to measure the lubrication characteristics and friction properties for such bearings. Range of the velocity is 0.01– 10 m/s, range of the external load is 0–1000 N. Fig. 7(a) shows the schematic line drawing of the test apparatus, it presents more detailed information about the whole system. The shaft is supported by two hydrostatic bearings between the coupling and the tested bearing. These two hydrostatic bearings stiffen the rotor system, decrease the deformation of the shaft when vertical load is applied on the tested bearing. They decrease the influence of overhung load and improve the accuracy of the system. For the deflection of the shaft, the structural analysis has been performed. For the external load 204 N, 320 N and 553 N, the maximum deflection of the shaft is 0.22 lm, 0.35 lm, 0.48 lm, respectively. Even for the maximum external load, the maximum deflection of the shaft is only 0.98 lm. It is small compared to the bearing clearance, which is several tens of microns. Therefore, influences of deformation of the shaft on the bearing lubrication regimes transition can be neglected. Fig. 7(b) presents the side view of the tested bearing unit. The water tank contains the lubricant water, and the tested bearing is fully submerged into the water to ensure sufficient lubrication. Vertical external loads are exerted on the bearing through the leading bar. Four displacement sensors are mounted on the tested bearing along the circumferential direction. The tested bearing is subjected to external load through the vertical load unit. Fig. 8(a) shows the physical map of the vertical external load unit of the whole system. Subparts have been marked clearly in the figure. There are two force sensors and four displacements sensors. External load is measured by force sensor 1 # in the vertical direction; tangential force is measured by force sensor 2 # in the circumferential direction. Four displacement sensors are distributed along the circumferential direction. Shaft vibration displacement and film thickness are measured by the four displacement sensors. The tested bearing and the shaft are fully submerged in water. In order to show this part clearly, Fig. 8(b) presents the enlarged physical map of the vertical external load unit.
Fig. 5. Relationship between the friction coefficient and the lubrication regimes transition.
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Fig. 6. Physical map of whole bearing-rotor test rig system.
Fig. 7. Schematic line drawing of the test rig. 1-coupling; 2-shaft; 3-hydrostatic bearing; 4-tested bearing; 5-sleeve; 6-leading bar; 7-force sensor; 8displacement sensors; 9-water tank; 10-water; 11-base.
Fig. 9 is presented to illustrate how to apply vertical load on the bearing. Fig. 9(a) is the test bearing, Fig. 9(b) and (c) are the vertical load unit. The purpose of component 1 is to embed the bearing into the rolling element bearing. The functions of rolling element bearing are: 1) applying radial external load to the bearing; 2) helping to measure the tangential force. The force arm is fixed on the bearing surface. Therefore, the bearing would not rotate with the rotating journal. The purpose of component 2 (the force arm) is to measure the tangential force to calculate the friction torque; purpose of component 3 is to install the leading bar (as shown in Fig. 7) and apply vertical load on the bearing. For the exact positioning of the bearing, the method is as follows. In the experiment, four displacement sensors are exploited (As can be seen in Fig. 10(a) and (b)). The accuracy is 0.1 lm. The nonlinear error is less than ± 0.1%, the response frequency is 10 kHz.
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Fig. 8. Enlarged physical map of the vertical load part.
Fig. 9. Physical map of the vertical load part.
For example, the exact value of radial clearance is C 0 ¼ 50lm. Before the experiment, we assume the initial bearing clearance is C. With the initial parameters, for a certain operating conditions, the film thickness distribution h and the pressure distribution p can be calculated theoretically. And 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 jd10 d1 j 6 0:5lm; jd20 d2 j 6 0:5lm; jd30 d3 j 6 0:5lm; jd40 d4 j 6 0:5lm, the radial clearance of the bearing is C. Else, modify the clearance, and restart the loop until it is convergent. The convergent criteria are 0.5 lm. Once the positioning of the bearing and shaft are determined, the eccentricity and attitude angle of the bearing can also be obtained. During the test, parameters need to measure includes: external load measured by the force sensor 1 #, tangential force T f measured by the force sensor 2 #. The friction torque equals to the tangential force T f multiplies the force arm R. Rotating
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Fig. 10. Layout of measuring points for the tested bearing.
speed N can be directly read in the controlling cabinet. Structural parameters of the bearing such as the diameter D, the length L and gravity G can be found in Table 1. F is the total vertical load, it includes the bearing gravity G, so the force applied on the bearing F r equals to:
Fr ¼ F G
ð7Þ
Therefore, the friction coefficient:
f ¼
2 Tf R Fr D
ð8Þ
Here, R ¼ 205mm; D ¼ 62mm; G ¼ 15kg, then we can get:
f ¼
205T f 31F r
ð9Þ
In the experiment, at first, the data acquisition software must be balanced. It is to guarantee the values equal to 0 when the shaft is on the equilibrium position. For every operating condition, friction is recorded for 3 min. The friction values fluctuate periodically with time. For more detailed information about the test rig, please refer to the references [27,28]. 3.2. Test bearing Four different bearings are manufactured for the experiments. Structural parameters of the bearings can be found in Table 1. Dry friction coefficient between the steel and the bushing material is 0.13. In order to verify the influences of surface roughness on the lubrication regimes transition, the combined surface roughness parameters are different. Surface roughness are 0.6 lm, 1.2 lm, 1.8 lm and 2.4 lm, respectively. Machining accuracy can meet the experimental requirements and engineering criteria. In order to investigate the influences of surface topography on the lubrication regimes transition, four samples with different surface roughness are manufactured to do the friction and wear test. The friction and wear test are performed on the latest universal tribometer manufactured. It can measure the friction and wear properties, as well as the 3D surface topography online during the experiment at the same time. The samples to be measured can be moved automatically in the friction wear/mechanical detection area, and can also move automatically in the surface topography analysis area. The friction, wear rate, wear volume, surface roughness as well as the 3D morphology can then be measured (See Fig. 11). As most of the machined surfaces belongs to Gaussian surface roughness, in the experiment, four test bearings (with the same surface roughness type) are processed for the test. Strictly speaking, it cannot represent all the roughness types. However, the paper emphasis on the influences of characteristic parameters of surface roughness for a certain roughness type on
Table 1 Basic parameters of water lubricated bearing. Description
Symbol
Value
Dimension
Bearing width Bearing diameter L/D ratio Radial clearance Relative clearance
L D L/D c 2c/D
80 62 1.30 0.03, 0.05, 0.07 0.96‰, 1.61‰, 2.25‰
mm mm – mm –
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Fig. 11. 3D model and physical map of the tested bearing.
the lubrication regimes transition. For a certain roughness type, experimental results of four test bearing samples is enough. We have examined the surface roughness after the repeating start and stop in the test. For the four tested bearings, characteristic parameters (Sa, Sq, Vmp and Vvv) of the surface topography decreases vividly, especially, the Vmp (Peak material volume), which means that the part will be worn out in the test, decreases sharply during the experiments.
4. Results and discussion 4.1. Measurement of the surface topography Fig. 12(a)–(d) show the 2D morphology distribution of the four tested samples. Table 2 gives the characteristic parameters of the surface topography for the four tested samples. Among them, Sa is the Arithmetic mean height; Sq is the Root mean square height; Vmp is the Peak material volume; Vvv is the Pit void volume.
Fig. 12. 2D morphology distribution of the four tested samples.
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Table 2 Characteristic parameters of measured surface topography. Before/After experiment
Unit
S1
S2
S3
S4
Sa
Before After
mm mm
1.439(±0.0288) 1.222(±0.0244)
2.046(±0.0409) 1.914(±0.0383)
2.107(±0.0421) 1.853(±0.0371)
3.134(±0.0627) 2.238(±0.0448)
Sq
Before After
mm mm
1.995(±0.0399) 1.668(±0.0334)
2.656(±0.0531) 2.216(±0.0452)
2.821(±0.0564) 2.029(±0.0406)
4.028(±0.0842) 2.870(±0.0574)
Vmp
Before After
mm3/mm2 mm3/mm2
0.183(±0.0037) 0.141(±0.0028)
0.197(±0.0039) 0.158(±0.0032)
0.230(±0.0046) 0.208(±0.0041)
0.323(±0.0065) 0.305(±0.0061)
Vvv
Before After
mm3/mm2 mm3/mm2
0.171(±0.0034) 0.165(±0.0033)
0.206(±0.0041) 0.162(±0.0032)
0.254(±0.0051 0.211(±0.0042)
0.259(±0.0052) 0.213(±0.0043)
One thing to note, for scientific validity, more than one sample extract per surface should be investigated. In the experiment, for one surface, dozens of samples have been extracted. For each sample, characteristic surface parameters are measured. There exist mathematic algorithms inside the Universal Profilemeter that can automatically deduce the characteristic surface parameters for the surface. From Table 2 we can see the variation of the real surface topography for the four tested samples. For sample 1 #, 2 #, 3 # and 4 #, the Arithmetic mean height and the Root mean square height increase correspondingly, and has small errors with the machine tolerance. Furthermore, in the start-up and shut-down stage of the bearing-rotor coupled system, the film is thin (less than 10 lm in most cases). In some extreme operating conditions, the film thickness is just a few microns. It is almost the identical number of magnitude with bushing interface topography (machining accuracy of composite material bushing >1.6 lm), film thickness ratio is very small. Micro-asperities contacts will take place under certain conditions. These will undoubtedly affect the micro interface fluid hydrodynamic performances as well as the lubrication regimes. Therefore, lubrication regimes transitions of the bearing will be experimentally verified.
4.2. Effect of operating conditions on friction curve The accuracy for force sensors can reach as high as 0.1 N, while 0.1 lm for displacement sensors. Sampling frequency is 1000 Hz for the displacement sensors. Before the data acquisition, the bearing should run in for serval minutes until the friction tends to be stable. During the experiment, the data acquisition records the friction coefficient data for about 180 s per working mode. The friction values fluctuate periodically with time with small amplitudes. In order to investigate the effects of operating conditions on the friction coefficient, Fig. 13 presents the friction coefficient curves with time under different external loads. Among them, the external load F1, F2, F3, F4, F5 increases in turn, F1 = 182 N, F2 = 380 N, F3 = 450 N, F4 = 500 N, F5 = 530 N. From the figure we can see that the friction values fluctuate with the time, and the friction increases with the external load. When the load is relatively small, the friction fluctuates severely. But when the load is large, the friction fluctuates mildly and tends to be stable more quickly. Fig. 14 shows the friction coefficient with time under different radial clearances, C1 = 0.07 mm, C2 = 0.05 mm. Under the same conditions, the larger the radial clearances, the bigger the friction coefficient. The friction decreases with the radial clearance.
Fig. 13. Effect of external loads on friction-time curves for the tested bearing.
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Fig. 14. Effect of radial clearances on friction-time curves for the tested bearing.
Fig. 15 gives the friction coefficient with time under different surface roughness, and Ra1 = 1.2 lm, Ra2 = 1.8 lm, Ra3 = 2.4 lm, respectively. Under the same conditions, the rougher the test surface, the bigger the friction coefficient. The friction increases with the surface roughness. 4.3. Effect of surface roughness on the lubrication regimes transition Fig. 16(a) presents comparison between the theoretical and experimental friction coefficients under different surface roughness. Among them, r1, r2 and r3 represent the combined surface roughness 2.15 lm, 1.79 lm and 1.28 lm, respectively. The test results show the same changing trend with the theoretical values. Test results agree rather well with the theoretical results for the surface roughness 1.79 lm in the whole velocity range, while deviation exists in the low velocity range (i.e. the boundary lubrication regime, BL) and the medium velocity range (i.e. the mixed lubrication regime, ML), and agree rather well in the high velocity range (i.e. the hydrodynamic lubrication regime, HL) for the surface roughness 2.15 lm and 1.28 lm. Fig. 16(b)–(d) gives the effects of different surface roughness on the lubrication regimes transition. The critical velocity that lubrication regime changes from mixed lubrication regime (ML) to hydrodynamic lubrication regime (HL) is defined as the lift-off speed. The friction coefficient corresponding to the lift-off speed is defined as the critical friction coefficient. Transition points from ML to HL are clearly marked in the figures. Lift-off speeds and the critical friction coefficients are listed in the table below. Table 3 gives the characteristic parameters of the lubrication regime transitions for different surface roughness. From Table 3 we can find that, for different surface roughness, the lift-off speed from ML to HL varies much. For surface roughness r1 = 2.15 lm, the theoretical and test lift-off speeds are about 2.1 m/s and 1.45 m/s, with the relative error 30.9%. The relative error is significantly large. However, at the same time, the critical friction coefficients for the theory calculation
Fig. 15. Effect of surface roughness on friction-time curves for the tested bearing.
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Fig. 16. Effect of different surface roughness on the lubrication regimes transition.
Table 3 Characteristic parameters of the lubrication regimes transition for different surface roughness. Item
Surface roughness
r1 = 2.15 lm
r2 = 1.79 lm
r3 = 1.28 lm
Lift-off speed/m/s
Theory Test Relative error
2.100 1.450 30.90%
0.900 0.820 8.80%
0.320 0.780 –
Critical friction coefficient
Theory Test Relative error
0.034 0.032 6.25%
0.028 0.028 0
0.026 0.026 3.70%
and test are 0.034 and 0.032, with the relative error 6.25%; For surface roughness r2 = 1.79 lm, the theoretical and test liftoff speeds are about 0.90 m/s and 0.82 m/s, with the relative error 8.8%. The critical friction coefficients for the theoretical calculation and test are the same, the values both are 0.028; For surface roughness r3 = 1.28 lm, the theoretical lift-off speed is about 0.32 m/s, while the test lift-off speed is more than two times bigger than that of the theoretical lift-off speed, with the value 0.78 m/s. The relative error is also very large. However, on the other side, the critical friction coefficients for the theoretical calculation and test are 0.026 and 0.027, with rather small relative error 3.70%. Furthermore, for the three different surface roughness, make horizontal contrast we can find that, significant changes exist for the lift-off speeds and the critical friction coefficients. For the theoretical lift-off speed, the value changes from 2.10 m/s to 0.90 m/s to 0.32 m/s while for the tested lift-off speed, the value changes from 1.45 m/s to 0.82 m/s to
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0.78 m/s. The theoretical critical friction coefficient changes from 0.034 to 0.028 to 0.026, while the test critical friction coefficient changes from 0.032 to 0.028 to 0.027. The circumstance illustrates that the lubrication regimes transition from ML to HL is seriously influenced by the surface roughness. From the analysis above, we can conclude that interface topography has significant influences on the lubrication regimes transition of water lubricated bearing. 4.4. Effect of external load on the lubrication regimes transition Fig. 17(a) shows the comparison between the theoretical and experimental friction coefficients under different external loads. Among them, F1, F2 and F3 represent the external loads 204 N, 320 N and 553 N, respectively. Fig. 17(b)–(d) gives the effects of different external loads on the lubrication regimes transition. Transition points from ML to HL are clearly marked in the figures. Lift-off speeds and the critical friction coefficients are listed in the table below. Table 4 gives the characteristic parameters of the lubrication regimes transition for different external loads. From Table 4 we can find that, for different external loads, the lift-off speed from ML to HL varies much. For external load F1 = 204 N, the theoretical and test lift-off speeds are about 0.88 m/s and 0.83 m/s, with the relative error 5.68%. At the same time, the critical friction coefficients for the theoretical calculation and test are 0.026 and 0.025, with the relative error 3.85%; For external load F1 = 320 N, the theoretical and test lift-off speeds are about 0.95 m/s and 0.84 m/s, with the relative error 11.57%. The relative error is significantly large. However, the critical friction coefficients for the theoretical calculation and test are 0.027 and 0.026, with the relative error 3.70%; For external load F1 = 320 N, the theoretical and test lift-off speeds are about 0.90 m/s and 0.82 m/s, with the relative error 8.88%. The relative error is also rather large. However, the
Fig. 17. Effect of different external loads on the lubrication regimes transition.
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Table 4 Characteristic parameters of the lubrication regimes transition for different external loads. Item
External load
F1 = 204 N
F2 = 320 N
F3 = 553 N
Lift-off speed/m/s
Theory Test Relative error
0.880 0.830 5.68%
0.950 0.840 11.57%
0.900 0.820 8.88%
Critical friction coefficient
Theory Test Relative error
0.026 0.025 3.85%
0.027 0.026 3.70%
0.025 0.025 0
critical friction coefficients for the theoretical calculation and test are the same, the values both are 0.025, with no relative error. Make comparison between Section 4.3 and Section 4.4, we can find that, as for friction coefficients and lift-off speeds, the agreement between theoretical results and tested results for the external load behaves better than the surface roughness. This is due to the mechanical load exploited in the experiment. The load is easy to control exactly. Therefore, influence of uncontrollable factors on the lubrication regime is relatively limited. Furthermore, for the three different external loads, make horizontal contrast we can find that, very limited changes for the lift-off speeds and the critical friction coefficients. For the theoretical lift-off speed, the value changes from 0.88 m/s to 0.95 m/s to 0.90 m/s while for the tested lift-off speed, the value changes from 0.83 m/s to 0.84 m/s to 0.82 m/s. The theoretical critical friction coefficient also changes from 0.026 to 0.027 to 0.025, while the tested critical friction coefficient changes from 0.025 to 0.026 to 0.025. It almost keeps constant, especially for the tested lift-off speed and tested critical friction coefficient. The circumstance illustrates that the lubrication regimes transition from ML to HL is slightly influenced by the external load. 4.5. Effect of radial clearance on the lubrication regimes transition Fig. 18(a) shows the comparison between the theoretical and experimental friction coefficients under different radial clearances. Among them, C1, C2 and C3 represent the different radial clearances 0.07 mm, 0.05 mm and 0.03 mm, respectively. Fig. 18(b)–(d) give the effects of different radial clearances on the lubrication regimes transition. Transition points from ML to HL are clearly marked in the figures. Lift-off speeds and the critical friction coefficients are listed in the table below. Table 5 gives the characteristic parameters of the lubrication regime transition for different radial clearances. From Table 5 we can find that, for different radial clearances, the lift-off speed from ML to HL varies much. For radial clearance C1 = 0.07 mm, the theoretical and test lift-off speeds are about 2.100 m/s and 1.400 m/s, with the relative error 33.33%. At the same time, the critical friction coefficients for the theoretical calculation and test results are 0.018 and 0.028, with the relative error 55.50%. For them, the relative errors both are large; For radial clearance C2 = 0.05 mm, the theoretical and test lift-off speeds are the same, with the value 0.800 m/s and no relative error. The critical friction coefficients for the theoretical calculation and test results are 0.020 and 0.024, with the relative error 20.00%; For radial clearance C3 = 0.03 mm, the theoretical and test lift-off speeds are 0.270 m/s and 0.400 m/s, with the relative error 48.1%. The critical friction coefficients for the theoretical calculation and test results are 0.026 and 0.034, with the relative error 30.76%. From the analysis we can find that, for lubrication regimes transition, the two characteristic parameters lift-off speed and critical friction coefficient do not show a certain direct relationship. For instance, for surface roughness r2 = 1.79 lm, the theoretical and test lift-off speeds are 0.90 m/s and 0.82 m/s, with the relative error 8.8%. The critical friction coefficients for the theoretical calculation and test are the same; for the case radial clearance C2 = 0.05 mm, the theoretical and test lift-off speeds are the same, but the critical friction coefficients for the theoretical calculation and test results are 0.020 and 0.024. Relative error reaches to 20.00%. Furthermore, for the three different radial clearances, make horizontal contrast we can find that, significant changes for the lift-off speeds and the critical friction coefficients. For the theoretical lift-off speed, the value changes from 2.10 m/s to 0.80 m/s to 0.27 m/s while for the tested lift-off speed, the value changes from 1.4 m/s to 0.80 m/s to 0.40 m/s. The theoretical critical friction coefficient also changes from 0.018 to 0.020 to 0.026, while the tested critical friction coefficient changes from 0.028 to 0.024 to 0.034. The circumstance illustrates that the lubrication regimes transition from ML to HL is seriously influenced by the radial clearance. 4.6. Effect of acceleration and deceleration on lubrication regimes transition In the experiment, even for the same tested bearing, the same speed and the external load, the acceleration and deceleration of the shaft have influences on the lubrication regimes transition. This differs from the theoretical analysis. Consequently, the effects of acceleration and deceleration process on the lubrication regimes transition are further investigated. For each acceleration and deceleration process, it needs 5 min.
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Fig. 18. Effect of different radial clearances on the lubrication regimes transition.
Table 5 Characteristic parameters of the lubrication regimes transition for different radial clearances. Item
Radial clearance
C1 = 0.07 mm
C2 = 0.05 mm
C3 = 0.03 mm
Lift-off speed/m/s
Theory Test Relative error
2.10 1.40 33.33%
0.800 0.800 0
0.270 0.400 48.10%
Critical friction coefficient
Theory Test Relative error
0.018 0.028 55.50%
0.020 0.024 20%
0.026 0.034 30.76%
Fig. 19 shows the effects of acceleration and deceleration process on the lubrication regimes transition under two different external loads, and F1, F2 are 550 N, 350 N, respectively. The radial clearance is 0.07 mm. From the whole changing trend of the figure, the friction coefficients show the same changing rule for acceleration and deceleration process of the shaft, but the specific values are different for different external loads. At the same time, Fig. 19 marks the transition points of lubrication regime, the lift-off speeds and the critical friction coefficients. Table 6 gives the characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different external loads. For different external loads, the lift-off speed varies much. For example, for external load F1 = 550 N, the lift-off speeds are both 1.10 m/s, the relative error 0. The critical friction coefficients are 0.049 and 0.060, respectively, the relative error reaches 18.33%. For external load F2 = 350 N, the lift-off speeds are both 0.72 m/s, the relative error is 0. The
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Fig. 19. Effect of acceleration and deceleration on the lubrication regimes transition under different external loads.
Table 6 Characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different external loads. Item
Operation condition
F1 = 550 N
F2 = 350 N
Lift-off speed/m/s
Acceleration Deceleration Relative error
1.10 1.10 0
0.72 0.72 0
Critical friction coefficient
Acceleration Deceleration Relative error
0.049 0.060 18.33%
0.040 0.038 5.26%
critical friction coefficients are 0.040 and 0.038, respectively, the relative error reaches 5.26%. The circumstances illustrate that acceleration and deceleration process have little influences on the lift-off speeds but significant influences on the critical friction coefficients under different external loads. Further investigation indicates that when the external load is small, the lift-off speed is small, the whole curves of friction coefficient shift towards the left side. Therefore, the mixed lubrication regime and hydrodynamic lubrication regime region have been broadened, which illustrates that it has improved the interface lubrication regime. On the contrary, the heavyload conditions deteriorate the interface lubrication regime. Fig. 20 shows the effects of acceleration and deceleration process on the lubrication regimes transition under two different radial clearances, and C1, C2 are 0.05 mm, 0.07 mm, respectively. From the whole changing trend of the figure, the friction coefficients show the same changing rule for acceleration and deceleration process of the shaft, but the specific values are different for different radial clearances. At the same time, Fig. 20 marks the transition points of lubrication regimes, the lift-off speeds and the critical friction coefficients.
Fig. 20. Effect of acceleration and deceleration on the lubrication regimes transition under different radial clearance.
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Table 7 gives the characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different radial clearances. For different radial clearances, the lift-off speed varies. For example, for radial clearances C1 = 0.05 mm, the lift-off speeds are both 0.80 m/s, the relative error is 0. The critical friction coefficients are 0.030 and 0.034, respectively, the relative error reaches 11.76%. For radial clearances C2 = 0.07 mm, the lift-off speeds are both 1.41 m/s, the relative error is 0. The critical friction coefficients are 0.018 and 0.020, respectively, the relative error reaches 10.00%. The circumstances illustrate that the acceleration and deceleration process have little influences on the lift-off speeds but significant influences on the critical friction coefficients under different radial clearances. Fig. 21 shows the effects of acceleration and deceleration process on the lubrication regimes transition under different surface roughness. The combined surface roughness of bearing 1 # is Ra1 = 0.6 lm while Ra4 = 2.4 lm for bearing 4#. From the whole changing trend of the figure, the friction coefficients show the same changing rule for acceleration and deceleration process of the shaft, but the specific values are different for different surface roughness. At the same time, Fig. 21 marks the transition points of lubrication regimes, the lift-off speeds and the critical friction coefficients. Table 8 gives the characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different surface roughness. For different surface roughness, the lift-off speed varies much. For example, for surface roughness Ra1 = 0.6 lm, the lift-off speeds are both 0.81 m/s, with the relative error 0; The critical friction coefficients are 0.036 and 0.040, respectively, the relative error reaches 10.00%. For surface roughness Ra4 = 2.4 lm, the lift-off speeds are both 1.10 m/s, with the relative error 0. The critical friction coefficients are 0.049 and 0.060, respectively, the relative error reaches 18.33%. The circumstances illustrate that acceleration and deceleration process have little influences on the lift-off speeds but significant influences on the friction coefficients under different surface roughness.
Table 7 Characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different radial clearances. Item
Operation condition
C1 = 0.05 mm
C2 = 0.07 mm
Lift-off speed/m/s
Acceleration Deceleration Relative error
0.80 0.80 0
1.41 1.41 0
Critical friction coefficient
Acceleration Deceleration Relative error
0.030 0.034 11.76%
0.018 0.020 10.00%
Fig. 21. Effect of acceleration and deceleration on the lubrication regimes transition under different surface roughness.
Table 8 Characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different surface roughness. Item
Operation condition
Ra4 = 2.4 lm
Ra1 = 0.6 lm
Lift-off speed/m/s
Acceleration Deceleration Relative error
1.10 1.10 0
0.81 0.81 0
Critical friction coefficient
Acceleration Deceleration Relative error
0.049 0.060 18.33%
0.036 0.040 10.00%
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Therefore, from the analysis above, for different external loads, different radial clearances and different surface roughness, the acceleration and the deceleration of the shaft have certain influences on the lubrication regimes transition. It has little effects on the lift-off speeds while significant influences on the critical friction coefficients. 4.7. Explanation of the acceleration and deceleration One thing to note, in Figs. 19–21, the curves of friction coefficients do not coincide for the acceleration and deceleration process, especially in the medium velocity region. The reason that friction shows large differences in the medium velocity region for acceleration and deceleration process can be explained as follows: (1) velocity perturbation. As can be seen in Fig. 22, the red line is the friction-time curve, in the medium velocity region, the bearing is in mixed lubrication regime. For a certain time, the velocity is V 0 , the corresponding friction is f 0 . Exert a positive velocity perturbation þ DV on the shaft (i.e. the acceleration process), the hydrodynamic effect is enhanced, therefore it will undoubtedly cause the decreases of the friction. Conversely, exert a negative velocity perturbation DV (i.e. the deceleration process), the hydrodynamic effect is weakened, and will cause the increases of the friction; (2) The fluid inertia effect. From another view, we can find that in the medium velocity region, the velocity of the fluid is also V 0 (no slip boundary condition). If decreases the velocity, then the fluid velocity keeps the same due to the inertia effect. Then negative relative velocity will exist between the shaft and the velocity, it will weaken the hydrodynamic effect of the bearing and thus increase the friction. Conversely, it will decrease the friction. In the high velocity region, the bearing has already been in the full film hydrodynamic lubrication regime, the friction all come from the ‘‘viscous effect”, therefore, velocity perturbation will not change the friction. Therefore, the friction-time curve almost keeps the same for acceleration and deceleration process in high velocity region. Furthermore, other factors also contribute to the phenomenon, such as, the temperature rise effect of the bearing liner, the orientation of machine marks in the counter-face and the orientation of the surface patterns. For further investigation, more attention will be focused on this question. To sum up, the lubrication regimes transition from ML to HL is slightly affected by the external load, while significantly affected by the surface roughness and radial clearance. For water lubricated bearing, once the dimension is determined, the processing and the surface finish is completed. In fact, external load is the operating conditions, surface roughness and radial clearance are the structural parameters. It means that for the lubrication regimes transition, the influence of structural parameters is larger than operating conditions. 4.8. Error analysis The tested data agrees rather well with the theoretical calculation, but also exists some errors. The main errors come from the following four aspects: 1. System errors, including the measurement accuracy of the displacement sensors and the force sensors. System errors cannot be avoided in the process. At the same time, torque due to the friction of rolling element bearing is neglected, which also brings system errors to the test results. 2. Installation errors. During the actual installation of the bearing, it is easy to offset due to the small radial clearance. The initial installation clearance is different from the theoretical value and has impacts on the experiment. In addition, there is no intermediate bearing support for the shaft, only two hydrostatic bearings. The principle of the structure is similar to that of the cantilever beam, radial runout of the shaft is large under heavy load conditions, which is also out of consideration in the experiment.
Fig. 22. Effect of acceleration and deceleration on the Stribeck curve.
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3. The effects of turbulence in bearing are without consideration in the theoretical analysis model. In actual test, with the increase of the rotating speed, the amplitude of the radial runout of the shaft increases correspondingly. The fluid flow is no longer pure laminar flow, and turbulence exists in local zones. It is not considered in the theoretical model and may results in errors. 5. Conclusions The paper experimentally investigates the influences of surface topography and operating conditions on the lubrication regimes transition. Simulations are performed under the same settings and operating conditions. Calculated friction coefficients are compared with tested results. The calculated and experimental results agree well. Effects of surface roughness, external loads and radial clearances on the lubrication regimes transition are also further investigated. Characteristic parameters of the lubrication regimes transition are summarized. Reasonable explanations are given to illustrate the results as well as the relative errors. From the analysis, we can obtain the following conclusions: 1. On the whole trend, experimental data is consistent with theoretical results. Deviations exist under some operating conditions; 2. For lubrication regimes transition, the lift-off speed and critical friction coefficient do not show a certain direct relationship. The lubrication regimes transition from ML to HL is slightly influenced by the external load, while significantly influenced by the surface roughness and radial clearance; 3. For different external loads, different radial clearances and different surface roughness, the acceleration and the deceleration process of the shaft have influences on the lubrication regimes transition. It has little influences on the lift-off speed while has significant influences on the critical friction coefficient. This paper has certain guiding significance for further step investigation on lubrication regimes transition. These conclusions are useful for the structure design, analysis and optimization of such bearings. For the future research, accuracy of the test rig will be improved. The mathematical model also need to consider multi-coupled factors. Acknowledgements The authors thank the editors and anonymous reviewers for their comments and suggestions. This work is financially supported and funded by China Postdoctoral Science Foundation (No. 2019M650257) and the Fundamental Research Funds for the Central Universities (No. JB190411). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
H. Hirani, M. Verma, Tribological study of elastomeric bearings for marine propeller shaft system, Tribol. Int. 42 (2) (2009) 378–390. W. Huang, Y. Xu, Y. Zheng, The tribological performance of ti (c, n)-based cermet sliding against si 3 n 4 in water, Wear 270 (9) (2011) 682–687. J. Wang, F. Yan, Q. Xue, Tribological behavior of ptfe sliding against steel in sea water, Wear 267 (9) (2009) 1634–1641. N. Wang, Q. Meng, P. Wang, Experimental research on film pressure distribution of water-lubricated rubber bearing with multiaxial grooves, J. Fluids Eng. 135 (8) (2013) 084501. W. Litwin, Experimental research on water lubricated three layer sliding bearing with lubrication grooves in the upper part of the bush and its comparison with a rubber bearing, Tribol. Int. 82 (2015) 153–161. D. Cabrera, N. Woolley, D. Allanson, Film pressure distribution in water-lubricated rubber journal bearings, Proc Inst Mech Eng Part J: J Eng Tribol 219 (2) (2005) 125–132. B. Heberley, Advances in Hybrid Water-Lubricated Journal Bearings for Use in Ocean Vessels, Massachusetts Institute of Technology, 2013. B. Majumdar, R. Pai, D. Hargreaves, Analysis of water-lubricated journal bearings with multiple axial grooves, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 218 (2) (2004) 135–146. X. Wang, K. Adachi, K. Otsuka, Optimization of the surface texture for silicon carbide sliding in water, ApSS 253 (3) (2006) 1282–1286. X. Zhang, Z. Yin, G. Gao, Determination of stiffness coefficients of hydrodynamic water-lubricated plain journal bearings, Tribol. Int. 85 (2014) 37–47. W. Litwin, Influence of surface roughness topography on properties of water-lubricated polymer bearings: experimental research, Tribol. Trans. 54 (3) (2011) 351–361. J. Wang, B. Chen, N. Liu, Combined effects of fiber/matrix interface and water absorption on the tribological behaviors of water-lubricated polytetrafluoroethylene-based composites reinforced with carbon and basalt fibers, Compos. A Appl. Sci. Manuf. 59 (2014) 85–92. B. Zhao, S. Zhang, J. Man, A modified normal contact stiffness model considering effect of surface topography, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. (2014), 1350650114558099. R. Pai, R. Pai, Stability of four-axial and six-axial grooved water-lubricated journal bearings under dynamic load, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 222 (5) (2008) 683–691. D. Zhu, Q. Wang, On the k ratio range of mixed lubrication, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 226 (12) (2012) 1010–1022. C. Yeo, R. Katta, J. Lee, Effect of asperity interactions on rough surface elastic contact behavior: hard film on soft substrate, Tribol. Int. 43 (8) (2010) 1438–1448. N. Patir, H. Cheng, Application of average flow model to lubrication between rough sliding surfaces, J. Tribol.-Trans. ASME 101 (2) (1979) 220–229. N. Patir, H.S. Cheng, An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication, J. Tribol.Trans. ASME 100 (1) (1978) 12–17. F. Sahlin, R. Larsson, P. Marklund, A mixed lubrication model incorporating measured surface topography. Part 2: roughness treatment, model validation, and simulation, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 224 (4) (2010) 353–365. F. Sahlin, R. Larsson, A. Almqvist, A mixed lubrication model incorporating measured surface topography. Part 1: theory of flow factors, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 224 (4) (2010) 335–351. J. Lin, C.H. Hsu, C. Lai, Surface roughness effects on the oscillating squeeze-film behavior of long partial journal bearings, Comput. Struct. 80 (3) (2002) 297–303.
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[22] T.C. Hsu, J.H. Chen, H.L. Chiang, Lubrication performance of short journal bearings considering the effects of surface roughness and magnetic field, Tribol. Int. 61 (2013) 169–175. [23] T. He, D. Zou, X. Lu, Mixed-lubrication analysis of marine stern tube bearing considering bending deformation of stern shaft and cavitation, Tribol. Int. 73 (2014) 108–116. [24] B. Lorentz, A. Albers, A numerical model for mixed lubrication taking into account surface topography, tangential adhesion effects and plastic deformations, Tribol. Int. 59 (2013) 259–266. [25] Y.Z. Hu, D. Zhu, A full numerical solution to the mixed lubrication in point contacts, J. Tribol.-Trans. ASME 122 (1) (2000) 1–9. [26] D. Zhu, Y.Z. Hu, A computer program package for the prediction of ehl and mixed lubrication characteristics, friction, subsurface stresses and flash temperatures based on measured 3-d surface roughness, Tribol. Trans. 44 (3) (2001) 383–390. [27] Z.L. Xie, Z.S. Rao, L. Liu, Theoretical and experimental research on the friction coefficient of water lubricated bearing with consideration of wall slip effects, Mech. Ind. 17 (1) (2016) 106–119. [28] Z.L. Xie, Z.S. Rao, N. Ta, Investigations on transitions of lubrication states for water lubricated bearing. Part I: determination of friction coefficients and film thickness ratios, Ind. Lubr. Tribol. 68 (3) (2016) 404–415. [29] Z.L. Xie, Z.S. Rao, N. Ta, Investigations on transitions of lubrication states for water lubricated bearing. Part II: Further insight into the film thickness ratio lambda, Ind. Lubr. Tribol. 68 (3) (2016) 416–429. [30] Z.L. Xie, Ta. Na, Z.S. Rao, The lubrication performance of water lubricated bearing with consideration of wall slip and inertial force, J. Hydrodyn. Ser. B 29 (1) (2017) 52–60. [31] Z.L. Xie, Z.S. Rao, H.L. Liu, Effect of surface topography on the lubrication performance of water lubricated bearing: theoretical and experimental study, Coatings 9 (1) (2019) 23. [32] A. Kraker, R. Ostayen, D. Rixen, Calculation of stribeck curves for (water) lubricated journal bearings, Tribol. Int. 40 (3) (2007) 459–469. [33] P. Isaksson, D. Nilsson, R. Larsson, The influence of surface roughness on friction in a flexible hybrid bearing, Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 225 (10) (2011) 975–985. [34] Z.L. Xie, Z.S. Rao, T. Na, Investigation on the lubrication regimes and dynamic characteristics of hydro-hybrid bearing of two-circuit main loop liquid sodium pump system, Ann. Nucl. Energy 115 (2018) 220–232. [35] Y. Jiang, W. Huang, J. Luo, et al, An improved dynamic model of defective bearings considering the three-dimensional geometric relationship between the rolling element and defect area, MSSP 129 (2019) 694–716. [36] J. Lv, J. Yu, Average combination difference morphological filters for fault feature extraction of bearing, MSSP 100 (2018) 827–845.
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Experimental research on the interface lubrication regimes transition of water lubricated bearing Zhongliang Xie, Huanling Liu ⇑ School of Electro-Mechanical Engineering, Xidian University, No. 2 Taibai South Road, Xi’an 710071, China
a r t i c l e
i n f o
Article history: Received 8 March 2019 Received in revised form 14 July 2019 Accepted 10 November 2019
Keywords: Water lubricated bearing Surface topography effects Lubrication regimes transition Lift-off speed Friction coefficient Tribological characteristics
a b s t r a c t This paper aims to experimentally investigate the influences of surface topography and operating conditions on lubrication regimes transition of water lubricated bearing in submarine. Simulations are also performed under the same settings and operating conditions. The calculated and experimental results agree well. Effects of surface roughness, external loads, radial clearances, the acceleration and deceleration process on the lubrication regimes transition are further investigated. Characteristic parameters of the lubrication regimes transition are summarized. Reasonable explanations are given to illustrate the results as well as the relative errors. It has certain guiding significance for further step investigation on influences of surface topography on lubrication regimes transition of such bearings. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The acoustic stealth performance is an important indicator to evaluate the operational capability of underwater submarines. Reducing the radiated noise and the harmonic target strength can not only improve the acoustic stealth of naval underwater submarine, but also can effectively improve the capability of the first enemy attack. With the improvement of the equipment’s low-noise performance and the hydrodynamic performance of the underwater submarines, the effects of the stern structure coupled vibration radiated noise on the acoustic stealth performance have become increasingly prominent. One of the main sources of the coupled vibration is the water lubricated plain journal stern tube bearing, which plays crucial roles in the ship’s propulsion system (as can be seen in Fig. 1). The mechanical properties of the bearing have significant influences on the dynamic response of the rotor system and the acoustic stealth performances. Recent years have witnessed the broad application of water lubricated bearings, with their unique advantages over the conventional bearings. Problems occur in the practical application and have drawn the research interests of many researchers. The former scholars have done researches in the lubrication mechanism [1–3], lubrication performances [4–6], optimum design [7–9] as well as the dynamic characteristics [10] of the bearings. Researchers [11–15] have also paid attentions to the modeling and simulating roughness effects on lubrication regimes. Among them, only several literatures [16–21] focus on the lubrication regimes transition. For instances, classical Reynolds equation was modified by Patir in 1978 [17,18] to accommodate the case of fluid flow associated with rough surfaces. Sahlin [19,20] and Hsu [21,22] investigate the lubrication performances of mixed lubrication with consideration of measured surface topography with modified mixed lubrication model. ⇑ Corresponding author. E-mail address: [email protected] (H. Liu). https://doi.org/10.1016/j.ymssp.2019.106522 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.
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Nomenclature c Rb ; Rj e r1 ; r2 L; D p hmin dh h h !0 P
x q l G w Ob ; Oj h
e
d1 ; d2
r k dgroov e E /s /c /x ; /z hs ; hf T Tf
radial clearance = Rb Rj bearing and journal radii eccentricity RMS surface roughness of two surfaces width and diameter of bearing hydrodynamic pressure the minimum film thickness macroscopic elastic deformation of bushing real film thickness nominal film thickness total external load angular velocity = 2pN lubricant density lubricant viscosity bearing gravity clearance ratio bearing, journal center angular coordinate eccentricity ratio, e=c roughness height of two surfaces combined surface roughness film thickness ratio fluid depth in the groove combined elastic modulus shear flow factor contact factor pressure flow factors the start angle and the end angle of the groove the bushing thickness tangential force
He [23] investigates the marine stern tube bearing with consideration of shaft deformation and cavitation, as well as the lubrication regimes transition from mixed lubrication (ML) to hydrodynamic lubrication (HL) transition. The modified model incorporates surface topography effects and covers all the three lubrication regimes. He also calculates the friction coefficient and establishes the relationship between friction and lubrication regimes transition. Critical speed that transition from mixed lubrication to hydrodynamic lubrication are explored. Lorentz [24] studies the influences of surface roughness on lubrication performances and lubrication regimes. Hu [25] and Zhu [26] develop deterministic method with consideration of surface topography effects to simulate the three lubrication regimes, the boundary lubrication regime (BL), the mixed lubrication regime (ML) and the hydrodynamic regime (HL). They also study the lubrication regimes transition with experimental results under different operating conditions. Lubrication regimes transition has significant effects on the operating stability and lifetime of the bearing-rotor coupled system. For the actual bearing and rotor surface, undoubtedly, lubrication regime on the interface are extremely complicated
Fig. 1. Schematic diagram of submarine stern structure with water lubricated bearing [1].
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and always in dynamic fluctuation states under different operating conditions. Lubrication regimes transition occurs in the process. Xie [27–31] determines the calculation of friction coefficient through theoretical method, experimentally verifies the simulation results. Stribeck curve is an important tool to analyze the tribological characteristic. Based on this, he has made attempts to establish the relationship between the friction coefficient and the lubrication regimes transition (BL to ML, ML to HL). Kraker [32] simulates the friction coefficient and surface-contact pressure through mixed-EHL lubrication model. He also investigates the lubrication regimes transition. Isaksson [33] studies the effects of surface threedimensional topography on the tribological characteristics with three different lubrication regime (BL, ML and HL). However, researches on the effects of surface topography and the lubrication regimes transition are relatively insufficient. Little literature is available. Only a few of them have addressed the importance of effects of surface topography [34–36] on lubrication regimes transition for such bearings. Fig. 2(a) and (b) show schematic diagram for practical water lubricated bearings with surface topography effects. From Fig. 2 we can see, surface topography effects are important factors for the lubrication regime analysis, especially in the thin film thickness region. This paper experimentally investigates the influences of surface topography and operating conditions on the lubrication regimes transition. In order to observe the micro interface topography vividly, surface roughness parameters are measured with high-resolution electron microscope. A set of specially designed experimental devices for the bearing are exploited. Simulations are also done under the same settings and operating conditions. Calculated friction coefficients are compared with tested results. The calculated and experimental results agree rather well. Effects of surface roughness, external loads and radial clearances on the lubrication regimes transition are further investigated. Characteristic parameters (lift-off speed & the critical friction coefficient) of the lubrication regimes transition are summarized. Reasonable explanations are given to illustrate the results as well as the relative error. The symbols are summarized in the Nomenclature section and discussed in details in the following section. 2. Theoretical model Fig. 3(a) presents the film thickness in rough surfaces contact, Fig. 3(b) presents the enlarged drawings of the film thickness in the load carrying area. For water lubricated bearing, the modified Reynolds equation [28,29]:
! ! @ qh3 @p @ qh3 @p @ ð/c qhT Þ @ ðq/s Þ @ ð/c qhT Þ þ ¼ 6U /x /z þ 6U rs þ 12 @x @z @x @x @t l @x l @z
ð1Þ
The modified fluid film thickness is obtained through:
h ¼ h0 þ dh þ d1 þ d2
Fig. 2. Schematic diagram for practical water lubricated bearings with surface topography [23].
ð2Þ
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Fig. 3. Film thickness in rough surfaces contact (enlarged drawings of film thickness).
dh is the macroscopic elastic deformation of the bushing. Macroscopic elastic deformation due to the normal hydrodynamic effect is calculated by Boussinesq formula:
dh ¼
2 pE0
ZZ X
pðn; fÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dndf ðx nÞ2 þ ðy fÞ2
ð3Þ
If lubrication groove is considered (as shown in Fig. 4), then the film thickness should be modified:
h¼
h0 þ dh þ d1 þ d2 ; h0 þ dh þ d1 þ d2 þ dgroov e ;
0 6 h 6 hs ; hf 6 h 6 2p hs < hf < h
ð4Þ
where dgroov e is the fluid depth in the groove. dgroov e ¼ d0 C ð1 þ cosðh p=2ÞÞ.
! The equilibrium equation for the hydrodynamic force, contact force and external load P are calculated by:
! ! ! P þ F fluid þ W asp ¼ 0
ð5Þ
R 1 R 2p R 1 R 2p ! ! where the hydrodynamic force F fluid ¼ 0 0 pdhdz, the contact force W asp ¼ 0 0 pasp dhdz, and the film pressure is determined by Eq. (1). A general formula of friction coefficient in water lubricated bearing is:
f total ¼ a1 f asp þ a2 f fluid
ð6Þ
a1 þ a2 ¼ 1; a1 ; a1 2 ½0; 1
Fig. 4. Lubrication groove of the bearing [31].
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where f asp is the friction coefficient due to micro-asperities contacts effect. f fluid is the friction coefficient due to viscous effect, i.e. the shearing stress of the fluid molecules. Fig. 5 shows the relationship between the friction coefficient and the micro-asperities contacts effect and the fluid viscous effect. In the low velocity region (V < V 1 , f fluid ! 0, f total f asp ), the micro-asperities contacts effect dominates the total friction coefficient; while in the high velocity region (V > V 2 , f asp ! 0, f total f fluid ), the fluid viscous effect dominates the total friction coefficient. For the actual bearing, we can obtain the Stribeck curve and determine the lubrication regimes transition point P1 and P2. Because for different lubrication regimes, the dynamic models for the bearing differs. Therefore, the lubrication regime significantly influences the dynamic characteristics of the whole bearing-rotor system. There actually exists turning points for the lubrication regimes transitions in the operating process. In the research process, lubrication regimes transition point from ML to HL is at the vicinity of the minimum friction coefficient point. It has been verified by references [28,29]. 3. Experiment introduction 3.1. Experimental equipment Fig. 6 gives physical map of the test rig. It is a multi-function bearing rotor system test equipment. It can test the water lubricated bearing, rolling element bearing as well as the thrust bearing according to the experimental design. Its main functions are to measure the lubrication characteristics and friction properties for such bearings. Range of the velocity is 0.01– 10 m/s, range of the external load is 0–1000 N. Fig. 7(a) shows the schematic line drawing of the test apparatus, it presents more detailed information about the whole system. The shaft is supported by two hydrostatic bearings between the coupling and the tested bearing. These two hydrostatic bearings stiffen the rotor system, decrease the deformation of the shaft when vertical load is applied on the tested bearing. They decrease the influence of overhung load and improve the accuracy of the system. For the deflection of the shaft, the structural analysis has been performed. For the external load 204 N, 320 N and 553 N, the maximum deflection of the shaft is 0.22 lm, 0.35 lm, 0.48 lm, respectively. Even for the maximum external load, the maximum deflection of the shaft is only 0.98 lm. It is small compared to the bearing clearance, which is several tens of microns. Therefore, influences of deformation of the shaft on the bearing lubrication regimes transition can be neglected. Fig. 7(b) presents the side view of the tested bearing unit. The water tank contains the lubricant water, and the tested bearing is fully submerged into the water to ensure sufficient lubrication. Vertical external loads are exerted on the bearing through the leading bar. Four displacement sensors are mounted on the tested bearing along the circumferential direction. The tested bearing is subjected to external load through the vertical load unit. Fig. 8(a) shows the physical map of the vertical external load unit of the whole system. Subparts have been marked clearly in the figure. There are two force sensors and four displacements sensors. External load is measured by force sensor 1 # in the vertical direction; tangential force is measured by force sensor 2 # in the circumferential direction. Four displacement sensors are distributed along the circumferential direction. Shaft vibration displacement and film thickness are measured by the four displacement sensors. The tested bearing and the shaft are fully submerged in water. In order to show this part clearly, Fig. 8(b) presents the enlarged physical map of the vertical external load unit.
Fig. 5. Relationship between the friction coefficient and the lubrication regimes transition.
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Fig. 6. Physical map of whole bearing-rotor test rig system.
Fig. 7. Schematic line drawing of the test rig. 1-coupling; 2-shaft; 3-hydrostatic bearing; 4-tested bearing; 5-sleeve; 6-leading bar; 7-force sensor; 8displacement sensors; 9-water tank; 10-water; 11-base.
Fig. 9 is presented to illustrate how to apply vertical load on the bearing. Fig. 9(a) is the test bearing, Fig. 9(b) and (c) are the vertical load unit. The purpose of component 1 is to embed the bearing into the rolling element bearing. The functions of rolling element bearing are: 1) applying radial external load to the bearing; 2) helping to measure the tangential force. The force arm is fixed on the bearing surface. Therefore, the bearing would not rotate with the rotating journal. The purpose of component 2 (the force arm) is to measure the tangential force to calculate the friction torque; purpose of component 3 is to install the leading bar (as shown in Fig. 7) and apply vertical load on the bearing. For the exact positioning of the bearing, the method is as follows. In the experiment, four displacement sensors are exploited (As can be seen in Fig. 10(a) and (b)). The accuracy is 0.1 lm. The nonlinear error is less than ± 0.1%, the response frequency is 10 kHz.
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Fig. 8. Enlarged physical map of the vertical load part.
Fig. 9. Physical map of the vertical load part.
For example, the exact value of radial clearance is C 0 ¼ 50lm. Before the experiment, we assume the initial bearing clearance is C. With the initial parameters, for a certain operating conditions, the film thickness distribution h and the pressure distribution p can be calculated theoretically. And 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 jd10 d1 j 6 0:5lm; jd20 d2 j 6 0:5lm; jd30 d3 j 6 0:5lm; jd40 d4 j 6 0:5lm, the radial clearance of the bearing is C. Else, modify the clearance, and restart the loop until it is convergent. The convergent criteria are 0.5 lm. Once the positioning of the bearing and shaft are determined, the eccentricity and attitude angle of the bearing can also be obtained. During the test, parameters need to measure includes: external load measured by the force sensor 1 #, tangential force T f measured by the force sensor 2 #. The friction torque equals to the tangential force T f multiplies the force arm R. Rotating
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Fig. 10. Layout of measuring points for the tested bearing.
speed N can be directly read in the controlling cabinet. Structural parameters of the bearing such as the diameter D, the length L and gravity G can be found in Table 1. F is the total vertical load, it includes the bearing gravity G, so the force applied on the bearing F r equals to:
Fr ¼ F G
ð7Þ
Therefore, the friction coefficient:
f ¼
2 Tf R Fr D
ð8Þ
Here, R ¼ 205mm; D ¼ 62mm; G ¼ 15kg, then we can get:
f ¼
205T f 31F r
ð9Þ
In the experiment, at first, the data acquisition software must be balanced. It is to guarantee the values equal to 0 when the shaft is on the equilibrium position. For every operating condition, friction is recorded for 3 min. The friction values fluctuate periodically with time. For more detailed information about the test rig, please refer to the references [27,28]. 3.2. Test bearing Four different bearings are manufactured for the experiments. Structural parameters of the bearings can be found in Table 1. Dry friction coefficient between the steel and the bushing material is 0.13. In order to verify the influences of surface roughness on the lubrication regimes transition, the combined surface roughness parameters are different. Surface roughness are 0.6 lm, 1.2 lm, 1.8 lm and 2.4 lm, respectively. Machining accuracy can meet the experimental requirements and engineering criteria. In order to investigate the influences of surface topography on the lubrication regimes transition, four samples with different surface roughness are manufactured to do the friction and wear test. The friction and wear test are performed on the latest universal tribometer manufactured. It can measure the friction and wear properties, as well as the 3D surface topography online during the experiment at the same time. The samples to be measured can be moved automatically in the friction wear/mechanical detection area, and can also move automatically in the surface topography analysis area. The friction, wear rate, wear volume, surface roughness as well as the 3D morphology can then be measured (See Fig. 11). As most of the machined surfaces belongs to Gaussian surface roughness, in the experiment, four test bearings (with the same surface roughness type) are processed for the test. Strictly speaking, it cannot represent all the roughness types. However, the paper emphasis on the influences of characteristic parameters of surface roughness for a certain roughness type on
Table 1 Basic parameters of water lubricated bearing. Description
Symbol
Value
Dimension
Bearing width Bearing diameter L/D ratio Radial clearance Relative clearance
L D L/D c 2c/D
80 62 1.30 0.03, 0.05, 0.07 0.96‰, 1.61‰, 2.25‰
mm mm – mm –
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Fig. 11. 3D model and physical map of the tested bearing.
the lubrication regimes transition. For a certain roughness type, experimental results of four test bearing samples is enough. We have examined the surface roughness after the repeating start and stop in the test. For the four tested bearings, characteristic parameters (Sa, Sq, Vmp and Vvv) of the surface topography decreases vividly, especially, the Vmp (Peak material volume), which means that the part will be worn out in the test, decreases sharply during the experiments.
4. Results and discussion 4.1. Measurement of the surface topography Fig. 12(a)–(d) show the 2D morphology distribution of the four tested samples. Table 2 gives the characteristic parameters of the surface topography for the four tested samples. Among them, Sa is the Arithmetic mean height; Sq is the Root mean square height; Vmp is the Peak material volume; Vvv is the Pit void volume.
Fig. 12. 2D morphology distribution of the four tested samples.
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Table 2 Characteristic parameters of measured surface topography. Before/After experiment
Unit
S1
S2
S3
S4
Sa
Before After
mm mm
1.439(±0.0288) 1.222(±0.0244)
2.046(±0.0409) 1.914(±0.0383)
2.107(±0.0421) 1.853(±0.0371)
3.134(±0.0627) 2.238(±0.0448)
Sq
Before After
mm mm
1.995(±0.0399) 1.668(±0.0334)
2.656(±0.0531) 2.216(±0.0452)
2.821(±0.0564) 2.029(±0.0406)
4.028(±0.0842) 2.870(±0.0574)
Vmp
Before After
mm3/mm2 mm3/mm2
0.183(±0.0037) 0.141(±0.0028)
0.197(±0.0039) 0.158(±0.0032)
0.230(±0.0046) 0.208(±0.0041)
0.323(±0.0065) 0.305(±0.0061)
Vvv
Before After
mm3/mm2 mm3/mm2
0.171(±0.0034) 0.165(±0.0033)
0.206(±0.0041) 0.162(±0.0032)
0.254(±0.0051 0.211(±0.0042)
0.259(±0.0052) 0.213(±0.0043)
One thing to note, for scientific validity, more than one sample extract per surface should be investigated. In the experiment, for one surface, dozens of samples have been extracted. For each sample, characteristic surface parameters are measured. There exist mathematic algorithms inside the Universal Profilemeter that can automatically deduce the characteristic surface parameters for the surface. From Table 2 we can see the variation of the real surface topography for the four tested samples. For sample 1 #, 2 #, 3 # and 4 #, the Arithmetic mean height and the Root mean square height increase correspondingly, and has small errors with the machine tolerance. Furthermore, in the start-up and shut-down stage of the bearing-rotor coupled system, the film is thin (less than 10 lm in most cases). In some extreme operating conditions, the film thickness is just a few microns. It is almost the identical number of magnitude with bushing interface topography (machining accuracy of composite material bushing >1.6 lm), film thickness ratio is very small. Micro-asperities contacts will take place under certain conditions. These will undoubtedly affect the micro interface fluid hydrodynamic performances as well as the lubrication regimes. Therefore, lubrication regimes transitions of the bearing will be experimentally verified.
4.2. Effect of operating conditions on friction curve The accuracy for force sensors can reach as high as 0.1 N, while 0.1 lm for displacement sensors. Sampling frequency is 1000 Hz for the displacement sensors. Before the data acquisition, the bearing should run in for serval minutes until the friction tends to be stable. During the experiment, the data acquisition records the friction coefficient data for about 180 s per working mode. The friction values fluctuate periodically with time with small amplitudes. In order to investigate the effects of operating conditions on the friction coefficient, Fig. 13 presents the friction coefficient curves with time under different external loads. Among them, the external load F1, F2, F3, F4, F5 increases in turn, F1 = 182 N, F2 = 380 N, F3 = 450 N, F4 = 500 N, F5 = 530 N. From the figure we can see that the friction values fluctuate with the time, and the friction increases with the external load. When the load is relatively small, the friction fluctuates severely. But when the load is large, the friction fluctuates mildly and tends to be stable more quickly. Fig. 14 shows the friction coefficient with time under different radial clearances, C1 = 0.07 mm, C2 = 0.05 mm. Under the same conditions, the larger the radial clearances, the bigger the friction coefficient. The friction decreases with the radial clearance.
Fig. 13. Effect of external loads on friction-time curves for the tested bearing.
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Fig. 14. Effect of radial clearances on friction-time curves for the tested bearing.
Fig. 15 gives the friction coefficient with time under different surface roughness, and Ra1 = 1.2 lm, Ra2 = 1.8 lm, Ra3 = 2.4 lm, respectively. Under the same conditions, the rougher the test surface, the bigger the friction coefficient. The friction increases with the surface roughness. 4.3. Effect of surface roughness on the lubrication regimes transition Fig. 16(a) presents comparison between the theoretical and experimental friction coefficients under different surface roughness. Among them, r1, r2 and r3 represent the combined surface roughness 2.15 lm, 1.79 lm and 1.28 lm, respectively. The test results show the same changing trend with the theoretical values. Test results agree rather well with the theoretical results for the surface roughness 1.79 lm in the whole velocity range, while deviation exists in the low velocity range (i.e. the boundary lubrication regime, BL) and the medium velocity range (i.e. the mixed lubrication regime, ML), and agree rather well in the high velocity range (i.e. the hydrodynamic lubrication regime, HL) for the surface roughness 2.15 lm and 1.28 lm. Fig. 16(b)–(d) gives the effects of different surface roughness on the lubrication regimes transition. The critical velocity that lubrication regime changes from mixed lubrication regime (ML) to hydrodynamic lubrication regime (HL) is defined as the lift-off speed. The friction coefficient corresponding to the lift-off speed is defined as the critical friction coefficient. Transition points from ML to HL are clearly marked in the figures. Lift-off speeds and the critical friction coefficients are listed in the table below. Table 3 gives the characteristic parameters of the lubrication regime transitions for different surface roughness. From Table 3 we can find that, for different surface roughness, the lift-off speed from ML to HL varies much. For surface roughness r1 = 2.15 lm, the theoretical and test lift-off speeds are about 2.1 m/s and 1.45 m/s, with the relative error 30.9%. The relative error is significantly large. However, at the same time, the critical friction coefficients for the theory calculation
Fig. 15. Effect of surface roughness on friction-time curves for the tested bearing.
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Fig. 16. Effect of different surface roughness on the lubrication regimes transition.
Table 3 Characteristic parameters of the lubrication regimes transition for different surface roughness. Item
Surface roughness
r1 = 2.15 lm
r2 = 1.79 lm
r3 = 1.28 lm
Lift-off speed/m/s
Theory Test Relative error
2.100 1.450 30.90%
0.900 0.820 8.80%
0.320 0.780 –
Critical friction coefficient
Theory Test Relative error
0.034 0.032 6.25%
0.028 0.028 0
0.026 0.026 3.70%
and test are 0.034 and 0.032, with the relative error 6.25%; For surface roughness r2 = 1.79 lm, the theoretical and test liftoff speeds are about 0.90 m/s and 0.82 m/s, with the relative error 8.8%. The critical friction coefficients for the theoretical calculation and test are the same, the values both are 0.028; For surface roughness r3 = 1.28 lm, the theoretical lift-off speed is about 0.32 m/s, while the test lift-off speed is more than two times bigger than that of the theoretical lift-off speed, with the value 0.78 m/s. The relative error is also very large. However, on the other side, the critical friction coefficients for the theoretical calculation and test are 0.026 and 0.027, with rather small relative error 3.70%. Furthermore, for the three different surface roughness, make horizontal contrast we can find that, significant changes exist for the lift-off speeds and the critical friction coefficients. For the theoretical lift-off speed, the value changes from 2.10 m/s to 0.90 m/s to 0.32 m/s while for the tested lift-off speed, the value changes from 1.45 m/s to 0.82 m/s to
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0.78 m/s. The theoretical critical friction coefficient changes from 0.034 to 0.028 to 0.026, while the test critical friction coefficient changes from 0.032 to 0.028 to 0.027. The circumstance illustrates that the lubrication regimes transition from ML to HL is seriously influenced by the surface roughness. From the analysis above, we can conclude that interface topography has significant influences on the lubrication regimes transition of water lubricated bearing. 4.4. Effect of external load on the lubrication regimes transition Fig. 17(a) shows the comparison between the theoretical and experimental friction coefficients under different external loads. Among them, F1, F2 and F3 represent the external loads 204 N, 320 N and 553 N, respectively. Fig. 17(b)–(d) gives the effects of different external loads on the lubrication regimes transition. Transition points from ML to HL are clearly marked in the figures. Lift-off speeds and the critical friction coefficients are listed in the table below. Table 4 gives the characteristic parameters of the lubrication regimes transition for different external loads. From Table 4 we can find that, for different external loads, the lift-off speed from ML to HL varies much. For external load F1 = 204 N, the theoretical and test lift-off speeds are about 0.88 m/s and 0.83 m/s, with the relative error 5.68%. At the same time, the critical friction coefficients for the theoretical calculation and test are 0.026 and 0.025, with the relative error 3.85%; For external load F1 = 320 N, the theoretical and test lift-off speeds are about 0.95 m/s and 0.84 m/s, with the relative error 11.57%. The relative error is significantly large. However, the critical friction coefficients for the theoretical calculation and test are 0.027 and 0.026, with the relative error 3.70%; For external load F1 = 320 N, the theoretical and test lift-off speeds are about 0.90 m/s and 0.82 m/s, with the relative error 8.88%. The relative error is also rather large. However, the
Fig. 17. Effect of different external loads on the lubrication regimes transition.
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Table 4 Characteristic parameters of the lubrication regimes transition for different external loads. Item
External load
F1 = 204 N
F2 = 320 N
F3 = 553 N
Lift-off speed/m/s
Theory Test Relative error
0.880 0.830 5.68%
0.950 0.840 11.57%
0.900 0.820 8.88%
Critical friction coefficient
Theory Test Relative error
0.026 0.025 3.85%
0.027 0.026 3.70%
0.025 0.025 0
critical friction coefficients for the theoretical calculation and test are the same, the values both are 0.025, with no relative error. Make comparison between Section 4.3 and Section 4.4, we can find that, as for friction coefficients and lift-off speeds, the agreement between theoretical results and tested results for the external load behaves better than the surface roughness. This is due to the mechanical load exploited in the experiment. The load is easy to control exactly. Therefore, influence of uncontrollable factors on the lubrication regime is relatively limited. Furthermore, for the three different external loads, make horizontal contrast we can find that, very limited changes for the lift-off speeds and the critical friction coefficients. For the theoretical lift-off speed, the value changes from 0.88 m/s to 0.95 m/s to 0.90 m/s while for the tested lift-off speed, the value changes from 0.83 m/s to 0.84 m/s to 0.82 m/s. The theoretical critical friction coefficient also changes from 0.026 to 0.027 to 0.025, while the tested critical friction coefficient changes from 0.025 to 0.026 to 0.025. It almost keeps constant, especially for the tested lift-off speed and tested critical friction coefficient. The circumstance illustrates that the lubrication regimes transition from ML to HL is slightly influenced by the external load. 4.5. Effect of radial clearance on the lubrication regimes transition Fig. 18(a) shows the comparison between the theoretical and experimental friction coefficients under different radial clearances. Among them, C1, C2 and C3 represent the different radial clearances 0.07 mm, 0.05 mm and 0.03 mm, respectively. Fig. 18(b)–(d) give the effects of different radial clearances on the lubrication regimes transition. Transition points from ML to HL are clearly marked in the figures. Lift-off speeds and the critical friction coefficients are listed in the table below. Table 5 gives the characteristic parameters of the lubrication regime transition for different radial clearances. From Table 5 we can find that, for different radial clearances, the lift-off speed from ML to HL varies much. For radial clearance C1 = 0.07 mm, the theoretical and test lift-off speeds are about 2.100 m/s and 1.400 m/s, with the relative error 33.33%. At the same time, the critical friction coefficients for the theoretical calculation and test results are 0.018 and 0.028, with the relative error 55.50%. For them, the relative errors both are large; For radial clearance C2 = 0.05 mm, the theoretical and test lift-off speeds are the same, with the value 0.800 m/s and no relative error. The critical friction coefficients for the theoretical calculation and test results are 0.020 and 0.024, with the relative error 20.00%; For radial clearance C3 = 0.03 mm, the theoretical and test lift-off speeds are 0.270 m/s and 0.400 m/s, with the relative error 48.1%. The critical friction coefficients for the theoretical calculation and test results are 0.026 and 0.034, with the relative error 30.76%. From the analysis we can find that, for lubrication regimes transition, the two characteristic parameters lift-off speed and critical friction coefficient do not show a certain direct relationship. For instance, for surface roughness r2 = 1.79 lm, the theoretical and test lift-off speeds are 0.90 m/s and 0.82 m/s, with the relative error 8.8%. The critical friction coefficients for the theoretical calculation and test are the same; for the case radial clearance C2 = 0.05 mm, the theoretical and test lift-off speeds are the same, but the critical friction coefficients for the theoretical calculation and test results are 0.020 and 0.024. Relative error reaches to 20.00%. Furthermore, for the three different radial clearances, make horizontal contrast we can find that, significant changes for the lift-off speeds and the critical friction coefficients. For the theoretical lift-off speed, the value changes from 2.10 m/s to 0.80 m/s to 0.27 m/s while for the tested lift-off speed, the value changes from 1.4 m/s to 0.80 m/s to 0.40 m/s. The theoretical critical friction coefficient also changes from 0.018 to 0.020 to 0.026, while the tested critical friction coefficient changes from 0.028 to 0.024 to 0.034. The circumstance illustrates that the lubrication regimes transition from ML to HL is seriously influenced by the radial clearance. 4.6. Effect of acceleration and deceleration on lubrication regimes transition In the experiment, even for the same tested bearing, the same speed and the external load, the acceleration and deceleration of the shaft have influences on the lubrication regimes transition. This differs from the theoretical analysis. Consequently, the effects of acceleration and deceleration process on the lubrication regimes transition are further investigated. For each acceleration and deceleration process, it needs 5 min.
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Fig. 18. Effect of different radial clearances on the lubrication regimes transition.
Table 5 Characteristic parameters of the lubrication regimes transition for different radial clearances. Item
Radial clearance
C1 = 0.07 mm
C2 = 0.05 mm
C3 = 0.03 mm
Lift-off speed/m/s
Theory Test Relative error
2.10 1.40 33.33%
0.800 0.800 0
0.270 0.400 48.10%
Critical friction coefficient
Theory Test Relative error
0.018 0.028 55.50%
0.020 0.024 20%
0.026 0.034 30.76%
Fig. 19 shows the effects of acceleration and deceleration process on the lubrication regimes transition under two different external loads, and F1, F2 are 550 N, 350 N, respectively. The radial clearance is 0.07 mm. From the whole changing trend of the figure, the friction coefficients show the same changing rule for acceleration and deceleration process of the shaft, but the specific values are different for different external loads. At the same time, Fig. 19 marks the transition points of lubrication regime, the lift-off speeds and the critical friction coefficients. Table 6 gives the characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different external loads. For different external loads, the lift-off speed varies much. For example, for external load F1 = 550 N, the lift-off speeds are both 1.10 m/s, the relative error 0. The critical friction coefficients are 0.049 and 0.060, respectively, the relative error reaches 18.33%. For external load F2 = 350 N, the lift-off speeds are both 0.72 m/s, the relative error is 0. The
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Fig. 19. Effect of acceleration and deceleration on the lubrication regimes transition under different external loads.
Table 6 Characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different external loads. Item
Operation condition
F1 = 550 N
F2 = 350 N
Lift-off speed/m/s
Acceleration Deceleration Relative error
1.10 1.10 0
0.72 0.72 0
Critical friction coefficient
Acceleration Deceleration Relative error
0.049 0.060 18.33%
0.040 0.038 5.26%
critical friction coefficients are 0.040 and 0.038, respectively, the relative error reaches 5.26%. The circumstances illustrate that acceleration and deceleration process have little influences on the lift-off speeds but significant influences on the critical friction coefficients under different external loads. Further investigation indicates that when the external load is small, the lift-off speed is small, the whole curves of friction coefficient shift towards the left side. Therefore, the mixed lubrication regime and hydrodynamic lubrication regime region have been broadened, which illustrates that it has improved the interface lubrication regime. On the contrary, the heavyload conditions deteriorate the interface lubrication regime. Fig. 20 shows the effects of acceleration and deceleration process on the lubrication regimes transition under two different radial clearances, and C1, C2 are 0.05 mm, 0.07 mm, respectively. From the whole changing trend of the figure, the friction coefficients show the same changing rule for acceleration and deceleration process of the shaft, but the specific values are different for different radial clearances. At the same time, Fig. 20 marks the transition points of lubrication regimes, the lift-off speeds and the critical friction coefficients.
Fig. 20. Effect of acceleration and deceleration on the lubrication regimes transition under different radial clearance.
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Table 7 gives the characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different radial clearances. For different radial clearances, the lift-off speed varies. For example, for radial clearances C1 = 0.05 mm, the lift-off speeds are both 0.80 m/s, the relative error is 0. The critical friction coefficients are 0.030 and 0.034, respectively, the relative error reaches 11.76%. For radial clearances C2 = 0.07 mm, the lift-off speeds are both 1.41 m/s, the relative error is 0. The critical friction coefficients are 0.018 and 0.020, respectively, the relative error reaches 10.00%. The circumstances illustrate that the acceleration and deceleration process have little influences on the lift-off speeds but significant influences on the critical friction coefficients under different radial clearances. Fig. 21 shows the effects of acceleration and deceleration process on the lubrication regimes transition under different surface roughness. The combined surface roughness of bearing 1 # is Ra1 = 0.6 lm while Ra4 = 2.4 lm for bearing 4#. From the whole changing trend of the figure, the friction coefficients show the same changing rule for acceleration and deceleration process of the shaft, but the specific values are different for different surface roughness. At the same time, Fig. 21 marks the transition points of lubrication regimes, the lift-off speeds and the critical friction coefficients. Table 8 gives the characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different surface roughness. For different surface roughness, the lift-off speed varies much. For example, for surface roughness Ra1 = 0.6 lm, the lift-off speeds are both 0.81 m/s, with the relative error 0; The critical friction coefficients are 0.036 and 0.040, respectively, the relative error reaches 10.00%. For surface roughness Ra4 = 2.4 lm, the lift-off speeds are both 1.10 m/s, with the relative error 0. The critical friction coefficients are 0.049 and 0.060, respectively, the relative error reaches 18.33%. The circumstances illustrate that acceleration and deceleration process have little influences on the lift-off speeds but significant influences on the friction coefficients under different surface roughness.
Table 7 Characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different radial clearances. Item
Operation condition
C1 = 0.05 mm
C2 = 0.07 mm
Lift-off speed/m/s
Acceleration Deceleration Relative error
0.80 0.80 0
1.41 1.41 0
Critical friction coefficient
Acceleration Deceleration Relative error
0.030 0.034 11.76%
0.018 0.020 10.00%
Fig. 21. Effect of acceleration and deceleration on the lubrication regimes transition under different surface roughness.
Table 8 Characteristic parameters of the lubrication regimes transition for acceleration and deceleration under different surface roughness. Item
Operation condition
Ra4 = 2.4 lm
Ra1 = 0.6 lm
Lift-off speed/m/s
Acceleration Deceleration Relative error
1.10 1.10 0
0.81 0.81 0
Critical friction coefficient
Acceleration Deceleration Relative error
0.049 0.060 18.33%
0.036 0.040 10.00%
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Therefore, from the analysis above, for different external loads, different radial clearances and different surface roughness, the acceleration and the deceleration of the shaft have certain influences on the lubrication regimes transition. It has little effects on the lift-off speeds while significant influences on the critical friction coefficients. 4.7. Explanation of the acceleration and deceleration One thing to note, in Figs. 19–21, the curves of friction coefficients do not coincide for the acceleration and deceleration process, especially in the medium velocity region. The reason that friction shows large differences in the medium velocity region for acceleration and deceleration process can be explained as follows: (1) velocity perturbation. As can be seen in Fig. 22, the red line is the friction-time curve, in the medium velocity region, the bearing is in mixed lubrication regime. For a certain time, the velocity is V 0 , the corresponding friction is f 0 . Exert a positive velocity perturbation þ DV on the shaft (i.e. the acceleration process), the hydrodynamic effect is enhanced, therefore it will undoubtedly cause the decreases of the friction. Conversely, exert a negative velocity perturbation DV (i.e. the deceleration process), the hydrodynamic effect is weakened, and will cause the increases of the friction; (2) The fluid inertia effect. From another view, we can find that in the medium velocity region, the velocity of the fluid is also V 0 (no slip boundary condition). If decreases the velocity, then the fluid velocity keeps the same due to the inertia effect. Then negative relative velocity will exist between the shaft and the velocity, it will weaken the hydrodynamic effect of the bearing and thus increase the friction. Conversely, it will decrease the friction. In the high velocity region, the bearing has already been in the full film hydrodynamic lubrication regime, the friction all come from the ‘‘viscous effect”, therefore, velocity perturbation will not change the friction. Therefore, the friction-time curve almost keeps the same for acceleration and deceleration process in high velocity region. Furthermore, other factors also contribute to the phenomenon, such as, the temperature rise effect of the bearing liner, the orientation of machine marks in the counter-face and the orientation of the surface patterns. For further investigation, more attention will be focused on this question. To sum up, the lubrication regimes transition from ML to HL is slightly affected by the external load, while significantly affected by the surface roughness and radial clearance. For water lubricated bearing, once the dimension is determined, the processing and the surface finish is completed. In fact, external load is the operating conditions, surface roughness and radial clearance are the structural parameters. It means that for the lubrication regimes transition, the influence of structural parameters is larger than operating conditions. 4.8. Error analysis The tested data agrees rather well with the theoretical calculation, but also exists some errors. The main errors come from the following four aspects: 1. System errors, including the measurement accuracy of the displacement sensors and the force sensors. System errors cannot be avoided in the process. At the same time, torque due to the friction of rolling element bearing is neglected, which also brings system errors to the test results. 2. Installation errors. During the actual installation of the bearing, it is easy to offset due to the small radial clearance. The initial installation clearance is different from the theoretical value and has impacts on the experiment. In addition, there is no intermediate bearing support for the shaft, only two hydrostatic bearings. The principle of the structure is similar to that of the cantilever beam, radial runout of the shaft is large under heavy load conditions, which is also out of consideration in the experiment.
Fig. 22. Effect of acceleration and deceleration on the Stribeck curve.
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3. The effects of turbulence in bearing are without consideration in the theoretical analysis model. In actual test, with the increase of the rotating speed, the amplitude of the radial runout of the shaft increases correspondingly. The fluid flow is no longer pure laminar flow, and turbulence exists in local zones. It is not considered in the theoretical model and may results in errors. 5. Conclusions The paper experimentally investigates the influences of surface topography and operating conditions on the lubrication regimes transition. Simulations are performed under the same settings and operating conditions. Calculated friction coefficients are compared with tested results. The calculated and experimental results agree well. Effects of surface roughness, external loads and radial clearances on the lubrication regimes transition are also further investigated. Characteristic parameters of the lubrication regimes transition are summarized. Reasonable explanations are given to illustrate the results as well as the relative errors. From the analysis, we can obtain the following conclusions: 1. On the whole trend, experimental data is consistent with theoretical results. Deviations exist under some operating conditions; 2. For lubrication regimes transition, the lift-off speed and critical friction coefficient do not show a certain direct relationship. The lubrication regimes transition from ML to HL is slightly influenced by the external load, while significantly influenced by the surface roughness and radial clearance; 3. For different external loads, different radial clearances and different surface roughness, the acceleration and the deceleration process of the shaft have influences on the lubrication regimes transition. It has little influences on the lift-off speed while has significant influences on the critical friction coefficient. This paper has certain guiding significance for further step investigation on lubrication regimes transition. These conclusions are useful for the structure design, analysis and optimization of such bearings. For the future research, accuracy of the test rig will be improved. The mathematical model also need to consider multi-coupled factors. Acknowledgements The authors thank the editors and anonymous reviewers for their comments and suggestions. This work is financially supported and funded by China Postdoctoral Science Foundation (No. 2019M650257) and the Fundamental Research Funds for the Central Universities (No. JB190411). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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