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1、 Kazama 1/6 NUMERICAL SIMULATION OF A SLIPPER MODEL FOR WATER HYDRAULIC PUMPS/MOTORS IN MIXED LUBRICATION Toshiharu KAZAMA Department of Mechanical Systems Engineering Muroran Institute of Technology 27-1, Mizumoto-cho, Muroran, 0508585, Japan (E-mail: kazamammm.muroran-it.ac.jp) ABSTRACT This paper

2、 presents a time-dependent mathematical model of a hybrid (hydrostatic/hydrodynamic) thrust pad bearing as a slipper of swash-plate type axial piston pumps and motors for the use of tap water under mixed and fluid film lubrication condition. Effects are examined of the load eccentricity, time-lag of

3、 changes in the supply pressure and load, surface roughness, recess volume, and the revolution radius. The bearings motion is simulated three-dimensionally, including roughness interaction and asperity contact. Solutions are obtained regarding the friction, flow rate, power loss, and stiffness. Calc

4、ulations indicate that the eccentric load causes local contacts. The preceding change in the load poses a larger motion of the bearing. The hydrodynamic effect becomes marked as the revolution radius increases. As the recess volume increases, the bearing stiffness decreases. KEYWORDS Water-hydraulic

5、s, Tribology, Hybrid/Hydrostatic bearings, Mixed lubrication, Piston pumps/motors NOMENCLATURE a : Ratio of recess radius = R1/R2 H : Reference clearance h : Clearance = h/H h0s : Pad center clearance = h0/ I : Moment of inertia =I(H/R2)3=H3I/(6R26) K : Bulk modulus of lubricant L : Power loss = L/(

6、ps0R23So) M : Moment-load = M/(ps0R13So) p : Pressure = p/(ps0So) r p : Recess pressure = pr/(ps0So) s p : Supply pressure = ps/(ps0So) Q : Flow rate = Q/(R23) 0 R : Radius of revolution = R0/R2 R2 : Outer radius of bearing r, , z: Coordinates = r/R2, , z/H w r : Eccentric ratio of load = rw/R2 So :

7、 Parameter = 0 S(R2/H)2 = 6(R2/H)2/ps0 T : Friction torque = T/(ps0R23So) r V : Recess volume = r V(R2/H)2 = 6Vr/(H2KR2) W : Load = W/(ps0R22So) X, Y, Z: Coordinates x, y, z : Coordinates : Inclination of pad = R2/H : Parameter of restrictor =(H/R2)3=4H3lc/(3rc4) : Time lag of load = t 0 : Parameter

8、 = 2W0loga/(1a2)ps0R22 : Viscosity : Surface roughness = (12+22)1/2 : Time = t yx, : Angle = x, yR/H : Azimuth of pad : Reference angular velocity 2C4-5 Proceedings of the 6th JFPS International Symposium on Fluid Power, TSUKUBA 2005 November 7-10, 2005 509Copyright 2005 by JFPS, ISBN 4-931070-06-X

9、Kazama 2/6 : Angular velocity of sliding plane : Angular velocity of pad Subscripts: a : Asperity f : Fluid m : Mean value x, y, z : x, y, z axes, respectively 0 : Reference (High-pressure period) INTRODUCTION As a power transfer medium of hydraulic systems, nothing in nature is more human compatibl

10、e and environmentally compatible than tap water: it is non- toxic, non-flammable, clean, easily disposable and readily available. In recent years, economic and environmental forces have helped water to regain its prominence as a hydraulic fluid 12. Nevertheless, disadvantages and challenges of tap-w

11、ater hydraulic systems remain: leakage resulting from waters low viscosity, erosion because of waters higher vapor pressure, and corrosion because of waters chemical reaction, in addition to bacterial growth and limited operating temperatures. In particular, lower viscosity and a lower viscosity-pre

12、ssure index engender inferior lubrication. As with oil-hydraulic pumps and motors, tap-water hydraulic pumps and motors are expected to function suitably in widely variant operating conditions. The efficiency and durability of the equipment are conspicuously influenced by tribological characteristic

13、s of the bearing and seal parts. Swash-plate type axial piston pumps and motors take advantage of high power density based on high pressure and compactness, and high efficiency of variable displacement machinery. Main bearings and seal parts of the pumps and motors form a slipper, which functions as

14、 a hydrostatic bearing 3. It is suitable for lubrication with low viscous fluids. Shute and Turnbull undertook a pioneering study 4; later, Iboshi and Yamaguchi 3 as well as Hooke, et al. 56 contributed to study in this area by investigating characteristics of slippers and by discussing effects of o

15、perating conditions and geometry. Typical hydrostatic bearings are usually designed and used under the full film lubricating condition 78. However, hydraulic pumps and motors strongly demand less leakage and compactness, which enforces smaller clearance of the same order as the roughness height. The

16、 bearings of real-life pumps and motors are thus required to operate in a mixed lubrication regime. Yamaguchi and Matsuoka developed a mixed lubrication model that is applicable to bearing and seal parts of hydraulic equipment 9. This model is based on a combination of the asperity-contacting model

17、proposed by Greenwood and Williamson (GW model) 10 and the average flow model by Patir and Cheng (PC model) 11 12. The former is a model for a contact mechanism of non-lubricated stationary rough surfaces. The latter is an approach using fluid film lubrication to the contact phenomena of surface asp

18、erities. They fill the gap separating these two extreme models. Moreover, they include effects of adsorption of lubricants, elastohydrodynamic lubrication (EHL) and cavitation around asperities. Comparison of numerical results using the model and many experimental data for thrust washers and hydrost

19、atic bearings 1314 shows good agreement. Subsequently, Yamaguchi, et al. 15 expanded the model to include plastic-to-metal contact under water lubricated conditions, whereas the primary model specifically describes metal-to-metal contact under oil lubricated conditions. In this study, the author app

20、lies the mixed lubrication model 9 to unsteady analysis 16 of a water-lubricated hybrid/hydrostatic thrust pad bearing, which is a model of a slipper of swash-plate type axial piston pumps and motors. Three-dimensional motion of the bearing is simulated and the tribological characteristics are discu

21、ssed. Effects of interference in roughness asperities, contact between the surfaces and the time lag of the dynamic load are examined. THEORY Theoretical model A hybrid thrust bearing that functions as hydrostatic and hydrodynamic bearings, lubricated with tap water, is shown in Fig. 1. Surface roug

22、hness, asperity contact, eccentric loads, and changes in the supply pressure and the load are considered. Heat generation resulting in a change in the physical properties of the lubricant and thermal distortion as well as elastic deformation of the bearing parts are neglected. The load acting on the

23、 bearing and seal parts of hydraulic pumps and motors fundamentally changes cyclically and synchronously with the supply pressure. However, the load change can be slightly preceded or delayed because of factors such as friction and inertia of the parts in the equipment. Thus, the supply pressure ps

24、and the load W (0) at /2, the downward force becomes larger and the pad is suppressed vice versa. During the high-pressure period, the surfaces remain in contact ( 0 /AAcnt =1) resulting in larger torque T because of asperity-contact. On the other hand, during the low- pressure period, the area rati

25、o 0 /AAcnt is less than unity or equal to zero and out Q and T are infinitesimal. Effect of radius of revolution of pad The radius 0 R of the revolution of the pad corresponds to the respective pitch radius of the cylinder-bores of piston pumps and motors. Figures 8 and 9 show the effect of 0 R on t

26、he bearing. As 0 R increases, the maximum max0 h of the center clearance increases, especially in the low- pressure periods, because of the hydrodynamic (wedge) effect; the minimum min0 h changes less. The increase in hydrodynamic load-carrying capacity lessens the cavitation period cav/2 and contac

27、ting period cnt/2 in a cycle. Effect of recess volume or liquid bulk modulus The effect of the normalized recess volume r Von pad fluctuation is examined in Fig. 10. As r V increases, the bearing-stiffness decreases. By contrast, the moment stiffness M remains almost constant. The mean power loss m

28、L is determined by the power loss Tm Lbecause of friction torque and the losses decrease slightly with increased r V. It is noteworthy that parameter V r is equivalent to the inverse number of the bulk modulus K of the liquid. Therefore, if K is reduced, e.g. by containing bubbles in the liquid, the

29、 pad would fluctuate distinctly and the bearing-stiffness would decrease. Fig. 5 Changes in center clearance hs0 in terms of time lag of the load Fig. 6 Changes in leakage flow rate out Q and friction torque T in terms of time lag of the load Fig. 7 Changes in cavitation area 0 /AAcav and contact ar

30、ea Acnt/A0 in terms of time lag of the load Fig. 8 Changes in center hs0 and minimum hs min clear- ances in terms of revolution radius 0 R Fig. 9 Effect of revolution radius 0 R on maximum clearance max0 h , minimum clearance min0 h , cavitation period cav/2 and contacting period cnt/2 0 0.5 1 0 0.5

31、 1 02 rad Acav / A0 0.5 rad 0.2 0 0.2 0.5 Acnt / A0 Acav / A0 Acnt / A0 0 0.005 0.01 0 0.01 02 rad T 0.5 rad 0.2 0 0.2 0.5 Qout Qout T 0 4 8 02 rad hs h0s R0 0.25 0.3 0.35 hmin s 2.533.5 0 2 4 0 0.5 1 R0 h0 ho max ho min cnt / 2 cav / 2 / 2 0 4 8 02 rad h0s 0.5 rad 0.2 0 0.2 0.5 h0s 513Copyright 200

32、5 by JFPS, ISBN 4-931070-06-X Kazama 6/6 Fig. 10 Effect of normalized recess volume r V ) on mean power loss m L, mean friction power loss Tm L , dynamic stiffness , and moment-stiffness M CONCLUDING REMARKS A hybrid thrust bearing as a slipper model of water hydraulic piston pumps and motors was de

33、veloped in this study. It includes uneven contact because of alignment in mixed and fluid film lubrication, based on the GW model and the PC model. Numerical solutions of the time- dependence problem are obtained over a wide range of operation under water lubricated conditions. The following conclus

34、ions were obtained: i) As the eccentric ratio of the load increases, the inclination of the pad increases, thereby causing local asperity-contact pressure. ii) The time lag between the supply pressure and the load influences the bearings motion and tribological characteristics. In particular, the pr

35、eceding change in loads engenders a larger motion of the pad. iii) The radius of revolution of the pad influences the bearing performance because of hydrodynamic wedge effect. iv) As the recess volume increases or the bulk modulus decreases, the bearing motion is enlarged, especially in the low-pres

36、sure period, decreasing the bearing stiffness. ACKNOWLEDGEMENTS The author would like to thank Professor Emeritus Atsushi Yamaguchi of Yokohama National University and Emeritus Professor Mitsuru Fujiwara of Muroran Institute of Technology for their thoughtful encouragement. REFERENCES 1._Yamaguchi,

37、A., Tap Water Possibility as a Hydraulic Fluid, Journal of Japan Hydraulic and Pneumatic Society, 1978, 94, pp. 205210. (in Japanese) 2._Modern Water Hydraulics Our Choice for the Future, 1995, National Fluid Power Association. 3._Iboshi, N. and Yamaguchi, A., Characteristics of a Slipper Bearing fo

38、r Swash Plate Type Axial Piston Pumps and Motors (1st Report), Bulletin of Japan Society of Mechanical Engineers, 1982, 25, pp. 19211930; (2nd Report), ibid., 1983, 26, pp. 1583 1589; (4th Report), ibid., 1986, 29, pp. 25392546. 4._Shute, N.A. and Turnbull, D.E., Minimum Power Loss of Hydrostatic Sl

39、ipper Bearings for Axial Piston Machines, Proceedings of International Convention on Lubrication and Wear, Institution of Mechanical Engineers, 1963, pp. 314. 5._Hooke, C.J. and Kakoullis, Y.P., Effects of Non- Flatness on the Performance of Slippers in Axial Piston Pumps, Proceedings of Institution

40、 of Mech- anical Engineers, Part C, 1983, 197, pp.239247. 6._Hooke, C.J. and Li, K.Y., The Lubrication of Overclamped Slippers in Axial Piston Pumps- Centrally Loaded Behaviour, Proceedings of Institution of Mechanical Engineers, Part C, 1988, 202, pp. 287293. 7._Williams, J.A., Engineering Tribolog

41、y, 1996, Oxford Science Publications. 8._Hamrock, B.J., Fundamentals of Fluid Film Lubrication, 1994, McGraw-Hill. 9._Yamaguchi, A. and Matsuoka, H., A Mixed Lubrication Model Applicable to Bearing and Seal Parts of Hydraulic Equipment, Journal of Tribology, Transactions of ASME, 1992, 114, pp. 1161

42、21. 10._Greenwood, J.A. and Williamson, J.B.P., Contact of Nominally Flat Surfaces, Proceedings of Royal Society, London, Series A, 1966, 295, pp. 300319. 11._Patir, N. and Cheng, H.S., An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication,

43、Journal of Lubrication Technology, Transactions of ASME, 1978, 100, pp. 1217. 12._Patir, N. and Cheng, H.S., Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces, Journal of Lubrication Technology, Transactions of ASME, 1979, 101, pp. 220230. 13._Kazama, T. and Yamaguchi,

44、A., Application of A Mixed Lubrication Model for Hydrostatic Thrust Bearings of Hydraulic Equipment, Journal of Tribology, Transactions of ASME, 1993, 115, pp. 686691. 14._Kazama, T. and Yamaguchi, A., Experiment on Mixed Lubrication of Hydrostatic Thrust Bearings for Hydraulic Equipment, Journal of

45、 Tribology, Transactions of ASME, 1995, 117, pp. 399402. 15._Yamaguchi, A., Okamoto, Y. and Wang, X., Friction Characteristics of PEEK for Bearing and seal Parts of Water Hydraulic Equipment, Transactions of Japan Fluid Power System Society, 2003, 34-2, pp. 4045. (in Japanese) 16._Kazama, T., Yamagu

46、chi, A. and Fujiwara, M., Motion of Eccentrically and Dynamically Loaded Hydrostatic Thrust Bearings in Mixed Lubrication, Proceedings of 5th JHPS International Symposium on Fluid Power, 2002, pp. 233238. 17._Johnson, K.L., Greenwood, J.A. and Poon, S.Y., Simple Theory of Asperity Contact in Elastohydrodynamic Lubrication, Wear, 1972, 19, pp. 91108. 0 0.5 1 1.5 Vr Lm M Lm x1011 510 x103 4.6 4.4 LTm 4.8 514Copyright 2005 by JFPS, ISBN 4-931070-06-X