Thermal Considerations in Tribology 155 O.Oo80 I I I I I I I I I Lubricated - d0Wr solid Dry - both 80lid 8UrfaGW 0.0075 - - Lubrlcated - faster 80lld A 0 / 0 0 4 q = ~-in/in*-s = frictional power intensity 0 0 0 0 0 - 0.0070 - T-r'F 0 0 - 0 0 1 0 0 5 0 0 & 0.0065 - c 0 0 # 0 - 0 # 0 0 0 * 40- 0- p' i O.Oo60 - 0 0- I- =. 7-7 * \ ____ .__ - * - 0.0055 - - 0.0050 I I I I I I I I I 1 I I I I . 0.0 0.2 0.4 0.6 0.8 1 .o siderable difference between the film and solid temperatures. Figure 5.27 represents the case of a metallic solid with an insulative surface layer in contact with another layered solid having an opposite combination of ther- mal properties. It can be seen from the figure that the heat partition is highly dependent on the ratio Hd/H,, where h02 + ho1 Hd = h2 -hol and H,, =- 2 The latter is kept constant to show the main influence of the difference in surface layer thicknesses. The existence of surface layers strongly deviated the heat partition from the dry sliding condition. This phenomenon could be explained by the cooling mechanism in the contact zone by a shallow region near the surface, which mainly incorporates the layer thickness. Figure 5.27 also demonstrates the possibilities for equalizing the heat partition between the moving solids by controlling both thermal properties and thicknesses of surface layers. Negligible sliding is assumed in this case. Figure 5.28 shows the variation in the maximum temperature rise in the contact zone and the solid surfaces with respect to Hd/Haw. The contact I I I I I I I I I 1 1 Film Slower Surface Fader Surface 8 10 12 14 16 18 20 P, (1 0' psi) Figure 5.26 Maximum temperature rise versus maximum pressure for 50% slid- ing (steel-oil-steel, rolling velocity = 400 in./sec, RI = R2 = 1 in.). 0.4 0.3 -1 .o -0.5 0.0 0.5 1 .o Ha ' H, Figure 5.27 Heat partition versus Hd/H,,,, (rolling velocity = 2000 in./sec, 1=.02 in., K~=K~~, K~~=K*=.I K~, H,,,=~o~ in.). Thermal Considerations in Tribology IS 7 0.0080 A 0.0057 z E E d > 0.0033 - 0.M)lO -1 .o -0.5 0.0 0.5 1 .o H*'H, Figure 5.28 KI = KO2, Kol = K2 = .I K1, Ha,, = 104 in.). Tmax/q, versus Hd/Hm(rolling speed = 2000 in./sec, 1 = .02 in., zone temperature is almost identical to the solid surface temperature, which carries the conductive surface layer. The generalized equation for heat partition in lubricated line contact problems, which has been derived for steady-state conditions, is applicable to all metallic solids. It can be deduced from this equation that the deviation in heat partition from that calculated by Jaeger and Blok is highly influenced by the conductivity and thickness of the lubricant film. The existence of the lubricant film tends to equalize heat partition between the rolling/sliding solids independent of their thermal properties and surface speeds. It is inter- esting to note that the maximum temperature rise for each moving solid is directly proportional to the heat partition coefficient, yl, y2, the ratio of the trailing edge penetration depth to thermal conductivity, D1 /kl, D2/k2, and the total heat flux, q,/l. The difference between the maximum film and surface temperatures is also controlled by the lubricant film thickness and its conductivity. This can be attributed to the fact that convection is not important in this case. Although the problem of layered surfaces is appropriately modeled in the developed finite difference program, an evaluation of the effect of the different system parameters on the temperature distribution would be extre- mely difficult. It was, therefore, imperative to limit the parametric analysis I58 Chapter 5 to single layered solids for two specific regimens of thermal properties, layer thicknesses, width of contact and operating speeds. The following can be concluded from the dimensionless equations developed for the considered cases: 1. The surface temperature of the solid decreases with increasing con- ductivity of the layers and their thickness due to the convection influence under such conditions. For conductive surface layers, the slide/roll ratio has little influence on the maximum solid surface temperatures, while for insulative surface layers, the slide/roll ratio has a significant influence on the maximum temperature. For conductive surface layers with equal thicknesses, increasing the thickness decreases the maximum surface tempratures for both the solid and the surface layers. For insulative surface layers with equal thicknesses, increasing the thickness slightly decreases the maximum solid surface tempera- tures and increases the maximum surface layer temperatures. For a conductive solid, kl, with an insulative surface layer, kol, contacting an insulative solid, k2, with conductive layer, k02, assuming that kl = k02 and k2 = kol, there is a signficant deviation of the partition from the unlayered case, as shown in Fig. 5.27. It can also be seen that with the above combination of properties it is possible to attain equal heat partition by proper selection of the layer thicknesses (at Hd/Have = -0.325 in this case). For the above case, the maximum contact temperature is approximately equal to the maximum temperature in the insulative solid with conductive layer for all thickness ratios (Fig. 5.28). The maximum temperature for the conductive solid with insulative layer is significantly lower than the interface temperature. 2. 3. 4. 5. REFERENCES 1. Cheng, H. S., “Fundamentals of Elastohydrodynamic Contact Phenomena,” International Conference on the Fundamentals of Tribology, Suh, N. and Saka, N., Eds., MIT Press, Cambridge, MA, 1978, p. 1009. Blok, H., “The Postulate About the Constancy of Scoring Temperature,” Interdisciplinary Approach to the Lubrication of Concentrated Contacts, P. 3. Sakurai, T., “Role of Chemistry in the Lubrication of Concentrated 2. M. Ku, Ed., NASA SP-237, 1970, p. 153. Contacts,” ASME J. Lubr. Technol., 1981, Vol. 103, p. 473. Thermal Considerations in Tribology 159 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. Alsaad, M., Blair, S., Sanborn, D. M., and Winer, W. O., “Glass Transitions in Lubricants: Its Relation to EHD Lubrication,” ASME J. Lubr. Technol., 1978, Vol. 100, p. 404. Blok, H., “Theoretical Study of Temperature Rise at Surfaces of Actual Contact Under Oiliness Lubricating Conditions,” Proc. Gen. Disc. Lubrication, Institute of Mechanical Engineers, Pt. 2, 1937, p. 222. Jaeger, J. c., “Moving Sources of Heat and the Temperature at Sliding con- tacts,” Proc. Roy. Soc., N.S.W., 1942, Vol. 56, p. 203. Cheng, H. S., and Sternlicht, B., “A Numerical Solution for the Pressure, Temperature, and Film Thickness Between Two Infinitely Long, Lubricated Rolling and Sliding Cylinders under Heavy Loads,” ASME J. Basic Eng., Vol. 87, Series D, 1965, p. 695. Dowson, D., and Whitaker, A. V., “A Numerical Procedure for the Solution of the Elastohydrodynamic Problem of Rolling and Sliding Contacts Lubricated by a Newtonian Fluid,” Proc. Inst. Mech. Engrs, Vol. 180, Pt. 3, Ser. B, 1965, p. 57. Manton, S. M., O’Donoghue, J. P., and Cameron, A., “Temperatures at Lubricated Rolling/sliding Contacts,” Proc. Inst. Mech. Engrs, 1967-1 968, Vol. 1982, Pt. 1, No. 41, p. 813. Conry, T. F., “Thermal Effects on Traction in EHD Lubrication,” ASME J. Lubr. Technol., 1981, Vol. 103, p. 533. Wang, K. L., and Cheng, H. S., “A Numerical Solution to the Dynamic Load, Film Thickness, and Surface Temperatures in Spur Gears; Part 1 Analysis,” ASME J. Mech. Des., 1981, Vol. 103, p. 177. Knotek, O., “Wear Prevention,” International Conf. on the Fundamentals of Tribology, Suh, N., and Saka, N., Eds., MIT Press, Cambridge, MA, 1978, p. 927. Torti, M. L., Hannoosh, J. G., Harline, S. D., and Arvidson, D. B., “High Performance Ceramics for Heat Engine Applications,” AMSE Preprint No. Georges, J. M., Tonck, A., Meille, G., and Belin, M., “Chemical Films and Mixed Lubrication,” Trans. ASLE, 1983, Vol. 26(3), p. 293. Poon, S. Y., “Role of Surface Degradation Film on the Tractive Behavior in Elastohydrodynamic Lubrication Contact,” J. Mech. Eng. Sci., 1969, Vol. I1(6), p. 605. Berry, G. A., and Barber, J. R., “The Division of Frictional Heat - A guide to the Nature of Sliding Contact,” ASME J. Tribol., 1984, Vol. 106, p. 405. Shaw, M. C., “Wear Mechanisms in Metal Processing,” Int. Conf. on the Fundamentals of Tribology, Suh, N., and Saka, N., Eds., MIT Press, Cambridge, MA., 1978, p. 643. Burton, R. A., “Thermomechanical Effects on Sliding Wear,” Int. Conf. on the Fundamentals of Tribology, Suh, N., and Saka, N., Eds., MIT Press, Cambridge, MA., 1978, p. 619. Ling, F. F., Surface Mechanics, J. Wiley, New York, NY, 1973. 84-GT-92, 1984. 160 Chapter S 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Klaus, E. E., “Thermal and Chemical Effects in Boundary Lubrication,” Lubrication Challenges in Metalworking and Processing, Proc. 1 st Int. Conf., IIT Res. Inst., June, 1978. Winer, W. O., “A Review of Temperature Measurements in EHD Contacts,” 5th Leeds-Lyon Symposium, 1978, p. 125. Townsend, D. P., and Akin, L. S., “Analytical and Experimental Spur Gear Tooth Temperature as Affected by Operating Variables,” ASME J. Lubr. Technol., 1981, Vol. 103, p. 219. Rashid, M., and Seireg, A., “Heat Partition and Transient Temperature Distribution in Layered Concentrated Contacts. Part I : Theoretical Model, Part 2: Dimensionless Relationships, ASME J. Tribol., July 1987, Vol. 109, p. 496. Taylor, E. S., Dimensional Analysis for Engineers, Oxford University Press, London, England, 1974. David, F. W., and Nolle, H., Experimental Modeling in Engineering, Butterworths, London, England, 1974. Arpaci, V. S., Conduction Heat Transfir, Addison-Wesley, Reading, MA, 1966, p. 474. Hamrock, B. J., and Jacobson, B. O., “Elastohydrodynamic Lubrication of Line Contacts,” ASLE Trans., 1984, Vol. 27, p. 275. Wilson, W. R. D., and Sheu, S., “Effect on Inlet Shear Heating Due to Sliding on Elastohydrodynamic Film Thickness,” ASME J. Lubr. Technol., 1983, Vol. 105, p. 187. Suzuki, A., and Seireg, A., “An Experimental Investigation of Cylindrical Roller Bearings Having Annular Rollers,” ASME J. Lubr. Technol., 1976, Vol. 98, p. 538. 6 Design of Fluid Film Bearings 6.1 HYDRODYNAMIC JOURNAL BEARINGS Fluid film bearings are a common means of supporting rotating shafts in rotating machinery. In such bearings a pressurized fluid film is formed with adequate thickness to prevent rubbing of the mating surfaces. There are two main types of fluid film bearings: hydrostatic and hydrodynamic. The first type relies on an external source of energy to supply the lubricant with the necessary pressure. In the second type, pressure is developed within the bearing as a result of the relative motion between shaft and bearing. The pressure is influenced by the geometry of the fluid wedge, which is formed to sustain the load, as illustrated in Fig 6.1. 6.1.1 Hydrodynamic Equations The basic equation governing the behavior of the fluid film in the hydro- dynamic case is Reynolds’ equation [ I]. Assuming isothermal, incompressi- ble flow this equation is derived by consideration of the Newtonian shear - velocity gradient relationship, the equilibrium of the fluid element (Fig. 6.2) in the x- and z-directions and the continuity equation. Accordingly: 161 162 W Chapter 6 Figure 6.1 Bearing geometry. where p = film pressure h = film thickness p = oil viscosity (assumed constant throughout the film) U = tangential velocity in the journal The solution of this equation for any eccentricity, e, would result in expres- sions for the quantity of flow required, the frictional power loss and the pressure distribution in the oil film. The latter determines the load-carrying capacity of the bearing. A closed-form solution of Reynolds’ equation can be obtained with either of the following assumptions: 1. Assume the bearing to be long compared to its diameter. This is generally known as the Sommerfeld bearing. In this case the change in pressure in the axial direction can be neglected com- pared to the change in the circumferential (x) direction. Accordingly: Design of Fluid Film Bearings Y 2 de ay ax sdx dy dz =-dx dy dt Figure 6.2 Equilibrium of fluid element in the x-direction. and Eq. (6.1) reduces to: Referring to Fig. 6.1 with the condition: 163 dy dr p=pmax at h = hl 164 Chapter 6 and p=po at 8=0 and 8=2n Equation (6.2) can be integrated to yield: 2C(1 - E*) 2 + E2 hi = 6p UR !)=PO+- E sin 8(2 + E cos 0) c* ((2 + &2)(1 + ECOSO) 12XE F (s) - 4n(l + 2~~) - PLU R 2+&2Ji-z2 (6.3) (6.4) (6.5) where W = bearing load F = tangential load h, = film thickness at peak pressure C = radial clearance R = radius of the bearing E = = eccentricity ratio 8 = angle measured from negative direction of x-axis (rads) C Equation (6.5) can be rewritten in terms of a dimensionless num- ber, S, called the Sommerfeld number: (2 + S= 12dE so that s = -(-)*= pUL R R 'pN p nW c where N, P =journal rotational speed and average pressure, respectively [...]... 2.5 01 K,,r = -0.4 81 6 ~-0 .88 63+0 .19 27(L/D) s-0.37 + 1. 7006 + 2. 219 8S(L’D’ I3+0 .14 76(~/~) - 0.9335s + 1 1.6940S2 - 16 .3368s I 73 Design of Fluid F l Bearings im Kyy = 2. 5 18 1 0 .15 5 S 5 1: K,, = 1. 12 51 (i) -0.3236 (g) -0.6746 ~0.4954-0.4007(L/D) s-0 .8 17 9+0.469I(L/D) s-0.5 584 + 1. 013 1(LID) - 1. 290s -0 I794 Kyy = 2.2202(;) + 1. 053S2 - 12 .272s ~0. 314 5-0.277I(L/D) s 2 1. 0: - 0.3 413 s + 0.3436s K~vx 2.32 58 - 1. 212 0... 3.521S2+ 19 .89 2s Cx,, = 0.46 41 + - 0. 685 2s + 0.3 214 S2 - Cy, = 0.4 619 + 2.0240($) - 0. 514 5s + 0. 416 8S2- 2 .16 85 S($) Cyy= -7 .85 2 + 10 . 414 - 15 . 18 5s + 3.576S2 + 10 . 084 s s > 1. OL C,, = 26.424 - 25.646 - 11 .2353- 0.044S2+ 43.391s Cxy= 1. 353+ 1. 109(;) +0.376S+0.027S2 - 1. 341s Cyx= 4.059 - 2. 013 - 0.231s - 0.004S2 + 1. 83 9s Cy,, = 4.735 - 5. 019 - 7.506s - 0. 089 S2 + 0.25 5 L j D < 0.50, S 5 0 .10 : C,, = 15 1.70... 220.05(;) +472 . 18 S - 86 7.94S2 + 59.59s Cxy= 9.272 - 14 .11 7 - 16 .80 3s - 21. 363S2 + 44 .80 2s C,.- = 0.250 - 0.0677 - 6 ’ 3 60 .89 4S2 + 63. 71 1 4 5 7+ s s Cyy= 10 . 58 - 14 .77 63.233 - 87 .31S2 - 28. 93s 0 .10 < s 5 1. 0: C,, = 45.534 - 57.339 + 5.6523 - 7.791S2 + 61. 060s Cxy= 8. 035 - 10 . 785 0 .13 4s- 0 .83 67S2 + 6.204s Cy.x 5.3 48 - 6.560 = Cyy= 1. OOO - 0.6 41 - 0 .10 5S2 + 5.534s + 1. 9 81 7 S(L/D) Chapter 6 I76 s > 1. 0: C,,... 28. 43 i - 672.00s + 2329.06S2+ 337.77s et:,* -2 .87 2 =z + 537; ( 8) - 2M892S + 83 3.223S2 + 0.05 < 5’ 5 0.25: C,, = 26.402 - 19 . 317 (;) - 5.201s + 55.945S2 + 17 .10 2s + C, = 1. 712 6 ~ ~ 8 7 0 ( ~ ) - 9.72445 + 10 .0934S2 + Cy,y I 612 5 - 0.2 01 == - 5.4768s + 0 13 44S2 + c]v4 = 1. 587 - ~ 0 8 3 ( ~- 18 .793S+ ) 13 .86 5S2 + - 22 .1 IS($) Design o Fluid Film Bearings f I75 0.25 < S 5 1. Or C,, = 6.357 - 2.2 98 - 17 .2943+... 2.32 58 - 1. 212 0 = 1. 1442 K.,, = 8 .15 15(:) 9.4 387 +0.47 I7(L/ D) Ky = 12 .2356 - 12 . 48 91 + 1. 2669s + 0.0756S2+ 2.2224s - 10 .9395S(L/D) K.vy = 1. 916 5($) -0.2674 K,.v = 16 .74 - 26.53 s-0.6257+0.6357(L/D) + 13 .63s - 38. 70S2 +43.72S 0 .15 < S 5 1. 0: -0.6076 K.x.x 1. 189 7($) = Ky.v = 4.4333 - 6.04 98 S 5 0.50: s-0.7 I 52+O.u)53(LID) + 3.9 980 s - 4. 213 3S2 + 0 .88 48s I 74 Chapter 6 + ~ 0 4 7 3 ( ~- 1. 964W-2.039S2... KY = -7.027 + 8. 995s(L/D’ S 2 0.50: Kyy = 3. 417 1 ($) 0.4507 S-0~2406+0.mS(t/D) s > 1. 0: -0.5053 K , = f 2702(:) $4 .8 19 9+0.9723(L/D) 92659+0 .8 17 3(L/D) K,, = 27.92 - 3 2 7 1 0 - 1. 21s - 0.404S2+ ~9.34(;) 1y?.,, = 3 .19 03($) 0. 386 0 9 . 18 13-0.3994(L/D) Damping Coefficients 0.5 5 LID 5 1. 0, S 5 0.05: C,, = -8. 090 + 30.59 - 16 47S6S+ 70 78. 46S2 + 739 .86 5 Cry =z - 21. 1 I + 27 .88 - 705,62S + 2327.28S2 + 353.02D... bearing length The solution of Eq (6.9) yields: (2 $) (1 + E COS q3 &dx2 (1 * ) + 16 c2 -E pULR 2n (6 .10 ) (6 .11 ) (6 .12 ) F = ( 7 ) - (6 .13 ) where 4 = attitude angle between the load line and the line of enters D =journal diameter Equation (6 .1 1) can be rewritten in terms of the Sommerfeld number, S,as: 2 '(i) (1 - E 2 y = m,/z2 (1- E +1 ~ ) 6 ~ ~ (6 .14 ) 16 6 6 .1. 2 Chapter 6 Numerical Solution The development... in this chapter These equations are: 0. 01 0. 01 I L 0 .1 Sommerfeld Number, S Figure 6.9 Minimum film thickness I 1 A 0-L 0 - Volume Flow Variable, Q/(RNCL) I 1 Friction Coefficient, F= f x (Wc) 1 I b 16 8 Chapter 6 Sommerfeld Number, S Figure 6.6 Peak pressure J = Joule'r equivalent of heat t 1 0. 01 I I 0 .1 1 Sommerfeld Number, S Figure 6.7 Temperature variable Design of Fluid Film Bearings 1. 2... 1. 2 2.0 2 .1 169 17 0 Chapter 6 The following notation is used in the analysis: C = radial clearance (in. ) D =journal diameter (in. ) e =journal eccentricity (in. ) f = coefficient of friction ho = minimum film thickness (in. ) L = bearing length (in. ) N =journal rotational speed (rps) P = bearing average pressure (lb /in. 2) P, = maximum oil film pressure (lb /in. 2) ,, Q = quantity of oil fed to bearing ( i... rotor in the x-direction Expressions for calculating the stiffness and damping coefficients k.v.v,C.rx, k.vy,C,,., k,, CYy, and C,, are given in the following They are derived k,,, by approximate curve fitting from Ref 14 as a function of S , which is evaluated at any instant from the instantaneous eccentricity ratio Stiffness Coefficients 0.5 5 LID 5 1. 0, S 5 0 .15 : (g) (g) K.r.r= 0.5979 - I. 0 18 1 -0.2 . 6. 7. 8. 9. 10 . 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . Alsaad, M., Blair, S., Sanborn, D. M., and Winer, W. O., “Glass Transitions in Lubricants: Its Relation to EHD Lubrication, ”. s 2 1. 0: K~vx = 2.32 58 - 1. 212 0 - 0.3 413 s + 0.3436s 1. 1442 K.,, = 8 .15 15(:) 9.4 387 +0.47 I7(L/ D) Ky. = 12 .2356 - 12 . 48 91 + 1. 2669s + 0.0756S2 + 2.2224s - 10 .9395S(L/D). Engrs, 19 67 -1 9 68, Vol. 19 82 , Pt. 1, No. 41, p. 81 3 . Conry, T. F., “Thermal Effects on Traction in EHD Lubrication, ” ASME J. Lubr. Technol., 19 81 , Vol. 10 3, p. 533. Wang, K. L., and Cheng,