FIGURE 12.Two elliptical oil lift pockets located at quarter points along a journal bearing axis. Removing the plug allows the supply passages to be flushed. with size. A L , the area of a single bearing pocket, is usually made about 1.5% of Ain a bearing of this type. Once the load is lifted, the pocket pressure falls because pressurized oil is distributed over the whole bearing area. The flow, oil film thickness, and other quantities can be calculated for this condition as a pure hydrostatic bearing (before rotation starts). The lift is the following function of the purnp flow: (14) This is the corrected form of Equation (12–19) in Reference 12. Anumerical example will illustrate the use of the above equations, where Q = 0.07 ᐉ/ sec (1.1 gpm), D = 533.4 mm (21 in.), L = 304.8 mm (12 in.), C D = 0.711 mm (0.028 in.), W = 382,500 N (86,000 lb), n = 2 pockets, A L = 2580 mm 2 (4 in. 2 ), and μ = 0.058 Pa·sec (8.4 µreyn). Solving Equation 13, assuming K BA = 3, gives a breakaway pressure of 21.1 MPa (3072 psi). The oil pump should be sized to give at least this much pressure with some margin (say, 50 to 100%) to allow for performance deterioration over a period of time. Employing appropriate units, Equation 14 gives an eccentricity ratio of 0.63. Assuming that the shaft moves straight up, the lift equals the minimum film thickness, (1 – ⑀)(C D /2), or 0.132 mm (0.005 in.). This is a realistic value which will allow for some misalignment or shaft deflection. In thrust bearings somewhat more pocket area has historically been used than in journal bearings, about 5%of the bearing area. The pad in Figure 13 is from a bearing with an outside diameter of 2286 mm (90 in.) with a design load of 4.1 MPa (600 psi). At breakaway the pocket pressure rises to 12.4 MPa (1800 psi). It then falls back to a steady-state value of 4.8 MPa (700 psi). ‘The lift (h) may be estimated by assuming a circular pressure distribution similar to the bearing in Figure 3. For the bearing pad in Figure 13 the following values may be assigned to the variables: Q = 0.025 ᐉ/sec (0.4 gpm), P p = 4.8 MPa (700 psi), μ = 0.056 Pa·sec (56 cP), (ISO VG 68 oil at 38°C), R = 286 mm (11.3 in.), and R o = 51 mm (2.0 in.). Solving Equation 9 for h, using consistent units, gives a lift of 0.099 mm (0.004 in.), 118 CRC Handbook of Lubrication Copyright © 1983 CRC Press LLC L c = Length of capillary tube n = Number of oil-lift pockets in journal bearing P BA = Breakaway pressure in oil lift P o = Ambient pressure around bearing sealing land P s = Pressure in bearing pocket P s = Supply pressure Q = Flow Q c = Capillary tube flow Q k = Constant flow Q o = Orifice flow Q – = Flow rate parameter, Reference 3 R = Outside radius of thrust bearing (Figure 3) R o = Pocket radius of thrust bearing (Figure 3) w = Width of land normal to direction of flow W = Load ⑀ = Eccentricity ratio = e Ϭradial clearance ρ = Fluid density θ = Angle subtended by circumferential land in journal bearing (Figure 10) φ = Direction of loading in journal bearing (zero is toward a pocket) (Figure 10) μ = Fluid viscosity η = Efficiency REFERENCES 1. Anon., Floating shoes form big bearings, Mach. Design, 49(27), 37, 1977. 2. Rippel, H. C., Cast Bronze Hydrostatic Bearing Design Manual, Cast Bronze Bearing Institute, Chicago, 1975. 3. Stout, K. J. and Rowe, W. B., Externally pressurized bearings — design for manufacture. III. Design of liquid externally pressurized bearings for manufacture including tolerancing procedures, Tribol. Int., 7(5), 195, October 1974. 4. Rowe, W. B., O’Donoghue, J. P., and Cameron, A., Optimization of externally pressurized bearings for minimum power and low temperature rise, Tribology, 3(4), 153, August 1970. 5. Sneck, H. J., A survey of gas-lubricated porous bearings, Trans. ASME, Ser. F, 90(4), 804, October 1968. 6. Fuller, D. D., Hydrostatic lubrication, in Standard Handbook of Lubrication Engineering, O’Connor, J. J., Boyd, J., and Avallone, E. A., Eds., McGraw-Hill, New York, 1968, 3-17. 7. Szeri, A. Z., Hydrostatic bearings, in Tribology: Friction, Lubrication, and Wear, Szeri, A. Z., Ed., McGraw-Hill, New York, 1980, 47. 8. Elwell, R. C. and Sternlicht, B., Theoretical and experimental analysis of hydrostatic thrust bearings, Trans. ASME, Ser. D, 82(3), 505, September 1960. 9. Raimondi, A. A. and Boyd, J., Hydrostatic journal bearings (compensated), in Standard Handbook of Lubrication, O’Connor, J. J., Boyd, J., and Avallone, E. A., Eds., McGraw-Hill, New York, 1968, 5- 66. 10. Hunt, J. B. and Ahmed, K. M., Load capacity, stiffness and flow characteristics of a hydrostatically lubricated six-pocket journal bearing supporting a rotary spindle, Part 3N, Proc. I.M.E., 182, 53, 1967-8. 11. O’Donoghue, J. P. and Rowe, W. B., Hydrostatic bearing design, Tribology, 2(1), 25, February 1969. 12. Wikock, D. F. and Booser, E. R., Bearing Design and Application, McGraw-Hill, New York, 1957. 120 CRC Handbook of Lubrication Copyright © 1983 CRC Press LLC SQUEEZE FILMS AND BEARING DYNAMICS J. F. Booker INTRODUCTION This chapter covers transient behavior of viscous lubricant films under loads which may be fixed or variable in magnitude and/or direction. Since it takes time for such films to be squeezed out from between surfaces, bearings can often carry surprisingly high peak loads as compared to those they might sustain in steady-state operation. “Squeeze-film” action is often of interest because of the damping it provides. Occasionally such special devices as dampers for turbomachinery are involved; more often, as in recip- rocating machinery, the damping action is provided by conventional bearings. The following analysis begins with treatment of the normal approach of planar bearings. It proceeds with examination of cylindrical bearings in one- and two-dimensional translation, both without and with accompanying rotation. Finally, by way of an example for connecting- rod bearings, analysis is supplemented by a parametric design study and correlation of a failure criterion with field experience. GENERALREYNOLDS EQUATION In its general form the incompressible Reynolds equation derived in an earlier chapter can be written in rectangular coordinates x, y or in polar coordinates, r, θ For an important class of normal approach “squeeze film” problems, the average tan- gential surface velocity U — has negligible effect, leaving only the squeeze rate ∂h/∂t as an effective driving term. PLANAR BEARINGS IN NORMALAPPROACH For isoviscous planar normal approach with uniform film thickness, the Reynolds equation simplifies in rectangular coordinates to or in polar coordinates Special Formulation forCircularSection As an example, consider Figure 1 in which a film is squeezed by the normal approach Volume II 121 121-137 4/11/06 12:28 PM Page 121 Copyright © 1983 CRC Press LLC where, for a constant load For particular fixed values µ, R, and F, the above relations give an approach rate slowing asymptotically as final closure is approached. This qualitative behavior is typical of all “squeeze films” in response to time integral (impulse), not instantaneous values of loading. General Formulation The relations derived for the circular section are also valid for general geometries if expressed in terms of area Aand dimensionless shape factors Pand K. Thus, Shape factor Pis a measure of the sharpness or nonuniformity of the pressure distribution, K the dynamic stiffness or damping rate of the lubricant film as a whole. Circular Section The circular section in the example has area A =πR 2 and shape factors P= 2 and K = 3 — 2π = 0.477 Elliptical Section An elliptical section with major and minor diameters Land B has area A =πLB/4 and shape factors as shown in Figure 2 P= 2 and 1/K = (B/L+ L/B) π /3 Note the reduction to the circular section result as slenderness ratio B/L → 1. Rectangular Section Arectangular section with sides Land B has area A = LB and shape factors Pand K as shown in Figure 3. Though these results have been computed from an exact series solution, 1 they are quite accurately fit by the optimum approximate Warner solution 2,3 expressions Volume II 123 121-137 4/11/06 12:28 PM Page 123 Copyright © 1983 CRC Press LLC “Narrow-Section” Formulas The previous results for rectangular sections show the asymptotic behavior K →L/B and P→3/2 while holding A = LB as B/L→0 These relations, which correspond to a one-dimensional parabolic pressure distribution (usually attributed to Sommerfeld), are applicable to any narrow section. For example, they hold in the limit for an annular ring with relatively similar inner and outer radii, corresponding to many simple thrust bearings. “Broad-Section” Formulas The previous results for elliptical sections can be expressed as in terms of area and polar moment These relations, which hold exactly for elliptical (and circular) sections, are also applicable approximately to any broad section. Application of this approximation (usually attributed to Saint Venant) to the rectangular section studied previously gives P≈2 and I/K ≈(B/L + L/B) (π/3) 2 so that for a square section P≈2 and K ≈(1/2) (3/π) 2 = 0.456 as compared to the approximate values computed from the Warner solution above P≈2.167 and K ≈0.419 and the numerically exact series values plotted in Figure 3 P = 2.100 and K = 0.421 Similarly, application of the approximations to an equilateral triangular section gives as compared to exact values P = 20/9 = 2.222 and K = √ – 3/5 = 0.346 Volume II 125 121-137 4/11/06 12:28 PM Page 125 Copyright © 1983 CRC Press LLC Other Sections and Surfaces Though the literature 1,4-6 contains exact formulas for normal approach of many other special planar sections (including complete and annular circles and sectors), the results given here should be entirely adequate for most purposes. The literature 1,4-8 also contains results for normal approach of a variety of nonplanar surfaces, including plates with small curvature (single and double), cones (complete and truncated), and spheres of various extents. CYLINDRICALJOURNALBEARINGS 9-13 The “squeeze-film” behavior of nonrotating cylindrical bearings in one-dimensional radial motion is qualitatively quite similar to that for planar bearings in normal approach, and generalization to two-dimensional motion is conceptually straightforward. Remarkably, even the addition of journal rotation causes no real difficulties. Thus solution of general cylindrical journal bearing dynamics problems rests on an understanding of “squeeze-film” behavior in simpler nonrotating cases. One-Dimensional Motion Without Rotation Figure 4 shows a nonrotating journal moving radially downward into a cylindrical half- sleeve. As before, rigorous analysis proceeds from the general Reynolds equation in rec- tangular coordinates wrapped around the journal circumference, a procedure justified by the clearance ratio h/R < < 1. Tangential surface velocities are neglected. Fully flooded ambient boundary conditions assumed at the axial and circumferential ends of the bearing film complete specification of the problem. Solution for pressure, etc., can be numerical or semianalytical. 11 In the latter case, com- putations are facilitated by special tables 23 for the “journal bearing integrals” which arise. General Formulation Relations analogous to previous ones can be expressed in terms of dimensional geometrical and material factors µ, L, D, R, and C and dimensionless functions P, Q, W, M, and J of dimensionless slenderness ratio L/D and dimensionless eccentricity ratio < 1. (Recall that previous dimensionless quantities for planar bearings were constants.) Thus, ∋ 126 CRC Handbook of Lubrication 121-137 4/11/06 12:28 PM Page 126 Copyright © 1983 CRC Press LLC so and For liquid films, which will not support significant negative pressures without rupturing, the short-bearing results given here for the half-sleevebearing of Figure 4 apply equally well to radial motion of the full-sleeve bearing of Figure 6. 128CRC Handbook of Lubrication FIGURE 5.Characteristics for cylindrical bearings in one-dimensional motion (short-bearing film model). (a) Pressure vs. eccentricity, (b) impedance and mobility vs. eccentricity, and (c) impulse vs. eccentricity. 121-137 4/11/06 12:28 PM Page 128 Copyright © 1983 CRC Press LLC vector or scalar quantities plotted over the clearance space of all possible eccentricity ratios. Figure 8 allows a comparison of typical maps 9-13 for the liquid-film short-bearing model (in which film pressure is positive throughout the bearing half with normally approaching surfaces and vanishes in the other). The maps are oriented to velocity or force directions as shown. Dashed/solid curvilinear families indicate magnitude/direction of pressure ratio, mobility, and impedance vectors in Figures 8a, b, and c, respectively. Though the same basic data are displayed in both impedance and mobility maps, each point on one map corresponds to a (different) point on the other. In particular, the sample points indicated in Figures 8b and c do not correspond. 130 CRC Handbook of Lubrication FIGURE 7. Coordinate axes and vectors for two-dimensional motion. FIGURE 8. Characteristics for cylindrical bearing in two-dimensional motion (short- bearing film model). 9-3 (a) Pressure vs. eccentricity, (b) mobility vs. eccentricity, and (c) impedance vs. eccentricity. 121-137 4/11/06 12:28 PM Page 130 Copyright © 1983 CRC Press LLC Generally, such maps are specific to a particular slenderness ratio; for the short-bearing film model vector Pis entirely independent of ratio L/D, while vector W(or M) varies with its square. One-dimensional Figures 5a and b correspond to the midlines of two-dimensional Figures 8a, b, and c. Similarly, the two-dimensional short-bearing approximations are generalizations of the one-dimensional approximations given earlier. More exact map data are available else where. 9-13,24,25 Application of the map data to nonrotating bearings is straightforward: specification of e and e . allows direct determination of F via W; specification of e and F allows direct deter- mination of p* (or e . ) via P (or M). Transformations are required if (as is often the case) the map frames x, y and/or x′,y′ do not coincide instantaneously with the computation frame X, Y. (Graphically, this simply requires rotating maps.) General Formulation — With Rotation For extension of these procedures to problems involving rotation of journal and/or sleeve, consider an “observer” fixed to the sleeve center but rotating at the average angular velocity ω − of journal and sleeve (positive CCW). The absolute journal center velocity e . abs seen in the “fixed” computation frame, X, Y and the relative velocity e . rel apparent to the observer are related to journal eccentricity e and ω − by the simple kinematic expression Since the average angular velocity of journal and sleeve (fluid entrainment velocity) apparent to this observer would vanish, maximum pressure p* and resultant force F would seem to be related solely to the relative (squeeze) velocity e . rel in exactly the same way as for the nonrotating bearings considered previously. Thus extension of the previous procedures to general problems requires only use of the kinematic relation above before the impedance procedure for finding force from (relative) velocity and/or after the mobility procedure for finding (relative) velocity from force; the procedure for finding maximum pressure from force requires no modification, however. The impedance and mobility methods are perfect complements. Both provide for efficient storage of bearing characteristics based on any suitable film model. Because pressure dis- tributions are not calculated, both methods permit efficient computation. In appropriate applications the resulting equations of motion are in explicit form, and iterative calculations can thus be avoided in most system simulation studies. Since the impedance formulation is appropriate to cases in which instantaneous force is desired, it seems most suited to problems in rotating machinery, particularly with damper bearings. Since the mobility formulation is appropriate when instantaneous force is known, it has found widest application in reciprocating machinery. By giving instantaneous journal center velocity, the mobility method provides a basis for predicting an entire journal center path by numerical extrapolation (while allowing simultaneous prediction of maximum film pres- sure). Numerical implementation of the mobility method is straightforward; simplified ver- sions require as few as 50 steps on programmable calculators. A digital computer program which accepts tabulated duty cycle data can be compiled from about 200 FORTRAN statements. 12,13 132 CRC Handbook of Lubrication 121-137 4/11/06 12:28 PM Page 132 Copyright © 1983 CRC Press LLC [...]... Figure 10 13, 21 summarizes 12 0 different minimum film thickness predictions for connecting-rod bearings Copyright © 19 83 CRC Press LLC 12 1 -13 7 4 /11 / 06 12 :28 PM Page 13 5 Volume II 13 5 Table 1 DANGER LEVELS FOR FILM THICKNESSES PREDICTED BY SHORT BEARING FILM MODEL FOR CONNECTING-ROD BEARINGS13 D(typical) mm(in.) Automotive (Otto) Automotive (Diesel) Industrial (Diesel) h(dangerous) µm(µin.) 50(2) 75 10 0... believed to be representative, they are offered for information only It is also interesting to compare these limiting values of predicted oil film thickness with peak-to-valley estimates of surface finish Copyright © 19 83 CRC Press LLC 12 1 -13 7 4 /11 / 06 12 :28 PM 13 6 Page 13 6 CRC Handbook of Lubrication NOMENCLATURE A I L B D R C U ω µ h P P q Q F K W M J e ∋ r ᐉ mrec mrot t r, θ x,y x′,y′ X,Y · * – Area Area... mobility method J Lubr Technol., 93, 16 8 and 315 , 19 71 13 Booker, J F., Design of dynamically loaded journal bearings, in Fundamentals of the Design of Fluid Film Bearings, Rohde, S M., Maday, C J., and Allaire, P E., Eds., American Society of Mechanical Engineers, New York, 19 79, 31 14 Barwell, F T., Bearing Systems, Oxford University Press, Oxford, 19 79, 2 61 15 Campbell, J., Love, P P., Martin, F... value Copyright © 19 83 CRC Press LLC [L2] [L4] [L] [L] [L] [L] [L] [LT–l [T 1] [FL–2T] [L] [FL–2] [–] [L3T 1] [–] [F] [–] [–] [–] [–] [L] [–] [L] [L] [M] [M] [T] [L,–] [L,L] [L,L] [L,L] [T 1] 12 1 -13 7 4 /11 / 06 12 :28 PM Page 13 7 Volume II 13 7 REFERENCES 1 Gross, W A., Matsch, L A., Castelli, V., Eshel, A., Vohr, J H., and Wildmann, M., Fluid Film Lubrication, John Wiley & Sons, New York, 19 80, chap 8 2 Warner,... so long as the maximum bearing load due to Copyright © 19 83 CRC Press LLC 12 1 -13 7 4 /11 / 06 13 4 12 :28 PM Page 13 4 CRC Handbook of Lubrication FIGURE 10 Connecting-rod bearing under inertia loading: minimum film thickness/maximum journal displacement .13 firing alone is no more than about seven times the maximum bearing load due to inertia alone For a particular medium-speed Diesel this means that firing... Technol., 91, 534, 19 69 10 Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic applications (with discussion by Booker, J F.), J Lubr Technol., 99, 19 8, 19 77 11 Booker, J F., Dynamically loaded journal bearings: mobility method of solution, J Basic Eng., 87, 537, 19 65 12 Booker, J F., Dynamically loaded journal bearings: numerical application of the mobility.. .12 1 -13 7 4 /11 / 06 12 :28 PM Page 13 3 Volume II 13 3 FIGURE 9 Connecting-rod bearing polar diagrams .12 ,13 ,15 (a) Journal loading cycle (four-stroke combustion), and (b) journal displacement cycle (short-bearing film model) Connecting-Rod Bearing Example13 Orbit Computation by the Mobility Method Connecting-rod bearings have complex... Bearings for reciprocating machinery: a review of the present slate of theoretical, experimental and service knowledge (with discussion by Booker, J F.), Proc Inst Mech Eng., 18 2(3A), 51, 19 67 16 Blok, H., Full journal bearings under dynamic duty: impulse method of solution and flapping action (with discussion by Booker, J F.), J Lubr Technol., 97, 16 8, 19 75 17 Hays, D F., Squeeze films: a finite journal... discussion by Phelan, R M.), J Basic Eng., 83, 579, 19 61 18 Donaldson, R R., Minimum squeeze film thickness in a periodically loaded journal bearing, J Lubr Technol., 93, 13 0, 19 71 19 Booker, J F Analysis of Dynamically Loaded Journal Bearings: The Squeeze Film Considering Cavitation, Ph.D thesis, Cornell University, Ithaca, N.Y., 19 61 20 New, N H., The use of computer design techniques applied to IC engines,... 533, 19 65 24 Moes, H and Bosnia, R., Mobility and impedance definitions for plain journal bearings, J Lubr Technol., 10 3, 468 , 19 81 25 Goenka, P K., Analytical curve fits for solution parameters of dynamically loaded journal bearings, ASME PaperNo 83-Lub-33, J Tribology, in press 26 Martin, F A., Developments in engine bearings design, in Tribology International, 16 , 14 7, 19 83, from Tribology of Reciprocating . [L,L] · Time derivative [T 1 ] * Special value – Average value ∋ 13 6 CRC Handbook of Lubrication 12 1 -13 7 4 /11 / 06 12 :28 PM Page 13 6 Copyright © 19 83 CRC Press LLC REFERENCES 1. Gross, W. A., Matsch,. about 200 FORTRAN statements. 12 ,13 13 2 CRC Handbook of Lubrication 12 1 -13 7 4 /11 / 06 12 :28 PM Page 13 2 Copyright © 19 83 CRC Press LLC Connecting-Rod Bearing Example 13 Orbit Computation by the Mobility. ratio < 1. (Recall that previous dimensionless quantities for planar bearings were constants.) Thus, ∋ 12 6 CRC Handbook of Lubrication 12 1 -13 7 4 /11 / 06 12 :28 PM Page 12 6 Copyright © 19 83 CRC