According to Lubkin (1962), stresses in the x, y, and z directions are principal stresses, and are defined by the maximum pressure at the center of the contact area as defined by the expression p max = 3F/(2 π a 2 ) (10.36) For an element on the z-axis the stresses are: (10.37) (10.38) The equations given above also apply to the contact of a sphere and a plane surface or to a sphere and an internal spherical surface. For a plane surface use d =∞. For an inter- nal surface the diameter is expressed as a negative quantity. A ball bearing’s inner raceway may be considered to be the segment of a sphere. The rolling elements have a mating surface. Hence, the two contacting surfaces may be ide- alized as spheres loaded against each other. If the raceway diameter = 2.25 in, ball diam- eter = 0.775 in, and contact force F = 15 lb, determine the contact area and stress levels. Use Young’s modulus E 1 = E 2 = 30 × 10 6 lb/in 2 and Poisson’s ratio n 1 = n 2 = 0.3 for both bodies. Solution The pressure within each sphere has a semielliptical distribution, as shown in Fig. 10.48. Maximum pressure occurs at the center of the contact area. For an internal surface, the diameter is negative. Then the contact area and maximum pressure are cal- culated to be Figure 10.49 provides variation in the stress components for a distance 3a below the surface. Note that shear stress t reaches a maximum value slightly below the surface. Maximum shear stress is considered to be the leading cause of surface fatigue failure in contacting elements. A crack initiating at the point of maximum shear stress below the surface and lubricant pressure flowing into the crack region may be enough to dislodge metal chips. When the contacting surfaces are cylindrical the contact area is a narrow rectangle of half width b, given by the equation (10.39) where l is the length of the contact area. The pressure distribution across the width 2b is elliptical, and maximum pressure is given by p max = 2F/pbl. When applied to a cylinder and a plane surface, for the plane use d =∞. For an inter- nal cylinder d is negative. To evaluate stresses, select the origin of a reference system at the center of the contact area with x parallel to the axes of the cylinders, y perpendic- ular to the plane formed by the two cylinder axes, and z in the plane of the contact force. b FE E ld d = − ( ) [] +− ( ) [] + 21 1 11 12 12 nn p 1 2 2 2 // [( / ) ( / )] a p =× ×− ×+− × − + = =× ×× = {( / ) [( . )/ ( . )/ ]/( / . / . )} . /( . ) , / max 3 15 8 1 0 3 30 10 1 0 3 30 10 1 2 25 1 0 775 0 007425 3 15 2 0 00739 129 895 26 26 13 2 in. lb/in. 2 π σ z p a az =− + max 2 22 σσ ν xy p z a a z a az == + − + + − max ()11 1 2 1 2 22 Tan BEARINGS AND SEALS 423 424 COMPONENT DESIGN FIGURE 10.49 Stress components below the surface of contacting spheres. FIGURE 10.50 Stress components below the surface of contacting cylinders. For elements on the z axis, principal stresses σ x , σ y , and σ z exist. Figure 10.50 shows a plot of the stresses for depths up to 3b below the contact surface. For the contacting spheres three different shear stresses are created, given by (10.40) This shear stress is also shown in Fig. 10.49, labeled t max . It reaches a peak value slightly below the surface, similar to what is seen in the case of the contacting spheres. Problem 10.2 The life of a bearing is defined as the total number of revolutions, or the number of hours at a fixed speed, of bearing operation required for the failure cri- teria to develop. Under ideal conditions the first evidence of fatigue failure will con- sist of a spalling of the load carrying surfaces. The Timken Company goes by the failure criterion of pitting or spalling of an area of 0.01 in. 2 In testing a group of bear- ings the objective is to establish the median and the L 10 , or rated, life. The L 10 implies that 90 percent of identical bearings operating at a constant speed and load will com- plete or exceed the test before the failure criterion develops. When a number of batches of bearings are under test, the median life is usually between four and five times the L 10 life. The concept of probable survival of a batch of bearings also needs examination. If a machine uses N bearings with each bearing having the same reliability R, then the relia- bility of all the bearings is (R) N . The distribution of bearing failures can be approximated by the Weibull procedure. By making adjustments to the Weibull parameters (Mischke, 1965), the distribution of bearing failures takes the form (10.41) If a certain application requires a reliability of 98 percent for the bearing to last for 2500 h, determine the rated life. Solution Using L = 2500 and R = 0.98 in the above equation Then the rated bearing life L 10 is calculated to be 10,262 h. Problem 10.3 Experiments indicate that identical bearings acting under different radial loads F 1 and F 2 and operating at speeds n 1 and n 2 have lives L 1 and L 2 according to the relation (10.42) where a = 3 for ball bearings and 10/3 for roller bearings. A roller bearing can safely accept a load of 4.5 kN at 650 rpm for an L 10 life of 1400 h. Determine its life if the load is reduced to 3.75 kN and the speed is increased to 725 rpm. FF nL nL a 21 1 2 1/ = 1 2 098 2500 6.84 1.17 . exp=− L 10 R L L =− exp . . 684 117 10 ττ σσ σσ xz yz xz yz == − = − 22 BEARINGS AND SEALS 425 Solution In equation (10.42) F 1 = 4.5 kN, F 2 = 3.75 kN, n 1 = 650 rpm, n 2 = 725 rpm, L 1 = 1400 h, and a = 10/3. Then L 2 = 2305 h. Problem 10.4 Equations (10.41) and (10.42) may be combined to obtain specified levels of load, speed, life, and reliability factor. Then (10.43) Determine the load rating F 2 if the bearing is to have a reliability level of 95 percent. Solution Substituting the values gives F 2 = 4.51 kN. Problem 10.5 Data for a journal bearing are as follows: viscosity m = 3.95 × 10 −6 reyn, speed N = 1800 rpm, radial load W = 525 lb, radius R = 0.875 in, clearance c = 0.0014 in, and length l = 1.625 in. Determine the bearing’s characteristic number, min- imum film thickness, and its angular location. Solution Bearing l/d = 0.929 and pressure P = (W/2Rl) = 184.6 lb/in. 2 Equation (10.9) provides the bearing Sommerfeld, or characteristic, number S = (.875/.0014) 2 × {3.95 × 10 −6 × 1800/(184.6 × 60)} = 0.251. From charts for minimum film thickness and eccen- tricity ratio (Raimondi and Boyd, 1958), the ratio of minimum film thickness and clear- ance h o /c = 0.54 and eccentricity ratio e = 0.46, and since clearance c = 0.0014, minimum film thickness h o = 0.00076 in and eccentricity e = 0.00064 in. For no load, eccentricity e = 0.0 and film thickness h = 0.0014. As the load is increased the journal is forced downward. Figure 10.51 shows the distribution of hydrodynamic pressure in the lubricant film. FF nL nL R a a 21 1 2 1 1 1.17 (6.84) 1 [(1)] = 1 2 / / ln / 426 COMPONENT DESIGN FIGURE 10.51 Hydrodynamic pressure distribution in fluid film journal bearing (Raimondi and Boyd, 1958). Problem 10.6 In Prob. 10.5, determine the coefficient of friction, lubricant flow, film pressure, and temperature rise. Solution From the chart for coefficient of friction (Raimondi and Boyd, 1958), the value is (R/c)f = 5.2. Hence, the coefficient f = 0.00832. The torque required to over- come this frictional loss is T = fWR = 3.82 in⋅lb, which is equivalent to HP = TN/63000 = 0.1092 hp. From the chart for lubricant flow, 60Q/RcNl = 4.08 and Q = 0.244 in 3 /s. Leakage at the two ends of the bearing is obtained from the charts, where Q s /Q = 0.54, so Q s = 0.132 in 3 /s. Heat generated by friction is dissipated by conduction, convection, and radiation, and also carried away by the oil flow. A conservative assumption calls for all the heat to be extracted by the oil. The temperature rise in degrees Fahrenheit is given by the expression The maximum pressure developed in the film is obtained from the charts in the form of pressure ratio P/P max = 0.47. Since P = 184.6, P max = 392.8 lb/in. 2 REFERENCES Allaire, P. E., Li, D. F., and Choy, K. C., “Transient unbalance response of four multi-lobe journal bearings,” Journal of Lubrication Technology, 1980. Allison G., Turbine Division, TM # 55-2840-231-23, 1981. ASME Report—Pressure/Viscosity in Rolling Element Bearings, Vol. II, ASME Report, New York, 1954. Bailey, J. K., and Galbato, A. T., “Evaluating bearings for high speed operation,” Machine Design, October 1981. Childs, D., Turbo-Machinery Rotor Dynamics, John Wiley & Sons, New York, 1993. Childs, D., and Kleynhans, G., “Experimental rotor dynamic and leakage results for short (L/D = 1/6) honeycomb and smooth annular pressure seals,” Proceedings of the 5th International Conference on Vibrations in Rotating Machinery, Institute of Mechanical Engineers, London, 1992. Darden, J. M., Earhart, E. M., and Flowers, G. T., “Comparison of dynamic characteristics of smooth annular seals and damping seals,” ASME Paper # 99-GT-177, New York, 1999. Ehrich, F. F., “The influence of trapped fluids on high speed rotor vibrations,” Journal of Engineering for Industry 89(4):806–812, 1967. Ehrich, F. F., Handbook of Rotor Dynamics, Krieger Publishing Co., Malabar, FL, 1999. Ferguson, J., “Brushes as high performance gas turbine seals,” ASME Paper # 88-GT-182, New York, 1988. Forster, N. H., “High temperature lubrication of rolling contacts with lubricants delivered from the vapor phase and as oil mists,” Ph.D. Thesis, University of Dayton, Ohio, 1996. Friswell, M. I., and Penny, J. E. T., “The choice of orthogonal polynomials in the rational fraction poly- nomial method,” International Journal of Analytical Experimental Modal Analysis 8(3):257–262, 1993. Greathead, S., and Bostow, P., “Investigations into load dependent vibrations of high pressure rotor on large turbo-generators,” Proceedings of the Conference on Vibrations in Rotating Machinery, Institute of Mechanical Engineers, Cambridge, pp. 279–286, 1976. Gunter, E. J., Barrett, L. E., and Allaire, P. E., “Design and application of squeeze film dampers for turbo-machinery stabilization,” Proceedings of the 4th Turbo-Machinery Symposium, Texas A & M University, College Station, Tex., pp. 127–141, 1975. ∆T P Rf c Q Q Q RcNl Fs =×× − =×× − × =° { . ( / )} / {[ / ( / )][ / ]} . . . /[( . / ) . ] . 0 103 1 1 2 60 0103 1846 52 1 0542 408 33 2 F. BEARINGS AND SEALS 427 Hagg, A. C., and Sankey, G. O., “Some dynamic properties of oil film journal bearings with reference to unbalance vibration of rotors,” Journal of Applied Mechanics 23(2):302–306, 1956. Harris, T. A., Rolling Bearing Analysis, John Wiley & Sons, New York, 1984. Hertz, H., “The contact of elastic solids,” J. Reine Angew Math. 92:156–171, 1881. Jones, A. B., “Analysis of stress and deflections,” New Departure Engineering Data, Bristol, Conn., 1946. Holmes, R., and Box, S., “On the use of squeeze film dampers in rotor support structures,” Machine Vibration 1:71–92, 1992. Kirk, R. G., “Oil seal dynamics: Considerations for analysis of centrifugal compressors,” Proceedings of the 15th Turbo-Machinery Symposium, Texas A & M University, College Station, Tex., 1986. Kirk, R. G., “A Method for Calculating Labyrinth Seal Inlet Swirl Velocity,” Rotating Machinery Dynamics, Vol. 2, pp. 345–350, ASME, New York, 1987. Liao, N. T., and Lin, J. F., “A new method for the analysis of deformation and load in a ball bearing with variable contact angle,” ASME Journal of Mechanical Design, New York, NY., 1999. Lubkin, J. L., “Contact problems,” in W. Flugge (ed.), Handbook of Engineering Mechanics, Sec. 42-1, McGraw-Hill, New York, 1962. Lund, J. W., “Spring and damping coefficients for the tilting pad journal bearing,” Transactions 7(4):342–352, 1964. Lund, J. W., and Thomsen, K. K., “A calculation method and data for the dynamic coefficients of oil lubricated journal bearings,” Topics in Fluid Film Bearing and Rotor Bearing System Design and Optimization, ASME, New York, pp. 1–28, 1978. Marquette, O. R., Childs, D. W., and San Andres, L., “Eccentricity effects on rotor dynamic coeffi- cients of plain annular seals: Theory versus experiment,” Journal of Tribology 119:443–448, 1997. Mischke, C., “Bearing reliability and capacity,” Machine Design 37(22):139–140, September 1965. Orcutt, F. K., “The steady state and dynamic characteristics of the tilting pad journal bearing in lami- nar and turbulent flow regimes,” Journal of Lubrication Technology 89(3):392–404, 1967. Padavala, S., Palazzolo, A. B., Vallely, D. P., and Ryan, S. G., “Application of an improved Nelson-Nguyen analysis to eccentric arbitrary profile liquid annular seals,” Workshop on Rotor Dynamic Instability Problems in High Performance Turbo-machinery, Texas A & M University, Tex., pp. 113–115, 1993. Raimondi, A. A., and Boyd, J., “A solution for the finite journal bearing and its application to design and analysis, Parts I, II, and III,” Transactions of ASLE, Vol. 1, Lubrication Science and Technology, Pergamon, New York, No. 1, pp. 159–209, 1958. Redmond, I., “Rotor dynamic modeling utilizing dynamic support data obtained from field impact tests,” Proceedings of the 6th International Conference on Vibrations in Rotating Machinery, Paper # C500/055/96, Oxford, 1996. Santhanan, C. K., and Koerner, J., “Transfer function synthesis as a ratio of two complex polynomi- als,” IEEE, Transactions Automatic Control, SME, Bethel, Conn., pp. 56–68, 1963. Santiago, O., San Andres, L., and Oliver, J., “Imbalance response of a rotor supported on open end integral squeeze film dampers,” ASME Paper # 98-GT-6, New York, 1998. Sawyer, T., Gas Turbines, Vols. I, II, and III, International Gas Turbine Institute, ASME, Atlanta, 1982. Sedy, J., “Improved performance of film-riding gas seals through enhancement of hydrodynamic effects,” ASLE Transactions 23(1):35–44, 1979. Shemeld, D., “A history of development in rotor dynamics—A manufacturers viewpoint,” Rotor Dynamic Instability Problems in High Performance Turbo-Machinery, NASA Report # CP 2443, pp. 1–18, 1986. Soto, E. A., and Childs, D. W., “Experimental rotor dynamic coefficient results for (a) a labyrinth seal with and without shunt injection and (b) a honeycomb seal,” ASME Paper # 98-GT-8, New York, 1998. Spakovszky, Z. S., Paduano, J. D., Larsonneur, R., Traxler, A., and Bright, M. M., “Tip clearance actu- ation with magnetic bearings for high-speed compressor stall control,” ASME Paper # 2000-GT-528, New York, 2000. 428 COMPONENT DESIGN Stephenson, R. W., and Rouch, K. E., “Generating matrices of the foundation structure of a rotor sys- tem from test data,” Journal of Sound and Vibrations 154(3):467–484, 1992. Stribeck, R., “Ball bearing for various loads,” Vol. 29, ASME, New York, pp. 420–463, 1947. Van Treuren, K. W., Barlow, D. N., Heiser, W. H., Wagner, M. J., and Forster, N. H., “Investigation of vapor phase lubrication in a gas turbine engine,” ASME Paper # 97-GT-3, New York, 1997. Vazquez, J. A., Barrett, L. E., and Flack, R. D., “Flexible bearing supports using experimental data,” ASME Paper # 00-GT-404, New York, 2000. Weigl, H., Paduano, J., Frechette, L., Epstein, A., Greitzer, E., Bright, M., and Strazisar, A., “Active stabilization of rotating stall and surge in a transonic single stage axial compressor,” ASME Journal of Turbo-Machinery 120:625–636, 1998. Wilcox, D. F., and Booser, E. R., Bearing Design and Application, Mc-Graw-Hill, New York, 1957. Zeidan, F. Y., San Andres, L., and Vance, J. M., “Design and application of squeeze film dampers in rotating machinery,” Proceedings of the 25th Turbo-Machinery Symposium, Texas A & M University, College Station, Tex., 1998. BIBLIOGRAPHY AFBMA Standards for Ball and Roller Bearings, Revision # 4, June 1972. American Petroleum Institute, “Centrifugal compressors for petroleum, chemicals and gas service industries,” API Standard, Vol. 617, 6th ed., 1995. Bently, D. E., “Monitoring rolling element bearings” 3(3):2–15, 1982. Boto, P. A., “Detection of bearing damage by shock pulse measurement,” Ball Bearing Journal, 1971. Eshelman, “The role of sum and difference frequencies in rotating machinery fault diagnosis,” Paper # C272/80, Institute of Mechanical Engineers, London, 1980. Lees, A. W., Friswell, M. I., Smart, M. G., and Prells, U., “The identification of foundation vibration parameters from running machine data,” Proceedings of the 7th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, ISROMAC-7, SME, Honolulu, Bethel, Conn., pp. 715–724,1998. Mathew, J., and Alfredson, R. J., “The condition monitoring of rolling element bearings using vibra- tion analysis,” ASME Journal of Vibration and Acoustics 106:447–453, 1984. Monk, R., “Vibration measurement gives early warning of mechanical fault,” Process Engineering, 135–137, November 1972. Pinkus, O., and Sternlicht, B., Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961. Shigley, J. E., and Mitchell, L. D., Mechanical Engineering Design, 4th ed.; McGraw-Hill, New York, 1983. Yu, J. J., Bently, D. E., Goldman, P., Dayton, K. P., and Van Slyke, B. G., “Rolling element bearing defect detection and diagnostics using displacement transducers,” ASME Paper # 01-GT-028, New York, 2001. BEARINGS AND SEALS 429 This page intentionally left blank. MATERIALS AND MANUFACTURE P ● A ● R ● T ● 3 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use. This page intentionally left blank. [...]... the particle because of the increased flexibility of the dislocations as they interact with coarser particles Dislocation pairs interacting with the particles can occur when one of them just shears the particle while the other is pulled forward by the boundary remaining in all particles cut by the first dislocation This may be expected to happen at long aging times In the dispersion form of imparting... (Sims, Stoloff, and Hagel, 1987) Hardening of austenitic alloys by particles is affected by the strain energy, differences in elastic modulii, and stacking fault energy between the particle and the matrix, creating additional particle–matrix interface and lattice resistance of particles with temperature When a dislocation cuts an ordered particle, the force on the dislocation must balance the antiphase... do not have identical yield strength in compression and tension, the difference depending on the orientation Creep rupture is also affected by the size of the g ′ particle (Gell and Duhl, 1986), as shown in Fig 11 .12 for PWA-1480 Smaller g ′ particles permit the dislocations to climb around them and reduce the creep strength, but the strength is maximized when the dislocations cut through With even larger... Studies of precipitate morphology of cobalt base alloys indicate a similar behavior with nickel base alloys Flow stresses are relatively insensitive to temperature when the particles are sheared by paired dislocation But as the particles are redissolved on aging with test temperatures above 930°F, the flow stress drops rapidly Increased volume fraction causes the flow stress to rise in both aged cobalt... dislocations do not alter to obtain the right shearing sequence, and deformation is a consequence of slippage alone A constant volume fraction causes larger particles to be more beneficial in restricting primary creep, as the line tension precludes penetration of the particles Factors that control steady-state creep resistance in single-phase crystalline solids are diffusivity, elastic modulus, temperature, stacking... second-phase particles the activation energy for creep is higher than for self-diffusion The difference can be minimized by including the temperature dependence of the elastic modulus Development of a substructure during primary creep and after substantial strain hardening helps in raising the steady-state creep in MAR-M 200 at 1400°F Dislocations forming between the g and g ′ are limited from moving by the particle... creep 11.3 NICKEL BASE ALLOYS Nearly half the total weight of aircraft engines comprises parts made from nickel base alloys The alloys are favored in elevated temperature regions of the power plant, in spite of the complex physical metallurgy The tensile- and creep-rupture strength up to 5000 h of these alloys in the 120 0 to 2000°F temperature zone makes them the prime candidates for turbine blades Industrial... solutions, with air-cooling causing finer particles that dissolved in the subsequent phase Aging causes the g ′ to grow again The final age results in a combination of moderate tensile strength and rupture life required for longer lasting turbine airfoils Cast alloys may be heat treated to a simpler cycle After cooling in the mold, aging may take place for 12 h at 1400°F More complex alloys may require... interaction of dislocations within the faults, especially when second phase particulate occur during service exposure But ductility is likely to be minimized in the temperature transition range Addition of nickel alleviates the potential phase instability associated with temperature cycling, and raises the stacking fault energy to reduce partial dislocations Thermomechanical processing controls the microstructure... stability of the materials at high temperatures Addition of fine particles of inert and thermodynamically stable ThO2 or Y2O3 provide excellent creep rupture strength to temperatures nearing the melting point of the base Evaluation of fracture behavior of cast and heat-treated MM-509 indicates that fracture initiates in the large carbide particles and eutectic concentrations with the onset of plastic . of squeeze film dampers for turbo- machinery stabilization,” Proceedings of the 4th Turbo- Machinery Symposium, Texas A & M University, College Station, Tex., pp. 127 –141, 1975. ∆T P Rf c Q. “Evaluating bearings for high speed operation,” Machine Design, October 1981. Childs, D., Turbo- Machinery Rotor Dynamics, John Wiley & Sons, New York, 1993. Childs, D., and Kleynhans, G., “Experimental. 1 11 12 12 nn p 1 2 2 2 // [( / ) ( / )] a p =× ×− ×+− × − + = =× ×× = {( / ) [( . )/ ( . )/ ]/( / . / . )} . /( . ) , / max 3 15 8 1 0 3 30 10 1 0 3 30 10 1 2 25 1 0 775 0 007425 3 15 2 0 00739 129