22.1 FIGURE 1 Shaft and 6-spoked bearing system hav- ing three rotor masses. (Product Engineering.) SECTION 22 BEARING DESIGN AND SELECTION Determining Stresses, Loading, Bending Moments, and Spring Rate in Spoked Bearing Supports 22.1 Hydrodynamic Equations for Bearing Design Calculations 22.6 Graphic Computation of Bearing Loads on Geared Shafts 22.13 Shaft Bearing Load Analysis Using Polar Diagrams 22.17 Journal Bearing Frictional Horsepower Loss During Operation 22.21 Journal Bearing Operation Analysis 22.22 Bearing Type Selection of a Known Load 22.23 Shaft Bearing Length and Heat Generation 22.28 Roller-Bearing Operating-Life Analysis 22.31 Roller-Bearing Capacity Requirements 22.32 Radial Load Rating for Rolling Bearings 22.32 Roller-Bearing Capacity and Reliability 22.34 Porous-Metal Bearing Capacity and Friction 22.35 Hydrostatic Thrust Bearing Analysis 22.37 Hydrostatic Journal Bearing Analysis 22.39 Hydrostatic Multidirection Bearing Analysis 22.42 Load Capacity of Gas Bearings 22.46 DETERMINING STRESSES, LOADING, BENDING MOMENTS, AND SPRING RATE IN SPOKED BEARING SUPPORTS Spoked bearing supports are used in gas turbines, large air-cooling fans, electric- motor casings slotted for air circulation, and a variety of other applications. For the shaft and 6-spoked bearing system in Fig. 1 having three rotor masses and these parameters and symbols, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Source: HANDBOOK OF MECHANICAL ENGINEERING CALCULATIONS 22.2 DESIGN ENGINEERING SI values P ϭ 10,000 lb (at either bearing) 4 I ϭ 25 in S 4 I ϭ 0.3 in R 2 A ϭ 7 in (for ring also) 6 E ϭ E ϭ 10 ϫ 10 psi SR L ϭ 10 in R ϭ 12 in C ϭ 0.40 in R (44,480 N) (1040.6 cm 4 ) (12.5 cm 4 ) (45.2 cm 2 ) (68,900 MPa) (25.4 cm) (30.5 cm) (8.9 cm) (1.02 cm) Symbols SI values A ϭ spoke cross-section area, in 2 (cm 2 ) C S ϭ distance, neutral axis to extreme fiber (of spoke), in (cm) C R ϭ distance, neutral axis to extreme fiber (of ring), in (cm) E R , E S ϭ elasticity moduli (ring and spoke), psi (kPa) R R1 ϭ axial loading in inclined spokes, lb; (ϩ for upper two, Ϫ for lower two) (N) F R2 ϭ axial loading in vertical spokes, lb; (ϩ for top, Ϫ for bottom) (N) F T ϭ tangential load at OD of inclined spokes, lb; (clockwise on left side, counterclockwise on right side) (N) I R ϭ outer-ring moment of inertia about neutral axis pependicular to plane of support, in 4 (cm 4 ) I S ϭ spoke moment of inertia about neutral axis perpendicular to plane of support, in 4 (cm 4 ) k ϭ spring rate with respect to outer shell, lb/in (N/cm) L ϭ spoke length, in (cm) M ϭ max bending moment (6-spoked support) in outer ring at OD of vertical spokes, in-lb; ( ϩ at inner-fiber upper point and outer-fiber lower point) ϭ max bending moment at OD of all spokes in 4-spoked support (Nm) P ϭ bearing radial-load, lb (N) R ϭ ring radius, in (cm) T ϭ axial loading in outer ring at OD of vertical spokes in 6-spoked support; all spokes in 4-spoked support ( Ϫ at upper points, ϩ at lower points) lb (N) ϩ denotes tension Ϫ denotes compression find (a) the maximum bending moment in the outer ring of the support, (b) the axial loading in the outer ring, (c) the total stress in the outer ring at the top vertical spoke, Fig. 2, (d) the total axial loading in one of the inclined spokes, (e) the bending moment in the spoke at the hub, and (f) the spring rate of the spoked bearing support. Use the free-body diagram, Fig. 3, to analyze this bearing support. Calculation Procedure: 1. Find the maximum bending moment in the outer ring of the 6-spoke bearing support Use the relation Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION BEARING DESIGN AND SELECTION 22.3 FIGURE 2 Typical 6-spoke bearing support having the mount load at the top for an aircraft gas turbine; in a stationary plant, mount load would be at the bottom of the support. (Product Engineering.) 3 IE R SS ϭ abscissae parameter, Fig. 4 ͩͪͩͪͩͪ ILE RR where the symbols are as shown above. Substituting, we find the parameter ϭ 144, from: 3 25 12 1 ͩͪͩͪͩͪ 0.3 10 1 Using the M curve in Fig. 4 for a parameter value of 144 gives 100M ϭ 1.45 PR Solving for M,wehave 1.45 PR 1.45(10,000)(12) M ϭϭ ϭ1740 in/lb (196.6 Nm) 100 100 2. Determine the axial loading in the outer ring at the outside diameter (OD) Find T, the axial loading from Fig. 4 for the 144 parameter as 10T ϭ 1.55 P Substituting, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION 22.4 DESIGN ENGINEERING FIGURE 3 Free-body diagram for 6- and 4-spoke bearing supports. (Product Engineering.) 1.55P (1.55)(10,000) T ϭϭ ϭ1550 lb (6894 N) 10 10 3. Compute the total stress in the outer ring at the top vertical spoke Use the relation T (0.4) 1550 2 M(C /I ) ϩϭ1740 ϩϭ2320 ϩ 220 ϭ 2540 lb/in (17501 kPa) RR A (0.3) 7 4. Find the total axial loading in one of the inclined spokes Using the F R1 curve in Fig. 4 for the same parameter, 144, 10F 0.82 P (0.82)(10,000) R1 ϭ 0.82 F ϭϭ ϭ820 lb (3647 N) R1 P 10 10 Also, 10F (1.47)(P) (1.47)(10,000) T ϭ 1.47 F ϭϭ ϭ1470 lb (6539 N) T P 10 10 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION BEARING DESIGN AND SELECTION 22.5 FIGURE 4 Six-spoke bearing-support parameter curves. (Product Engineering.) 5. Determine the bending moment in the spoke at the hub The bending moment in the spoke at the hub is (F T )(L) ϭ (1470)(10) ϭ 14,700 in- lb (1661 Nm) ϭ M S . Then, the total stress in this section is CF (3.5) 820 SR1 M ϩϭ(14,700) ϩ ͩͪ S IA 25 7 S 2 ϭ 2060 ϩ 120 ϭ 2180 lb /in (15,020 kPa) 6. Find the spring rate of the spoked bearing support For the 144 parameter, 3 6 kR (2850) IE (2850)(0.3)(10)(10 ) RR ϭ 2850, whence k ϭ k ϭ 33 IE R (12) RR 6 ϭ 4.93 ϫ 10 lb/in (8633 kN /cm) Related Calculations. Figure 5 gives values for 4-spoke bearing supports. Use it in the same way that Fig. 4 was used in this procedure. With more jet aircraft being built, an increase in the use of aero-derivative gas turbines in central-station and industrial power plants, wider use of air conditioning throughout the world, and construction of larger and larger electric motors, the Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION 22.6 DESIGN ENGINEERING FIGURE 5 Four-spoke bearing-support parameter curves. (Product Engineering.) spoked bearing support is gaining greater attention. The procedure presented here can be used for any of these applications, plus many related ones. In calculations for spoked bearings the rings are assumed supported by sinuso- idally varying tangential skin-shear reactions. Spokes are also assumed pinned to the ring but rigidly attached to the hub. This procedure is the work of Lawrence Berko, Supervising Design Engineer, Walter Kide & Co., as reported in Product Engineering magazine. SI values were added by the handbook editor. HYDRODYNAMIC EQUATIONS FOR BEARING DESIGN CALCULATIONS Bearing design also requires a number of hydrodynamic formulas involving hy- draulics, fluid flow, power, pressure head, torque, fluid viscosity, and fluid density. Table 1 presents useful formulas for the hydrodynamic design of bearings in both USCS and SI units. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION 22.7 TABLE 1 Hydrodynamic Equations for Bearing Design Name Unit Symbol Formula or value System Mass Slugs M 2 lb ϫ s ft USCS Kilogram mass Metric slug M 2 kg ϫ s m SI Gram mass 2 dyn ϫ s cm M 2 dyn ϫ s cm SI Gravitational constant ft 2 s g 32.174 (in London) USCS m 2 s g 9.807 (in Paris) SI Force dyn P 1 g 981 SI Poundal P 1 lb 32.174 USCS Pressure head ft H For water, 1 ft equals 0.433 lb / in 2 or 62.335 lb/ft 2 USCS Rated work hp N 550 ft ⅐ lb / s or 33,000 ft ⅐ lb / min USCS hp N 3 ft head ϫ sp gr ϫ or s 8.8 USCS gal head ϫ sp gr ϫ min 3960 hp N 3 ft lb 1 ϫϫ or 2 min ft 33,000 USCS gal lb 1 ϫϫ 2 min in 1714 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION 22.8 TABLE 1 (Continued ) Name Unit Symbol Formula or value System Torque lb ⅐ ft T hp ϫ 33,000 5250 ϭ rpm ϫ 2 ␲ rpm USCS Density Mass Unit volume ␳ 2 lb s slugs ϫϭ 33 ft ft ft SI Mass Unit volume ␳ 2 gs ϫ 3 cm cm Absolute viscosity in USCS units Mass Length ϫ Time ␮ slugs lb ⅐ s ϭ 2 ft ⅐ sft USCS 1 unit of ϭ 178.69 P slugs ft ⅐ s ␮ A lb ⅐ min ϭ 4,136,000 P 2 in poundal ⅐ s ϭ 14.88 P 2 ft Absolute viscosity in SI units P 1 dyn ⅐ s 2 cm SI cP Z 1 P 100 g ⅐ s 2 cm 981 P Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION 22.9 Kinematic viscosity Area Time ␯ ␮ absolute viscosity ␯ ϭϭ ␳ density USCS and SI St, 2 cm s 1P Density SI cSt 1 St 100 Saybolt universal seconds V For conversion of SUS units into centistokes SI When SUS Ϲ 100 cSt ϭ 0.226 SUS. Ϫ 195 / SUS. When SUS. ϭ 100 cSt ϭ 0.220 SUS. Ϫ 135 / SUS. Specific viscosity Dimensionless Ratio of absolute viscosity of any fluid to that of water at a temperature of 20 ЊC Absolute value Viscosity of water at a temperature of 20 ЊC cP ZZ ϭ 1cP SI Fluidity Length ϫ time Mass ␾␾ ϭ inverse value of absolute viscosity 1 ␮ SI and USCS Reynolds number Dimensionless N R ␳ vd vd N ϭϭ R ␮␯ v ϭ velocity of fluid ␳ ϭ density d ϭ pipe diameter ␮ ϭ absolute viscosity ␯ ϭ kinematic viscosity Absolute value Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION 22.10 TABLE 1 (Continued ) Name Unit Symbol Formula or value System Critical Reynolds number for pipe flow Dimensionless N e N e ϭ 2320 ϭ Reynolds number which represents the separation point between laminar and turbulent flow Absolute value For N R Ͼ 2320, turbulent flow For N R Ͻ 2320 laminar flow From N R ϭ 1920 to 4000, instability flow Friction loss formula for pipe flow ft or m H ƒ H ƒ ϭ ƒ ϫ 2 1 v ϫ d 2g SI and USCS v ϭ velocity, ft / s or m / s d ϭ pipe dia., ft or m l ϭ pipe length, ft or m 22 g ϭ gravity constant, ft / s or m / s ƒ ϭ flow coefficient Flow coefficient for laminar (viscous) flow Dimensionless ƒ 64 ƒ ϭ N R Absolute value Grade or roughness of pipe is immaterial Flow coefficient for clean cast- iron pipe—circular section Dimensionless ƒ 0.214 ƒ ϭ turbulent flow 5 N ͙ R 64 ƒ ϭ laminar flow N R Absolute value Flow coefficient for very smooth pipe, circular section Dimensionless ƒ 0.316 ƒ ϭ turbulent flow 4 N ͙ R 64 ƒ ϭ laminar flow N R Absolute value Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. BEARING DESIGN AND SELECTION [...]... given at the website BEARING DESIGN AND SELECTION BEARING DESIGN AND SELECTION 22 . 23 FIGURE 15 Various zones of possible lubrication for a journal bearing by the bearing, W ϭ friction factor (P)(␲)(bearing diameter, in / 12 in / ft)(rpm) Substituting, W ϭ 0. 02( 1000)(␲) (2. 25 / 12) (20 0) ϭ 23 5 6 ft-lb / min ( 53. 2 W) 3 Compute the work of friction and total heat generated (d) The work of friction, w ϭ W /... Vr ϭ 3. 1416(8. 12) (1800) ϭ 37 70 ft / min (1149 m / min) 2 Compute the rubbing surface area The rubbing surface area of a journal bearing, Ar ϭ (d / 2) (␣)(␲)(L), where d ϭ bearing diameter, in; ␣ ϭ optimum bearing angle, degrees; L ϭ bearing length, in Substituting, Ar ϭ (8 / 2) (140 / 180)(␲)(9) ϭ 87.96 in2 (567.5 cm2) 3 Calculate the bearing pressure With a total load of 20 ,000 lb (9080 kg), the bearing... situation in which the life of the bearing is of greater importance than its size Such a situation is common when the reliability of a machine is a key factor in its design A dynamic rating of a given amount, say 72, 400 lb ( 32 2, 051 .3 N), means that if in a large group of bearings of this size each bearing has a 72, 400-lb ( 32 2, 051 .3- N) load applied to it, 90 percent of the bearings in the group will complete,... website BEARING DESIGN AND SELECTION BEARING DESIGN AND SELECTION 22 .29 FIGURE 17 Bearing temperature limits (Product Engineering. ) ϭ allowable mean bearing pressure, lb / in2 [ranges from 25 to 25 00 lb / in2 (1 72. 4 to 17, 23 7 .5 kPa) for normal service and up to 8000 lb / in2 (55,160.0 kPa) for severe service], on the projected bearing area, in2 ϭ ld; d ϭ shaft diameter, in Thus, for this bearing, assuming... Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 06 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website BEARING DESIGN AND SELECTION 22 .20 DESIGN ENGINEERING SI Values 24 00 lb (10,675 N) 28 0 lb ( 124 5 N) 1800 lb (8006 N) (2. 54 cm = 533 8 N) FIGURE 13 Polar diagram of sheave drive showing... AND SELECTION 22 .34 DESIGN ENGINEERING load RT is the sum of the belt, shaft, and pulley loads, or RT ϭ 720 ϩ 145 ϭ 865 lb (38 47.7 N) 7 Compute the required radial capacity of the bearing The required radial capacity of a bearing RC (865)(1 .34 0)(1 . 32 ) (2. 0)(0.86) ϭ 2 630 lb (11,698.8 N) ϭ RTƒLƒOƒBƒS ϭ 8 Select a suitable bearing Enter the manufacturer’s engineering data at the shaft rpm (30 0 r / min for... value System BEARING DESIGN AND SELECTION Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 06 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 22 . 12 BEARING DESIGN AND SELECTION BEARING DESIGN AND SELECTION 22 . 13 GRAPHIC COMPUTATION OF BEARING LOADS ON GEARED SHAFTS Geared shafts having... Terms of Use as given at the website BEARING DESIGN AND SELECTION BEARING DESIGN AND SELECTION 22 .25 TABLE 3 Key Characteristics of Rolling and Sliding Bearings* 5 Evaluate rolling bearings Rolling bearings have lower starting friction (coefficient of friction ƒ ϭ 0.0 02 to 0.005) than sliding bearings (ƒ ϭ 0.15 to 0 .30 ) Thus, the rolling bearings is preferred for applications requiring low staring... requirements of sleeve and rolling-element bearings to carry the same diameter shaft (Product Engineering. ) TABLE 5 Oil-Film Journal Bearing Characteristics* The running friction of rolling bearings is in the range of ƒ ϭ 0.001 to 0.0 02 For oil-film sliding bearings, ƒ ϭ 0.0 02 to 0.005 Rolling bearings are more susceptible to dirt than are sliding bearings Also, rolling bearings are inherently noisy Oil-film bearings... have, FT ϭ 2( 100) / 1.75 ϭ 114 lb (508 N) Then, the torque on the center shaft ϭ D(FT) ϭ 2( 114) ϭ 22 8 lb / in (25 .8 Nm) The gear loads are computed from FA ϭ (percent torque on shaft) (2) (torque on center shaft, lb / in)(sec of pressure angle on the gear) / D, where FA ϭ gear load, lb (N), on gear A, Fig 7 Substituting using the given and computed values, FA ϭ 0.4 (2) (22 8)(sec 20 Њ) / 2 ϭ 97 lb ( 431 .5 N) . Bearing Frictional Horsepower Loss During Operation 22 .21 Journal Bearing Operation Analysis 22 .22 Bearing Type Selection of a Known Load 22 . 23 Shaft Bearing Length and Heat Generation 22 .28 Roller-Bearing. 22 .28 Roller-Bearing Operating-Life Analysis 22 .31 Roller-Bearing Capacity Requirements 22 . 32 Radial Load Rating for Rolling Bearings 22 . 32 Roller-Bearing Capacity and Reliability 22 .34 Porous-Metal Bearing. and Friction 22 .35 Hydrostatic Thrust Bearing Analysis 22 .37 Hydrostatic Journal Bearing Analysis 22 .39 Hydrostatic Multidirection Bearing Analysis 22 . 42 Load Capacity of Gas Bearings 22 .46 DETERMINING