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ROLLING-CONTACT BEARINGS 18.10
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BEARINGS AND LUBRICATION
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FIGURE 18.13 Survival function representing endurance tests on rolling-contact bearings from data accumulated by SKF Industries, Inc. (From Ref. [18.2].)
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If LR is in hours and nR is in revolutions per minute, then L10 = 60LRnR. It follows that C10 = FD LDnD LRnR
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(18.6)
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where the subscript D refers to desired (or design) and the subscript R refers to rating conditions.
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18.3 SURVIVAL RELATION AT STEADY LOAD
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Figure 18.14 shows how reliability varies as the loading is modified [18.2]. Equation (18.5) allows the ordinate to be expressed as either F/C10 or L/L10. Figure 18.14 is based on more than 2500 SKF bearings. If Figs. 18.13 and 18.14 are scaled for recovery of coordinates, then the reliability can be tabulated together with L/L10. Machinery applications use reliabilities exceeding 0.94. An excellent curve fit can be realized by using the three-parameter Weibull distribution. For this distribution the reliability can be expressed as R = exp x x0 x0
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where x = life measure, x0 = Weibull guaranteed life measure, = Weibull characteristic life measure, and b = Weibull shape factor. Using the 18 points in Table 18.3 with x0 = 0.02, = 4.459, and b = 1.483, we see that Eq. (18.7) can be particularized as R = exp L/L10 0.02 4.439
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(18.8)
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
ROLLING-CONTACT BEARINGS 18.11
ROLLING-CONTACT BEARINGS
FIGURE 18.14 Survival function at higher reliabilities based on more than 2500 endurance tests by SKF Industries, Inc. (From Ref. [18.2].) The three-parameter Weibull constants are = 4.459, b = 1.483, and x0 = 0.02 when x = L/L10 = Ln/(LRnR).
For example, for L/L10 = 0.1, Eq. (18.8) predicts R = 0.9974.
18.4 RELATING LOAD, LIFE, AND RELIABILITY GOAL
If Eq. (18.3) is plotted on log-log coordinates, Fig. 18.15 results. The FL loci are rectified, while the parallel loci exhibit different reliabilities. The coordinates of point A are the rating life and the basic load rating. Point D represents the desired (or design) life and the corresponding load. A common problem is to select a bearing which will provide a life LD while carrying load FD and exhibit a reliability RD. Along line BD, constant reliability prevails, and Eq. (18.4) applies:
TABLE 18.3 Survival Equation Points at Higher Reliabilities
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
ROLLING-CONTACT BEARINGS 18.12
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FIGURE 18.15 Reliability contours on a load-life plot useful for relating catalog entry, point A, to design goal, point D.
FB = FD
xD xB
(18.9)
Along line AB the reliability changes, but the load is constant and Eq. (18.7) applies. Thus R = exp x x0 x0
(18.10)
Now solve this equation for x and particularize it for point B, noting that RD = RB. xB = x0 + ( x0) ln Substituting Eq. (18.11) into Eq. (18.9) yields FB = C10 = FD xD x0 + ( x0)[ln (1/RD)]1/b
1 RD
(18.11)
(18.12) 1 R and Eq.
For reliabilities greater than 0.90, which is the usual case, ln (1/R) (18.12) simplifies as follows: C10 = FD xD x0 + ( x0)(1 R)1/b
(18.13)
The desired life measure xD can be expressed most conveniently in millions of revolutions (for SKF). Example 1. If a ball bearing must carry a load of 800 lb for 50 106 and exhibit a reliability of 0.99, then the basic load rating should equal or exceed
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
ROLLING-CONTACT BEARINGS 18.13
ROLLING-CONTACT BEARINGS
C10 = 800
50 0.02 + (4.439)(1 0.99)1/1.483
= 4890 lb This is the same as 21.80 kN, which corresponds to the capability of a 02 series 35mm-bore ball bearing. Since selected bearings have different basic load ratings from those required, a solution to Eq. (18.13) for reliability extant after specification is useful: R=1 xD x0(C10/FD)a ( x0)(C10/FD)a
(18.14)
Example 2. If the bearing selected for Example 1, a 02 series 50-mm bore, has a basic load rating of 26.9 kN, what is the expected reliability And C10 = (26.9 103)/445 = 6045 lb. So R=1 50 0.02(6045/800)3 (4.439)(6045/800)3
= 0.9966
The previous equations can be adjusted to a two-parameter Weibull survival equation by setting x0 to zero and using appropriate values of and b. For bearings rated at a particular speed and time, substitute LDnD/(LRnR) for xD. The survival relationship for Timken tapered-roller bearings is shown graphically in Fig. 18.16, and points scaled from this curve form the basis for Table 18.4. The survival equation turns out to be the two-parameter Weibull relation: R = exp x
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