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Bearing cage fracture: failures expected in the next 12 months from a fleet age table (AFWAL Weibull Analysis Handbook, Table 3.3)

Tool	Warranty Forecasting
Source	Abernethy, R. B., Breneman, J. E., Medlin, C. H., and Reinman, G. L. (1983). Weibull Analysis Handbook. AFWAL-TR-83-2079, Air Force Wright Aeronautical Laboratories, Wright-Patterson Air Force Base, OH. Section 3.10, Figure 3.5 and Table 3.3, pp. 43–47.
Level	Professional

Problem

Six bearing cages have fractured, at 230, 334, 423, 990, 1,009 and 1,510 hours, in a fleet whose 1,703 unfailed bearings have between 50 and 2,050 hours each. The handbook's Weibull plot of these data gives a shape of 2.3 and a characteristic life of 6,487 hours. Each bearing gains 25 hours a month and failed bearings are not replaced. How many more fractures should be expected in the next 12 months?

Inputs to Enter

Age table (units, hours): 288 at 50, 148 at 150, 125 at 250, 112 at 350, 107 at 450, 99 at 550, 110 at 650, 114 at 750, 119 at 850, 128 at 950, 124 at 1,050, 93 at 1,150, 47 at 1,250, 41 at 1,350, 27 at 1,450, 12 at 1,550, 6 at 1,650, 1 at 1,850, 2 at 2,050
Weibull shape β = 2.3, characteristic life η = 6,487 h (entered); usage 25 h per period; 12 forecast periods; no warranty limit, no added units; confidence 90 %

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Answers

Each unit's risk [F(t + 300) − F(t)] / [1 − F(t)]: 0.001198 at 50 h (book 0.0012), 0.01055 at 950 h (book 0.0106), 0.02575 at 2,050 h (book 0.0258)
Failures expected in the next 12 months: 11.56 (book 11.613; the book's table was computed with unrounded parameters, and β = 2.2975, η = 6,485 h reproduce every cell within one unit of its last digit and give 11.61)
Month by month the expected failures rise from 0.76 to 1.17; 90 % bounds on the 12-month count 6.6 to 17.6 (chance variation only, the parameters being entered)
Present risk, the sum of N F(t): 12.26 failures expected to date against 6 observed
Case-study answers: F(1,000 h) = 1.35 % or 23 of 1,703 bearings (book about 1.3 %, 22); F(4,000 h) = 0.28, 477 bearings (book 0.28, 477)
Leaving the shape and scale blank fits the six failures and 1,703 suspensions by maximum likelihood: β = 2.03, η = 11,855 h, 5.1 failures in 12 months with 90 % bounds 1.1 to 11.5

Notes

Choose Fleet ages: AFWAL bearing cage, Table 3.3 in the Data and Example select; it loads these inputs, press Submit. The tool opens on a different case, an illustrative Nevada chart of shipments and returns simulated from a known Weibull distribution, for which no printed example exists. The same select loads the handbook's Tables 3.2, 3.5 and 3.6 (tool 0.2508, 56.24 and 42.75 failures; book 0.2506, 56.2352 and 42.7432). In Table 3.6 the last F(t + u) is printed 0.9660 where the distribution and the row's own risk give 0.9600.