Question 24.13: The life of a ball bearing is a normally distributed random ...

\text { Given } \quad \mu=10000 h \quad \hat{\sigma}=500 h \quad R=90 \% .

Step I Standard variable Z_{0} for 90 % reliability The probability of survival of the bearings or reliability is 90% or 0.9. As shown in Fig. 24.19, the shaded area below the normal curve from Z=Z_{0}  to Z = + ∞ should be 0.9. The area below the normal curve from Z = 0 to Z = + ∞ is 0.5.
Therefore, the area below the normal curve from Z = 0 to Z = -Z_{0} should be 0.4. From Table 24.6, the corresponding value of Z_{0} is approximately 1.28.
Therefore,

Table 24.6 Areas under normal curve from 0 to Z

Z_{0}=-1.28 .

Step II Rated life of bearings

Z_{0}=\frac{X_{0}-\mu}{\hat{\sigma}} \text { or }-1.28=\frac{X_{0}-10000}{500} .

\text { or } X_{0}=9360 h .

Therefore 90% of the bearings will complete or exceed a life of 9360 h before fatigue failure.