Determination of a Ball Bearing’s Strength at the Lower L₁₀ Life Limit and Its 1, 2 and 3 Sigma Percentiles

In this paper, based on the Hertzian contact analysis, we determine the minimum dynamic load of a ball bearing using a Weibull probabilistic approach. The lower L₁₀ life limit was determined considering a confidence interval, the sample size that corresponds to the L₁₀ life (90% reliability), and the standard deviation of the related Weibull scale parameter η. Then, we determine a functional relationship between the normal and the Weibull distributions to determine the corresponding $1\mathsf{\sigma }$ , $2\mathsf{\sigma }$ , and $3\mathsf{\sigma }$ Six Sigma indices of the ball-bearing life. Finally, a standard method was established to determine the minimum dynamic load rating required for a ball bearing in such a way that it achieves 90% reliability at the lower bound of its ${\mathrm{L}}_{10}$ life. In the case study, the SKF 6009 ball bearing was initially selected, with a dynamic load rating of 22.1 kN, exhibiting a lower bound reliability of 0.86 under a nominal Weibull scale parameter of strength of 910 MPa. After applying the proposed analysis, it was replaced with the SKF 71909 ball bearing, which provides a nominal Weibull scale parameter strength of 1174.54 MPa, which ensures 90% reliability at the lower bound of its L₁₀ life. Its reliability indices at the $1\mathsf{\sigma }$ , $2\mathsf{\sigma }$ , and $3\mathsf{\sigma }$ Six Sigma levels were $\mathrm{R}\left(\mathrm{t}\right)=0.9174$ , $\mathrm{R}\left(\mathrm{t}\right)=0.8880$ , and $\mathrm{R}\left(\mathrm{t}\right)=0.8534$, respectively.