# Fatigue Life Analysis of Cylindrical Roller Bearings Considering Elastohydrodynamic Lubrications
## 3 Calculation of Bearing Fatigue Life
### Modified Bearing Fatigue Life Model  
The conventional L-P theory-based fatigue life model fails to account for lubricant effects in practical applications. Therefore, the established cylindrical roller bearing fatigue life model is modified by integrating elastohydrodynamic lubrication theory.  
A systematic methodology grounded in the I-H theoretical framework is employed to enhance the traditional L-P model's bearing service life calculation. This improved model incorporates the influences of material properties, lubrication characteristics, lubricant contamination, and fatigue limits. The modified formula is expressed as follows [3]:  
\[\mathrm{L}=\mathrm{A_{I}}\mathrm{A_{ISO}}\mathrm{L_{P}} \tag{19}\]  
In this formula,\(\mathrm{A_{I}}\) is the reliability coefficient when the probability of bearing availability is 0.9 and \(\mathrm{A_{ISO}}\) is the total life modification factor.  
The support bearings on the generator's main shaft primarily operate under elastohydrodynamic lubrication conditions under normal operating conditions [26]. To determine the total life modification factor, this section requires the integration of the oil film thicknessdistribution obtained from the elastohydrodynamic lubrication theoretical model to derive the roller-raceway film parameter.  
The expression for the film parameter is given as follows:  
\[\mathrm{C}=\frac{h_{\mathrm{min}}}{\left(\mathrm{s}_{\mathrm{I}}^{2}+\mathrm{s}_ {\mathrm{II}}^{2}\right)^{\frac{1}{2}}} \tag{20}\]  
where \(\mathrm{s}_{\mathrm{I}}\mathrm{ands}_{\mathrm{II}}\) represent the root mean square surface roughness values of the roller and raceway, respectively, with \(\mathrm{s}_{\mathrm{I}}=\mathrm{s}_{\mathrm{II}}=1.25\mathrm{R}_{\mathrm{a}}\), and \(\mathrm{R}_{\mathrm{a}}\) denotes the arithmetic average surface roughness.  
The composite life modification factor \(\mathrm{A}_{\mathrm{ISO}}\) achieves precise bearing life evaluation by incorporating multiple influencing parameters. Its computational model encompasses critical factors including material properties, lubrication conditions, environmental contaminant ingress, fatigue limit stress of contact materials, and operational loading conditions. This study focuses on elucidating the mechanistic effects of lubrication on fatigue life; thus, quantitative calculations of lubrication parameters are prioritized, while other influencing factors are set as fixed parameters. The mathematical expression of this modification factor is formulated as follows:  
\[\mathrm{A}_{\mathrm{ISO}}=0.1\left[1-\left(\mathrm{c}_{1}-\frac{\mathrm{c}_{2} }{\kappa^{\alpha_{1}}}\right)^{\alpha_{2}}\left(\frac{\mathrm{C}_{\mathrm{L}} \mathrm{Q}_{\mathrm{lim}}}{\mathrm{Q}_{\mathrm{e}}}\right)^{\alpha_{3}}\right]^ {\alpha_{4}} \tag{21}\]  
where \(\kappa\) is the viscosity ratio of the lubricant, with \(\kappa=\mathrm{C}^{1.3}\); \(\mathrm{C}_{\mathrm{L}}\) is the contamination coefficient, determined as 0.6 from standard cleanliness tables; \(\mathrm{Q}_{\mathrm{lim}}\) is the fatigue limit stress of the contact materials; \(\mathrm{c}_{\mathrm{I}}\mathrm{and}\mathrm{c}_{2}\) are empirical constants (see Table 1); and \(\alpha_{1},\alpha_{2},\alpha_{3}\), and \(\alpha_{4}\) are exponents (see Table 1).  
The values of the constants and exponents in the mathematical expression of the comprehensive life correction coefficient are given below [25] in Table 1.