# Fatigue Life Analysis of Cylindrical Roller Bearings Considering Elastohydrodynamic Lubrications
## 2 Quasi-Static Modeling of Cylindrical Roller Bearings Considering Elastohydrodynamic Lubrication
### Establishment of the Quasi-Static Equations  
After numerically solving the elastohydrodynamic lubrication equations and obtaining the oil film thickness and pressure distribution, the dimensionless film thickness parameter H is substituted into Equation (9). Through a nonlinear iterative algorithm, the numerical solution of the elastic deformation \(\delta\) can then be determined.  
The contact characteristics of rolling element bearings satisfy the following three fundamental assumptions: (1) the contacting bodies undergo solely elastic deformation, exhibiting an elastic response that obeys [PERSON]'s law; (2) the contact surfaces are ideally smooth and subjected exclusively to normal loads, with no tangential friction acting; and (3) the dimensions of the contact zone are significantly smaller than the radii of curvature of the contacting bodies.  
Therefore, according to [PERSON]'s contact theory, the relationship between the contact load and the elastic deformation is as follows [25]:  
\[\rm\delta=1.36\frac{(kQ)^{n}}{10.8} \tag{10}\]  
Figure 3: Geometric approximation diagram.  
where Q is the contact load between the cylindrical roller and the inner ring or the outer ring; k is the load-deformation constant between the cylindrical roller and the inner or outer ring, which is related to the geometric characteristics and the material; n is an exponential, where for point contact, n = 3/2 and, for line contact, n = 10/9; \(\delta\) is the contact elastic variables between the cylindrical roller and the inner or outer ring; and l is the effective length of the rollers.  
According to the formula, the contact load Qij between the j-th cylindrical roller and the inner ring, and the contact load Qoj between the j-th cylindrical roller and the outer ring are obtained sequentially.  
As shown in Figure 4, three equilibrium equations for cylindrical rollers under the condition of constant rotation can be obtained [6]:  
\[\mathrm{P_{ij}+T_{ij}-P_{oj}-T_{oj}\pm F_{mj}=0} \tag{11}\]  
\[\mathrm{Q_{jj}+F_{c}-Q_{oj}\pm f_{m}F_{mj}=0} \tag{12}\]  
\[\mathrm{T_{ij}+T_{oj}-f_{m}F_{mj}=0} \tag{13}\]  
In this equation, rollers in the unloaded zone are denoted by superscript symbols, while those in the loaded zone are designated with subscript symbols. \(f_{m}\) is the friction between the cage and the cylindrical rollers; \(P_{ij}\) and \(P_{oj}\) are the hydrodynamic pressure acting on the cylindrical rollers through the oil film; \(T_{ij}\) and \(T_{oj}\) are frictional forces acting on a cylindrical roller through the oil film; \(F_{c}\) is the centrifugal force on the cylindrical rollers; and \(F_{mj}\) is the normal force between the cage and the cylindrical rollers.  
Under the condition of constant rotation, the bearing ring is balanced under the radial load and the cylindrical roller load, and the balance equation of the ring is obtained:  
\[\mathrm{F_{r}-\sum_{j=1}^{z}Q_{ij,oj}cos\varphi_{j}=0} \tag{14}\]  
where \(F_{r}\) is the radial force exerted on the bearing ring; \(\varphi_{j}\) is the positional angle of each cylindrical roller; and \(z\) is the number of cylindrical rollers.  
Figure 4: Roller force analysis diagram.