# Fatigue Life Analysis of Cylindrical Roller Bearings Considering Elastohydrodynamic Lubrications
## 4 Result Calculation and Analysis
### Numerical Calculation Method of the Model  
Therefore, in order to reduce the number of variables in the equation and eliminate the influence of dimensional differences between the original indexes, it is necessary to dimensionlessly transform the basic equation of Reynolds elastic flow lubrication and its boundary conditions, and the parameters of some dimensionless equations are defined as follows:  
Film thickness parameters \(\mathrm{H}=h\mathrm{R_{x}}/\mathrm{b^{2}}\); entrainment velocity parameters \(\mathrm{U}=\eta_{\mathrm{o}}\mathrm{u_{s}}/\mathrm{ER_{x}}\); load parameters \(\mathrm{W}=\mathrm{w}/\mathrm{ER_{x}}\mathrm{l}\); coordinate parameters \(\mathrm{X}=\mathrm{x}/\mathrm{b}\); pressure parameters \(\mathrm{P}=\mathrm{p}/\mathrm{p_{H}}\); lubricant density parameters \(\rho^{*}=\rho/\rho_{\mathrm{o}}\); lubricant viscosity parameters \(\eta^{*}=\eta/\eta_{\mathrm{o}}\); material parameters \(\mathrm{G}=\alpha\mathrm{E}\); \(\mathrm{p_{H}}\) is the maximum Hertz pressure, where for line contact, \(\mathrm{p_{H}}=2\mathrm{w}/\pi\mathrm{b}\mathrm{l}\); and \(\mathrm{b}\) is half width of the Hertz contact area, with \(\mathrm{b}=\sqrt{8\mathrm{w}\mathrm{R_{x}}/\pi\mathrm{EI}}\).  
Building upon the quasi-static analysis and integrated with the elastohydrodynamic lubrication model, the specific computational workflow of the developed analytical model is illustrated in Figure 6. The detailed calculation procedures are as follows:  
1. First, input the key geometric parameters of the cylindrical roller bearing, including the roller diameter, quantity, and length and the inner/outer raceway diameters, and define the operational conditions such as radial load. Subsequently, by establishing the initial positions of the races and rollers, solve the geometric approach quantity and entrainment velocity between contacting pairs.
2. Employ the multigrid method to solve the elastohydrodynamic lubrication model for cylindrical rollers. During computation, set grid layers \(\mathrm{M}=5\) with \(\mathrm{W}\)-cycle iterations, performing all calculations in non-dimensional form. For the roller-raceway lubrication analysis, configure the finest grid layer with 256 nodes and non-dimensional boundary coordinates \(\mathrm{X_{in}}=-4.0\) and \(\mathrm{X_{out}}=1.5\). The convergence criterion requires that the relative errors for both pressure and load are less than \(10^{-4}\). Upon satisfying this criterion, the film thickness and pressure distribution are ultimately determined.
3. Substitute the non-dimensional film thickness parameter into Equation (9) to determine the numerical solution of elastic deformation through nonlinear iterative algorithms. Use these results to calculate the final contact load distributions.
4. With the obtained contact loads, sequentially perform numerical solutions for each rolling element sequentially from Equation (11) through Equation (13). If the results satisfy the predefined convergence accuracy thresholds, output the contact load distribution data. For non-convergent cases, refine the roller coordinate position parameters via the bisection method and reintroduce them into the computational loop for iterative optimization.
5. Input the contact loads of all rolling elements into Equation (14) for iterative computation. If the results exceed the allowed convergence tolerance, adjust the race spatial coordinate parameters using the bisection algorithm and restart the computational loop. When the convergence criteria are met, output the optimized contact load datasets and concurrently initiate a quantitative evaluation of bearing fatigue life.