Calculating Bearing Load Capacity Using Finite Difference Method for Reynolds Equation and Composite Simpson's Numerical Integration

The finite difference method for solving bearing Reynolds equation represents an efficient numerical approach for computing hydrodynamic bearing performance. This technique discretizes continuous differential equations into difference equations, enabling numerical solutions on computers. Specifically, the Reynolds equation describes lubricant film pressure distribution, and the finite difference method transforms this complex partial differential equation into a system of linear equations suitable for iterative solutions. In code implementation, this typically involves creating a 2D grid structure and constructing coefficient matrices using central difference approximations.

During the solution process, appropriate grid discretization strategies must first be established, with uniform grids commonly employed to enhance computational efficiency. Subsequently, partial derivatives are approximated using central difference schemes to convert the Reynolds equation into discrete form. Boundary condition handling is equally critical, typically assuming zero pressure at bearing edges or periodic boundary conditions. Code implementation requires careful indexing management and boundary value assignment procedures to ensure numerical stability.

After obtaining pressure distribution, bearing load capacity calculation requires numerical integration. The composite Simpson's method provides high-precision numerical integration suitable for smooth function integration. This approach divides the integration interval into multiple subintervals, applies Simpson's rule to each subinterval, and sums the results. Compared to the trapezoidal rule, composite Simpson's method offers superior convergence rates and computational accuracy, enabling more precise estimation of bearing load capacity. Algorithm implementation involves iterative summation with weight coefficients (1-4-2-4-...-2-4-1 pattern) across subintervals.

In conclusion, the combination of finite difference method with composite Simpson's numerical integration provides a stable and efficient numerical approach for solving bearing Reynolds equations and calculating load capacity, making it suitable for practical engineering applications. The complete computational workflow can be programmed using matrix operations for finite differences and iterative loops for numerical integration.