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

In mechanical engineering, the finite difference method serves as a fundamental numerical analysis technique widely employed for solving various differential equations, including the bearing Reynolds equation. The composite Simpson's numerical integration method provides an efficient approach for approximating integral values, which is crucial for determining bearing load capacity. By combining these two methodologies, we can effectively solve the bearing Reynolds equation and compute the bearing's load-carrying capacity to ensure stable operation of mechanical equipment. Implementation approach involves discretizing the Reynolds equation using central difference schemes for second-order derivatives, followed by solving the resulting system of linear equations. The pressure distribution obtained is then integrated using composite Simpson's rule with adaptive segment division for accuracy. Key functions would include gradient calculation for pressure derivatives and iterative solvers for convergence. The algorithm typically requires mesh generation for the bearing surface and proper boundary condition handling to achieve physically meaningful results.