An exact analytical solution of the Reynolds equation for the finite journal bearing lubrication

The Reynolds equation for the pressure distribution of the lubricant in a journal bearing with finite length is solved analytically. Using the method of separation of variables in an additive and a multiplicative form, a set of particular solutions of the Reynolds equation is added in the general solution of the homogenous Reynolds equation thus a closed form expression for the definition of the lubricant pressure is presented. The Reynolds equation is split into four linear ordinary differential equations of second order with non-constant coefficients and together with the boundary conditions they form four Sturm–Liouville problems with the three of them to have direct forms of solution and one of them to be confronted using the method of power series. The mathematical procedure is presented up to the point that the application of the boundaries for the pressure distribution yields the final definition of the solution with the calculation of the constants. The current work gives in detail the mathematical path with the help of which the analytical solution is derived, and ends with the pressure evaluation and a comparison with past numerical solutions and an approximate analytical solution for a finite bearing. The resultant pressure distribution presents slight differences compared to this of the numerical solution and the approximate analytical solution in the values of maximum and minimum pressure but also in the domain of lower values of pressure.