Analytical Model for Nonlinear Fluid Film Forces of Hydrodynamic Journal Bearing with Axial Grooves

An analytical model for nonlinear fluid film forces of hydrodynamic journal bearing with axial grooves was proposed in this study. Based on the Sturm-Liouville theory, the pressure distribution of nonlinear oil film was solved using the cavitation boundary condition. In order to solve the Reynolds equation of hydrodynamic lubrication of oil film conveniently, the oil film pressure function was decomposed into an additive form of a particular solution and a homogeneous solution, the rupture locations of lubricating oil film were determined by the continuity condition. Based on the method of separation of variables, the pressure distribution of the particular solution was divided into an additive form of the circumferential separation function and the axial separation function, and then the circumferential separation function was solved by the Sommerfeld transform. The pressure distribution of the homogeneous solution was decomposed into a multiplicative form of the circumferential separation function and the axial separation function. By using variable substitution, the circumferential separation function equation was transformed into Sturm-Liouville equation, the eigenvalues and eigenfunctions were obtained using the boundary conditions, and then the circumferential pressure distribution of the homogeneous solution was obtained. By solving the differential equation, the solution of axial separation function was expressed as a hyperbolic tangent function with eigenvalues. In complete oil film field, the analytical expression of the oil film pressure distribution was integrated to obtain the nonlinear oil film forces of finite length journal bearing with axial grooves. The results by the proposed method were in good agreement with the finite difference method, the proposed analytical model was validated.