Necessary parameter setting of the finite‐difference method for determining selected rotor‐dynamics characteristics

Abstract

The aim of the study is to analyse the adequacy of the finite difference-method (FDM) for determining the stability boundary of a rigid rotor mounted on short hydrodynamic plain journal bearings through the solution of the Reynolds partial differential equation (PDE). The Reynolds equation, obtained by coupling the equation of motion with the continuity equation, governs the pressure distribution in a lubricant film. The quasi stationary form of this equation requires normalization prior to solving, for the numerical error of the FDM to be minimized. Except for numerical benefits, the dimensionless form of the PDE is universally applicable to a bride spectrum of tribological problems, including hydrodynamic plain journal bearings. The present study first examines the adequacy of solver settings, including the computational grid, through comparison of the corresponding numerical results with an analytical solution obtained by means of the Short Bearing Theory (SBT). Two characteristic tribological quantities are selected as the basis for comparison: the modified Sommerfeld number and the bearing attitude angle. Further, dimensionless stiffness and damping tensors, valid for any short hydrodynamic plain journal bearing, are determined from the numerically obtained distribution of the dimensionless hydrodynamic pressure.These two dimensionless tensors allow for the subsequent estimation of the stability boundary, which marks the dimensionless speed at which a given rigid rotor becomes dynamically unstable. The results of the current study demonstrate sensitivity of the FDM results to the numerical step size. The corresponding numerical error is verified using the SBT and presented as function of the dimensionless (relative) journal eccentricity together with rotor start-up curves. As a major finding, a dimensionless range of the step size is identified for the error to remain below 5%. K E Y W O R D Splain journal bearing, Reynolds PDE, rotor-dynamics, Short Bearing Theory, solver settings, stability, tribology, verification PHYSICAL AND NUMERICAL BACKGROUND OF THE REYNOLDS PDEMost mechanical systems contain rotating components, such as shafts, which are supported by stationary parts, or bearings. In general, bearings are classified into three types: fluid film, dry or semi-lubricated bearings, or rolling element bearings. Accordingly, the supporting action is applied to the shaft through a thin film of fluid, dry contact of surfaces, or through rolling elements. Fluid film bearings are furthermore categorised according to direction of load transmission. "Journal bearings" only transfer radial loads, whereas "thrust bearings" are designed to support axial loads. The former are the subject of the present