Analytical and numerical calculations for thermal elastohydrodynamic lubrication problems of helical W-N gears

Abstract By adopting the pressure-viscosity coefficient perturbation method, the numerical calculations required to solve elastohydrodynamic lubrication (EHL) problems are greatly reduced. In addition, the initial gaps between tooth flanks and the instantaneous velocities of tooth profiles are deduced. The inlet and outlet boundaries of the lubricant are also discussed. The finite difference method (FDM) is used to determine perturbation solutions. It can be seen during the setting up of the difference equation, that higher grades of solution can be achieved with no iteration on the basis of the α 0 grade of solution, i.e. the isoviscosity solution, which is easier to obtain owing to its small non-linearity of pressure and film thickness. In addition, all matrices of coefficients of higher grades of the difference equation are the same except for the constant item of the equation, which will obviously make programming and calculation very convenient. Solutions to thermal EHL problems are converted into a superposition of the solution of isothermal EHL and higher grades of solution by the temperature-viscosity coefficient perturbation method. This greatly reduces the amount of numerical calculation because solutions of isothermal EHL will have already been obtained. The FDM is adopted to determine the perturbation solutions to thermal EHL problems. During the setting up of the difference equation it can be seen that higher grades of solution can be solved much more easily on the basis of the β 0 grade of solution, i.e. the solution of the isothermal EHL problem. In addition, the numerical calculation pattern of the Reynolds equations is the same as that of isothermal EHL. All matrices of coefficients of higher grades of difference equation (both Reynolds and energy equations) are the same except for the constant item of the equations. Groups of thermal EHL problems (totalling 1536) of helical W-N gears under different conditions are calculated analytically and numerically. Results are given in the form of figures and formulae from which many useful conclusions can be obtained. From a group of non-dimensional formulae calculating the maximum hydrodynamic lubrication pressures between tooth flanks under different conditions, the ratios of maximum oil film pressure under the condition of testing the limit contact stresses of tooth flanks and under different working conditions have been obtained. The ratios are taken as the coefficients modifying the limit contact stresses of tooth flanks considering the influence of thermal EHL because experimental determination of limit stresses is difficult. According to the traditional concepts, the coefficients are separated into coefficients of lubrication and velocity, which have been compared with relative coefficients of involute gears.