A Computer Program for the Computation of Running Gear Temperatures Using Green's Function

摘要

A new technique has been developed to study two dimensional heat transfer problems in gears. When compared with the finite element or finite difference methods, this new technique can provide substantial improvements in computational speed. It is also more adaptable for use in small lubricated concentrated contacts, such as gears, since high resolution can be obtained without using large numbers of nodes or elements. The new technique consists of transforming the heat equation into a line integral equation with the use of Green's theorem. The equation is then expressed in terms of eigenfunctions that satisfy the Helmholtz equation, and their corresponding eigenvalues for an arbitrarily shaped region of interest. It is shown that the discretized version of the eigenfunction can be obtained by solving a set of homogenous simultaneous equations. Once the eigenfunctions are found, the transient temperature is expanded in terms of the eigenfunctions with unknown time dependent coefficients that can be solved by using Runge-Kutta methods. The time integration is extremely efficient, therefore, any changes in the time dependent coefficients or source terms in the boundary conditions do not impose a great computational burden on the user. This result is very important when performing accurate scoring analysis in gears so that the gear temperature can be accurately determined. The gear, out of mesh surface temperature is not a constant along the tooth profile at steady state running conditions as is usually assumed by gear engineers.