Mathematical modelling and analytical solution for workpiece temperature in grinding
The authors consider the temperature field in a metal body (workpiece) which is grinding by a rotating wheel whose axis is fixed, while the grinding body moves with a constant velocity. During grinding, most of the mechanical energy is transformed into heat which is accumulated in the contact zone between the grinding device and the workpiece. It is assumed that both the wheel and the workpiece are rigid. A fluid flows between the wheel and the workpiece lubricating and cooling the contact surface and removing grinded material. The larger region over which the grinding wheel contacts the workpiece is due to the curvature of the wheel. This region is assumed to be of given length and remains constant during the process of grinding. If the workpiece is of finite length it is assumed that the grinding wheel is equiped with a mechanism that moves it away from the workpiece surface periodically, i.e., the the interaction between the grinding wheel and the workpiece takes place in repeated cycles. In order to study only a two-dimensional thermal field in the workpiece, the authors use a simplified mathematical model in the form of a linear partial differential equation which is of the convection-diffusion type, together with convective boundary conditions which are also linear. A laborious mathematical method is used and a closed form solution is obtained using the Laplace and Fourier transforms and the Green's function method. The content of the paper can be of considerable use in practical applications in which the transformation of mechanical energy into the heat is required.
variable coefficient boundary-value PDE problem
scientific article; zbMATH DE number 5312880