The prediction of tool and workpiece shapes in electro-chemical machining.

The primary object of the work was to develop mathematical and other methods for predicting (a) tool shapes to machine defined work shapes, and (b) work shapes machined by defined tool shapes, operating under equilibrium conditions. Existing methods were examined, the essence of which was a "Cos theta" theory. This theory, based on an analysis of machining between plane parallel electrodes normal to the direction of motion, was found to provide a useful approximate method for designing tool surfaces. However, the theory was shown to be inadequate for curved surfaces whose inclination to the direction of motion of the cathode was less than 350, particularly when used to predict work shapes. Equations relating inter-electrode gap and time, obtained from the plane parallel electrode analysis, were shown also to represent the surface produced by a flat cathode inclined to its direction of motion. Work and tool shape prediction procedures making use of these equations are described and a modification to the equations to allow for overpotential is included. The current and potential distribution in the inter-electrode gap was then studied as a field problem. Two boundary conditions at the workpiece surface were identified and analytical solutions were obtained for tool shapes to produce semi-cylindrical and hemispherical work shapes, including modifications to account for overpotential at the work surface. Solutions were also obtained by an electrolytic tank analogue apparatus and by two numerical methods. A finite difference method was used successfully to predict work shapes but was found to be unsuitable for designing tool shapes. A second method was developed specifically for this purpose using a specified work shape and its boundary conditions to build up a field solution step by step. The computations for both methods were performed by digital computer. Experimental work was undertaken to provide shapes to compare with the theoretical predictions and good agreement was obtained. An attempt was made to measure anode overpotential for various metals under actual machining conditions and readings less than 1 volt were recorded. Discussion of the merits of the various methods and their applicability to three dimensional problems, conclusions and suggestions for future studies complete the work.