Making a Workpiece with Spiral Turns by Means of Forming Cutters

Abstract

Workpieces with spiral turns (e.g. milling cutters, borers) are produced by means of forming cutters, generators or abrasive wheels. For this purpose, a cylinder of the desired raw material is passing the rotating tool with the correct twist. Through this process, parts of the slug are milled away and one cog of the workpiece is formed. Our problem is, given the desired shape of the workpiece, to construct (compute) a tool which forms a cog of this workpiece. It is of course important to select shape and position of the forming tool in such a way that already correctly made parts of the workpiece are not damaged in a later stage of the production process.

First, we describe the traditional construction of a solution on the drawing board and formulate it mathematically. We show that the problem is ill-posed in the sense that small data changes can lead to an unsolvable problem. The discretization of the traditional approach has the property that local errors in the data result in global changes in the solution, which is not desirable.

Then we describe a new solution technique, which is computationally more efficient and whose discretization has the property that data errors propagate into the solution only locally. We can show continuous dependence of the solution on the data and give error bounds for a technically important special case. Numerical results will be presented.

This work has been partially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (project S32/03) and the Austrian Federal Ministry for Science and Research. The practical part of this project was performed for VOEST-ALPINE Werkzeuge- und Präzisionstechnik GmbH, Ferlach, Austria, while both authors were at the University of Klagenfurt