The Swept Volume of a Circular Paraboloid End Mill Moving Along a Helical Path

Abstract

To obtain complex part geometries at one pass using machining processes, it is important to employ the tools with non-conventional geometries. A circular paraboloid is a solid of revolution, which can be obtained by rotating a parabola. The swept volume of an end mill can be defined as the unification of all sets of points on the tool for every instant as it moves, and its derivation is an obligation to determine the machined part geometry prior to an actual machining process. After derivation of the swept volume of the tool, machined part geometry is obtained by subtracting the swept volume of the tool from the volume of the initial workpiece. However, derivation of the swept volume of the tool is not a straightforward task. In this work, an analytical model was introduced to derive a complete set of points on the machined part by means of well-defined and constrained tool geometry and tool path. In the model, a plane that passes through the screw axis was used to observe the instant cross-section of the tool as it moves along the helical path. By overlapping the instant cross-sections of the tool in the plane, the final cross-section was derived. Since all cross-sections that pass through the screw axis are identical, the method gives an entire set of points on the machined surface. To validate the model, a computer-aided design program was utilized.