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With the wide use of complex helical surfaces in screw components, it becomes more and more urgent to investigate efficient ways for the design and manufacture of components of this kind. This study establishes the mathematical model of helical surface based on discrete points on its axial section profile using cubic spline curve fitting method, and presents a theoretical approach to calculate the tool path for the whirling process which aims to produce the helical surface through enwrapping movements of standard cutters inserted in the tool ring. The approach is implemented in MATLAB environment focusing on the calculation of the cutter location points. After that, the theoretical machining error is analyzed focusing on the scallop height, which can be divided into axial and cross-sectional errors. Finally, a case study is provided, together with numerical simulation demonstrating the validity of the proposed approach.

As helical surfaces, in their many and varied forms, are finding more and more applications in engineering, new approaches to their efficient design and manufacture are desired. To that end, the helical projection method that uses curvilinear projection lines to map a space object to a plane is examined in this paper, focusing on its mathematical model and characteristics in terms of graphical representation of helical objects. A number of interesting projective properties are identified in regard to straight lines, curves, and planes, and then the method is further investigated with respect to screws. The result shows that the helical projection of a cylindrical screw turns out to be a Jordan curve, which is determined by the screw's axial profile and number of flights. Based on the projection theory, a practical approach to the modeling of screws and helical surfaces is proposed and illustrated with examples, and its possible application in screw manufacturing is discussed.