Helical grooves for drill flutes, milling cutters, gears, screw mechanisms, and so on, are usually machined from cylindrical stock by grinding or milling using profiled disc-type cutters as illustrated in Figure I. During machining of the helical groove, the cutter rotates about its axis, Z^sub 2^, as the workpiece moves along and rotates about its axis, z. The combined motion of the cutter and workpiece generates the helical groove. Due to the nonrectilinear motion of the cutter along the helical cutting path, the cutter profile defined in the cutter axial section (section AA) in Figure 1 is different from the workpiece profile viewed in the same cross section. This makes it difficult to find either the cutter profile to generate a required shape of the helical groove or the helical groove shape generated by a given cutter profile. Furthermore, it is first necessary to specify the machine setup condition, which includes the separation distance, s, between the cutter and workpiece axes and the setting angle, α, as shown in Figure 1. Each machine setup can lead to a different solution for the required cutter or workpiece profile.
Historically, cutter profiles for helical grooves have been designed by graphical trial-and-error methods (Dudley and Poritsky 1943). To overcome the inherent inaccuracy of this approach, analytical methods were developed to design the required cutter profile or to find the helical groove. These methods can be divided into two classes. The first class is based on graphic reasoning (Friedman, Boleslavski, and Meister 1972; Kaldor and Messinger 1988; Veliko, Nankov, and Kirov 1998), which considers the trajectories of selected points on the cutter profile. The results are visually presented as the envelope of the trajectories in the workpiece transverse section. However, this method cannot be directly applied to finding the cutter profile to generate a desired nelical groove shape. The second class is based on the condition that there is a common normal vector at the points defining the line of contact betweeni the cutter and the helical groove surfaces during cutting (Agullo-Batlle, Cardona-foix, and Vinas-sanz 1985; Sheth and Malkin 1990; Ehmann 1990; King, Ehmann, and Lin 1996a, 1996b). After each contact point is found, the corresponding points on the cutter and helical groove profiles are mathematicalfy related to each other. While the desired cutter or workpiece profile is obtained directly, the application of this method is complicated by the need to calculate the derivative of the given profile.
When applying these methods, it is often found that no valid cutter profile can be obtained for a given machine setup and helical groove profile. In such cases, the machine setup must be adjusted to obtain a valid cutter profile (Agullo-Batlle, Cardona-foix, and Vinas-saiiz 1985). However, no general method has been presented on how to adjust the machine setup. Some ressarchers statistically related the machine setup to a critic al portion of the required helical groove profile, but this inevitably leads to some errors in the resulting profile (Ekambaram and Malkin 1993).
The present paper is concerned with selecting valid machine setup conditions for helical groove machining. Beginning with an analytical model for the cutter profile design, criteria for identifying valid machine setups for a given helical groove are analyzed. The mathematical model for machine setup is then further developed for profiles with singularities. These criteria and the mathematical model are then applied to finding the range of valid machine setup conditions. Some examples are presented to illustrate the method.
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