The imaginary profile wheel is composed of the path of the blade in the virtual machining process, and the number of teeth is not necessarily an integer. The extended epicycloid formed by the tool path is shown. Figure 1 describes the relationship between the imaginary generating wheel and the machined object. The rotation angle between the frame and the workpiece is fixed in the generative process, but not fixed in the forming process, although the imaginary generating wheel is collinear with the rotation axis of the frame.
In Fig. 2, the coordinate systems st (XT, YT, ZT) and SD (XD, YD, ZD) are fixedly connected to the cutter head and the imaginary generating wheel respectively. Point q is the center point of the tool on the machine plane. The auxiliary coordinate systems SA, Sb and SC describe the relative offset between the cutter head and the imaginary profile wheel. The rotation axis point oi of the inner cutter is located at point Q, and the rotation axis point of the outer cutter head is OA. The relative position of OI and OA is determined by the eccentricity exz and the direction angle ψ e. the projection of point OA on the machine plane is point OA ‘.
Through the coordinate transformation, the tooth surface equation of the imaginary gear can be obtained from the tool path in the process of machining, that is, from the cutter head coordinate system st to the imaginary gear coordinate system SD