# Mathematical model of machining hypoid gear wheel by forming method

The basic parameters for machining hypoid gear wheel by forming method include: horizontal tool position H2, vertical tool position V2, correction value xg2 of hypoid gear wheel axis and root cone angle of machine tool γ m. Bed and vertical wheel position are zero. In addition, there are cutter head diameter and cutter tooth angle. According to the forming method of the hypoid gear wheel, the coordinate system shown in the figure is established to determine the relative position relationship between the hypoid gear wheel and the cutter head, and the coordinate systems S2, SM and SG of the hypoid gear wheel, machine tool and cutter head are taken respectively.

SM = [om; XM, YM, ZM] is the fixed coordinate system of the gear cutting machine tool. The origin om of the coordinate system is located in the center of the gear cutting machine tool, and the plane xmomym is located in the plane of the gear cutting machine tool. Since the coordinate system is a fixed coordinate system of the machine tool, the direction of each coordinate axis remains unchanged during the whole machining process of the hypoid gear wheel.

SG = [og; XG, YG, ZG] is the coordinate system of the cutter head. The origin og of the coordinate system is located in the center of the cutting cutter head, and the plane xgogyg is located in the plane of the cutting cutter tip. When machining the hypoid gear wheel by forming method, the shaking table of the machine tool is fixed. Therefore, the coordinate system SG of the cutting cutter head is also fixedly connected to the coordinate system sm of the machine tool, where H2 is the horizontal tool position and V2 is the vertical tool position.

S2 = [O2; X2, Y2, Z2] is the fixed coordinate system of the hypoid gear wheel, the origin O2 is located at the design intersection of the processed hypoid gear wheel, the axis of the processed hypoid gear wheel coincides with the coordinate axis Z2 of the coordinate system S2, and the coordinate axis Z2 is in line with the coordinate axis XM of the fixed coordinate system sm of the machine tool γ M angle, γ M angle is equal to the root cone angle of the machine tool. Where, omo2 = xg2.

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