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(b) Point is moving in plane
In the manufacturing of worm gear, the center of hob cutter can be set at the fixed point M (Fig. 2). However, when the hobbing machine is used to manufacture spur or helical gears, the motion of hob cutter is in plane. Therefore, the motion of point (a point on the hob cutter axis) is a curve and is moving in plane.When velocities and have a specific relation in the manufacturing process, the locus of point can be expressed as follows:
Then, the tangent direction can be obtained by
where represents the angle formed by the tangent vector of hob cutter path and -axis. A general quadratic equation is chosen for the hob cutter path and represented as follows:
Then, the angle expressed in Eq. (24) becomes
This is due to the fact that some axes of a 6-axis CNC hobbing machine may be considered to be fixed or have specific relations in gear generation processes. As shown in this case, axes A and Y are fixed (i.e., = 0 and setting angle is constant), and axes X and Z have specific relationship shown in Eqs. (25) and (26). The relation between the work piece and the axes of CNC hobbing machines, shown in Eq. (15), can be rewritten as follows:
where is the lead of the generated gear. The second term expressed in Eq. (27) is the additional rotation due to the helical angle of generated gears. Equation (27) shows the work piece rotation in terms of two independent variables and Vz. When substituting Eq. (27) into equation (14), two equations of meshing can be obtained as follows:
By simultaneously solving Eqs. (28) and (29), and the locus of hob cutter represented in the coordinate system (i.e., Eq. (6)), the gear tooth surfaces can be obtained. When point is moving in the plane, several types of gears can be generated:
(i) Spur gears manufacturing
When spur gears are manufactured, the parameters expressed in equation (29) can be specified, i.e. the lead = and angle = 0°. The equation of meshing shown in Eq. (29) is simplified as follows:
Equation (30) indicates the contact lines of the hob cutter and the generated gear are independent of parameters of and the position of hob cutter in the direction. Then, the manufacturing of spur gears is reduced to a two-dimensional problem and the equation of meshing shown in Eq. (28) is simplified as:
The tooth surfaces of a spur gear can be obtained by considering the locus of hob cutter represented in the coordinate system and the equation of meshing shown in Eq. (31), simultaneously. Therefore, Eqs. (6) and (31) represent a spur gear's tooth surfaces.
(ii) Noncircular gears manufacturing
The generation of noncircular gears can also be considered a two-dimensional problem. However, the distance shown in Fig. 2 is not a constant in the generation process of noncircular gears. In this case, is equal to the distance between the center of rotation of noncircular gears and the axis of hob cutters . A noncircular gear's tooth surface can be obtained by considering the locus of hob cutter represented in the coordinate system and the equation of meshing shown in Eq. (31), simultaneously. Therefore, Eqs. (6) and (31) represent the tooth surface of noncircular gears. Because the generation of noncircular gears can be considered a two-dimensional problem, the rack cutters can be used to develop the mathematical model of noncircular gears. Based on the cutting mechanism and geometry of rack cutter, Chang et al. (1995) proposed a complete mathematical model and performed undercutting analysis of elliptical gears. However, the CNC hobbing machine is inappropriate for the manufacturing of noncircular gears with convexconcave pitch curves. The shaper cutter should be applied to manufacture such a type of noncircular gears. Moreover, Chang and Tsay (1995) have already developed the mathematical models of elliptical gears.