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Mechanism and Kinematic Relation of a CNC Hobbing Machine
A 6-axis CNC hobbing machine can be used to manufacture different types of gears owing to the movement of its axes with multi-degree of freedom. The generation process of gears by applying a CNC hobbing machine is quite complex. Figure 1 presents a schematic drawing of a 6-axis CNC hobbing machine, where axes X, Y and Z are the radial axis, the tangential axis, and the axial axis, respectively; axes A, B and C are the hob swivel axis, the hob spindle axis and the work table axis, respectively.However, in today's CNC hobbing machines, axes A and Y are for set-up only and rotational ratio between axes B and C is a constant. Some machines allow an interrelationship between axes X and Z; therefore, today's CNC hobbing machines are 3-axis machines. The proposed 6-axis hobbing machine may be built in the future. Figure 2 illustrates the kinematic relationship among those axes. Coordinate system is attached to the hob cutter while coordinate system is attached to the work piece which is cut by the hob cutter.Coordinate system is the fixed coordinate system attached to the machine housing and coordinate system is the reference coordinate system. and are rotation angles of the hob cutter and gear blank, respectively; and is the setting angle of hob cutter. Also, kinematic relation between different coordinate systems can be obtained by applying coordinate transformation matrix equations. The transformation matrix
transforms the coordinates from coordinate system to . Based on the relationship in Fig. 2, matrices , , and can be represented as follows:
where = , = ,and = .According to the mechanism of a 6-axis CNC hobbing machine, the loci of hob cutter represented in coordinate systems , and can be easily obtained by using matrix transformation equations. If the position vector equation and unit normal to the hob cutter are and ,respectively, the locus and unit normal of the hob cutter, represented in coordinate system , can be obtained by applying the following matrix transformation equations:
where matrices , and are vector transformation matrices, and can be obtained by deleting the last row and last column of matrices and , respectively.
Similarly, the locus and unit normal of the hob cutter, represented in coordinate system , which is attached to the work piece, can be obtained by applying the following matrix transformation equations:
where vector transformation matrix can be obtained by deleting the last row and last column of matrix expressed in equation (3).
Relative Velocity Between Hob Cutter and Work Piece
Based on the kinematic relationship of the CNC hobbing machine mechanism, the relative velocity between the hob cutter and work piece can be obtained. The relationship between hob cutter and work piece was set up and shown in Figs. 1 and 2. As indicated in Fig. 2, point f is a common point to both hob cutter and work piece. The surface coordinates of the hob cutter can be transformed to the fixed coordinate system , and expressed as follows:
The velocity of point P attached to the work piece can be obtained as follows:
where = is the angular velocity of the work piece. Also, the velocity of point P attached to the hob cutter can be obtained as follows:
After some mathematical operations, Eq. (10) can be simplified as follows:
According to Eqs. (9) and (11), the relative velocity represented in coordinate system can be obtained as
Equation (12) shows the relative velocity of the hob cutter and the work piece (generated gear tooth surfaces) at their common point P. At the point of contact, their common surface unit normal is perpendicular to their relative velocity . Therefore, the following equation must be observed;