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Equation (13) is the so called equation of meshing. Substituting equation (12) into equation (13) yields
This equation relates the motion parameter of the generated gear to the surface parameter of the hob cutter surface. Gear manufactured by using CNC hobbing machines, the motions of the hob cutter shown in Figs. 1 and 2 may be considered independent motions in the derivation of tooth profile. Actually, most of the commercial hobbing machines are semi or 3-axis machines. Some axes of the CNC hobbing machine are fixed while some axes have specific relationship during the manufacturing process. For instance, the axis A should be adjusted according to the helix angle of the generated gear and is fixed in most cases. That is, when a right-hand helical gear with lead angle is manufactured by the right-hand hob cutter with lead angle , the setting angle of the hob cutter is equal to ( — ). In the manufacturing of helical and spur gears, axes X and A are fixed; in the manufacturing of worm gears, axes A, X and Z are fixed. To simulate all possible manufacturing processes for gears generated by CNC hobbing machines, the relation between work piece and motions of axes can be written as follows:
where , is the number of start of the hob cutter (i.e. number of the hob cutter teeth) and is the number of the generated gear teeth. However, when gear profile modification is considered, then is a variable. Substituting Eq. (15) into Eq. (14) and rearranging in terms of four independent variables , , and yield the following equation:
Since , , and are independent variables, the terms in braces are all equal to zero. Therefore, four equations of meshing are obtained and represented in coordinate system as follows:
Notably, Eqs. (17) through (20) do not need to simultaneously exist in the manufacturing different types of gears. When some axes are fixed or have a specific relationship, those four equations of meshing may be reduced which depends on the real motion of hob cutters. By considering the equations of meshing shown in Eqs. (17) to (20) and the locus of hob cutter represented in the coordinate system which is attached to the work piece, a general mathematical model of gear tooth surfaces generated by a 6-axis CNC hobbing machine is obtained. In the following section, we apply the proposed gear mathematical model to different types of gear generation by specifying the parameters expressed in the equations of meshing.
General Gear Mathematical Model Applicable to Different Types of Gears Generation
A 6-axis CNC hobbing machine can be used to manufacture spur gears, helical gears, worm gears, and noncircular gears. In this section, we discuss different gears manufactured by a 6-axis CNC hobbing machine. Besides, a novel type of gear named "Helipoid" is investigated. Examples are chosen to illustrate the gear tooth surfaces obtained by considering multi-equation of meshing. As shown in Fig. 2, axis is the rotation axis of hob cutter. Point , is the intersection point of axis and plane.
(a) Point is fixed
As indicated in Figs. 1 and 2, when axes A,X, Y, and Z of a 6-axis CNC hobbing machine are fixed and the reference center point of hob cutter O;, is located at point M, then a worm gear is manufactured. In this case
By substituting Eq. (21)intoEq. (16), the equation of meshing can be simplified as follows:
where = , is the number of start of the hob cutter and is the number of worm gear teeth. The worm gear tooth surface can be obtained by considering the locus of hob cutter represented in coordinate system and the equation of meshing expressed in Eq. (22), simultaneously. Therefore, Eqs. (6) and (22) represent the worm gear tooth surface.