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(iii) Helipoid gears manufacturing
Hypoid gears and crossed helical-gears are widely used in the crossed axes power transmissions. However, special machines and tool-setting are necessary to produce hypoid gears with proper contact ellipses and locations. Higher cost and welltrained manufacturers are also necessary and requested. The crossed helical-gear is an another effective alternative for the crossed axes power transmission owing to its low cost. However, low contact ratio and load capacity limit the application and working life of crossed helical-gears. In the theory of gearing, two meshing surfaces in crossed axes are hyperboloids and the contact line is a screw axis (Litvin, 1989). In order to increase the contact ratio and load capacity of the crossed helical- gear, a novel type named Helipoid gears is proposed based on the concept of two meshing hyperbolids. Consider that the equations of the instantaneous screw axis , as shown in Fig.3, are known. The axis of screw motion may be represented by the following equations:
where b = and l= . The matrix representation of the hyperbolid represented in coordinate system is expressed by
By substituting Eqs. (32) into Eq. (33), the equations of the hyperbolid can be written as follows:
where and are parameters of the hyperbolid. When the "Helipoid" gears are manufactured with a worm type hob cutter, point of the hob cutter moves in the plane equidistant to the axial section of hyperbolid, and the equidistance is where is the radius of pitch circle of the hob cutter. The axial section of the hyperbolid can be obtained by setting = 0. Consequently,
Equation (35) can also be expressed in quadratic form as follows:
Equation (36) shows that the axial section of hyperbolid is indeed a hyperbola curve. The hob cutter's motion can thus be obtained and expressed as follows:
Because the path of the hob cutter is parallel to the axial section of the hyperbolid, angle can be obtained by considering Eqs. (26) and (36), simultaneously. Therefore,
By substituting Eqs. (37) and (38) into Eqs. (28) and (29), the equations of meshing for the Helipoid gear generation are obtained. The tooth surfaces of Helipoid gears can be obtained by considering the locus of hob cutter represented in the coordinate system , which is attached to the Helipoid gears, and the equations of meshing shown in Eqs. (28) and (29), simultaneously. An illustrative example is given below to show the generation of Helipoid gears manufactured by the ZN type hob cutter.
Example: A right-hand ZN worm-type hob cutter is used to manufacture the right-hand novel Helipoid gear. Table 1 summaries the fundamental data of the hob cutter and Helipoid gear. The mathematical model of the ZN-type worm surface is well-known in industry and the ZN worm-type hob cutter is widely used in the gear manufacturing. The hob cutter's surfaces can be generated by a blade with straight-lined shape, performing a screw motion with respect to the hob cutter's axis (Fig. 4). The surface equation of the ZN-type worm proposed by Litvin (1994) are adopted and rewritten as follows:
The surface unit normal of the ZN-type worm is given as follows:
The upper sign indicates the surface generated by the generating line I shown in Fig. 4(b and the lower sign indicates the surface generated by the generating line II. In equation (39), parameter p is the screw parameters of the worm. Parameters u determines the location of current point A (or A') on the generating line; where u = (or u= for the generating line II) as shown in Fig. 4. By applying the general gear mathematical model developed here, the Helipoid gears' tooth surface can be specified. By substituting equations ( 3 7 ) –