Perfect Conjugacy in Hypoid Gearsets

<p style="text-align: center;"><img src="/ueditor/php/upload/image/20260609/1781019782485871.png" title="1781019782485871.png" alt="1.png"/></p><p><span style="font-size: 14px;">It begins to become more problematic for hypoid gears. Frequently, the pitch elements of crossed axes hypoid gears are drawn as cones. Even though the face cones of hypoid gears and pinions are machined conically, the pitch elements are hyperboloids. Ernest Wildhaber and Arthur Stewart described their invention of hypoid gearing in 1926 (Ref. 3). Boris Shtipelman published in 1978 the relationships and derivations required to understand hypoid gears and their hyperbolic pitch elements (Ref. 4). Figure 9 offers a graphical interpretation of the hyperbolic pitch elements and their generator. The pitch surface generator is a line that winds on the surface of a cylinder beginning at the crossing point of the axes, equal to the first contact point of the pinion and gear pitch surfaces. The pitch surface generator is developed by the connecting line between the pinion and gear pitch surfaces (nop-nog).by shifting the connecting line along the pinion and gear axes. The connecting line (nop-nog) is normal to the pitch elements. Point P is one point of the pitch surface generator.</span></p><p><span style="font-size: 14px;">Although the pitch elements are hyperbolic and not conical, it is possible to use conical faces for the blanks of pinion and gear. If point P in Figure 9 was chosen at the center of the face width, then line (nop-nog) can be used as a normal vector to define the face angle of a blank with straight face cones if the hypoid set was manufactured by face hobbing, which implies parallel depth teeth.</span></p><p><span style="font-size: 14px;">Straight face cones will merely influence the top root clearance of the gearset in the range of 30 to 60 microns. Using straight face cones will not change the form of the pitch surface, nor will it influence the base surface (and root surface). Those functional surfaces are given by kinematical relationships and must be considered when thinking about the shape of the surface of action. No plane of action can exist between two hyperbolic base elements.</span></p><p><span style="font-size: 14px;">Conjugacy Between Meshing Flanks The term conjugate is used in mathematics for two or more surfaces that contact each other along a line. Since the 1970s, the term conjugate has also been employed in gear technology literature to define the “exact” gear pair that presents a triple plurality of line contact between two gear flanks during the meshing process (Ref. 5): The flanks contact along a line (contact line), which is only limited by the boundaries of the teeth, i.e., the working area The line contact between the flanks exists within the entire area of engagement in every mesh position Line contact is maintained in the entire area of engagement if the pinion and ring gear are rotated by angular increments, where: (angular pinion increment) / (angular ring gear increment) = (transmission ratio) The Ease-Off is a three-dimensional graphic of the flank deviations from a conjugate pair. It is calculated by the transformation of a pinion flank “into” the gear coordinate system according to the first gearing law, resulting in a virtual gear flank that is conjugate to the actual pinion flank. This conjugate gear flank will then be compared to the present gear flank, where all differences in arc length are plotted point by point in ordinate direction into the Ease-Off graphic. If both mating bevel gears have conjugate manufacturing data, then the Ease-Off graphic has no deviations in the ordinate direction. Also, if the pinion flanks and the gear flanks have spiral-angle and pressure-angle errors of equal amounts, the Ease-Off graphic will not show any deviation. Although the individual gears are considered incorrect in this case, they will roll conjugate with each other, which subsequently leads to an Ease-Off without any ordinate values. Figure 11 shows the analysis results of a typical conjugate hypoid gearset. The Ease-Off graphics have zero crowning magnitudes in the ordinate direction. The motion graph has, next to some numerical entrance and exit variation, zero motion error. The contact bearings show line contact within the entire working area. The coast side contact ends at a toe root undercut (section f).</span></p><p><span style="font-size: 14px;">Each spiral bevel gearset with uniform tooth depth has a conjugate base design. This applies to all face-hobbed and some face-milled gearsets. Hypoid gearsets can only be conjugate with a non-generated gear that meshes with a generated pinion. For the calculation and manufacturing process, the hyperbolic pitch elements are calculated for the gear first. Then a suitable blade profile (gear cutter in Figure 12) is chosen and positioned in a face cutter head. The cutter head is positioned to create the desired spiral angle. With this procedure, a non-generated gear can be created by computer simulation, and it can be manufactured with a bevel gear cutting machine (Ref. 6). A pinion cutter (see Figure 12) is positioned in a mathematical model or in a bevel gear cutting machine such that it represents one tooth of the non-generated gear by rotating around its axis. An additional simultaneous rotation around the pinion generating gear axis results in this pinion cutter becoming the generating gear of a conjugate pinion. If the pinion is positioned with the same offset “a” that was used to determine the pitch surfaces (Figure 9), then the cutter rotation around the pinion generating gear axis will form a pinion that is perfectly conjugate to the non-generated gear. The tooth contact analysis in Figure 11 has been obtained from such a non-generated hypoid gearset and therefore shows perfect conjugacy.</span></p><p><span style="font-size: 14px;">More complicated is the generation of a conjugate bevel or hypoid gearset with tapered depth teeth (see Figure 13). If the generating gear axes have identical axes of rotation that are perpendicular to the pitch line, the rotating cutter heads and their blades will not follow the root line of a tapered depth tooth. Tilting the cutter head to follow the root line would violate the first kinematic coupling requirement for teeth that are congruent to the slots of the mating member. The following solution was developed in the 1940s (Ref. 7). If the cutting edges are adjusted in the cutting machine such that the tooth reference profile and depth are matched at midface, and if an axial motion of the cradle is introduced that guides the blades along the tapered root line while the generating roll progresses along the face width, then the requirements of congruent teeth and slots are fulfilled with the result of perfect conjugacy. However, in the case of hypoids, the gear must be non-generated, and the pinion must be generated with helical motion to achieve conjugacy.</span></p><p><span style="font-size: 14px;">The process configuration and kinematics in Figure 13 are called duplex completing. Today, all face-milled and ground spiral bevel and hypoid gears are manufactured with the duplex completing process. The axial cradle movement in this process is called helical motion and was first introduced with mechanical bevel gear machines in the late 1940s. The helical motion of the days of mechanical machines required an additional change gearbox which actuated a cam that moved the sliding base during the generation process. Today’s CNC-controlled Phoenix free form machines have the helical motion capability automatically through their interpolated axes movement.</span></p>