Perfect Conjugacy in Straight Bevel Gearsets

Bevel gears with intersecting axes are the topic of a series of three papers published between October 2014 and January 2015 (Ref. 2). A straight bevel gearset with skew teeth was modeled, and a sample was manufactured. This publication addressed two points: the design of a gearset with a low tooth count and the solution for perfect conjugacy, which was successfully achieved. Also, Coniflex straight bevel gears used since the 1940s can achieve perfect conjugacy when the machine root angle is equal to the pitch angle of the manufactured gear (generated on the pitch line). This principle applies to any tooth count combination. Straight bevel gears, such as Coniflex, have tapered depth teeth, where the pitch cones roll on each other, and the pitch apexes of the pinion and gear match the crossing point of the axes. In the standard case, the face and root cone apexes also match the crossing point. In such a standard Coniflex design, the base elements are also cones with cone apexes that match the crossing point of the axes. The involute development in Figure 3 can be applied to an infinite number of normal sections along the face width of a straight bevel gear, which allows an involute development like that for cylindrical gears. The conical base elements of both members can be connected with a straight line (the line of action) in each section along the face width, whereas the plurality of all lines of action forms a plane (the plane of action).

This principle is shown in Figure 5. The two cones in Figure 5 are base cones of a straight bevel or a spiral bevel gearset. In the right two graphics, the view is directed such that the plane of action appears as a line that is tangential to the two cones enveloping surfaces. The left side graphic in Figure 5 shows the plane of action three-dimensionally and how it connects the two base elements.

The plane of action cannot be extended beyond its tangential contacting line with the base elements, as shown in Figure 5. The plane of action only exists where tooth engagement is possible, and it is different than the generating gear plane (more specifically explained below). There is, however, one difference from the true involute of cylindrical gears. The rotation of the pinion and gear does not occur in the normal plane but in the transverse plane. Because of this difference, the flank profile of straight bevel gears (and all other bevel gear types) is called Octoide. The Octoide is the analog function of an involute, and it provides bevel gears with the same advantages as an involute provides to cylindrical gears. Those advantages are constant ratio, center distance insensitivity, and ease of manufacturing. Like cylindrical gears, bevel gears also have a trapezoidal generating profile. The straight rack of cylindrical gears becomes a ring, as shown in Figure 6. It is required to establish certain conditions to make the ring rack the generating gear for a pinion and a ring gear that will mesh perfectly together with zero transmission error and line contact identical to cylindrical gears. Those conditions are postulated in the kinematic coupling requirements: The flank surfaces of the generating gears of the two mating bevel gears are congruent (same shape but mirror images, as given in the example of Figure 6) The generating gears of the two mating bevel gears require identical axes of rotation (the top and bottom of the generating gear in Figure 6 form the same generating gear, which rotates in both cases around the same axis and therefore satisfies condition 2) The surface of engagement of pinion and ring gear must be identical to the surface of engagement between pinion and generating gear, and to the one between ring gear and generating gear (without detailed knowledge of the surfaces of engagement, the global condition in Figure 6 seems to satisfy this requirement)

The generating gear principle must be understood as the ultimate vehicle to form the teeth of two mating gears. The first fundamental law of gearing is fully executed by choosing trapezoidal profiles and by applying the kinematic coupling requirements. Gears are designed and manufactured to mesh with each other. What better way to manufacture them than by way of a generating gear? The generating gear is represented by the manufacturing machine; it forms the teeth of the gear (at the bottom in Figure 6) while meshing with this gear perfectly. If the pinion is manufactured with the same generating gear but on the opposite side (at the top in Figure 6) and if the generating gear is imagined infinitely thin, then the result is a pinion that perfectly meshes with the gear having zero motion error. It is also given in such a case that line contact between the pinion and gear flank surfaces exists along the entire face width. Coniflex Pro designed straight bevel gears are manufactured with peripheral cutters, where the tangent to the cutter tip circle is aligned with the root line of the tapered tooth, and the blade profile is aligned with the profile on one side of a generating gear tooth (Figure 6). With this process, it is required that, first, e.g., all left flanks are machined. In a second step, the cutter changes its orientation such that the blade profile aligns now with the second side of a generating gear tooth, and e.g., all right flanks are machined. This way, the generating roll, which is a rotation around the generating gear axis, is repeated for each slot twice. This kinematic condition satisfies the requirement from Figure 6 and fulfills the kinematic coupling conditions. Coniflex is the fastest straight bevel gear manufacturing process, although each slot is addressed twice to generate both flanks. An example straight bevel gearset computer model is shown in Figure 7. The solid model in Figure 7 has been generated by using standard Gleason basic settings, based on the generating gear approach, and by applying a standard Coniflex Plus cutter head as used on Phoenix bevel gear manufacturing machines. The Coniflex straight bevel gear calculation for conjugate contact must be conducted with a dish angle of zero degrees and no root angle correction (DGammaM = 0). The dish angle is creating the length crowning, and the profile crowning is commonly generated with a DGammaM (machine root angle correction). With a dish angle of zero degrees and a DGammaM of zero degrees, flank lead lines are straight lines, and the profile is a true Octoide (involute equivalent).

A contact analysis of the gearset in Figure 7 is shown in Figure 8. The top of the figure shows the Ease-Offs of the left and right flanks (called the coast and drive side in the graphic). The center of the figure shows the motion transmission errors of the pinion and gear flank pairs. The two bottom graphics are the representation of the tooth contact pattern. The contact pattern graphics are axial projections of the flank surfaces and the contact lines in the same plane where a two-dimensional part print would show the tooth area. The contact analysis in Figure 8 confirms the full line contact in each roll position (lower graphics) as well as the zero motion error (only numerical static) in the center graphic, which makes this example a perfectly conjugate straight bevel gearset. In the lower graphic, the path of contact was calculated as a zig-zag line, which indicates an undefined contact path. This means that due to the conjugacy, every point along each contact line is a path of contact point which makes the analysis program pick random points.

The Ease-Off base plane (top graphics in Figure 8) defines the conjugate state of a flank surface pair. Because the Ease-Off graph of the calculated flank pairs matches the presentation plane (base plane) precisely, that is proof that a conjugate and precisely rolling gearset was the input of this contact analysis calculation. The above experiment, creating a conjugate straight bevel gearset, is strictly academic. Conjugacy is the basis of all gearsets manufactured in high volume on dedicated manufacturing machines. A conjugate bevel gearset cannot be used for power transmission because manufacturing tolerances and load-affected deflections, as well as material expansions and deformations under high operating temperatures, will result in edge contact and high load concentrations. The load concentrations already start with a moderate load and cause material damage and considerable noise emission. Although conjugacy is used as a reference for each design, predetermined amounts of length and profile crowning are applied. The right amount of crowning makes a gearset quiet and gives it the required load-carrying capacity. The crowning is shown in the Ease-Off graphics with the conjugate reference always being present as the Ease-Off base plane. Several Ease-Off examples of a gearset with length and profile crowning are shown in the proceedings of this chapter.