technical Non-Involute Gearing, Function and Manufacturing Compared to Established Designs Dr. Hermann J. Stadtfeld and Jasmin K. Saewe

Introduction Positive Profile Shift The standard profile form in cylindri- Zero Profile Shift cal is an involute. Involutes are generated with a trapezoidal rack — the basis for easy and production-stable manufacturing (Fig. 1). However, the Work Roll Work Roll resulting involute shape not only fulfills the law of gearing for a smooth trans- x.m mission of rotation and torque; it also Pitch presents the most robust mathematical Circles profile function regarding center dis- Line of Engagement Work Line of Engagement Work tance changes and other misalignments. Pitch Line Involutes can be modified with a profile x.m shift in order to avoid undercut and to provide a strength balance between pin- Generating Rack ion and gear. The profile shift can also Datum Line Generating Shift x. m = 0 Pitch Line be used to accommodate certain cen- ter distances that are considerably afar Figure 1 Generating principle of involute gearing. from the theoretical center distance for a standard gearset. Involute gearing has line contact between the two mating members. Under load, the contact lines change to elliptical contact areas. The elasto-hydrodynamic of invo- lute gears is well known and easily opti- mized. The pitch line (pitch point in Fig. 2) separates the addendum and the dedendum of the tooth profiles. The sliding and rolling velocities move the profile contact from the top of the gear tooth to the pitch point. The direction of the sliding velocity changes at the pitch point but the rolling velocity direction remains constant. The sliding velocity directional change at the pitch point causes a zero sliding velocity condition Figure 2 Profile sliding and rolling of involute profiles. in an infinitesimally small area “around” the pitch point. In other words, only a relative rolling without any sliding exists For Related Articles Search between the two mating profiles. tooth shapes The absence of sliding presents a criti- at www.geartechnology.com cal caused by the deteriorating hydro- dynamic conditions around the pitch point, or pitch line. High load and low speed reduce the ability of the lubricant

This paper was first presented at the Gleason Corp-hosted 5th (2014) U.S./ WZL Gear Conference.

[www.geartechnology.com] 42 GEAR TECHNOLOGY | January/February 2015 to maintain a surface separating film. Good gear design and modern, high-per- formance oils can eliminate this pitch line phenomenon (Ref. 1). Scientists and inventors frequently introduce new, non-involute gear profiles. The major target of those profile or tooth forms is the reduction of the relative cur- vature between the two contacting flanks in every roll position. Many of the pro- posed systems seem to have a close rela- tionship to the cycloidal tooth profile as it is used in clocks and watches — or to the Wildhaber-Novikov design, first intro- duced by Ernest Wildhaber in 1910. The main argument — often quoted as explanation — as to why those gear types have not been discussed and exam- ined much earlier is the fact that current computation technology and mechanical Figure 3 Cycloidal generating rack profile development. tools of the past were incapable of utilizing both the complex tools as meshing gears to the correct center dis- to manufacture cycloidal tooth profiles well as the complicated machine motions tance location (utilizing the axes play of uses a pointed tool, which follows the needed in order to create the non-invo- the gears in watches). cycloidal profile, guided either by cams or lute profiles. A development of a cycloidal generat- interpolating axes motions. This paper discusses the proposed ing rack tooth profile is shown in Figure The tools to manufacture cycloidal non-involute and involute-related profiles 3. The reference line or pitch line sepa- gears are either special and, therefore, with their advantages and disadvantages. rates the addendum and dedendum of expensive, or the manufacturing process The criteria of the discussion are: the tooth. One roll circle, that is located is very slow. The center distance-main- • Design calculation above the pitch line, rolls to the left and taining feature is only of interest if the • Analysis and optimization possibility generates — with one fixed point — the axes’ position can float. In regular power • Ease of tool manufacturing addendum of the right flank. A second transmissions, where the center distance • Accuracy of tool geometry roll circle rolls on the pitch line from is rigid, but might vary from the theo- • Production-stable manufacturing below and generates the dedendum of the retical center distance, motion error and • Robust operating performance right flank (Fig. 3). The two profile sec- vibration are generated. After an analysis of alternative cylin- tions meet at the pitch line, where both Wildhaber-Novikov gears. The Swiss- drical gear tooth profiles, some new bevel cycloids have an infinitely high curvature. born Ernest Wildhaber invented in and hypoid gear geometries that have The cycloidal tooth profile is “S-shaped,” 1926, shortly after his immigration to been proposed during the past decade are which achieves in the rolling interaction the United States and his employment as discussed as possible replacements for the between mating flanks a large contact Gleason scientist, a helical gear profile traditional face and face milling area on the flank surfaces. The convex which was cut with a circular arc pro- systems. addendum has constant rolling contact file rack cutter (Ref. 2). Thirty years later, Involute-based cylindrical gears — as with the concave dedendum. Surface Mikhail Novikov invented in Russia a well as bevel and hypoid gears — went stress is greatly reduced compared to similar system which featured tooth pro- through a multiplication of their power involute gears due to this arrangement, files that are circular in the transverse density during the past 20 years. In order while the root bending stress can poten- plane (Ref. 3). Because of the similarity to create a fair discussion with objective tially be somewhat lower because of the of the two independent developments, comparisons, the last part of this paper concave dedendum profile, which blends cylindrical gears with circular tooth pro- is devoted to present the latest advance- without curvature reversal into the root file have been called Wildhaber-Novikov ments of “traditional” involute gearing fillet radius. gears. Not only are the tooth profiles cir- using the example of asymmetric cylin- Cycloidal gear profiles can be gener- cular, but the slots of the gear have the drical gear designs. ated with generating rack tooth profiles, appearance of half-circles and remind in Cycloidal gears. Cycloidal gearing has like the one shown in Figure 3.Those pro- its shape of a . Figure 4 reveals been described as center distance main- files must be calculated and manufac- that the gear slots have profiles with taining. The S-shaped profile of the gen- tured dependent upon the individual gear two equal radii, connected with a small- erating rack teeth will form S-shaped pair. Standard generating profiles — like er radius in the root, which blends with teeth and generate a force in the center the straight line in involute gearing — are the flank radii. The profile radii distance direction in order to move the not possible. Another possible process are smaller than the profile radii of the

January/February 2015 | GEAR TECHNOLOGY 43 technical gear, and are connected at the top with a straight line. In order to transmit a constant ratio, the contacting point between the two cir- cular flank surfaces must be kept at the same profile location, which is preferably a point at mid-profile height with a desir- able pressure angle. This point is called the “profile reference point” (Fig. 4). The different radii of pinion and gear flanks

(p1 and p2) have to be oriented normal to the profile reference point, but with a larger gear radius value and a smaller pinion radius value (Fig. 4, left). Keeping the contacting point in the same initial profile location (Fig. 4, left) during an incremental pinion (and gear) rotation can only be realized with the introduc- tion of a helix angle; the helix angle must be defined such that for a certain pinion rotation the profile rotates back to the initial position. Subsequently, the gear profile also has to rotate back into the ini- tial position by an angular amount equal Figure 4 Wildhaber-Novikov tooth profile design. to the angle of the pinion rotation, divid- ed by the ratio between the two members. The method of corrective rotation of the tooth profile in order to transmit a constant ratio is visualized (Fig. 5). The first point of contact between the mesh- ing teeth is shown in the upper graphic as the connecting point of R1 and R2. As the gears rotate to an advanced angu- lar position, the radii R3 and R4 would transmit a different ratio because the vec- tor length — as well as the normal vec- tor direction — changed. The cylinder in the lower graphic demonstrates that if a flank line with a particular helix angle is used that rotates the contacting point for incremental rotations into the horizon- tal axis connecting plane, then contact movement (Fig. 6) can be expected. The ratio will remain constant in this case because the radii — and the normal vec- tors in the horizontal plane along the ref- erence cylinders — remain constant. A simple example: A pitch angle of 24° Figure 5 Line of engagement is kept at centerline connecting plane. and a contact ratio of 1.0. If the pinion is rotated by 24° and the initial contact at the profile reference point is located at the front face of the teeth, then the back rotation of the profile has to occur as the contact moves along the face width to the back face of the teeth.

[www.geartechnology.com] 44 GEAR TECHNOLOGY | January/February 2015 The helix angle at the reference radius is calculated: ßR = arctan (RR ∙ P/F) Example: Given: Ratio i = 2 Face width F = 50 mm Pinion pitch radius RP1 = 38 mm Pinion reference radius RR1 = 41mm Pinion angular pitch P1 = 24° Gear pitch radius RP2 = 76 mm Gear reference radius RR2 = 73 mm Gear angular pitch P1 = 12° Wanted helix angle along pitch line in case of face contact ratio = 1.0: Pinion helix angle at Ref. rad.: ßR1 = arctan (41 ∙ 24°/180° ∙ π/50) = 18.95° Pinion helix angle at pitch rad.: ß1 = arctan (tan (18.95°)/41 ∙ 38) = 17.66° Gear helix angle at Ref. radius: ßR2 = arctan (73 ∙ 12°/180° ∙ π/50) = 17.00° Gear helix angle at pitch radius: ß2 = arctan (tan (17.00)/73 ∙ 76) = 17.66° The example shows that a particular Figure 6 Contact movement maintains constant ratio. Wildhaber-Novikov gear design requires a particular helix angle that depends on the desired contact ratio and the face width. The explanation and the calcula- tion also show that the transverse contact ratio of Wildhaber-Novikov gears is zero and the modified contact ratio is equal to the face contact ratio. The advantages of circular tooth profile are the large contact area. Russian studies from the 1950s and 1960s report about three-to-five times the load-carrying capacity, without detrimental pitting or wear. The reports also state that the cir- cular wedge geometry between the con- tacting circles that moves along the face width, pumps the lubricant into the con- tact area and generates oil film thickness- es up to 10 times that of involute gear- ing, which should also improve efficiency (Chironis). Disadvantages include the more complicated rack tool geometry, the Figure 7 Extended version of Wildhaber-Novikov gears. high influence of the helix angle to the smoothness of , as well as Because of this constraint, Wilhaber- optimal is high stiffness at the root of the the sensitivity of center distance changes. Novikov teeth consist of less than a half- teeth and elasticity from mid-dedendum The helix angle in Wildhaber-Novikov circle, which results in tooth depths that to the tip. The elasticity contributes to a gears is not a design freedom like in invo- are about 1.2 times the module, com- reduced entrance impact during mesh- lute gearing, but is exactly given for each pared to 2.2 times the module for stan- ing at different loads and improves the particular design. The operating vibra- dard involute gears. The pressure angle load sharing between consecutive tooth tion and noise of Wildhaber-Novikov change along the profile-per-unit-length pairs. In order to account for those facts, gears has been reported to be higher than in case of less than a half-circle is a multi- Wildhaber, as well as Novikov, mentioned that of comparable involute gears. ple of the involute curvature change. This in their teachings the possibility to extend The original Wildhaber-Novikov tooth is an additional reason to the non-invo- their ideas to double-circular profiles that profiles are basically half-circles, which lute profile function for the high sensitiv- consist of a convex circle at the adden- call for a certain pressure angle change ity to center distance changes of pure cir- dum and a concave circle at the deden- between top and root. The end of the cular profile forms. The low-profile teeth dum (Fig. 7). internal circle towards the top and the show a very high stiffness, which often All variations of Wildhaber-Novikov external circle towards the root are given is falsely judged as an advantage and a gears can be manufactured by the hob- by a pressure angle that drops below 5°. contributor to high power density. More bing and shaping processes. Depending

January/February 2015 | GEAR TECHNOLOGY 45 technical upon whether the tooth normal profile the extended Wildhaber-Novikov profile nominal pressure angle at mid-adden- or the tooth transverse profile should be design (Ref. 4). Nagata bases the profile dum. of circular shape, the cutter rack profile definitions not on the gear teeth, but on Nagata shows in a theoretical evalua- must consist of modified curves (no cir- the rack cutter or reference profile. The tion a significant reduction in center dis- cles) to accommodate for the generating basic construction of this profile is rep- tance sensitivity of the improved profile motion between cutter and work. Also, resented in Figure 8. The new profile has combination of circular and involute ele- the use of circular-shaped rack cutters is a circular addendum and a dedendum ments. In a following study, Nagata tested mentioned in the literature, which would that consists of two involutes that blend a variety of Wildhaber-Novikov-Nagata of course generate complex non-circular at mid-dedendum. The upper dedendum gears and published the results with rec- profile curves. is developed with a cord — unrolled from ommendations for optimal parameters Profile development. Shigeyoshi the upper base circle — where the lower in 1985 (Nagata 85). The manufacture of Nagata, a professor at the University of dedendum is developed with a cord that Wildhaber-Novikov-Nagata gears is pos- Tokyo published a paper in 1981 where unrolls from the lower base circle. The sible by hobbing and shaping. All com- he discusses a proposed improvement of addendum is circular (Fig. 8) and has a mon hard-finishing methods can also be applied if the tool profile is formed accordingly. Like in standard Wildhaber- Novikov gears, the helix angle is required and differs depending on the individual gear design. Convoloid gearing. Bernard Berlinger and John Colbourne introduced in a paper, published in 2011, a tooth form called Convoloid (Ref. 5). The new tooth form appears optically very similar to the Wilhaber-Novikov-Nagata development (Fig. 9). The addendum has a convex shape, while the dedendum is concave. The transition zone at the pitch point seems to be “S-shaped” rather than the straight section of Nagata’s development. The authors report that the tooth profiles are computer calculated as a point cloud for each application case individually. One interesting conclusion of Figure 8 Wildhaber-Novikov-Nagata profile construction. Berlinger’s and Colbourne’s findings is the fact that while involute gearing fit well with traditional, mechanical , it is outdated for today’s engineering and manufactur- ing environment. Test rig investigations of Convoloid gears resulted in 20%-35% increased torque levels vs. involute gear- ing. The center distance insensitivity of involute gears seems not to be given, but the inventors state that the Convoloid gears can withstand the customary deflections given in modern gearboxes. An interesting aspect of Convoloid gears is that the tooth contact can move from root to top while maintaining the correct ratio. This makes Convoloid gears independent from the helix angle and allows a choice of suitable helix angles, depending on gearbox application requirements. The Convoloid profile of Figure 9 refers to the final teeth, not to the rack profile. In order to establish the rack cutter profile, the kinematic rela- Figure 9 Convoloid tooth profile.

[www.geartechnology.com] 46 GEAR TECHNOLOGY | January/February 2015 tionships, e.g., of a hobbing process, have to be employed to calculate a point-based cutting edge definition. S-shaped tool profile for spiral cutting. Stepan Lunin discussed a Wildhaber-Novikov-style profile for a bevel gear cutting tool in a paper pub- lished in 2001 (Ref. 6). The profile con- sists of radii and straight lines (Fig. 10). The pitch line of spiral bevel gears is in all cases of non-miter gears located towards the pinion root (positive profile shift) which will create profile sliding at the location of the transition wave. Profile sliding during the generating process will eliminate or mutilate the transition wave, which makes the intention of the mid- section of the proposed profile question- Figure 10 S-shaped cutting blade profile for spiral bevel gears. able. The goal of using such a profile might be the reduction of unit surface pressure with an increase of power density. This could be accomplished with the extended Wildhaber-Novikov profile that would also eliminate the complicated tool mid- section, which is not believed to serve any practical purpose. The paper doesn’t mention the relationship between profile and spiral angle. Because of the circular profile sections, it is assumed that the tool profile in Figure 10 requires — just as with the Wildhaber-Novikov gears — a particular spiral angle in order to main- tain the correct transmission ratio during the meshing of the mating members. This will make the gearsets generated by the tool profile in Figure 10 sensitive to hous- ing tolerances and deflections. Toroidal drive. The inventor M.R. Kuehnle developed the idea of a compact and high-power- density, high-reduction Figure 11 Upper-half of the toroidal “ring gear.” planetary unit. The “heart” of the unit is a toroid which consists of a mounted of the internal toroidal threads that ini- ponents to a high level of mechanical upper and a lower “half-toroid” (Figure tiate a slow cage rotation (cage as out- design and manufacturability. 11 shows the upper-half of the toroid ring put shaft). The principle of planet and Kuehnle stated his toroidal transmis- gear). The toroid has internal spherical cage rotation is shown in Figure 13. The sion will out-perform worm gear, plan- threads that are the grooves for balls that inventive properties of the toroidal gear- etary and cycloidal transmissions for connect the toroid with a sun gear via box are high reduction with compact medium to high ratios. Center worm planets and planet carrier. The arrange- gearbox (high power density) and low and planet units seem straightforward in ment between center unit (sun gear wear. manufacturing and assembly. However, worm), planets and planet carrier motion In order to streamline the design of the spiral grooves in the split, internal is shown in Figure 12. The sun gear unit the toroidal gearbox and verify the attri- toroidal gear components are difficult is a multi-start, spherical worm (similar butes of its functionality with scientific to manufacture and will, post-assembly, to a throated worm) (Ref. 7). data, the inventor consulted the Institute present a disturbance in the smooth roll- The toroid-shaped unit is normally of Machine Elements of the Technical ing of the balls at the fitting seam. used as the transmission housing. If the University of Aachen. The scientists and Globoidal gearing. Yakov Fleytman in sun unit is used as an input shaft, then engineers at the institute brought all com- 1999 invented a tooth form that most- the planets will rotate along the grooves ly applies to hypoid gears. Fleytman

January/February 2015 | GEAR TECHNOLOGY 47 technical claims the strength of his new hypoid ment. Flank profile and tooth lead form in 2007 a bevel gear with a sinusoidally gear design is three times that of tradi- seem different to common bevel and formed flank line with the claim of 40% tional hypoid gears (Ref. 8). It seems that hypoid gears because no customary pro- increased load-carrying capacity (Ref. 9). the creation of the gear tooth was done file requirements are mentioned and the The teeth of that system have some sim- by using simple straight or twisted sur- equivalent pitch elements are not related ilarity with the “herringbone” teeth in faces. The pinion flank surfaces might to the transmission ratio (Fig. 14). cylindrical gearing. The observer of the have been generated with simulation soft- Globoidal gears cannot be manufac- photographs in Figure 15 notices large ware, similar to Vericut. The gear can be tured using traditional manufacturing variations of the tooth thickness and a defined as a tool and the axis positions methods. Prototypes have been manu- curvature inflection point at each side of pinion and gear in a given hypoid gear factured with 5-axis machines, but no of the center. It appears that the reason box can be used as tool and work posi- reports of evaluation results have become why the inventor anticipates a higher tion in order to simulate the generat- public. strength could be given by the curved ing process of the pinion flank surfaces. Cosine gears. HPG in the Netherlands, profile in the center of the face width. Fleytman uses the kinematic a company that developed 5-axis machin- The reversal with a lowering of the spiral condition to generate the pinion teeth ing technologies for the soft and hard angle towards the ends of the teeth seems where the gear is used as a generating ele- manufacture of bevel gears, introduced unjustified and results basically in teeth with no spiral angle — which gives them similar properties to ZEROL bevel gears. Real herringbone gears are two connect- ed helical gears with opposite helix angle directions. The advantage is a complete cancellation of all axial forces, combined with a very smooth mesh characteristic in their operation. The two opposing sec- tions of herringbone gears are separated by a groove in order to eliminate the sin- gularity at the point of helix angle change. This common design takes advantage of the smooth mesh- ing and the high strength of helical gears without the disadvantage of rolling dis- turbance in the transition area between right- and left-hand sections. It appears that the developers of the gear in Figure 15 supposed that the sep- aration groove of herringbone gears is Figure 12 Worm (sun gear), planets and planet carrier cage motion. strictly the result of a manufacturing lim- itation of the traditional manufacturing methods and believed that the curved flank line at the center of the face width would increase the strength. The addi- tional reversal of the flank line curvature

Figure 13 Toroidal, spiral motion and cage rotation. Figure 14 Globoidal, angular gear pair.

[www.geartechnology.com] 48 GEAR TECHNOLOGY | January/February 2015 towards the end of the teeth is probably The problem is not only the manu- also only done because it wouldn’t be facturing of asymmetric spur gears. It possible in traditional manufacturing. seems to be the fact that additional opti- Meshing conditions of bevel gears mizations — like circular top relief — have without a spiral angle are best with to be applied in order to make the straight bevel gears and deteriorate with improvement visible. Fuentes developed ZEROL bevel gears. The smooth screw- first an optimal top relief for a baseline ing into mesh like in spiral bevel gearing design and then converted the gearset is not possible with the cosine teeth in to asymmetric profile. However, he cre- Figure 15. The face contact ratio is slight- ated comparable gearsets with symmetric ly above zero and will not contribute to and asymmetric profile and was able to improved strength. The cosine-shaped achieve contact stress reductions in the tooth form prevents elastic deformation vicinity of 10%. under load, which in spiral bevel gears is Several bevel gear manufacturers used to achieve a smooth meshing and to accepted the publications’ findings and increase the load-sharing between con- research in asymmetric spur gears and secutive tooth pairs. Hartmuth Müller applied moderate amounts (up to 4°) of published theoretical investigation results asymmetric pressure angles to their tra- of bevel gears with sinusoidally shaped ditionally symmetric spiral bevel gears flank lines and concluded that their in order to improve the load-carry- rolling performance — as well as their ing capacity of mostly unidirectional- strength properties — is far below any used angular transmissions. The posi- average set (Ref. 10). tive experiences with spiral bevel gears Asymmetric tooth gears. The two flank allow the conclusion that, not only spur profiles of involute gearing are generally gears, but also helical gears would benefit developed from a common base circle. from asymmetric profiles. Because of the In this case both flank profiles have the individually ground cutting blade pro- same pressure angle and are mirror imag- files, this technology seems to gain more es from each other. With the develop- acceptance in bevel gearing. For cylindri- ment of hypoid gears, Ernest Wildhaber cal gears the introduction of asymmetric in 1930 developed the asymmetric tooth profiles would require a departure from Figure 15 Ring gear and pinion with cosine profile (Ref. 11). Wildhaber’s develop- standard hobs and may require additional teeth (Source: hpg-nl.com). ment eliminated the conflict that the path profile corrections in order to work out of engagement of the drive-side flanks the full advantages of asymmetric gear- and the coast-side flanks was not equal ing, which seems to be a large step and if hypoid gears were manufactured with thus precludes ready acceptance in the equal pressure angles. industry. Today, asymmetric profiles that have been proposed for more than 20 years are discussed and applied to practical applications. The goal is not to extend the angle of engagement, as it was for Wildhaber, on the drive-side of hyp- oid gears. Extending the active line of engagement would require a reduction of the pressure angle. Alfonso Fuentes et al. (Ref. 11) proved in an analytic compari- son that especially flank surface stress is reduced on the driving side by increas- ing the pressure angle. A tooth mesh with asymmetric pressure angles is shown in Figure 16. Alexander Kapelevich promot- ed asymmetric cylindrical gears for many years (Ref. 12) and wrote calculation soft- ware that allows design calculations for optimized geometries.

Figure 16 Asymmetric tooth engagement.

January/February 2015 | GEAR TECHNOLOGY 49 technical

Summary to the properties of the “half-round” pro- performance advantages resulted in this The tooth shapes discussed here can be file of Figure 4. case in load-carrying capacity similar to sorted into those with interesting physical The Nagata development is very simi- straight bevel gears, and a rolling perfor- properties and those without sufficient- lar to the extended Wildhaber-Novikov, mance that cannot compare to standard ly proven functionality. It is quite true but requires rather more complicated tool spiral bevel gears. that validating a new system is difficult geometry. The “S-shaped” profile makes Asymmetric gears build upon the if the technical community rejects the it an additional version of the Cycloidal strength of involute gearing, with the idea based on solely subjective reasons. gear type with similar physical proper- acknowledgement that in case of a pre- This would prevent any private or public ties. Center distance sensibility and diffi- ferred driving direction, the properties research funding. However, many of the cult manufacturability also prevented the of the driving flanks can be improved new tooth profile solutions discussed in introduction of this gear type in power by taking away from the non-driving, this paper are based on brilliant ideas that transmissions. or coasting, flanks. The practical results failed to have a breakthrough for similar Convoloid gears, with their dedendum of transmissions with asymmetric gears reasons, as why, for example, the Wankel transition point, which is a “flat spot” show a significant increase of power den- engine never replaced the stroking piston in a finite profile section, cannot accept sity due to an improvement of root bend- engine. significant center distance changes. The ing strength and higher surface durabil- Cycloidal gears have been used for flat section of the two mating profiles ity. Investigations showed that the full watches and clocks. Even though they must be located precisely during the roll- advantage of asymmetric gearing vs. are center distance-sensitive, the watch ing process to separate addendum and gears with symmetric involutes can only designer allows rather large center toler- dedendum rolling. The separation may be realized if sensible tip relief modifica- ances in connection with axle bearing interrupt the hydrodynamic lubrication, tions are applied. This in turn creates a play. The “S-shaped” profiles of the mat- and misalignments may cause rolling problem for many manufacturers — even ing cycloidal gears develop a self-center- of the transition flat into addendum or those that mass-produce, say, automotive ing force that forces the gears to roll with dedendum, which will create roll inter- transmissions. The wealth of experience correct center distance. The advantage of ference. The transition flat prevents or that has been compiled over decades that this design was the manufacturing cost complicates the profile generation with found its way into the international stan- reduction due to large bearing position a reference profile on a tool like a hob. dards, as well as the material application tolerances and large axle play. The large Convoloid profiles seem to present the tables, has no or only limited use for the bearing play in turn would guarantee disadvantages of cycloidal profiles, with dimensioning and design of asymmetric low friction in the sleeve bearings, which increased vulnerability around the pitch gears. Also, the cutting tools are now not achieves one important objective of chro- line. standard tools anymore. Different pres- nometers. Manufacturing of the cycloidal The “S-shaped” bevel gear profile is sure angle offsets for different applica- requires specially formed tool profiles, or very similar to the Convoloid profile. tions and gear design parameters in con- pointed tools, that move along a cycloidal However, where the Convoloid profile nection with circular tip relieves of differ- path. The advantages cycloidal gears have is the result of an analytical mesh and ent amounts will not only make existing in watches cannot be duplicated in indus- contact area optimization, the singular- tooling obsolete but also eliminate today’s trial power transmissions. ity of the “S-shaped” bevel gear profile in standards in hobs and even shaper cut- Wildhaber-Novikov gears have proven mid-profile will not permit relative pro- ters which will result in increased cutting their increased load-carrying capacity, yet file sliding in this section; this makes this tool cost and contribute to longer tool- show significant center distance sensitiv- proposed profile unusable in practical ing lead times. The fact that asymmetric ity, are difficult to manufacture, require applications. gears have not had a breakthrough yet is non-standard tools and depend on a spe- Globoidal gears use the kinematic cou- the result of those obstacles. However, the cific, pre-determined helix angle in order pling conditions of formate gears. The symmetry offset, as well as tip reliefs or to transmit a constant ratio. Changes in profile of the gear member is basical- other corrections, could be standardized, center distance and axes inclinations due ly chosen to be straight and the pinion depending on module and application, to tolerances and deflections will intro- member is generated with respect to the which would remove the major obstacle duce significant motion errors and trans- desired ratio, shaft angle and center dis- for the broad application of asymmetric mission vibrations, which would increase tance. The principle of interacting pitch involute gears. the cost for shafts, bearings and/or hous- elements is not applied, which leads to ing in order to reduce tolerances and high relative sliding and can cause partial Conclusions deflections. Even then, the robustness mutilation to the flank surface. The motivation to change from involute and “mechanical intelligence” of involute Cosine gears have been discussed to profiles and proven straight or curved gearing that provides them with smooth also elaborate on flank line ideas which flank lines to alternative shapes is fueled rolling through many million cycles do not conform to the common mathe- by the logic that, in today’s time and age, could not be achieved. The extended ver- matical flank line functions. Cosine gears more than just incremental improve- sion of the Wildhaber-Novikov profile might have the ability to hold higher ments should be possible, applying new has an “S-shape” — just like the Cycloidal torques in a non-rotating, static condi- theories and sophisticated computation profile that adds the self-centering ability tion. The attempt to design a gearset with technology. Those dramatic improve-

[www.geartechnology.com] 50 GEAR TECHNOLOGY | January/February 2015 ments would consequently lead to Nevertheless, it seems fair to say: “The geometries that would appear exotic if involute is here to stay.” observed with the traditional viewpoint. The objective of this paper was to observe References the published alternative tooth forms 1. Stadtfeld, H.J. Gleason Bevel Gear Technology with an open mind in order to find objec- —The Science of Gear Engineering and mod- ern Manufacturing Methods for Angular tive answers to the question of why the Transmissions, Gleason Publication, ISBN 978-0- interesting and sophisticated tooth form 615-96492-8, 2014. proposals haven’t had a breakthrough in 2. Wildhaber, E. “Gears with Circular Tooth Profile the gear manufacturing industry. Similar to the Novikov System,” VDI Berichte, No. 47, 1961, Germany. In angular gear drives, the absence of 3. Novikov, M.L. USSR Patent No. 109,750, 1956. tight standards, as they exist for cylindri- 4. Nagata, S. And Y Ariga. “Development of a New cal gears, led to many gear types with dif- Wildhaber-Novikov Gear with a Basic Rack ferent profile and lead functions. The fact of Combined Circular and Involute Profile, ” International Symposium on Gearing & Power that bevel gear cutting tools have never Transmission, Tokyo, 1981. been standardized opened the door for 5. Berlinger, B.E. and J.R. Colbourne. “Convoloid this variety of different processes and Gearing Technology,” Gear Solutions, Media Dr. Hermann J. Stadtfeld in 1978 tooth forms. The downside is that bevel Solutions, INC Publishing, February 2012. 6. Lunin, S. “New Methods of Gear Geometry received his B.S. and in 1982 his M.S. in gear design and manufacturing is per- Calculation,” the JSME International Conference mechanical engineering at the Technical ceived as complicated, not straightfor- on Motion and Power Transmission, The Japan University in Aachen, Germany; upon receiving ward — and expensive. The standards Society of Mechanical Engineering, Fukuoka, his Doctorate, he remained as a research scientist at the University’s Machine Tool in cylindrical gears, which are based on Nov. 15, 2001. 7. Troeder, C. “Toroid-Getriebe Laboratory. In 1987, he accepted the position standard reference profiles and involutes, Forschungsvereinigung Antriebstechnik,” FVA of head of engineering and R&D of the Bevel make design and manufacturing trans- Information Congress, Karlsruhe, November Gear Machine Tool Division of Oerlikon Buehrle 1975. AG in Zurich and, in 1992, returned to academia parent and keep the cost of cylindrical as visiting professor at the Rochester Institute gear manufacturing relatively low. 8. Fleytman, Y. Worm Gear Transmission United States Patent No. 6,148,683, November 2000. of Technology. Dr. Stadtfeld returned to the The center distance insensitivity of 9. Schlossig H.P. Auf einfachem Weg zu guten commercial workplace in 1994 — joining The involute gears also applies in the rela- Zähnen Werkstatt und Betrieb, Carl Hanser Gleason Works — also in Rochester — first Publishing, Munich, April 2007. as director of R&D, and, in 1996, as vice tionship between generating rack and president R&D. During a three-year hiatus work gear and therefore makes hob- 10. Mueller H. “Function Oriented 5-Axes Machining of Bevel Gears” Seminar (2002-2005) from Gleason, he established a gear research company in Germany while bing a very robust process. The fact that “Innovations in Bevel Gear Technology,” WZL, simultaneously accepting a professorship every point along the involute has a nor- Technical University Aachen, March 2012. to teach gear technology courses at the 11. Fuentes, A. “On the Behaviour of Asymmetric mal direction that is tangent to the base University of Ilmenau. Stadtfeld subsequently Cylindrical Gears in Gear Transmissions” SAE- circle gives the involute its robustness returned to the Gleason Corporation in 2005, China and FISITA (eds., Proceedings of the where he currently holds the position of vice during manufacturing and in operation FISTA 2012 World Automotive Congress, DOI: president, bevel gear technology and R&D. A 10.1007/978-3-642-33744-4_13, Springer- (Fig. 17) (Ref. 13). All of the discussed prolific author (and frequent contributor to Gear non-involute gears are sensitive to cen- Verlag, Berlin, Heidelberg 2013. Technology), Dr. Stadtfeld has published more ter distance changes and require indi- 12. Kappelevich, A. “Asymmetric Gears:Parameter than 200 technical papers and 10 books on Selection Approach,” Gear Technology, June/July vidual tool designs. Standardization is bevel gear technology; he also controls more 2012, Pages 48-51, Randall Publishing Inc., Elk than 50 international patents on gear design, not possible or difficult, which presents Grove Village, Illinois. gear process, tools and machinery. an additional risk in design and manu- 13. Stadtfeld, H.J. “Operating Pressure Angle,” Gear Upon high school graduation Jasmin K. facturing. If higher load-carrying capac- Technology, May 2013, Pages 57–58, Randall Publishing Inc., Elk Grove Village, Illinois. Saewe, from April to June 2011, completed ity, lower noise and increased efficiency basic training in metalworking during are goals of new gear geometries, then the her internships with German-based TRW Automotive and Aleris Rolled Products. absence of standards and higher design In October 2011 she enrolled as a student and manufacturing cost might be accept- of mechanical engineering at the RWTH able in some cases. However, the physical Aachen University. At the Aachen University properties of involute gearing are superi- Jasmin became a student research assistant at the WZL, where she was involved with or to most of the discussed non-involute practical gear cutting trials and tool life gears. The potential of asymmetric, invo- investigations;this spurred her interest in an lute tooth profile will allow significantly internship with Gleason. Saewe in April 2014 improved cylindrical gears by maintain- began an internship in the R&D department at The Gleason Works in Rochester, New York. As ing the advantages of involutes. A broad- of January 2015, Jasmin continues her studies er interest on asymmetric gears could ini- at the RWTH-Aachen University in order to tiate the development of new standards finish her Bachelor’s degree, while continuing her employment as student research assistant and would — over time — allow the gear at the WZL. community to gain sufficient experience in this advanced system.

January/February 2015 | GEAR TECHNOLOGY 51