Charles D. Schultz
Today’s competitive industrial gear marketplace demands products with excellent reliability, high capacity and low noise. Surface-hardened, ground tooth gearing predominates, but the legacy tooth forms handicap further improvements in capacity and noise generation. Vehicle and aircraft equipment use tooth forms not found in the standard tables to achieve better performance — with little or no increase in cost. This paper will propose adopting these high-contact ratio forms to industrial use.
I first became aware of deeper-thanstandard tooth forms in 1979. The venerable company had been through tough times but its staff of engineers and designers came up with some creative solutions in the effort to remain competitive. When competitors started to shift to carburized gearing and invest in gear grinding equipment, the owners did not have the cash to follow suit. Some clever engineer decided to use teeth that were 20% deeper than standard and nitride them. The rating methods then in effect gave them competitive power densities with only the purchase of custom cutting tools.
The 1.2 addendum combined with the 25° pressure angle did not result in true high-contact ratio geometry (Fig. 1). Poor tool life, especially when cutting hard pre-nitriding blanks, made for some production challenges. Coming from a through-hardening background I was very skeptical, but over time found the tooth form provided good results in the field. Replacing the special hobs wasn’t possible in the reduced order volume of the early 1980s, however, and we did not use the 1.2 addendum system in new design standard products.
My next exposure to high-contact ratio gearing came 11 years later during a tour of the Saturn automobile plant in Spring Hill, Tennessee. The Society of Automotive Engineers (SAE) organized the event and we were keen to see the compact, integrated gear manufacturing cell that had been set up to produce all the components needed for a frontwheel drive transaxle. It was an impressive achievement in 1990 to begin with raw forgings at one end of the line and have complete carburized, hardened, and ground helical gears ready for assembly at the other end. General Motors spent plenty of money on the project and it challenged the best equipment builders in the world to participate.
The gear line included an automated inspection station after the gear grind operation. While watching the charting of parts in the cue, I noticed that the teeth were much deeper than “normal” but did not think to ask our guide a question about it. The equipment supplier gave out sample charts and when we debriefed back at our office we tried to run the geometry shown on it through our gear analysis software. The home-brewed code “blew up” at the dimensions entered and when we dug into the error codes it was found to have exceeded the “allowable” profile contact ratio of 1.99. We didn’t at first understand the significance of this limit in conventional gear design, but after scouring our engineering library we came across a great paper by J.C. Leming (Ref. 1) that explained things very well. Despite the many advantages of high-contact ratio gearing that Leming pointed out, we put the concept aside and continued to design products with “standard” teeth.
A couple years later, though, one of our salesmen asked us to help a potential customer resolve a noise problem with his equipment. Our firm had a welldeserved reputation as a supplier of high-quality ground tooth gears and we went to work reviewing a consultant’s telephone book-thick report on the customer’s “problem.” Unfortunately the solutions suggested were things we had tried before without much success and we told the salesman we did not think the project was worth pursuing. But this salesman was a very persistent man and he refused to take no for an answer. Under the guise of giving the client a tour of our facility, he arranged for a couple of engineers to meet with my boss and me. We explained our dismal prognosis for quieting his gearbox and figured we were done with the matter. These engineers were just as persistent as our salesman and they knew we wouldn’t be able to resist a well-argued challenge — especially after they told us their project motto was “We won’t fail because we didn’t spend enough money.”
During the brainstorming that followed the Saturn tour, the Leming article came up. While I went to retrieve the reference book with the Leming paper in it, my boss committed me to designing a set of high-contact ratio gears in less than a week. There was, after all, a three-day weekend coming up and there would be fewer distractions. Six days later we met again and reviewed the proposed design. We had no way of predicting the possible noise reduction but the geometry worked out and we were ready to make drawings. The customer started expediting delivery of prototypes before the review meeting was over. We thought perhaps two weeks after the hobs arrived, maybe eight to ten weeks total.
But this was not acceptable and the customer promised to use his influence to get the hobs made more quickly. The next day, when the drawings were done, he called back to report that there could be no rush hob delivery. What other options were there? Jokingly reminding him of his project motto, we suggested wire cutting the parts. He didn’t find the attempted humor funny and asked for blanks to be ready for his pick-up in two days. Said blanks were back to us three days later with Q9 quality teeth cut in them using tooth plots we provided. The sample gearbox was put on test two weeks later and the results were excellent. Noise reduction goals were easily met with no tooth modifications required.
Knowledgeable observers could not let go of the long, thin teeth appearing to be so delicate; surely those skinny teeth will break, they insisted. Upon completion of the sound tests the prototype gearbox was subjected to the same breakage test used many years earlier to approve the previous gearbox for production. It was still running flawlessly after completing the test three times. The conventional gearbox seldom survived extended testing. A modified version of the high-contact ratio gearbox has now been in production for over 20 years.
Tooling budgets and production schedules prevented me from often using high-contact ratio tooth forms while a gear company engineer. We managed to purchase a few HCR hobs for specific projects where there simply was not enough room for conventional gears to transmit the load but, regrettably, there was not the will to implement this technology in a widespread way. Now that I have my own consulting firm I hope to change that situation and assist clients in developing HCR-geared products.
The History of High-Contact Ratio Gearing
The official “history” of high-contact ratio gears begins with aircraft gearboxes in World War II. Leming’s excellent summary of the development work on aircraft systems was published in 1977 but there is also some unofficial history dating back much further that bears study. We take the “standard” involute tooth forms for granted as they were adopted long before any of today’s working engineers were born. The 14½° “full-depth” involute was the first to gain official recognition in April of 1921; but even back then there was an effort to switch to 20°, first at stub-depth and shortly thereafter at full-depth, to meet increasing load requirements for automobiles and trucks. A “composite” 14½° system that combined an involute and cycloidal form into a single reference rack was also adopted in the 1920s, a recognition that not everyone was completely sold on the involute system either.
So where did the “standard” form come from? If you look at old photographs or drawings you will see a variety of tooth proportions, especially prior to the widespread use of hobbing and shaping machines in the late 1880s. Many gears had cast teeth and there is some evidence that the 14½° system became popular in part because the sine of 14½° is 0.25 and that makes it easier to draw the tooth shape into the pattern than other pressure angles. A more plausible reason, based upon my limited foundry experience, is that 14½° teeth have wider top-lands, which would be easier to maintain in the foundry conditions of that time.
In research for this paper I purchased a reprint of the American Machinist Gear Book (Ref. 2). Originally published in 1915 (pre-dating AGMA), this volume is a time capsule of our trade. Six different involute tooth systems are described as a prelude to discussing the need for a “standard” tooth form (Table 1). Wilfred Lewis’s 1900 speech to the American Society of Mechanical Engineers (ASME) is quoted at length. When he started in gears in 1870 cycloidal teeth were predominant. By 1875 he was sold on the advantages of the involute system but he didn’t like the 14.5° and 15° systems proposed. He went with 20° as, “I did not at the time have the courage of my convictions that the obliquity should be 22.5° or one-fourth of a right angle.” I mention this as evidence that there is nothing magic about the tooth forms we have settled on as “standard.” Using Lewis’s dates we have a timeline of involute teeth coming into common use in 1875; a committee being assigned to adopt a standard form in 1891 with ASME; AGMA being formed in 1914; and the 14.5° full-depth tooth not being enshrined as standard until the 1921 AGMA annual meeting. Even in 1921 there was enough debate so that the 20° stub, composite cycloidal/involute rack, and 20° full-depth form were put “on track” for later standardization.
It is reasonably safe to say that the 14.5° form was not selected for its dynamic characteristics, as the 1921 debate recognized the more favorable sliding characteristics of the 20° stub system, along with its purported greater strength. I say “purported” based upon some instances I observed many years later where shaker screen gears were actually found to resist tooth breakage better at 14½° than even 25°. This puzzled us until we discovered the profile contact ratio was 2.47 with the legacy tooth form, and only 1.63 with the supposedly stronger 25° tooth. The same part with 20° full-depth teeth had a 1.93 profile contact ratio and it too suffered tooth breakage in the field. This situation points out the need to avoid single tooth contact entirely when designing HCR sets; the profile contact ratio has to remain over 2.00 at all times, regardless of tip relief or center distance fluctuation.
Many pressure angle and tooth depth systems were in use prior to “standardization” and they continued to be popular long after the 1920s. None had an addendum that exceeded the familiar 1/transverse diametrical pitch until Buckingham (Ref. 3, Section 2, Spur and Internal Gears) proposed a 1.35/NDP system for instrument gears (Fig. 2). I confess to using this book for many years and not noticing this gear tooth system until I started researching this paper. Buckingham does not discuss profile ratio in his presentation, despite developing the rack offsets needed to use the tooth form on spur pinions down to 5 teeth.
This is not to say that high-contact ratio gears were not used prior to 1935. One example of very non-standard tooth proportions that I am personally familiar with dates to 1895-vintage Hulet unloading machines. These revolutionary devices created an amazing reduction in the cost of unloading bulk products from the holds of ships on the Great Lakes and are considered national landmarks in Cleveland, Ohio and Wisconsin. The drive mechanism used “finger gears” to allow for a big change in center distance (on the order of 1 inch); finger gears (Fig. 4) were so named because they looked like fingers. The pressure angle was very low, around 8°, but the whole depth was on the order of 5 inches divided by the nominal DP. We were contracted to make spare pinions using our 1916-vintage gear milling machine. As I recall the tooth space was so deep and narrow we had to use three different milling cutters to get the shape and, because of accuracy limitations of the technology, hand-file the transitions to get relatively smooth operation.
Most of the manufacturing techniques in use today were available 100 years ago; the machines were far less accurate and they were a great deal slower. Metallurgy and heat treating were not as sophisticated; bearings were of much lower capacity and quality. Every aspect of machinery was slower and our predecessors, being very practical people, reserved gear grinding for applications where it was the only way to get the gearbox to work. The 14½° full-depth form was still adequate for most applications in 1921, but designers could see that the 20° form, first in stub-depth and later in full-depth, offered advantages for the future.
My purpose in bringing this topic into the discussion of high-contact ratio teeth is simply this: The old answers were based on old conditions. We have different conditions in effect today. Many of the old technology and cost limitations are no longer in effect. We are under great commercial pressure to produce lighter, more compact, longer lasting gearboxes at lower prices. The design rules have to change to help us respond to those commercial pressures.
Design Concerns with HCR Teeth
Since the publication of Leming’s paper, high-contact ratio (HCR) gears have been used in many aircraft, defense, and vehicle applications. They have yet to be featured in “catalog” gearboxes, despite the following advantages:
These advantages, while noteworthy, have been overshadowed by concerns about susceptibility to scoring or other lubrication failures; lower efficiency; narrow top-lands; limited bearing capacity; gearbox thermal limitations; tooling costs; and uncertainty over rating methods. During these past decades many papers have been presented on these concerns as our aircraft and vehicle designing colleagues investigated the best ways to use the immerging technology. I claim no great breakthroughs in this paper but hope to alleviate a few fears and suggest a path forward.
The most difficult advantage to quantify for HCR designs is reduced noise level. In every application of HCR gearing that I know of, the noise level was lower than the conventional gearing it replaced. While mathematical models have been developed to determine optimum tooth modifications for conventional gears, those models require specific information on the load and speed for which noise reduction is needed. Catalog products are sold “off the shelf” with only limited load and speed information. A recent paper on the use of HCR timing gears in diesel truck engines (Ref. 4) revealed that the best noise performance was obtained with gears having little or no profile modification. The worst performers had modifications closer to what conventional math models suggested was “optimum.” My own experience with un-modified HCR profiles leads me to believe HCR gearing can be successfully used in catalog gearboxes with no tip or root relief at all.
With regard to the durability rating of HCR gears, the theoretical basis of current AGMA and ISO contact stress formulas has no restrictions on profile contact ratio. The increased capacity of HCR teeth is a matter of tooth curvature and the length of the line of contact. Depending upon the addendum factor chosen for the HCR tooth form, durability ratings can increase from 25 to 50% over similar-sized conventional gears; lab testing has confirmed these results (Ref. 5).
Our current tooth bending strength models are based upon single tooth contact. True HCR designs never see single tooth loading, so a new stress calculation formula will ultimately be needed to accurately predict the success of any HCR tooth form. Photo elastic modeling and finite element analysis results indicate that HCR teeth experience between 57 and 63% of the bending load of conventional gearing. Further testing will be needed before an HCR bending strength formula can be adopted.
Math Modeling HCR Gears
For the purposes of this paper I have selected two different size cataloged, parallel shaft, double-reduction speed reducers for study. Since specific design details are proprietary, I began by designing conventional gear, normal contact ratio (NCR) sets that would fit within the housing envelope and then selecting suitable taper roller bearings. These conventional 25° pressure angle helical sets were then rated for durability and strength to confirm that they were capable of the published catalog ratings. The catalog ratings and simulated gear geometry were used to calculate L10 gearing life using the advanced method (a23 factor).
The next step was to design HCR gear sets for the same conditions and repeat the durability and strength calculations before revisiting the bearing life issue. Durability was calculated using the AGMA 2001 method; strength was calculated using the standard method but the result was divided by 0.60 to reflect the load sharing reported in FEA modeling. As there is no “standard” HCR tooth form, I elected to use the 1.35 addendum 20° NPA system that Buckingham proposed for instrument gearing. Occasional minor, warning notes were received from the rating software for top-lands less than 0.250/NDP, but the rating process was otherwise unimpeded. Narrow top-lands are thought to contribute to tooth bending failures; the same warnings were received on some NCR 25° pressure angle sets.
While proposals have been advanced to achieve profile contact ratios of 1.95 or more using standard 20° full-depth tooling (Ref. 6), I chose to study only deeper than standard depth tooth forms. The use of standard depth tools on HCR gears results in reduced operating pressure angles and increased risk of undercutting, without the increased durability rating offered by the deeper tooth form. Catalog ratings are determined by the lowest capacity in a number of categories. Back in the through-hardened days it was expected that products would be durabilitylimited and that strength ratings would generally be 40 to 50% higher. When we moved to carburized and hardened gearing we found that the durability and strength ratings both came into play in establishing catalog ratings.
The use of standard depth tooling to achieve HCR profile overlaps would return us to durability-limited catalog ratings; overall ratings would probably not increase at all. Contrast this with the move to deeper than standard teeth where durability capacity will increase by 25 to 50% and strength ratings may double. Commercial success comes with high-quality products at lowest prices; high power density contributes to lower prices, as you are more likely to be able to meet a specific application with a “one size smaller” gearbox than a competitor.
Tables 2 through 5 show the results of the two-unit NCR/HCR rating comparison. HCR designs achieved a durability rating increase of 28 to 29%. HCR strength ratings were 44 to 48% more than comparable NCR designs. Greater improvements may be possible with more flexibility in the choice of center distance combinations and stage ratios. These particular examples were chosen to illustrate the potential for HCR redesigns of existing products using existing housing dimensions.
Many existing product lines are also bearing life-limited; the 25° normal pressure angles needed to obtain high bending strengths also increase the forces on the bearings. Space limitations and bearing availability prevent squeezing in more bearing capacity. The lower pressure angles used in the HCR designs have lower bearing forces, but the packaging problem may prevent utilization of increased rating capacity. Allowable “bearing horsepower” for each of the units studied is shown in Tables 6 through 9. With the space available for bearings in the current design units, I was not able to obtain a 10,000-hour L10 on every bearing with the published catalog ratings. Since few gearboxes are sold at a unity service factor, this is not a surprise.
Converting existing gearbox designs to HCR will reduce noise levels and provide additional service factors. To best leverage the technology, however, more flexibility in center distance sequences and ratio combinations will be needed. This is not unprecedented; a review of parallel-shaft gearbox catalogs shows that pre-1964 designs had far different proportions than more recent designs. The first-stage center distance in those through-hardened units is typically 50 to 62% of the second stage. The lowspeed gear ratio in those units may be as high as 6.5:1. These are a reflection of the rating methods in effect at the time they were designed. Up until 1964, for example, the durability rating was calculated based upon pinion pitch diameter and pinion rotational speed. This, along with the favorable treatment of allowable stress for second and third reductions, encouraged higher ratios on the output set.
When the “modern” rating method was adopted via AGMA 218 in the 1980s, the durability rating formula changed to the pinion pitch diameter squared, and the favorable treatment of second and third reductions went away. This change in rating method is reflected in the design of newer parallel shaft units. The first-stage center distances are now typically 70 to 80% of the second stage. Output stage gear ratios seldom exceed 5:1. Just as the adoption of carburized and ground gearing motivated that shift, HCR designs may also require a different approach to these fundamental design parameters.
With regard to the lubrication concerns with HCR gears, scoring and wear probabilities were calculated for the modeled gears using commercial software. Unfortunately the program wouldn’t accept gearing with profile contact ratios over 2.00, so the outside diameters of the HCR gears were reduced to obtain a 1.99. With the surface finish expected for form ground gears (22 AA) and required lubricant conditions (ISO 320EP at 160°F bulk temperature), all sets had scoring and wear probabilities of less than 5%. Efficiency testing, in conjunction with thermal rating development, would be necessary to determine whether HCR gearing has any disadvantage compared to similar-sized NCR gearing. A review of the factors involved with operating efficiency and thermal limitation shows that the longer line of action and slightly larger outside diameters of the HCR designs could increase power loss. On the other hand, the higher power density of HCR gearing would make the drives smaller in size and potentially render the overall efficiency equal. The author is not privy to the test results of automotive gearbox builders, but doubts they would have moved to HCR designs if efficiency were a problem.
The way forward. The advantages of HCR gear designs are ripe for commercial adoption. Tougher noise restrictions are inevitable and HCR technology has amply demonstrated its ability to reduce noise levels in vehicles. The opportunity to increase power density — whether for overall commercial advantage or just to raise ratings in specific situations, at only a slight increase in material cost — is very attractive in today’s competitive market.
Early adopters of any technological change have to temper enthusiasm with common sense. A well thought out test program will be needed to verify the rating advantages and validate the thermal capacity of the products. Theoretical work is needed to support a new high-contact ratio bending strength rating method, along with laboratory testing of HCR sets under standardized conditions.
The industry has devoted much of the last 140 years to exploiting the “standard,” full-depth tooth form. We moved from simple cast teeth to highly modified carburized and ground ones as market demands grew and technology evolved. An opportunity now exists to increase the capacity of our products by 25% or more, while simultaneously meeting stringent noise standards, through the adoption of a deeper than “full-depth” tooth geometry that has already been successful in aerospace and vehicle equipment.
Charles D. Schultz, PE is Chief Engineer for Beyta Gear Service (email@example.com) in Winfield, Illinois, and a Technical Editor for Gear Technology and Power Transmission Engineering magazines. He is also a longtime AGMA member, having served on or chaired a number of its committees over the years. And now you can follow Chuck’s new Gear Technology blog every Tuesday and Wednesday at geartechnology.com.