Designing Asymmetric Gears

Harnessing Direct Gear Design for asymmetric gears results in low weight-to-output torque ratio and reduced levels of noise and vibration.

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In propulsion gear transmissions, the tooth load on one flank is significantly higher and is applied for longer periods of time than for the opposite one. An asymmetric tooth shape reflects this functional difference. The design intent of asymmetric gear teeth is to improve performance of the primary drive profiles at the expense of the performance for the opposite coast profiles. The coast profiles are unloaded or lightly loaded during a relatively short work period. Asymmetric tooth profiles also make it possible to simultaneously increase the contact ratio and operating pressure angle beyond the conventional gears’ limits.

The main advantage of asymmetric gears is contact stress reduction on the drive flanks, resulting in higher torque density (load capacity per gear size). These benefits of the gears with asymmetric tooth profiles for unidirectional torque transmission are well known. There are many publications on this subject written by authors from different countries [1-8]. All these and other related articles, however, are dedicated to the mathematical modeling or lab specimen testing and don’t present any practical implementations of the gears with asymmetric tooth profiles.

In most of this research [2-4, 7, 8] the asymmetric teeth profiles are defined by the pre-selected asymmetric generating gear rack parameters (Figure 1), a common method for conventional gears with symmetric teeth. The advantages of this method for symmetric gears include interchangeability, wide availability of standard tooling, and great analytical and testing database for the gears with symmetric teeth. However, all these advantages are not applicable to the gears with asymmetric teeth, because the standards for this kind of gears do not exist and analytical and testing database is not sufficient for practical definition of the asymmetric rack that could be considered for standardization.

This paper presents the nontraditional Direct Gear Design® (DGD) method of the asymmetric gears. It also demonstrates an example of the gearbox where the asymmetric gears are used which passed all development stages and currently is in production.

Gear Geometry

Gear Tooth Parameters
DGD is a practical realization of the Gearing Theory of Generalized Parameters developed by Prof. E.B. Vulgakov [1]. He also has applied this theory to the gears with asymmetric involute teeth, presenting the asymmetric tooth as a combination of two halves of different symmetric teeth without using any generating rack. Later A.L. Kapelevich [5, 6] defined the asymmetric tooth by two involutes from different base diameters that drastically expanded previously achievable range of the gear mesh parameters, such as the operating pressure angle and contact ratio.

DGD presents the asymmetric tooth form by two involutes of two different base circles with the arc distance between them and tooth tip circle describe the gear tooth (Figure 2). The equally spaced teeth form the gear. The fillet between teeth is not in contact with the mating gear teeth. However, this portion of the tooth profile is critical because this is the area of the maximum bending stress concentration. The fillet profile is designed independently and a subject of optimization providing minimum bending stress concentration and sufficient clearance with the mating gear tooth tip in mesh.

Gear Mesh Parameters
The asymmetric gear mesh (Figure 3) presents to two different drive and coast flank meshes with different pressure angles and contact ratios. The operating pressure angle aw and the contact ratio ea for the gear with asymmetric teeth are defined by the following formulae:

For external gearing
inv awd + inv awc = [inv n1d + inv n1c + u * (inv n2d + inv n2c) – 2 * p / n1)] / (1 + u),

ead = n1 * [tan aa1d + u * tan aa2d – (1 + u) * tan awd] /(2 * p),

eac = n1 * [tan aa1c + u * tan aa2c – (1 + u) * tan awc] / (2 * p).

For internal gearing
inv awd + inv awc = [u * (inv n2d + inv n2c) – inv v1d -inv v1c] / (u – 1)],

ead = n1 * [tan aa1d –u * tan aa2d + (u – 1) * tan awd] /(2 * π),

eac = n1 * [tan aa1c –u * tan aa2c + (u – 1) * tan awc] / (2 * π),

where u = n2 / n1 is a gear ratio, aa = acos(db/da) – involute profile angle at the tooth tip circle diameter.

Tooth Fillet Profile Optimization
The tooth fillet design begins when the involute flank parameters are completely defined. The initial fillet profile is a trajectory of the mating gear tooth tip in the tight (zero backlash) mesh. The fillet optimization process [9] utilizes three methods: random search method locating fillet points; trigonometric functions for fillet profile approximation; and FEA for stress calculation.

The first and the last fillet profile points of the initial fillet profile lay on the form diameter circle (Figure 4) and cannot be moved during an optimization process. The random search method is used to move the fillet nodes along the beams that pass through the approximately defined fillet center and the nodes of the initial fillet profile. The bending stresses are calculated for every new fillet profile points’ combination. If the maximum bending stress is reduced the program continues searching in the same direction, if not it steps back and starts searching the different direction. After the given number of iteration steps (or given optimization time) the optimization process stops, resulting with an optimized fillet profile that provides minimum bending stress concentration. Figure 5 and Table 1 present a comparison of the rack generating fillets with the optimized fillet, which provides minimum bending stress concentration.

As for asymmetric vs. symmetric gears, Table 2 presents a comparison of traditionally designed gear pairs with conventional and high gear ratio, generated by the full radius 25º and 20º racks accordingly with similar gear pairs created by Direct Gear Design with asymmetric teeth.

Example of Implementation
An example of implementation of the gears with asymmetric tooth profiles is the two stage planetary gearbox of the TV7-117S turboprop engine [9, 10]. This engine has been used in the Russian airplane IL-114 for several years and is going to be used in IL-112, MIG-110, TU-136 airplanes.

Gearbox Data and Arrangement
In conventional planetary gear systems, the planet gear is usually in simultaneous contact with the sun and planet gears. The tooth load and number of load cycles are equal for both flanks of the planet gear. However, one flank of the planet gear is in mesh with the concave tooth flank of the ring gear with internal teeth. The resulting contact stress in this mesh is much lower in comparison with contact stress of the convex tooth flanks in sun-planet gear contact, which defines the load capacity and size of the gears. In order to reduce this contact stress the higher operating pressure angle was chosen for the sun-planet gear contacting tooth flanks. This choice is in compliance with the ANSI/AGMA 6123-B06 standard "Design Manual for Enclosed Epicyclic Gear Drives," which states: “Best strength to weight ratio is achieved with high operating pressure angles at the sun to planet mesh and low operating pressure angles at the planet to ring gear mesh.” Figure 6

The drive tooth flanks of the sun-planet gear mesh have increased the contact curvature radii resulting with the greater hydrodynamic oil film thickness, lower flash temperature, and scuffing probability. Main parameters of the TV7-117S gearbox are presented in the Table 3. The TV7-117S gearbox arrangement (Figure 7) is proved to provide maximum power transmission density for the required total gear ratio.

The first planetary-differential stage has three planet gears. The second coaxial stage has five planet (idler) gears and stationary carrier. Part of transmitted power goes from the first stage carrier directly to the propeller shaft. The rest of the transmitted power goes from the first stage ring gear to the second stage sun gear and then through the planets to the second stage ring gear also connected to the propeller shaft.

Gear Parameters
Basic gear geometry parameters are presented in Table 4. Direct Gear Design of the asymmetric tooth profiles also allows shaping the coast flanks and fillet independently from the drive flanks, managing the tooth bending strength and stiffness, and improving load sharing while keeping a desirable pressure angle and contact ratio on the drive profiles. This allows increasing tooth tip deflection, damping tooth mesh impact, leading to gear noise and vibration reduction.

The number of teeth is selected in order to increase the drive operating pressure angle in the planet-ring gear mesh in comparison with the coast operating pressure angle in the sun-planet gear mesh:

awd 2-3 = arccos [cos (a wc 1-2) * (n3 – n2) / (n1 + n2)],
where
awc 1-2 – coast operating pressure angle in the sun-planet gear mesh;
n1 – sun gear number of teeth;
n2 – planet gear number of teeth;
n3 – ring gear number of teeth.

Gear Manufacturing and Assembly
All gears are made of the forged blanks of the steel 20KH3MVF (EI-415). Its chemical composition includes: Fe – base material, C – 0.15-0.20 percent, S – <0.025 percent, P – <0.030 percent, Si – 0.17-0.37 percent, Mn – 0.25 – 0.50 percent, Cr – 2.8-3.3 percent, Mo – 0.35-0.55 percent, W – 0.30-0.50 percent, Co – 0.60-0.85 percent, and Ni – < 0.5 percent.

Machining of the directly designed sun and planet gears with asymmetric teeth requires custom gear cutters. The gear hob rack profile and the shaper cutter profile and are defined by reversed generation of the gear profile. The gear blanks position during machining must provide the asymmetric teeth point in a certain direction (clockwise or counterclockwise). Otherwise the drive flank of one gear will be positioned in contact with the coast profile of the mating gear that makes the assembly impossible.

After preliminary tooth cutting, the gears are carburized and heat-treated to achieve tooth surface hardness > 59 HRC with the case depth of 0.6-1.0 mm. The core tooth hardness is 33-45 HRC. Final gear machining includes tooth grinding and honing. Asymmetric gear flanks also require special setup for both these operations.

Assembly of the gearbox includes selection of planet gears and their initial orientation, which is based on transmission error function of every gear. All planet gears are classified by transmission error (TE) function in several groups. Each group has planet gears with similar TE function. Then during assembly, position and orientation of the each planet gear depends on its TE function profile, providing better engagement of the driving flanks and load distribution between planet gears [11]. Some of the TV7-117S turboprop engine gearbox components and assemblies are presented in Figure 7, Figure 8, Figure 9, Figure 10.

Summary
This paper describes an alternative Direct Gear Design approach for the asymmetric gear design, demonstrating the basic gear tooth and mesh parameter definition. It also familiarizes with a proprietary tooth fillet profiles optimization technique, providing minimum bending stress concentration.

Application of the asymmetric gears in the TV7-117S turboprop engine gearbox resulted in extremely low weight to output torque ratio, about 50 percent lower in comparison with the gearboxes of its predecessors AI-20 and AI-24 turboprop engines. It also significantly reduced noise and vibration level, cut down duration and expense of operational development.

The new design and technological approaches that have found their realization in the TV7-117S engine gearbox were recommended for the development of gearboxes for advanced aviation engines.  

References:
1) E. B. Vulgakov, Gears with Improved Characteristics, Mashinostroenie, Moscow, 1974 (in Russian).
2) C. A. Yoerkie, and A. G. Chory, “ Acoustic Vibration Characteristics of High Contact Ratio Planetary Gears”, The Journal of American Helicopter Society, Vol. 40, 1984, pp. 19 – 32.
3) I. A. Bolotovsky, O. F. Vasil’eva, V. P. Kotelnikov, Involute gears with asymmetric teeth, Vestnik Mashinostroeniya, Issue 4, 1984, pp. 15-17 (in Russian).
4) G. DiFrancesco and S. Marini, Structural Analysis of Asymmetrical Teeth: Reduction of Size and Weight, Gear Technology, September/October, 1997, pp. 47 – 51.
5) A. L. Kapelevich, Synthesis of asymmetric involute gearing gears, Mashinovedenie, (1) (1987) 62-67 (in Russian).
6) A.L. Kapelevich, “Geometry and design of involute spur gears with asymmetric teeth”, Mechanism and Machine Theory, 2000, Issue 35, pp. 117-130.
7) G. Gang, T. Nakanishi, Enhancement of Bending load Carrying Capacity of Gears Using An Asymmetric Involute Tooth, The JSME International Conference on Motion and Transmissions (MPT2001-Fukuoka), Fukuoka, JAPAN, pp.513-517, 15-17/11/2001.
8) F. Karpat, K. Cavdar, and F.C. Babalik, Computer aided analysis of involute spur gears with asymmetric teeth, VDI Berichte, 1904 I, 2005, pp. 145-163.
9) A. L. Kapelevich, Y. V. Shekhtman, Tooth Fillet Profile Optimization for Gears with Symmetric and Asymmetric Teeth, AGMA Fall Technical Meeting, San Antonio, Texas, USA, October 12-14, 2008, (08FTM06)
10) E. B. Vulgakov, A. L. Kapelevich, The gearbox for the TV7-117S turbo-prop engine, Vestnik Mashinostroeniya, 2000, Issue 11, pp. 13 – 17 (in Russian). 11.
11) A.S. Novikov, A.G. Paikin, V.L. Dorofeyev, V.M. Ananiev, A.L. Kapelevich, Application of Gears with Asymmetric Teeth in Turboprop Engine Gearbox, Gear Technology, January/February 2008, 60 – 65.
12) E. B. Vulgakov et al., Aviation Gearboxes, Handbook, Mashinostroenie, Moscow, 1981 (in Russian).

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operates the gear-design consulting firm AKGears, LLC in Shoreview, Minnesota. He has a master’s degree from Moscow Aviation Institute and a Ph.D from Moscow State Technical University and 40 years of experience with custom gear drive development. He is author of the book, “Direct Gear Design,” and many technical articles.