Contact Ratio Optimization of Powder-Metal Gears

August 11, 2017

Price and the ability to offer a much larger design window are making manufacturers look to powder metal gears as a solution for high-performance gears.

Powder metal (PM) alloys are becoming more of a solution for high-performance gears, not only because of part price but also because the technology offers a wider design window. Since powder metal alloys have a lower modulus of elasticity and Poisson ratio — the factors that amplify gear-tooth deflections — the design window opens up further.

The contact ratio is a critical gear-mesh parameter that greatly affects gear-drive performance, including load capacity, noise, and vibration. Gear-drive operating load produces bending and contact tooth deflections, which increase the actual effective contact ratio.

This article describes the analysis and gear macro-geometry optimization of powder-metal gears with transitional nominal contact ratio εα = 1.7-1.85. Under the operating load, the effective contact ratio of these gears is increased to εαe ≥ 2.0 creating greater load sharing, as well as providing stress and transmission error reduction.

PM gear technology has the inherent ability to reduce the weight and inertia of the gear wheel, thus reducing mass and energy losses.

When designing PM gears, special attention has to be paid to the use of the correct material properties, meaning the modulus of elasticity and Poisson’s ratio. Designers also can improve weight and dynamics by understanding the possibilities that PM offers through its unique production methods. The PM process route also offers a direct reduction of the number of manufacturing steps, leading to improved cost performance.

The modulus of elasticity and Poisson’s ratio can be empirically calculated as a function of density by equations 1 and 2 from [1].

42403.jpg Equation 1
42403.jpg Equation 2


In a spur gear mesh, the effective contact ratio can be defined as the ratio of the tooth engagement angle to the angular pitch. The tooth engagement angle is the gear’s rotation angle from the start of the tooth engagement with the mating gear tooth to the end of the engagement. The effective contact ratio is [2]

42403.jpg Equation 3


φ1 and φ2 —pinion and gear engagement angles.

z1 and z2 — pinion and gear numbers of teeth.

360/z1 and 360/z2 — pinion and gear angular pitches.

The gear mesh load is

42403.jpg Equation 4

where T1 — pinion operating torque in Nm, dbd1 — pinion drive flank base diameter in mm.

The load-sharing factor is

42403.jpg Equation 5

where Fmax — maximum contact load in the single tooth contact. If the effective contact ratio εαde ≤ 2.0, the load sharing factor L = 100%. If the effective contact ratio εαde > 2.0, the load sharing factor L < 100%.

The transmission error is [3]

42403.jpg Equation 6


θ1 and θ2 — driving pinion and driven gear rotation angles.

rb2 — driven gear base radius.

A typical transmission error chart for a spur gear pair with the effective contact ratio 1.0 < εαe < 2.0 is shown in Figure 1.

The effective contact ratio and transmission error are influenced by manufacturing tolerances, assembly misalignments, and operating conditions — including the gear’s and other gearbox components’ deflections under load, their thermal expansion or shrinkage, etc. In this article, only the bending and contact tooth deflections are considered for the definition of the effective contact ratio and transmission error. Each angular position of the driven gear relative to the driving gear is iteratively defined by equalizing the sum of the tooth contact load moments of each gear to its applied torque. The related tooth contact loads are also iteratively defined to conform to tooth bending and contact deflections, where the tooth bending deflections in each contact point are determined based on the FEA-calculated flexibility, and the total contact deflection is calculated based on the Hertz equation [4].


Table 1 and Figure 2 present comparison of the traditionally designed gear pair and the gear pair defined by the Direct Gear Design® method [5]. The traditionally designed gear pair has tooth macrogeometry used for high-performance gear transmissions. It is generated by the full tip radius tooling rack with a 25-degree pressure angle. The directly designed gear pair has the same pressure angle and the same tooth thicknesses at the pitch diameters as in the traditionally designed gears. The assumed average friction coefficient in both gear sets is equal to 0.05. Unlike in traditional gear design, Direct Gear Design allows selecting a desirable nominal contact ratio, setting its value as an input parameter. In this case, the nominal contact ratio is chosen to achieve an effective contact ratio under operating load that is equal to or slightly greater than 2.0. In addition, the directly designed gears have optimized tooth root fillets [5] to minimize bending stress concentration. The gear material is Höganäs Astaloy Mo 0.25%C with a modulus of elasticity of 160,000 MPa (density 7.27 g/cm3) and Poisson’s ratio of 0.28. Because of the lower modulus of elasticity and Poisson’s ratio, the traditionally and directly designed PM gears from Table 1 have bending and contact tooth deflections that are 27 percent to 30 percent greater than if they were made out of the case-harden gear steel with a modulus of elasticity of 206,000 MPa and Poisson’s ratio of 0.3. As a result of combining the PM alloy properties with the gear tooth macrogeometry optimized by Direct Gear Design, the transitional nominal contact ratio gears, under the operating load, become the high effective contact ratio gears — with the load shared between two or three pairs of teeth, reducing bending and contact stresses and transmission error.

Comparable Gear Analysis

Table 2 and Table 3 present gear parameters of six gear sets with different numbers of teeth. These gear sets are optimized by the Direct Gear Design method to satisfy the following conditions: center distance — 150 mm; gear ratio — 2:1; pinion and gear face widths — 32 mm; tooth tip thickness — about 0.30·module; effective contact ratio at the 1,500 Nm pinion torque is equal to 2.0; assumed average friction coefficient — 0.05; pinion and gear material is Höganäs Astaloy Mo 0.25%C with the modulus of elasticity – 160,000 MPa and Poisson’s ratio — 0.28. Operating load range of all gear sets lays between 1,500 Nm and 2,500 Nm of driving pinion torque. All gears have optimized tooth root fillets.

Overlays of the pinion and gear tooth profiles of the analyzed six sets are shown in %%0817-PM-3%%. It indicates a significant difference in gear tooth sizes.

Table 4 and Table 5 present the effective contact ratios, pinion and gear bending stress, contact stress, and transmission error of the analyzed six gear sets under various pinion torque values.

The effective contact ratio vs. the pinion torque chart is presented in Figure 4. When the pinion torque is zero, the effective contact ratio values are equal to the nominal contact ratio values because tooth deflections are zero. Increasing pinion torque T1 increases the effective contact ratio that reaches its value of 2.0 at T1 = 1,500 Nm. Further pinion torque growth provides load sharing between two or three pairs of teeth and an effective contact ratio εαe > 2.0.

When εαe > 2.0, the single tooth maximum load is reduced below 100 percent, typical for a conventional gear mesh that has εαe < 2.0 (Figure 5). The higher the pinion’s torque, the lower the single tooth maximum load. The tooth deflections become lower, respectively reducing the transmission error (TE). Figure 6 charts indicate that with εαe > 2.0 the transmission error decreases, stays flat, and then gradually increases beyond the operating load range.

Figure 7 and Figure 8 present the pinion and gear bending stress vs. pinion torque charts, which show the high stress increase gradient for low pinion torque values that produce an effective contact ratio εαe < 2.0. When pinion torque values provide an εαe > 2.0, the stress increase gradient is lower because the share of the maximum tooth load becomes lower. This allows for reduced bending stress in gears with transitional nominal contact ratio gears when compared to low nominal contact ratio gears. A similar effect of the transitional contact ratio is observed for the contact stress with growing pinion torque (Figure 9).


This article describes the analysis of powder metal gears with a transitional nominal contact ratio εα = 1.7-1.85. A combination of the PM alloys’ properties, the PM compaction technology, and tooth macro-geometry optimized by Direct Gear Design allows for increased bending and contacting tooth deflections. Under the operating load, these gears become high contact ratio gears with an effective contact ratio εαe ≥ 2.0, reducing bending and contact stresses and transmission error. Unlike solid steel gears, whose design is typically based on the rack generation process producing the trochoidal tooth, the PM gear compaction technology allows us to use an optimized root fillet profile, additionally reducing bending stress.

The article compares several gear sets with a transitional contact ratio and different numbers of teeth, while keeping the same gear ratio, center distance, gear face widths, and PM material. This comparison helps to select a suitable gear set depending on the gear drive performance priorities and limitations.

Results of this study might be useful for automotive transmissions, where application of the powder metal alloy gears is considered prospective. 


  1. Flodin, A. Forden, L., Root and contact stress calculations in surface densified PM gears, Proceedings from World PM2004 Conference Vol.2. (2004), pp 395-400.
  2. Kapelevich, A., Shekhtman, Y., Analysis and Optimization of Contact Ratio of Asymmetric Gears, Proceedings of the International Conference on Power Transmissions ICPT 2016-Chongqing, October 27-30, (2016).
  3. D.P. Townsend, Dudley’s Gear Handbook. 1967. 2nd edition. McGraw Hill. New York: 1040
  4. Young, W.C., Budynas, R. G., Roark’s Formulas for Stress and Strain, Seventh Edition, McGraw-Hill (2002).
  5. Kapelevich, A., Direct Gear Design. CRC Press. (2013).

About The Authors

Alexander Kapelevich

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.

Anders Flodin

received his doctoral degree in 2000 from KTH in Stockholm on the topic wear modeling of tooth flanks of cylindrical gears. He is with Höganäs, Sweden, and is working with developing powder metal gear technology for automotive applications. Flodin has been involved with transmission development for helicopters, ships, and cars and has 15 years experience with PM gears.