Cases 1 and 2

The gear teeth are symmetric and their surface durability is identical for both drive and coast flanks. Case 1 presents the traditionally designed 25° pres-sure angle gear pair with the full radius fillet. This case is considered a base-line, and its Hertz contact stress, bearing load, and specific sliding velocity are assumed as 100% for comparison with other gear pairs. This type of gear profile is used in the aerospace industry because it provides better bending strength and flank surface endurance in comparison with the standard 20°

pressure angle gears typical for commercial applications. Case 2 is the high 32° pressure angle symmetric gears, optimized by the Direct Gear Design method. Its Hertzian contact stress is about 8% lower and its specific sliding velocity is about 6% lower than those for the baseline gear pair. This should provide better flank tooth surface pitting or scoring resistance. However, the bearing load is 7% higher.

TABLE 5.1 Gear Geometry Parameters for High Pressure Angle and High Contact Ratio Gears TypeHigh-Pressure AngleHigh Contact Ratio (HCR) Tooth FormSymmetricAsymmetricSymmetricAsymmetric Tooth profile Number of teeth Pinion and gear z1,235353535 Pressure angleαwd30°42°22°26° αwc30°14°22°12° Asymmetry factor K1.01.31.01.09 Transverse contact ratioεαd1.391.252.042.02 εαc1.392.022.042.16

TABLE 5.2 Gear Parameters of Bidirectional and Unidirectional Gear Drives Case No.12345 Load transmissionBidirectionalMostly unidirectionalUnidirectional Loaded flanksBothBothDrive, lower coast loadDrive, very low coast loadDrive flank only Tooth profileSymmetric (baseline)SymmetricAsymmetricAsymmetricAsymmetric Gear mesh Pressure angleαwd25°32°40°46°60° αwc25°32°24°10°—* Asymmetry coefficient K1.01.01.01.191.42—* Transverse contact ratioεαd1.351.21.21.21.2 εαc1.351.21.441.0—* Hertz contact stress, %Drive flank10092888694 Coast flank10092102150—* Bearing load, %Drive flank100107118130181 Coast flank1001079992—* Specific sliding velocity, %Drive flank10094756849 Coast flank1009410897—* Source:Graphics from Kapelevich, A.L., Gear Technology, June/July 2012, 48–51. With permission. * Coast flank mesh does not exist.

5.1.2.2 Case 3

These asymmetric gears are for mostly unidirectional load transmission with a 40° pressure angle driving tooth flanks providing 12% contact stress and 25% sliding velocity reduction. At the same time, the contact stress and slid-ing velocity of the coast flanks are close to these parameters of the baseline gears and should provide a tooth surface load capacity similar to that for the baseline gears. This type of gear may find applications for drives with one main load transmission direction, but it should be capable to carry a lighter load for shorter periods of time in the opposite load transmission direction.

5.1.2.3 Case 4

These asymmetric gears have a 46° drive pressure angle that allows reduc-tion of the contact stress by 14% and sliding velocity by 32%. The disadvan-tage of such gear teeth is a high (+30%) bearing load. These types of gears are only for unidirectional load transmission. Their 10° coast pressure angle flanks have insignificant load capacity. They may find applications for drives with only one load transmission direction that may occasionally have a very low load coast flank tooth contact, like in the case of a tooth bouncing in high-speed transmissions.

5.1.2.4 Case 5

These asymmetric gears have only driving tooth flanks with the extreme 60° pressure angle with no involute coast tooth flanks at all. As a result, the bearing load is significant.

There are many applications, as described in a Case 3, where a gear pair transmits load in both load directions, but with significantly different mag-nitude and duration (Figure  5.1). In this case, the asymmetry factor K for a gear pair is defined by equalizing potential accumulated tooth surface damage defined by operating contact stress and number of tooth flank load cycles. In other words, the contact stress safety factor S_H should be the same for the drive and coast tooth flanks. This condition can be presented as

SH HPd

where σHd and σHc are the operating contact stresses for the drive and coast tooth flanks, and σHPd and σHPc are the permissible contact stresses for the drive and coast tooth flanks that depend on the number of load cycles.

Then from (5.1)

The contact stress at the pitch point [48] is

is the zone factor that for the directly designed spur gears is zH

w

= 2

2

sin( α ) ; (5.4)

ZE is the elasticity factor that takes into account gear material properties (modulus of elasticity and Poisson’s ratio); Z_ε is the contact ratio factor, its conservative value for spur gears is Zε = 1.0; Zβ is the helix factor, for spur external gearing and – for external gearing.

Then for the directly designed spur gears the contact stress at the pitch point can be presented as

σ^{H E} w w αw

Asymmetric gear pair, T1d and T1c, pinion torque applied to the drive and coast tooth flanks.

(From Kapelevich, A.L., Gear Technology, June/July 2012, 48–51. With permission.)

Some parameters of this equation, Z_E, d_w1, b_w, and u, do not depend on the rota-tion direcrota-tion, and Equarota-tion (5.2) for the pitch point contact can be presented as

sin( )

where a parameter A is

A T

According to [48], “the permissible stress at limited service life or the safety factor in the limited life stress range is determined using life factor Z_NT.” This allows replacement of the permissible contact stresses in Equation (5.7) for the life factors

When parameter A is defined and the drive pressure angle is selected, the coast pressure angle is calculated by Equation (5.6) and the asymmetry coefficient K from a common solution of (5.6) and (2.93):

K A wd

If the gear tooth is equally loaded in both the main and reversed load appli-cation directions, then both the coefficient A and the asymmetry factor K are equal to 1.0 and gear teeth are symmetric.

Example 1

The drive pinion torque T1d is two times greater than the coast pin-ion torque T_1c. The drive tooth flank has 10⁹ load cycles, and the coast tooth flank has 10⁶ load cycles during the life of the gear drive. From the S-N curve [48] for steel gears an approximate ratio of the life factors Z_NTd/Z_NTc = 0.85. Then the coefficient A = 0.85²/2 = 0.36. Assuming the drive pressure angle is αwd = 36°, the coast pressure angle from Equation (5.6) is αwc = 10° and the asymmetry factor from Equation (5.9) is K = 1.22.

In many unidirectional gear drives like in, for example, propulsion sys-tem transmissions that seem irreversible, the coast tooth flanks are loaded because of the system inertia during the drive system deceleration or the tooth

bouncing in the high RPM drives. This coast tooth flank load can be significant and should be taken in consideration while defining the asymmetry factor K.

If the gear drive is completely irreversible and the coast tooth flanks never transmit any load (Case 4), the asymmetry factor is defined only by the drive flank geometry. In this case, increase of the drive flank pressure could be lim-ited by a minimum selected contact ratio and a separating load applied to the bearings. Application of a very high drive flank pressure angle results in the reduced coast flank pressure angle and possibly its involute profile undercut near the tooth root. Another limitation of the asymmetry factor of the irre-versible gear drive is growing compressive bending stress at the coast flank root. Usually for conventional symmetric gears compressive bending stress does not present a problem, because its allowable limit is significantly higher than for the tensile bending stress. However, for asymmetric gears it may become an issue, especially for gears with thin rims.

In the unidirectional chain gear drives (Figure 5.2), the idler gear transmits the same load by both tooth flanks. This arrangement seems unsuitable for asym-metric gear application. However, in many cases, the idler’s mating gears have significantly different numbers of teeth. This allows equalizing contact stresses on opposite flanks of the asymmetric teeth to achieve maximum load capacity.

Equation (5.5) is used to define the pitch point contact stress in the pinion/idler gear mesh,

Chain gear arrangement: 1 - input pinion; 2 - idler gear; 3 - output gear. (From Kapelevich, A.L., Gear Technology, June/July 2012, 48–51. With permission.)

and in the idler/output gear mesh,

or, ignoring gear mesh losses,

σ^H z^E w w αw

where bw12 and bw23 are the contact face widths in the pinion/idler gear and the idler/output gear meshes, accordingly; u12 = z2/z1 is the gear ratio in the pin-ion/idler gear mesh; u23 = z3/z2 is the gear ratio in the idler/output gear mesh;

and z1, z2, and z3 are the number of teeth of the input, idler, and output gears.

Numbers of the idler gear tooth load cycles and permissible contact stresses in this case are equal in both meshes, and Equation (5.2) can be presented as σH12 = σH23. Then considering that all gears are made from the same material, the idler gear pressure angle ratio is defined by

sin( )

is a parameter that reflects the gear ratios u12 and u23, and contact face widths bw12 and bw23 in the pinion/idler gear and idler/output gear meshes, accordingly.

Then considering Equation (2.93) the asymmetry factor K can be presented as

K B wd application of asymmetric gears can be considered.

Example 2

The pinion number of teeth is n₁= 9, the idler gear number of teeth is n₂ = 12, the output gear number of teeth is n₃ = 20, the contact face width ratio is b_w12/b_w23 = 1.2. This makes parameter B = 0.82. Then assuming the pinion/idler gear mesh pressure angle is αw12 = 35°, the idler/output pres-sure angle from (5.13) is 25.32° and the asymmetry factor from Equation (5.15) is K = 1.10.

Similarly, the contact stress equalization technique can be applied for the unidirectional epicyclic gear stage (Figure 5.3), because the planet gear can be considered as the idler gear engaged with the sun gear and ring gear.

In this case, the asymmetry factor K is also defined by Equation (5.15), where parameter B is

B b

b_w12 and b_w23 are the contact face widths of the sun/planet gear and planet/ring gear meshes; u₁₂ = z₂/z₁ is the gear ratio in the sun/planet gear mesh; u₂₃ = z₃/z₂ is the gear ratio in the planet/ring gear mesh; and z1, z2, and z3 are the number of teeth of the sun, planet, and ring gears.

In a typical epicyclic gear stage z2 = (z3 – z1)/2. This allows simplification greater than 1.0, which makes an epicyclic gear stage suitable for asymmetric gear application.

Planetary gear arrangement: 1 - sun gear; 2 - planet gear; 3 - ring gear. (From Kapelevich, A.L., Gear Technology, June/July 2012, 48–51. With permission.)

Example 3

The sun gear number of teeth is n₁ = 9, the planet gear number of teeth is n₂ = 12, the output gear number of teeth is n₃ = 33, and the contact face width ratio is b_w12/b_w23 = 1.8. This makes parameter B = 0.49. Then assum-ing the pinion/idler gear pressure angle is αw12= 40°, the idler/output pressure angle from (5.13) is 14.5° and the asymmetry factor from Equation (5.15) is K = 1.26.