The following results on tooth bending strength are presented in the literature:

An asymmetric gear pair improves the tooth-root bending load carrying capacity of the pinion and wheel gear at higher pressure angles on the coast side compared to a conventional symmetric gear. The optimum profile shift values increases with an increase in the speed ratio and number of teeth in the pinion, and increasing the asymmetric factor and pressure angles on the drive side improves the tooth-root bending capacity. When the speed ratio increases, the optimum maximum fillet stress increases very slightly compared to that of optimum profile shift factor for pinion [1] .

Asymmetric involute-type teeth were studied, since the non-involute teeth application has a number of disadvantages. The concept of one-sided involute asymmetric spur gear teeth is to increase the load carrying capacity of the driving involute. The literature concludes that the load carrying capacity can increase to 28% higher than that of standard 20˚ involute teeth [2] .

The advantage of using proposed asymmetric design in gearboxes is increased bending strength, pitting resistance, without changing the dimension or number of teeth in the gearbox [2] .

An alternative method to increase the tooth bending strength of involute gear teeth is positive modification of addendum (positive shifting) the pinion and, in some cases, mating wheel. This method produces well-running teeth, but both the pitting resistance and scoring resistance are reduced due to the positive shifting [2] .

A smaller pressure angle causes to produce undercut for a given number of teeth. However, the contact ratio increases, and load carrying capacity may be improved [3] .

Tooth profile modification is an effective parameter for optimising the geometric design parameters of gears. A numerical study found that the application of positive profile modification results in reduced tooth bending stress and increased safety factor for tooth bending stress [4] .

The gearbox mechanism is shown in Figure 1. Where Z_1p, Z_2p, Z_3p and Z_4p denotes the 1^st speed pinion gear, the 2^nd speed pinion gear, the 3^rd speed pinion gear and the 4^th speed pinion gear respectively. Z_Cp and Z_Rp denote the constant speed pinion gear and the rear speed pinion gear. Z_g1, Z_g2, Z_g3 and Z_g4 denotes the 1^st speed wheel gear, the 2^nd speed wheel gear, the 3^rd speed wheel gear and the 4^th speed wheel gear respectively. Z_Cg and Z_Rg denote the constant speed wheel gear and the rear speed wheel gear. S₁, S₂ and S₃ denote synchronisers.

The contact ratio (CR) is defined as the average number of teeth in contact during the gear rotation. The transverse contact ratio, ε_α is calculated as follows [8] .

where g_α is the path length of the contact line [mm], and p_et is the base pitch [mm], d_a1 is the addendum circle diameter of the pinion gear [mm], d_b1 is the base circle diameter of the pinion gear [mm], d_a2 is the addendum circle diameter of the wheel gear [mm], d_b2 is the base circle diameter of the wheel gear [mm], a_d is the centre distance [mm], α_t is the transverse pressure angle [˚], and m_t is the transverse module [mm].

The addendum circle diameter of the pinion gear, d_a1, is calculated as follows [8] .

where m_n is the normal module [mm], z is the number teeth [-], and β is the helix angle [˚].

The base circle diameter of the pinion gear, d_b1, is calculated as follows [8] .

The addendum circle diameter of the wheel gear, d_a2, is calculated as follows [8] .

The base circle diameter of the wheel gear, d_b2, is calculated as follows [9] .

The centre distance, a_d, is calculated as follows [8] .

The overlap ratio, ε_β is calculated as follows [8] .

where U is the action length [mm], p_t is the transverse pitch [mm], b is the face width [mm], and m_n is the normal module [mm].

The total contact ratio, ε_γ is calculated as follows.

where ε_α is the transverse contact ratio and ε_β is the overlap ratio. Helical gears have higher contact ratio than spur gears thus, they have also higher load carrying capacities than spur gears.

The bending stress in distribution are shown in Figure 5. The real tooth-root stress, is calculated as follows [7] [8]

The permissible bending stress, , is calculated as follows [7] [8] .

where all the responsible parameters for the tooth bending stress are given in Table 1. The safety factor for bending stress is calculated as follows [7] [8]

The contact stress, distribution is shown in Figure 6. The real contact stress, is calculated as follows [7] [8]

The permissible contact stress, is calculated as follows [7] [8] :

where all the responsible parameters for the tooth contact stress are given in Table 2.

The safety factor for contact stress, , is calculated as follows [7] [8] :

Tooth bending stresses are considered as objective functions, during the optimisation study. The flowchart of the optimisation procedure of geometric design parameters is shown in Figure 7. The following objective function was used:

Following minimum tooth bending stress is defined as objective function:

Thus, the module m, the number of teeth z, and the helix angle β, are the design parameters to be determined. During the constrained optimisation, the following optimisation problem is solved:

Subject to: and (19)

where LB is lower bound and UB is upper bounds on the design parameter vector. The iterations start with the initial values of design parameters such as, m_0, z₀, β₀, and b₀. Initial design parameters X0 are varied during the optimisation process, where G (X) ≤ 0 is the nonlinear inequalities.

During constraint optimisation, the tooth contact stress and constant distance between gear centres are considered as the constraint function as follows:

where is the real contact stress [N/mm²] and is the permissible contact stress [N/mm²].

where a₁ is the centre distance of the 1^st speed, a₂ is the centre distance of the 2^nd speed, a₃ is the centre distance of the 3^rd speed, a₄ is the centre distance of the 4^th speed, a₅ is the centre distance of the 5^th speed and a_R is the centre distance of the rear speed.

The contact ratio and bending stress relation for the 1^st speed is shown in Figure 8. As the contact ratio increases from 1.3862 to 1.7836, the bending stress reduces from 987.300 [N/mm²] to 822.2394 [N/mm²]. Thus, increasing the contact ratio 28.66% results in a 20.07% reduction in tooth bending stress.

The contact ratio and bending stress relation for the 2^nd speed is shown in Figure 9. As the contact ratio for the 2^nd speed increases from 1.4086 to 1.8255, the bending stress reduces from 817.7000 [N/mm²] to 679.0009 [N/mm²]. Thus, increasing the contact ratio 29.59% reduces the tooth bending stress 20.42%.

The contact ratio and bending stress relation for the 3^rd speed is shown in Figure 10. As the contact ratio for the 3^rd speed increases from 1.4150 to 1.8423, bending stress reduces from 836.2000 [N/mm²] to 692.3186 [N/mm²]. Thus, increasing the contact ratio 30.19%, results a 20.78% reduction in tooth bending stress.

The contact ratio and bending stress relation for the 4^th speed is shown in Figure 11. As the contact ratio for the 4^th speed increases from 1.4321 to 1.8895, the bending stress reduces from 680.0000 [N/mm²] to 558.4413 [N/mm²]. Thus, increasing the contact ratio 31.93% reduces the tooth bending stress 21.76%.

The contact ratio and bending stress relation for the 5^th speed is shown in Figure 12. As the contact ratio for the 5^th speed increases from 1.3970 to 1.7963, the bending stress reduces from 377.0000 [N/mm²] to 314.6310 [N/mm²]. Thus, increasing the contact ratio 28.58%, results in a 19.82% reduction in the tooth bending stress.

The contact ratio and bending stress relation for the rear speed is shown in Figure 13. As the contact ratio for the rear speed increases from 1.4295 to 1.8913, the bending stress reduces from 1033.1000 [N/mm²] to 847.6631 [N/mm²]. Thus, increasing the contact ratio 32.30%, reduces the tooth bending stress 21.87%.

The contact ratio and pressure angle relation for the 1^st speed is shown in Figure 14. As the pressure angle for the 1^st speed reduces from 22 [˚] to 12 [˚], the contact ratio increases from 1.3862 to 1.7836. Thus, decreasing the pressure angle 83%, results in a 28.66% increase in the contact ratio.

The contact ratio and pressure angle relation for the 2^nd speed is shown in Figure 15. As the pressure angle for the 2^nd speed reduces from 22 [˚] to 12 [˚], the contact ratio increases from 1.4086 to 1.8255. Thus, decreasing the pressure angle 83%, increases the contact ratio 29.59%.

The contact ratio and pressure angle relation for the 3^rd speed is shown in Figure 16. As the pressure angle for the 3^rd speed reduces from 22 [˚] to 12 [˚], the contact ratio increases from 1.4150 to 1.8423. Thus, decreasing the pressure angle 83%, results in a 30.19% increase in the contact ratio.

The contact ratio and pressure angle relation for the 4^th speed is shown in Figure 17. As the pressure angle for the 4^th speed reduces from 22 [˚] to 12 [˚], the contact ratio increases from 1.4321 to 1.8895. Thus, decreasing the pressure angle 83%, result in increases the contact ratio 31.93%.

The contact ratio and pressure angle relation for the 5^th speed is shown in Figure 18. As the pressure angle for the 5^th speed reduces from 22 [˚] to 12 [˚], the contact ratio increases from 1.3970 to 1.7963. Thus, decreasing the pressure angle 83%, results in a 28.58% increase in the contact ratio.

The contact ratio and pressure angle relation for the rear speed is shown in Figure 19. As the pressure angle for the rear speed reduces from 22 [˚] to 12 [˚], the contact ratio increases from 1.4295 to 1.8913. Thus, decreasing the pressure angle 83%, results in a 32.30% increase in the contact ratio.

The tooth profile modification and bending stress relation for the 1^st speed is shown in Figure 20. While the profile modification factor increase from 0 to 0.3, the bending stress reduces from 1171.5000 [N/mm²] to 927.4486 [N/mm²].

The tooth profile modification and bending stress relation for the 2^nd speed is shown in Figure 21. As the profile modification factor increase from 0 to 0.3, the bending stress reduces from 854.7000 [N/mm²] to 712.5610 [N/mm²].

The tooth profile modification and bending stress relation for the 3^rd speed is shown in Figure 22. While the profile modification factor increase from 0 to 0.3, the bending stress reduces from 873.9000 [N/mm²] to 728.3622 [N/mm²].

The tooth profile modification and bending stress relation for the 4^rd speed is shown in Figure 23. As the profile modification factor increase from 0 to 0.3, the bending stress reduces from 709.5000 [N/mm²] to 591.5892 [N/mm²].

The tooth profile modification and bending stress relation for the 5^th speed is shown in Figure 24. As the profile modification factor increase from 0 to 0.3, the bending stress reduces from 394.4000 [N/mm²] to 328.8225 [N/mm²].

The tooth profile modification and bending stress relation for the rear speed is shown in Figure 25. As the profile modification factor increase from 0 to 0.3, the bending stress reduces from 1074.800 [N/mm²] to 899.1084 [N/mm²].

A flowchart of the optimum design of effective parameters based on pressure angle is shown in Figure 26.

The safety factor for bending stress, S_F, and safety factor for contact stress S_H, are the basic selection criteria used by the Optimum Design. The Selective Optimum Design is shown in Table 4.

Although, obtained optimised geometric design parameters are significant for all constraints, the best solutions, based on pressure angle are determined from the obtained optimum solutions for each speed.

The geometric design parameters are optimised simultaneously for each given gearbox speed. However, it is not necessary to choose a single solution that changes with respect to the pressure angle. Therefore, all effective geometric design parameters can be determined independently for each speed from obtained optimum solutions.