Increasing the helix angle β, results in increasing the overlap contact ratio εβ. Thus, increasing the helix angle β results in increasing the total contact ratio εγ.

ε α = 0.5 ⋅ ( d a 1 2 − d b 1 2 + d a 2 2 − d b 2 2 ) − a d ⋅ sin α t π ⋅ m t ⋅ cos α t (2)

The maximum Von Mises stress of the pinion gear reaches 112, 900 [N/mm2] on the pinion tooth root for a helix angle β = 22 [˚], as shown in Figure 13.

The helix angle and axial force relation is shown in Figure 11. As the helix angle decreases from 22 [˚] to 12 [˚], the axial force is increased from 2319 [N] to 3280 [N].

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Optimisation of effective design parameters to reduce the tooth bending stress in an automotive transmission gearbox is presented. Therefore, the contact ratio effect on the tooth bending stress by changing the contact ratio with respect to the pressure angle is analysed [1] . It is concluded that a higher contact ratio results in reduced tooth bending stress, while a higher pressure angle and decreased contact ratio caused an increase in tooth bending stress and contact stress [1] .

The maximum Von Mises stress of the pinion gear reaches 110, 348 [N/mm2] on the pinion tooth root for a helix angle β = 28 [˚], as shown in Figure 16.

The maximum Von Mises stress of the pinion gear reaches 112, 340 [N/mm2] on the pinion tooth root for a helix angle β = 24 [˚], as shown in Figure 14.

The helix angle and total contact ratio relation are shown in Figure 8. As the helix angle increases from 22 [˚] to 32 [˚], the total contact ratio increases from 2.52 [-] to 2.88 [-].

Helical gears have higher load carrying capacities than spur gears because their contact ratios are larger than those of spur gears.

A second pair of mating teeth should come into contact before the first pair is out of contact during pinion and wheel gear running [6] .

S H = σ H p σ H (12)

The contact ratio is an important parameter for successful gear design. The helix angle is considered to be an effective parameter to increase the contact ratio of a helical gear. Thus, it is possible to increase the helical gear load carrying capacity, including the tooth bending stress and tooth contact stress.

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The applied load was considered in 6 different pinion gears that had 6 different helix angles. The Von-Mises stresses are shown in Figures 12-17.

results. By considering the tooth bending stress, the analytically obtained results and finite element method results only differ by 5%.

During simulation, the tooth bending stress and tooth contact stress were calculated according to ISO 6336. The effects of the helix angle on the tooth bending stress and tooth contact stress are analysed by varying the helix angle. The tooth bending stress and tooth contact stress parameters are shown in Table 1.

where gα is the path length of the contact line [mm], pet is the base pitch [mm], da1 is the addendum circle diameter of the pinion gear [mm], db1 is the base circle diameter of the pinion gear [mm], da2 is the addendum circle diameter of the wheel gear [mm], db2 is the base circle diameter of the wheel gear [mm], ad is the centre distance [mm], αt is the transverse pressure angle [˚], and mt is the transverse module [mm].

It was concluded that a helix angle increase had significant effects on the tooth-root bending stress and tooth compressive stress. Moreover, it was observed that when the helix angle increased from 0 [˚] to 22.5 [˚], both the bending stress and compression stress were reduced approximately 10% [3] .

ε β = b ⋅ tan β p t (4)

Solid (CAD) modelling of a pinion gear was obtained using SOLIDWORKS software. A solid model is essential for finite element method (FEM) analysis [13] [14] . Solid (CAD) modelling of a pinion gear is shown in Figure 6.

ε β = b ⋅ sin β π ⋅ m n (5)

The dimensions of the helical gear are shown in Figure 2, and the contact line of the helical gear is shown in Figure 3.

where εα is the transverse contact ratio and εβ is the overlap ratio. Helical gears have higher load carrying capacities than spur gears because their contact ratios are larger than those of spur gears.

The Von Mises stress obtained by finite elements analyses are shown in Table 3. A comparison between Von Mises Stress with FEM and analytical static stress depending on the helix angle is shown in Figure 12. By observing obtained finite elements method results, it is concluded that 45% increased helix angle result in 6.5% decreased Von Mises stress.

The helix angle and tooth bending stress relation are shown in Figure 9. As the helix angle decreases from 22 [˚] to 12 [˚], the bending stress is reduced from 365 [N/mm2] to 233 [N/mm2].

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

ε α = g α p e t (1)

Increasing the helix angle β results in an increase in the axial force Fa. Thus, one of the disadvantages of increasing the helix angle is the increase of axial forces on the helical gear mechanism.

The contact ratio for a helical gear pair increases with the helix angle, which generates the screwed surface of the tooth face [5] .

The aim of this study is to investigate the helix angle effect on the helical gear load carrying capacity, including the bending and contact load carrying capacity. During the simulation, the transverse contact ratio is calculated with respect to the constant pressure angle. By changing the helix angle, both the overlap contact ratio and total contact ratio are calculated and simulated. The bending stress and contact stress of a helical gear are calculated and simulated with respect to the helix angle. Solid (CAD) modelling of a pinion gear was obtained using SOLIDWORKS software. The analytically obtained results and finite elements method results are compared. It is observed that increasing the helix angle causes an increase of the contact ratio of the helical gear. Furthermore, increasing the contact ratio reduces the bending stress and contact stress of the helical gear. However, with a constant transverse contact ratio, it is possible to improve the total contact ratio depending on the helix angle. It is concluded that a higher helix angle increases the helical gear bending and contact load carrying capacity.

If the gear contact ratio is equal to 1, one tooth is leaving contact just as the next tooth is beginning contact. If the gear contact ratio is larger than 1, load sharing among the teeth is possible during pinion and wheel gear running [7] .

σ H = F t b m n u + 1 u Z H Z E Z ε Z β K A K V K H β K H α (11)

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It is well-known that the average number of teeth that are in contact as the gears rotate is the contact ratio (CR). The contact ratio is calculated from the following equation.

F a = F t tan β (8)

[1] Bozca, M. (2017) Optimisation of Effective Design Parameters for an Automotive Transmission Gearbox to Reduce Tooth Bending Stress. Modern Mechanical Engineering, 7, 35-56. https://doi.org/10.4236/mme.2017.72004 [2] Ventkatesh, B., Prabhakar, Vattikuti, S.V. and Deva Prasad, S. (2014) Investigate the Combined Effect on Gear Ratio, Helix Angle, Facewidth and Module on Bending and Compressive Stress of Steel Alloy Heical Gear. Procedia Material Science, 6, 1865-1870. https://doi.org/10.1016/j.mspro.2014.07.217 [3] Zhan, J.X. and Fard, M. (2018) Effects of Helix Angle, Mechanical Errors, and Coefficient of Friction on the Time-Varying Tooth-Root Stress of Helical Gears. Measurement, 118, 135-146. https://doi.org/10.1016/j.measurement.2018.01.021 [4] Pedrero, J.I., Pleguezuelos, M. and Munoz, M. (2011) Contact Stress Calculation of Undercut Spur and Helical Gear Teeth. Mechanism and Machine Theory, 46, 1633-1646. https://doi.org/10.1016/j.mechmachtheory.2011.06.015 [5] Kang, J.S. and Choi, Y.-S. (2008) Optimisation of Helix Angle for Helical Gear System. Journal of Mechanical Science and Technology, 22, 2393-2402. https://doi.org/10.1007/s12206-008-0804-z [6] Juvinall, R.C. and Marshek, K.M. (2006) Fundamentals of Machine Component Design. John Wiley & Sons, Inc., Hoboken. [7] Norton, R.L. (2011) Machine Design. Prentice Hall, Upper Saddle River, NJ. [8] ISO 6336-5: Calculation of Load Capacity of Spur and Helical Gears-Part 5: Strength and Quality of Materials. [9] ISO 6336-3: Calculation of Load Capacity of Spur and Helical Gears-Part 3: Calculation of Tooth Bending Strength. [10] Matek, R. (2005) Maschinenelemente. Vieweg & Sohn Verlag/Fachverlage, GmbH, Wiesbaden. [11] Decker (2009) Maschinenelemente. Carl Hanser Verlag, München. [12] Naunheimer, H., Bertsche, B., Ryborz, J. and Novak W. (2011) Automotive Transmissions. Springer-Verlag, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16214-5 [13] Moaveni, S. (2003) Finite Element Analysis, Theory and Application with ANSYS. Prentice Hall, Upper Saddle River, NJ. [14] Chandrupatla, T.R. and Belegundu, A.D. (2002) Introduction to Finite Elements in Engineering. Prentice Hall, Upper Saddle River, NJ.

The contact ratio consists of two parts, such as the transverse contact ratio, εα, and the overlap or face contact ratio, εβ.

Static structural analysis of the pinon gear was completed for the applied load considering the Von Mises stress. The Von Mises stress is written as follows.

The aim of this study is to investigate the helix angle effect on the helical gear load carrying capacity, including the bending and contact load carrying capacity. For this aim the analytically obtained results and finite elements method results are compared. It is concluded that a higher helix angle increases the helical gear bending and contact load carrying capacity.

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For a given number of teeth, a smaller pressure angle may produce an undercut. However, the contact ratio increases, so the load carrying capacity may improve as the load is distributed along a longer line of contact [4] .

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S F = σ F p σ F (10)

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The maximum Von Mises stress of the pinion gear reaches 111, 236 [N/mm2] on the pinion tooth root for a helix angle β = 26 [˚], as shown in Figure 15.

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During simulation, tooth bending stress and tooth contact stress were calculated according to ISO 6336. Solid (CAD) modelling of a pinion gear was completed

The obtained Solid (CAD) model is used to obtain the finite element method (FEM) model using the SOLIDWORKS finite element tool.

The real contact stress, σH is calculated as follows [8] [9] [10] [11] [12] . The contact stress at the tooth flank is shown in Figure 5.

The helix angle and contact stress relation are shown in Figure 10. As the helix angle decreases from 22 [˚] to 12 [˚], the bending stress is reduced from 1265 [N/mm2] to 1128 [N/mm2].

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Gears are widely used to mechanically transmit power in automotive transmissions. The aim of the gears is to couple two shafts together; the rotation of the drive-shaft is a function of the rotation of the drive-shaft in the gear mechanism. Therefore, determining the geometric design parameters of gears is crucial.

where Ft is the nominal tangential load [N], b is the face width [mm], mn is the normal module [mm], YF is the form factor [-], Ys is the stress correction factor [-], Yε is the contact ratio factor [-], KA is the application factor [-], KV is the internal dynamic factor [-], KFβ is the face load factor for tooth-root stress [-] and KFα is the transverse load factor for tooth-root stress [-]. The safety factor for bending stress SF is calculated as follows [8] [9] [10] [11] [12] .

σ v M = σ 2 + 3 τ 2 (13)

σ F = F t b m n Y F Y S Y ε Y β K A K V K F β K F α (9)

where u is the gear ratio [-], ZH is the zone factor [-], ZE is the elasticity factor [ ], Zε is the contact ratio factor [-], Zβ is the helix angle factor [-], KHβ is the face load factor for contact stress [-] and KHα is the transverse load factor for contact stress [-].

When the helix angle is increased from 15 [˚] to 35 [˚], the corresponding bending stress and compression stress decrease [2] .

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The helix angle and overlap contact ratio relation are shown in Figure 7. As the helix angle increases from 22 [˚] to 32 [˚], the overlap contact ratio increases from 1.01 [-] to 1.43 [-].

The real tooth-root stress, σF is calculated as follows [8] [9] [10] [11] [12] . The bending stress of the tooth-root is shown in Figure 4.

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The maximum Von Mises stress of the pinion gear reaches 108, 242 [N/mm2] on the pinion tooth root for a helix angle β = 30 [˚], as shown in Figure 17.

F t = 2 ⋅ T L d 1 (7)

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The maximum Von Mises stress of the pinion gear reaches 105, 984 [N/mm2] on the pinion tooth root for a helix angle β = 30 [˚], as shown in Figure 18.

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The effect of the helix angle on the tooth bending stress and tooth contact stress was analysed by varying helix angle, and the following conclusions are drawn.

σ F T = F t b ⋅ m Y F (14)

If the gear profile contact ratio is less than 2.0, it is called the Low Contact Ratio (LCR). If he gear profile contact ratio equals 2.0 or greater, it is called the High Contact Ratio (HCR).