The helix angle, β, is the angle between the helix line and horizontal axis, as shown in Figure 1.

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

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] .

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] .

When the contact ratio is equal to 2 or more, at least two pairs of teeth are theoretically in contact [7] .

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).

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

1) Transverse contact ratio ε_α

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.

The transverse contact ratio, ε_α, is calculated as follows [8] [9] [10] [11] .

ε α = g α p e t (1)

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)

where g_α is the path length of the contact line [mm], 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].

2) Overlap ratio ε_β

The overlap ratio, ε_β, is calculated as follows [8] [9] [10] [11] .

ε β = U p t (3)

ε β = b ⋅ tan β p t (4)

ε β = b ⋅ sin β π ⋅ m n (5)

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].

3) Total contact ratio ε_γ

The total contact ratio, ε_γ is calculated as follows.

ε γ = ε α + ε β (6)

where ε_α is the transverse contact ratio and ε_β 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 nominal tangential load F_t is calculated as follows.

F t = 2 ⋅ T L d 1 (7)

where T_L is the applied torque [N・mm] and d₁ is the base diameter of the tooth diameter [mm].

Axial load F_a is calculated as follows.

F a = F t tan β (8)

where β is helix angle [˚].

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

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 F_t is the nominal tangential load [N], b is the face width [mm], m_n is the normal module [mm], Y_F is the form factor [-], Y_s is the stress correction factor [-], Y_ε is the contact ratio factor [-], K_A is the application factor [-], K_V is the internal dynamic factor [-], K_Fβ is the face load factor for tooth-root stress [-] and K_Fα is the transverse load factor for tooth-root stress [-]. The safety factor for bending stress S_F is calculated as follows [8] [9] [10] [11] [12] .

S F = σ F p σ F (10)

where σ_Fp is the permissible bending stress.

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

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)

where u is the gear ratio [-], Z_H is the zone factor [-], Z_E is the elasticity factor [ ], Z_ε is the contact ratio factor [-], Z_β is the helix angle factor [-], K_Hβ is the face load factor for contact stress [-] and K_Hα is the transverse load factor for contact stress [-].

The safety factor for contact stress, S_H is calculated as follows [8] [9] [10] [11] [12] .

S H = σ H p σ H (12)

where σ_Hp is the permissible contact stress.

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.

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 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 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 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.

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 108, 242 [N/mm²] on the pinion tooth root for a helix angle β = 30 [˚], as shown in Figure 17.

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.