Keywords

1 Introduction

Optimisation of the geometric design parameters of a five speed gearbox for an automotive transmission is studied. The purpose of this study is to determine the geometric design parameters of a five-speed gearbox for an automotive transmission by minimising the tooth bending stress.

By optimising the geometric parameters of the gears, such as, the module, number of teeth, helix angle, and face width, it is possible to obtain light-weight gearbox structures. Optimised geometric design parameters satisfy all constraints, and the best solutions are selected from the obtained optimum solutions for each given speed.

1.1 Gearbox Mechanism

The gearbox mechanism includes pinion gears, wheel gears, an input shaft, an output shaft, a lay shaft, a bearing support, and synchronisers, as shown in Fig. 1.

Fig. 1
figure 1

A five speed gearbox for an automotive transmission

Full size image

All pinion and wheel gears are helical, and all gears are made of 16MnCr5.

2 Contact Ratio

The average number of teeth in contact as the gears rotate together is the contact ratio (CR) [1].

The total contact ratio, εγ, is calculated as follows.

$$ {\varepsilon}_{\gamma }={\varepsilon}_{\alpha }+{\varepsilon}_{\beta } $$
(1)

where εα is the transverse contact ratio and εβ is the overlap ratio.

3 Profile Modification

Profile modification is given as follows.

$$ V=x.m $$
(2)

where x is the profile modification factor [−] and m is the module [mm]. When x is positive, it is called positive profile modification, and when x is negative, it is called negative profile modification [2].

4 Calculating the Load Capacity of Helical Gears

4.1 Tooth Bending Stress

The real tooth-root stress, σ F , is calculated as follows [2–4]

$$ {\sigma}_F=\frac{F_t}{b{m}_n}{Y}_F{Y}_S{Y}_{\varepsilon }{Y}_{\beta }{K}_A{K}_V{K}_{F\beta }{K}_{F\alpha } $$
(3)

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 [2–4]

$$ {S}_F=\frac{\sigma_{Fp}}{\sigma_F} $$
(4)

where σ Fp is permissible bending stress.

4.2 Tooth Contact Stress

The real contact stress, σ H , is calculated as follows [2–4]

$$ {\sigma}_H=\sqrt{\frac{F_t}{b{m}_n}\frac{u+1}{u}}{Z}_H{Z}_E{Z}_{\varepsilon }{Z}_{\beta}\sqrt{K_A{K}_V{K}_{H\beta }{K}_{H\alpha }} $$
(5)

where d 1 is the reference diameter of the pinion [mm], u is the gear ratio [−], Z H is the zone factor [−], Z E is the elasticity factor [\( \sqrt{N/m{m}^2} \)], 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 [2–4]

$$ {S}_H=\frac{\sigma_{Hp}}{\sigma_H} $$
(6)

where σ Hp is the permissible contact stress.

5 Optimisation of Gearbox Design Parameters

Constrained optimisation approaches are applied to the gears system. All programs are developed using MATLAB and in all optimisation studies, the sequential quadratic programming (SQP) method is employed.

To find the optimum design parameter, the initial design parameters of the gear system including m, z, β, and b, are varied. Four design parameters are optimised simultaneously using the programs developed. During optimisation, different initial value vectors are used to identify the global minimum solution of the objective function, σ(m, z, β, b).

The following objective function was employed:

$$ F= \min \left(\sigma \right) $$
(7)

Minimum tooth bending stress min(σ), is defined as follows:

$$ \min \left({\sigma}_F\right)= \min \left(\frac{F_t}{b{m}_n}{Y}_F{Y}_S{Y}_{\varepsilon }{Y}_{\beta }{K}_A{K}_V{K}_{F\beta }{K}_{F\alpha}\right) $$
(8)

The tooth contact stress is considered as the constraint functions during optimisation. The following are considered as the constraint functions.

$$ {\sigma}_H-{\sigma}_{Hp}\le 0 $$
(9)

6 Numerical Example

The contact ratio and bending stress relations are shown in Fig. 2. Increasing of the contact ratio, results in reduced tooth bending stress and reduced contact stress.

Fig. 2
figure 2

Contact ratio and bending stress relation

Full size image

The contact ratio and pressure angle relations are shown in Fig. 3. Increasing of the pressure angle, result in reduced the contact ratio and increased the tooth bending stress and contact stress.

Fig. 3
figure 3

Pressure angle and contact ratio relation

Full size image

The profile modification factor and bending stress relations are shown in Fig. 4. Increasing of the profile modification factor, results in a reduction in tooth bending stress.

Fig. 4
figure 4

Profile modification factor and bending stress relations

Full size image

7 Conclusion

Optimisation of the geometric design parameters of a five speed gearbox for an automotive transmission is studied. The following conclusions are drawn.

By optimising the geometric design parameters of the gearbox, including the module, number of teeth, helix angle and face width, it is possible to reduce the tooth bending stress and obtain a light-weight gearbox structures.

Increasing of the contact ratio, results in reduced tooth bending stress and contact stress. In contrast, increasing the pressure angle reduces the contact ratio and increases the tooth bending stress and contact stress.

Profile modification is an effective parameter to reduce tooth bending stress. Increasing of the profile modification factor, results in a reduction in tooth bending stress.

All geometric parameters can be selected independently for each speed inside the obtained optimum solutions. The best solutions are selected from the obtained optimum solutions for each speed (Table 1).

Table 1 Optimisation results
Full size table