Keywords

1 Introduction

Mechanical drive systems are the most common type among many types of drive systems such as electric drive, pneumatic drive, hydraulic drive, and so on. This is due to its straightforward structure, dependable operation, and low cost. A typical mechanical drive system includes a motor, a gearbox, and two couplings, or a coupling and a V-belt or chain drive (Fig. 1). Of the mechanical drive system elements, the gearbox is undeniably the most important because it is the main component to reduce speed and torque from the motor shaft to the working shaft. Therefore, optimal design of gearboxes is an urgent research topic.

There have been numerous studies on the optimal design of gearboxes up to this point. In [1] gearbox geometric design parameters were optimized to reduce rattle noise in an automotive transmission using a torsional vibration model approach. The authors of [2] presented a study on multi-objective optimization for the drivetrain design and gear shifting control of internal combustion engine vehicles, with the goal of minimizing fuel consumption, exhaust emissions, and gearbox power losses. The optimum partial gear ratios to minimize the cost of a three-stage helical gearbox were determined in [3]. The problem of constrained multi-objective non-linear optimization of planetary gearboxes using a hybrid metaheuristic algorithm was reported in [4]. In [5] a modal-based design optimization of a gearbox housing using Finite Element Analysis was reported. In [6] a multi-objective optimization of a two-stage helical gearbox with a variety of constraints was described. The optimum gear ratios of different types of gearboxes have been found such as helical gearboxes [6,7,8,9], bevel gearboxes [10,11,12] and worm gearboxes [13,14,15,16,17]. According to the results of the above analysis, despite the fact that many studies on the optimization of gearbox parameters have been conducted, no optimization study has determined key design parameters (Fig. 1).

Fig. 1.
figure 1

Schema of a mechanical drive system: (1) Motor; (2) Coupling; (3) Gearbox; (4) Chain drive; (5) Belt conveyor

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This paper presented an optimization study to determine the optimum main design parameters for a two-stage helical gearbox to obtain the shortest gearbox length. A simulation experiment was carried out using the Taguchi method and the Minitab R19 software. The effect of the main design parameters on gearbox mass was investigated. The best values for the five most important design factors have been assigned.

2 Methodology

2.1 Calculation of Gearbox Length

The length of a two-stage helical gearbox Lgb can be detemined by (see Fig. 2):

$${L}_{gb}=\frac{{d}_{w11}}{2}+{a}_{w1}+{a}_{w2}+\frac{{d}_{w22}}{2}+20$$
(1)
Fig. 2.
figure 2

Calculating schema

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In Eq. (1), dw11 is the diameter of the drive gear of stage 1; dw22 is the diameter of the driven gear of stage 2; These diameters are found by [18]:

$${d}_{w11}=2\cdot {a}_{w1}/\left({u}_{1}+1\right)$$
(2)
$${d}_{w22}=2\cdot {a}_{w2}\cdot {u}_{2}/\left({u}_{2}+1\right)$$
(3)

In Eqs. (2) to (3), \({u}_{1}\) and \({u}_{2}\) are the gear ratios of stage 1 and stage 2; \({u}_{2}={u}_{g}/{u}_{1}\); with \({u}_{g}\) as the gearbox ratio; \({a}_{w1}\) and \({a}_{w2}\) are the center distances of stage 1 and stage 2 which are determined by [18]:

$${a}_{w1}={k}_{a}\cdot ({u}_{1}+1)\cdot \sqrt[3]{{T}_{11}\cdot {k}_{H\beta }/({\mathrm{AS}}_{1}^{2}\cdot {u}_{1}\cdot {X}_{ba})}$$
(4)
$${a}_{w2}={k}_{a}\cdot ({u}_{2}+1)\cdot \sqrt[3]{{T}_{12}\cdot {k}_{H\beta 2}/({\mathrm{AS}}_{2}^{2}\cdot {u}_{2}\cdot {X}_{ba2})}$$
(5)

where, \({k}_{H\beta }\) is the contacting load ratio for the pitting resistance; \({k}_{H\beta }\) = 1.05 ÷ 1.27 [18] and it was chosen as \({k}_{H\beta }=1.16\); AS1 and AS2 is the allowable contact stress of the first and the second stages (MPa); ka = 43 is material coefficient (for steel gear) [18]; Xba1 and Xba2 are the wheel face width coefficients of stage 1 and stage 2; \({T}_{11}\) and \({T}_{12}\) are the torques on the drive gear of stage 1 and stage 2 (Nmm):

$${T}_{11}={T}_{out}/\left({u}_{g}\cdot {\upeta }_{hg}^{2}\cdot {\eta }_{b}^{3}\right)$$
(6)
$${T}_{12}={T}_{out}/\left({u}_{2}\cdot {\eta }_{hg}\cdot {\eta }_{be}^{2}\right)$$
(7)

In which, Tout is the output torque (N.mm); ηhg=0.96 ÷ 0.98 is the efficiency of a helical gear unit [18]; ηbe = 0.99 ÷ 0.995 is the rolling bearing efficiency [18].

2.2 Optimization Problem

From the above analysis, the optimization problem is describes as:

$$ \text{Minimize} {L}_{gb}$$
(8)

With

$${m}_{gb}=f\left({u}_{1}; {X}_{ba1};{X}_{ba2};{AS}_{1};{AS}_{2}\right)$$
(9)

And with the following constraints:

$${1\le u}_{1}\le 9; {1\le u}_{2}\le 9$$
(10)

3 Simulation Experiment

To investigate the impact of main design factors on gearbox length, a simulation experiment was carried out. The following design parameters were investigated in this experiment: u1, Xba1, Xba2, AS1, and AS2. Table 1 defines these parameters and their levels. For the experimental design and data analysis, the Minitab R19 software and the Taguchi method were exploited.

Table 1. Main design parameters and their levels
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To reduce computer programming workload, the impact of the main design parameters on the length of the gearbox was investigated using gear ratio values of 5, 10, 15, 20, 25, and 30. Furthermore, for this experiment, a 5-level for 5 factors Taguchi design (L25) was chosen, so the simulation experiment was conducted based on 25 test runs with each of the above values of gear ratios. Table 2 shows the test plan and the output results (the gearbox length) for the gear ratio of 5.

Table 2. Experimental plan and output results (Lgb) for ugb = 5
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4 Results Discussion

To evaluate the effect of the main design factors on Lgb for ugb = 20, the Analysis of Variance (ANOVA) method is used in accordance with Minitab R19 software. The signal-to-noise ratio, or S/N number, is calculated for each experiment to find the impact of each main design factor on the output results. The S/N ratios are calculated to minimize gearbox length by:

$$\mathrm{S}/\mathrm{N}=-10{\mathrm{log}}_{10}\left(\frac{1}{n}\sum_{i=1}^{n}{y}_{i}^{2}\right)$$
(11)

The average SN ratio for each parameter and level is calculated after calculating the SN ratio for each experiment. Table 3 and Fig. 2 show how the main design factors affect Lgb. Figure 2 shows that Lgb is inversely proportional to all five major design parameters. Furthermore, Table 1 displays that AS2 has the greatest influence on Lgb (49.09%), followed by AS1 (20.2%), Xba2 (18.22%), and Xba1 (17.22%). (11.59%). Additionally, u1 has almost no effect on Lgb (0.89%). The influence order of the main design factors on the gearbox length is shown in Table 4.

Table 3. Analysis of variance for means
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Table 4. The order of impact of main design parameters on gearbox length
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Using the objective function in Equation, the S/N value is maximized for each major design factor to obtain the shortest gearbox length (8). The best main design factors are discovered after analyzing the effect of each factor on the S/N ratio in the plot in Fig. 3: u1 = 2.2; Xba1 = 0.33; Xba2 = 0.4; AS1 = 420 (MPa); AS2 = 420 (MPa) (Fig. 4).

Fig. 3.
figure 3

Influence of main design parameters on gearbox length

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Fig. 4.
figure 4

Main effects plot for S/N ratios

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Table 5. Optimum values of main design parameters
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Fig. 5.
figure 5

Optimum gear ratio of stage 1 versus gearbox ratio

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Continue as before for the remaining ugb values of 10, 15, 20, 25, and 30. The best values for the main design parameters are shown in Table 5. The following findings were obtained from Table 5 and Fig. 5:

  • The optimal Xba1 and Xba2 values are their maximum values: Xba1 = 0.33 and Xba2 = 0.4. In order to obtain a minimum gearbox length, the coefficients Xba1 and Xba2 must be as large as possible to minimize the center distances of stage 1 and stage 2 (Eqs. (4) and (5)).

  • The optimal AS1 and AS2 values are also their maximum values. The reason for this is that in order to have the shortest possible gearbox length, the AS1 and AS2 values must be as large as possible to minimize the center distance of the gear stage 1 and stage 2 (Eqs. (4) and (5)).

  • The optimal values of the gear ratio of stage 1 (u1) have a first-order relationship with ugb (Fig. 4). Also, the following regression equation (with R2 = 0.9967) is for determining the optimum values of u1:

    $${u}_{1}=0.2559\cdot {u}_{gb}+1.0453$$
    (12)

After obtaining u1, the optimum values of u2 is calculated by u2 = ugb/u1.

5 Conclusions

The findings of a study on optimizing a two-stage helical gearbox to obtain the smallest gearbox length are introduced in this paper. The gear ratio of the first stage, the coefficient of wheel face width of stages 1 and 2, and the allowable contact stress of stages 1 and 2 were investigated in this study. In addition, a simulation experiment with a Taguchi L25 type of design was carried out to solve the optimization problem. The effect of main design factors on gearbox length was also investigated. Furthermore, the following optimum values of the main design factors were proposed, as well as a regression model for calculating the optimum values of u1: Xba1 = 0.33; Xba2 = 0.4; AS1 = 420 (MPa); AS2 = 420 (MPa); u1 is calculated by Eq. (12).