Keywords

1 Introduction

The most important component of a mechanical drive system is the gearbox. Its purpose is to reduce the speed and torque transmitted from the motor shaft to the working machine shaft. It is widely used in practice due to its simple structure, stable operation, reliability, and low cost in comparison to electric, hydraulic, or pneumatic reduction systems. As a result, the optimal design of the gearbox is a constant concern.

Until now, the optimal design of the gearbox has been carried out on a variety of subjects. The created model is used in [1] to perform vibration and noise analysis on the gearbox housing. The authors in [2] introduced an investigation into the potential for improving energy efficiency and lowering lifecycle costs of electric city buses with multispeed gearboxes. A two-speed dual clutch gearbox and a continuously variable transmission were investigated and compared to a reference fixed gear ratio powertrain in this study. In [3] investigates the optimization of gearbox geometric design parameters to minimize rattle noise in an automotive transmission using an empirical model approach. The selection of the coefficient of wheel face width for multi-speed helical gear transmissions was introduced in [4]. However, the coefficient values are not optimal; they were only reasonably chosen based on some advice. In [5], the modal characteristic of gearbox housing with applied load was investigated. The problem of determining optimum gear ratios has received considerable attention. It was done for helical gearboxes with two [6, 7], three [8, 9] or four [10] stages. Although there have been numerous studies on optimizing gearbox parameters, no research has been conducted to identify the optimal design parameters simultaneously.

This paper introduces a study to determine the optimum main design parameters for a two-stage helical gearbox. The objective function of the optimization problem is the minimum gearbox cross-section. To do that, a simulation experiment with the application of the Taguchi method and the Minitab R19 software was conducted. The influence of main design parameters on the objective function was evaluated. Optimal values of main design parameters have been proposed.

2 Methodology

2.1 Determining Gearbox Cross-Section Area

The cross-section area of the gearbox Agb can be calculated by (Fig. 1):

Fig. 1.
figure 1

Schema for determining cross-section gearbox area

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$$ A_{gb} = L \cdot B_1 $$
(1)

In which, L, and \({B}_{1}\) can be calculated by:

$$L=\frac{{d}_{w11}}{2}+{a}_{w1}+{a}_{w2}+\frac{{d}_{w22}}{2}+20$$
(2)
$$ B_1 = b_{w1} + 4 \cdot S_G $$
(3)
$$ S_G = 0.005 \cdot L^{\prime} + 4.5\;\;\;[11] $$
(4)

In Eqs. (2) and (3), bw1 is the gear width of the first stage; \({d}_{w12}, {d}_{w21},\) and \({d}_{w22}\) are the pitch diameters of the pinion and the gear sets which can be determined by [4]:

$$ d_{w21} = 2 \cdot a_{w1} \cdot u_1 /\left( {u_1 + 1} \right) $$
(5)
$$ d_{w12} = 2 \cdot a_{w2} /\left( {u_2 + 1} \right) $$
(6)
$$ d_{w22} = 2 \cdot a_{w2} \cdot u_2 /\left( {u_2 + 1} \right) $$
(7)
$$ b_{w1} = X_{ba1} \cdot a_{w1} $$
(8)

In Eqs. (5)–(8), \({u}_{1}\) and \({u}_{2}\) are the gear ratios of the first and the second stages; \({u}_{2}={u}_{g}/{u}_{1}\); with \({u}_{g}\) is gearbox ratio; \({a}_{w1}\) and \({a}_{w2}\) are the center distances of the first and the second stages which are calculated by [4]:

$$ a_{w1} = k_a \cdot \left( {u_1 + 1} \right) \cdot \sqrt[3]{{T_{11} \cdot k_{H \beta } /\left( {{\text{AS}}_1^2 \cdot u_1 \cdot X_{ba} } \right)}} $$
(9)
$$ a_{w2} = k_a \cdot \left( {u_2 + 1} \right) \cdot \sqrt[3]{{T_{12} \cdot k_{H\beta 2} /\left( {{\text{AS}}_2^2 \cdot u_2 \cdot X_{ba2} } \right)}} $$
(10)

where, \({k}_{H\beta }\) is the contacting load ratio for pitting resistance; \({k}_{H \beta } = 1.05 \div 1.27\) [4] 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); \({k}_{a}\) is the material coefficient; As the gear material is steel, \({k}_{a}\)= 43 [4]; \({X}_{ba1}\) and \({X}_{ba2}\) are the wheel face width coefficients of the first and the second stages; \({T}_{11}\) and \({T}_{12}\) are the torques on the pinions of the first and the second stages (Nmm):

$$ T_{11} = T_{out} /\left( {u_g \cdot {{ \eta }}_{hg}^2 \cdot \eta_b^3 } \right) $$
(11)
$$ T_{12} = T_{out} /\left( {u_2 \cdot \eta_{hg} \cdot \eta_{be}^2 } \right) $$
(12)

wherein, Tout is the system output torque (N.mm); ηhg is the efficiency of a helical gear unit (\({\eta }_{hg}=0.96 \div 0.98\) [4]; ηb is the efficiency of a rolling bearing pair (ηh = 0.99 ÷ 0.995 [4]).

2.2 Optimization Problem

Based on the preceding analysis, the optimization problem can be defined as follows:

$$ {\text{Minimize}}\,\,A_{gb} $$
(13)

With

$${A}_{gb}=f\left({u}_{1}; {X}_{ba1};{X}_{ba2};{AS}_{1};{AS}_{2}\right)$$
(14)

And with the following constraints:

$$ 1 \le u_1 \le 9;\;\;\;1 \le u_2 \le 9 $$
(15)

3 Simulation Experiment

A simulation experiment was carried out to investigate the influence of main design paremeters on the objective function (across-section area). Besides, the Minitab R19 software and the Taguchi method were applied for experimental design and analysis the results. In addition, five main design factors including u1, Xba1, Xba2, AS1, and AS2 were investigated. Table 1 describes these parameters and their level.

For the convenience of the survey, the influence of these parameters on the across-section area was evaluated with each value of the gearbox ratios: 5, 10, 15, 20, 25 and 30. For each above value of the gearbox ratio, 25 test runs for the simulation experiment were performed because the 5-level design type of Taguchi design (L25) was selected. Table 2 presents the experimental plan and the output response (the gearbox cross-section area) when the gearbox ratio is 5.

Table 1. Main design parameters and their levels
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Table 2. Experimental plan and output response (Agb) when ugb = 5
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4 Results Discussion

The Analysis of Variance (ANOVA) method is used with Minitab R19 software to evaluate the influence of the main design parameters when ugb = 5 on Agb. The signal-to-noise ratio, or S/N number, must be calculated for each experiment to determine the effect of each parameter on the output. The calculated S/N number will determine the best parameters and which parameter has the most influence on the outcome. The following S/N ratio Eq. (16) should be used to minimize performance characteristics:

$$\mathrm{S}/\mathrm{N}=-10{\mathrm{log}}_{10}\left(\frac{1}{n}\sum_{i=1}^{n}{y}_{i}^{2}\right)$$
(16)

After calculating the SN ratio for each experiment, the average SN ratio for each parameter and level is calculated. The effect of main design factors on Agb was described in Table 3 and Fig. 2. From Fig. 2 it is easy to see that Agb is proportional to all five main design parameters. Besides, Table 3 shows that AS2 has the greatest influence on Agb (59.19%); followed by Xba2 (22.46 percent) and u1 (10.86%). AS1 and Xba1 have little effect on Agb (4.62% and 2.81%). Table 4 shows the order of the main parameters' influence on gearbox across-section area.

To obtain the smallest gearbox cross-section area using the objective function in Eq. (13), the S/N value is maximized for each main design parameter. As a result of the analysis of the effect of the parameters on the S/N ratio in the chart in Fig. 3, the optimum main design parameters were discovered as follows: u1 = 2.22; Xba1 = 0.33; Xba2 = 0.4; AS1 = 420; AS2 = 420.

Table 3. Analysis of variance for means
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Fig. 2.
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Effect of main design factors on Agb

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Table 4. The order of influence of the parameters on gearbox across-section area
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Fig. 3.
figure 3

Main effects plot for S/N ratios

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Table 5. Optimum values of main design parameters
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Fig. 4.
figure 4

Gear ratio of the first stage versus gearbox ratio

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For other values of ugb including 10, 15, 20, 25 and 30, proceed in the same way as above. Table 5 shows the optimal values of main design parameters. From Table 5, the following observations were reported:

  • Both Xba1 and Xba2 optimal values are their maximum values. The reason for this is that in order to have a small gearbox cross-section, the coefficients Xba1 and Xba2 must be as large as possible in order to reduce the center distance of the gear sets, resulting in a smaller size L (Eq. (2) and Agb (formula (1)).

  • The optimal values of AS1 and AS2 also take their maximum values. The reason is the same as above, in order to have the minimum gearbox cross-section, the AS1 and AS2 values need to be taken as large as possible to reduce the axial distance of the gear transmissions, leading to a reduction in the size L (formula (2)) and Agb ((formula (1)).

  • The gear ratios of the first stage and the gearbox have a first-order relationship (Fig. 4). From this relationship, the following regression formula (with R2 = 0.9947) is proposed to determine the optimum values of u1:

    $$ u_1 = 0.2515 \cdot u_{gb} + 1.1673 $$
    (17)

After having u1, the optimum values of u2 can be easily found by u2 =  ugb/u1.

5 Conclusions

The results of a study on optimization of a two-stage helical gearbox to get the smallest gearbox cross-section are presented in this paper. In this study, 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 the five main design parameters of the gearbox that were optimized. A simulation experiment with Taguchi L25 type was designed and carried out to solve this work. The effect of main design factors on gearbox cross-section area was also investigated. Furthermore, the optimum values of the main gearbox parameters and a regression model to calculate the optimum gear ratios of the first stage have been proposed as follows: Xba1 = 0.33; Xba2 = 0.4; AS1 = 420 (MPa); AS2 = 420 (MPa); u1 is calculated by Eq. (17).