#### 4.1. The Influence of Input Parameters and Their Interactions

The evolution of the optimum gear ratio of the second step (

${u}_{2}$) as functions of each input parameter is presented in

Figure 4. It is observed that

${u}_{2}$ increases when Total gearbox ratio

$({u}_{t}),$ Allowable contact stress of stage 2 (

$A{S}_{2}$), and Cost of shafts (

${C}_{s}$) increase also. Nevertheless, for this tendency it is realized that Total gearbox ratio

$({u}_{t}$) has greater influence than that of other factors. Conversely,

${u}_{2}$ decreases with the growth of Allowable contact stress of stage 1 and 3 (

$A{S}_{1}$ and

$A{S}_{3}$), and Output torque (

${T}_{out})$. Moreover, it is shown that Coefficients of wheel face width of stage 1, 2, and 3

$({X}_{ba1},{X}_{ba2}$ and

${X}_{ba3}$) do not have influence on

${u}_{2}$. Regarding to the case of

${u}_{3}$ (c.f.

Figure 4b), the experimental results reveal that Cost of gears

$\left({C}_{g}\right)$ and Cost of shafts (

${C}_{s}$) have a significant effect on the value of

${u}_{3}$. It means that

${u}_{3}$ develops when Cost of shafts rises or Cost of gears declines. Furthermore, the first five input parameters mentioned in

Table 2 do not have impact on the evolution of

${u}_{3}$. It is noticed that the influence investigation of the input factors on response as previously mentioned do not take their interactions into account. This will be considered in the next part of the current study.

Figure 5a displays the interactions between the input parameters on the response of

${u}_{2}$. It is observed that the interactions between

u_{t} and some input parameters such as AE (

u_{t}*AS_{1}), AF (

u_{t}*AS_{2}), AG (

u_{t}*AS_{3}), AK (

u_{t}*C_{g}), and AL (

u_{t}*C_{s}) in both values of 30 and 100 has the most significant influence on the

${u}_{2}$ response, while the stable tendency is observed for the interactions between

${u}_{t}$ and remaining input parameters, e.g., AB (

u_{t}*X_{ba1}), AC (

u_{t}*X_{ba2}), AD (

u_{t}*X_{ba3}), AH (

u_{t}*T_{out}), and AJ (

u_{t}*C_{gh}). Similarly, we can realize the interactions which have significant influence on the

${u}_{2}$ response but are lesser than the ones of

u_{t} like BF (

X_{ba1}*A_{S2}), BE (

X_{ba1}*A_{S1}), CL (

X_{ba2}*C_{s}), CK (

X_{ba2}*

C_{g}), CF (

X_{ba2}*AS_{2}), FH (

AS_{2}*

T_{out}), FG (

AS_{2}*

AS_{3}), FL (

AS_{2}*C_{s}), FK (

AS_{2}*C_{g}), FL (

AS_{2}*C_{s}), FJ (

AS_{2}*C_{gh}), HK (

T_{out}*C_{g}), GL (

AS_{3}*C_{s}), GJ (

AS_{3}*C_{gh}), and GH (

AS_{3}*T_{out}). Referring to the case of the response

${u}_{3}$ (cf.

Figure 5b), it is visualized that the interactions JK (

C_{gh}*C_{g}), JL (

C_{gh}*C_{s}), GK (

AS_{3}*C_{g}), GL (

AS_{3}*C_{s}), FK(

AS_{2}*C_{g}), FL(

AS_{2}*C_{s}), KL (

C_{g}*C_{s}), BH (

X_{ba1}*T_{out}), BL (

X_{ba1}*C_{s}), DG (

X_{ba3}*AS_{3}), EG (

AS_{1}*AS_{3}), and EK (

AS_{1}*C_{g}) have significant influences on the

${u}_{3}$ response.

Figure 6 presents the Normal Plot of the standardized effects in which the relationship between the responses (

${u}_{2}$ and

${u}_{3}$) and the input parameters as well as their interactions are exposed. Based on the results presented in the figure, it is seen that

u_{t} and

$A{S}_{2}$ have the greatest influence on the response

${u}_{2}$ as previously documented. Furthermore, it is realized that, in addition to single input parameters as early presented (

u_{t},

AS_{2},

C_{s},

C_{g},

AS_{1}, and

AS_{3}) the interactions of some input parameters also have both positive and negative impacts on the response of

${u}_{2}$. For instance, the increase in the interactions of AK, EL, AJ, AF, BH, HK, and GK leads to the augment of the

${u}_{2}$ response. Conversely, the decrease in the interactions of AL, KL, EK, AE, JL, and AG causes the reduction of

${u}_{2}$ response. Considering the case of

${u}_{3}$, the results anew reveal that Cost of gears (

${C}_{g}$) and Cost of shafts (

${C}_{s}$) have dominant impact on the value of

${u}_{3}$ as mentioned above. Besides, the interactions between the input parameters also have influence in both positive and negative trends. For example, the response of

${u}_{3}$ is positively influenced by the interactions of JK, EK, and BL, while being negatively affected by those of KL, GK, EG, and BH. Based on the results shown in the Normal Plot of the Standardized Effects, the parameters or interactions with insignificant influence can be eliminated, while those with strong impact are remained. The testing process can go further and in more detailse with the remained parameters. In these situations, the remained parameters are listed in

Table 4 and

Table 5 in the case of

${u}_{2}$ and

${u}_{3}$, respectively.

#### 4.2. Proposed Regression Model of the Response

In order to achieve equations of the response

${u}_{2}$ and

${u}_{3}$, a regression process with two interaction factors is carried out using

[email protected] The significance of this regression is α = 0.05. The estimated effects and the coefficients for

${u}_{2}$ response are exhibited in

Table 4 where the factors with no influence on them are eliminated. It is noticed that if the effect of each input parameter or interaction has

p-value higher than the significance of α, it does not strongly impact the response. For example, the factor of

X_{ba1} has

p-value of 0.111 superior to α = 0.05, which means that

X_{ba1} is not significant to the response

${u}_{2}$. The regression equation of the

${u}_{2}$ response is described by following model (Regression Equation in Uncoded Units):

It can be said that the experimental data are greatly consistent with the proposed model when the minimum value of R-square is approximately 98% (all of them are more than 98%).

In the case of

${u}_{3}$ response, the results obtained from regression process show the difference from those of

u_{2} response. Indeed,

${u}_{t}$ and

X_{ba2} have no influence on the response, moreover, only eight interactions between input parameters have impact on it (cf.

Table 5). It is observed that the factors of B, D, E, and H have

p-value of 0.078, 0.184, 0.146, and 0.052 respectively, larger than significance α (0.05). Hence, these parameters have little influence on the

${u}_{3}$ response. However, the interactions of BH, BL, DG, EG, and EK have

p-value inferior to α. For this reason, they strongly influence the response of

${u}_{3}$. The regression equation of this response can be presented as following model (Regression Equation in Uncoded Units):

The results in

Table 5 also report that the experimental data are highly consistent with the proposed model when the minimum value of R-square approximately 92.02% (all of them are more than 92.02%). However, this is less reliable when compared to that of

${u}_{2}$ response.

Based on previous analysis, it can be said that the proposed models of ${u}_{2}$ and ${u}_{3}$ can be utilized to get the optimum gear ratio of the second and third stages. As a consequence, the optimum gear ratio of the first stage can be obtained by ${u}_{1}={u}_{t}/\left({u}_{2}\xb7{u}_{3}\right)$.

#### 4.3. Analysis of Variance—ANOVA

In order to quantitatively conclude the impact of each parameters and their interactions on the responses, Analysis of Variance is necessary.

Table 5 reveals the Analysis of Variance in case of

${u}_{2}$ response. It is observed that F-values of some parameters of A, F, K, E, L, G, AK, AL, EK, AE, KL, H, EL, AF, AJ, and JL exhibit the F-value higher than 50, and it can be concluded that these parameters have static significance. The R-square value in this case is high when the lowest R-square approaches

$92\%$. In a similar way, we can also identify the high F-value of parameters in case of the

${u}_{3}$ response, such as K, L, G, KL, F, and J. the lowest value of R-square reaches 92%.

#### 4.4. Validation of Proposed Model

The estimation of errors resulting from the difference between experiments and model of

${u}_{2}$ is qualitatively described in

Figure 7. From the Normal Probability Plot, it is observed that the contribution of errors is similar to normal distribution. The Versus fits graph discloses that the relation between residual and fitted value of model is random. Moreover, the Versus Order also exhibits the random relationship between residual and order of data point. The identical tendency is also noted when comparing experiments and proposed model in case of

${u}_{3}$ response. The observed phenomena given by the graphs one more time show the reliability of the proposed model which is highly fitted for the experiments.

Another way to validate the approximation of data is probability plot exhibited in

Figure 8. The Anderson–Darling test in

[email protected] which is a statistical test to validate the data set come from a specific distribution, e.g., the normal distribution or not. In this way, the data set is representative by blue points. There are three straight lines in the plot where the middle line presents the probability of normal distribution, while two lines in the left and the right refer to limiting boundary with significance of 95%. It is observed that all data set for both case of

${u}_{2}$ and

${u}_{3}$ are sited inside two limiting line when the

p-value of 0.289 and 0.097 are greater than α value of 0.05. This indicates that the data set follows the distribution.