3.5.2. Steepest Ascent Test of the Xylem

Based on the Plackett–Burman test for the xylem to derive the significant influences on the stacking angle, the steepest ascent test was carried out with the xylem–Q235A steel rolling friction coefficient (*X*3 ), xylem–xylem static friction coefficient (*X*5 ), and xylem– xylem rolling friction coefficient (*X*6 ) as the independent variables. The test results are shown in Table 12. The results show that as the values of *X*<sup>3</sup> , *X*<sup>5</sup> , and *X*<sup>6</sup> increased, the relative error between the simulated stacking angle and the physical stacking angle first decreased and then increased. The minimum relative error was obtained when the third level was selected, indicating the existence of the optimal value range for the third level. Therefore, the stacking angle test result of level 3 was taken as the center point, and the stacking angle test results of levels 2 and 4 were taken as the low and high levels for the subsequent response surface design. The optimization ranges of the xylem–Q235A steel rolling friction coefficient (*X*3 ), xylem–xylem static friction coefficient (*X*5 ), and xylem– xylem rolling friction coefficient (*X*6 ) were determined to be 0.028–0.074, 0.22–0.45, and 0.013–0.028, respectively.

**Table 12.** Results of the steepest ascent test of the xylem.


#### *3.6. Response Surface Design Results*

3.6.1. Regression Model Establishment and Experimental Results Analysis of the Variance of the Regression Model in Phloem

To investigate the impact of the static friction coefficient (*X*5) and rolling friction coefficient (*X*6) between the ramie phloem and phloem on the phloem stacking angle (*Y*1) during the response surface optimization test, we utilized the range of values obtained from the steepest ascent test. The central composite design test was then conducted using Design-Expert to optimize the response surface. Tables 13 and 14 display the factor encoding level values and central composite design experimental results, respectively. A total of thirteen parameter combinations were tested, of which five were center-level repeats.

**Table 13.** Factor level codes for the phloem central composite design test.



**Table 14.** The phloem central composite design test scheme and results.

By employing the Design-Expert software, a second-order polynomial equation was developed through a multiple regression analysis of the central composite design experimental outcomes. The equation was used to fit the phloem stacking angle and achieve the multivariate nonlinear regression model fitting of the static friction coefficient (*X*5) and rolling friction coefficient (*X*6) related to the phloem stacking angle (*Y*1). Furthermore, the model and coefficients were subjected to a significance test, and the regression equation is shown in Equation (8):

$$Y\_1 = 39.54 + 1.64X\_5 + 0.81X\_6 - 0.23X\_5X\_6 - 0.74X\_5^2 - 0.54X\_6^2 \tag{8}$$

Table 15 shows the results of the variance analysis. The significance test of the regression model for the stacking angle of the ramie phloem indicated *p* < 0.0001, a lack of fit of *p* = 0.1123, a determination coefficient of 0.9976, and an adjusted determination coefficient of 0.9958. The regression model was extremely significant, the lack of fit was nonsignificant, and the model was effective. The high value of the determination coefficient (close to 1) indicates a good fit of the regression equation. The coefficient of variation was only 0.45%, and the adequate precision was 68.357, indicating a high correlation between the actual and predicted values and the high reliability of the experimental results. As shown in Table 15, the static friction coefficient (*X*5) and rolling friction coefficient (*X*6) of the ramie phloem, as well as their quadratic terms (*X*<sup>5</sup> <sup>2</sup> and *X*<sup>6</sup> 2), all have an extremely significant effect on the equation. Combined with the linear regression equation, the order of the factors affecting the stacking angle is the static friction coefficient (*X*5) > rolling friction coefficient (*X*6) of the ramie phloem.


**Table 15.** ANOVA results of the phloem stacking angle.

\*\*, Indicate significance at the 0.01 level.

To visually analyze the model's reliability, Design-Expert software's Diagnostics module was used to obtain a quadratic model residual diagnostic plot, as shown in Figure 14. Figure 14a shows a normal plot of the residual, which can be observed to be linearly distributed on both sides of the line for each test group, indicating that the model describes the relationship between the influencing factors and phloem simulation stacking angle with sufficient reliability. Figure 14b is the residual plot of the equation and the predicted values. The random dispersion of the residuals shows an irregular distribution, indicating a good prediction of the equation. Figure 14c shows the distribution of the ratio of the predicted and experimental values of the phloem simulation stacking angle, and the linear distribution indicates a good fit of the model. Overall, these results indicate that the model's reliability is extremely high.

**Figure 14.** Diagnostic plot of the phloem quadratic model residuals: (**a**) normal plot; (**b**) residual vs. predicted; (**c**) predicted vs. actual.

Analysis of the Variance of the Regression Model in Xylem

Based on the interval of the values of the xylem–Q235A steel rolling friction coefficient (*X*3 ), xylem–xylem static friction coefficient (*X*5 ), and xylem–xylem rolling friction coefficient (*X*6 ) among the ramie xylem obtained from the steepest ascent test in order to investigate the effect of the influencing factors (i.e., *X*<sup>3</sup> , *X*<sup>5</sup> , and *X*<sup>6</sup> ) on the xylem stacking

angle (*Y*2) of the response surface optimization test, the central composite design test was carried out using Design-Expert. The stacking angle simulation test was conducted for 23 sets of parameter combinations of which three sets were repeated at the central level. The results of their factor coding level values and central combination tests are shown in Tables 16 and 17.


**Table 16.** Factor level codes for the xylem central composite design test.

**Table 17.** The xylem central composite design test scheme and results.


A multivariate regression analysis was conducted on the results of the central composite design test using Design-Expert software. After eliminating the insignificant factors while ensuring the model significance and insignificance of the lack-of-fit terms, the secondorder regression model was optimized to obtain a new regression equation:

$$Y\_2 = 28.06 + 0.69X\_3 + 0.70X\_5 + 1.09X\_6 - 0.12X\_5X\_6 - 0.15X\_3^2 - 0.32X\_5^2 - 0.33X\_6^2 \tag{9}$$

The results of the variance analysis are shown in Table 18. The *p*-value (*p* < 0.0001) of the model confirms its significance within the 95% confidence interval. The *p*-value of the lack-of-fit term was 0.6381, less than 0.05, indicating the effectiveness of the second-order model for the xylem stacking angle. In addition, the determination coefficient and adjusted determination coefficient were 0.9899 and 0.9829, respectively, both close to 1, indicating good agreement between the calculated model and experimental data. The difference between the adjusted determination coefficient and the predictive determination coefficient of 0.9657 was less than 0.2, indicating a good fit, and the adequate precision was 46.594, indicating a high correlation between the actual and predicted values.


**Table 18.** ANOVA results of the xylem stacking angle.

\*\*, \* Indicate significance at the 0.01 and 0.05 levels, respectively.

Based on the Diagnostics module in the Design-Expert software, the residual diagnostic plots of the quadratic model were obtained, as shown in Figure 15. Figure 15a shows a normal plot of the residual, which can be observed to be linearly distributed on both sides of the line for each test group, indicating that the model describes the relationship between the influencing factors and the xylem simulation stacking angle with sufficient reliability. Figure 15b is the residual plot of the equation and the predicted values. The random dispersion of the residuals shows an irregular distribution, indicating a good prediction of the equation. Figure 15c shows the distribution of the ratio of the predicted and experimental values of the xylem simulation stacking angle. The linear distribution indicates a good fit for the model. Overall, these results suggest that the model's reliability is extremely high.

**Figure 15.** Diagnostic plot of the xylem quadratic model residuals: (**a**) normal plot; (**b**) residual vs. predicted; (**c**) predicted vs. actual.

3.6.2. Analysis of the Interaction Effects among the Factors

Interaction Effects of the Factors in the Phloem on the Simulation Stacking Angle

Based on the variance analysis of the phloem simulation stacking angle, the static friction coefficient (*X*5) and rolling friction coefficient (*X*6) between the phloem and phloem had an extremely significant impact on the phloem simulation stacking angle. Therefore, the Design-Expert software was used to analyze the nonlinear relationship between *X*5, *X*6, and the phloem simulation stacking angle. Figure 16 shows the response surface of the interaction between *X*<sup>5</sup> and *X*<sup>6</sup> on the phloem simulation stacking angle. When the static friction coefficient among the phloem was constant, the phloem simulation stacking angle increased with the rise in the rolling friction coefficient between the phloem and phloem. When the rolling friction coefficient between the phloem and phloem was constant, the phloem simulation stacking angle increased with the rise in the static friction coefficient between the phloem and phloem. However, the contour slope of *X*<sup>5</sup> was steeper than *X*6, indicating that the static friction coefficient between the phloem and phloem (*X*5) had a more significant impact on the phloem simulation stacking angle than the rolling friction coefficient between the phloem and phloem (*X*6). Therefore, the order of the effects of each factor on the phloem simulation stacking angle is *X*<sup>5</sup> > *X*6, which is consistent with the results of the variance analysis.

**Figure 16.** Response surface of the interaction effects of the factors in the phloem on the stacking angle.

Interaction Effects of the Factors in the Xylem on the Simulation Stacking Angle

Based on the analysis of the variance of the xylem simulation stacking angle (*X*<sup>2</sup> 3 ), *X*<sup>2</sup> 5 and X<sup>2</sup> 6 had an extremely significant impact on the simulation stacking angle. *X*<sup>5</sup> and *X*<sup>6</sup> significantly affected the simulation stacking angle, while *X*<sup>3</sup> and *X*<sup>5</sup> and *X*<sup>3</sup> and *X*<sup>6</sup> had no significant effect on the simulation stacking angle. Therefore, Design-Expert software was used to analyze the relationship between the static friction coefficient (*X*5 ), rolling friction coefficient (*X*6 ), and xylem and the xylem simulation stacking angle. Figure 17 shows the response surface of the interaction between *X*<sup>5</sup> and *X*<sup>6</sup> on the simulation stacking angle. When *X*<sup>5</sup> was constant, the simulation stacking angle increased with the increase in *X*<sup>6</sup> . When *X*<sup>6</sup> was constant, the simulation stacking angle increased with the rise in *X*<sup>5</sup> . However, the contour slope of *X*<sup>6</sup> was steeper than that of *X*<sup>5</sup> , indicating that *X*6 had a more significant impact on the simulation stacking angle. Therefore, the order of the effects of each factor on the simulation stacking angle is *X*<sup>6</sup> > *X*<sup>5</sup> > *X*3´, which is consistent with the results of the variance analysis.

**Figure 17.** Response surface of the interaction effects of the factors in the xylem on the stacking angle.

3.6.3. Parameter Optimization and Validation Optimization of the Parameters in the Phloem

Based on the results of the Plackett–Burman test and the steepest ascent test in the phloem, the ranges of *X*<sup>5</sup> and *X*<sup>6</sup> were 0.35–0.52 and 0.053–0.084, respectively. Taking the physical test value of the stacking angle of the phloem as the optimization objective, using the parameter Optimization module built into the Design-expert software, the nonsignificant factors were taken as the physical test values, and the rest were taken as the middle values of the steepest ascent test level to determine the optimal combination of the static friction coefficient (*X*5) and rolling friction coefficient (*X*6) of the phloem–phloem; the optimization objective function and constraints are shown in Equation (10):

$$\begin{cases} \text{tarY}\_1 = 37.93\\ \quad 0.35 \le X\_5 \le 0.52\\ \quad 0.053 \le X\_6 \le 0.084 \end{cases} \tag{10}$$

After solving, 44 sets of optimized solutions were obtained. The simulated results of the optimized parameter group were compared with the physical test results. The optimized solution with the most similar shape of the cylindrical lifting physical test stacking angle was found. The static friction coefficient (*X*5) between the phloem and phloem particles was determined to be 0.41, and the rolling friction coefficient (*X*6) between the phloem and phloem particles was 0.056.
