# Force Prediction and Cutting-Parameter Optimization in Micro-Milling Al7075-T6 Based on Response Surface Method

^{*}

## Abstract

**:**

_{z}), axial cutting depth (a

_{p}), spindle speed (n) and tool extended length (l)), a rotatable center composite experiment of micro-milling straight micro-groove in the workpiece of Al7075-T6 were designed, based on second-order response surface methods. According to the experiment results, the least square method was used to estimate the regression coefficient corresponding to the cutting parameters. Simultaneously, the response prediction model of micro-milling was established and successfully coincide the predicted values with the experiment values. The significance of the regression equation was tested by analysis of variance, and the influence of micro-milling cutting parameters on force and top burrs morphology was studied. The experiment results show that in a specific range of cutting parameters, a

_{p}and f

_{z}have a significant linear relation with the micro-milling force and the top burrs width. According to the optimal response value, the optimized cutting parameters for micro-milling obtained as: n is 11,393 r/min, f

_{z}is 6 µm/z, a

_{p}is 11 μm and l is 20.8 mm. The research results provide a useful reference for the selection of cutting parameters for micro-milling.

## 1. Introduction

## 2. Experimental

#### 2.1. Micro-Milling Experiment Setup

#### 2.2. Experimental Design

^{2}).

- (1)
- A series of center points (the center point of rectangle in Figure 2) provide information on whether there is a curved surface in the model or information about pure errors: including groups 1–7 experiments in the central point (0,0,0,0);
- (2)
- The factor points (the vertices of the cube in Figure 2) are mainly used to estimate the linear and interactive terms: the 8–23 groups experiment with 2 full-factor part experiment points;
- (3)
- The axial point parts (the star point in Figure 2) are used to estimate the quadratic term: the 24–31 groups are experiments of the axial point part and the axial point of each factor is −2 or 2. There were 8 groups of experiments in which 4 factors were combined.

_{z}is 2−18 μm/z, the range of a

_{p}is 10–50 μm, the range of n is 8000–16,000 r/min and the range of l is 17–33 mm. To facilitate later experimental design analysis, the main factor coding and level settings are shown in Table 2.

## 3. Experimental Results

_{1}and b

_{2}, respectively.

_{x}), feed-direction force (F

_{y}) and normal direction force (F

_{z}). In micro-milling, the amplitude of the normal direction force is smaller than the other two directions, which is because the force in the normal direction is mainly affected by the elastic recovery of the material and it is less affected by the cutting parameters. Therefore, only studying F

_{x}and F

_{y}, using the 10-point average method simplifies the calculation of the corresponding response value of F

_{x}and F

_{y}. The response values (b

_{1}and b

_{2}, F

_{x}and F

_{y}) can be used as the evaluation indicators of the micro-milling workpiece quality. The experimental results of the actual measurement are summarized in the response value section of Table 1.

## 4. Discussion

#### 4.1. Micro-Milling Force Analysis

_{x}and F

_{y}quadratic response surface prediction model are as follows:

_{z}or increasing l is likely to decrease F

_{x}. Equation (3) can be used to preliminarily determine the factors that have significant effects on F

_{x}as f

_{z}, a

_{p}and l, and Equation (4) can preliminarily determine the factors that have significant effects on F

_{y}as f

_{z}, a

_{p}.

_{x}and F

_{y}. It can be seen from Figure 4a,b that the predicted force value and measured force value are basically in the same waveform, which further verifies the accuracy of the prediction model about micro-milling quadratic response surface obtained from Equations (3) and (4). The accuracy of the predictions is excellent, but it is found that F

_{x}has a higher fitting degree than F

_{y}between the predicted value and the measured value. In addition, the pros and cons of the prediction model can be evaluated by the goodness of fit (R-sp). R-sp refers to the ratio of the sum of squared regressions to the sum of squared deviations. The fitting degree of the model is better when the R-sp value is closer to one. The quadratic response surface prediction model of the micro-milling force in this experiment obtained R-sq about F

_{x}and F

_{y}are 89.48% and 86.41% in turn, which shows that it fits well with the measurement results after the experiment and has high reliability. Further significant analysis of the regression model is carried out to validate the ability of the prediction model on reflecting the relationship between cutting parameters and the micro-force. P-value of the micro-milling force regression model is zero, which indicates the significance of independent variables. F-value is a statistic used to judge the significance of the regression model. It shows that the regression model is significant, indicating that there is a linear significant relationship between some cutting parameters and the micro-force.

_{x}and F

_{y}, which is usually used to analyze the primary and secondary influence of cutting parameters on the output response, and the F-value is used as a key indicator to measure the influence level of cutting parameters on the response surface. Further, the significance of the significant results can be 95% when the p-value is less than 0.05, which indicates that the main effect, secondary effect or interaction effect of the cutting parameters have a significant effect on the response. It is known from Table 3 and Table 4 that the p-values of f

_{z}and a

_{p}are both 0, which shows that f

_{z}and a

_{p}have a significant linear effect on F

_{x}and F

_{y}. In addition, the response values (F

_{x}and F

_{y}) contain the same significant interaction terms f

_{z}*a

_{p}and f

_{z}*l and multiple significant quadratic terms f

_{z}

^{2}, a

_{p}

^{2}, n

^{2}appear in F

_{x}, which indicates that the effect of changes in cutting parameters in F

_{x}, compared with F

_{y}, is more significant. This may be due to the micro-milling tool’s radial runout in the vertical feed direction during micro-milling. As shown in Figure 3a, the measured width of the micro-groove is 1005 µm, which is slightly larger than the theoretical diameter of the micro-milling tool, 1000 µm.

_{x}:

- (1)
- Individual effect: a
_{p}> f_{z}> l > n; - (2)
- Interaction effect: f
_{z}*l > f_{z}*a_{p}> f_{z}*n > a_{p}*l > n*l > a_{p}*n; - (3)
- Quadratic effect: n
^{2}> f_{z}^{2}> a_{p}^{2}> l^{2}.

_{y}:

- (1)
- Individual effect: f
_{z}> a_{p}> l > n; - (2)
- Interaction effect: f
_{z}*l > f_{z}*a_{p}> n*l > a_{p}*l > f_{z}*n > a_{p}*n; - (3)
- Quadratic effect: a
_{p}^{2}> n^{2}> f_{z}^{2}> l^{2}.

_{z}*a

_{p}and f

_{z}*l with significant interaction effects can be quantitatively analyzed by the response surface graph of F

_{x}and F

_{y}. Figure 7 and Figure 8 show that reducing both f

_{z}and a

_{p}can effectively reduce the micro-milling force F

_{x}. Under the experimental conditions of spindle speed n = 12,000 r/min and l = 25 mm, the response value F

_{x}is more sensitive to the change of a

_{p}than f

_{z}. Figure 7 and Figure 8 show that reducing both f

_{z}and l can effectively reduce the value of the response result F

_{x}. Under the experimental conditions of n = 12,000 r/min and a

_{p}= 30 µm, the response value F

_{y}is more significant to the change of f

_{z}compared to l. It is particularly noteworthy that the tool extended length and micro-milling force have a profound influence on the tool wear and the workpiece surface quality, which is also proposed in previous studies [32]. In summary, the goal of reducing the micro-milling force can be achieved by reducing f

_{z}, a

_{p}and l. The relationship between the micro-milling force and the cutting parameters is mainly a linear effect and there are certain interaction effects and secondary effects. Among them, F

_{x}, F

_{y}and f

_{z}, a

_{p}have a linear positive correlation and f

_{z}, a

_{p}affects the micro-milling force the effect increases in turn.

#### 4.2. The Top Burrs Morphology Analysis

_{1}and b

_{2}quadratic response surfaces are as follows:

_{1}and b

_{2}) are f

_{z}, a

_{p}and l. Figure 9 is a comparison of the predicted and measured values of the top burrs width after each group of experiments. Figure 9a shows that the change waveform of the predicted value and the measured value about b

_{1}are basically the same, but the fitting degree is poor, which indicates that the quadratic response surface model of b

_{1}is in a statistically insignificant state and the accuracy of the prediction model is average. The comparison between the measured value and the predicted value about b

_{2}in Figure 9b shows that the established response regression model has a high fitting degree, which indicates that quadratic response surface model of b

_{2}has high credibility and can be preferentially used for the top burr analysis.

_{1}and b

_{2}are 62.67% and 85.50%, respectively based on the quadric response surface prediction model of the top burr width on the up-milling side in this study, which indicates that the quadric response prediction model of b

_{2}response is in a significant state. The model of b

_{2}fits well with the experimental results and is highly reliable. The R-sq of b

_{1}is 62.67% smaller than 70%. Therefore, the quadric surface prediction model of the top burrs width on the down-milling side needs to be used carefully, and the correlation between the cutting parameters and the top burrs width on the down-milling side is weak, which may be because the response value (b

_{1}) is mainly affected by the main linear effect, while the secondary effect and the interaction effect are not significant.

_{1}and b

_{2}, where the p-values of f

_{z}and a

_{p}are both less than 0.05, indicating that f

_{z}and a

_{p}have a significant first-order linear effect on b

_{1}and b

_{2}. Reducing f

_{z}or a

_{p}means reducing the burrs width at the top micro-groove. Both b

_{1}and b

_{2}contain the same quadratic term n

^{2}, but multiple quadratic terms f

_{z}*a

_{p}, f

_{z}*n and a

_{p}*n appear in b

_{2}, which indicates that b

_{2}is more sensitive to changes in cutting parameters than b

_{1}. This may be because the chip outflow direction is opposite to the tool rotation direction on the micro-groove up-milling side compared to the micro-groove down-milling side, and some chips do not escape when flowing out along the micro-groove edge, which is more likely to form long burrs at the top, resulting in a large change in top burrs width.

_{2}) decreased with the increase of per-feed tooth. As shown in Figure 10b, when per-feed tooth was fixed at 10 µm/z, spindle speed was fixed at 12,000 r/min and tool extended length was fixed at 25 mm, b

_{2}increased with the increase of axial cutting depth. The per-feed tooth is the most significant factor contributing to the width of top burrs. Figure 11a,b are photomicrograph of the top burrs under Figure 10a conditions when f

_{z}is 2 µm/z and 18 µm/z, respectively.

_{z}, a

_{p}, n, l) have no obvious effect on b

_{1}, but have a significant effect on b

_{2}. As can be seen in Figure 13, the pairwise interactions (f

_{z}*a

_{p}, f

_{z}*n and a

_{p}*n) have the most significant impact on b

_{2}. In addition, the pairwise interactions (f

_{z}*a

_{p}, f

_{z}*n and a

_{p}*n) with the most significant interaction effects can be quantitatively analyzed by the response surface graph of b

_{2}. Figure 14a shows that decreasing a

_{p}and increasing n can effectively reduce b

_{2}. Under the experimental conditions of f

_{z}= 10 µm/z and l = 25 mm, b

_{2}is more sensitive to the change of a

_{p}than n. Figure 14b shows that increasing f

_{z}and a

_{p}can effectively reduce the value of response b

_{2}. Under the experimental conditions of n = 12,000 r/min and l = 25 mm, the change of a

_{p}has a more significant effect on b

_{2}than f

_{z}. Figure 14c shows that choosing moderate f

_{z}and n can effectively reduce the value of response b

_{2}. Under the experimental conditions of a

_{p}= 30 µm and l = 25 mm, the change of n has a more significant effect on b

_{2}than f

_{z}.

#### 4.3. Cutting-Parameter Optimization

_{x}quadratic response surface prediction model with the highest credibility was selected and the combined minimum response micro-milling force (F

_{x}and F

_{y}) and width of top burrs (b

_{1}and b

_{2}) were the constraints. According to the prediction model of Equation (3) can be output minimum response value combination: F

_{x}= 2.31 N, F

_{y}= 2.13 N, b

_{1}= 59 µm, b

_{2}= 75 µm. In addition, a set of corresponding optimized combinations of cutting parameters can be obtained: n = 11,394 r/min, f

_{z}= 5.8 µm/z, a

_{p}= 11.6 µm, l = 20.9 mm. The machined micro-groove morphology using a set of optimized cutting parameters is shown in Figure 15. Simultaneously, the roughness value of the micro-groove bottom was 0.38 µm, which was acceptable. The top burrs were analyzed in conjunction with Figure 3 and it was found that the number of micro-groove top burrs were significantly reduced. This shows that the optimized combination of cutting parameters, solved by the quadratic response surface model, used in actual micro-milling, could achieve the purpose of improving the workpiece surface quality.

## 5. Conclusions

- (1)
- The change of cutting parameters had a significant effect on the micro-milling force and the width of up-milling side top burrs. The prediction model of the quadratic response surface around micro-milling force (F
_{x}and F_{y}) and the width of burrs on the up-milling side (b_{2}) was in a significant state. The experimental measured value and the predicted value had a high fitting degree; - (2)
- During micro-milling workpiece material Al7075-T6, a
_{p}and f_{z}show a significant linear effect on force and width of top burrs. The response values (F_{x}, F_{y}, b_{1}and b_{2}) were mainly affected by a_{p}, followed by was f_{z}, but n and l had few significant effects; - (3)
- In addition, mainly considering the linear effects of a
_{p}and f_{z}, the optimization of cutting parameters also needs to consider the interaction effects and secondary effects between each cutting parameter. Simultaneously reducing f_{z}and a_{p}or simultaneously reducing f_{z}and l could actively reduce the micro-milling force, while reducing a_{p}and increasing n or simultaneously increasing f_{z}and a_{p}could effectively reduce the top burrs; - (4)
- The reasonable setting of cutting parameters could improve the quality of machined surface. According to the quadratic response surface model, the optimal response value could be obtained by optimizing combination of cutting parameters: n = 11,394 r/min, f
_{z}= 5.8 µm/z, a_{p}= 11.6 µm and l = 20.9 mm.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**,

**b**) Schematic diagram of micro-milling; (

**c**) micro-milling tool; (

**d**) force–component directions of the dynamometer (Kistler-9119AA1).

**Figure 3.**(

**a**) Micro-slot morphology and measurement of top burr width; (

**b**) micro-milling force with time.

**Figure 4.**Comparison of measured value and predicted value of micro-milling force. (

**a**) F

_{x}; (

**b**) F

_{y}.

**Figure 7.**Interactive influences between per-feed tooth and axial cutting depth on micro-milling force: (

**a**) F

_{x}; (

**b**) F

_{y}.

**Figure 8.**Interactive influences between per-feed tooth and tool extended length on micro-milling force: (

**a**) F

_{x}; (

**b**) F

_{y}.

**Figure 9.**Comparison of measured value and predicted value of top burrs width: (

**a**) b

_{1}; (

**b**) b

_{2}.

**Figure 10.**Effect of the significant single cutting parameter on width of top burrs. (

**a**) f

_{z}; (

**b**) a

_{p}.

**Figure 14.**(

**a**) Interactive influences between per-feed tooth and axial cutting depth on width of up-milling side top burrs; (

**b**) interactive influences between per-feed tooth and spindle speed on width of up-milling side top burrs; (

**c**) interactive influences between axial cutting depth and spindle speed on width of up-milling side top burrs.

No. | Variables | Response Value | ||||||
---|---|---|---|---|---|---|---|---|

${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ | ${x}_{4}$ | b_{1} (μm) | b_{2}(μm) | F_{x} (N) | F_{y} (N) | |

1 | 0 | 0 | 0 | 0 | 94 | 104 | 5.193 | 3.681 |

2 | 0 | 0 | 0 | 0 | 60 | 103 | 2.992 | 2.854 |

3 | 0 | 0 | 0 | 0 | 88 | 103 | 3.042 | 3.027 |

4 | 0 | 0 | 0 | 0 | 64 | 94 | 2.994 | 3.091 |

5 | 0 | 0 | 0 | 0 | 60 | 82 | 2.975 | 3.091 |

6 | 0 | 0 | 0 | 0 | 64 | 94 | 2.954 | 2.998 |

7 | 0 | 0 | 0 | 0 | 73 | 77 | 2.818 | 3.084 |

8 | −1 | −1 | 1 | −1 | 79 | 109 | 3.182 | 2.628 |

9 | −1 | 1 | 1 | −1 | 122 | 207 | 5.332 | 4.46 |

10 | 1 | 1 | −1 | −1 | 91 | 104 | 8.338 | 7.246 |

11 | −1 | −1 | 1 | 1 | 84 | 92 | 5.033 | 5.37 |

12 | 1 | 1 | 1 | −1 | 73 | 143 | 8.338 | 7.246 |

13 | 1 | −1 | 1 | −1 | 75 | 91 | 4.66 | 3.28 |

14 | 1 | −1 | 1 | 1 | 60 | 88 | 3.562 | 3.437 |

15 | −1 | 1 | 1 | 1 | 102 | 194 | 4.66 | 3.278 |

16 | −1 | −1 | −1 | −1 | 77 | 106 | 3.415 | 2.983 |

17 | 1 | −1 | −1 | −1 | 68 | 115 | 4.871 | 4.825 |

18 | −1 | 1 | −1 | −1 | 100 | 131 | 3.312 | 2.604 |

19 | 1 | −1 | −1 | 1 | 64 | 126 | 3.476 | 3.136 |

20 | −1 | 1 | −1 | 1 | 110 | 143 | 4.626 | 3.382 |

21 | −1 | −1 | −1 | 1 | 79 | 109 | 3.242 | 3.024 |

22 | 1 | 1 | 1 | 1 | 70 | 97 | 5.751 | 4.95 |

23 | 1 | 1 | −1 | 1 | 82 | 124 | 5.925 | 5.286 |

24 | 0 | 0 | −2 | 0 | 88 | 101 | 4.824 | 3.375 |

25 | 0 | 0 | 0 | −2 | 60 | 103 | 4.688 | 2.958 |

26 | 0 | 2 | 0 | 0 | 70 | 98 | 7.712 | 6.056 |

27 | 0 | −2 | 0 | 0 | 63 | 73 | 1.345 | 1.471 |

28 | 0 | 0 | 2 | 0 | 86 | 122 | 5.729 | 3.978 |

29 | 2 | 0 | 0 | 0 | 82 | 101 | 6.668 | 5.192 |

30 | 0 | 0 | 0 | 2 | 60 | 106 | 3.988 | 3.577 |

31 | −2 | 0 | 0 | 0 | 87 | 154 | 2.503 | 1.555 |

Parameter | Notation | Unit | Levels | ||||
---|---|---|---|---|---|---|---|

−2 | −1 | 0 | 1 | 2 | |||

Per-feed tooth (f_{z}) | ${x}_{1}$ | μm/z | 2 | 6 | 10 | 14 | 18 |

Axial cutting depth (a_{p}) | ${x}_{2}$ | μm | 10 | 20 | 30 | 40 | 50 |

Spindle speed (n) | ${x}_{3}$ | r/min | 8000 | 10,000 | 12,000 | 14,000 | 16,000 |

Tool extended length (l) | ${x}_{4}$ | mm | 17 | 21 | 25 | 29 | 33 |

Coefficient | f_{z} | a_{p} | n | l | f_{z}^{2} | a_{p}^{2} | n^{2} | l^{2} | f_{z}*a_{p} | f_{z}*n | f_{z}*l | a_{p}*n | a_{p}*l | n*l |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p-value | 0 | 0 | 0.187 | 0.006 | 0.032 | 0.029 | 0.002 | 0.055 | 0.018 | 0.312 | 0.005 | 0.885 | 0.37 | 0.85 |

F value | 32.52 | 55.04 | 1.90 | 3.88 | 6.27 | 5.78 | 13.86 | 4.28 | 6.91 | 1.09 | 10.46 | 0.02 | 0.85 | 0.04 |

Significance level | 2 | 1 | 10 | 9 | 6 | 7 | 3 | 8 | 5 | 11 | 4 | 14 | 12 | 13 |

Coefficient | f_{z} | a_{p} | n | l | f_{z}^{2} | a_{p}^{2} | n^{2} | l^{2} | f_{z}*a_{p} | f_{z}*n | f_{z}*l | a_{p}*n | a_{p}*l | n*l |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p-value | 0 | 0 | 0.345 | 0.085 | 0.200 | 0.054 | 0.074 | 0.273 | 0.045 | 0.691 | 0.011 | 0.816 | 0.549 | 0.504 |

F value | 77.71 | 29.98 | 0.95 | 3.37 | 1.79 | 4.31 | 3.65 | 1.29 | 4.74 | 0.16 | 8.36 | 0.06 | 0.38 | 0.47 |

Significance level | 1 | 2 | 10 | 7 | 8 | 5 | 6 | 9 | 4 | 13 | 3 | 14 | 12 | 11 |

Coefficient | f_{z} | a_{p} | n | l | f_{z}^{2} | a_{p}^{2} | n^{2} | l^{2} | f_{z}*a_{p} | f_{z}*n | f_{z}*l | a_{p}*n | a_{p}*l | n*l |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p-value | 0.024 | 0.015 | 0.88 | 0.436 | 0.079 | 0.949 | 0.049 | 0.564 | 0.134 | 0.58 | 0.606 | 0.631 | 0.882 | 0.361 |

F value | 6.19 | 7.47 | 0.02 | 0.64 | 3.52 | 0 | 4.52 | 0.35 | 2.5 | 0.32 | 0.28 | 0.24 | 0.02 | 0.8 |

Significance level | 2 | 1 | 12 | 7 | 4 | 14 | 3 | 8 | 5 | 9 | 10 | 11 | 12 | 6 |

Coefficient | f_{z} | a_{p} | n | l | f_{z}^{2} | a_{p}^{2} | n^{2} | l^{2} | f_{z}*a_{p} | f_{z}*n | f_{z}*l | a_{p}*n | a_{p}*l | n*l |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p-value | 0.001 | 0 | 0.19 | 0.71 | 0.002 | 0.827 | 0.027 | 0.084 | 0.004 | 0.021 | 0.962 | 0.003 | 0.766 | 0.06 |

F value | 16.01 | 21.64 | 1.87 | 0.14 | 14.49 | 0.05 | 5.96 | 3.4 | 11.56 | 6.6 | 0 | 11.78 | 0.09 | 4.11 |

Significance level | 2 | 1 | 10 | 11 | 3 | 13 | 7 | 9 | 5 | 6 | 14 | 4 | 12 | 8 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, M.; Chen, Y.; Zhang, G.
Force Prediction and Cutting-Parameter Optimization in Micro-Milling Al7075-T6 Based on Response Surface Method. *Micromachines* **2020**, *11*, 766.
https://doi.org/10.3390/mi11080766

**AMA Style**

Zhou M, Chen Y, Zhang G.
Force Prediction and Cutting-Parameter Optimization in Micro-Milling Al7075-T6 Based on Response Surface Method. *Micromachines*. 2020; 11(8):766.
https://doi.org/10.3390/mi11080766

**Chicago/Turabian Style**

Zhou, Menghua, Yinghua Chen, and Guoqing Zhang.
2020. "Force Prediction and Cutting-Parameter Optimization in Micro-Milling Al7075-T6 Based on Response Surface Method" *Micromachines* 11, no. 8: 766.
https://doi.org/10.3390/mi11080766