*2.4. Constraint and Optimization Model*

With the grinding speed constrained, the grinding wheel speed *vs* must meet the following requirement:

$$v\_{\rm s} \in \left[\frac{\pi d\_0 n\_{\rm min}}{1000} \le v\_{\rm s} \le \frac{\pi d\_0 n\_{\rm max}}{1000}\right] \tag{21}$$

where *n*min and *n*max represent the minimum and maximum speeds of the CNC grinding machine spindle, respectively.

The tangential feed amount *vr* of the grinding wheel needs to be within the range of the maximum and minimum values of the spindle feed of the machine tool, that is

$$
\upsilon\_{r\text{min}} \le \upsilon\_r \le \upsilon\_{r\text{max}} \tag{22}
$$

The workpiece rotation speed *vw* needs to be within the range of the maximum and minimum values of the workpiece speed of the CNC grinding machine, that is

$$
v\_{w\text{min}} \le v\_w \le v\_{w\text{max}}\tag{23}$$

With the cutting force constrained, the grinding wheel's tangential force needs to be less than the maximum cutting force to protect the grinding wheel and the surface quality of the part, namely

$$F\_t = F\_p \upsilon\_s^x \upsilon\_r^y \upsilon\_w^z \le F\_{t \text{max}} \tag{24}$$

Power constraint, the calculated power needs to be less than the maximum power of the CNC grinding machine, namely

$$P\_m = \frac{F\_\text{t} \cdot v\_\text{s}}{75 \times 1.36 \times 9.81 \times \eta} \le P\_{\text{max}}\tag{25}$$

The surface quality of the machine is constrained. The surface roughness value needs to be greater than the minimum machining roughness value of the machine tool. It is also an important condition for restraining the linear speed *vs* of the grinding wheel and the feed rate *vw* of the table. According to the Ono theory [41], the surface roughness expression of the external grinding is

$$Ra = 0.975 \gamma^{1.2} \times (\cot \varphi)^{0.1} \times (\frac{v\_w}{v\_s} \sqrt{\frac{1}{2r} + \frac{1}{d\_s}})^{0.4} \ge Ra\_{\text{min}} \tag{26}$$

where γ is the cutting edge spacing considered by volume density; ϕ is half of the cutting edge angle; *ds* is the wheel diameter.

*Processes* **2020**, *8*, 3

In summary, the low-carbon and low-cost parameter multi-objective optimization model in the grinding process is

$$\begin{cases} \min f(\boldsymbol{\ell}\boldsymbol{\ell}) = \left(\min W, \min \mathbf{C}\right)^{T} \\ \text{s.t.} \quad \left(v\_{s}, v\_{\text{m}}\right)^{T} \\ v\_{s} \in \left[\frac{\pi d\_{0} v\_{\text{min}}}{1000} \le v\_{s} \le \frac{\pi d\_{0} v\_{\text{max}}}{1000}\right], \\ v\_{\text{min}} \le v\_{r} \le v\_{r\text{max}}, \\ v\_{\text{wmin}} \le v\_{w} \le v\_{\text{wmax}}, \\ F\_{t} = F\_{p} v\_{s}^{\text{v}} v\_{r}^{\text{v}} v\_{w}^{\text{v}} \le F\_{\text{max}}, \\ P\_{m} = \frac{F\_{t} v\_{r}}{75 \times 136.981 \times 0.81 \eta} \le P\_{\text{max}}, \\ Ra = 0.975/^{1.2} \times (\cot \eta)^{0.1} \times \left(\frac{v\_{w}}{v\_{r}} \sqrt{\frac{1}{2\tau} + \frac{1}{d\_{s}}}\right)^{0.4} \le Ra\_{\text{max}}. \end{cases} (27)$$

#### **3. Parameter Optimization Based on Improved NSGA-II Algorithm**

#### *3.1. Improved NSGA-II Algorithm*

To solve the multi-objective optimization problems, the weighted summation method is used to assign weights to each target value, and the multi-objective problem is simplified to a single-objective problem. However, there is no standard for the assignment of target weights, and the objective functions usually have different dimensions. If the weight value cannot be determined between the carbon emissions and the cost cash, such as in this paper, it will have a greater impact on the calculation results. The algorithm used by a multi-objective function in the MATLAB program is an improved multi-objective optimization algorithm based on the non-dominated sorting genetic algorithm (NSGA-II) with an elite strategy, which can effectively solve the multi-objective optimization problem.

The characteristics of the genetic algorithm include determining the dominance and non-inferiority of the individual, and comparing and judging the better target individual. Based on the dominating judgment order value, the individuals in the population are assigned to different front-ends according to the size, and the higher the front, the stronger the dominance. The crowding distance is used to calculate the distance between a certain body in a front-end and other individuals in the front-end, and to characterize the degree of crowding between individuals—the greater the distance, the better the diversity of the population. The improved algorithm introduces the optimal front-end individual coefficients unique to the gamultiobj function, defines the proportion of individuals in the optimal front-end in the population, and also directly determines the number of individuals retained during the pruning process.

The algorithm flow is to first determine the constraint type of the optimization problem, generate the initial population, and judge whether the algorithm can be exited. And if it exits, get the Pareto optimal solution. If not, the population will evolve into the next generation. In the process of evolution, the gamultiobj function only uses the tournament selection method based on the order value and the crowded distance. The selected individual is assigned to several front-ends to generate the parent population. The parent population crosses, and the mutation produces the children. The gamultiobj function allows the elite to automatically retain, and the scaling function is no longer needed. The parent and the child are merged, and the individuals in the population are sorted by the non-dominated sorting function so that all the merged individuals are assigned to different front-ends. Then the crowded distance is used to calculate the distance between each body in a front-end and its neighbors. According to the optimal front-end individual coefficient, individuals equal to the size of the population are pruned in the population twice as large as the parent–child mergence to obtain a new parent population, and it is judged whether the iteration is terminated or whether the algorithm can be exited. The algorithm flow is shown in Figure 3.

**Figure 3.** Algorithm flow chart.

#### *3.2. Optimization Target Solving*

The air compressor crankshaft blank of 45# steel was selected, and the second main journal of the crankshaft is finely ground by a face cylindrical grinding machine, M-181. The diameter of the main journal is ϕ 37.96–37.944, the root radius is R 1.4–1.7, the outer diameter of the main shaft is 0.015, and the roughness of the outer circle is Ra 0.8. The process drawing is shown in Figure 4, and the actual processing is shown in Figure 5. The grinding machine uses a 100# resin-bonded diamond grinding wheel to grind the outer end surface of the shaft parts, and the grinding precision and smoothness are high; the longitudinal movement of the grinding wheel frame is driven by the gear oil pump, and the movement is stable to ensure the uniform feeding speed of the grinding wheel; double-paired high-precision rolling bearings can make high rotation accuracy and rigidity. The relevant parameters of the machine tool are shown in Table 1.

**Figure 4.** Fine-grinding the second main journal of the crankshaft.

**Figure 5.** Actual machining of the crankshaft. **Table 1.** CNC cylindrical grinding machine parameters.


Control the grinding machine standby, air cutting, and other stages of operation. At each stage, use the three-phase four-wire wiring method to lap the power recorder and collect the power value of each stage of the machine tool. At the same time, the current-monitoring equipment is used to monitor the data reliability of the power meter. Finally, when the second main journal of the crankshaft is ground, a top force-measuring instrument is installed to detect and collect the grinding force at different cutting speeds in real-time. The experiment for the grinding force is carried out to obtain the parameters in Table 2, and the parameters are introduced into the model of the second part. The specific experimental instruments are shown in Figures 6 and 7, and the corresponding parameter values are shown in Table 3.

**Table 2.** Grinding force experiment.


**Figure 6.** Grinding machine power recorder.

**Figure 7.** Grinding machine current monitor.


**Table 3.** Grinding parameters.

Set the optimal front-end individual coefficient to 0.3, the population size to 100, the maximum evolutionary algebra to 300, the stop algebra to 300, and the fitness function deviation to 0.001 to calculate the results. Since the initial population of the algorithm is randomly generated, the operation results obtained each time are different. The Pareto front-end values obtained from the result of a certain operation is shown in Figure 8. Table 4 shows the specific parameters of the 30 optimal front individuals, each row representing an individual speed of the grinding wheel outer circle, the cutting feed rate, the rotational speed of the workpiece, the individual's carbon emissions, and costs.

**Figure 8.** First front-end individual distribution map.


**Table 4.** Pareto front-end values obtained from a certain operation.

#### *3.3. Fuzzy Matter Element-Based Decision-Making*

Multi-objective decision-making schemes include use things, features, and fuzzy magnitudes to quantitatively analyze and calculate things. Assume that the thing is the plan *Ki*, and the feature is the evaluation index *Oi* (for the purposes of this paper, the evaluation index is the wheel speed, the table feed speed, the carbon emission, and the cost), and the fuzzy value gives the value of *xij*, thereby constituting the matter element. For the optimization model of this paper, take the smaller and better decision, and the metric is determined by the correlation coefficient λ*ij*.

$$
\lambda\_{ij} = \frac{\mathbf{x}\_{ij} - \mathbf{min} \mathbf{x}\_{ij}}{\mathbf{max} \mathbf{x}\_{ij} - \mathbf{min} \mathbf{x}\_{ij}} \tag{28}
$$

According to the scheme, the characteristics and the correlation coefficient, it can construct the complex fuzzy matter element of the dimension correlation coefficient of a thing.

$$R\_{\lambda} := \begin{bmatrix} K\_1 & K\_2 & K\_3 & \dots & K\_{30} \\ O\_1 & 0.994 & 0 & 1 & 0.976 \\ O\_2 & 0.741 & 1 & 0 & 0.264 \\ O\_3 & 0.040 & 1 & 0 & 0.007 \\ O\_4 & 0.001 & 1 & 0 & 0.001 \\ O\_5 & 0.946 & 0 & 1 & 0.968 \end{bmatrix} \tag{29}$$

*Rw* is the weighted composite element for each decision-making indicator, and the weight of the *<sup>i</sup>*-th evaluation indicator for each scenario is *Wi* <sup>=</sup> *<sup>n</sup> j*=1 <sup>λ</sup>*ji*/ *<sup>n</sup> j*=1 *m i*=1 λ*ji*.

$$R\_w = \begin{bmatrix} O\_1 & O\_2 & O\_3 & O\_4 & O\_5 \\ W\_i & 0.285 & 0.249 & 0.153 & 0.092 & 0.222 \end{bmatrix} \tag{30}$$

The weighted average centralized processing is used to construct the correlation fuzzy matter element, namely

$$R\_k = \begin{bmatrix} K\_1 & K\_2 & K\_3 & \cdots & K\_{30} \\ 0.4994 & 0.4933 & 0.5067 & \cdots & 0.5593 \end{bmatrix} \tag{31}$$

The minimum value is obtained by sorting the degree of association, and the optimal solution is *K*21. The results in the single target case are obtained and compared, as shown in Table 5.

**Table 5.** Comparison of optimization results.


The best variable for the second main spindle refining of the crankshaft with low carbon and low cost is the grinding wheel linear velocity of 24.235 m/s, the spindle feed speed is 0.00016 m/s, the workpiece rotation speed is 14.0428 m/s, and the carbon emission is 0.0252 kg/piece. The processing cost is 0.825 yuan/piece. With low carbon as the goal, the grinding wheel's linear speed is high, the processing time is short, but the grinding wheel wear and cutting fluid usage are significant and the cost is high; With low cost as the target, the low linear speed of the grinding wheel reduces the amount of wear and the amount of cutting fluid, but the processing time is long and the carbon emission is high.

### **4. Conclusions**

This paper systematically analyzes the energy consumption characteristics and parts-manufacturing costs of various stages of machine tools in grinding. An optimization model for the external grinding parameters with the minimum carbon emission and the optimal cost as the multi-objective is established. The use of auxiliary tools and the division of the whole process are considered in the modeling process. Considering the dynamic change of cutting fluid and the service life of the grinding wheel, an adjustment function is introduced based on the linear speed of the grinding wheel and the feed rate of the working table. The optimized grinding parameters are calculated by using the NSGA-II algorithm, and these parameters are evaluated through the fuzzy matter-element decision method. The machining process is fitted in a single grinding depth during the modeling process, but during the actual production, one part often requires multiple-time grinding. According to different grinding depth and surface roughness values, applying them in the multi-objective optimization dynamic model can realize step-by-step optimization in machining and be referred to for the selection of grinding process parameters.

**Author Contributions:** Conceptualisation, M.H. and Q.G.; Investigation and experiment, Y.S., S.T. and Y.W.; Visualisation, writing—review and editing, Y.S., M.H. and Q.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The work described in this paper was supported by the 58th postdoctoral science foundation program of China (Grant No. 2015M581301), the National Natural Science Foundation of China (Grant No. 51775392 and 51675388), the Educational Commission of Hubei Province (Grant No. B2018069), the National Science Foundation of Hubei province (Grant No. 2019CFB384), and the Key Laboratory of Automotive Power Train and Electronics (Grant No. ZDK1201802). These financial contributions are gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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