**5. Thermal modeling and optimization of elastic abrasive cutting of structural steels**

The general form of the models describes the dependency between the workpiece temperature *Tw*, cut-off wheel temperature *Ts* and cut piece temperature *Td* in elastic abrasive cutting of C45 and 42Cr4 steels, and the operating conditions of the process (cut-off wheel diameter *ds*, cut-off wheel compression force *F* exerted on the workpiece, and workpiece rotational frequency *nw*), is as follows:

$$y\_g = b\_0 + \sum\_{i=1}^{3} b\_i X\_i + \sum\_{i=1}^{3} b\_{ii} X\_i^2 + \sum\_{i$$

where: *yg*—studied response variables (*g* ¼ 1 � 6): *y*<sup>1</sup> ¼ *Tw*,*<sup>C</sup>*45, *y*<sup>2</sup> ¼ *Ts*,*<sup>C</sup>*45, *y*<sup>3</sup> ¼ *Td*,*<sup>C</sup>*45, *y*<sup>4</sup> ¼ *Tw*,42*Cr*4, *y*<sup>5</sup> ¼ *Ts*,42*Cr*4, *y*<sup>6</sup> ¼ *Td*,42*Cr*4; *X*<sup>1</sup> ¼ *ds*, *X*<sup>2</sup> ¼ *F*, *X*<sup>3</sup> ¼ *nw*—control factors (**Table 1**).

To build the models (4), multi-factor experiments were conducted using an orthogonal central-composite design with a number of trials *<sup>N</sup>* <sup>¼</sup> <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>15</sup> (*n* ¼ 3 is the number of control factors). Three observations were made for each experiment. The variation levels of the factors, were chosen on the basis of preliminary conducted experimental studies on the cut-off wheel performance [13, 31] and thermal flux distribution in the workpiece, chip, cut-off wheel, and cut piece in elastic abrasive cutting [25, 44], are presented in **Table 1**.

The models (4) were built using the measured values of the workpiece maximum instantaneous temperature, cut-off wheel maximum contact temperature, and cut piece temperature at the end of the cut-off cycle.

After statistical analysis of the experimental results by applying the multi-factor regression analysis method and QstatLab software [48], the following regression models for the workpiece temperature, cut-off wheel temperature, and cut piece temperature were built:

• when machining С45 steels:

$$T\_{s,42Cr4} = 89.928 + 3.887d\_s + 5F - 0.394n\_w - 0.011d\_s^2$$

$$T\_{s,C45} = 103.865 + 0.447d\_s + 8F - 0.201n\_w \tag{5}$$

$$T\_{d,C45} = -88.426 + 3.69d\_s + 4.667F - 0.359n\_w - 0.011d\_s^2$$


**Table 1.** *Factor levels in the experimental design.* • when machining 42Cr4 steels:

$$\begin{aligned} T\_{w,42Cr4} &= 2965.024 - 21.143d\_s - 140.028F + 0.063d\_s^2 + 37.403F^2 + 0.006n\_w^2; \\ T\_{s,42Cr4} &= -89.928 + 3.887d\_s + 5F - 0.394n\_w - 0.011d\_s^2 \\ T\_{d,42Cr4} &= -89.928 + 3.887d\_s + 5F - 0.394n\_w - 0.011d\_s^2 \end{aligned} \tag{6}$$

The models built extremely accurately describe the dependency between the variables and control factors. The values of the determination coefficients are *<sup>R</sup>*^<sup>2</sup> *g* ¼ 0*:*849 � 0*:*965 and they were determined at a significance level of *α* ¼ 0*:*05.

In **Figures 8** and **9** the bar diagrams of measured and calculated maximum temperatures are presented, at different parameters of the abrasive cutting process

#### **Figure 8.**

*Temperature bar diagrams for different parameters of the elastic abrasive cutting of 42Cr4.*

**Figure 9.**

*Temperature bar diagrams for different parameters of the elastic abrasive cutting of C45.*

#### *Remote Nondestructive Thermal Control of Elastic Abrasive Cutting DOI: http://dx.doi.org/10.5772/intechopen.103115*

for both materials with the same workpiece diameter. The maximum temperatures are the averages of five measurements for the cut-off wheel, the workpiece, and the cut piece. The error of the calculated values does not exceed 2% (under 20°C) in the worst case.

The analysis of the models built makes possible the evaluation of the effect of the operating conditions on the temperatures of the workpiece, cut-off wheel, and cut piece:


Each studied temperature parameter of the elastic abrasive cutting process has a specific meaning yet is insufficient for its optimum control. The optimum values of the temperatures of the cut piece, cut-off wheel, and workpiece for each material being machined will be obtained at different combinations of values of control factors (cut-off wheel diameter, compression force, and workpiece rotational frequency). Therefore, optimization by one parameter is irrelevant. Multi-objective optimization provides much more information so as to make a justified decision on the selection of optimum elastic abrasive cutting conditions. There are various algorithms for its implementation, which differ in the type and number of target parameters, as well as in the method for determining the optimal solution [45, 49]. To determine the optimum elastic abrasive cutting conditions, multi-purpose optimization was implemented as the area where the temperature parameters under study obtain minimum values were determined. The optimization problem is reduced to solving the following system of inequalities:


#### **Table 2.**

*Optimum conditions of elastic abrasive cutting.*

$$\begin{cases} d\_{s,l} \le d\_s \le d\_{s,u} \\ F\_l \le F \le F\_u \\ n\_{w,l} \le n\_w \le n\_{w,u} \\ T\_d = f(d\_s, F, n\_w) \to \min \\ T\_s = f(d\_s, F, n\_w) \to \min \\ T\_w = f(d\_s, F, n\_w) \to \min \end{cases} \tag{7}$$

where *ds*,*<sup>l</sup>*, *ds*,*<sup>u</sup>*, *Fl*, *Fu*, *nw*,*<sup>l</sup>*, *nw*,*<sup>u</sup>* are respectively the top and bottom levels of the control factors of the elastic abrasive cutting process (cut-off wheel diameter *ds*, compression force *F* and workpiece rotational frequency *nw*)—**Table 1**.

Functions *Td* ¼ *f ds* ð Þ , *F*, *nw* , *Ts* ¼ *f ds* ð Þ , *F*, *nw* and *Tw* ¼ *f ds* ð Þ , *F*, *nw* reflecting the correlation dependencies of the temperatures of workpiece, cut-off wheel, and cut piece on the control factors of elastic abrasive cutting process are described by Eqs. (5) and (6) for the two materials being machined—С45 and 42Cr4 steels.

The optimum conditions of elastic abrasive cutting, providing the best combination of minimum values of the temperatures of workpiece, cut-off wheel, and cut piece, were determined by applying two methods—genetic algorithm and random search method with increasing density. The optimization problem was solved upon machining of С45 and 42Cr4 steels by using QStatLab software [48].

The defined optimum conditions of the elastic abrasive cutting process are presented in **Table 2**.

#### **6. Conclusions**

This chapter considers the specifics of implementing the process of elastic abrasive cutting and analyzes the conditions for stabilizing the dynamic thermal phenomena accompanying it. The processes of heat generation and heat removal in abrasive cutting are generally analyzed, as well as the methods and tools applied to investigate temperature and thermal fluxes. An innovative approach to nondestructive thermal measurement and control of elastic abrasive cutting experimented for two types of structural steels by applying the methodology of planned experiment and multi-objective optimization has been proposed.

Latest trends show that there is a need to apply an automatic smart system for controlling thermal fluxes in the cutting zone so as to ensure a higher quality of machined surfaces and longer cutting tool life. This is also linked to the design of a new approach to non-destructive thermal control of abrasive cutting when developing a smart thermographic system.
