3.1.1. Roughness, Ra

Equations (5)–(7) provide the regression models for Ra in the rough, semi-finish and finish operation respectively.

Ra, rough = 2.64 − 0.00570 Gs + 0.0017 De + 0.00630 Pr + 0.0459 Vt − 0.394 Vl − 0.000848 Deˆ2 − 0.000007 Prˆ2 + 0.00653 Vlˆ2 + 0.000546 Gs·De + 0.000145 Gs·Vt <sup>−</sup> 0.000283 Gs·Vl <sup>−</sup> 0.000085 Pr·Vt + 0.000249 Pr·Vl (5)

Ra, semi-finish = −2.869 + 0.0712 Gs − 0.0500 De + 0.000231 Pr − 0.0053 Vt + 0.1704 Vl − 0.000683 Gsˆ2 + 0.000866 Deˆ2 − 0.00444 Vlˆ2 + 0.000019 Gs·Pr + 0.000301 Gs·Vt − 0.000036 De·Pr + 0.000387 De·Vt + 0.000047 Pr·Vl − 0.000896 Vt·Vl (6)

Ra, finish = 1.165 − 0.07211 Gs − 0.01334 De − 0.000773 Pr − 0.01099 Vt + − 0.00007 Vl + 0.001133 Gsˆ2 + 0.000627 Gs·De + 0.000036 Gs·Pr + 0.000150 Gs·Vt + 0.000013 Pr·Vt (7)

> Figure 6 depicts the significant terms for the roughness parameter, Ra, in the rough, semifinish and finish operations.

**Figure 6.** Significant terms for Ra in the rough, semi-finish and finish operations.

In the rough honing operation, the main factor influencing roughness is grain size, Gs, followed by Vt, Pr and Vl. The higher the grain size, tangential speed and pressure, the higher roughness is. Conversely, the lower the linear speed, the higher roughness is. The interaction between grain size and density is significant, as has been observed in previous works [10]. The higher the grain size, the higher density should be in order to assure the correct cutting operation. Other significant interactions are Pr·Vl, Pr·Pr and De·De. Lawrence and Ramamoorthy [22] found that rotational speed was the most influential factor on the Rz parameter, followed by oscillatory speed, honing time and pressure. These results are in accordance with the present work, considering that they did not vary grain

size nor abrasive density. Gunay and Korkmaz [35] also reported a higher influence of grit size compared to linear speed in honing processes.

In the semi-finish operation, the most significant term becomes De·De, followed by Pr, while the term Gs·Gs is also important. This suggests that, although roughness depends directly on pressure, it is also influenced by grain size and density. The fact that pressure influences roughness is in accordance with the results of Kanthababu et al. [5].

In the finish operation, grain size and pressure seem to be the only factors that have an influence on roughness, while density appears in the Gs·De interaction. Gs·Gs, Gs·Pr and Pr·Vt are also influential. Conversely, Bai et al. [13] found that surface roughness depends on tangential speed. In plateau honing processes, Gunay and Korkmaz [35] observed that roughness depended mainly on grain size, linear speed and number of strokes. A grain size of 150, a linear speed of 7 m/min and four strokes are recommended in order to minimize Ra.

In summary, in rough honing processes it is important to select low grain size and low density to ensure low roughness. In addition to grain size and density, pressure also becomes important in the semi-finish operation. The lower the pressure, the lower roughness is. In the finish operation, the main factor to be considered is grain size, followed by pressure. Thus, in finish honing processes, the density of the abrasive is not so important as in rough and semi-finish processes.

As an example, Figure 7 shows a roughness profile for Experiment 2 on (a) rough honing, (b) semi-finish honing and (c) finish honing.

**Figure 7.** Examples of roughness profiles and Abbott–Firestone curves: (**a**) rough honing with grain size 181, (**b**) semi-finish honing with grain size 76 and (**c**) finish honing with grain size 30.

All the profiles present sharp peaks and rounded valleys, with an irregular shape that is characteristic of abrasive machining processes. As expected, the higher the grain size, the higher roughness is. The Abbot–Firestone curves have the s-shape that is characteristic of the abrasive machining processes.

3.1.2. Material Removal Rate

Equations (8)–(10) correspond to the regression models for the material removal rate, Qm, in the rough, semi-finish and finish operations, respectively:

Qm, rough = −0.419 − 0.000830 Gs + 0.00801 De + 0.001799 Pr − 0.00387 Vt − 0.00962 Vl − 0.000154 Deˆ2 − 0.000002 Prˆ2 + 0.000017 Gs·De + 0.000027 Gs·Vt + 0.000104 De·Vl + 0.000006 Pr·Vt + 0.000016 Pr·Vl (8)

Qm, semi-finish = −0.3379 + 0.01439 Gs − 0.00465 De + 0.000007 Pr − 0.002380 Vt + 0.000350 Vl <sup>−</sup> 0.000119 Gsˆ2 + 0.000063 Deˆ2 + 0.000033 De·Vt + 0.000004 Pr·Vt (9)

Qm, finish = 0.1621 − 0.001941 Gs − 0.001733 De − 0.000108 Pr − 0.001910 Vt − 0.00958 Vl + 0.000182 Vlˆ2 + 0.000088 Gs·De + 0.000002 Gs·Pr + 0.000002 Pr·Vt + 0.000003 Pr·Vl + 0.000051 Vt·Vl (10)

> Figure 8 corresponds to the models of material removal rate, Qm, in rough, semi-finish and finish processes, respectively.

**Figure 8.** Significant terms for Qm in the rough, semi-finish and finish operations.

In the rough honing operation, the most significant term influencing the material removal rate is Pr·Pr, followed by Vt, Pr, De·De and Gs. Thus, pressure seems to be crucial to ensure a sufficient material removal rate in this operation, as has been previously observed [11], although the other parameters are also important in this case. In honing processes with diamond stones of grain size 181 and 151, respectively, Vrac et al. [7] found that cutting speed greatly influenced material removal rate, while pressure was less relevant.

As for the semi-finish operation, the main terms are Gs·Gs, Pr and De·De. This suggests that, as the quantity of material to be removed decreases in subsequent honing operations, the importance of pressure is reduced because the cutting operation becomes easier to perform.

In the finish operation, different factors show a similar impact: Pr, Gs, Vt and Vl. Vl·Vl is also significant, and density appears in the Gs·De interaction.

In summary, all factors influence the material removal rate in rough honing. In semifinish honing, mainly pressure, grain size and density should be considered, while in finish honing, all factors except density are important.

3.1.3. Tool Wear

Equations (11)–(13) show the regression models for tool wear, Qp, in the rough, semifinish and finish operations respectively.

```
Qp, rough = −0.001659 + 0.000001 Gs + 0.000037 De + 0.000003 Pr + 0.000025 Vt
     + 0.000005 Vl − 0.000000 Deˆ2 − 0.000000 De·Pr − 0.000000 De·Vt (11)
```

```
Qp, semi-finish = −0.000317 + 0.000021 Gs − 0.000015 De + 0.000002 Pr − 0.000019 Vt
− 0.000034 Vl − 0.000000 Gsˆ2 + 0.000000 Gs·De − 0.000000 Gs·Pr + 0.000000 Gs·Vl − 0.000000 De·Pr +
                                   0.000000 De·Vt + 0.000001 Vt·Vl
                                                                                                           (12)
```

```
Qp, finish = 0.000906 − 0.000095 Gs + 0.000019 De − 0.000001 Pr − 0.000000 Vt
    + 0.000002 Gsˆ2 − 0.000001 Gs·De + 0.000000 Gs·Pr + 0.000000 Gs·Vt -
                                0.000000 De·Vt
                                                                                              (13)
```
Figure 9 depicts the main terms influencing tool wear, Qp, in the rough, semi-finish and finish operations.

**Figure 9.** Significant terms for Qp in the rough, semi-finish and finish operations.

The main factor affecting tool wear in rough honing is the density of the abrasive, De, with a negative impact on tool wear. This suggests that a lower density favors the removal of grains from the bond, which restores the stones' ability to cut but at the cost of increasing tool wear. Other important factors are pressure and the interaction between density and pressure.

In the semi-finish honing operation, density and pressure are still the most important factors. However, a new Gs·Gs starts to influence tool wear with a negative impact: higher grain size leads to lower tool wear.

In the finish operation, grain size is the most important factor influencing tool wear, with the terms Gs and Gs·Gs, followed by pressure and by the interaction between grain size and pressure.

In summary, the density of the abrasive is a crucial factor in rough honing. However, in semi-finish and finish honing, the grain of the abrasive becomes more important. In all the honing steps, pressure is a factor to be considered.

When using cBN tools, tool wear is characterized by low values. For this reason, tool wear has only a small influence on the performance of the present tests, in which honing time was relatively short.

#### *3.2. Multi-Objective Optimization*

The main results of the optimization step are presented in Sections 3.2.1, 3.2.2 and 3.3.3 for the rough, semi-finish and finish phases, respectively.

#### 3.2.1. Rough Honing

Figure 10 presents the results of the multi-objective optimization for the rough honing operation.

**Figure 10.** Multi-objective optimization of the rough honing operation.

The combination that minimizes tool wear and roughness while maximizing the material removal rate is presented in Table 3.

**Table 3.** Results of the multi-objective optimization in the rough honing operation.


This corresponds to medium grain size and high values for the rest of the factors. In rough honing, a high grain size would be recommended in order to provide a high material removal rate, but a low grain size would provide a better surface finish [36]. Thus, medium grain size optimizes both responses.

#### 3.2.2. Semi-Finish Honing

Figure 11 corresponds to the results of the multi-objective optimization for the semifinish honing operation.

**Figure 11.** Multi-objective optimization of the semi-finish honing operation.

The combination that minimizes roughness and tool wear and maximizes the material removal rate is shown in Table 4.

**Table 4.** Results of the multi-objective optimization in the semi-finish honing operation.


This combination includes a low grain size, while the rest of the variables are kept at their high values.

#### 3.2.3. Finish Honing

Figure 12 shows the results for the finish phase.

Table 5 presents the results of the multi-objective optimization in the finish honing operation.

**Table 5.** Results of the multi-objective optimization in the finish honing operation.


Recommended values for the variables are: low grain size (close to the lower limit of 15), high density, low pressure (close to the lower limit of 400), high tangential speed and high linear speed.

In all the honing phases, high linear and tangential speed values are to be selected.

#### *3.3. Sensitivity Analysis*

The results of the sensitivity analysis for the rough, semi-finish and finish phase are presented in Sections 3.3.1–3.3.3, respectively.

**Figure 12.** Multi-objective optimization of the finish honing operation.
