3.1 Grinding forces and force ratio
Figure 4 shows the effects of processing parameters on normal force Fn and tangential force Ft while grinding the hardened steel surface. The normal force and tangential force of UVAG tend to decrease compared with those of CG under all experimental grinding parameters. From Fig. 4a, the normal and tangential forces are reduced from 193.56 N to 130.65 N and from 97.22 N to 67.97 N in CG, respectively, and from 177.80 N to 112.38 N and from 85.88 N to 58.70 N in UVAG, respectively, as the wheel speed (vs) rises from 15 m·s− 1 to 30 m·s− 1 at a feed speed vw of 8 m·min− 1 and a cut depth ap of 15 µm. Ultrasonic vibration reduces the tangential grinding forces by up to 13.64% and the normal grinding forces by up to 14.08% in UVAG compared with those in CG. However, the difference in forces between CG and UVAG changes minimally with increasing wheel speed, suggesting that grinding speed is not sensitive to the effect of ultrasonic vibration on grinding forces, as reported by Wang et al. [10] and Cao et al. [12]. As the workpiece speed of vw increases from 4 m·min− 1 to 10 m·min− 1, Fn of CG increases from 88.67 N to 154.69 N, and Ft of CG increases from 49.63 N to 77.96 N, as shown in Fig. 4b. The tendency of Fn and Ft in UVAG is similar to that in the CG process, but Fn and Ft are always lower by 16.76 and 10.18 N, respectively in contrast to those in CG.
The effect of ultrasonic vibration in reducing grinding forces decreases with increasing workpiece speed, indicating that the reduced workpiece feed more satisfies UVAG, as stated by Huang et al. [2] and Zhao et al. [11]. The reason is that the ultrasonic vibration increases the relative feed speed more obviously at a small feed speed [25]. As illustrated in Fig. 4c, Fn and Ft of CG and UVAG rise with the increase in cut depth ap, i.e., from 100.04 N to 214.05 N and from 53.74 N to 107.57 N in CG, respectively, and from 86.92 N to 198.76 N and from 43.49 N to 96.14 N in UVAG, respectively. As the material removal rate increases, the grinding force also increases significantly; therefore, more energy is required to remove the material. UVAG demonstrates lower values of Fn and Ft by 14.08% and 19.07% than those of CG, respectively. This phenomenon implies that the employment of ultrasonic vibration contributes to the reduction in grinding forces owing to the ultrasonic softening and lubrication effects [26]. Meanwhile, the impact of ultrasonic vibration increases the microbreakage of the wheel grain and maintains the sharpness of the grain cutting edge during the grinding process [27]. Figure 4d presents the grinding force ratio with the different material removal rates (Q'w) applied in CG and UVAG.
$$Q_{{\text{w}}}^{\prime }={a_\text{p}}{v_\text{w}}$$
4
As Q'w rises from 1 mm3·(mm·s)−1 to 2.5 mm3·(mm·s)−1, the force ratio varies in the range of 1.79–2.02 during CG, while the UVAG force ratio on a scale of 1.82–1.89 is more stable. UVAG can reduce the grain wear of wheels effectively and maintain their sharpness constantly. This is also an important proof of the reduction in grinding forces in UVAG; the grinding wheel is constantly kept sharp by the microshattering of the abrasive grains under ultrasonic impact to create new cutting edges.
Specific grinding energy es plays a vital role in estimating the power for removing materials, which is calculated from grinding forces and machining parameters [27]. The es of hardened surfaces under CG and UVAG can be expressed by material removal rate (\({Q^{\prime}_\text{w}}\)), as shown below [29]:
![](data:image/png;base64,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)
From Fig. 5, the specific grinding energy decreases as Q'w increases under different grinding conditions. Under the grinding condition of Q'w = 1 mm3·(mm·s)−1, the specific grinding energy in CG and UVAG is 124.08 J·mm− 3 and 100.14 J·mm− 3, respectively. The proportion of cutting processes rapidly increases with ultrasonic vibration; thus, the energy consumption caused by sliding and ploughing processes decreases. In this case, the specific grinding energy in UVAG can be reduced compared with that in CG under the same condition. This is because ultrasonic vibration leads to thicker and shorter chips at a low material removal rate [30]. When the material removal rate Q'w increases from 1 mm3·(mm·s)−1 to 2.5 mm3·(mm·s)−1, es values in CG and UVAG processing are reduced by 26.77% and 10.24%, respectively, indicating that the energy consumption per unit decreases as the material removal rate increases owing to the size effect [31].
3.2 Ground surface roughness
Surface roughness is an important component of surface quality and plays a key role in machining accuracy and fatigue life [32]. Figure 6 shows the influence of machining parameters on the surface roughness profile Ra during CG/UVAG processes. With all processing parameters, Ra of UVAG is always less than that of CG. In Fig. 6a, when the wheel speed (vs) rises from 15 m·s− 1 to 30 m·s− 1 with constant values vw = 7.5 m·min− 1 and ap = 15 µm, the surface roughness Ra decreases in the range of 0.392 − 0.357 µm under CG and in the range of 0.376 − 0.357 µm under UVAG. That is, the ultrasonic vibration reduces the surface roughness by up to 9.47% compared with CG (Fig. 6a). As the vw value rises from 4 m·min− 1 to 10 m·min− 1, with vs = 25 m·s− 1 and ap = 15 µm, the Ra value of CG rises from 0.317 to 0.389 (i.e., by 22.71%), and that of UVAG increases from 0.298 to 0.356 (i.e., by 19.46%), as depicted in Fig. 6b. When ap rises from 10 µm to 25 µm, Ra of CG increases by 17.56% from 0.354 to 0.415, and that of UVAG increases by 10.72% from 0.345 to 0.382 (Fig. 6c). UVAG shows the most significant improvement in surface integrity at 25 µm cutting depth because the ultrasonic vibration allows the grinding wheel to undergo microbreakage while retaining the sharpness as the cutting depth increases. As the material removal rate increases from 1 mm3·(mm·s)−1 to 3.33 mm3·(mm·s)−1, the CG surface roughness increases from 0.317 µm to 0.415 µm (i.e., an increase of 30.9%), while the UVAG surface roughness is always lower than the CG value (i.e., up to 9.47%). Surface roughness is related to the height of the material remaining on the machined surface30. The repeated ironing effect of the grinding wheel on the processed surface by ultrasonic vibration reduces the residual height on the surface [33], resulting in better Ra of UVAG.
3.3 Parameter optimization
From the normal grinding force data obtained from the 10 sets of grinding tests, 9 sets are regarded as sample data. After statistical regression, the empirical equations for CG and UVAG are obtained as follows:
Similarly, with surface roughness as the target, the empirical equations for CG and UVAG are obtained as follows:
Table 5 shows the adjusted determination coefficient (Adj. R2) and significance F of the above regression equations. The Adj. R2 of different regression equations are all greater than 0.8, which is very close to 1. Therefore, each characteristic parameter is meaningful. The significance F indicates that the lowest confidence of Ra regression equation of CG is 97.92%, and the grinding force is 99.99%. Thus, these regression equations can be used for later analysis.
Table 5
Adjusted determination coefficient (Adj. R2) and significance F of regression analysis.
|
Fn
|
Ra
|
CG
|
UVAG
|
CG
|
UVAG
|
Adj. R2
|
0.917
|
0.954
|
0.859
|
0.804
|
Significance F
|
3.63×10− 4
|
6.39×10− 5
|
0.0207948
|
0.00685
|
As listed in Table 6, the last 3 set of experimental parameters is substituted into the empirical model, and the calculated values are within 5% error compared with the experimental values, indicating that the established model for normal grinding forces and ground surface roughness profiles is reliable. The error in CG is smaller than that in UVAG, probably due to the random fragmentation of the abrasive grain caused by the ultrasonic vibration.
Table 6
Comparative results between the calculated and experimental values of the formula.
Parameters
|
Value
|
Fn/N
|
Ra/µm
|
CG
|
UVAG
|
CG
|
UVAG
|
vs = 25 m·s− 1
vw = 8 m·min− 1
ap = 15 µm
|
Calculated
|
142.67
|
125.42
|
0.377
|
0.361
|
Experimental
|
140.98
|
121.13
|
0.376
|
0.368
|
Error/%
|
1.20
|
3.54
|
0.26
|
1.90
|
vs = 30 m·s− 1
vw = 6 m·min− 1
ap = 25 µm
|
Calculated
|
167.08
|
142.46
|
0.383
|
0.357
|
Experimental
|
162.24
|
143.08
|
0.376
|
0.344
|
Error/%
|
2.90
|
0.435
|
1.82
|
3.64
|
vs = 15 m·s− 1
vw = 4 m·min− 1
ap = 10 µm
|
Calculated
|
90.10
|
75.36
|
0.325
|
0.304
|
Experimental
|
94.57
|
78.32
|
0.315
|
0.305
|
Error/%
|
4.97
|
3.93
|
3.08
|
0.33
|
The empirical equations, i.e., Equations 6–9, are substituted into Eq. 1, and the boundary conditions are substituted into Eq. 2. Programming is conducted in MATLAB, setting the inertia weight factor w = 0.5, the constant values c1 = 2 and c2 = 2, and the maximum number of iterations k ≤ 300. The optimization results are listed in Fig. 7. The particles keep moving toward the lower left corner of Fig. 7, i.e., in the direction of lower surface roughness and grinding forces. The Pareto solution contains 15 sets of noninferior solutions, and the Pareto optimal frontier is obtained, as shown in the right panel of Fig. 7. The optimized wheel speed is the maximum boundary value, indicating that increasing the grinding speed has a better effect on reducing grinding forces and improving surface quality. Meanwhile, increasing the cutting depth while decreasing the workpiece speed will help achieve a smaller surface roughness. Accordingly, the MOPSO algorithmic model law is consistent with the grinding mechanism and is useful for optimizing the machining process.
3.4 Ground surface and chip morphologies
Figure 8 shows the surface morphology of the workpiece end face (cutout surface) after profile grinding of CG and UVAG (with constant processing values of vs=30 m·s− 1, vw=6 m·min− 1, and ap =25 µm). CG shows various defects, such as material adhesion and microfracture on the end face, whereas no defects are evident on UVAG. Clear edges and no extruded burrs in the root area of the curved part in profile grinding of CG and UVAG are observed in Fig. 8a. There is material adherence on the end face in the area of crest Ⅰ under profile grinding of CG because the chips at the crest are thinner in profile grinding [34], and the heat is difficult to carry away during the machining of the crest. The accumulation of grinding heat makes the residual material of the thinner edge chips be cold welded on the end face. The crest II area of the tooth shows microfracture under profile grinding of CG, suggesting a reduction in the accuracy of the grinding wheel shape at crest II due to the abrasive particles from grinding wheel wear at a higher grinding temperature in the tooth area. On the contrary, as illustrated in Fig. 8b, none of these defects appear at crest I and crest II during the profile grinding of UVAG, and excellent performance is achieved in the dimensional accuracy and surface profile of formed parts. The vibration cuts the workpiece surface, resulting in shorter chips and more chip breakage than that in CG; therefore, no chip adhesion occurs, as demonstrated in Fig. 10. The average chip length (LCGmax) is 493 µm and the average width (WCgmax) is 25 µm under CG, while the average chip length (LUVAGmax) is 164 µm and the average width (WUVAGmax) is 11 µm under UVAG. Meanwhile, profile grinding of UVAG is subjected to alternating grinding forces and the abrasive grains are susceptible to micro-breakage and maintain sharp, excellent form accuracy on formed surfaces consequently, as reported by Cao et al. [35].
Figure 9 depicts the surface morphologies of the root and tip of the workpiece after the machining process with optimized parameters. In Figs. 9a and 9c, the grinding texture on the root surface of the workpiece is clearer than that on the tip surface (Figs. 9b and 9d) under both grinding processes. The mechanical interaction between the abrasive grit and the workpiece surface is greater than that in the concave arc [36] because more material has to be removed in the convex arc of the grinding wheel. Deep abrasion marks tend to form on the bottom area of the workpiece consequently, and surface damage from material breakage even occurs on the CG root surface. In the tip region of the workpiece, due to the accumulation of grinding heat and the increased material plasticity, material coating of the machined surface occurs in profile grinding of CG and UVAG. Less pits occur under UVAG due to the ability of ultrasonic vibration to further reduce the hardness of the workpiece material [37]. These results demonstrate the effectiveness of the optimized parameters in producing shaped workpieces with no fractured tooth edges and no surface defects.