## 3.1 Grinding forces

Grinding forces as an important indicator has a great influence on the grinding specific energy, grinding temperature, grinding wheel performance, and ground surface quality [23]. Fig. 2 shows the influence of grinding parameters on the normal force *F*n and tangential force *F*t during CG and UVAG processes. When grinding speed *v*s rises from 15 to 30 m/s, the normal force of CG decreases from 103.95 N to 63.91 N, tangential force of CG decreases from 46.17 N to 29.45 N as illustrated in Fig. 2a. A similar varying tendency with the *F*n and *F*t can be observed in UVAG process compared to the other one, while as the normal and tangential forces are always lower than the other one by 2.69%-6.98% and 4.87%-7.91%, respectively. However, the difference is small with the change of grinding speed, indicating that grinding speed had little significant effect on the ability of UVAG in terms of the reduction of grinding forces. In addition, the *F*n, *F*t of CG/UVAG rises with the increases of workpiece speed *v*w (Fig. 2b). Meanwhile, the maximum difference between UVAG and CG normal force and tangential force occurs at the maximum feed speed *v*w of 9 m/min, which are 8.94% and 12.19% respectively. Due to the increase of workpiece feed speed, the intermittent grinding phenomenon between the abrasive particles and workpiece is more likely to occur. As illustrated in Fig. 2c, the *F*n, *F*t of CG and UVAG increase with the rise of depth of cut *a*p. However, the difference of *F*n, *F*t between UVAG and CG is most obvious at the smaller cutting depth (*a*p=10 µm), and decreases by 16.44% and 17.44%, respectively, indicating that the UVAG process with soft WA wheel is suitable for the smaller cutting depth. As for the large cutting depth, there is little difference between grinding forces for UVAG and CG processes. The main reason can be summarized that although ultrasonic vibration improve the ability of self-sharpening effect for grains [24] and it will accelerate wheel wear under a large depth of cut, so that the grinding force of UVAG and CG is similar at a large depth of cut.

## 3.2 Specific grinding energy

Specific grinding energy *e*s as an important indicator to evaluate the degree of difficulty in removing materials and measure the generated heat during grinding processes, presents the consumed energy by removal materials per volume, which can be expressed as follows:

(2)

Where the grinding width *b* of workpiece is fixed at 10 mm.

In order to investigate the specific grinding energy clearly, the maximum undeformed chip thickness should be studied [25]. Here, as mentioned by Pahlitzsch et al [26], the maximum undeformed chip thickness *a**g*max can be given using the following expression (Eq. 3) in consideration of the adjacent abrasive grains with the same spacing of grinding wheels.

(3)

Seen from Fig. 3, the evolution of specific grinding energy *e*s affected by the grinding wheel speed, feed speed, depth of cut was plotted, respectively. As the grinding speed *v*s increases from 15 to 30 m/s under the feed speed *v*w of 7.5 m/min and cutting depth *a*p of 15 µm, the specific grinding energy *e*s of CG increases from 36.94 J/mm3 to 47.12 J/mm3, by 27.56% and UVAG increases from 35.14 J/mm3 to 43.39 J/mm3, by 23.48%, respectively (Fig. 3a). Seen from Fig. 3b, as the *v*w value increases from 4.5 m/min to 9 m/min with the *v*s of 25 m/s and *a*p of 15 µm, the *e*s value of CG decreases from 49.22 J/mm3 to 40.56 J/mm3, by 17.59% and UVAG decreases 45.58 J/mm3 to 35.61 J/mm3, by 21.87%, respectively. As *a*p increases from 10 to 25 µm, the *e*s value of CG decreases by 35.68% from 55.04 J/mm3 to 35.40 J/mm3, and UVAG decreases by 24.14% from 45.44 J/mm3 to 34.47 J/mm3, respectively (Fig. 3c). Fig. 3d demonstrates the *e*s decreases with the rise of *a**g*max according to Eq. 3. It can be found that the *e*s of UVAG fitting curves is reduced in range of 5.30%-11.29% compared with CG. Meanwhile, the decreasing tendency of *e*s with the increase of *a**g*max reveals that UVAG has better grindability than CG, lower grinding energy consumption and grinding heat generation. This phenomenon can be explained in view of different material removal stages (e.g. scratching, ploughing and cutting) of abrasive grains. There are more consumed energy in scratching and ploughing due to the size effect according to Eq. 3. Here, the *a**g*max is increased with the decrease of *v*s and rise of *v*w and *a*p, which reflected in the reduction of *e*s.

Aim at investigate the influence of ultrasonic vibration parameters on material cutting processes clearly, the *a**g*max model of UVAG methods was established in basis of traditional grinding processes, which was illustrated in Fig. 4. Here, the *a**g*max value (Eq. 4) could be rewritten using the line of EH.

$${a_{gmax}}=EH=EF \cdot \sin \theta$$ 4

According to the coordinate system in Fig. 4, the moving trajectory of two continuous abrasive grains with a distance of *λ*s during UVAG processes could be expressed as follows:

(5)

Where *A* is ultrasonic amplitude, *f* is ultrasonic frequency, *t* is current time, *R* is the radius of grinding wheel. Seen from Eq. 5, the following equation could be presented:

\(E{F_u}={x_{u,i - 1}} - {x_{u,i}}=\frac{{{\lambda _s}{v_w}}}{{{v_s}}}+A\sin (\omega (t+\frac{{{\lambda _s}}}{{{v_s}}})) - A\sin (\omega t)\) (6)

In this case, the *a*gmax needed to be calculated within a period, that is, *t*=-*λ*s/(2*v*s). Here, the maximum value of *EF**u* should be:

\(E{F_{u,\hbox{max} }}=\frac{{{\lambda _s}{v_w}}}{{{v_s}}}+2A\sin (\omega \frac{{{\lambda _s}}}{{2{v_s}}})\) (7)

\(\sin \theta =\frac{{\sqrt {{d_s}^{2} - {{({d_s} - 2{a_p})}^2}} }}{{{d_s}}}=\frac{{\sqrt {4{d_s}{a_p} - 4{a_p}^{2}} }}{{{d_s}}}\), after simplification: \(\sin \theta =2\sqrt {\frac{{{a_p}}}{{{d_s}}}}\) (8)

Therefore, the maximum undeformed chip thickness *a*u,*g*max of UVAG could be concluded as:

\({a_{u,g\hbox{max} }}=2(\frac{{{\lambda _s}{v_w}}}{{{v_s}}}+2A\sin (\frac{{\omega {\lambda _s}}}{{2{v_s}}}))\sqrt {\frac{{{a_p}}}{{{d_s}}}}\) (9)

According this line of consideration based on Eq. 9, the maximum undeformed chip thickness *a*u,*g*max was remarkably affected by the ultrasonic vibrating and grinding parameters. Moreover, the phenomenon of intermittent cutting processes between the abrasive grain and workpiece could be concluded in aspects of the *a*u,*g*max model (see Eq. 9) and schematic diagram (see Fig. 4). Fig. 5 shows the relationship between the *a*u,*g*max and ultrasonic amplitude as well as specific grinding energy. The actual value of *a*u,*g*max of UVAG is larger than that of CG and the entering speed of cutting stage of the former one is faster than the later one. Under the grinding parameters (e.g. grinding speed of 25 m/s, feed speed of 7.5 m/min, and cutting depth of 15 µm) as well as the ultrasonic amplitude of 4 µm, the *a*u,*g*max of UVAG increased by 25.4% compared with CG according to Eq. 9. Seen from fitting curves in Fig. 5b, a slight higher of the *e*s value for UVAG than that of CG with the increase of *a*u,*g*max value. This can be explained that the ability of the UVAG to reduce *e*s decreases with the increase of *a**u,g*max, because the ultrasonic vibration process has a more obvious effect on the smaller *a**u,g*max value.

## 3.3 Grinding temperature and coolant heat transfer coefficient

During grinding processes, amount of energy was required to remove materials and then most of those energy would be converted into heat. Here, the heat would be transmitted into the workpiece surface, resulting in grinding burns and affecting the service life of workpiece eventually [27]. Fig. 6 illustrates the relationship between grinding temperatures and grinding parameters for both grinding methods. As the grinding wheel speed rise from 15 to 30 m/s, the grinding temperature of CG rise from 87.80℃ to 100.68℃, and the grinding temperature of UVAG increases from 77.22℃ to 90.25℃. Here, the grinding temperature of UVAG is generally lower than that of CG, with a decrease of 10.36–12.48%. The grinding temperature rises with the rise of feed speed *v*w, as seen from Fig. 6b. The tendency of grinding temperature of UVAG and CG was similar, but the temperature of UVAG decreased by 10.82%-19.01% compared with CG. In addition, the grinding temperature of CG and UVAG increases with the rise of *a*p (see Fig. 6c), and the grinding temperature of UVAG is always lower than CG, ranging from 5.84%-17.45%. This phenomenon reveals that the grinding temperature can be effective reduced by employing ultrasonic vibrating techniques into traditional grinding processes, owing to the smaller *e*s value under the same *a*u,*g*max.

During grinding processes, the total grinding heat flux (*q*t) was transmitted into various parts, including workpiece (*q*w), grinding wheel (*q*s), chip (*q*ch), and coolant (*q**f*), respectively [15]. Here, the total heat flux is deduced as:

\({q_t}={q_w}+{q_s}+{q_{ch}}+{q_f}\), where \({q_t}=\frac{{{F_t}{v_s}}}{{b{l_c}}}\) (10)(11)

As an effective method to avoid grinding burns, the coolants heat transfer coefficient *h**f* of can be expressed via the following formulas.

At first, in basis of the "fluid wheel" hypothesis, the thermal performance *β**f* of coolants is considered and the coolant is not boiling [20]. In this case, the heat transfer coefficient can be written as:

\({h_{f,\operatorname{FW} }}=0.94{\beta _f}\sqrt {\frac{{{v_s}}}{{{l_c}}}}\) (12)

Then, when the coolant flowing through the grinding arc zone is regarded as a laminar outward sweep plate, the thermal conductivity and specific heat coefficient of water-based coolant are considered, based on the similarity principle of fluid mechanics [28]:

\({h_{f,\operatorname{LW} }}=0.759\sqrt {\frac{{{v_s}}}{{{l_c}}}}\) (13)

Finally, the coolants heat transfer coefficient by backtracking the experimental data is confirmed, which is also the method adopted in this paper. Here, the convective heat transfer coefficient of coolant *h**f,exp* is presented as [29]:

,

(14)

Where *l*c is contact length, *ρ*w is workpiece density, *c*w is specific heat conductivity, *T*ch is chip melting temperature, *β*w is thermal performance of workpiece, *k*g is thermal conductivity coefficient of abrasive particles, C is temperature factor, and *r**g* is the effective contact radius of abrasive particles (20 µm in this article) [20].

Figure 7 shows the effect of grinding speed *v*s and depth of cut *a*p on coolant heat transfer coefficient according to Eq. 14. The coolant heat transfer coefficient increases with the rises of wheel speed and decreases of cutting depth, which shows the same rule as the "fluid wheel" model (see Eq. 12) and laminar flow model (see Eq. 13). Compared with the bar chart, suggesting the coolant heat transfer coefficient of UVAG is higher than CG, by 1.87%-11.2%. Meanwhile, the strengthening effect of ultrasonic vibration on heat transfer coefficient is weakened as the grinding speed increases. Here, the reduction of grinding temperature caused by UVAG is that ultrasonic vibration improves the convective heat transfer capacity of coolant, leading to the reduction of heat flux allocated to the workpiece. Another important reason for the lower grinding temperature of UVAG shows that the ultrasonic vibration improves the convective heat transfer capacity of the coolant, leading to a decrease in heat flux allocated to the workpiece.

During grinding processes, the coolant in the grinding arc zone is normally regarded as the laminar fluid with the same grinding speed [28, 29], and its convective heat transfer capacity is usually measured with the thickness of the thermal boundary layer. Generally, the heat transfer coefficient with ultrasonic vibrating effects increases 20%-400% compared to traditional processes due to the strengthening effect on fluid heat transfer of vibrations. Here, the effect degree is depended on the vibration intensity and vibration system [30–32]. Fig. 8 illustrates the convection heat transfer mechanism of coolants under the influence of ultrasonic vibrations. The implosion of cavitation bubbles near the solid-liquid interface will destroy the thermal boundary layer and velocity boundary layer due to the cavitation or acoustic flow under low-frequency ultrasonic vibration of 20-100 kHz, reducing the thermal resistance and producing micro-turbulence. Meanwhile, the wall of workpiece vibration will disturb the laminar coolant flow boundary layer near the grinding arc surface and then increase the turbulence intensity. In this case, the fluid thermal boundary layer thickness in the grinding arc zone becomes thinner and the heat transfer capacity of the coolant in the grinding arc zone is thus enhanced. However, the strengthening effect of vibration is not obvious under a higher grinding speed due to the thinner wall of thermal boundary layer and greater ability of heat transfer, resulting from the larger flow Reynolds number and turbulence of the coolant. This reveals that the ability of ultrasonic cooling fluid heat transfer will decreases as the grinding speed raises. Besides, Fig. 4 and Eq. 5 show that "cutting- separating" phenomenon exists during the removal process of abrasive grains. At the separating stage, the gap between grinding wheel and workpiece increases, and thus the coolant flows into the sliding surface between the grain and workpiece, contributing to the heat dissipation of the contacting interface.

## 3.4 Wear surface topography

Figure 9 describes the typical SEM morphology and associated schematic images of the wheel wear surface under CG and UVAG processes, aiming at revealing the effect of ultrasonic vibrating process on wear behaviors of grinding wheels. Seen from the Figs. 9a and 9b, the material adhesion can be clearly observed under CG processes, and the top surface of the abrasive grains is adhered with the cloud-like grinding material. Here, the chip and adhesive are mixed together. As illustrated in Fig. 9c, the long chips generated during CG processes are more difficult to remove from the wheel-workpiece interface than the shorter chips. In addition, the cutting point temperature of abrasive particles is much higher than the average grinding temperature during removing materials [33]. In this case, the adhesion of chips is easy to observed on the top of abrasive grains under the action of thermal and mechanical. On the other hand, when the ultrasonic vibration is applied in grinding processes, no adhesion wear is found on the top of abrasive grains. However, the grain micro-fracture can be observed, as shown in Figs. 9c and 9d. As depicted in Fig. 9f, the short chips are easier be produced and escaped from the grinding zone due to the “intermittent cutting” of abrasive particles. Meanwhile, the coolant can enter and cool the machined surface during the separation period for UVAG, and the cutting point temperature of abrasive particles is much lower than that of CG, consequently. Moreover, the multiple cutting edges on the top of abrasive grains are prone to produced owing to the influence of alternating loads, which can effectively keep the sharpness of the abrasive grain and thus reduce the grinding temperature.

## 3.5 Ground surface topography

Figure 10 shows the typical ground surface topography detected by SEM for both grinding processes under the grinding condition: *v*s=25 m/s, *v*w=7.5 m/min, *a*p=25 µm. As illustrated in Figs. 10a and 10b, there are many surface defects for CG processes under the influence of mechanical and thermal actions, such as scratches, fracture, smearing, and redeposited materials. However, none of these surface defects can be observed on the surface after UVAG processes in Figs. 10c and 10d. Moreover, there are obvious traces of side flow, ploughing, and striation on the surfaces of UVAG. Because of the squeezing action between abrasive grains and workpiece, the long chip and large broken grain appear easily, and thus attributes to the formation of grinding scratches, fracture, and redeposited material on the surface of CG. However, the grinding temperature of UVAG is lower, the adhesion and diffusion wear are slight, and the abrasive particles are prone to form micro-fractures under ultrasonic impacts of workpiece [24]. Therefore, the grinding sharpness of abrasive grains for UVAG processes can be remained stable for a long term and thus the ground surface quality is greater than that of CG processes owing to the fewer grinding surface defects. In addition, the multiple micro ploughing and scratches can be produced due to the axial movement of the workpiece under multi-mode vibration, attributing to reduce the accumulation of removed materials and thus improve the grinding quality [34].