# Research on Grinding Force of Ultrasonic Vibration-Assisted Grinding of C/SiC Composite Materials
## 3 Result and Discussion
### Refinement of Grinding Force Empirical Formula  
According to traditional cutting theory, there is a complex exponential relationship between grinding force and grinding process parameters. The classic grinding force formula is (1).  
Figure 11: End grinding: the influence of feed speed changes on the grinding force.  
Figure 12: (a) Side grinding force ratio. (b) End grinding force ratio.  
\[F_{t}=F_{p}a_{p}^{x}v_{s}^{y}v_{w}^{z} \tag{1}\]  
where \(F_{p}\) is a coefficient that combines the material to be processed and the grinding conditions. The exponential model of Equation (1) is nonlinear. In order to facilitate regression analysis, linearization is carried out first. After the regression, the regression equation can be obtained as (2):  
\[y=b_{0}+b_{1}x+b_{2}y+b_{3}z \tag{2}\]  
Encode the grinding amount level; the large value is \(+1\), and the small value is \(-1\). The grinding force test values of common grinding and UVAG in the side grinding and end grinding tests are all taken as natural logarithms. The results are shown in Tables 6-9, respectively.  
From the test data in Table 6, the four coefficient values of \(b0\), \(b1\), \(b2\), and \(b3\) in Equation (2) can be obtained through uniformity design software: \(b_{0}=1.36186,b_{1}=0.62,b_{2}=-0.3\), \(b_{3}=0.11\). Then, we find the variable values in Equation (2): \(x=0.62,y=-0.3,z=0.11\), \(F_{p}=10^{1.36186}=23.01\). Finally, the empirical formula of grinding force for common grinding during side grinding is (3).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Serial Number**} & \multicolumn{4}{c}{**Parameters Used for Grinding**} & \multicolumn{2}{c}{**Measured Value**} \\  & \multicolumn{2}{c}{**Parameters Used for Grinding**} & \multicolumn{2}{c}{**of Tangential Force**} \\ \cline{2-6}  & \(a_{\text{p}}\) (\(\mu\)m) & \(v_{\text{S}}\) (m/s) & \(v_{\text{W}}\) (mm/min) & \(F_{\text{f}}\) (N) & \(\log F_{\text{f}}\) \\ \hline Condition 1 & 10 & 1.256 & 50 & 0.878 & \(-0.05651\) \\ Condition 2 & 10 & 12.56 & 500 & 0.5729 & \(-0.2419\) \\ Condition 3 & 40 & 1.256 & 500 & 2.6882 & 0.4295 \\ Condition 4 & 40 & 12.56 & 50 & 1.047 & 0.01995 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Grinding force test data for side-common grinding.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Serial Number**} & \multicolumn{4}{c}{**Parameters Used for Grinding**} & \multicolumn{2}{c}{**Measured Value**} \\  & \(a_{\text{p}}\) (\(\mu\)m) & \(v_{\text{S}}\) (m/s) & \(v_{\text{W}}\) (mm/min) & \(F_{\text{f}}\) (N) & \(\log F_{\text{f}}\) \\ \hline Condition 1 & 5 & 1.256 & 50 & 0.9725 & \(-0.01211\) \\ Condition 2 & 5 & 12.56 & 500 & 0.5198 & \(-0.2841\) \\ Condition 3 & 20 & 1.256 & 500 & 3.3307 & 0.5225 \\ Condition 4 & 20 & 12.56 & 50 & 1.0247 & 0.0106 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Grinding force test data for end-common grinding.  
\[F_{t}=23.01a_{p}^{0.62}v_{s}^{-0.3}v_{w}^{0.11} \tag{3}\]  
In the same way, according to the test data in Tables 7-9, the UVAG force during side grinding, the common grinding force during end grinding, and the UVAG force during end grinding can be obtained in turn. The empirical formulas are (4)-(6).  
\[F_{t} =21.48a_{p}^{0.64}v_{s}^{-0.27}v_{w}^{0.13} \tag{4}\] \[F_{t} =58.75a_{p}^{0.69}v_{s}^{-0.39}v_{w}^{0.12}\] (5) \[F_{t} =42.95a_{p}^{0.71}v_{s}^{-0.27}v_{w}^{0.15} \tag{6}\]