## 5 Result Analysis
### FRF Analysis  
In the machining, the amount of material removal by each cutting step and the amount of the total material removal can be obtained from Figure 9. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Condition Number** & **Rotating Speed (rpm)** & **Axial Depth (mm)** & **Radial Depth (mm)** \\ \hline C1 & 5000 & 8 & 0–2 \\ C2 & 10,000 & 8 & 0–2 \\ C3 & 16,000 & 8 & 0–2 \\ C4 & 5000 & 12 & 0–2 \\ C5 & 10,000 & 12 & 0–2 \\ C6 & 16,000 & 12 & 0–2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cutting conditions. Figure 9: The amount of material removal changes with the cutting path. In Figure 9, the relationship between the amount of material removal \(m(\gamma)\) and the cutting path \(\gamma\) can be expressed as  
\[m(\gamma)=5.05\times 10^{-7}\gamma^{2}. \tag{17}\]  
The original frequency response functions at the midpoint of each cutting step before and after machining are obtained from the modal experiment, excluding the influence of the acceleration sensor on the frequency response function, so the modal parameters of each cutting step before and after machining are obtained, as shown in Table 3. According to the data in the table, the modal mass, modal damping, and modal stiffness of each cutting step can be obtained, as shown in Figure 10 by the Equation (11). As can be seen from Table 3, the natural frequency at the midpoint of the each cutting step before and after the machining remains constant, before machining the natural frequency is 1253 Hz, the after machining natural frequency is 1359 Hz. This shows that the structure of the workpiece is unchanged. Figure 10: The relationship between the dynamic characteristics of the workpiece and the tool path. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Position Coordinates**} & \(\omega_{l0}\) & \(\omega_{l\mu}\) & \(m_{l\mu}\) & \(c_{l0}\) & \(c_{l\mu}\) & \(c_{l\mu}\) & \(k_{l0}\) & \(k_{l\mu}\) \\  & (Hz) & (Hz) & (kg) & (kg) & (Ns/m) & (Ns/m) & (N/m) & (N/m) \\ \hline
**P1** & 1253 & 1359 & 0.0240 & 0.0192 & 1.42 & 0.94 & 37,477.2 & 35,408.3 \\
**P2** & 1253 & 1359 & 0.0312 & 0.0248 & 1.87 & 1.34 & 49,024.4 & 45,880.0 \\
**P3** & 1253 & 1359 & 0.0334 & 0.0261 & 2.04 & 1.49 & 52,424.6 & 48,185.8 \\
**P4** & 1253 & 1359 & 0.0312 & 0.0236 & 1.87 & 1.34 & 49,024.4 & 43,546.4 \\
**P5** & 1253 & 1359 & 0.0240 & 0.0178 & 1.42 & 1.03 & 37,747.2 & 32,841.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Modal parameters at different locations before and after machining. From the modal stiffness values at the midpoint of each cutting step before machining, it can be found that the modal stiffness value (37,747.2 N/m) on both sides of the workpiece is less than the modal stiffness values (52,424.6 N/m) in the middle position. It is well explained that the deformation of the workpiece on both sides is greater than the deformation of the workpiece in the middle position of the phenomenon. Comparing the value of the stiffness of the same position before and after machining, it can be found that the greater the amount of material removal there is (the workpiece becomes thinner), the more the modal stiffness of the workpiece decreases. At point P1, the modal stiffness value is 37,747.2 N/m before machining, and the value is 35,408.3 N/m after machining, with a stiffness decrease of 2338.9 N/m. At point P5, the modal stiffness value is 37,747.2 N/m before machining, the modal stiffness value is 32841.8 N/m after machining, and the amount of change is 4905.4 N/m. In addition, the variation of the modal mass values and the modal damping values before and after machining is basically the same along the machining path. In other words, the modal mass values and the modal damping values at the midpoint of the workpiece have reached the maximum and gradually decrease to both sides.