## 4 Simulation and Experimental Validation
### Experimental Validation of Force Fluctuation Characteristics  
In this section, a milling experiment with a variable axial depth of cut is presented to verify the proposed theory of uniformity and periodicity and symmetry characteristics of force fluctuations. The milling experiment was conducted using the VMC-C50 machine tool (Shanghai TOP Numerical Control Technology Co., Ltd., Shanghai, China). The milling scheme is given in Table 2 and the experimental parameters are given in Table 3. In order to make the experiment more accurate, the values of the spindle speed and \(K_{t}\) given in Table 2 were revised to an average of repeated measurements. As shown in Figure 9a, the 9171A rotary dynamometer (Kistler, Winterthur, Switzerland) was used to collect the force fluctuation data of \(F_{t}\) and the sampling rate was 2 kHz. In order to avoid the noise from other signals, the dynamometer used a low-pass filter with a cut-off frequency of 300 Hz. The milling cutter had six helical-edges with an effective length of 40 mm, a radial rake angle of 8\({}^{\circ}\) and an axial rake angle of 12\({}^{\circ}\). The workpiece was made of aluminum alloy (A5052), which was pre-processed into a shape of 200 \(\times\) 10 \(\times\) 50 mm. Its chemical composition is presented in Table 4. The milling process was side milling and the other parameters used in the experiments are similar to the parameters in Section 4.2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Processing Parameters** & **Value** & **Cutter Parameters** & **Value** \\ \hline \(f_{t}\) & 0.02 (mm/tooth) & \(N_{t}\) & 6 \\ \(A_{pox}\) & 5.236 (mm) & \(\beta\) & 45\({}^{\circ}\) \\ \(A_{p}\) & 2.618–5.236 (mm) & \(R\) & 5 (mm) \\ \(A_{e}\) & 1.5, 2.5, 3.5 (mm) & \(K_{t}\) & 474 (N/mm\({}^{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the force fluctuation simulation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & \(A_{e}\) (mm) & \(A_{p}\) (mm) & **Feed Speed** (mm/min)** & **Spindle Speed (rpm)** \\ \hline
1 & 2.5 & 1 & 240 & 2279.8 \\
2 & 2.5 & 2.618 & 240 & 2279.8 \\
3 & 2.5 & 4.236 & 240 & 2279.8 \\
4 & 2.5 & 5.236 & 240 & 2279.8 \\
5 & 2.5 & 6.236 & 240 & 2279.8 \\
6 & 2.5 & 0–10 & 240 & 2279.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Milling scheme of the experiment. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Processing** & **Value** & **Cutter Parameters** & **Value** \\ \hline Milling type & Down milling & \(N_{t}\) & 6 \\ Cutting speed & 71.6 (m/min) & \(\beta\) & 45\({}^{\circ}\) \\ \(f_{t}\) & 0.017 (mm/tooth) & \(R\) & 5 (mm) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental conditions of the milling test. In particular, the change in the axial position of the milling cut presented in Figure 9b is demonstrated only for the purpose of easier understanding. During the milling experiment, the milling cutter was fixed at the position of 10 mm and the change in the value of \(A_{p}\) was caused by the shape of the workpiece, which was an oblique triangle shape and gradually became higher during milling process. Then, the five marking points of \(A_{p}\) = 1 mm, \(A_{p}\) = 2.6 mm, \(A_{p}\) = 4.2 mm, \(A_{p}\) = 5.2 mm and \(A_{p}\) = 6.2 mm were used to verify the theory of milling force fluctuation characteristics. The force fluctuations of the fast estimation method and the process of the milling experiment are presented in Figure 9b. As mentioned in Section 4.1, the cutting area varies with the axial depth of cut from zero to 10 mm, containing two periods, where each period is composed of the ascending and descending parts. The milling force fluctuates to the maximum on the symmetry axis and fluctuates to the minimum on the periodic axis. Then, a comparative experiment between the fast estimation method and the test force data was performed. The corresponding experimental data are shown in Figure 10. The frequency of the milling force was analyzed by the FFT method.