## 4 Simulation and Experimental Validation
### Experimental Validation of Force Fluctuation Characteristics
As shown in Figure 10a, there are two main frequency bands, one is the tooth passing frequency at 227.8 Hz and the other is the tool runout frequency at 37.9 Hz. The difference between two frequency is 6 times, which is exactly equal to the number of cutting edges [25]. For the purpose of the obvious display of force fluctuations generated by each cutting edge, the force fluctuation data at tool pass frequency were extracted by a band-pass filter with a frequency range of 220 Hz to 235 Hz and the force results are shown in Figure 10b. The experimental results were consistent with those of the fast estimation method. The maximum fluctuation of the milling force was 23.12 N and it was observed at two positions on the symmetry axis, at \(A_{p}\) = 2.62 mm and \(A_{p}\) = 7.85 mm; meanwhile, the minimum fluctuation was 1.08 N and it was observed at the position of \(A_{p}\) = 5.23 mm on the period axis. Due to the accuracy of the experiment, it is acceptable that the minimum value is not zero. Both the fast estimation results of Figure 10a and the experimental results of Figure 10b show the proposed force fluctuation characteristics. Furthermore, the experimental results in Figure 10b show that the force fluctuation characteristics with the axial depth of cut from zero to 10 mm can be expressed by the part with the axial depth of cut from zero to 2.6 mm. Therefore, the experimental result also verified that 75% of the experimental workload in Figure 9b can be reduced. By \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Chemical Composition (\%)**} & \multicolumn{1}{c}{**Webster Hardness**} \\ \hline
**Si** & **Fe** & **Cu** & **Mn** & **Mg** & **Cr** & **Zn** & **Al** & **(HW)** \\ \hline \textless{}0.25 & \textless{}0.4 & \textless{}0.1 & \textless{}0.1 & 2.2–2.8 & 0.15–0.35 & \textless{}0.1 & remain & \(11\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Chemical composition and hardness of the A5052 aluminum alloy. Figure 9: Experimental validation of the force fluctuation characteristics: (**a**) experimental environment and (**b**) description of the milling process and the fast estimation method. applying the proposed force fluctuation theory, the milling force fluctuations of all depths of cut can be represented by the milling force fluctuation characteristics of the first half period (\(A_{poc}/2\)). Then, as the research range of \(A_{p}\) increases, more testing workload can be saved. When the axis position of the force fluctuation on the force curve was determined in Figure 10b, the positions of the marking points in the force curve were determined based on the time interval and axis position. The values 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 on the curve corresponded to the times of 13.2, 18.7, 24.1, 27.4 and 30.6 s, respectively. With the use of the same time label, the marked points were projected onto the force fluctuation data of the milling experiment, as shown in Figure 10c. The milling force in Figure 10c was used to determine the difference between the simulation method and the experimental test. The results showed that the average error of the simulation method was less than 3 N and this method was more accurate than the fast estimation method. In order to reduce the force fluctuation caused by the cutter runout, the collected data were filtered with a band-pass filter of a cutter runout frequency of 33 to 42 Hz. The milling force near the marked points was extracted and the obtained results of uniformity, periodicity and symmetry characteristics of force fluctuations are displayed in Figure 11a-c respectively. The results presented in Figure 11a verify the uniformity characteristics of the milling force fluctuation theory. The minimum value of the milling force fluctuation with \(A_{poc}\), which was calculated by Equation (14), was compared with the actual measured milling force fluctuation.