## 4 Simulation and Experimental Validation
### Accurate Simulation Method of Milling Force Fluctuation  
In order to predict the milling force fluctuations more accurately, a method for accurate simulation of the milling force fluctuations is proposed. The flowchart of this method is shown in Figure 7.  
As shown in Figure 7, at the beginning of the simulation, before calculation, it is required to determine the milling parameters, including the milling cutter parameters and processing parameters. The cutter parameters include the number of helical edges, the cutter radius, helix angle and cutter rotation angle, and the processing parameters include the radial depth of cut, axial depth of cut, feed per tooth and force coefficients. Next, the milling type is determined according to these parameters. The entrance and exit angles of different milling types are calculated using Equations (4) and (5), respectively. Then, two parallel calculations are carried out to improve calculation efficiency. By using different initial angles, the force generated at each angle is calculated by Equation (8) and stored in a database. After that, the force fluctuation of one period is calculated by Equation (9). Further, by using different initial values of \(A_{p}\), another parallel computation is performed to show the characteristics of force fluctuation at different axial depths of cut. Finally, the curves of force fluctuations and axial cutting depth are drawn based on the recorded data.  
Figure 7: Simulation process of force fluctuation.  
The error between the fast estimation model and the accurate simulation method is verified using the parameters given in Table 1. As mentioned below Equation (25), the research range of axial depth of cut is \((0,A_{pc}/2)\). But the zero point of \(A_{p}\) cannot be verified in the simulation; the range of \(A_{p}\) is selected to be at the descending area of the first period, where the range of \(A_{p}\) is \([A_{pc}/2,A_{pc}]\). To measure the influence of the radial depth of cut, three \(A_{e}\) of 1.5, 2.5 and 3.5 mm were selected in the experiment. The comparison of the results of the two proposed methods are presented in Figure 8, where it can be seen that the error of the fast estimation method is related to \(A_{e}\) and \(A_{p}\). At the values of \(A_{e}\) of 1.5, 2.5 and 3.5 mm, the maximum error of the fast estimation method has the values of 0.65%, 5.32% and 9.81%, respectively. Thus, the error is gradually increasing, which makes the fast estimation method more suitable for the smaller radial depth of cut. In addition, in Figure 8, the relationship between the error and \(A_{p}\) can be clearly observed. The smallest error is always at two points, \(A_{pc}\) and \(A_{pc}/2\). As shown in Figure 8a, as the value of \(A_{p}\) moves away from the symmetry axis, the maximum error appears at \(A_{p}\) = 3.63 mm and \(A_{p}\) = 4.86 mm. However, different conclusions can be drawn based on the results presented in Figure 8b,c, where the fluctuation error has only one maximum value at \(A_{p}\) = 4.4 mm. The reasons for the inconsistency of results presented in Figure 8a-c can be the extremely small estimation error and unsatisfactory simulation accuracy. Finally, based on the three comparison results, it can be concluded that the overall error of the fast estimation method is not more than 10%. Therefore, if the value of \(A_{e}\) is small and the accuracy requirement is low, then the fast estimation method is suitable and effective. Otherwise, the simulation prediction method will be more appropriate.  
Figure 8: Comparison of the fast estimation method and the simulation method: (**a**) \(A_{e}\) = 1.5 mm; (**b**) \(A_{e}\) = 2.5 mm; and (**c**) \(A_{e}\) = 3.5 mm.