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(1) Since the space curve of the cutting edge of the ball end mill can be subdivided into a number of tiny units, and each small unit can be regarded as a straight edge cutting tool, the cutting of the curved cutting edge of the ball end mill can be regarded as Many straight edge cutters are cutting at an angle. In bevel cutting, the cutting force of the cutting edge can be divided into two parts: 1 Cutting force related to the cutting area; 2 Cutting force related to the length of the cutting edge. In order to calculate the cutting forces of these two parts, the corresponding cutting parameters on the tiny unit can be calculated according to the cutting conditions, and the calculation coefficient of the corresponding cutting force can be calculated from the cutting database. The calculation equation is 
(2) For each micro-cutting unit, the three components dFpij, dFqij and dFrij of the diagonal cutting force can be obtained by using the above-mentioned calculation coefficients. 
(3) Switch the cutting forces in three directions to the X, Y, and Z axes, respectively. 
(4) A cutting force on one tooth can be obtained by summing the cutting force components formed by all (m) micro-cutting units, and then the cutting forces of all the cutting teeth on the milling tool can be accumulated to obtain the ball-end milling. The cutting force and torque of a knife at a certain moment can be expressed by the general formula 
(5) 
(6)
Table Numerical simulation of cutting conditions Item Content Workpiece material S1214 Easy-cutting steel tool material Coating HSS method Forward angle 0°, 5°, 10° Tool radius (mm) 8,10, 12.5 Number of teeth 2, 3, 4 axes Cutting depth 0.25R, 0.5R, R, 1.25R Radial cutting width 0.5R, 2R Cutting speed (r/mm) 200, 250, 315 Feed rate (in/min) 2, 3, 4 3 Numerical simulation Cutting conditions The numerical simulation study of the computer-prediction model of the cutting force of the ball-nose cutter described above can be used to further clarify the influence of various process parameters and structural parameters on the cutting force components, the momentary torque value and the mean value of the milling cutter. In order to make the prediction research of the ball-end cutter consistent with the actual production and representative, we have selected the corresponding cutting conditions according to the recommended parameters of the relevant manual and the information provided by the tool manufacturer, see the right table. Using the above groups of cutting conditions, the influence laws and characteristics of various cutting parameters on the instantaneous value and average value of the cutting force components and torque of the milling cutter can be studied. 4 Analysis of numerical simulation results Based on the above cutting conditions, the established mathematical model was used to determine the geometric parameters and cutting parameters at each tiny unit on the cutting edge, and the calculation coefficients of the two part cutting forces of the small unit were calculated from the cutting database. K value; then find the three components of the miter cutting force of the small unit; and finally substituting the formula (4)-(6), you can obtain the three components of the cutting force and torque of the cutting edge of an instant ball end mill. . Average cutting force and torque change trends The average cutting force component and the average torque are obtained by averaging all the instantaneous values ​​of the ball-nose cutter within a week. Their trend of change with the main process parameters is shown in Figure 1. In order to study the trend of the average cutting force and torque of ball-end cutters, we set a set of nominal cutting conditions and independently change each of them. The nominal cutting conditions are: milling cutter method forward angle gn=0°, cutter radius R=10mm, number of teeth Nt=3, axial depth ap=5mm, radial width ar=5mm, nominal depth aw = 5.886 mm, feed per tooth ft = 0.054 mm, cutting speed 315 r/min (Vmax = 17.14 m/min). From Figures 1a, b, and c, it can be seen that the average cutting force and torque of the milling cutter are linearly related to the feed per tooth ft, the number of teeth Nt and the depth of cut ap. When these three parameters increase, the average cutting force and torque are basically the same. It also showed a linear increase. 
Figure 1 The trend of average cutting force and torque under ideal conditions From Figure 1d we can see that when the radial cut width ar increases, the torque and Fy, Fz will increase with it, and Fx will decrease after a period of increase, reason It is at this moment that there are two cutters that participate in the cutting at the same time and the direction of the cutting force of the two cutter teeth is opposite in the X direction, resulting in that some of the cutting forces cancel each other out. From figures 1e, f, and g, it can be seen that when the tool radius R, the normal angle gn, and the cutting speed V increase, the average cutting force and torque will decrease, which is consistent with the bevel cutting theory. Cutting force and torque waveforms are numerically simulated using a computer-based prediction model of a ball-end milling cutter to predict the shape and amplitude of the cutting force and torque waveforms for each tooth of the milling cutter and for the entire milling cutter. The cutting force and torque characteristics of the cutter teeth are very useful. Figure 2 shows the three force components of the cutting force in the X, Y, and Z directions and the torque waveform. The curve 1 in the figure shows the waveform of the unilateral cutting mode (ar=5mm). There is a maximum area of ​​cutting force and torque near the cutting angle of each tooth. When the tooth is not in contact with the workpiece, the cutting force and torque are minimized. And zero; curve 2 represents the waveform of the slot milling mode. Sometimes two blades cut at the same time, sometimes only one blade cuts, which results in a significant change in cutting force and torque waveform. 
Fig. 2 Cutting Force and Torque Waveforms of Ball End Mill Under Ideal Conditions Variations in Cutting Force Waveform Characteristics The influence of cutting conditions on the maximum, minimum, mean, force region, and fluctuation index of cutting force for ball-end milling cutters is studied. It is of great significance to improve the cutting performance of ball end mills. Figure 3 shows the influence of different process parameters on the Fx and its fluctuation when the axial depth of cut changes continuously. The number of teeth Nt, the forward angle gn, and the radius R of the milling cutter are parameters of discrete change, while the axial Depth ap is a continuously varying parameter. The change pattern of Fy and Fz is similar to this and will not be repeated. 
Fig.3 The influence of various process parameters on the Fx component fluctuations.5 Conclusion The numerical simulation study of the cutting force, torque and cutting power of the flat rake face ball-nose cutter under different cutting conditions has achieved the ball under different cutting conditions. Average cutting force and average torque value of head cutters. Trend and regularity. Through the simulation of the shape and amplitude of the cutting force and the torque of each cutter tooth and the whole milling cutter of the ball-end milling cutter, the characteristics of the cutting force and the torque of each cutter are obtained and the changing rules are obtained. Through the numerical simulation study, we further understand the characteristics and trends of different cutting conditions on the maximum, minimum, average, force region and fluctuation index of ball-end milling cutter cutting force. The above research work has further improved and improved the computerized prediction model of the ball-end milling cutter. It is of great significance to the optimization of the cutting amount and the monitoring of cutting parameters in the cutting process. This research also has research on the same type of ball-end milling cutter. Reference value.
Numerical Simulation of Computer Prediction Model for Cutting Force of Ball End Milling Cutter
1 INTRODUCTION The flat rake face ball-end cutter has the advantages of simple structure, easy manufacture, and easy re-grinding. It has been widely used in the milling of workpieces with complex three-dimensional surfaces. Based on computer prediction models of cutting force, torque and cutting power of ball-end cutters under different cutting conditions, we conducted numerical simulations on the cutting performance of flat-face ball-end cutters. By analyzing the influence of various parameters on the cutting performance, the cutting force and torque characteristics of the ball-end cutter under different cutting conditions and the changing trend were obtained. 2 The mathematical model of the computer forecasting model is based on the rectangular coordinate system and the mutual geometric relationship of the flat rake face ball-end milling cutter. The equation for the cutting edge with the forward angle gn is established as