# Natural Frequency Prediction Method for 6R Machining Industrial Robot
## 4 Discussion  
Once the prediction model of natural frequency is constructed and verified, the prediction result can be used as optimization parameter to improve the machining performance. An application case of the prediction model is explained as follows.  
\(a_{2}\) and \(a_{3}\) control the two longest links, by which the basic configuration of robot is determined. Through VIP values, items involving \(q_{2}\) and \(q_{3}\) are demonstrated to act dominated role in prediction model. \(a_{4}\) and \(a_{5}\) decide the direction of robot flange, which affect the configuration less than \(a_{2}\) and \(a_{3}\). Taking M\({}_{2}\) for an example, the influence of \(q_{2}\) and \(q_{3}\) on natural frequency is specifically studied and illustrated in Figure 11. When \(q_{2}=-55^{\circ}\) and \(q_{3}=40^{\circ}\) (P\({}_{\rm max}\)), the maximum first order natural frequency is obtained, and the minimum appears when \(q_{2}=-2^{\circ}\) and \(q_{3}=-56^{\circ}\) (P\({}_{\rm min}\)).  
Three configurations near P\({}_{\rm max}\), P1 (\(q_{2}=-48^{\circ}\), \(q_{3}=50^{\circ}\)), P2 (\(q_{2}=-38^{\circ}\), \(q_{3}=45^{\circ}\)), and P3 (\(q_{2}=-28^{\circ}\), \(q_{3}=40^{\circ}\)) are chosen for milling test (as shown in Figure 12. The first order natural frequencies decrease in turn from P1 to P3 according to model M\({}_{2}\). The setup of milling test is shown in Figure 13. High speed  
Figure 11: Variation of natural frequency with \(q_{2}\) and \(q_{3}\).  
Figure 10: Construction quality of M\({}_{6400}\).  
milling with short and straight path are executed, so that robot configuration can be ignored in one milling path. Milling parameters are as follows: the spindle rotation frequency is 800 Hz, \(f\) is 2.4 mm/s, \(a_{e}\) is 1 mm, and \(a_{p}\) is 4 mm. Acceleration signals and milled surface are analyzed.  
Milling acceleration signals are treated with short-time fourier transform (STFT), spectrograms of the range 5-100 Hz are displayed in Figure 14. For all three configurations, several peaks are conspicuous around 50-90 Hz during the whole milling process. In low frequency stage, a peak about 16 Hz can be found in P2 and a 15 Hz peak for P3, which are corresponding to their predicted natural frequencies. That no conspicuous can be found in P1, may because the certain frequency is not significantly impacted by the milling parameters.  
As for the milled surface, an obvious tool recession appears at P3, as shown in the red frame in Figure 15, while the situations are better in at P1 and P2. In addition, the quality of milled surfaces at P3 is the poorest compared with P1 and P2, and the vibration phenomenon on milled surface is getting severer from P1 to P3, indicating that configuration with higher first order frequency may lead to better milling performance, as mentioned in [20]. Whether there exits practical relevance between the value of natural frequency and machining performance can be a further research topic to develop more application of natural frequency prediction.  
Figure 14: Short-time fourier transform (STFT) of vibration signals of P1–P3.  
Figure 12: Chosen robot configurations for milling test.  
Figure 13: Milling test system setup.