## 4 Conclusions  
The dynamic behaviors of the flexible rotor-bearing system for ball end milling were first numerically investigated by using the GPEM. The tool-tip responses have been made more realistic due to consideration of the coupling effect of the flexible rotor-bearing system. Thanks to the gyroscopic effect of the flexible rotor, the important whirl map can be drawn. Then, it can be observed from the whirl map that, the critical speeds, which are directly proportional to the natural frequency and inversely proportional to the number of flutes, due to some natural modes, can be excited to significantly affect the tool responses. The flexible isotropic rotor-bearing system with rotating speed can excite the forward and backward modes along the excitation line in the whirl map to form two different critical speeds. In addition, the cutting forces, which depend, in part, on the tool-tip response, combine with the stiffness matrix of the rotor system to complicate the issue. To the best of the authors' knowledge, most previous studies on ball end milling always ignore the rotor system or assume that the rotor is rigid or discretized to a few degrees of freedom, thereby ignoring the gyroscopic effect of the flexible rotor. Unlike the previous literature, the natural frequency in this study varies with the spindle speed. Thanks to the gyroscopic effect of the flexible rotor, which of the different modes are included in the tool-tip response depend mainly on the spindle speed. The model established by considering the coupling effect of the flexible rotor and using GPEM can be more efficient and realistic in analyzing the dynamic behavior of the tool-tip. The important new findings of the present study, by using numerical analysis, are obtained as follows:  
1. In milling at a higher spindle speed (2160-19,950 rpm), the tool-tip oscillates with the cutting-force frequency combined with a longer cycle vibration of the first backward mode due to the coupling effects of the flexible rotor-bearing system, and the period of the long cycle vibration enlarges as the spindle speed increases. From the FFT plot, it can be seen that as the spindle speed becomes larger, the peak amplitude at the first backward mode (long-period vibration) becomes larger compared to the peak at the cutting force frequency.
2. Under lower spindle speed milling (960, 1320 rpm), the vibration trajectory of the tool-tip in the co-plane expands clockwise gradually around the origin until arriving at a stable loop. This is because only the first backward mode of the rotor-bearing system is excited.  
The authors of [18] calculated the critical speeds of the rotor-bearing system for ball end milling and verified the numerical results with the experimental data. However, the rotor was assumed to be rigid and the lateral rotation was neglected, so the gyroscopic effects of the system could not be included. Therefore, the critical speed was independent of the spindle speed and only one mode was calculated. In this study, the critical speeds of the flexible rotor-bearing system vary with the spindle speed due to the gyroscopic effect. The vibration responses of the tool-tip, including forward and backward modes, are numerically calculated. However, it is also recommended that future work is required to add a comparison by setting up an experiment to validate the presented results.  
Conceptualization, C.-J.-H., J.-R.C., T.-N.S. and K.-H.C.; formal analysis, C.-J.H. and J.-R.C.; investigation, K.-H.C.; methodology, C.-J.H., T.-N.S. and K.-H.C.; software, K.-H.C.; supervision, T.-N.S.; validation, [PERSON]-J.H. and [PERSON]-R.C.; writing--original draft, J.-R.C. and K.-H.C.; writing--review and editing, C.-J.H., [PERSON]-[PERSON] and T.-N.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.