# Analysis on Free Vibration and Stability of Rotating Composite Milling Bar with Large Aspect Ratio
## 4 Numerical Examples
### Stability Analysis
Figure 10 shows the displacement response of the points of A, B and C in Figure 9 at different internal damping. The divergence and convergence results of the displacement responses indicate the existence of the unstable zone, which can be initiated by internal damping at an extremely high cutting speed. If rotation is not considered, this phenomenon will not occur. Therefore, the instability of internal damping at high speed is caused by the interaction of rotation and internal damping. Figure 8: The variation of composite bar’s phase difference between successive undulations with the rotating speed. Figure 7: The variation of composite bar’s chatter frequency with the rotating speed. Figure 9: The influence of internal damping and external damping on stable zone for composite milling bar. Figure 11 shows the variation of instability zones with the values of external viscous damping when \(c_{i}=0\). The results indicate that both the isotropic milling bar and the anisotropy milling bar have the same trend. In addition, as the damping ratio increases, the stable cutting depth is larger. At all rotating speeds, when the external viscous damping intensified from \(\zeta=0.015\) to \(\zeta=0.02\), the stable cutting depth increased by approximately 25%. Figure 10: Bending displacement response curves of Point A, B and C: (**a**) Point A (\(\Omega=2000\) rad/s); (**b**) Point B (\(\Omega=5000\) rad/s ); (**c**) Point C (\(\Omega=7500\) rad/s). It is worth noting that some peculiar peaks appeared near 450 rad/s in the cutting depth curve of the composite milling bar with the damping ratio 0.025 in Figure 11a and near 750 rad/s in the cutting depth curve of steel milling bar in Figure 11b. A similar phenomenon can be seen in Figure 6. This phenomenon might be caused by the following two factors: (1) the accuracy of calculation method; (2) the stability change at different values of damping parameters. The influence of damping parameters on the stability is similar to the influence of radial immersion [35]. The exact reason needs experimental verification. If the second factor is the reason for these peculiar peaks, the accuracy of the time domain method is higher than that of the frequency domain method. Figure 12 shows the effects of ply angle on the composite bar with \(L/2b=10\), \(c_{i}/(c_{e}+c_{i})=0.5\) and lamination schemes \([\eta]_{36}\). It can be seen that when \(\eta=0\), the composite bar has the largest stable zone. This result can be explained by the fact that when the fiber orientation is closer to 0, the elastic modulus of composite is higher (as shown in Table 1), which results in greater stability due to the intensification of dynamic stiffness of the composite bar. However, composite milling bar with the ply angle of 0 might fail during cutting. The analysis results in this paper only focused on the bending deformation, but torsion should also be considered in practical work. The milling bar with the ply angle of 0 has the weakest strength of torsion deformation, which may cause damage in machining. The above stability analysis is based on a fixed natural frequency value at the rotating speed of zero. When the influence of tool bar rotation is considered, the natural frequency value is variable. Figure 13 shows the stability analysis results when the natural frequency changes without considering the internal damping. It can be seen that when the natural frequency is used as the backward whirl  
Figure 11: The influence of viscous damping on stable zone for (**a**) composite milling bar and (**b**) steel milling bar. Figure 6.0 The influence of ply angle on stable zone. frequency, the cutting depth is significantly reduced. Therefore, the rotation effect plays a negative damping role in the stability analysis.