# Analysis on Free Vibration and Stability of Rotating Composite Milling Bar with Large Aspect Ratio
## 4 Numerical Examples
### Stability Analysis  
The bars made of composite and steel as shown in Figure 3 with the ratio of \(L/2b=10\) were analyzed. \(K_{\rm s}\) is set to a typical value in milling process, i.e., \(K_{\rm s}=556.4\) Mpa [33]. Meanwhile, other parameters  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(L/2b\) & **Timoshenko Beam (Hz)** & **Euler Bernoulli (Hz)** & **Deviation (\%)** \\ \hline
5 & 1468.8373 & 1659.73402 & 11.50165 \\
6 & 1055.19671 & 1152.17921 & 8.41731 \\ \hline
7 & 792.19311 & 846.316 & 6.395116 \\
8 & 615.42  
are set as follows: \(\beta_{0}=73.3^{\circ}\), \(\varsigma_{\text{steel}}=0.01\), \(\varsigma_{\text{composite}}=0.015\). In addition, the damping values were assigned based on concepts from an oscillator with single-degree of freedom, that is  
\[c_{e}+c_{i}=\frac{2\zeta\omega_{n}M_{11}}{\int_{0}^{L}{(\xi_{1})^{2}\text{d}z}} \tag{40}\]  
When \(c_{i}/(c_{e}+c_{i})=0.5\), the stability lobes of both steel bar and composite milling bar are shown in Figure 6.  
Figure 6 shows that after composite material is used, the cutting depth limit is increased due to its high stiffness. At the lowest point of the rightmost lobe of the steel bar, the corresponding cutting depth and rotation speed are 0.063 mm and 647 rad/s, respectively. For the composite material, the corresponding cutting depth and rotation speed are 0.073 mm and 1124 rad/s, respectively. It can be seen from the results in Figure 6 that at a large aspect ratio, the cutting depth of the cutter bar made of composite material (material properties are listed in Table 1) is larger than the cutter bar made of steel by 15.87%. The increase in the cutting depth by the use of composite material is not significant. However, when the composite material is used, the increase in the rotating speed is up to 73.7% (from 647 rad/s to 1124 rad/s), which can significantly improve the cutting efficiency. The results in frequency domain shown in Figure 6 are calculated based on the theory in Reference [27] proposed by [PERSON]. The calculation results using the methods in this paper and in the reference are consistent, which demonstrates the reliability of the calculation results in this paper. Figures 7 and 8 show that as the value of \(\omega_{\text{c}}T\) jumps, chatter frequencies have abrupt changes. Physically speaking, \(\omega_{\text{c}}T\) represents the total angular displacement of the vibrating milling bar. During one rotation period, the milling bar vibrates at the chatter frequency. Meanwhile, \(k\) is the number of complete waves traversed by the tool. Therefore, \(\omega_{\text{c}}T\) is directly related to the phase difference between successive undulations on the work piece surface. Due to its larger natural frequency, the composite material has higher chatter frequency than steel, as shown in Figure 7. Meanwhile, at the same phase difference, the composite bar has larger rotating speed than the steel bar. Figure 9 shows the effects of the internal damping of materials. From the figure, when the internal damping is not considered, each lobe in the lobe diagram has the same lowest cutting depth. When the internal damping is considered, the lowest cutting depth increases with the rotating speed of the milling bar, as shown in the dot line. Furthermore, at low rotating speed, the influence of internal damping on stability is negligible. As the rotating speed increases, the influence of internal damping progressively becomes noticeable. At an extremely high rotating speed, the internal damping can induce an unstable zone. On the contrary, at a lower rotating speed, the stable zone can be obtained by larger internal damping. Therefore, the internal damping can improve the stability of the bar at a low cutting speed but cause instability at a high cutting speed. This conclusion is consistent with the results  
Figure 6: The effects of material on stability. in Reference [34], which indicates that the internal damping maybe unstable, depending on rotating speed.