# Analysis on Free Vibration and Stability of Rotating Composite Milling Bar with Large Aspect Ratio
## 4 Numerical Examples  
### Free Vibration Analysis  
Using the motion equations derived in this paper, the rotating bar's free vibration characteristics were analyzed, as shown in Figure 3. The material parameters are listed in Table 1. Meanwhile, two milling bars made of different materials, i.e., the isotropy (steel) and anisotropy (boron-epoxy, (0\({}_{6}\)/90\({}_{3}\)/ \(\pm\) 20\({}_{3}\)/90\({}_{3}\)/0\({}_{6}\))) material were analyzed. The fundamental natural frequencies, which were critical to the stability, were calculated and compared. Figure 4 shows the variation of fundamental natural frequencies of bars with aspect ratio (_L_/_2_b_). The results were also compared to the reports by [PERSON] in 1999 based on [PERSON] beam theory [29].  
As shown in Figure 4, the fundamental natural frequencies of steel bar and composite bar gradually decreased with the increase of _L_/_2\(b\) ratio. In addition, the natural frequencies of composite bar were always higher than the steel bar. For isotropic steel bar, when _L_/_2\(b\) is larger than 5, the calculation results by Euler Bernoulli beam theory have high accuracy. However, for the anisotropic composite bar, in order to ensure the calculation accuracy, _L_/_2\(b\) should be larger. As shown in Table 2, for composite material, _L_/_2\(b\) should be larger than 8 to ensure the error less than 5%, which is the acceptable error of calculation results in many mechanical engineering fields. When the value of _L_/_2\(b\) is small, the results of the two beam models are not in a close agreement, which is due to the significant effect of the  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Boron-Eposxy** & **Steel** \\ \hline E\({}_{11}\) (GPa) & 211 & 207 \\ E\({}_{22}\) (GPa) & 24.1 & 207 \\ G\({}_{12}\) = G\({}_{13}\) (GPa) & 6.9 & 80 \\ G\({}_{23}\) (GPa) & 6.9 & 80 \\ \(\mu_{12}\) & 0.36 & 0.2975 \\ \(\rho\) (kg/m\({}^{3}\)) & 1967 & 7700 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of composite materials [30].  
Figure 4: The influence of aspect ratio on fundamental frequency (\(\Omega\) = 0).  
Figure 3: Composite milling bar (\(b\) = 15.875 mm, \(t\) = 11.1125 mm).  
transverse shear at small aspect ratio. Therefore, Euler beam theory should be used to analyze the composite milling bars with a larger aspect ratio.  
When the effects of transverse shear are not considered, the established model by Galerkin method in this paper has good convergence. The natural frequencies of the bar (\(L/2b=15\)) with different number of modes were calculated, as listed in Table 3. The results of the first three modes are very close, indicating that only one mode shape is enough for convergence.  
Due to the influence of rotating inertial force and gyroscopic force, the natural vibration frequency can be divided into forward whirl frequency and backward whirl frequency. When the backward whirl frequency is equal to 0, the rotating speed is the critical speed of the cutter bar. When the \([\eta]_{36}\) (\(\eta\) is shown in Figure 1) laying mode is adopted, the natural frequency of the cutter bars with differentply angles changes with the rotation speed, as shown in Figure 5. When the ply angle is 0\({}^{\circ}\), the critical speed of the cutter bar has the largest value.