## 3 Analysis of Vibration Mode Shapes  
The vibration mode shapes of the shear boring bar are compared with those of the non-shear boring bar. A carbon composite material was selected as the material to construct the boring bar. The macroscopic mechanical parameters of the carbon composite are shown in Table 2. The boring bar length L was set to 0.6 m, the inside radius \(R_{i}\) was set to 0.043 m, and the outside radius \(R_{o}\) was set to  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(i\) & **1** & **2** & **3** & **4** \\ \hline \(\lambda_{i}L\) & 1.8751 & 4.6941 & 7.8548 & 10.9955 \\ \(\Omega_{i}\) & 3.1914 & 14.026 & 30.066 & 46.018 \\ \hline \hline \end{tabular}
\end{table}
Table 1: \(\lambda_{i}L\) and \(\Omega_{i}\) for non-shear boring bar and shear boring bar, respectively.  
Figure 2: Relationship between the material principal direction 1 and the Z axis of boring bar coordinate system.  
0.045 m. The ply mode was set to the same angle and contained eight layers (denoted as \([\theta]_{8}\)). The ply angle \(\theta\) was set to 45\({}^{\circ}\). The thickness of each layer was set to 0.25 mm.  
According to Equation (23), after 14 iterations, the accuracy was adequate. The result was \(\Omega_{1}=3.379317\). The accuracy was calculated as follows:  
\[\left|\Omega_{1}^{14}-\Omega_{1}^{13}\right|=0.000008\leq 0.00001 \tag{33}\]  
By substituting \(\Omega_{1}\) into Equation (25), one can obtain Equation (34). To meet the accuracy requirement, we selected the first 14 terms to compose the modal function, as follows:  
\[\begin{array}{l}\widetilde{Y}_{1}^{14}(Z)=0.35497\mathrm{Z}+2.93914Z^{2}-1.3 9423Z^{3}-0.11458Z^{4}+6.27048\times 10^{-2}Z^{5}\\ \hskip 14.226378 pt+8.48415\times 10^{-2}Z^{6}-1.75834\times 10^{-2}Z^{7}-1.40259 \times 10^{-3}\times Z^{8}+3.25191\times 10^{-4}Z^{9}\\ \hskip 14.226378 pt+1.78538\times 10^{-4}Z^{10}-2.39683\times 10^{-5}Z^{11}-1.83 380\times 10^{-6}Z^{12}+2.64660\times 10^{-7}Z^{13}\\ \hskip 14.226378 pt+8.03156\times 10^{-8}Z^{14}\end{array} \tag{34}\]  
Under the same settings of parameters (boring bar length L, radius \(R_{i}\) and \(R_{o}\), ply angle \(\theta\), etc.), the vibration mode function of non-shear boring bar is calculated by Equation (28). Therefore, based on the Equations (28) and (34), we plotted the mode shapes of the non-shear boring bar and shear boring bar using the MATLAB software, respectively. The four vibration mode shapes are shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & **E1 (Gpa)** & **E2 (Gpa)** & **G12 (Gpa)** & **G23 (Gpa)** & **v12** & **v23** & \(\rho\) (kg/m3)** \\ \hline Carbon & 181.0 & 10.3 & 7.17 & 3.78 & 0.28 & 0.30 & 1.76 \\ Aramid & 76 & 11.5 & 6.3 & 4.4 & 0.30 & 0.34 & 1.44 \\ Graphite & 25.8 & 8.7 & 3.5 & 3.5 & 0.34 & 0.32 & 1.672 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Macroscopic mechanical parameters of composite materials.  
Figure 3: Four vibration mode shapes of the shear boring bar and non-shear boring bar. (**a**) First-order mode shape; (**b**) Second-order mode shape; (**c**) Third-order mode shape; (**d**) Fourth-order mode shape.  
Figure 3 shows that the first vibration mode shapes of the shear boring bar almost overlapped with the first-order non-shear mode shape of the boring bar. However, for the second-, third-, and fourth-order modes, the spacing between the curves of the two kinds of boring bars increased gradually, and the curves no longer coincided. Thus, it can be concluded that the vibration mode shapes of the boring bar are influenced by the shear deformation and rotational inertia.