## 5 Analysis of Chatter Stability of Two Boring Bar Models
### Comparison of Chatter Stability with Different Stacking Sequences  
To analyze the influence of different stacking sequences on the chatter stability of the two boring bar models, we designed six stacking sequences (see Table 4). Each sequence contained eight layers. In the first two sequences, there were two 0 deg layers, two 45 deg layers, and four 90 deg layers. The latter four sequences included four 0 deg layers, two 45 deg layers, and two 90 deg layers. The chatter stabilities of the two boring bar models under these six stacking sequences were calculated. The resulting lobe curves are drawn and shown in Figure 9.  
Figure 8: Chatter stability lobes of the two models with different ply angles.  
Figure 9: Chatter stability lobes with different stacking sequences. (**a**) Chatter stability lobes of six typical sequences; (**b**) Chatter stability lobes of the latter four sequences.  
Figure 9a shows four curves. Under the same boring bar model, the differences between the chatter stability lobes of sequences 1 and 2 were very small. Moreover, the differences between the chatter stability lobes of sequences 3-6 were also relatively small. Figure 9b shows an enlarged section (section highlighted by dashed rectangle in Figure 9a of the 3 rd-6 th sequences of the lobe curves for the shear boring bar in Figure 9a.  
In Figure 9a, the chatter stability lobes of the two boring bar models are different for the different stacking sequences. The chatter stability curves of the 1 st-2 nd sequences (including four 90\({}^{\circ}\) layers) are below, and the chatter stability curves of 3 rd-6 th sequences (including four 0\({}^{\circ}\) layers) are above. The greater the number of 0\({}^{\circ}\) layers was, the greater the cutting depth was. Furthermore, the chatter stability lobes of the non-shear boring bar model are at the top, whereas the chatter stability lobes of the shear boring bar model are at the bottom. These results illustrated that the cutting depth decreased when the shear deformation and rotational inertia were considered. As shown in Figure 9b, though all the 3 rd, 4 th, 5 th, and 6 th stacking sequences included the same number of layers (i.e., eight layers), the cutting depths of the composite boring bars are different. The best chatter stability lobe was obtained by the 6 th sequence, followed by the 5 th sequence, the 3 rd sequence, and the 4 th sequence.  
The ultimate cutting depth in Figure 9a was obtained and is listed in Table 4. As shown in Table 4, the ultimate cutting depths of the shear boring bar model were significantly lower than those of the non-shear boring bar model. Comparing the ultimate cutting depths of the 1 st and 2 nd sequences, the value of the 2 nd sequence was larger (i.e., better). Similarly, for stacking sequences 3-6, the best ultimate cutting depth was for the 6 th sequence, which is consistent with the result shown in Figure 9b.  
Figure 10 shows that the chatter stability lobes of the two composite boring bar models were different for different L/D ratios. The lobe curve for the L/D ratio of 4.4 is at the top, the lobe curve for the L/D ratio of 8.9 is at the bottom, and the lobe curve for L/D = 6.7 is in the middle. Moreover, the chatter stability curve of the shear boring bar model (which is labeled with the subscript S) is lower than that of the non-shear boring bar model when the shear deformation and rotational inertia are taken into account. In particular, the larger the L/D ratio is, the smaller the difference between the chatter stability lobes of the shear model and the non-shear model become. When the L/D ratio is 8.9, the chatter stability lobes of the shear model and the non-shear model almost overlap. Thus, it can be concluded that the smaller the L/D ratio of boring bar, the greater the influence of shear deformation and rotational inertia on the chatter stability of boring bar is.