# Chapter and Stability Analysis of the Slender Composite Boring Bar with Constrained Damping Layer
## 4 Analysis and Discussion  
### Rigidity Analysis  
According to the above analysis, the lobe curves moved downwards with the increasing length-to-diameter ratio of the boring bar, accompanied by a decline in the minimum cutting depth and the increase of the corresponding rotating speed in the stable region. As the length of the boring bar increased, the static rigidity of the boring bar decreased, and the cutter head on the end of the boring bar cannot bear great force to avoid heavy deformation. Therefore, the cutting depth should be set as a low value to ensure safety. When the cutting depth remained unchanged, the required rotating speed of the main shaft in stable cutting decreased with the increase of the length-to-diameter ratio. However, the stable region expanded towards the region with high rotating speed. Accordingly, the conditional cutting depth range expanded with the increase of the rotating speed of the boring bar. It can also be observed from the lobe curves in Figure 4 that the boring bar with a composite base exhibited a greater cutting depth and a wider stable region than that with a metal base. However, the cutting depths of these boring bars dropped with the increasing length-to-diameter, while the areas of the stable regions increased with the increase of the rotating speed of the spindle. In particular, the carbon fiber/epoxy composite boring bar operated best in stability. According to Equation (31), the minimum cutting depth \(b_{\text{lim}}\) can be calculated. It was found that the minimum cutting depth was in direct proportion to the product of inherent frequency, the damping ratio, and the modal mass. The stability of the composite also depended on the ply angle of the carbon fiber beam. As shown in Figure 6, the stability of the composite dropped with the increasing ply angle, which can be attributed to the established mechanical model of the boring bar based on the Euler-Bernoulli beam theory. The boring bar was only subjected to horizontal force, and the carbon fiber beam can bear great bending stress at a ply angle of 0\({}^{\circ}\). Therefore, only the most stable condition at a ply angle of 0\({}^{\circ}\) was selected in this study for calculation. The minimum cutting depth is related to the related parameters in Equation (31) and the dynamic rigidity during the rotation of the boring bar. Previous studies have demonstrated that the maximum cutting depth is proportional to the product of the structure's damping ratio and the static bending rigidity when the angle of the cutting edge and the materials to be processed were the same. In other words, the maximum cutting depth was in direct proportion to the dynamic rigidity. The dynamic rigidity, denoted as \(K_{d}\), can be calculated according to Equation (32). \[K_{d}=0.2427L\xi_{1}\alpha_{1}^{2}(\rho A)_{\text{equiv}} \tag{32}\]  
Next, the stability characteristics of the boring bar were analyzed from the perspective of dynamic rigidity. Table 7, Table 8, Table 9, and Table 10 list the calculated dynamic rigidities of various boring bars. Through comparison, the dynamic rigidity of the boring bar with the carbon/epoxy base layer was greater than that of the boring bar with the graphite/epoxy base layer. The boring bar using Teflon damping materials also exceeded the boring bars with PMMA and POM damping materials in dynamic rigidity. The dynamic rigidity of the boring bar at a ply angle of 0\({}^{\circ}\) was greatest, while the dynamic rigidity of the boring bar at a ply angle of 90\({}^{\circ}\) was smallest. After the addition of the constrained layer, the dynamic rigidity of the boring bar was enhanced by 2.8 times, i.e., the boring bar with the constrained layer exhibited better stability.