# Chapter and Stability Analysis of the Slender Composite Boring Bar with Constrained Damping Layer
## 5 Conclusions  
This study focuses on the chattering stability of the composite boring bar with a constrained damping layer during the boring process. Based on the Euler-Bernoulli beam theory, the regenerative chattering mechanical model of the composite boring bar with a constrained damping layer was established. Using the equivalent bending rigidity, the equivalent surface density, and the damping calculation method of the constrained damping boring bar, the expressions of the critical cutting width \(b_{\rm lim}\) and the rotating speed of the spindle \(n\) were derived. The main conclusions are described below.  
(1) The composite boring bar with a constrained damping layer exhibited a greater limit cutting width than the metal-boring bar. When using the slender boring bar for small and deep-hole processing, the base layer with different materials and thicknesses significantly affected the cutting stability of the boring bar even though the diameter (i.e., the thickness) of the base layer was quite small. The boring bar with the thicker base layer was more stable.  
Both the ply angle of the base and the thickness of the constrained layer imposed an obvious effect on the cutting stability of the composite boring bar. The boring bar with the constrained layer was stable at a ply angle of \(0^{\circ}\).  
(2) The material of the damping layer was also the main factor that affected the stability of the composite boring bar. The material of the damping layer significantly affected the cutting stability of the boring bar, even when the diameter of the damping layer was fairly small.  
Figure 9: Influence of the thickness of the restraint layer and the damping layer on chatter stability.  
(3) When various materials and the overall geometrical size of the constrained damping composite boring bar were unchanged, the stability of the boring bar first increased and then dropped with the increasing thicknesses of various layers. Accordingly, there exists an optimal thickness proportion of the thicknesses of various layers in the composite boring bar so as to reach the most favorable stability.  
(4) When using the boring bar with a large diameter-to-length ratio for deep-hole boring, the rotating speed of the spindle and the unconditional cutting depth should be set as low values to ensure boring stability. Meanwhile, the conditional cutting depth increased. Furthermore, the above conclusions were still applicable to the increasing length-to-diameter ratio.  
This work is part of a project that focuses on the study of chatter stability in cutting processes with a composite tooling system. The experimental verification of the presented model using real boring bars will be the subject of a study in the future.  
See Table 11 for the important parameters in the paper.  
**Author Contributions:** [PERSON] proposed the main idea and the validation method; [PERSON] and [PERSON] derived the equations in the model, coded the computer program, and obtained the calculations; [PERSON] wrote the manuscript and drew the figures; [PERSON] and [PERSON] helped perform the analysis with constructive discussions. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the National Natural Science Foundation of China (Grant No.11672166), and China Postdoctoral Science Foundation Funded Project (Grant No.2016M5922165).  
**Conflicts of Interest:** The authors declare that there is no conflict of interest regarding the publication of this paper.