# Chapter and Stability Analysis of the Slender Composite Boring Bar with Constrained Damping Layer
## 4 Analysis and Discussion
### Estimation of the Optimal Thicknesses of Various Layers  
According to the analysis in Section 4.1, the stability of the boring bar is closely related to the change of the dynamic rigidity. Based on the theory of structural loss factor and viscous damping theory of the constrained damping beam, the loss factors is twice of the first-order damping coefficient \(\xi_{1}\), a factor that affects the change of dynamic rigidity. \[\eta=2\xi_{1} \tag{33}\]  
Substituting Equation (33) into Equation (32), the expression of dynamic stiffness is obtained. \[K_{d}=0.12135L\eta a_{1}^{2}(\rho\text{A})_{\text{equivv}} \tag{34}\]  
Substitute Equation (18) into Equation (34), and combine Equation (19) to get Equation (35). \[K_{d}=0.12135L\omega_{1}^{2}(\rho\text{A})_{\text{equivv}}\alpha\sqrt{\frac{1} {1+\alpha^{2}}}\frac{1}{(1+\frac{2}{\xi_{2}})\sqrt{1+S_{2}}} \tag{35}\]  
By substituting Equation (20) into Equation (35), the dynamic stiffness related to the geometric dimension of the composite boring, the material parameters, and the vibration characteristics can be expressed. \[K_{d}=0.12135L\omega_{1}^{2}(\rho\text{A})_{\text{equivv}}\alpha\sqrt{\frac{1 }{1+\alpha^{2}}}\frac{1}{(1+\frac{E_{1}R_{1}^{4}+E_{3}(R_{2}^{4}-R_{2}^{4})}{ 2E_{3}(R_{3}-R_{2})\left(\frac{E_{3}+R_{2}}{2}\right)^{3}})}\sqrt{1+\frac{4E_{ 3}(R_{3}-R_{2})\left(\frac{E_{3}+R_{2}}{2}\right)^{3}}{E_{1}R_{1}^{4}+E_{3}(R_ {3}^{4}-R_{2}^{2})}} \tag{36}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **\(L\)(m)** & \(\xi_{1}\) & \(\omega_{1}\)**(rad/s)** & **(\(\rho A\))\({}_{\text{equivv}}\)(kg)** & \(K_{d}\)**(N/m)** \\ \hline Carbon/Epoxy & 0.1 & 0.024528 & 1756.2 & 0.0266 & 48.8185 \\ Graphite/Epoxy & 0.1 & 0.02357 & 1694.8 & 0.0270 & 44.3639 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The dynamic characteristics of different matrix materials. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **\(L\)(m)** & \(\xi_{1}\) & \(\omega_{1}\)**(rad/s)** & **(\(\rho A\))\({}_{\text{equivv}}\)(kg)** & \(K_{d}\)**(N/m)** \\ \hline PMMA & 0.1 & 0.024518 & 1737.3 & 0.0257 & 46.1570 \\ Teflon & 0.1 & 0.024528 & 1756.2 & 0.0266 & 48.8185 \\ POM & 0.1 & 0.01228 & 1578.8 & 0.0300 & 22.2866 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The dynamic characteristics of damping materials. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\(\theta\)** & **\(L\)(m)** & \(\xi_{1}\) & \(\omega_{1}\)**(rad/s)** & **(\(\rho A\))\({}_{\text{equivv}}\)(kg)** & \(K_{d}\)**(N/m)** \\ \hline
0\({}^{\circ}\) & 0.1 & 0.024528 & 1756.2 & 0.0266 & 48.8185 \\
30\({}^{\circ}\) & 0.1 & 0.024553 & 1732.4 & 0.0266 & 47.5721 \\
60\({}^{\circ}\) & 0.1 & 0.024583 & 1704.5 & 0.0266 & 46.1084 \\
90\({}^{\circ}\) & 0.1 & 0.02458 Through the above analysis, when the maximum size of the boring bar \(R_{3}\) was unchanged, the change ranges of the thicknesses of the base layer and the damping layer (\(R_{1}\) and \(R_{2}\)) were determined, i.e., the optimization of the thickness of various layers can be transformed from Equation (22) into solving the optimal value of \(\max\{K_{\rm d}\}\) at the boundary described in Equation (37). \[\left\{ \begin{array}{l}0\leq R_{1}\leq R_{3}\\ R_{1}\leq R_{2}\leq R_{3}\end{array} \right. \tag{37}\]  
When the hanging length \(l\) of boring bar is 100 mm and the outer diameter \(R_{3}\) is 3 mm, the radii of the base layer and the damping layer in the composite boring bar are calculated as \(R_{1}\) = 0.4828 and \(R_{2}\) = 1.0172 mm, respectively. The optimal thicknesses of various layers can be calculated as \(H_{1}\) = 0.4828, \(H_{2}\) = 0.5344, and \(H_{3}\) = 0.4828 mm. In a proportional format, \(H_{1}\):\(H_{2}\):\(H_{3}\) = 1:1.11:1. As shown in Figure 9, when the thicknesses of the constrained layer and the damping layer are changed simultaneously, but the sum is fixed, the lobe curves of the chattering stability are different.