# Chapter and Stability Analysis of the Slender Composite Boring Bar with Constrained Damping Layer
## 2 Dynamic Modeling of Composite Boring Bar with Constrained Damping Layer  
The structural model of a composite boring bar with a constrained layer is shown in Figure 1, wherein \(L\) is the overhanging length of boring bar; \(R_{1}\), \(R_{2}\), and \(R_{3}\) are the radii of the base layer, the damping layer, and the constrained layer respectively. The dynamic modeling of a composite boring bar with a constrained damping layer is shown in Figure 2.  
The vibration differential equation of a boring bar with a constraint layer based on the Euler-Bernoulli beam theory can be obtained. \[\frac{\partial^{2}}{\partial x^{2}}\left((EI)_{\text{equiviv}}\frac{\partial^ {2}y(x,t)}{\partial x^{2}}\right)+\left(\rho A\right)_{\text{equiviv}}\frac{ \partial^{2}y(x,t)}{\partial t^{2}}=f(x,t)=\Delta F(t)\delta(x-L) \tag{1}\]  
where \(y(x,t)\) is the displacement of the cross-section on the boring bar from the origin \(o\) to \(x\) at time \(t\); \((EI)_{\text{equiviv}}\) and \((\rho A)_{\text{equiviv}}\) is the equivalent bending stiffness and surface density of the constrained damping structure, respectively; \(\delta\) denotes the Dirac delta function;\(f(x,t)\) is the distributed force; \(\Delta F(t)\) is the equivalent concentrated force of \(f(x,t)\). \[\left\{ \begin{array}{l}\left(EI\right)_{\text{equiviv}}=\left(EI\right)_{1}+ \left(EI\right)_{2}+\left(EI\right)_{3}\\ \left(\rho A\right)_{\text{equiviv}}=\left(\rho A\right)_{1}+\left(\rho A \right)_{2}+\left(\rho A\right)_{3}\end{array} \right. \tag{2}\]  
Figure 1: Structure diagram of the boring bar of composite material with damping of the constrained layer. (**a**) The constrained layer; (**b**) the damping layer; (**c**) the base layer. Figure 2: Dynamic model of a boring rod of composite material with damping of the constrained layer. where \((EI)_{1}\), \((EI)_{2}\), and \((EI)_{3}\) are the bending stiffness of matrix, damping layer, and constraint layer of boring bar, respectively; \((\rho A)_{1}\),\((\rho A)_{2}\), and \((\rho A)_{3}\) denote the surface density of matrix, damping layer, and constraint layer of boring bar, respectively. The bending stiffness and surface density of the composite matrix are as follows,  
\[\left\{ \begin{array}{l}\left(EI\right)_{1}=\frac{\pi}{4}\overline{Q}_{11} \Big{(}r_{\mathrm{N+1}}^{4}-r_{1}^{4}\Big{)}\\ \left(\rho A\right)_{1}=\pi\rho_{1}\Big{(}r_{\mathrm{N+1}}^{2}-r_{1}^{2}\Big{)} \end{array} \right. \tag{3}\]  
where \(\overline{Q}_{11}\) is the off-axis stiffness coefficient of composite matrix [20]; \(r_{\mathrm{N+1}}\) and \(r_{1}\) are the radii of the \(N\)th layer and the first layer in the matrix layer, respectively; \(\rho_{1}\) is the material density of the matrix layer. Similarly, the bending stiffness and surface density of the damping and constraint layer of boring bar are given by,  
\[\left\{ \begin{array}{l}\left(EI\right)_{2}=\frac{\pi E_{2}\left(\left(2R_{2} \right)^{4}-\left(2R_{1}\right)^{4}\right)}{64}\\ \left(\rho A\right)_{2}=\pi\rho_{2}(R_{2}^{2}-R_{1}^{2})\end{array} \right. \tag{4}\]  
\[\left\{ \begin{array}{l}\left(EI\right)_{3}=\frac{\pi E_{3}\left(\left(2R_{3} \right)^{4}-\left(2R_{2}\right)^{4}\right)}{64}\\ \left(\rho A\right)_{3}=\pi\rho_{3}(R_{3}^{2}-R_{2}^{2})\end{array} \right. \tag{5}\]  
where \(E_{2}\) and \(E_{3}\) are the elastic modulus of the damping layer and the constrained layer, respectively; \(\rho_{2}\) and \(\rho_{3}\) are the material density of the damping layer and the constraint layer, respectively. The main mode shape of the boring bar is expressed as  
\[y(x,t)=\sum_{i=0}^{\infty}Y_{i}(x)\sin(\omega t+\phi)=\sum_{i=0}^{\infty} \left(C_{1}\sin\beta x+C_{2}\cos\beta x+C_{3}sh\beta x+C_{4}ch\beta x\right) \sin(\omega_{i}t+\phi) \tag{6}\]  
where \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\) and \(\phi\) are constants.