## 2 Modelling of Boring Bar
### Model of Boring Bar with Shear Deformation and Rotational Inertia  
Based on the AMDM method [25], a model of a boring bar considering shear deformation and rotational inertia can be defined as follows [26; 27; 28]:  
\[\rho A\frac{\partial^{2}y}{\partial t^{2}}-\kappa AG\left(\frac{\partial^{2}y }{\partial z^{2}}-\frac{\partial\psi}{\partial z}\right)=0 \tag{1}\]  
\[EI\frac{\partial^{2}\psi}{\partial z^{2}}+\kappa AG\left(\frac{\partial y}{ \partial z}-\psi\right)-\rho l\frac{\partial^{2}\psi}{\partial t^{2}}=0 \tag{2}\]  
where \(\rho\) is the mass density of the boring bar material, \(A\) is the cross-sectional area of the boring bar, \(\kappa\) the shear correction factor of the boring bar, \(G\) represents the shear modulus of the boring bar material, \(E\) represents the Young's modulus of the boring bar material, \(I\) is the area moment of inertia of the boring bar, and \(y=y(z,t)\) and \(\psi=\psi(z,t)\) are the transverse deflection of the boring bar and the angle of rotation of the cross-section due to the bending of the boring bar at position \(z\) and time \(t\), respectively. According to Figure 1, the boundary conditions at z = 0 (i.e., the fixed end, at which the position, transverse deflection, and angle of rotation are zero) are defined as follows:  
\[\kappa AG\left[\frac{\partial y(z,t)}{\partial z}-\psi(z,t)\right]-k_{TL}y(z,t)=0 \tag{3}\]  
\[EI\frac{\partial\psi(z,t)}{\partial z}-k_{RL}\psi(z,t)=0 \tag{4}\]  
The boundary conditions at z = L (i.e., the free end, at which the position, bending moment, and shear force are zero) are defined as follows:  
Figure 1: Boring bar model. \[\kappa AG\left[\frac{\partial y(z,t)}{\partial z}-\psi(z,t)\right]=0 \tag{5}\]  
\[EI\frac{\partial\psi(z,t)}{\partial z}=0 \tag{6}\]  
With the following definitions, Equations (1) and (2) can be written in matrix form as Equation (8). \[X=\frac{z}{l},\ Y(Z)=\frac{Y(z)}{l},\ \Psi(Z)=\psi(z),\ \Omega^{2}=\frac{\rho A \omega_{s}^{2}l^{4}}{EI},\ \eta=\frac{I}{AI^{2}},\ \xi=\frac{\kappa GAl^{2}}{EI}, \tag{7}\]  
\[\left[ \begin{array}{c}Y^{\prime\prime}(Z)\\ \Psi^{\prime\prime}(Z)\end{array} \right]+\left[ \begin{array}{c}0-1\\ \xi 0\end{array} \right]\left[ \begin{array}{c}Y^{\prime}(Z)\\ \Psi^{\prime}(Z)\end{array} \right]+\left[ \begin{array}{cc}\frac{\Omega^{2} }{\xi}&0\\ 0-\xi+\eta\Omega^{2}\end{array} \right]\left[ \begin{array}{c}Y(Z)\\ \Psi(Z)\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array} \right] \tag{8}\]  
Equation (8) can be simplified as follows:  
\[u^{\prime\prime}(Z)+Bu^{\prime}(Z)+Du(Z)=0 \tag{9}\]  
where \(u(Z)=\left[ \begin{array}{c}Y(Z)\\ \Psi(Z)\end{array} \right]\), \(u^{\prime}(Z)=\left[ \begin{array}{c}Y^{\prime}(Z)\\ \Psi^{\prime}(Z)\end{array} \right]\), \(u^{\prime\prime}(Z)=\left[ \begin{array}{c}Y^{\prime\prime}(Z)\\ \[PERSON]^{\prime\prime}(Z)\end{array} \right]\), \(B=\left[ \begin{array}{c}0-1\\ \xi 0\end{array} \right]\) and \(D=\left[ \begin{array}{c}\frac{\Omega^{2}}{\xi}&0\\ 0&\eta\Omega^{2}-\xi\end{array} \right]\). According to the AMDM [28] solution of Equation (9) can be approximated by the first \(k\) terms, as follows:  
\[U(Z)=\left[ \begin{array}{c}Y(Z)\\ \Psi(Z)\end{array} \right]=\left[ \begin{array}{c}\sum\limits_{j=0}^{k-1}C_{ 1,j}Z^{j}\\ \sum\limits_{j=0}^{k-1}C_{2,j}Z^{j}\end{array} \right]=\sum\limits_{j=0}^{k-1} C_{j}Z^{j}=\Phi(Z)+\sum\limits_{j=2}^{k-1}C_{j}Z^{j} \tag{10}\]  
where:  
\[\Phi(Z)=C_{0}+C_{1}Z=U(0)+U^{\prime}(0)Z \tag{11}\]  
\[C_{0}=U(0)=\left[ \begin{array}{c}Y(0)\\ \Psi(0)\end{array} \right] \tag{12}\]  
\[C_{1}=U^{\prime}(0)=\left[ \begin{array}{c}Y^{\prime}(0)\\ \Psi^{\prime}(0)\end{array} \right] \tag{13}\]  
\[C_{j}=\frac{-1}{j(j-1)}\Big{[}(j-1)BC_{j-1}+DC_{j-2}\Big{]},\ j=2,3,4,... \tag{14}\]  
The boundary conditions of Equations (3) and (4) at z = 0 can be written in matrix form as follows:  
\[\left[ \begin{array}{c}\xi 0\\ 0\ 1\end{array} \right]\left[ \begin{array}{c}Y^{\prime}(Z)\\ \Psi^{\prime}(Z)\end{array} \right]+\left[ \begin{array}{cc}-K_{TL}&-\xi\\ 0&-K_{RL}\end{array} \right]\left[