## 2 Equation Formulation Using GPEM  
In this study, the GPEM is used to solve the dynamic behavior of the flexible rotor-bearing system proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (1998); [PERSON] and [PERSON] (1999). The lateral displacements are assumed to be very small and all the deflections are assumed to be parallel to the \(x\)-\(y\) plane. In addition, the shear effect, and the torsional and axial vibrations are neglected. Denote the displacements of two translations and two rotations at any cross section of the shaft as \((u,\,v)\) and \((B,\,\Gamma)\). \(u\) and \(B\) are along the \(x\)-axis. \(v\) and \(\Gamma\) are along the \(y\)-axis. The deflections of the flexible rotor are described as functions of the time \(t\), the shaft length \(l\), and the axial coordinate \(z\), as follows:  
\[\begin{split} u(\frac{z}{l},t)=\sum\limits_{n=1}^{N_{\text{p}}}a _{n}(t)(\frac{z}{l})^{n-1},\,v(\frac{z}{l},t)=\sum\limits_{m=1}^{N_{\text{p}}} b_{m}(t)(\frac{z}{l})^{m-1}\\ B(\frac{z}{l},t)=-\frac{\partial v(\frac{z}{l},t)}{\partial(\frac{z }{l})}=-\sum\limits_{m=2}^{N_{\text{p}}}(m-1)(\frac{z}{l})^{m-2}b_{m}(t)\\ \Gamma(\frac{z}{l},t)=\frac{\partial u(\frac{z}{l},t)}{\partial \frac{z}{l}}=\sum\limits_{n=2}^{N_{\text{p}}}(n-1)(\frac{z}{l})^{n-2}a_{n}(t) \end{split} \tag{1}\]  
where \(a_{n}(t)\) and \(b_{m}(t)\) represent the corresponding generalized coordinates, that is, the coefficients of the polynomial, and \(N_{\text{p}}\) represents the total number of terms of the polynomial.  
The kinetic energy of the shaft and disk, and the strain energy of the shaft and bearing are first composed. Using the Lagrangian approach, the system equation of motion can be expressed as:  
\[\begin{split}\begin{bmatrix}M&0\\ 0&M\end{bmatrix}\begin{bmatrix}\frac{\cdot\cdot}{a}\\ \cdot\frac{\cdot\cdot}{b}\end{bmatrix}+\Omega\begin{bmatrix}0&G\\ -G&0\end{bmatrix}\begin{bmatrix}\frac{\cdot\cdot}{a}\\ \cdot\frac{\cdot\cdot}{b}\end{bmatrix}+\begin{bmatrix}C_{\text{xx}}&C_{\text{yy }}\\ \cdot\frac{\cdot\cdot}{b}\end{bmatrix}\begin{bmatrix}\frac{\cdot\cdot}{a}\\ \cdot\frac{\cdot\cdot}{b}\end{bmatrix}+\begin{bmatrix}K_{\text{s}}+K_{\text{xx} }&K_{\text{xy}}\\ K_{\text{yx}}&K_{\text{s}}+K_{\text{yy}}\end{bmatrix}\begin{bmatrix}\frac{ \cdot\cdot}{a}\\ \cdot\frac{\cdot\cdot}{b}\end{bmatrix}=\begin{bmatrix}\frac{\cdot\cdot\cdot}{F_ {a}}\\ \cdot\frac{\cdot\cdot}{F_{b}}\end{bmatrix}+\begin{bmatrix}\frac{\cdot\cdot\cdot \cdot}{F_{a}}\\ \cdot\frac{\cdot\cdot}{F_{a}}\\ \cdot\frac{\cdot\cdot}{F_{b}}\end{bmatrix}\end{split} \tag{2}\]  
where \(\overset{\cdot\cdot}{a}=\left\{a_{1}\quad a_{2}\quad\ldots\quad a_{N_{\text{p} }}\right\}^{T}\), \(\overset{\cdot\cdot}{b}=\left\{b_{1}\quad b_{2}\quad\ldots\quad b_{N_{\text{p} }}\right\}^{T}\), \(\Omega\) denotes the constant rotating speed and \(M\), \(G\), \(C_{\text{xx}}\), \(C_{\text{yy}}\), \(C_{\text{yy}}\), \(K_{\text{s}}\), \(K_{\text{xx}}\), \(K_{\text{xy}}\), and \(K_{\text{yy}}\) are the \(N_{\text{p}}\times N_{\text{p}}\) component matrices shown in Appendix A. The first and the second vector terms on the right side of Equation (2) represent the linear and nonlinear forces. It is noted that \(G\) is the gyroscopic matrix caused by the rotation of the shaft section. Therefore, the natural frequencies of the system vary with the rotational speed of the spindle. The whirl map, including forward and backward modes, can be established. The system response at any axial position can be solved by using Equations (1) and (2). In this study, the GPEM is employed to analyze the tool-tip vibration for ball end milling combined with the rotor-bearing system. The left side of Equation (2) will include the mass, damping and stiffness effects, and gyroscopic effects of the overall tool spindle system, while the force terms generated by the ball end mill, discussed in the next section, will be placed on the right side.