# Dynamic Modeling of Motorized Spindle System with Unbalanced Mass and Spindle Inclination
## 2 Dynamic Modeling
### Dynamic Modeling of Motorized Spindle without Fault  
As shown in Figure 2a, the forces of the left and right sections of the rigid thin disc have a bending moment of \(M^{L}\) and \(M^{R}\) and a shear force of \(Q^{R}\) and \(Q^{L}\). The equations of inertia force and moment of inertia are as follows [25]:  
\[F=m\omega^{2}y \tag{2}\]  
\[M=\left(J_{d}-J_{p}\frac{\Omega}{\omega}\right)\omega^{2}\theta \tag{3}\]  
where \(y\) is the deflection of the disc, \(\theta\) is the rotation angle of the disc, \(\Omega\) is the rotation angular velocity of the rotor, and \(\omega\) is the whirling angular velocity, \(J_{d}\) and \(J_{p}\) are the polar and equatorial moment of inertia, respectively. When the node is unsupported, the inertia force is zero. The D'Alembert equation can be established as follows:  
\[\left\{ \begin{aligned} & y^{R}=y^{L}\\ &\theta^{R}=\theta^{L}\\ & M^{R}=M^{L}-\left(J_{d}-J_{p}\frac{\Omega}{\omega}\right)\omega ^{2}\theta+k_{\theta}\theta\\ & Q^{R}=m\omega^{2}y+Q^{L}-k_{r}y\end{aligned} \right. \tag{4}\]  
where \(k_{r}\) is the radial stiffness of the bearing and \(k_{\theta}\) is the angular stiffness of the bearing. Equation (4) is expressed in the form of a matrix to obtain  
\[\left[ \begin{matrix}y\\ \theta\\ M\\ Q\end{matrix} \right]_{i}^{R}=\left[ \begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&k_{\theta}-\left(J_{d}-J_{p}\frac{\Omega}{\omega}\right)\omega^{2}&1&0\\ m\omega^{2}-k_{r}&0&0&1\end{matrix} \right]_{i}\left[ \begin{matrix}y\\ \theta\\ M\\ Q\end{matrix} \right]_{i}^{L} \tag{5}\]  
where \(y\), \(\theta\), \(M\), and \(Q\) are the deflection, rotation angle, bending moment, and shear force of the disc, respectively. The superscript \(R\) or \(L\) indicates the right side or left side of the disc. The subscript \(i\) indicates the disc of the \(i\)th unit. Therefore, when the disc left side  
Figure 1: Schematic diagram of high-speed motorized spindle: (**a**) description of what is contained in the first panel; (**b**) structure of motorized spindle; (**c**) lumped mass model of high-speed motorized spindle. cross-sectional state of the \(i\)th unit is known, Equation (5) can be used to solve its right side cross-sectional state. Similarly, as shown in Figure 2b, the left end of the elastic shaft connects the disc's right side of the same unit, and the right end of the shaft connects the left side of the disc which belongs to the next disc -shaft unit. Therefore, the shear force and bending moment at the left end of the shaft are superscripted as \(R\) and subscripted as \(i\). The shear force and bending moment at the right end are superscripted as \(L\) and subscripted as \(i+1\). It is also known from the bending deformation conditions of shafts in the mechanics of materials that  
\[\left\{ \begin{array}{c}y_{i+1}=y_{i}+\theta_{i}l_{i}+\frac{M_{i+1}^{L}l_{i}^ {2}}{2E_{i}l_{i}}-Q_{i+1}^{L}\left(\frac{l_{i}^{3}}{3E_{i}l_{i}}+\frac{k_{i}l _{i}}{G_{i}^{2}A_{i}}\right)\\ \theta_{i+1}=\theta_{i}+\frac{M_{i+1}^{L}l_{i}}{E_{i}l_{i}}-Q_{i+1}^{L}\frac{ l_{i}^{2}}{2E_{i}l_{i}}\\ M_{i+1}^{L}=M_{i}^{R}+Q_{i}^{R}l_{i}\\ Q_{i+1}^{L}=Q_{i}^{R}\end{array} \right. \tag{6}\]  
where \(k_{s}\) is the shape factor of the section, for the round solid shaft \(k_{s}=10/9\). \(G_{i}\) is the shear modulus of the \(i\)th subsegment unit, \(A_{i}\) is the sectional area of the \(i\)th subsegment unit, \(E_{i}\) is the modulus of elongation of the subsegment, and \(l_{i}\) is the inertia moment of the \(i\)th subsegment. Similarly, by changing the above equation into matrix form, the following formula can be obtained:  
\[\left[ \begin{array}{c}y\\ \theta\\ M\\ Q\end{array} \right]_{i+1}^{L}=\left[ \begin{array}{cccc}1&l&\frac{l^{2}}{2 EI} &\frac{l^{3}}{6 EI}(1-v)\\ 0&1&\frac{l}{EI}&\frac{l^{2}}{2 EI}\\ 0&0&1&l\\ 0&0&0&1\end{array} \right]_{i}\left[ \begin{array}{c}y\\ \theta\\ M\\ Q\end{array} \right]_{i}^{R} \tag{7}\]  
where \(v=k_{s}(6E_{i}l_{i})/\left(G_{i}A_{i}l_{i}^{2}\right)\) is called the shear effect coefficient.