## 2 Modeling and Blade Modal Identification of AMB-Rotor System  
### Dynamic Modeling of AMB-Rotor System  
A structural diagram of the MSTMP is shown in Figure 1. The object of this research is the MSTMP, which is supported by a magnetically suspended bearing and driven by a high-speed motor. It has the remarkable advantages of no lubricating oil pollution or friction loss, and active vibration control, and is very suitable for high-speed, vacuum, and ultra-clean applications. This MSTMP is self-developed in our laboratory and has a pumping speed of 4500 L/s (pumping speed). For the turbo rotor, which is composed of multi-stage turbo blades, a shaft, and a thrust disk, \(m\) is the mass of the turbo rotor, \(J_{r}\) are the equatorial moments of inertia, \(J_{z}\) are the polar moments of inertia, \(\Omega\) is the rotation speed, \(\alpha\) and \(\beta\) are the tilting angles around the \(x\)-axis and the \(y\)-axis, and \(x\) and \(y\) are the displacement in the \(x\)-axis and the \(y\)-axis direction, respectively. If \(O\) is defined as the turbo rotor center of gravity, then \(a_{m}\) and \(b_{m}\) are the distances from \(O\) to the center of the radial magnetic bearing at End-A and End-B and \(a_{s}\) or \(b_{s}\) are the distances from \(O\) to the center of the radial sensor at End-A and End-B, respectively. Gyro technical equations are the core formulas of the dynamic model of the magnetically suspended rotor [3]. According to [PERSON]'s law and [PERSON]'s law, the dynamic equations of the AMB-rotor system can be obtained as follows:  
\[\left\{ \begin{array}{l}m\ddot{x}=f_{\text{AX}}+f_{\text{BX}}\\ J_{r}\ddot{y}-J_{z}\Omega\dot{x}=f_{\text{AX}}a_{m}-f_{\text{BX}}b_{m}\\ m\ddot{y}=f_{\text{AY}}+f_{\text{BY}}\\ J_{r}\ddot{\alpha}+J_{z}\Omega\dot{\beta}=-f_{\text{AY}}a_{m}+f_{\text{BY}}b_{ m}\end{array} \right.. \tag{1}\]  
Figure 1: Structural diagram of the MSTMP with a pumping speed of 4500 L/s (pumping speed). (**a**) Coordinate system and component structure. (**b**) Turbo rotor. The four radial channels, labeled AX, AY, BX, and BY, have the same control method, and the control is decentralized and independent, so the control method in the AX and BX channels (X direction) is analyzed as an example. According to [3], in the direction of X, the magnetic force \(f_{\rm X}\) is the binary quadratic function of current \(i_{\rm X}\) and displacement \(x\)  
\[f_{\rm X}=\mu_{0}N^{2}A_{p}\cos\left(22.5\cdot\frac{\pi}{180}\right)\left[ \left(\frac{i_{0}+i_{\rm X}}{p_{0}-x}\right)^{2}-\left(\frac{i_{0}-i_{\rm X}}{ p_{0}+x}\right)^{2}\right] \tag{2}\]  
where \(\mu_{0}\) is the magnetic permeability of a vacuum, \(N\) is the number of coils, \(A_{p}\) is the magnetic pole cross-sectional area, \(i_{0}\) is the bias current, and \(p_{0}\) is the magnetic gap when the center of the rotor and the center of the radial magnetic bearing coincide. For radial magnetic bearings, \(k_{ia}\) and \(k_{ib}\) are defined as the current stiffnesses of the radial magnetic bearing at End-A and End-B and \(k_{ha}\) and \(k_{hb}\) are defined as the displacement stiffnesses of the radial magnetic bearing at End-A and End-B, respectively. Under normal working conditions, the rotor center of the radial magnetic bearing is located in the geometric center of the stator. To simplify the expression of the magnetic force \(f_{\rm X}\), it can be Taylor-expanded by ignoring high-order items at the equilibrium point \(i_{\rm X}=0,x=0\)[3]. Therefore, the linearized equation of the magnetic force can be obtained as follows  
\[\left\{ \begin{array}{l}f_{\rm AX}=k_{ia}i_{\rm AX}+k_{ha}m_{\rm AX}\\ f_{\rm BX}=k_{ib}i_{\rm BX}+k_{hb}m_{\rm BX}\end{array} \right.. \tag{3}\]  
Introducing the general linearized expression \(f=k_{i}i+k_{h}m\) of the magnetic force \(f_{\rm AX},f_{\rm BX}\) into (1), we can obtain  
\[\left\{