## 2 Modeling and Blade Modal Identification of AMB-Rotor System
### Blade Modal Identification by Sinusoidal Sweep Excitation  
Modes are intrinsic vibrational characteristics of the turbo rotor, and each mode has a specific natural frequency, damping ratio, and vibration pattern. Finite element analysis (FEA) was performed using ANSYS Workbench. The turbo rotor model used 5-mm-sized elements, resulting in a mesh with 354,509 nodes and 173,934 elements. Th elastic modulus scaling factors were 0.01 for clearance fit, 0.1 for nut tightening, and 1 for interference fit. The first 40 natural frequencies were calculated to capture the main vibration characteristics. The following assumptions were made during the FEA:  
1. Material homogeneity and isotropy. 2. Linear elastic behavior. 3. Neglecting temperature effects. Setting fluid model and boundary conditions for FEA of the MSTMP's turbo rotor. Figure 2: Block diagram of the AMB-rotor control system. 1. Fluid model: The \(k-\varepsilon\) turbulence model is chosen, which is suitable for high Reynolds number flows. 2. Inlet boundary condition: A velocity inlet is defined with a speed set at 573 m/s. 3. Outlet boundary condition: A pressure outlet is specified with a pressure set at 0 Pa (relative pressure). 4. Wall boundary condition: No-slip conditions are applied to simulate the actual interaction between the fluid and the wall surfaces. The finite element simulation results of the turbo rotor are shown in Figure 3. The frequency response of the blade mode is characterized by second-order underdamping, and the multi-stage-blade modes have a superposition effect. So taking the multi-stage-blade modes into account, the AMB-rotor model \(G_{r}(s)\) can be replaced by \(G_{m}(s)\)  
\[G_{m}(s)=\prod_{i=1}^{n}\frac{k_{ia}A\Big{(}s^{2}+\sigma_{i}w_{fi}s+w_{fi}^{2} \Big{)}}{(s^{2}-k_{ha}B)\Big{(}s^{2}+\lambda_{i}\sigma_{i}w_{fi}s+w_{fi}^{2} \Big{)}} \tag{12}\]  
where \(w_{fi}\) is the \(i\)-th, \((i=1,2,3,4,5,6)\) blade modal frequency, and \(\lambda_{i}\) and \(\sigma_{i}\) are the peak amplitude factor and peak width factor of the \(i\)-th blade modes, respectively, which can be identified by sinusoidal sweep excitation. In order to obtain the multi-stage-blade modal frequencies during static suspension, a sinusoidal excitation signal is added to the channel of BY. The sampling frequency is set as 10 kHz, the sweep frequency range as 1-2 kHz, and the excitation frequency interval as 1 Hz. The test result of sinusoidal sweep excitation is shown in Figure 4a. The labels (419 Hz, \(-63.87\) dB. 525 Hz, \(-60.02\) dB. 583 Hz, \(-37.06\) dB, 674 Hz, \(-30.56\) dB, 710 Hz, \(-38.15\) dB. 1598 Hz, and \(-65.82\) dB) at the locations of the peaks are indicative of multi-stage-blade modes during static suspension (0 rpm). Aiming for an attenuation to \(-80\) dB, the peak bandwidth \(H_{\text{wid}}\) and the peak amplitude \(E_{\text{amp}}\) are defined as shown in Figure 4a. According to \(E_{\text{amp}}\) and \(H_{\text{wid}}\), the peak amplitude factor \(\lambda_{i}\) and the peak width factor \(\sigma_{i}\) can be obtained, respectively  
\[\lambda_{i}=10^{-\frac{E_{\text{amp}}}{20}}, \tag{13}\]  
\[\sigma_{i}=\sqrt{\frac{2-\sqrt{\frac{16\pi^{2}(H_{\text{wid}})^{2}}{w_{fi}^{ 2}}+4}}{2\times\left(10^{-\frac{E_{\text{amp}}}{20}}\right)^{2}-1}}. \tag{14}\]  
Firstly, the theoretical frequencies of multi-stage-blade mode self-excited oscillations are obtained by finite element simulation. Secondly, the actual frequencies of blade modes are obtained by a sinusoidal sweeping excitation test. Finally, multiple superimposed  
Figure 3: Results of finite element simulation. (**a**) One blade mode with a frequency of 419.53 Hz. (**b**) Another blade mode with a frequency of 583.75 Hz. second-order damped oscillation terms are introduced based on the AMB-rotor model \(G_{r}(s)\) in Section 2.1. Thus, the whole process from simulation analysis to experimental verification and obtaining the transfer function model is realized for multi-stage-blade mode self-excited oscillations.