## 3 High-Speed PMSG Characteristics with STS420-J2 Shaft Material  
### Mechanical Characteristics  
In high-speed rotating machines, there exists the problem of damage and loss of the rotor caused by the centrifugal force. Therefore, it is essential to consider it in the design. Generally, a sleeve is used to prevent PMs from scattering of high-speed rotating machines thus avoiding damage [17]. The thickness of the sleeve was selected considering the electrical characteristics and mechanical stress of the machines. The sleeve thickness of the rotor used in this study was selected as 4 mm, based on the electromagnetic and mechanical conditions. Additionally, it is crucial to accurately predict the natural frequency mode of a rotating body for high-speed machines.  
Critical speed causes high-frequency vibration as the vacuum space rapidly increases due to resonance, causing shaft displacement, vibration, and damage to bearings and systems directly [18,19]. Therefore, the critical speed of the rotating body must be accurately predicted, and the equation of motion of the rotating body can be calculated as: [20]  
\[m_{e}\ddot{q}\ +\ g_{e}\dot{q}\ +\ k_{e}q\ =\ f_{e} \tag{1}\]  
where \(m_{e}\) is the mass, \(g_{e}\) is the gyroscopic, \(k_{e}\) is the stiffness matrix, \(q\) is the displacement, and \(f_{e}\) is the vector component of the force. The rotating body can be classified into rigid body mode and bending mode according to its shape, and the design should be developed avoiding the bending mode as much as possible. Generally, the resonant frequency at the critical speed is predicted using the Campbell diagram.  
With the rotational speed as the horizontal axis and the natural frequency as the vertical axis, the intersection of the rotational speed and the natural frequency can be identified as the resonance area. Figure 2 shows the critical velocity analysis result of the STS420-J2 shaft material through FEM using the value derived from Equation (1). Consequently, the critical speed closest to the rated speed was 41873 rpm. Since modal analysis directly affects the electromagnetic and mechanical vibration of a rotating machine,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Rated power & 100 [kW] \\ Rated speed & 36000 [rpm] \\ Rated Line to Line voltage & 380 [V\({}_{rms}\)] \\ Thickness of sleeve & 4 [mm] \\ Operation temperature & 180 [\({}^{\circ}\)C] \\ Material of PM & Sm\({}_{2}\)Co\({}_{17}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design requirement condition.  
Figure 1: Shape of high-speed generator according to shaft material.  
it is important to check the natural frequency and mode shape. Figure 3 shows the results of modal analysis when rotating at rated speed. At the rated speed of 36,000 rpm, the resonant frequencies are 1173 Hz and 1188.1 Hz in the 1 st and 2 nd modes respectively, and the resonant frequency intersection occurs at 3295 Hz and 3352 Hz in the 3 rd and 4 th modes. Frequency means the natural vibration frequency according to each mode, and rigid body mode and bending mode occur at similar frequencies. Therefore, the frequency is derived differently according to the rigid body mode and the bending mode. However, as a result of the analysis, it was confirmed that there was no problem in the design of the rotating body because there was no bending mode intersection at the rated speed.