## 4 Numerical Analysis and Experimental Verification
### Experimental Verification and Real-Time Diagnosis  
The model RK 4 Bently Nevada experimental rig depicted in Figure 6 was used for the rotor fault experiment. This consisted of a motor attached to a single-disk rotor. The motor ran counterclockwise as viewed from the motor side. The rotor shaft was upheld by simple, identical bearings of an unknown stiffness. The diameter of the rotor shaft was 10 mm. A disk of 75 mm in diameter and 800 g in mass was mounted on the rotor shaft by radial screws. There were 16 tapped holes symmetrically placed on each side of the disk with flat faces at \(e=30\) mm to attach any desired amount of imbalance mass. Memstec Glory Laser CD3S-30 and CD3S-50 sensors were precisely positioned on the corresponding disk along the X and Y axes, which were positioned at a separation angle of 90 degrees. After installation, the sensors were integrated into the power supply system to ensure a reliable and uninterrupted power source to ease the measurement operation. The rotor in the study encountered flexible support conditions, whereby the stiffness and damping coefficients were determined by the combination of the shaft and bearing components in the rotor system, as shown in Figure 7. The support stiffness consisted of the combination of \(K_{\text{s}}\) (shaft stiffness) and \(K_{\text{b}}\) (bearing stiffness). It is important to acknowledge that the mass of the shaft, \(m_{s}\), cannot be ignored and it must be lumped into a certain fraction with the mass of the disk, \(M_{d}\). To determine the amount of shaft mass to be lumped with the disk mass, calculations can be carried out using an equivalent system. This entails establishing an equivalent system by analyzing the total kinetic energy (KE) of the shaft under the assumption of vibration in the static deflection mode. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Fault Components**} & \multicolumn{3}{c}{**Average RMSE**} \\ \cline{2-4}  & \(\boldsymbol{\zeta<1}\) & \(\boldsymbol{\zeta=1}\) & \(\boldsymbol{\zeta>1}\) \\ \hline \(U_{x}\) & \(1.17\times 10^{-5}\) & \(2.93\times 10^{-5}\) & \(1.05\times 10^{-5}\) \\ \(U_{y}\) & \(1.35\times 10^{-5}\) & \(2.43\times 10^{-5}\) & \(1.62\times 10^{-5}\) \\ \(s_{x}\) & \(8.75\times 10^{-5}\) & \(8.21\times 10^{-5}\) & \(8.66\times 10^{-5}\) \\ \(s_{y}\) & \(6.81\times 10^{-5}\) & \(9.15\times 10^{-5}\) & \(7.27\times 10^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison of RMSE values for fault components under different damping ratios. Figure 6: The rotor experiment platform. Subsequently, the effective mass of the shaft that should be lumped onto the disk is 0.48 \(m_{s}\). Additionally, the mass of the disk (\(M_{d}\)) is increased by 0.48 \(m_{s}\). Therefore, the value of \(M_{eq}\) would be 0.96 kg. The system's overall stiffness is determined by the series connection of the shaft and the bearings, as shown in Figure 7. The stiffness of the shaft under simple supports can be easily calculated to be \(K_{s}=\left(48E\text{I}/L^{3}\right)\). Nonetheless, the bearing stiffness \(K_{b}\) is difficult to calculate theoretically; therefore, a practical approach is used to run the rotor experimentally to identify its critical speed (\(\omega_{\sigma}\)), which was found to be 2300 rpm for the present case. The identified system critical speed \(\omega_{cr}=\sqrt{K_{eq}/M_{eq}}\) can be used in reverse to evaluate the bearing stiffness \(K_{b}\). The critical speeds associated with X and Y vibrations for an asymmetric rotor are different. However, the asymmetry in the rotor rig is too small to be differentiable in the vibration measurements. While evaluating the critical speeds, it was assumed that the first recorded critical speed pertains to the Y direction. Additionally, it was noted that the critical speed in the X direction, as referenced in prior research [48], is approximately 5% greater than in the Y direction.