## 4 Numerical Analysis and Experimental Verification
### Experimental Verification and Real-Time Diagnosis
The damping ratios in the X and Y directions of the fundamental (1 st) mode can be calculated using the logarithmic decrement (LD) from the measured transient response after shutting down the motor from steady operation, as shown in Figure 8. Nevertheless, the LD is designed for a simple 1-dof vibratory system. The shaft-disk-bearing system on the test rig, however, contains an infinite number of modes. The participation of higher modes in the transient response inevitably interferes with the fundamental wave, as seen in the saw-toothed waves. The rotor involves light damping, such that a slight interference at the peak will make the succeeding peak greater than the previous. To avoid the noise from higher frequencies, we picked two adjacent waves with the least interference in the peaks and smoothed them out for the damping calculation. The values obtained after averaging for \(\zeta_{x}\) and \(\zeta_{y}\) were 0.5% and 0.47%, respectively. Table 4 summarizes all model parameters of the rotor-bearing system in the test rig. The experimental rotor was connected to a computer, enabling the acquisition of data via sensors, which were then analyzed using LabVIEW Version 20.0.1f1 (32-bit) software. The rotor's vibration signals were recorded by the sensors located on the rotor disk, measuring the vibrations in both the X and Y axes. The rotor was subjected to single-frequency excitation and the steady state response, as expected, showed an almost simple harmonic wave, unlike the transient response in Figure 8. Figure 9 demonstrates one example of a recorded sensor response, in which the red-colored dots represent the key phasor pulses in the X and Y directions. The response phase angle \(\varphi\) is defined as the angular displacement in degrees from the key phasor pulse to the first positive peak of the vibration amplitude. From Figure 9, the estimated value of \(\varphi_{x}\) is roughly \(110^{\circ}\), whereas \(\varphi_{y}\) is approximately \(200^{\circ}\), with \(\varphi_{y}\) lagging behind the X probe sensor by \(90^{\circ}\). This phase difference somewhat verifies  
Figure 7: Comparison of the Jeffcott rotor to an equivalent 1-D discrete system. the correctness of the data sequences, as the Y sensor is precisely positioned 90\({}^{\circ}\) behind the X sensor. The vibration responses from the sensor outputs can be expressed as:  
\[ \begin{array}{l}u(t)=A_{x}\cos(\Omega t-\varphi_{x})=A_{x}\cos(\varphi_{x}) \cos(\Omega t)+A_{x}\sin(\varphi_{x})\sin(\Omega t)\\ v(t)=A_{y}\cos(\Omega t-\varphi_{y})=A_{y}\cos(\varphi_{y})\cos(\Omega t)+A_{y} \sin(\varphi_{y})\sin(\Omega t)\end{array}  \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**C** & \(M_{eq}\) **(kg)** & \(\zeta\) & \(K_{s}\) **(kN/m)** & \(K_{b}\) **(kN/m)** & \(K_{eq}\) **(kN/m)** & \(\omega_{n}\) **(rad/s)** \\ \hline Y & 0.96 & 0.47\% & 96.908 & 108.92 & 51.282 & 230.1 \\ X & 0.96 & 0.5\% & 96.908 & 135.72 & 56.538 & 241.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimated model parameters of the rotor-bearing system in the test rig. Figure 8: Transient responses in the damping evaluation. Figure 9: Phase angle and amplitude from the measured sensor responses. where the vibration amplitudes and phases in the X and Y directions can be obtained from Figure 9. The four feature responses (\(f_{1},f_{2},f_{3}\), and \(f_{4}\)) from the experimental study can be assessed using the equations shown above and Equations (20)-(23). This response can be fed into a trained FNN implemented into the monitoring system for real-time fault component (\(U\), \(\alpha\), \(s\), \(\theta\)) diagnosis. The disk in the test rig features 16 holes with an angular separation of \(22.5^{\circ}\) between each hole, facilitating the determination of the imbalance phase angle \(\alpha\) measured from the key phasor. The formation of a small permanent shaft bow is difficult to control because the shaft must be heated and bent to reach the plastic deformation region.