## 4 Numerical Analysis and Experimental Verification
### Experimental Verification and Real-Time Diagnosis
Therefore, only two shaft bows were employed in the study, each with residual bowing of \(0.5\) mm and \(4\) mm, as shown in Figure 10. These shafts were installed on the rotor kit and the mass imbalances on the disk were varied before initiating the rotor operation. The operational parameters were deliberately set to 1600 rpm and 3200 rpm, corresponding to the sub-critical (\(\tau=0.69\)) and trans-critical speeds (\(\tau=1.39\)), respectively, given initial bowing of \(0.5\) mm. In the interest of safety, the machinery was deliberately operated at a speed far below the critical threshold during the initial bowing phase of \(4\) mm. This experimental validation approach is used to determine an FNN's capability in identifying fault components related to imbalances and shaft bowing (\(U,\alpha,s,\theta\)). However, the real fault component was employed with four different values while keeping \(s\) constant to estimate the fault components using the FNN. A detailed comparison between the real experimental data and the FNN's estimated values is given in Tables 5-7. Each table illustrates four scenarios of imbalance and shaft bowing combinations--case 1 reveals two fault angles in the same quadrant with a difference, case 2 demonstrates the two fault angles in the phase, case 3 depicts faults almost in the anti-phase, and case 4 demonstrates two faults almost perpendicular to each other, i.e., almost a \(90^{\circ}\) difference. In the following tables, the diagnosed errors are expressed as two types. First, they are expressed in terms of the fault amplitude percentage error and the phase difference, as shown in column 5. Columns 6 and 7, however, express the errors in terms of the in-line and vertical amplitude error percentages \(\Delta U_{j\prime}\), \(\Delta s_{j\prime}\) and \(\Delta U_{\perp}\), \(\Delta s_{\perp}\). The error in line with the fault vector indicates the amplitude error, while the vertical component can be viewed as the direction error. Based on the results presented in Table 5, it can be observed that under sub-critical speeds and with the amplitude error of concern, the imbalance and shaft bowing errors both exhibit the lowest values, as they are in-phase. The biggest errors for the two faults occur in case 4, in which the two faults are perpendicular to each other. Table 6 shows similar cases to Table 5 except that the rotor was running at a trans-critical speed of 3200 rpm. It can be observed that the lowest error value for the imbalance occurred in case 4 and the lowest bow error value occurred in case 3. A peculiar feature can be observed in Table 6, in which the shaft bowing exhibits significant error values of up to 38% for all cases except case 3, when two faults were in the anti-phase. This is because the bowing response diminishes at high speeds, such that the identification process generates larger errors. Table 7 illustrates the case of \(s=4\) mm at a very slow running speed. Under such conditions, the overall imbalance errors are bigger because the bow strongly dominates the  
Figure 10: (**a**) A shaft bow with \(s=0.5\) mm and (**b**) a shaft bow with \(s=4\) mm. response, such that the imbalance response becomes a fraction of the total response and generates a larger error during the data fitting process. The lowest imbalance error happens in the anti-phase situation. The overall shaft bow errors are small, except for case 4. However, an observable error disparity related to the imbalance and shaft bowing becomes apparent when the rotor is rotating over the critical speed, particularly above 3200 rpm. Investigating the vibration patterns caused by imbalance and shaft bow faults suggests that a smaller discrepancy in inaccuracy occurs for the imbalance when the frequency ratio surpasses one, which can be observed at 3200 rpm in Figure 5. Upon further examination, it is evident that the imbalance error at 3200 rpm exceeds the prediction based on the theoretical explanation.