## 5 Numerical Simulation Results and Discussion
### Practical Test 2: Imbalanced Vertical Shift with Friction and Fluid  
Figures 17a and 18a are almost similar to the behavior of that in the fluid. The frictional impact of higher resonance frequencies and larger subharmonic peaks occurring with the bounce of the outer drill string orbit can be seen in both Figures 17 and 18. In general, friction-induced noise and vibrations exhibit nonlinear, time-varying transient characteristics. For comparison, it is shown that the obtained experimental signal is highly consistent with the simulated defect signal. The extracted impulse features in the inviscid fluid are plotted as shown in Figure 17b. The proposed results can accurately display the impulse characteristics of the rubbing contact, even when the fault signal is buried under strong noise interference. **Observation 5:** For the case where faults such as imbalance and rotor-stator rub coexist, the response of the drill string orbit in the lateral direction and its FFT spectrum showed that the harmonics of 1 \(\times\) order (213.8 Hz) representing the imbalance exist in each response (with and without fluid). The FFT spectral plot also shows that higher frequencies, for example, 2X, 3X, 5X, etc., get excited as it occurs. The presence of a weak subharmonic resonance at 1/2 of the critical bending speed can be observed in the submerged imbalanced rubbing drill string. The experimental results show how the fluid also indirectly affects the frictional torque in the system. Figure 17: Experimental response of imbalance and rotor-casing rub (**a**) without water and (**b**) with water. Figure 18: Experimental orbit response of imbalanced rotor-casing rub (**a**) without water and (**b**) with water. In this section, since the rub fault is highly nonlinear and the fluid expression is always transient, the time-frequency representation (TFR) is properly used for this study using both IF and NWSST. Figures 19 and 20 show waveform plots of the IFs extracted by wavelet synchrosqueezing and the NWSST spectra without and with fluid, respectively. In general, aperiodic and chaotic windows can be observed in the experimental responses in Figure 19a, and a period-doubling route can be noted from the expanded diagram, as presented in Figure 20a and observed previously in Figures 13a and 14a. To further experimentally show the performance of the NWSST in analyzing the signal with multiple faults, the center frequencies of these fault components are confined to the interval [90 Hz to 500 Hz] where the TF characteristics of these two faults are captured by the NWSST (Figure 19a). The NWSST provides a time-frequency representation of the signal, which may require further analysis and interpretation to extract meaningful information about the dynamic behavior of the system. The corresponding IF is then used to filter these two fault signals. It can be seen in Figure 19b that the reconstructed IF with a denoised signal is highly consistent. With the increase in the impact, the large number of continuous components of the frequency spectrum of the system shows that there is a serious frictional impact phenomenon in the system. These frequency components can be used as characteristic frequencies for drill string rub impact fault diagnosis. Figure 21 shows the standard deviation trends corresponding to the defective cases of the selected signal segments, revealing an explicit energy level difference between the reference (air) and fluid medium (water). **Observation 6:** Figure 20b shows that the main difference in the dynamic response with and without the fluid-structure interaction model lies in the dynamic amplitude of the response and the major influence of the fluid in the side frequencies on the side mode shape. The overall trend is different for the two models when the vibration amplitude curve of the drill string without fluid is greater than that in water.