## 5 Numerical Simulation Results and Discussion
### Feature Extraction of the Rubbing Drill String System by Wavelet Synchrosqueezing Transform
Through time, it can be noted that a satisfactory energy concentration result is observed around the fundamental value of 400 Hz, which denotes a better time-frequency location and better characterization of the time-varying feature. The drill string makes contact with the borehole wall, which then leads to a sustained fluctuating frequency representation. It is worth mentioning, from Figure 13b, that it is difficult to identify the periodicity of a single peak after transforming the signal into the frequency domain. This explains the results of the chaotic behavior obtained in the three-dimensional orbit plot (Figure 10a). Figure 13a shows a distinct set of points that indicates the chaotic behavior of the distributed energy in a fluid medium. The corresponding estimated frequency spectrum (red line) extracted by the NWSST method shown in Figure 13b contains a regular breaking point that involves the damping effect of the fluid upon shock contact. In Figure 14a, the dominant frequency of the resonance mode of the drill string can be observed at a considerably lower frequency. If the evolution through time of the peak at 250 Hz is followed, it can be seen that there is no longer an oscillation around a defined frequency. However, the curve (circled area) shows  
Figure 13: No fluid-NWSST representation of the rotor-casing rubbing signal in the \(Y\) direction. (**a**) Time-frequency ridges from NWSST; (**b**) Estimated IF amplitude spectrum of the signal. sudden periodic jumps above that of the frequency resonance as if the behavior of the system changed from one mode to another, which corresponds to the minimized chaotic point observed in Figure 13a. In the presence of hydrodynamic forces, the rubbing intensity observed in Figure 13 is attenuated, but the nonlinear phenomena of the fluid introduced a disturbed cascade of peaks throughout the vibration. The IF of the imbalanced system under the frictional impact of the phenomenon is relatively obvious, as observed in Figure 14b, where a series of peak frequency components are aperiodically distributed by irregular low spectral lines representing the contact permanent. As fluid forces and frictional impacts occur, the wavelet scalogram gives a good display of time in the frequency region with an improved analysis of the time-frequency resolution; higher frequency components appear and make it possible to identify the time of occurrence, and the frequency of the friction caused impacts and a sharp decrease in the amplitudes of noiseless extracted IF components, which reveal clearly the time interval between successive high pulses. **Observation 4:** The useful information in the harmonic distribution is observed in Figure 12 at some specific disturbed periods illustrated by the high peak's shape disturbance throughout the signal, while in the spectrum bands, the diminution of the impact amplitude is prominent. It can also be seen from Figure 12 that there are good approximations of the signal's instantaneous frequency, which oscillates periodically and differs from the IF of the imbalanced drilling/casing system with the eliminated hydrodynamic forces. The damping imposed by the presence of fluid makes it possible to obtain a good prediction of the behavior of the drill string subjected to an impact. Adding a dissipative inviscid fluid in the system minimizes its vibration by significantly amortizing the shock. It is obvious that, if only the imbalance effect is taken into account, the IF remains unchanged, while the shock generated by the rotor-casing system oscillates gradually in amplitude. **6. Experimental Analysis and Discussion**  
In this section, an extracted signal contaminated with strong noise is used to evaluate the performance of the NWSST technique. The drill string vibration signal under various operating conditions is imported as an excel sheet into Matlab to perform the time-frequency analysis.