## 5 Numerical Simulation Results and Discussion
### Feature Extraction of the Rubbing Drill String System by Wavelet Synchrosqueezing Transform  
Parametric studies in the context of the dynamics of nonlinear systems require the use of robust and efficient methods for the study of periodic regimes and their stability, such as the use of time-frequency methods. The oscillation feature of the denoising vibration response signal using the wavelet hard-thresholding methodology is first extracted so that the time-frequency resolution is better. Then, the denoised signal is fed to the instantaneous frequency and is taken as an index to qualitatively describe the highly oscillated rotor-casing system and detect the feature of the rub impact in an inviscid fluid through the fluctuation of energy. The procedure followed to analyze the time signals and interpret the results is different based on the dissipative energy of the system. A direct comparison between conditions with and without friction and fluid is preferred to the main effect plot. Based on the NWSST representation, the influences of some dynamic parameters, including eccentric mass and rub impact under hydrodynamic forces, are investigated as shown in Figure 12. The mass of the fluid affects the frictional impact leading to high quasiperiodic motion. The energy distributions obtained based on the NWSST, as shown in Figure 12a,b, demonstrated that, despite the fluid mass added around the contact point, the friction phenomenon is still perceived by the appearance of some trivial lower amplitude harmonic interference components. The increase in the mass system causes a significant reduction in the bit fluctuation (Figure 12b) where in the severity of stick-slip is more apparent when operating in a dry environment. **Observation 3:** Note that the 3D plot scale is different; the high amplitudes of the \(1X\) and \(2X\) peaks (t = 0.3 s and 12 s) are lower than the levels observed without fluid impact. The existence of nonsynchronous components in the vibration spectrum is the source of  
Figure 12: No fluid-NWSST representation of the imbalanced drill string signal in the \(Y\) direction. (**a**) Extracted feature from NWSST; (**b**) Extracted impulse feature from NWSST immersed in fluid. axial direction and a red flag to the analyst that rubbing problems may exist. The severity of the stick-slip vibrations generates different frequencies based on its physical make-up, as observed in Figure 12, where, despite the presence of the fluid around the stick-slip and drill-wall contacts, the vibrations do not disappear for this drilling section. The drill string system first oscillates with a registering long period of bit sticking with a peak in large resonance amplitudes (Figure 12a), and a sharp decrease in amplitudes of the noiseless extracted IF components reveals the complexity of the time interval between successive high pulses. Figure 13 illustrates the feature of the rubbing contact when the dissipative inviscid fluid is not integrated into the drill string system. In Figure 13a, the time-frequency ridges from the nonlinear wavelet synchrosqueezing plot can be observed, and this representation is used to obtain a higher resolution for the energy concentration analysis. The maximum energy of the time-frequency ridge is extracted in cycles per sample of the wavelet synchrosqueezed transform. The aperiodic oscillations during the shock reflect the influence of the amplitude of the impact during the contact. During impact, the drill string oscillates at its first resonance frequency of 200 Hz where the frequency of the first resonance mode of the drill string can be observed. The second resonant frequency of the drill string with a high frequency of 400 Hz shows that the drill string, at this time, is excited in its first two modes. The peaks of the second resonance mode of the drill string can be observed with a slight shift below the resonance frequency (circled area) for the second mode, which may be sufficient to change the behavior of the drill string according to its rotational speed.