## 5 Numerical Simulation Results and Discussion
### Response of the Imbalanced Drilling System and Diagnosis of the Rubbing Effects  
Flexible coupling problems add multiple harmonics, such as imbalance and friction, producing a variety of symptoms in different environments; each case must be individually diagnosed. In the case of an imbalanced drill string with a rub, the effects of the fluid influence significantly the rub impact features. An interesting behavior observed in Figures 10 and 11 is the effect of the hydrodynamic forces on the system stability. As it is seen, the existence of rub on the imbalanced drill string causes instability in a highly disturbed drilling system, and this effect is intensified proportionally to the axial excitation force for a given rub clearance (\(\delta\) = 6 \(\times\) 10\({}^{-4}\) m selected). When the bit begins to interact, the system will inevitably be disturbed by the possibility that axial and lateral vibrations occur simultaneously, and the formation of their critical frequencies is noticeable. Eccentricity, in this case, produces a high vibration level at \(1X\) in the radial direction. Due to various nonlinear coupling terms, all critical frequencies are increased during contact conditions, and several resonance peaks are  
Figure 8: Response of the imbalanced drilling system while passing its first critical speed. (**a**) 3D imbalance response orbit in air; (**b**) FFT response of imbalanced drill in air. Figure 9: Response of the imbalanced drill string system immersed in an inviscid fluid at the first critical speed. (**a**) 3D imbalance response orbit in fluid; (**b**) FFT response of drill string in fluid. observed as shown in Figure 10b. The effect of the permanent impact of the drill string on the casing generates multiple unwanted critical speeds, occurring at 75 rpm, 175 rpm, and 225 rpm (\(1X\), \(2X\)... \(5X\)). These undesirable frequencies show that the rub on the vertical drill string with axial excitation forces would practically reinforce the high vibration of the system and cause its destruction. The results of the excitation due to the imbalanced drill string-casing contact and axial forces, in Figure 10a, generate a high unwanted fluctuation along with axial displacement (\(Z\)-axis), as shown in the 3D orbit plot where the rub impact is strongly irregular throughout the radial bilinear friction forces. From Figures 10a and 11a the 3D orbit vibration of the drill string experiences changes from being periodic to chaotic eventually. It is shown that, depending on the radial bilinear frictional forces and the axial friction, the drill string undergoes a transient phase marked by intermittent rebounds. It is apparent, comparing both Figures 10b and 11b, that when the hydrodynamic forces are applied, the vibration response decreases considerably with a reducing range of the first critical speed. The presence of dissipative effects such as hydrodynamic forces eventually settles into multiple limit cycle behaviors, as seen in Figure 11a. Figure 11a reveals that, depending on the radial bilinear frictional forces and the axial friction, the unstable contact is intermittent along the radial direction where the friction, despite the presence of fluid, remains  
Figure 11: Response of the imbalanced drill string-casing system with rubbing in an inviscid fluid. (a) 3D imbalanced drill string response orbit in fluid with rub; (b) FFT response of imbalanced and rubbed drill in fluid. Figure 10: Response of the imbalanced drill string-casing system with rubbing. (a) 3D imbalanced drill string response orbit in air with rub; (b) FFT response of imbalanced and rubbed drill in air. effective. This means that the hydrodynamic forces influence the motion of the drill string, as observed throughout the torsional deflection (Figure 7b), where the torsional motion becomes attenuated with a large amplitude in the presence of fluid. **Observation 2:** The spectrum of the frequency illustrates also the effect of attenuation of vibration features of the drilling system partially immersed in a fluid.