## 5 Model Parameters and Numerical Simulation
### Influence of Fluid Forces on the Vibration Response of the Rubbing Coupled System  
Based on the fundamental assumption that the mass of fluid surrounding the drivehaft always shows laminar flow behaviour dependent on the rotational speed of the input shaft, to avoid having a turbulent effect, the system response is simulated with the propeller shaft speed equal to twice the excitation frequency of the fluid in motion (\(\Omega=2\,\omega\)). In the case of the driveshaft immersed in liquid, the responses of the fluid-rotor system become more complicated when combining the nonlinear fluid forces with the nonlinear rotor-stator impact force. External damping fluid has the effect of reducing the system vibration response and changes aperiodically the fluid force's amplitudes that arise during driveshaft excitation (Figure 11a,b). Figure 10: The 3-D driveshaft frequency spectrum of angular velocity during acceleration. (**a**) motor speed variation frequency (**b**) angular velocity vibration of the unbalanced and rubbing driveshaft. The mutual interaction of the mass of fluid and the lateral-torsional motion becomes important for supercritical speeds, while the transition deflections are reduced, especially when the small-diameter shafts and the influence of torque loads are involved. Despite the presence of external damping in the system, the simulation in Figure 12a shows an even stronger fluctuating vibration generating two maximum circular orbit responses at a critical speed, which causes the circular orbit to attenuate. Figure 11: Frictional forces profile. (**a**) dry driveshaft-stator friction (**b**) wet driveshaft-stator rubbing effect. Figure 12: Dynamic response of driveline rotors with driveshaft-to-stator rubbing inside fluid (**a**), (**b**) orbit of the input and output shafts, (**c**) Fast Fourier Transform spectrum response (**d**) transient deflection response of shafts. However, the natural frequency of the rotating driven shaft generates an elliptical orbit, which in Figure 12b does not diverge, indicating that the rotor vibration is effectively suppressed by the coupling joint. It was mentioned in Section 5.1 that the classical critical speed is excited by the unbalanced mass. However, it is observed from Figure 12c that for isotropic rotors, the fluid effect may predict two disturbed regions; one is, in particular, above the critical speed of the rotor's system and the other is below the critical speed. Adding external damping fluid makes the system more stable (Figure 12d) and can be an effective method of suppressing the strong excitation of the driveshaft-stator rubbing motion in certain cases. Figure 13a shows the characteristics of periodic motion in a viscous fluid at \(\Omega=30\) rad/s through motor speed, frequency spectrum and amplitude. All of the modelled excitations of the driveshaft can be seen in the simulated spectrum maps. With increasing rotational speed, the system passes through a critical speed at \(\omega=120.6\) rad/s and enters regular motion. The amplitude of each frequency component varies nonlinearly and, at a certain speed, there is a continuous spectrum component evident in the three-dimensional spectrum. Contrary to the results in [5], the vibration response demonstrated that the second-order vibration and noise generated by the Cardan joint can be decreased effectively by considering the effects of the fluid forces. In summary, nonlinear fluid forces have an important influence on the stable operation and dynamic characteristics of the driveshaft system. Usually, very low vibration amplitudes are required over a large range of exciting forces and frequencies to keep dynamic stresses, noise, fatigue and other effects to acceptable levels. Some periodic forces are harmonic but, even if they are not, they can be represented as a series of harmonic functions using time-frequency spectrum analysis techniques. It can be seen from the results that the amplitude of both shaft's system vibration response decreases and changes periodically in the fluid domain.