## 5 Model Parameters and Numerical Simulation  
Taking a real Cardan shaft vehicle as an example, the parameters of the dynamic model such as stiffness, damping, mass and inertia, etc., (Figures 1, 3 and 4) are listed in Tables 1 and 2. The fluid damping, fluid stiffness and centrifugal fluid forces are estimated by Equations (32) and (33). The model of matrix differential Equation (37) is solved by using the model parameters in Tables 1 and 2. The torque profile resulting from rotor velocity and torque orbits are shown in Figure 6a,b, respectively.  
Considering only the torque and angular velocity relationships of the driving motor as they follow from Equations (39) and (40), and increasing linearly the driveshaft angular speed with time (Figure 6a), the applied torque may be required to vary unrealistically. A complete and accurate picture of the transition situation is obtained with a fairly straightforward analysis. The displayed result is likely to be valid as long as the operational driveshaft speed range investigated is below the maximum (Figure 6b). Moreover, Figure 5a indicates that for various shafts' inclination angle \(\beta>2\), there is a linear relationship between the phase input shaft angle and the joint perturbation function on which disturbance has a relatively significant effect when \(\beta\) increases.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Shaft Parameters** & **Value and Units** & **Shaft Parameters** & **Value and Units** \\ \hline Length of the shaft (L) & 570 mm & Shaft stiffness \(K_{1X}\) & 7.35 \(\times\) 10\({}^{5}\) Nm\({}^{-1}\) \\ Shaft Diameter (D) & 10 mm & Shaft torsional stiffness \(K_{1T}\) & 6.9 kN\({}^{-1}\)/rad \\ Shaft torsional damping (\(C_{1T}\)) & 90 Nms./rad & Stator stiffness \(K_{s}\) & 8 \(\times\) 10\({}^{6}\) Nm\({}^{-1}\) \\ Disc thickness (D\({}_{\rm disc}\)) & 10 mm & **Discs** & **Value and Units** \\ Disk moment of inertia (\(J_{D}\)) & 0.1861 kg m\({}^{2}\) & Mass (M) & 16.845 kg \\ Motor moment of inertia (\(J_{M}\)) & 10.1861 kg m\({}^{2}\) & Eccentricity mass (m\({}_{\rm u}\)) & 0.25 g \\ Friction coefficient (\(\lambda\)) & 0.2 & unbalance eccentricity (e) & 10 mm \\ Hooke’s joint angle (\(\beta\)) & 7\({}^{\circ}\) & Radial clearance (\(\delta\)) & 2.35 \(\times\) 10\({}^{-5}\) m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rotor system R-K 4 parameters.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Oil \(T^{\circ}\) & 20 \({}^{\circ}\)C \\ \(\omega\) & 36.86 rad/s \\ Viscosity (\(v\)) & 4.2 \(\times\) 10\({}^{-6}\) m\({}^{2}\) /s \\ Mass \(m_{\#}\) & 43.82 kg \\ Euler constant \(\Upsilon\) & 0.5772156 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Viscous fluid properties. (Oil type GST Oil 32 (Chevron)).  
Figure 6: Motor torque fluctuation (**a**) torsional torque transmitted to the driveshaft (**b**) driveshaft speed profile at a maximum speed of 1500 rpm.  
Equation (33) provides the time history of the reactive centrifugal fluid forces acting around the eccentric unbalanced disc 1 at various driveshaft speeds, as illustrated in Figure 5b. The profile of the centrifugal forces in the vicinity of disc 1 shows that when the rotating speed is linearly proportional to a critical value, the combination of disc and the incompressible liquid cause the formation of a harmonic sinusoidal fluid profile that is strongly dependent on the input rotational speed (Figure 5b). Thus, it is necessary to analyse the time-varying nature of the coupled systems partially immersed in a viscous fluid and to simulate the motion of the shafts during motion transmission.