## 5 Model Parameters and Numerical Simulation
### Vibration Analysis of the Unbalanced Rotors System  
The first dynamic response of the misaligned Cardan shaft is performed. To validate the proposed model in this study, the parameters for the numerical simulation are chosen to be the same as those of the experiment in [10], and the proportionality between \(\omega\) and \(\Omega\) is maintained for a wide range of \(\Omega\), which is the threshold (\(\omega=\Omega\)). To illustrate the numerical results, the following plots in Figures 7 and 8 are used. As the engine runs, the elliptical orbits of the driveshaft fluctuate with a change in shape (the waveform becomes disturbed closer to the critical speed). The orbital trajectories of the centre of the rotors in a steady state with some regular and intermittent loops are orbitally stable (Figure 7a,b). The operating point of the driven shaft is close to the limit of stability compared to the previous study on the balanced twin-rotor system in [2]. The frequency analysis (Figure 7c) is confirmed in both shafts, the presence of the main rotational frequency from 0 to 50 Hz and the frequency multiple of the rotation frequency of the drive shaft. Note that the value of the over-synchronous frequency is shifted, which reflects the action of the engine power on the dynamic behaviour of the unbalanced driveshaft. The comparative temporal analysis in the direction of the force of gravity \(Y_{1}\), \(Y_{2}\) (Figure 7d) shows the phase difference between the two shafts. As the speed increases, the system displacement response shows that the lateral deflection keeps on attenuating until constant vibration. However, for each period, the movement is identical in the case of the first shaft and practically nonlinear in the case of the output shaft with larger amplitudes. The system torsional deflection in Figure 8b shows the complexity of torsional deflection in the operating period of vibration. Conversely, after a transient vibration process with a much higher amplitude, it gradually becomes stable and the amplitude of vibration continues to subside until torsional vibration stops. This implies that the input excitation of the motor is responsible for strong torsional vibration and is a valuable approach to control the torsional vibration of the driveshaft. This is inferred in the following analysis: a time-frequency method is applied for measuring driveshaft fluctuation based on the engine speed and the shaft oscillation frequency, which does not involve any mechanical forces except for the measured unbalance. The measured pickup signal spectrum was analysed and yielded the driveshaft spectral response. The proposed method suitable to different system conditions is tailored for various modes of operation (unbalance, friction and fluid friction). The 3-D speed-frequency spectrum results presented in Figure 8a shows the effect of operating speeds on FFT plots to analyse the system behaviour. The frequency responses to the initial unbalance generated a randomly appearing subsynchronous frequency scattered around the resonance frequency, which reflects the action of the motor on the dynamic behaviour of the unbalanced driveshaft. It can be seen that the amplitude of instantaneous frequency is about 50 Hz and its fluctuating frequency is in the range of 0-100 Hz, which is consistent with the set parameters as shown in Figure 7c. The frequencies are more important in amplitude and present around the main frequency of rotation at a considerable level of disturbance (unstable vibration). This section finds that the centrifugal forces cause high vibration amplitude at a frequency equal to 1\(\times\) rpm in spectral data and waveform in the time domain of the shaft's lateral deflection. Misalignment of couplings typically produces raised frequencies at 1\(\times\) rpm and 2\(\times\) rpm that are not essential to distinguish unbalance from misalignment.