## 5 Discussion and Conclusions
### Conclusions  
In this paper, the vibration behaviour of a large class of machines such as ground vehicle propeller shaft connected by the Cardan joint or other inter-coupled machines with primary and secondary rotary shaft inertia is described by this simplified equivalent configuration. Numerical and experimental methods study a model for analysing the coupled lateral and torsional vibrations of unbalanced and cracked driveshaft systems inter-connected with a Hooke's joint. The application of the classical techniques and the time-frequency techniques for monitoring methods of the cracked driveshaft, including the nonlinear perturbation of the flexible Hooke's joint under several operational conditions, were theoretically analysed and verified through measurements. The results are summarized as follows:  
The nonlinear dynamic model constructed from the equations of vehicle dynamics driveline, including 7 DOFs, is established by the Lagrangian formalism. The perturbation factor of the flexible coupling, the breathing crack mechanism and the second-order lateral and torsional vibrations of the coupled shaft system are simulated numerically.  
Figure 13: 3D experimental NWSST of (**a**) cracked unbalanced driveshaft, (**b**) unbalanced driven shaft.  
The concept of transmission of motion through a Hooke's joint should also take into consideration the perturbation generates by the input shaft as the study revealed that the weakness of the cracked driveshaft contributes to the high disturbance of the secondary shaft. This implies that the mixed mode of the rigid and disturbing Hooke's joint noise should be considered when selecting the Time-Frequency techniques.  
The numerical and experimental responses validate that noise and vibration are effectively significant when the intersection angle of the Cardan joint and the torsional and bending vibration of connected components increased. Experimentally, this phenomenon also means that the vibration amplitude transmission of the driveshaft throughout the Hooke's joint decreases considerably, and thus the flexible coupling fluctuation has as well good damping performance properties which were proved using the IF estimation method.  
The applications of the nonlinear synchrosqueezing wavelet transform (NWSST) and IFs, which directly deals with non-stationary response signals, are in good agreement with each other and exhibit the same changing trend as well.  
The results simulated by the model ultimately seemed quantitatively close to those obtained by experimentation on a test bench. However, the bench used generated a lot of noise, and the correlation was difficult. In the future, it could be considered to validate the model from another fairly robust test bench in order, on the one hand, to reduce the possible inaccuracies due to the experimental dimension, and on the other hand, to be able to check the validity of taking the support into account.  
The topic of this research was conceptualized by [PERSON] and [PERSON]; formal analysis was performed by [PERSON]; the first version manuscript was prepared by [PERSON]; and this version of the manuscript was read and approved by [PERSON] and [PERSON], who also substantially contributed to it. Both authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
This work is based on the research supported in part by the Vaal University of Technology (VUT), South Africa, through the Department of Higher Education and Training University Capacity Development Grant.  
The authors declare no conflict of interest.