## 4 Simulation and Experiment on the Cardan Shaft under Unbalanced Condition
### Experimental Response and Fault Diagnosis of a Cracked Cardan Shaft  
Experiments using a four-bearing spin bench supporting two connected shafts (one of them exhibited a transverse breathing crack) were conducted to validate the simulation presented in Sections 4.1 and 4.2. The reduced diameter of the driveshaft (Section 3.3) has been designed so that at the reduced area, the crack would not propagate while the driveshaft is operating (Figure 5a) and at least one critical speed is in the 0-2000 rpm rotating range. Following the measurement of the high stresses in the cracked driveshaft, an assessment of the possibility of identifying a transverse crack by vibration monitoring was conducted and an estimate was made of the depth of the crack that could be detected as presented in Figure 9.  
The dynamic response obtained experimentally during start-up of a secondary shaft disc M\({}_{2}\) was clearly affected by the breathing crack but only at low frequencies. Time-domain response contained many tighten harmonics (Figure 9b); however, it may be seen that the amplitude of output time response of secondary shaft-disc M\({}_{2}\) was lower compared with the case shown in Figure 9a. The crack on the primary shaft M\({}_{1}\) gradually affected the disturbance of the shaft-disc M\({}_{2}\) and the appearance frequency value of the harmonics of the second shaft increased, which corresponded to a reduction of the maximum periods of the movement due to the crack of the main shaft. During the primary shaft deflection, a split at resonance occurred as a result of the crack impact (Figure 9a). For the multi-fault condition, the frequency response (extracted from the signals in Figures 8c and 9c,d) provided useful information about the resonance frequency of the rotor at the vicinity of its first critical speed. The behaviour of the cracked system during this period was highly non-linear, and standard features of a cracked shaft manifested by the occurrence  
Figure 8: Experimental response of the lateral-torsional Cardan shaft: (**a,b**) lateral deflection of shaft 1 and shaft 2, and (**c,d**) frequency spectrum of the drive and driven shafts. of a vibration component of sup-harmonic peaks at low amplitude were observed. The intensity of the sub-synchronous frequencies increased significantly, and its characteristics were found on the second shaft as well, thus destabilising the system. This observation has been recently reported for a coupled lateral and rigid-rotor rotation with neglected torsional deflection of the shaft [21]. Extraction of features of the secondary shaft M\({}_{2}\) contained harmonics of the main resonance phenomenon in a low-frequency amplitude, which can be used as a good indicator of the existence of cracks. However, the contribution of the crack is always present, the parametric nonlinearity of the Hooke's joint affects the total non-linear response and more particularly on the super-harmonic orders \(n\times\), which are directly proportional to the intensity of the rotor amplitudes. The evolution of orders 2\(\times\) and 3\(\times\) as a function of the speed of rotation of the rotor clearly showed peaks at the passages of sub-harmonic resonances (Figure 9c,d). This observation constitutes an essential point for the prediction of the rotor crack. Figure 9: Experimental response of lateral-torsional coupled Cardan shaft with a crack: (**a,b**) vertical deflection of drive and driven shaft and (**c,d**) frequency spectrum of shaft 1 and shaft 2. In conclusion, the use of super-harmonic order tracking makes it possible to identify the presence of a crack regardless of the intensity of the unbalance compared with the non-linear effects due to the presence of a rotating crack. Knowing, therefore, that a crack strongly influences the dynamic behaviour of a rotor, and that noise influences the analysis of the data collected, a more in-depth analysis is necessary because of the coupling joint and the speed ratio existing between the two rotors.