## 4 Simulation and Experiment on the Cardan Shaft under Unbalanced Condition
### Simulation Response and Fault Diagnosis of a Cracked Cardan Shaft  
In order to extract the features of the driveshaft with a breathing crack and analyse the transmission motion to the unbalanced driven shaft, the breathing shaft stiffness \(\epsilon_{1c}=0.3\) was allowed, and an initial unbalance with \(m_{1u}=0.25\) kg was adopted. The steady-state response of the coupled rotor system with a breathing crack and an unbalance is shown for the vertical deflection and the FFT spectrum (Figure 7). From the analysis below (Figure 7c), it can be concluded that the identification of initial unbalances from the FFT spectrum response contains the information on synthetic unbalance, which is composed of initial unbalanced drive and driven shaft at its first critical speed (56.83 Hz). The steady-state response of the coupled rotor system with a breathing crack and unbalance (Figure 7) permitted to extract the highly oscillating peaks induced by the \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Shaft Parameters** & **Value \& Units** & **Shaft Parameters** & **Values and Units** \\ \hline Length of the shaft (L) & 570 mm & Shaft stiffness K\({}_{0}\) & 7.35 \(\times\) 10\({}^{5}\) Nm\({}^{-1}\) \\ \hline Shaft Diameter (D) & 10 mm & Shaft torsional stiffness K\({}_{\text{T}}\) & 8.2 \(\times\) 10\({}^{4}\) Nm\({}^{-1}\) \\ \hline Density of the material (\(\varphi\)) & 7800 kg m\({}^{-3}\) & Damper c & 200 Ns/m \\ \hline Modulus of elasticity (E) & 2.11 \(\times\) 10\({}^{11}\) Pa & **Discs** & **Values and Units** \\ \hline Viscous damping ratio & 0.8 \(\times\) 10\({}^{-6}\) s\({}^{-1}\) & Mass (m) & 16.45 kg \\ \hline [PERSON]’s joint angle (\(\beta\)) & 7\({}^{\circ}\) & Eccentricity mass (m\({}_{\text{u}}\)) & 0.25 kg \\ \hline Lateral modes & 0.0287 & Mass eccentricity (e) & 10 mm \\ \hline Torsional mode & 0.0088 & Moment of inertia, J\({}_{\text{M}}\) & 10.610 kg m\({}^{2}\) \\ \hline Torsional mode & 0.01234 & Disc moment of inertia, J\({}_{\text{D1,2}}\) & 0.1861 kg m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rotor system R-K 4 parameters. transverse crack successfully. An increase in amplitudes of vibration was observed for cracked driveshaft with the occurrence of a resonance point at the critical driveshaft speed of \(\omega\) to 5\(\omega\) amplitudes (red ellipses in Figure 7a). Figure 7a shows the lateral deflection of the driveshaft, which indicates that the maximum deflection at the resonance point was increased by including the breathing crack into the unbalanced model. The driven shaft response is shown in Figure 7b in terms of lateral deflection time history; there was a main resonance point corresponding to the ordinary unbalance and the other ones developed as a result of the [PERSON]'s joint parametric excitation between the two shafts. It was observed that the motion transmitted through the [PERSON]'s joint caused consistent fluctuation, as observed in the lateral deflection response of the secondary shaft with the appearance of some resonance points at the rotational speed of \(\Omega\) = \(\omega\)/2, \(\Omega\) = \(\omega\), and \(\Omega\) = \(n\omega\) (Figure 7b). The input shaft underwent substantial vibrations, which could be easily captured by the rotational speed, which happened at \(\Omega\) = \(n\omega\). This produced synchronous vibrations similar to those caused by a cracked shaft at Lower amplitudes compared to the primary shaft response. These results were confirmed by analysis of the frequency spectrum of both shafts (Figure 7c). It was noted that the influence of the cracked shaft was important at a high speed, as a variety of frequency components was observed (presence of 1\(\times\), 2\(\times\),... 6\(\times\)); then, it faded to regain its dominance, contrary to what is commonly known in previous research [25; 26]. A closer zoom on the frequency spectrum of the second shaft (in black colour) showed that it FFT peaks decreased considerably in amplitude compared to the driveshaft frequency.