## 7 Results and Discussions  
In the present work, the natural frequencies of the slant-cracked EG rotor system are calculated to understand how the crack parameters and temperature gradients affect the dynamic behaviour of the EG rotor-bearing system. The steady-state and transient responses have been simulated to establish the slant crack detection methodology and to study the severity of the crack and the crack breathing frequency through the vibration responses of a slant-cracked EG rotor-bearing system. A Python code has been developed to determine the natural frequencies, steady-state and transient responses. The numerical results obtained in this section are based on the theory derived in this study with proper validations. An EG rotor-bearing system with a slant crack is considered. The inner core of the EG shaft is made of Stainless Steel, and the outer layer is made of Zirconia. The material gradation and temperature distribution varied radially across the cross-section using exponential law and ETD. Natural frequencies, steady-state responses and transient responses have been calculated using finite element analysis for flexural vibrations. Houbolt time marching technique has been employed to compute the time response, and the corresponding frequency spectra are obtained by using Fast Fourier Transform. Rotor-bearing data used in the present work are tabulated in Table 5. Natural frequencies of an uncracked and cracked EG rotor-bearing system are computed at 10, 14, 18, 20 and 22 finite elements in Table 6 to check the mesh convergence. The natural frequencies are found to be converged when the EG rotor is discretised into twenty finite rotor elements. Therefore, twenty finite elements have been considered in the present work. Effect of Crack Depth (\(\overline{\alpha}\)) on Natural Frequencies for Different Crack Locations (\(\overline{Lc}\))  
The natural frequencies of a slant-cracked EG rotor-bearing system at room temperature are plotted against normalised crack depths for different crack locations. It has been noted from Figure 6a, the natural frequencies of the rotor-bearing system decrease with an increase in crack depth. This phenomenon is caused due to the reduction in stiffness in the EG shaft with the increase in the crack depth. Natural frequencies of the slant-cracked EG rotor-bearing system, in Figure 6b, with the crack positioned near the bearings (\(\overline{Lc}\) = 0.5) turn out to be higher than those obtained when the crack position is far from bearings (near the disc) (\(\overline{Lc}\) = 9.5). The reason is that the crack effect is not significant when it is near to the mode's nodal point, namely the bearing location. From Figure 6a, it is observed that the curve (\(\overline{Lc}\) = 0.5) almost remains straight with the increase in crack depth. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Points** & **Present** & **Digitised Values [(29)]** & **l Difference** \(\mid\) \\ \hline
1 & 2.839 & 2.842 & 0.003 \\
2 & 1.059 & 1.063 & 0.004 \\
3 & 0.652 & 0.652 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The amplitude (mm) of the points represented on Figure 5b. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Points** & **Present** & **Digitised Values [(26)]** & **l Difference** \(\mid\) \\ \hline
1 & 0.019 & 0.020 & 0.001 \\
2 & 0.021 & 0.021 & 0 \\
3 & 0.013 & 0.014 & 0.001 \\
4 & 0.016 & 0.017 & 0.001 \\
5 & 0.015 & 0.015 & 0 \\
6 & 0.015 & 0.016 & 0. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No. of Finite** & **Uncacked (Hz)** & **a/D = 0.1 (Hz)** & **a/D = 0.3 (Hz)** \\ \hline
10 & 47.2127 & 47.2212 & 47.2214 \\
14 & 47.2127 & 47.1955 & 46.9362 \\
18 & 47.2127 & 47.1637 & 46.5665 \\
20 & 47.2127 & 47.1637 & 46.5672 \\
22 & 47.2127 & 47.1637 & 46.5672 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Natural frequencies of an uncracked and cracked EG rotor-bearing system for various finite elements to determine the mesh convergence. Figure 6: _Cont._