## 7 Results and Discussions
### Steady-State Responses of a Slant-Cracked EG Rotor-Bearing System  
Steady-state time and frequency responses of an uncracked and slant-cracked EG rotor-bearing system for normalised crack depth (\(\overline{\alpha}\)) = 0.3 at \(\overline{Lc}\) = 9.5 are shown in Figures 8 and 9. Steady-state time responses are obtained using the Houbolt method and the corresponding frequency responses have been obtained using Fast Fourier Transform (FFT). The crack features cannot be confirmed from the steady-state time signals since  
Figure 6: Variation in natural frequencies of a cracked EG rotor\(-\)bearing system for different: (**a**) crack depths; (**b**) crack locations. Figure 7: Variation in natural frequencies for different crack depths at different thermal gradients. the amplitudes of steady-state time responses, shown in Figures 8a and 9a, are similar. Therefore, the corresponding frequency spectra are used to detect the crack in the rotor. The first peak at 9.55 Hz, shown in Figures 8b and 9b, represents the rotor speed, and the second peak at 47.21 Hz and 46.41 Hz represents the critical speeds of the uncracked and cracked EG rotor-bearing system, respectively. The side lobes (subharmonic peaks) are present in Figure 9b due to the presence of a crack. These harmonic peaks are centred on 9.55 Hz (rotor speed of the shaft) with an interval frequency corresponding to the breathing torsional frequency \(\omega_{T}=40\) rad/s. The frequency of the harmonic peaks of the steady-state frequency spectra can be represented by \(\omega_{n}=\Omega\pm n\omega_{T}\); \(n=0\), \(1\)... [26]. However, the subharmonic peaks are not present in Figure 8b as the crack is not present in the EG shaft. \(\omega_{cr}\) is the critical speed of the slant-cracked EG rotor-bearing system. Figure 8: Steady\(-\)state (**a**) time response and corresponding (**b**) FFT of an uncracked EG rotor\(-\)bearing system with a rotor speed of 60 rad/s. The frequency spectra of a slant-cracked EG rotor-bearing system for different normalised crack depths are shown in Figure 10a-c. The subharmonic peaks, shown in Figure 10a-c, are centred on 9.55 Hz (60 rad/s) with an interval frequency of 6.37 Hz, which is the same as the torsional frequency \(\omega_{T}=40\) rad/s. Although the subharmonic peaks are centred on 9.55 Hz (60 rad/s), the amplitude of the subharmonics is not clearly visible on the frequency spectra when the crack depth \(\overline{\alpha}=0.1\). However, the amplitude of the subharmonics when the crack depth \(\overline{\alpha}=0.4\) is enormous compared to the sub-harmonic frequencies at \(\overline{\alpha}=0.1\), 0.2 and 0.3. The amplitude of the harmonic peaks increases with increase in the crack depth. Therefore, it is possible to determine the severity of the crack using frequency spectra of an EG rotor-bearing system. Steady-state responses of slant-cracked EG rotor-bearing systems have been computed for different torsional frequencies and are represented in Figure 11a,b. From the time responses, the frequency of the torsional vibration cannot be indicated, however it is  
Figure 9: Steady\(-\)state (**a**) time response and corresponding (**b**) FFT of a slant\(-\)cracked EG rotor\(-\)bearing system with a rotor speed of 60 rad/s, and torsional frequency (\(\omega_{T}\)) = 40 rad/s for \(\overline{\alpha}=0.3\). possible to identify the crack through the FFT plot. The subharmonics are centred on 9.55 Hz (60 rad/s), in Figure 11a,b, at an interval frequency of 3.18 Hz (20 rad/s) and 9.55 Hz (60 rad/s), respectively. Figure 10: Steady–state FFT of a slant–cracked EG rotor-bearing system with a rotor speed of 60 rad/s, and torsional frequency (\(\omega_{T}\)) = 40 rad/s for (a) \(\overline{x}=0.1\) (b) \(\overline{x}=0.2\) and (c) \(\overline{x}=0.4\), respectively. In Figure 12a,b, steady-state frequency responses of uncracked and cracked EG rotor-bearing systems have been depicted at \(\Delta\)T = 0 and \(\Delta\)T = 300 K. The amplitude of the FFT at \(\Delta\)T = 300 K is higher compared to the FFT at \(\Delta\)T = 0.