## 7 Results and Discussions
### Transient Responses of a Slant-Cracked EG rotor-Bearing System  
Detection of crack through transient response of a rotor system is as important as steady state response when the rotor starts or stops quite frequently, as in case of aircraft engines. The transient responses of the uncracked and cracked EG rotor-bearing systems passing through the critical speed are depicted in Figures 13 and 14. The transient time response is modelled using the Houbolt method with a time step of 0.001 s. The amplitude of vibration increases when the crack is present. However, no obvious symptoms of crack are observed. However, the crack features can be identified through the frequency response of the uncracked and cracked EG rotor-bearing system. The first critical speed of the uncracked EG rotor is 296.6 rad/s and slant-cracked EG rotor system for \(\overline{\alpha}=0.3\) is 291.593 rad/s. Therefore, the peaks, shown in Figures 13b and 14b, are exactly located at 47.21 Hz and 46.41 Hz, respectively. The subharmonic peaks, shown in Figure 14b, are centred on 46.41 Hz with an interval frequency of 10.34 Hz, which is equal to torsional breathing frequency of the crack, \(\omega_{T}=65\) rad/s. The side lobes in the frequency spectra confirm the presence of the crack. The frequency of harmonic peaks of the transient frequency spectra are represented by \(\omega_{n}=\omega_{cr}\pm n\omega_{T}\); \(n=0\), \(1\)... \(\omega_{cr}\) is the critical speed of the EG rotor, which confirm the published results of the uniform shaft having a slant crack [29]. The first critical speeds of the EG rotor when \(\overline{\alpha}\) = 0.1, 0.2 and 0.4 are 296.254 rad/s, 294.668 rad/s and 285.741 rad/s, respectively. Therefore, it has been observed from Figure 15a-c that subharmonic peaks are centred on (critical speeds of slant-cracked EG rotors) 47.15 Hz, 46.90 and 45.48 Hz, respectively, with an interval frequency of 10.34 Hz, which is equal to the torsional frequency \(\omega_{\rm T}\) = 65 rad/s. When the crack depth is \(\overline{\alpha}\) = 0.4, the amplitude of the subharmonics is high, whereas the amplitude of the harmonics is low when crack depth \(\overline{\alpha}\) = 0.1. Hence, it is also possible to detect the severity of the crack in the accelerated system, as in case of the unaccelerated system. Figure 13: Transient (**a**) time response and corresponding (**b**) FFT of an uncracked EG rotor\(-\)bearing system with an angular acceleration of 30 rad/s\({}^{2}\). Figure 14: Transient (**a**) time response and corresponding (**b**) FFT of a slant\(-\)cracked EG rotor\(-\)bearing system with an angular acceleration of 30 rad/s\({}^{2}\), and torsional frequency (\(\omega_{T}\)) = 65 rad/s for \(\overline{\kappa}\) = 0.3. Figure 15: Transient FFT of a slant–cracked EG rotor – bearing system with an angular acceleration of 30 rad/s\({}^{2}\), and torsional frequency (\(\omega_{T}\)) = 65 rad/s for (**a**) \(\overline{\alpha}\) = 0.1 (**b**) \(\overline{\alpha}\) = 0.2 and (**c**) \(\overline{\alpha}\) = 0.4, respectively. The transient frequency responses of the cracked EG rotor-bearing system have been computed in Figure 16a,b for torsional frequency = 45 rad/s and 85 rad/s. From Figure 16a,b, it is found that the crack breathing frequency is an important parameter to identify the crack in an FG rotor system through the FFT plot. The presence of a crack can be detected even when the EG shaft is subjected to temperature gradients. Transient frequency spectra of an uncracked and cracked EG rotor-bearing system have been represented in Figure 17a,b for different thermal gradients. The amplitude of the frequency spectra at \(\Delta\)T = 300 K is higher than the amplitude at \(\Delta\)T = 0. As the first critical speeds of the uncracked and cracked EG rotor-bearing system at \(\Delta\)T = 0 and \(\Delta\)T = 300 K are different, the split between the first peaks is observed in Figure 17a,b. The subharmonic peaks are visible in the frequency responses when the crack is present.