## 8 Conclusions  
The dynamic analysis of a slant-cracked EG rotor-bearing system has been carried out using the finite element method to investigate the influence of a slant crack on natural frequencies, steady-state and transient responses. Exponential law is used to grade the material properties along the radial direction of an EG shaft, and exponential temperature distribution law is used for temperature gradation across the cross-section. The effect of crack depth, thermal gradient, crack location, torsional frequency, rotor speed, and angular acceleration on eigenfrequencies and dynamic responses have been investigated, and the following important conclusions are drawn from the analysis.  
1. The decrease in natural frequencies with an increase in crack depth is significant when a crack is located near the disc or far from the bearings. The stiffness of the cracked EG rotor is reduced due to the presence of a slant crack.
2. The percentage decrease in the natural frequency is less when a slant crack is present near the bearings compared to the percentage decrease in natural frequency when a slant crack is near the disc. The reduction in stiffness of the EG shaft is compensated by the bearing stiffness when a crack is present near the bearings. Therefore, the  
Figure 17: Transient FFT of (**a**) uncracked (**b**) slant\(-\)cracked EG rotor-bearing system with an angular acceleration of 30 rad/s\({}^{2}\), torsional frequency (\(\omega_{T}\)) = 65 rad/s and, \(\overline{\alpha}=0.3\) at \(\Delta\)T = 0 and \(\Delta\)T = 300 K.  
natural frequencies of an EG rotor-bearing system are hardly affected when a slant crack is present near the bearings.
* Natural frequencies of the EG rotor-bearing system decrease with the increase in the crack depth as the stiffness is affected. The increase in temperature gradient of the EG rotor decreases the stiffness as the Young's modulus is affected. Therefore, the stiffness of a cracked EG rotor-bearing system is furthermore reduced due to an increase in the thermal gradients. The sudden drop in the natural frequencies for different crack depths at higher thermal gradients has also been observed for this reason.
* Subharmonic peaks are found to be centred on the operational speed at an interval frequency corresponding to the torsional crack breathing frequency in steady-state frequency spectra of a slant-cracked EG rotor-bearing system. The amplitude of the harmonic peaks is increased with an increase in crack depth. Hence the severity of the crack could be identified through the sub-harmonic peaks.
* The frequency spectrum of the transient response of a slant-cracked EG rotor-bearing system is found to have the subharmonic frequencies centred on the critical speed of the rotor system at an interval frequency corresponding to the torsional frequency corresponds to crack breathing frequency. The presence of subharmonics on the frequency spectra confirms the presence of the crack in the EG rotor.  
Since a slant crack may develop due to excessive torsional vibrations of rotor systems, which is possible during the start-up and shutdown of machines, it is essential to detect the crack through vibration response to avoid catastrophic failures. From the present study, a slant crack could be detected through steady state and transient responses of an EG rotor system by extracting the subharmonic peaks corresponding to crack breathing frequency. However, for crack depths below 0.1, other detection techniques need to be explored. The work presented in this paper is based on the theoretical validations, and experiments are being planned for the near future.  
Conceptualization, P.S; methodology, P.S. and A.B.; validation, P.S. and A.B.; writing--original draft preparation, A.B.; writing--review and editing, P.S., G.C. and R.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.