## 5 Conclusions  
The 3-D FEA cracked rotor model herein demonstrated that the neutral axis of bending for a cracked cross-section cannot remain horizontal. This preliminary investigation showed that the neutral axis inclination angle increases from horizontal as the crack rotates depending on the openness of the crack. In response to this finding, the new breathing model was developed using non-symmetric beam bending theory to determine the changes in the geometry of the breathing crack. The analytical model supports the preliminary findings regarding a shifting neutral axis and also that the peak inclination angle value increased as the crack became deeper. In fact, the neutral axis is only seen to be horizontal when the crack is at its starting position or fully closed. Due to the time-varying neutral axis a new formula for the crack closing angle \(\theta_{1}\) was derived out of necessity. It was found that values \(\theta_{1}\) calculated through the proposed formula diverged from an existing formula [12] as the crack deepened, where at a crack depth of \(\mu=1\) there was a significant difference of 33.6\({}^{\circ}\). As a result, for the proposed model, the crack begins closing later in its rotation and re-opening sooner and so the cracked shaft would be less stiff in bending when compared to the HNA model. This change in behaviour was most clearly manifested through the predicted \(I_{\overline{V}}\) values of the two models, where the proposed model had overall lower \(I_{\overline{V}}\) values for \(\mu=0.3\), 0.5 and 0.7 and particularly more so for the deeper cracks. As for the 2-D FE vibration analysis of the cracked rotor using the harmonic balance method, the vertical amplitudes of the first three harmonic components (1X, 2X and 3X) were focused on. The proposed crack breathing model was found to be highly similar to the HNA model for the case of no crack, \(\mu=0.2\) and \(\mu=0.5\) except at particular rotor speeds referred to as points of interest. Four points of interest were chosen and it was found that at two points of interest, the two models differed because of the numerical instability of either model and the other two points had largely different amplitudes without numerical instabilities. Having differences without numerical instability suggests that the difference in crack breathing behaviour between the two models has some value for predicting the vibration of cracked rotors with early-mid-stage cracks. Furthermore, the differing transverse trajectory predicted between the two models is suggested as a potential indicator for experimental validation of the proposed model. As a side note, both models predicted an appreciable increase in the amplitude of the 1X, 2X and 3X frequencies and an increase in amplitude peaks at discrete fractions of the first through third critical speeds as the crack deepened. Both of these factors are typical indicators for the presence of fatigue cracks in rotating shafts. The 1X, 2X and 3X amplitudes at \(\mu=0.8\) resulted in a substantial difference between the two models where virtually no common amplitude peaks occurred. The proposed crack breathing model may be useful to capture the dynamic behaviour of rotors with mid-to-late-stage cracks once it is experimentally validated, which could be part of future work for continuation. [PERSON] created the finite element models and conducted simulations to extract the numerical results as the Ph.D. student and also drafted the paper. [PERSON] and [PERSON] helped with data analysis, technical discussions and revision of the paper. All the authors contributed to results and discussions and commented on the paper. All authors have read and agreed to the published version of the manuscript. The first author was supported by the Postgraduate Scholarship Awards provided by the Western Sydney University. This support is greatly appreciated. The authors would like to show gratitude to [PERSON] and [PERSON] for their valuable discussions on devising the cracked rotor model in Abaqus/CAE.