###### Abstract  
Many studies on cracked rotors developed crack breathing models that assume that the neutral axis of bending always remains horizontal for simplification. These models may generate significant discrepancies and thus there is a need to develop more sophisticated models to look into the shifting of the neutral axis for a cracked rotor. Herein, a case study on the shifting of the neutral axis for a cracked rotor is firstly performed by using a three-dimensional finite element model to confirm that the neutral axis becomes inclined as the cracked rotor rotates. In response to this finding, non-symmetric bending principles are used to develop a new crack breathing model which has the advantage of being able to numerically calculate the inclination angle of the neutral axis. When compared to an existing crack model in the literature that assumes that the neutral axis remains horizontal (HNA model), the proposed model is relatively less stiff in bending as a result of an overall lower area moment of inertia. Using the harmonic balance method, a two-dimensional finite element vibration model of a cracked rotor was devised by employing the proposed crack breathing model and the HNA model for validation. It can be found that the vibration amplitudes of the first three frequency components are similar between the two models for shallow cracks and significantly differed for deep cracks. This result highlights the potential of the proposed model for modelling and detecting mid-to-late-stage cracks in rotors. c 2020 xxxxxxxx-xxxx-xxxxxx-xxxxxx-xxxxxx-xxxxxxxx-xxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxx-xxxxxxxxxx-xxxxxxxxxx-xxxxxxxxxx-xxxxxxxxxxxx-  
behaviour of breathing cracks was neglected. Additionally, switching crack models are associated with chaotic and quasi-periodic vibrations that are not seen in experimental testing [10]. As such, there was a significant rise in the number of approaches that describe the crack breathing mechanism of a rotating shaft with more complexity. Sinusoidal functions or other periodic waveforms [11; 12; 13; 14] provide a more intricate description of the intermediate crack behaviour between fully open and fully closed states. An overarching objective of crack breathing models is to quantify transient stiffness changes in a cracked shaft in order to study the dynamic behaviour of the entire cracked rotor. Strain energy release rate (SERR) theory and reduction of the second moment of inertia are examples of popular theories to describe the breathing mechanism as a function of shaft rotation and crack depth. Studies pertaining to SERR of cracked rotor systems [15; 16; 17] employed linear fracture mechanics and rotor dynamics to calculate the local compliance of the rotating crack. The change in stiffness of a rotating shaft due to a transverse fatigue crack can be modelled by considering the periodic change in the area moment of inertia of the cross-section immediate to the crack location [18]. As the shaft rotates, the area moment of inertia of the transversely cracked cross-section varies between maximum and minimum values, depending on the openness and orientation of the crack. The transient area moment of inertia of the cracked cross section in conjunction with the Euler-Bernoulli beam theory is useful for developing the time-varying stiffness matrix of cracked rotor, particularly with the finite element method [11; 12; 19; 20]. A great number of studies examined vibration patterns indicating the presence of a fatigue crack in a rotating shaft. In particular, virtually all vibration frequency analyses of cracked rotors with sufficiently deep fatigue cracks [21; 22; 23] reported highly detectable super-harmonic components (2X, 3X or 4X, etc.) at discrete fractions of a cracked rotor's critical speeds. For example, at one-third of critical speeds the 3X harmonic component is appreciable. Similarly, at one-half of critical speeds the 2X harmonic component is also appreciable.