## 3 Development of Analytical Crack Breathing Model  
The objective of this section is to develop an analytical model for determining the neutral axis position and to describe crack behaviour using non-symmetric bending principles. Understanding crack behaviour is useful for numerically determining the transient stiffness changes of the shaft local to the crack position and subsequently predicting the vibration behaviour of the shaft. Consider a transverse cut through a shaft of radius \(R\) at the location of a straight-front fatigue crack. This transverse slice can be fashioned as shown in Figure 7a, where the shaded region depicts a fully open crack of depth \(h\) and the remaining region \(A_{1}\) is the undamaged area. The normalised crack depth \(\mu\) is calculated by \(\mu=h/R\) and the location of the crack in Figure 7a marks the initial position of the shaft (\(\Omega t=0^{\circ}\)). Two Cartesian coordinate systems are established, the \(X\)-\(Y\) coordinate system remains fixed about the shaft centre O and the \(\overline{X}-\overline{Y}\) coordinate system whose centre remains coincident with the centroid of the non-cracked section \(C\). Throughout the rotation of the shaft, the \(\overline{X}\) and \(\overline{Y}\) axes remain parallel to the \(X\) and \(Y\) axes, respectively, and are therefore non-rotating. Moreover, the principal centroidal axes \(U\) and \(V\) are the first and second principal centroidal axes, respectively. The bending moment vector \(M\) acts upon \(C\) and its direction and magnitude are time-invariant as the model is assumed to be weight-dominant. In a weight-dominant model, only the weight of the rotor contributes to the breathing mechanism of the fatigue crack and any dynamic loads have a negligible contribution. The regions of the shaft above and below \(M\) are considered to be in compression and tension, respectively. Figure 6: Predicted neutral axis orientations with the crack at various positions. Furthermore, Figure 7 shows key milestones throughout the first half of the cracked shaft's rotation, which is useful for determining the transient change in area moment of inertia of the cracked cross-section. Figure 7a shows a fully open crack (shaded crack) in the initial position with the crack front parallel to the neutral axis. The value of the intact area \(A_{1}\) is calculated by Equation (1) [11],  
\[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]  
and the crack angle \(\alpha\)/2 is derived to be  
\[\frac{\alpha}{2}=cos^{-1}(1-\mu). \tag{2}\]  
Moreover, the distance between the \(X\) and \(\overline{X}\) axes, denoted \(e\), is given in [23] as  
\[e=\frac{2R^{3}}{3A_{1}}\gamma^{3} \tag{3}\]  
where \(\gamma=\sqrt{\mu(2-\mu)}\) is a convenience parameter. The position of the crack in Figure 7b depicts the instant that the crack begins closing; this angle is denoted as \(\theta_{1}\). As such, for angles \(0^{\circ}\leq\Omega t\leq\theta_{1}\), the crack is considered to be fully open. During this range of angles (excluding \(\Omega t=0^{\circ}\)), neither of the principal centroidal axes (\(U\) or \(V\) axes) is coincident with the bending moment vector, therefore, the neutral axis must be inclined relative to the bending direction, as shown. The angle between the first principal axis \(U\) and the bending moment vector, denoted \(\theta^{*}\), is given as  
\[\theta^{*}=\frac{\pi}{2}-\Omega t. \tag{4}\]  
Consequently, the angle between the first principal axis \(U\) and the neutral axis is given by Equation (5):  
\[\phi^{*}=tan^{-1}\Big{(}\frac{I_{U}}{I_{V}}tan\theta^{*}\Big{)} \tag{5}\]  
where \(I_{U}\) and \(I_{V}\) are the area moments of inertia of \(A_{1}\) about the principal centroidal axes [24]. When the crack is fully open, the values of \(I_{V}\) and \(I_{U}\) are equal to \(\bar{I}_{1}\) and \(\bar{I}_{2}\), respectively [12].