## 3 Development of Analytical Crack Breathing Model
\begin{table}
\begin{tabular}{c c} \hline \hline
**Crack Status** & **Range** \\ \hline Fully open & \(0\leq\Omega t\leq\theta_{1}\) \\ Partially open & \(\theta_{1}<\Omega t<\theta_{2}\) \\ Fully closed & \(\theta_{2}\leq\Omega t\leq 2\pi-\theta_{2}\) \\ Partially open & \(2\pi-\theta_{2}<\Omega t<2\pi-\theta_{1}\) \\ Fully open & \(2\pi-\theta_{1}\leq\Omega t\leq 2\pi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Crack state based on the shaft rotation angle. Figure 9: Percentage of the crack closure as the shaft rotates. Chosen crack depths of (**a**) \(\mu=0.5\), (**b**) \(\mu=0.7\) and (**c**) \(\mu=1\). \(R=6.35\) mm. Additionally, knowing the state of a crack is essential for determining the area moments of inertia of the cracked cross-section, which, in turn, is used to quantify stiffness changes in the shaft. When the crack is fully closed the area moment of inertia of the total non-cracked cross-section will be at a maximum (equal to \(\pi R^{4}\)) in the \(\overline{X}\) and \(\overline{Y}\) directions (\(I_{\overline{X}}\) and \(I_{\overline{Y}}\), respectively) and the product moment of area (\(I_{\overline{XY}}\)) is zero as the effective cross-section is a solid circle. (It should be noted that the product moment of area is linked to the local cross-coupling stiffness associated with the breathing crack.) When the crack is open to any degree, the area moment of inertia is non-maximum and product moment of area is non-zero and requires a more rigorous process to calculate. The variation in \(I_{\overline{X}}\), \(I_{\overline{Y}}\) and \(I_{\overline{XY}}\) over one period of the shaft rotation can be accurately approximated by [12]:  
\[I_{\overline{X}}(t)=\Big{(}I-\overline{I}_{1}\Big{)}f_{1}(t)+\overline{I}_{1}, \tag{13}\]  
\[I_{\overline{Y}}(t)=\Big{(}2I-\overline{I}_{1}-\overline{I}_{2}\Big{)}f_{2}( t)-\Big{(}I-\overline{I}_{1}\Big{)}f_{1}(t)+\overline{I}_{2}, \tag{14}\]  
\[I_{\overline{XY}}(t)=\Bigg{[}\frac{\overline{I}_{2}+\overline{I}_{1}}{2} \Bigg{]}f_{3}(t), \tag{15}\]  
where,\(f_{1}(t),f_{2}(t)\) and \(f_{3}(t)\), known as breathing functions, are shape functions that reflect the changes in area moment of inertia over time. The breathing functions were derived in the following form using Fourier series approximation:  
\[f_{1}(t)=1-\frac{1}{2^{p}}\Bigg{[}\Bigg{(} \begin{array}{c}p\\ p/2\end{array} \Bigg{)}+2\sum_{j=1}^{(p/2)}\Bigg{(} \begin{array}{c}p\\ j-1\end{array} \Bigg{)}\cos\Big{(}(p-2(j-1))\frac{\Omega t}{2}\Bigg{)}\Bigg{]}, \tag{16}\]  
\[f_{2}(t)=1-\bigg{(}\frac{\theta_{1}+\theta_{2}}{2\pi}\bigg{)}+\frac{2}{\pi( \theta_{2}-\theta_{1})}\sum_{j=1}^{p}\frac{cos(j\theta_{2})-cos(j\theta_{1})} {j^{2}}cos(j\Omega t), \tag{17}\]  
\[f_{3}(t)=\sum_{j=1}^{p}\frac{2\overline{\theta}_{2}sin(\pi-j\overline{\theta} _{2})}{\pi^{2}-j^{2}\overline{\theta}_{2}^{2}}sin(j\Omega t), \tag{18}\]  
where \(p\) is a positive even integer related to the number of harmonics in the Fourier series. Equation (18) provides the breathing function for the product moment of area [27], where the parameter \(\overline{\theta}_{2}\) is given as \(\overline{\theta}_{2}=0.8\times\theta_{2}\). Since these Fourier series approximations were based on data that assume that the neutral axis remains horizontal, the procedure must be reinvestigated to apply non-symmetric bending principles. Appendix A amends the iterative geometric procedure [12] to determine the inclination angle of the neutral axis, based on the area moments of inertia \(I_{\overline{X}}\), \(I_{\overline{Y}}\) and \(I_{\overline{XY}}\), repeatedly, until both are below a relative error threshold. Figure 10 shows the final neutral axis inclination angle at particular shaft rotation angles and for various crack depths.