## 2 Mathematical Model Descriptions
### The Equation of Motion of the Coupled Cardan Shaft
\begin{array}{l}m_{\varphi_{2}\theta_{1}}=(1-g)(J_{02}+m_{2\pi}e_{2}^{2});\;m _{\varphi_{2}\theta_{2}}=J_{02}+m_{2\pi}e_{2}^{2};\;m_{\varphi_{2}\theta_{1}}= 0;m_{\varphi_{2}\chi_{1}}=0;m_{\varphi_{2}\eta_{1}}=0\\ m_{\varphi_{2}\chi_{2}}=-m_{2\pi}(\epsilon_{2\pm 2}\cos\theta_{2}+\epsilon_{2\pm 2} \sin\theta_{2});\;m_{\varphi_{2}\eta_{2}}=m_{2\pi}(\epsilon_{2\pm 2}\cos\theta_{2}- \epsilon_{2\pm 2}\sin\theta_{2})\\ m_{\chi_{2}\theta_{1}}=m_{2\pi}(g-1)\big{[}(\epsilon_{2\pm 2}+\epsilon_{2\pm 2}\psi_{2}) \cos\theta_{2}+(\epsilon_{2\pm 2}-\epsilon_{2\pm 2}\psi_{2})\sin\theta_{2}\big{]};\;m_{ \chi_{2}\eta_{2}}=m_{\chi_{2}\chi_{1}}=0;\\ m_{\chi_{2}\theta_{2}}=-m_{2\pi}(\epsilon_{2\pm 2}\cos\theta_{2}+\epsilon_{2\pm 2} \sin\theta_{2});\;m_{\chi_{2}\chi_{2}}=(M_{2}+m_{2\pi});\;m_{\chi_{2}Y_{2}}=0 \\ m_{Y_{2}\varphi_{2}}=m_{2\pi}\big{(}\epsilon_{2\pm 2}\cos\theta_{2}-\epsilon_{2\pm 2} \sin\theta_{2}\big{)};\;m_{\chi_{2}Y_{2}}=(M_{2}+m_{2\pi});\;m_{\chi_{2}Y_{2}}=0 \\ m_{Y_{2}\varphi_{1}}=m_{Y_{2}X_{1}}=m_{Y_{2}Y_{1}}=m_{Y_{2}X_{2}}=0\end{array}  \tag{30}\]  
Based on partial Equation (22) and following Equation (23), the damping of the rotor system can be expressed by differentiating Rayleigh's dissipation function with respect to each generalised coordinate. The damping elements of the rotor system can be described by:  
\[\begin{array}{l}C_{\theta_{1}\theta_{1}}=C_{T2}(\frac{\theta_{1}(\pi-1)}{ \theta_{1}+\theta_{1}})^{2}\hat{C}_{\theta_{1}\theta_{1}}=C_{T_{2}}(\Gamma-\mu) \frac{|\theta_{1}(\Gamma-1)+\theta_{1}(\mu-1)|}{(\theta_{1}+\theta_{1})^{2}} \hat{\varphi}_{1}\hat{C}_{\theta_{1}\varphi_{2}}=C_{T_{2}}\frac{(\Gamma-\mu)}{ \theta_{1}+\theta_{1}}\hat{\varphi}_{1};\\ C_{\theta_{1}\theta_{1}}=C_{T_{2}}(\Gamma-\mu)\frac{|\theta_{1}(\Gamma-1)+\theta_{1}( \mu-1)|}{(\theta_{1}+\theta_{1})^{2}}\hat{C}_{\theta_{1}\varphi_{1}}=C_{T_{1}} +C_{T_{2}}\fraccross-sectional area A\({}_{1}\) of the cracked driveshaft about the fixed \(\overline{X}_{1}\) and \(Y_{1}\) axes and were given as:  
\[I_{\overline{X}_{1}}(t)\cong I-h_{1}(t)I_{11} \tag{34}\]  
\[I_{\overline{Y}_{1}}(t)\cong I-h_{1}(t)I_{11}+h_{2}(t)I_{22} \tag{35}\]  
and  
\[I_{\overline{X}_{1}\overline{Y}_{1}}(t)\cong-\left(\frac{I_{\overline{X}_{1}} (t)-I_{\overline{Y}_{1}}(t)}{2}+\frac{A_{1}e^{2}}{2}\right)\times\sum_{n=1}^{ p}\frac{2\overline{\phi}_{2}\sin(\pi-n\overline{\phi}_{2})}{\pi^{2}-n^{2} \overline{\phi}_{2}^{2}}\sin(n\Omega t) \tag{36}\]  
where \(I=\pi R^{4}/4\) is the cross-sectional area moment of inertia for the fully closed crack. The breathing functions \(h_{1}(t)\) and \(h_{2}(t)\) are defined by [18]:  
\[h_{1}(t)=\left(\cos(\frac{\Omega t}{2})\right)^{m}=\frac{1}{2^{m}}\left[ \left( \begin{array}{c}m\\ m/2\end{array} \right)+2\sum_{n=0}^{(m/2)-1}\left( \begin{array}{c}m\\ n\end{array} \right)\cos(\left((m-2n)\frac{\Omega t}{2}\right)\right] \tag{37}\]  
\[h_{2}(t)=\frac{1}{\pi}\left[-\frac{\phi_{1}+\phi_{2}}{2}+\frac{2}{\phi_{2}- \phi_{1}}\times\sum_{i=1}^{p}\frac{\cos(i\phi_{2})-\cos(i\phi_{1})}{i^{2}} \cos(i\Omega t)\right] \tag{38}\]  
Figure 2: (**a**) Schematic diagram and assembly of the interior of a driveshaft structure. (**b**) Deformed configurations of shaft discs 1 and 2.  
where \(i\), \(n\), and \(m\) are positive even numbers that control the deflection and shape of the breathing functions. As a result, the complete stiffness matrix \([K_{stiffness}]\) of the driveshaft with a breathing crack can be approximated by:  
\[[K_{crack}]=\left[ \begin{array}{cc}K_{X_{1}X_{1}}(t)&K_{X_{1}Y_{1}}(t)\\ K_{Y_{1}X_{1}}(t)&K_{Y_{1}Y_{1}}(t)\end{array} \right]\cong\frac{48E}{L^{3}} \left[ \begin{array}{cc}\hat{I}_{\overline{X}_{1}}(t)&\hat{I}_{\overline{X}_{ 1}\overline{Y}_{1}}(t)\\ \hat{I}_{\overline{Y}_{1}\overline{X}_{1}}(t)&\hat{I}_{\overline{Y}_{1}}(t) \end{array} \right] \tag{39}\]  
where \(E\) is the elastic modulus of elasticity, \(L\) is the length of the rotor.