## 2 Mathematical Model Descriptions
### The Equation of Motion of the Coupled Cardan Shaft
Since the cross-section of the cracked driveshaft is symmetric about the \(Y_{1}\) axis, so that \(I_{X_{1}Y_{1}}=0\), the cross-section \(\hat{I}_{\overline{X}_{1}\overline{Y}_{1}}(t)=\hat{I}_{\overline{Y}_{1} \overline{X}_{1}}(t)=0\). For the non-dimensional crack depth (\(E_{1c}=h/R=0.1\)), the time-varying stiffness, which represents the breathing mechanism of the transverse crack, and the torsional stiffness during the rotating process are shown in Figure 3.  
Figure 3a shows self-excited oscillations when cracks occurred on the driveshaft. The cracked shaft stiffness components oscillated periodically with an oscillation period equal to the rotating shaft period. The torsional stiffness of the driveshaft was significantly excited up at the start-up and then self-oscillated into harmonic and periodic motion before converging to stability. Compared with the time-varying stiffness in the \(X_{1}\) and \(Y_{1}\) directions, the amplitude of the torsional stiffness was much smaller; therefore, we experimentally neglected its effect. Upon introducing the breathing crack functions into the governing equation, the system equation of motion was found to be:  
\[\left[ \begin{array}{cccccc}m_{\theta_{1}\theta_{1}}&m_{\theta_{1}\varphi_{1} }&m_{\theta_{1}X_{1}}&m_{\theta_{1}Y_{1}}&m_{\theta_{1}\varphi_{2}}&m_{\theta_ {1}X_{2}}&m_{\theta_{1}Y_{2}}\\ m_{\varphi_{1}\theta_{1}}&m_{\varphi_{1}\theta_{1}}&m_{\varphi_{1}X_{1}}&m_{ \varphi_{1}Y_{1}}&0&0&0\\ m_{X_{1}\theta_{1}}&m_{X_{1}\varphi_{1}}&m_{X_{1}X_{1}}&0&0&0&0\\ m_{Y_{1}\theta_{1}}&m_{Y_{1}\theta_{1}}&0&m_{Y_{1}Y_{1}}&0&0&0\\ m_{\varphi_{1}\theta_{1}}&0&0&0&m_{\varphi_{1}\theta_{1}\theta_{1}}&m_{\varphi _{1}\theta_{1}}&m_{\varphi_{1}\theta_{1}}\\ m_{\varphi_{1}\theta_{1}}&0&0&0&m_{\varphi_{1}\theta_{1}}&m_{\varphi_{1} \theta_{1}}\\ m_{\varphi_{1}\theta_{1}}&0&0&0&m_{\varphi_{1}\theta_{1}}&m_{\varphi_{1} \theta_{1}}\\ m_{\varphi_{1}\theta_{1}}&0&0&0&0&0\\ \end{array} \right]\left[ \begin{array}{cccccc}C_{\theta_{1}\theta_{1}}&C_{ \theta_{1}\varphi_{1}}&0&C_{\theta_{1}\varphi_{2}}&0&0\\ C_{\theta_{1}\theta_{1}}&C_{\varphi_{1}\theta_{1}}&0&0&C_{\varphi_{1} \theta_{1}}&0&0\\ 0&0&C_{X_{1}X_{1}}&0&0&0\\ 0&0&0&C_{Y_{1}Y_{1}}&0&0&0\\ C_{\varphi_{1}\theta_{1}}&C_{\varphi_{1}\theta_{1}}&0&0&C_{\varphi_{1} \theta_{1}}&0\\ 0&0&0&0&C_{\varphi_{1}\theta_{1}}&0&0\\ C_{\varphi_{1}\theta_{1}}&C_{\varphi_{1}\theta_{1}}&0&0&C_{\varphi_{1} \theta_{1}}&0\\ 0&0&0&0&0&C_{\varphi_{1}\theta_{1}}&0\\ \end{array} \right]\left[ \begin{array}{cccccc}\theta_{1}\\ \psi_{1}\\ X_{1}\\ Y_{1}\\ Y_{2}\\ \end{array} \right]+ \tag{40}\]  
\[\left[ \begin{array}{cccccc}0&0&0&0&0&0\\ K_{\theta_{1}\varphi_{1}}&0&0&K_{\gamma_{1}}(\mu-1)&0&0\\ 0&K_{0}-K_{X_{1}X_{1}}(t)&K_{12c}&0&0&0\\ 0&K_{21c}&K_{0}-K_{Y_{1}Y_{1}}(t)&0&0&0\\ 0&K_{T_{2}}(\mu-1)&0&0&K_{\theta_{2}\varphi_{2}}&0&0\\ 0&0&0&0&K_{0}&0\\ 0&0&0&0&0&K_{0}\end{array} \right]\left\{ \begin{array}{c}\theta_{1}\\ \psi_{1}\\ X_{1}\\ Y_{1}\\ Y_{2}\\ \end{array} \right\}=\] \[\left\{\begin{array}{c}\par\par\par\par\par\par\par\par\par\par \par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par \par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par \par\par\par\par\par\par\par\par\par\[ \begin{array}{ll}\xi_{L_{2}}&=\xi_{L_{1}}=\frac{c_{X_{1}X_{1}}}{2\sqrt{K_{1}X_{1} \chi_{1}(M_{1}+m_{2})}};\ \ \ \text{and}\\ \xi_{T_{1}}&=\frac{c_{P_{1}}t_{1}}{2\sqrt{K_{1}\eta_{1}(10+m_{1}a_{2}^{2})}};\ \xi_{T_{2}}=\frac{c_{P_{ 1}\eta_{2}}}{2\sqrt{K_{2}\eta_{2}(10+m_{2}a_{2}^{2})}}\\ \omega_{L_{2}}&=\omega_{L_{1}}=\sqrt{\frac{K_{X_{1}X_{1}}}{M_{1}+m_{2}}};\ \omega_{T_{1}}=\sqrt{\frac{K_{ \eta_{1}\eta_{2}}}{(10+m_{1}a_{2}^{2})}};\ \omega_{T_{2}}=\sqrt{\frac{K_{\eta_{1}\eta_{2}}}{(10+m_{2}a_{2}^{2})}} \end{array}  \tag{43}\]  
The dynamic behaviour of the coupled lateral and torsional vibration of the driveline was then investigated numerically by integrating the non-linear state-space equations of motion (40).