## 2 Mathematical Model Descriptions
### The Equation of Motion of the Coupled Cardan Shaft  
Substituting the expression of kinetic and potential energy and Rayleigh's dissipation function into [PERSON]'s formulation with respect to \(\{q\}\), the vector of generalised coordinates  
\[\frac{d}{dt}\left(\frac{\partial G_{r}}{\partial\dot{q}}\right)+\frac{\partial R }{\partial\dot{q}}+\frac{\partial V}{\partial q}-\frac{\partial G}{\partial q}=T _{q};\quad q=\theta_{1},\psi_{1},\psi_{2},X_{1},Y_{1},X_{2},Y_{2} \tag{23}\]  
yielding the system vibration equation in the form:  
\[[M]\big{\{}\ddot{q}\big{\}}+[C]\big{\{}\dot{q}\big{\}}+[K]\big{\{}q\big{\}}=\big{ }\big{[}\mathrm{NL}_{q}\big{\}}+\big{\{}F_{q}\big{\}}+\big{\{}C_{q}\big{\}} \tag{24}\]  
[M], [C], and [K] are respectively the mass matrix, the damping matrix, and stiffness matrix. The terms on the right hand are obtained and defined in Section 3.1. Elements of the respective matrices and vectors have been determined based on Equations (20)-(22), and their final forms are:  
\[ \begin{array}{l}m_{\theta_{1}\theta_{1}}=(J_{01}+J_{M}+m_{1\pi}e_{1}^{2}(1+ \psi_{1}^{2}))+(J_{02}+m_{2\pi}e_{2}^{2}(1+\psi_{2}^{2}))(g-1)^{2};\;m_{\theta_{1 }\xi_{1}}=J_{01}+m_{1\pi}e_{1}^{2}\\ m_{\theta_{2}\chi_{1}}=-m_{1\pi}\big{(}[\epsilon_{1\pm 1}+\epsilon_{1\pm 1}\psi_{1} )\cos\theta_{1}+(\epsilon_{1\pm 1}-\epsilon_{1\pm 1}\psi_{1})\sin\theta_{1}\big{)}\\ m_{\theta_{1}\gamma_{1}}=m_{1\pi}\big{[}(\epsilon_{1\pm 1}-\epsilon_{1\pm 1}\psi_{1} )\cos\theta_{1}-(\epsilon_{1\pm 1}+\epsilon_{1\pm 1}\psi_{1})\sin\theta_{1} \big{]};\;m_{\theta_{1}\eta_{2}}=(J_{02}+m_{2\pi}e_{2}^{2})(1-g)\\ m_{\theta_{2}\chi_{2}}=m_{2\pi}(g-1)\big{[}(\epsilon_{2\pm 2}+\epsilon_{2\pm 2} \psi_{2})\cos\theta_{2}+(\epsilon_{2\pm 2}-\epsilon_{2\pm 2}\psi_{2})\sin\theta_{2} \big{]}\\ m_{\theta_{1}\gamma_{2}}=-m_{2\pi}(g-1)\big{[}(\epsilon_{2\pm 2}-\epsilon_{2\pm 2} \psi_{2})\cos\theta_{2}-(\epsilon_{2\pm 2}+\epsilon_{2\pm 2}\psi_{2})\sin\theta_{2} \big{]}\\ \end{array}  \tag{25}\]  
\[ \begin{array}{l}m_{\varphi_{1}\theta_{1}}=J_{01}+m_{1\pi}e_{1}^{2};\;m_{ \varphi_{1}\theta_{1}}=J_{11}+m_{1\pi}e_{1}^{2};\;m_{\varphi_{1}\chi_{1}}=-m_{1 \pi}(\epsilon_{1\pm 1}\sin\theta_{1}+\epsilon_{1\pm 1}\cos\theta_{1});\\ m_{\varphi_{1}\gamma_{1}}=m_{1\pi}(\epsilon_{1\pm 1}\cos\theta_{1}-\epsilon_{1\pm 1} \sin\theta_{1});\;m_{\varphi_{1}\gamma_{2}}=0;m_{\varphi_{1}\chi_{2}}=0;\;m_{ \varphi_{1}\gamma_{2}}=0\\ \end{array}  \tag{26}\]  
\[ \begin{array}{l}m_{\chi_{1}\theta_{1}}=m_{1\pi}\big{(}(\epsilon_{1\pm 1}+ \epsilon_{1\pm 1}\psi_{1})\cos\theta_{1}+(\epsilon_{1\pm 1}-\epsilon_{1\pm 1} \psi_{1})\sin\theta_{1}\big{)}\\ +m_{2\pi}(g-1)\big{(}(\epsilon_{2\pm 2}+\epsilon_{2\pm 2}\psi_{2})\cos\theta_{2}+( \epsilon_{2\pm 2}-\epsilon_{2\pm 2}\psi_{2})\sin\theta_{2}\big{)};\;m_{\chi_{1}\chi_{ 1}}=(M_{1}+m_{1\pi})\\ m_{\chi_{1}\varphi_{1}}=-m_{1\pi}(\epsilon_{1\pm 1}\cos\theta_{1}+\epsilon_{1\pm 1} \sin\theta_{1});\;m_{\chi_{1}\gamma_{1}}=m_{\chi_{1}\varphi_{2}}=m_{\chi_{1} \chi_{2}}=0\\ \end{array}  \tag{27}\]  
\[ \begin{array}{l}m_{Y_{1}\theta_{1}}=m_{1\pi}\big{[}(\epsilon_{1\pm 1}-\epsilon_{1 \pm 1}\psi_{1})\cos\theta_{1}-\big{(}\epsilon_{1\pm 1}\psi_{1}+\epsilon_{1\pm 1} \big{)}\sin\theta_{1}\big{]};\;m_{Y_{1}X_{1}}=m_{Y_{1}\phi_{2}}=0\\ m_{Y_{1}\psi_{1}}=m_{1\pi}\big{(}\epsilon_{1\pm 1}\cos\theta_{1}-\epsilon_{1\pm 1} \sin\theta_{1}\big{)};\;m_{Y_{1}Y_{1}}=(M_{1}+m_{1\pi});\;m_{Y_{1}X_{2}}=m_{Y_{ 1}Y_{2}}=0\\ \end{array}  \tag{28}\]  
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