## 2 Mathematical Model Descriptions
### The Lagrange Expression  
Two inertial reference frames, \(X_{1}\)\(Y_{1}\)\(Z_{1}\) and \(X_{2}\)\(Y_{2}\)\(Z_{2}\) (Figure 1) were adopted for the global representation of the lumped mass system. \(X_{1}\)\(Y_{1}\)\(Z_{1}\) is fixed to the motor with \(Z_{1}\), coincident with the motor output shaft axis, whereas \(X_{2}\)\(Y_{2}\)\(Z_{2}\) is attached to the left bearing of the secondary shaft, such that \(Z_{2}\) is parallel to the central axis of the bearing as shown in Figure 1. The vectors \(R_{e1}\) and \(R_{e2}\) represent the global position of \(m_{u1}\) and \(m_{u2}\), respectively. The pairs of vectors \(R_{e1}\), \(\phi_{1}\), and \(R_{e2}\), \(\phi_{2}\) represent the centres of the rotor masses \(M_{1}\) and \(M_{2}\), respectively. The kinetic energy of the system can be expressed as:  
\[G_{r1}=\frac{1}{2}J_{M}\dot{\theta}_{1}{}^{2}+\frac{1}{2}J_{D1}\Big{(}\dot{ \theta}_{1}+\dot{\psi}_{1}\Big{)}^{2}+\frac{1}{2}M_{1}\bigg{(}\dot{X}_{1}^{2} +\dot{Y}_{1}^{2}\bigg{)}+\frac{1}{2}m_{u1}\dot{R}_{e1}^{T}\dot{R}_{e1} \tag{1}\]  
\[G_{r2}=\frac{1}{2}J_{D2}\Big{(}\dot{\theta}_{2}+\dot{\psi}_{2}\Big{)}^{2}+ \frac{1}{2}M_{2}\bigg{(}\dot{X}_{2}^{2}+\dot{Y}_{2}^{2}\bigg{)}+\frac{1}{2}m_{ u2}\dot{R}_{e2}^{T}\dot{R}_{e2} \tag{2}\]  
where individual inertias and their net displacements are as follows: \(J_{M}\)--inertia of motor output shaft, which undergoes only rigid-body rotation \(\theta_{1}\); \(J_{D1}\)--elastic deformation of spring, \(k_{1}\), i.e., \(\psi_{1}\) superposed on the flexible rotation of \(J_{M}\); \(J_{D2}\)--elastic deformation of the spring \(k_{2}\), i.e., \(\psi_{2}\) superposed on the elastic rotation of the shaft two, i.e., \(\theta_{2}\); \(\dot{R}_{e1}\) and \(\dot{R}_{e2}\)--the velocity vectors of \(m_{u1}\) and \(m_{u2}\), respectively. The vectors \(R_{e1}\) and \(R_{e2}\) can be expressed as [23]:  
\[R_{e1} =[A(\theta_{1})][A(\psi_{1})]\mathrm{e}_{1} \tag{3}\] \[R_{e2} =[A(\theta_{2})][A(\psi_{2})]\mathrm{e}_{2} \tag{4}\]  
where, \(\mathrm{e}_{1},\gamma_{1}\) and \(\mathrm{e}_{2},\gamma_{2}\) represent locations of \(m_{u1}\) and \(m_{u2}\) in the disc's body coordinate systems \(x_{1}\), \(y_{1}\) and \(x_{2}\), \(y_{2}\). The angles \(\theta_{1}\) and \(\psi_{1}\) are the motor shaft elastic rotation and the torsional deformation angle of the primary shaft measured with respect to the motor coordinate system, respectively. The angles, \(\theta_{2}\) and \(\psi_{2}\) are the secondary bearing elastic rotation and the torsional deformation angle measured with respect to the bearing coordinate system, respectively. The matrices have the following significance: \([A(\theta_{1})]\)--rotational transformation from the motor coordinate system \(x_{1}^{m}\), \(y_{1}^{m}\) to the inertial reference frame \(X_{1},Y_{1}\); \([A(\psi_{1})]\)--rotational transformation from the disc coordinate systems \(x_{1}\), \(y_{1}\) to the motor coordinate system \(x_{1}^{m}\), \(y_{1}^{m}\); \([A(\theta_{2})]\)--rotational transformation of the secondary shaft-bearing coordinate system \(x_{2}^{m}\), \(y_{2}^{m}\) to the reference frame \(X_{2}\), \(Y_{2}\); \([A(\psi_{2})]\)--rotational  
Figure 1: Illustration of a vehicle with two propeller shafts interconnected by a single Hooke’s joint. transformational matrices from the disc coordinate system \(x_{2},y_{2}\) to the secondary shaft-bearing coordinate system \(x_{2}^{b}\), \(y_{2}^{b}\). The matrices \([A(\theta_{1})]\) and \([A(\psi_{1})]\) are given as:  
\[[A(\theta_{1})]=\left[ \begin{array}{cc}\cos\theta_{1}&-\sin\theta_{1}\\ \sin\theta_{1}&\cos\theta_{1}\end{array} \right]\text{, and }[A(\psi_{1})]= \left[ \begin{array}{cc}1&-\psi_{1}\\ \psi_{1}&1\end{array} \right] \tag{5}\]  
and analytical expressions for \([A(\theta_{2})]\) and \([A(\psi_{2})]\) are developed as follows:  
\[[A(\theta_{2})]=\left[ \begin{array}{cc}\cos\theta_{2}&-\sin\theta_{2}\\ \sin\theta_{2}&\cos\theta_{2}\end{array} \right]\text{, and }[A(\psi_{2})]= \left[ \begin{array}{cc}1&-\psi_{2}\\ \psi_{2}&1\end{array}