## 2 Modelling Approach and Governing Equation
### Mathematical Modelling of the System  
The inertial reference frames, \(X_{1}\), \(Y_{1}\), \(Z_{1}\) and \(X_{2}\), \(Y_{2}\), \(Z_{2}\), as shown in Figure 2, are considered in this work for the global representation of the lumped mass system. The system equations of motion can be obtained using the Energy principle, where the kinetic energy of the coupled rotor system is expressed as  
\[G=\frac{1}{2}\Big{(}J_{\text{MI}}+J_{\text{g}}+J_{\text{M2}}(1-g)^{2}\Big{)} \dot{\theta}_{1}^{2}+\frac{1}{2}M_{1}\bigg{(}\dot{X}_{1}^{2}+\dot{Y}_{1}^{2} \bigg{)}+\frac{1}{2}M_{2}\bigg{(}\dot{X}_{2}^{2}+\dot{Y}_{2}^{2}\bigg{)}+\frac {1}{2}m_{u1}\ddot{R}_{e1}^{T}\ddot{R}_{e1}+\frac{1}{2}m_{u2}\ddot{R}_{e2}^{T} \ddot{R}_{e2} \tag{7}\]  
where,\(\dot{R}_{e1}\) and \(\dot{R}_{e2}\) are the velocity vectors of unbalance masses \(m_{u1}\) and \(m_{u2}\), respectively, rotating with the moment of inertia of disc 1 and disc 2 [2]. After manipulation and calculation, the vectors \(\dot{R}_{e1}\) and \(\dot{R}_{e2}\) can be expressed as  
\[\dot{R}_{e1}=\left\{\begin{array}{ll}\dot{X}_{1}-\dot{\theta}_{1}e_{x1}\sin \theta_{1}-\dot{\theta}_{1}e_{y_{1}}\cos\theta_{1}\\ \dot{Y}_{1}+\dot{\theta}_{1}e_{x1}\cos\theta_{1}-\dot{\theta}_{1}e_{y1}\sin \theta_{1}\\ \dot{X}_{2}-\dot{\theta}_{1}e_{x2}\sin\theta_{2}-\dot{\theta}_{1}e_{y2}\cos \theta_{2}\\ \dot{Y}_{2}+\dot{\theta}_{1}e_{x2}\cos\theta_{2}-\dot{\theta}_{1}e_{y2}\sin \theta_{2}\\ \end{array}\right\} \tag{8}\]where the pairs \(e_{x1}\), \(e_{y2}\) and \(e_{x2}\), \(e_{y2}\) are components of \(e_{1}=\{1,0....0\}^{T}\) and \(e_{2}=\{1,0....0\}^{T}\) representing the locations of \(m_{u1}\) and \(m_{u2}\) in their respective disc's body coordinate systems \(x_{1},y_{1}\) and \(x_{2},y_{2}\), as indicated in Figure 2. Assigning and performing the appropriate substitutions of Equation (8) into Equation (7) leads to the total kinetic energy equation defined by  
\[\begin{array}{l}G=\frac{1}{2}\Big{(}J_{M1}+J_{g}+J_{M2}(\mu-1)^{2}\Big{)} \dot{\theta}_{1}^{2}+\frac{1}{2}(M_{1}+m_{u1})(\dot{X}_{1}^{2}+\dot{Y}_{1}^{2} )+\frac{1}{2}(M_{2}+m_{u2})(\dot{X}_{2}^{2}+\dot{Y}_{2}^{2})\\ \qquad-m_{u1}e(\sin 2\theta_{1}\dot{X}_{1}-\cos 2\theta_{1}Y_{1})\dot{\theta}_{1}-m _{u2}e(\mu-1)(\sin 2\theta_{2}\dot{X}_{2}-\cos 2\theta_{2}\dot{Y}_{2})\dot{ \theta}_{1}\end{array} \tag{9}\]  
Following the same formulation, the system potential energy comprises both shafts lateral and the torsional strain energy expressed as  
\[V=\frac{1}{2}K_{11}\theta_{1}^{2}+\frac{1}{2}K_{1X}X_{1}^{2}+\frac{1}{2}K_{1Y }Y_{1}^{2}+\frac{1}{2}K_{2X}X_{2}^{2}+\frac{1}{2}K_{2Y}Y_{2}^{2}+\frac{1}{2}K _{2T}\theta_{2}^{2} \tag{10}\]  
where \(K_{1X1X},K_{1Y1Y},K_{2X2X}\) and \(K_{2X2X}\), are the shaft stiffnesses, \(K_{1T}=K_{2T}\) is the torsional stiffness coefficient associated with the system degrees of freedom \(K_{1X1X}=K_{1Y1Y}\)=\(K_{2X2X}=K_{2X2X}=K_{2X2X}=K_{0}\).  
The total [PERSON]'s dissipation function is obtained by considering the effect of the damping coefficient, and then expressed as  
\[D=\frac{1}{2}C_{1T}\dot{\theta}_{1}^{2}+\frac{1}{2}C_{1X}\dot{X}_{1}^{2}+ \frac{1}{2}C_{1Y}\dot{Y}_{1}^{2}+\frac{1}{2}C_{2X}\dot{X}_{2}^{2}+\frac{1}{2} C_{2Y}\dot{Y}_{2}^{2}+\frac{1}{2}C_{2T}\dot{\theta}_{2}^{2} \tag{11}\]  
where \(C_{1X1X},C_{1Y1Y},C_{2X2X}\) and \(C_{2Y2Y}\) are the flexural vibration damping of the respective degree of freedom's damping coefficients, \(C_{1T}=C_{2T}\) is the torsional vibration damping of the first and second shaft, respectively. \(C_{1X1X}=C_{1Y1Y}=C_{2X2X}=C_{2Y2Y}=C_{0}\).