# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 2 The Dynamic Model of the MCS
### The Vibration Equations of the Tower and the Cable
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Designation** & **Parameter** \\ \hline \multirow{8}{*}{Tower} & Unit mass & \(m_{t}(z)\) \\  & Static modulus of elasticity & \(E_{t}\) \\  & Bending moment of inertia & \(I_{t}\) \\  & Damping coefficient & \(c_{t}\) \\  & Height & \(L_{t}\) \\  & Lateral vibration displacement & \(v_{t}(z,t)\) \\ \hline \multirow{8}{*}{Ci\# cable} & Unit mass & \(m_{ci}(x_{ci})\) \\  & Static modulus of elasticity & \(E_{ci}\) \\  & Cross-sectional area & \(A_{ci}\) \\  & Static alignment & \(w_{ci}(x_{ci})\) \\  & Vibration displacement in the chord direction & \(u_{ci}(x_{ci},t)\) \\  & Vibration displacement in the transverse direction & \(v_{ci}(x_{ci},t)\) \\  & Static tension in the chord direction & \(H_{ci}\) \\  & Dynamic tension in the chord direction & \(h_{ci}\) \\  & Static tension in the tangential direction & \(T_{ci}\) \\  & Dynamic tension in the tangential direction & \(T_{ci}\) \\  & Displacement of the cable in chord direction & \(L_{i}\) \\  & The length of the cable under the static state & \(L_{ci}\) \\  & Sag of the Ci\# cable at the midspan position & \(D_{ci}\) \\  & Damping coefficient & \(c_{ci}\) \\  & The angle of the cable with beam & \(\theta_{i}\) \\ \hline \multirow{4}{*}{Bi\# beam portion} & Mass & \(M_{bi}\) \\  & Static modulus of elasticity & \(E_{bi}\) \\  & Bending moment of inertia & \(l_{bi}\) \\  & Damping coefficient & \(c_{bi}\) \\ \hline \multicolumn{2}{c}{The generalized coordinates, limited with time} & \(V_{t}(t),\;V_{ci}(t),\;V_{bi}(t)\) \\ \multicolumn{2}{c}{Acceleration of gravity} & \(g\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The nomenclature list for the variable parameters. The parabolic alignment is selected as the initial alignment of the cable under the static state [9; 10]:  
\[w_{ci}(x_{ci})=\frac{m_{ci}\mathcal{B}\cdot L_{ci}{}^{2}\cdot cos\theta_{i}}{2H_{ci }}(\frac{x_{ci}}{L_{ci}}-\frac{{x_{ci}}^{2}}{L_{ci}{}^{2}}) \tag{4}\]  
\[L_{ci}=\int_{0}^{L_{i}}\biggl{(}\frac{ds_{i}}{dx_{ci}}\biggr{)}^{2}\cdot\frac{d \mathcal{s}_{0i}}{dx_{ci}}dx_{ci}\approx\int_{0}^{L_{i}}\biggl{(}\frac{d \mathcal{s}_{i}}{dx_{ci}}\biggr{)}^{3}dx_{ci}=L_{i}\cdot[1+8\cdot\biggl{(}\frac{ D_{ci}}{L_{i}}\biggr{)}^{2}] \tag{5}\]  
\[D_{ci}=\frac{m_{ci}\mathcal{B}\cdot L_{i}{}^{2}\cdot cos\theta_{i}}{8H_{ci}} \tag{6}\]  
According to the force relationship between cable balance and vibration, the vibration dynamic equation of the Cii\(\epsilon\) cable can be obtained based on the D'Alembert Principle [8; 20]:  
\[(H_{ci}+h_{ci})\cdot\frac{\partial^{2}x_{ci}(x_{ci}t)}{\partial x _{ci}{}^{2}}+h_{ci}\cdot\frac{\partial^{2}w_{ci}(x_{ci})}{\partial x_{ci}{}^{2}} \tag{7}\] \[=m_{ci}\cdot\frac{\partial^{2}w_{ci}(x_{ci}t)}{\partial t^{2}}+c_{ ci}\cdot\frac{\partial v_{ci}(x_{ci}t)}{\partial t}\]  
\[h_{ci}=E_{ci}A_{ci}\cdot\epsilon_{ci}\approx E_{ci}A_{ci}\cdot\left[U_{x_{ci}}+ \frac{\partial v_{ci}(x_{ci},t)}{\partial x_{ci}}\frac{\partial w_{ci}(x_{ci}, t)}{\partial x_{ci}}+\frac{1}{2}\biggl{(}\frac{\partial v_{ci}(x_{ci},t)}{ \partial x_{ci}}\biggr{)}^{2}\right] \tag{8}\]  
\[U_{x_{ci}}=u_{ci}(L_{ci},t)+u_{ci}(0,t) \tag{9}\]  
where \(\epsilon_{ci}\) represents the dynamic strain in the chord direction of the Cii\(\epsilon\) cable; \(U_{x_{ci}}\) represents the vibration displacement at both ends of the cable in the direction of \(x_{ci}\).