# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 3 The Mathematical Model of the MCS  
### The Equations of the MCS  
To avoid repetition, \(V_{t}(t)\), \(V_{ci}(t)\) and \(V_{bi}(t)\) are all reduced to the symbols \(V_{t}\), \(V_{ci}\) and \(V_{bi}\). Equations (1) and (15) are reduced to non-dimensional form equations by imposing the Galerkin Method. Summarizing Equation (22), the dynamic equations of the whole system can be obtained, as shown in Equation (23):  
\[\begin{array}{ll}\ddot{V}_{t}+\alpha_{1}\cdot V_{t}+\alpha_{2}&\cdot\dot{V} +\alpha_{3}\cdot V_{t}^{2}+\alpha_{4}\cdot V_{bi}+\alpha_{5}\cdot V_{bi}^{2}+ \alpha_{6}\cdot V_{ci}+\alpha_{7}\cdot V_{ci}{}^{2}+\alpha_{8}\\ &\cdot V_{bi}V_{t}+\alpha_{8}=0\end{array} \tag{23}\]  
\[  
In Table 2, without considering the influence of gravity on the cable's vibration, the calculation results in this paper are basically consistent with the results of related references. For further comparison, two conditions (S1 and S2) are considered:  
(i)   Ignoring the vibration displacement of the tower (\(V_{t}=0\) in Equation (24)), selecting the parameters in Reference [25] to satisfy the equation \(\sqrt{\beta_{1}}\approx 2\sqrt{\gamma_{1}^{1}}\
eq 2\sqrt{\gamma_{1}^{2}}\) and comparing with the Figure 3 of Reference [25];  
(ii)   Selected parameters in [31] to satisfy the equation \(\sqrt{\alpha_{1}}\approx\sqrt{\beta_{1}^{1}}\approx\sqrt{\beta_{1}^{2}}\approx \sqrt{\beta_{1}^{3}}\) and compare with the Figures 6-8 of [31]. The vibration displacement comparison of C1# cable is selected under different conditions, as shown in Figure 3.  
In Figure 3, although the phases and the maximum amplitudes of these two vibration displacement curves exhibit the difference to some extent, the curve change rules and coupling characteristics of the two calculation cases are quite similar. It is evident from the results that the reduced model proposed in this paper can accurately the simulate dynamic behavior of the MCS. Otherwise, during  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Provenance** & **Shorthand** & \multicolumn{2}{c}{**Dynamic Parameter (\(\sqrt{\beta_{1}^{1}}\), rad/s)**} \\  & & **RMS** & **PMS** \\ \hline The case of double-cable tructure in [25] & S1C1 & 1.8928 & 1.8928 \\  & S1C2 & 1.8925 & 1.8925 \\  & S2C1 & 3.0753 & 3.0731 \\ The case of three-cable structure in [31] & S2C2 & 3.0659 & 3.0747 \\  & S2C3 & 3.0803 & 3.0748 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The comparison of the self-vibration frequency calculated by different methods.  
Figure 3: The comparison of simulation results obtained from this paper with other researches: (**A**) compared with [25]: the vibration displacement comparison of Cable 1 when \(\sqrt{\beta_{1}}\approx 2\sqrt{\gamma_{1}^{1}}\
eq 2\sqrt{\gamma_{1}^{2}}\); (**B**) compared with [31]: the vibration displacement comparison from Cable 1 to Cable 3 when \(\sqrt{\alpha_{1}}\approx\sqrt{\beta_{1}^{1}}\approx\sqrt{\beta_{1}^{2}}\approx \sqrt{\beta_{1}^{3}}\).  
the verification process, it was found that the methods proposed in the [25,31] did not consider the interaction caused by the CEB when the adjacent cables resonated. More specifically, the numerical analysis of the full-bridge resonance in these two references is adopted under the theoretically given conditions. This is also the target problem addressed in this article.