# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 4 The Numerical Analysis
### The Numerical Analysis of the Dynamic Parameter  
In this part, sub-system #1 is selected as the research object to apply the numerical analysis. The Fast Fourier Transform method is applied to obtain the spectrogram of different components in sub-system #1. To further verify the effectiveness of the model proposed in this paper on the dynamic analysis, changing the dynamic parameter of each component with variable properties, the dynamic parameter ratio of the tower, cable and beam is deeply discussed through three working conditions which have been widely verified to be the parametric resonance conditions [8; 9; 11; 17; 28; 31]:  
1. The first working condition (W1): change \(m_{t}=85,000\) kg/m to satisfy the equation \(\sqrt{\alpha_{1}}\):\(\sqrt{\beta_{1}^{i}}\approx 1:1\), and the displacement and the spectrogram of each component is obtained, as shown in Figure 5A;
2. The second working condition (W2): change \(M_{bi}=70,000\) kg to satisfy the equation \(\sqrt{\beta_{1}^{i}}\):\(\sqrt{\gamma_{1}^{i}}\approx 1:2\), and the displacement and the spectrogram of each component is obtained, as shown in Figure 5B;
3. The third working condition (W3): change \(m_{t}\) and \(M_{bi}\) at the same time to satisfy the equation \(\sqrt{\alpha_{1}}\): \(\sqrt{\beta_{1}^{i}}\):\(\sqrt{\gamma_{1}^{i}}\approx 1:1\) : 2, and the displacement and the spectrogram of each component are obtained, as shown in Figure 5C.  
By applying the Fourth Runge-Kutta method on the undamped model, the initial displacement of each component is selected as 0.01 m, while the fixed-step is selected as 0.02 s. The analytical results of the above three conditions are shown in Figure 5.  
In Figure 5, it is illustrated that the resonance of the Tower-Cable or Cable-Beam would respectively be observed when the equation of \(\sqrt{\alpha_{1}}\):\(\sqrt{\beta_{1}^{i}}\)\(\approx\)1:1 or \(\sqrt{\beta_{1}^{i}}\):\(\sqrt{\gamma_{1}^{i}}\)\(\approx\)1:2 is satisfied. In Figure 5A, when \(\sqrt{\alpha_{1}}\):\(\sqrt{\beta_{1}^{i}}\)\(\approx\)1:1 is satisfied, the coupled resonance can be observed from the displacement curves between the tower and the C1# cable. A significant characteristic of the 'beat' can be observed on the cable displacement curve. From the spectrogram in Figure 5A, when the vibration frequency of the tower (\(f_{t}=0.8161\) Hz or or 0.8161 Hz) changes to the same value as the C1# cable (\(f_{C1\#}=0.8161\) Hz), the cable amplitude reaches the peak of the maximum, which is 9.148 times the initial amplitude; in Figure 5B, when \(\sqrt{\beta_{1}^{i}}\):\(\sqrt{\gamma_{1}^{i}}\approx 1:2\) is satisfied, the coupled resonance can be observed from the displacement curves of the C1# cable and the B1# beam portion. In addition, from the spectrogram  
Figure 4: The in-plane vertical natural vibration mode shapes of the MCS.  
in Figure 5B, when the vibration frequency of the C1# cable (\(f_{\text{C1#}}=0.7391\) Hz) changes to nearly half the value of the B1# beam portion (\(f_{\text{B1#}}=1.485\) Hz), the cable amplitude reaches the peak of the maximum, which is 13.36 times the initial amplitude. Particularly in Figure 5C, the condition is regarded as the special working condition of the parametric resonance. Under this condition, the total energy conversion trans between the tower, the cable and the beam. From the spectrogram in Figure 5C, it is clear that the resonance modes in the system are extremely rich. Different resonances influence and interfere with each other while obtaining a lower maximum amplitude, 7.289 times initial amplitude, than the first two conditions. In general, the parameter of \(\alpha_{1}\), \(\beta_{1}^{i}\) and \(\gamma_{1}^{i}\) characterize the ability of different components to participate in the global resonance to a certain extent and are evidently important parameters for the dynamic design of cable-stayed bridges.  
Figure 5: _Cont._  
Figure 5: The time and frequency domain diagrams of each degree-of-freedom of subsystem #1 under: (**A**) W1; (**B**) W2; (**C**) W3.