# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 5 The Influence Discussion on the Internal Resonance under Different Excitations  
To accurately analyze the influence of the different excitations from different components on the parametric resonance, the controlled variable method is applied and the conditions of W1 and W2 are selected. The initial velocity (\(v_{in}\)) or initial displacement (\(V_{in}\)) of the tower, the cable and beam in subsystem #1 is changed from 0 m to 0.02 m, respectively. It is noticeable that the subscript indicates the initial state of the parameter. The maximum value (\(V_{c1 max}\)) of the different components under these two conditions and the fitting curves are selected, as shown in Figure 6.  
From Figure 6, the correlation coefficients of the fitting \(R^{2}\)\(\in[0.9381,0.9999]\) show a significant relationship of linear increase. The initial velocity and initial displacement of different components will have different excitation effects on the cable when the parametric resonance is satisfied in this sub-system. Additionally, in Figure 6D, some curves seem to remain at 0 until \(V_{in}\) = 0.8 cm, while it is known that the internal resonance occurs only above a threshold amplitude. To further discuss the influence on the parametric resonance, the correlation coefficients of the influence (\(k\)-\(V_{in}\) or \(k\)-\(v_{in}\)), also regarded as the slopes of polynomial linear fitting curves, are selected, as shown in Figure 7.  
In Figure 7, the axis coordinate range locates at the interval of [-2,10]. The smaller the slope value, the closer to the center point, and the less significant the effect. Based on this, several priority phenomena of resonance excitation in the parametric resonance of MCS can be clearly observed: 1 the excitation effect of the initial displacement under the condition of CBR is generally larger than that under the condition of TCR; 2 the excitation effect of the initial displacement is generally  
Figure 6: The influence of initial conditions on the maximum vibration displacement of different components: (A) relationship between \(v_{in}\) and \(V_{c1 max}\) under W1 condition; (**B**) relationship between \(V_{in}\) and \(V_{c1 max}\) under W1 condition; (**C**) relationship between \(v_{in}\) and \(V_{c1 max}\) under W2 condition; (**D**) relationship between \(V_{in}\) and \(V_{c1 max}\) under W2 condition. larger than that of the initial velocity; \(\mathfrak{z}\) the excitation sequence of the internal components of the subsystem is: beam > tower > cable; \(\mathfrak{z}\) the excitation effect of the initial displacement under the condition of CBR has the most significant influence among the others, whose influence coefficient (\(k\)-\(V_{bin}\)) reaches the peak of 9.457. Additionally, the cable can also be excited by the indirect influence of adjacent cables under the whole condition of MCE. To analyze this effect, the initial displacement of the beam section under the W1 working condition, whose coupling 'beat' characteristic is relatively obvious, is applied to the background of the investigation object. The initial displacement of the B1# beam portion, B2# beam portion and B3# beam portion are selected as 0.01 m, while other parameters are selected as 0. The displacement of the C1# cable is shown in Figure 8.  
From Figure 8, the cable of the MCS can be excited by the components of other sub-systems through the medium effect of CEB or the tower. The originally static component would be excited to obtain an initial velocity or displacement due to the effect. Moreover, if the sub-system of the exciting cable satisfies the parametric resonance conditions, the effect will be converted into the parametric resonance excitation of the cable, resulting in a severe oscillation. In addition, the closer  
Figure 8: The displacement of the C1# cable under different initial conditions. Figure 7: The correlation coefficients of the influence (\(k\)-\(V_{in}\) or \(k\)-\(v_{in}\)) under W1 and W2. the beam portion of the resonant subsystem is, the larger the resonance amplitude of the cable.