# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 2 The Dynamic Model of the MCS
### The Reduced Model of the Beam Considering the CEB  
As shown in Figure 1A, the bridge beam is divided into rigid portions connected by springs and hinges. However, for a Euler-Bernoulli beam with distributed mass and distributed load, the plastic material properties hypothesis focused on a certain point does not represent the CEB as a whole structure, which also carries out onerous works on numerical simulation by establishing thousands of finite element model. To address this problem, the shear difference between the left and the right side of the beam section at each cable anchor position is applied to simulate the CEB [28; 29; 30]. The refined processes of the reduction are shown in Figure 2A: 1 according to the anchoring position of each cable, the single beam is reduced to N discrete beam portions connected by spring hinges. It is assumed that the interval displacement \(d\) of adjacent cables stays the same, resulting in that the equivalent mass and length \(\Delta d\) between independent beam sections are also the same. 2 The single beam can be reduced to an integrated system comprised of a number of discrete beam portions. In Figure 2B,C, to distinguish it from the mass symbol, where \(\overline{M}_{b(i-1)}\), \(\overline{M}_{bi}\) represent the bending moments on the left and right sides of the section between adjacent beam portions, respectively, and \(F_{b(i-1,i)}\), \(F_{b(i,i+1)}\) represent the shear force on the left and right sides of the beam portion at the cable's anchored position, respectively. Figure 2: The reduced model of the Bi# beam portion: (**A**) The reduced process of the single beam; (**B**) The schematic diagram of the middle section; (**C**) The schematic diagram of Bi# beam portion. After considering the equilibrium conditions, the vibration equation of the Bi\(\#\) beam portion can be obtained by the D'Alembert Principle:  
\[M_{bi}\cdot\ddot{V}_{bi}(t)-h_{ci}\cdot sin\theta_{i}+\Delta F_{i}+c_{bi}\cdot \dot{V}_{bi}(t)=0 \tag{16}\]  
where the shear difference \(\Delta F_{(i)}\) represents the influence between different beam portions due to the bending stiffness of a continuous single beam. To reduce the amount of the calculation, the finite difference is also applied to correct the vibration equation of each beam portion. The curvature \(\rho_{bi}\) and the bending moment \(\overline{M}_{bi}\) of the Bi\(\#\) beam portion can be approximated obtained by the second-order central difference:  
\[\overline{M}_{bi}=-E_{bi}I_{bi}\cdot(\frac{1}{\rho})_{bi}\approx\frac{-E_{bi}I _{bi}}{d^{2}}\Big{[}V_{b(i-1)}(t)-2V_{bi}(t)+V_{b(i+1)}(t)\Big{]} \tag{17}\]  
The shear at both sides of the micro-segment satisfies the following equations:  
\[\Delta F_{i}=-F_{b(i-1,i)}+F_{b(i,i+1)} \tag{18}\]  
\[F_{b(i-1,i)}=\frac{\overline{M}_{bi}-\overline{M}_{bi-1}}{d} \tag{19}\]  
\[F_{b(i+1)}=\frac{\overline{M}_{bi+1}-\overline{M}_{bi}}{d} \tag{20}\]  
Thus, the influence between different beam portions due to the bending stiffness of a continuous single beam satisfies the equation:  
\[\Delta F_{i}=\frac{EI}{d^{3}}\Big{[}-V_{b(i-2)}(t)+4V_{b(i-1)}(t)-6V_{bi}(t)+4 V_{b(i+1)}(t)-V_{b(i+2)}(t)\Big{]} \tag{21}\]  
It is noticeable that one end of the B1\(\#\) beam portion and B6\(\#\) beam portion is close to the simply supported end where the bending moment and displacement are equal to zero. Inserting \(\Delta F_{i}\) into Equation (16), by some simplifications, the vibration equation of the Bi\(\#\) beam portion can be obtained as follows:  
\[