# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 6 Conclusions  
This paper aims to simulate the CEB of the MCS and investigate the excitations of parametric resonance through the defined relationship of the dynamic parameters of different components. To make ODEs more accessible to the numerical simulation, the shear difference and the finite difference method are applied to simulate the CEB of the MCS, whose accuracy and effectiveness have been verified by the case study and the dynamic analysis of mode shapes. A few meaningful conclusions are obtained as follows:  
1. The CEB, including the effect of bending stiffness of the single beam or indirect influence of adjacent cables, does affect the parametric resonance. The effectiveness of the refined dynamic model considering CEB has been verified by case investigations. 2. The concept of dynamic parameters of each component, which affects the conditions of the parametric resonance, is proposed in this paper and deeply discussed with the influence effect on the parametric resonance. 3. The excitation effects of different initial conditions are discussed in this paper. The excitation of the beam's initial displacement under the condition of the CBR, by which the maximum displacement of the cable reaches the peak of 13.36 times the initial value, is relatively large among others. In addition, the different internal resonance behavior of the cable would interfere with each other, resulting in the relatively small maximum vibration displacement of the cable when the CBR and TCR occur at the same time. 4. The process of the energy conversion through the medium of the CEB or the tower has been simulated when the parametric resonance occurs. It is evident that the indirect coupling effect of adjacent cables through the beam or the tower cannot be ignored in the parametric resonance of the MCS. Hence, the dynamic analysis of CEB should be paid more attention to the engineering design of the cable-stayed bridge. The reduced model established in this paper provides an accessible method for investigating the dynamic characteristics of MCS. The next step of the research will focus on the refined simulation of the single beams and the high-order modal nonlinear resonance analysis of the cable. Conceptualization, S.H. and K.C. ; methodology, Y.S. and B.W. ; software, K.W. and W.H. ; formal analysis, K.C. ; investigation, Y.S. ; resources, S.H. ; writing--original draft preparation, K.C. and K.W. ; writing--review and editing, B.W. ; funding acquisition, S.H. and W.H. All authors have read and agreed to the published version of the manuscript. Financial supports from the Natural Science Foundation of Shaanxi Province. (Funded by W.H., no. 2021 JM-174) and the Fundamental Research Business Fees of Central Universities of Chang'an University (Funded by S.H., no. 310821161012) are gratefully acknowledged. The data of this study are available from the corresponding author upon request. On behalf of all authors, the corresponding author states that there is no conflict of interest. For the tower, the kinetic energy is:  
\[T_{i}=\frac{1}{2}\cdot\int_{0}^{L_{t}}M_{t}(Z)\cdot\left[\dot{v}_{t}(z,t) \right]^{2}dz=\frac{1}{2}\cdot\int_{0}^{L_{t}}M_{t}(Z)\cdot q_{t}(z)^{2}\cdot  
The damping work is:  
\[D_{t}=\frac{1}{2}\cdot\int_{0}^{L_{t}}c_{t^{*}}[\dot{v}_{t}(z,t)]^{2}dz\] (A3)  
According to the Lagrange Functions, the dynamic:  
\[ \begin{array}{ll}\int_{0}^{L_{t}}M_{t}(Z)\cdot\varphi_{t}(z)^{2}&dz\cdot\dot {V}_{t}(t)-0+\int_{0}^{L_{t}}E_{t}l_{t^{*}}[\varphi_{t}(z)^{*}]^{2}dz\cdot V_{ t}(t)\\ &+\int_{0}^{L_{t}}c_{t^{*}}\varphi_{t}(z)^{2}dz\cdot\dot{V}_{t}(t)=\sum\limits_{i=1 }^{6}(H_{i}+h_{i})\text{cos}\theta_{i}\cdot\delta(z-z_{i})\end{array} \] (A4)  
\(\delta(z-z_{i})\) represents the Dirac delta function, shown as follows:  
\[\delta(z-z_{i})=\left\{ \begin{array}{ll}\delta(z-z_{i})=0,&(z\
eq z_{i})\\ \int_{-\infty}^{+\infty}\delta(z-z_{i})dz=1,(z=z_{i})\end{array}