# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 2 The Dynamic Model of the MCS  
### The Vibration Equations of the Tower and the Cable  
The parametric resonance of the cable excited by the tower or the beam is the object of this investigation. To focus on the object, the dynamic model of a tower-six-cables-beam structure is established, as shown in Figure 1A. It is agreed that the relevant parameters of towers, cables, and beams are subscripted by t, c and b; the C1#-C6# cable corresponds to the 1#-6# subsystem, and the reduced model of the Ci# cable is shown in Figure 1B. The nomenclature list for the variable parameters is selected as shown in Table 1.  
In Figure 1B, the directions of \(x_{ci}\) and \(y_{ci}\) represents the chord and transverse direction of the cable vibration, respectively. To highlight the research emphasis, some assumptions are given as follows [15; 21]: (i) the influence of cable bending and torsion stiffness on vibration is negligible; (ii) the influence of the cable's gravity on the string tension in the chord direction is ignorable; (iii) the flexural, torsional, shear strain and the longitudinal inertia force of cables are ignorable; (iv) it is believed that the deformation constitutive relationship of the cable and the bridge tower satisfies [PERSON]'s law when it vibrates; and (v) the axial influence of stay cable on the tower and the beam segment is ignorable. Figure 1: The schematic diagram of the dynamic model: (**A**) the reduced model of a dynamic tower-six-cables–beam system; (**B**) the reduced model of the Ci# cable sub-system; (**C**) the micro-element section of the cable. In Figure 1, the bridge tower can be regarded as a cantilevered tower with a rigid foundation, cast-in boundary condition and upper-end tension without considering the influence of foundation deformation or the superstructure. It is noticeable that this paper aims to propose a refined model for investigating the simulation of the CEB on the parametric resonance of the whole dynamic system. On this basis, it is assumed that the cables are anchored at the same position of the tower. It is easy to obtain the partial differential equations (PDEs) of the tower as follows:  
\[M_{t}^{*}\cdot\ddot{v}_{t}(z,t)+\dot{c}_{t}\cdot\dot{v}_{t}(z,t)+K_{t}^{*}\cdot v _{t}(z,t)=F_{t}^{*} \tag{1}\]  
where \(M_{t}^{*}\), \(K_{t}^{*}\) and \(F_{t}^{*}\) represent the equivalent mass, equivalent stiffness and equivalent external force, respectively, and the detailed derivation process is shown in Appendix A. The in-plane lateral vibration displacement of the tower is as shown in Equation (2):  
\[v_{t}(z,t)=\sum_{n=1}^{n}\varphi_{tn}(z)\cdot V_{t}(t) \tag{2}\]  
where \(\varphi_{tn}(z)\) represents the mode shape of the nth-order. To reduce the amount of calculation, the following research only studies the first order of the mode shape (\(n=1\), \(\varphi_{t}(t)=\varphi_{t1}(t)\)). The boundary conditions of the tower are:  
\[v_{t}(0,t)=0,\;v_{t}(L_{t},t)=V_{t}(t) \tag{3}\]  
To avoid complex calculations, only the first mode shape of the tower which satisfies the geometry and the dynamic boundary conditions is selected as \(\varphi_{t}(z)=1-cos\frac{\pi z}{2L_{t}}\)[9; 13].