# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## 1 Introduction  
Complex spatial cable architecture, such as long-span cable-stayed bridges, exhibits a complex dynamic characteristic [1]. Due to the periodic change in the lateral restoring force of the cable, the lateral vibration of the cable is coupled with the horizontal/vertical vibration of the tower/beam. When the ratio of the local modal frequency of the cable to the overall modal frequency satisfies a certain relationship, 1:1 or 1:2, a strong and complex coupled internal resonance will occur in the combined structure [2, 3, 4]. So far, the same characteristic violent vibrations of the cable with a maximum lateral amplitude exceeding 1 m have been observed in real bridge monitoring in many countries [5, 6]. The failure of the dampers on the stay cables caused by this severe vibration will further reduce the structural stiffness and also seriously affect the safe operation of the bridge [7]. To address this problem, a number of researchers have carried out a wealth of theoretical and practical research. [PERSON] [8], [PERSON] [9] and [PERSON] [10] proposed a number of dynamic models reduced from the cable-stayed bridges according to different research objects and analyzed the parameter sensitivity of the parametric resonance by various numerical simulation methods, such as the Runge-Kutta Method, the Multi-Scale Method and others. Focusing on the different resonance conditions, 1:2 and 2:1, of cable-beams, investigations were conducted to deeply analyze the characteristics of the parametric resonance in cable-beam structures [11]. [PERSON]. [12] investigated the mechanism of resonance change caused by cable end angle changes and analyzed the conversion of low-frequencyvibration and high-frequency vibration from the perspective of energy transfer. [PERSON] [13] analyzed the influence factors, such as the frequency ratio of the cable-beam, the excitation amplitude, the cable force and cable dampers, on the characteristics of structural coupling by the numerical simulation of a proposed model. In addition, [PERSON] [14; 15] observed the severe vibration phenomena of 2:1 resonance through the model experiment of the cable-stayed bridge and studied the process of mode shape coupling under different order modes. [PERSON] [16] conducted the resonance interval of the frequency ratio between the cable to beam or tower through a field experiment of the existing bridges. Moreover, [PERSON] [17] concluded that the linear cable-deck interaction under the first two internal resonance modes was considered to be the key excitation source for the cable oscillation with a large amplitude through the experimental data on Guadiana Bridge. [PERSON] [18] investigated the structural dynamic instability phenomenon when the local mode and global mode were coupled. The influence of excitation amplitude and additional damping on the steady-state response of stay cables under parametric excitation was further discussed through finite element analysis and field experiments. It has been concluded that the parametric excitation polynomial in cable vibration differential equations is one of the main factors that cause global-local resonance in the combined systems of cable-supported bridges [12]. It is illustrated that the finite element model requires a large number of subdivisions to perform the fine analysis of the coupled parametric resonance well [19], and it is definitely difficult to analyze the dynamic characteristic of the full-bridge resonance accurately. Thus, limited by the difficulty of accurate finite element division and field testing of bridges, researchers have conducted a number of refined studies focusing on the parameter sensitivity analysis of the dynamic behavior of a single-cable system [20; 21; 22; 23].