# Vibration Response Analysis of Overhead System Regarding Train-Track-Bridge Dynamic Interaction
## 3 Coupling Vibration Analysis of Train-Track-Overhead System
### Influence of Damping on the Dynamic Response of Train Track Bridge System  
Due to track irregularities, wheel rail wear, and other factors, the train will have a dynamic impact on the bridge structure when it travels at a certain speed. The internal force and deformation of the bridge will be greater as a result of this impact than they would be under a static live load. The key concept is: the maximum dynamic deflection caused by excitation and wheel rail force to the maximum static deflection caused by applying the same train load at low speed or static state when the train crosses the bridge at a certain speed [28]. Its formula is as follows:  
\[\mu=\frac{Y_{d}}{Y_{p}}-1 \tag{11}\]  
where \(Y_{d}=\) the maximum dynamic deflection of the beam response when the train passes through the bridge structure at a given speed. \(Y_{p}=\) the maximum static deflection of the structure when the train is loaded. Damping refers to all forms of friction and other obstacles that lessen free vibration. In the fields of civil engineering, machinery, aerospace, and other related fields, the damping ratio is a crucial concept of structural dynamics. It describes the standard damping size of structures and expresses the relationship between damping coefficient and critical damping coefficient. The value of damping is determined by the structure's shape, material, type of support, and other factors. In practice, structural damping is expressed as the damping ratio, which is frequently measured using extremely complex experiments. The damping ratio can be calculated by definition, shown in Formula (12). \[ \begin{split}&\bar{\zeta}=C/C_{0}\\ & C_{0}=2 mw_{1}\end{split}  \tag{12}\]  
where: \(C=\) the damping coefficient, \(C_{0}\) = the critical damping coefficient, \(m\) = weight of object, \(w_{1}\) = natural frequency of structure. The damping ratio is defined in Reference [29] as the ratio between the actual resistance of damped vibration and the resistance necessary to produce critical damping. The damping ratio, which has no units and shows how the structure vibrates after being excited, is attenuated. It can be broken down into four categories: greater than 1, greater than 0, equal to 1, and between 0 and 1. All structures have typical damping ratios between 0 and 1. It is specified in Reference [30] that the structural damping ratio is 0.05. Reference [30] states the steel structure's seismic damping ratio should be between 0.01 and 0.1. The damping factor affects the vibration of the aerial body. When the damping value of the aerial body is large, the energy dissipation increases, and the corresponding vibration energy of the beam body decreases. The damping ratios are 2.5 percent, 5 percent, 7.5 percent, and 1 percent, respectively, under five groups of working conditions, according to research on the overhead system damping effect. The test is conducted using the piers (1 m in diameter) of the II aerial system structure, and a freight train traveling at 45 km/h is used to determine whether the irregularity level complies with AAR5. The chart displays various damping ratios under the influence of structure vibration response and vibration response. Figures 18-20 show that the free vibration of the train leaving the bridge primarily reflects the impact of damping on the overhead system. The repeating vibration has a minor amplitude with a large damping ratio, and a comparatively large amplitude with no damping. The vibration mode essentially stays the same, the wave peaks and troughs are comparable to how they were in the case of each damping ratio, and the damping has no discernible impact on the vibration frequency. Figure 19: Mid\(-\)span vibration response of structure with damping ratio of 0.05: (**a**) Mid\(-\)span deflection (mm). (**b**) Mid\(-\)span vertical acceleration (m/s\({}^{2}\)). (**c**) Mid\(-\)span transverse amplitude (mm). (**d**) Mid\(-\)span lateral acceleration (m/s\({}^{2}\)). Figure 18: Mid\(-\)span vibration response of structure with damping ratio of 0.025: (**a**) Mid\(-\)span deflection (mm).