## 1 Introduction  
The rapid development of urban rail transport has greatly facilitated travel for residents in large cities. However, the vibration and noise issues arising from trains passing through densely populated areas along rail transit lines have increasingly drawn public attention. Prolonged and frequent vibrations can lead to structural damage in tracks, harm buildings situated near the rail lines, and interfere with the operation of high-precision instruments, thus impacting people's daily lives and productivity. In order to reduce the vibration problem of urban rail transit, vibration-damping tracks have been implemented on numerous rail lines [1; 2]. The effects of mobile vehicle impact and damping on the vibration of rail-bridge systems were verified by experiments on test-bed method by [PERSON] [3]. [PERSON] analyzed the impact of vibration isolator effectiveness on the vibration-damping effect of floating plate rail by establishing a vehicle-rail steel-spring floating-plate rail interaction model [4]. [PERSON] studied the effects of three track structure forms, namely, ordinary monolithic track bed, slab track bed, and floating-slab track, on the vibration of rail transportation [5]. [PERSON] analyzed the effect of track irregularities on the dynamic response of bridges, taking into account the stiffness and damping of the soil base [6]. These studies have analyzed the vibration response transmitted to the bridge structure from various phenomena, such as vehicles, vibration isolators, and track structure forms. However, the studies have not systematically revealed the vibration transmission problem among the structures of the bridge line. As people's demand for higher operating speeds in urban rail transport increases, the intensity of vibration sources in urban rail transport also intensifies. The development of more targeted measures to dampen vibrations on the tracks has become an urgent requirement to address this issue. The key to solving this problem is identifying track structure components with significant vibration energy accumulation and understanding how changing structural parameters affects this energy accumulation. The power flow method, based on the theory of vibration isolation, has proven highly effective in analyzing energy transfer and accumulation within structures. This method offers a comprehensive reflection of vibration energy traits within the structure. Notably, its advantage lies in frequency domain analysis, allowing precise representation of vibration energy distribution across each layer of the structure. Following [PERSON]'s description and discourse on a continuum-dynamics-centered power flow analysis approach [7; 8; 9], the power flow theory finds extensive application in diverse fields like marine, aerospace, and machinery [10; 11; 12]. This approach is equally relevant in the context of rail transportation, constituting a continuous dynamics challenge. [PERSON] examined the impact of velocity on vibration power flow within simply supported beams subjected to multipoint dynamic loading, estimating bridge vibration power distribution [13]. [PERSON] explored the vibration energy characteristics of a double-block ballasted track through a power flow method rooted in a train-track-bridge interaction model [14]. [PERSON] proposed a subway tunnel vibration power flow calculation method, comparing power flow between ordinary monolithic track beds and floating-slab tracks [15]. The application of the vibration power flow method to address the dynamics issues of urban railway bridge lines allows for solving structural vibration problems involving multiple damping layers and components. This approach offers the opportunity to analyze the energy transfer dynamics of each section of the track structure on the bridge. Building on this foundation, more targeted rail transit vibration-damping products can be developed according to the vibration accumulation of each component, providing certain theoretical support for the subsequent design of vibration-damping track.