## 1 Introduction  
With the rapid development of high-speed railways and track-bridge infrastructure in China, train speeds have significantly increased. However, higher operating speeds tend to induce intense vibrations in track-bridge structures, which can threaten operational safety and reduce ride comfort. Therefore, the analysis and control of vibration in railway structures are of critical engineering significance. In the study of vibration control for track-bridge systems, dynamic response analysis is a fundamental step. Common numerical simulation methods include the FEM and Statistical Energy Analysis (SEA). Since the rail structure exhibits high-frequency vibrations and satisfies SEA assumptions, SEA is often preferred for its computational efficiency. However, applying FEM to such high-frequency scenarios results in a dense modal spectrum and a high number of degrees of freedom, making the equations difficult to solve. Conversely, bridge structures, which typically exhibit low-frequency vibrations, are more accurately analyzed using FEM. Neither method alone is sufficient for comprehensive modeling of systems. To overcome this limitation, [PERSON] et al. [1,2] proposed a hybrid FE-SEA method based on modal superposition and fuzzy structural theory, which enables full-frequency analysis of complex structural responses. [PERSON] et al. [3] applied the FE-SEA method to simulate a four-layer track-bridge structure, simplifying the vehicle model to four independent wheelsets with vertical motion, and analyzed vertical dynamic responses under random irregularities. [PERSON] et al. [4] used a hybrid model to study the noise characteristics of box girders and validated the results against experimental data. These studies confirm the applicability and accuracy of the FE-SEA method in analyzing railway structure vibrations, as well as its high efficiency, laying a solid foundation for vibration control investigations in such systems. Additionally, [PERSON] et al. [5] investigated the influence of overall ballast track fastening stiffness on the vertical vibration and sound power characteristics of rails using the spectral element method, while [PERSON] et al. [6] employed the 2.5D finite element method to study unsaturated ground vibrations induced by high-speed train loads. These wave-based models offer clear advantages in the analysis of high-frequency responses in track structures. In contrast, the hybrid FE-SEA method, benefiting from its simplicity in modeling and high computational efficiency, holds a significant position in time-domain vibration response analysis of track systems and serves as an effective and convenient computational platform for the multi-objective optimization conducted in this study. Vibration control strategies for track-bridge structures include structural parameter optimization and integration of vibration mitigation devices such as Tuned Mass Dampers (TMDs) and inerters. [PERSON] et al. [7] explored the seismic damping mechanism and performance of CRTS III slab ballastless track on an isolation system extradosed cable-stayed bridge during rare earthquakes, while [PERSON] et al. [8,9] experimentally studied TMD configurations in box girders. In recent years, inerter-based systems have gained attention for their structural simplicity, adaptability, and low maintenance. Studies have examined their effectiveness in vibration control across mechanical, automotive, and civil engineering applications [10, 11, 12, 13, 14]. [PERSON] et al. [15] analyzed the vertical dynamic performance of curved tracks using various dampers, finding that TID and AM-TMD outperform conventional TMD. [PERSON] et al. [16] developed an Inerter-Enhanced Dynamic Vibration Absorber (IDVA) for floating slab tracks, which reduced relative displacements by over 50%. [PERSON] et al. [17] applied TIDs to floating slab tracks and performed multi-objective optimization, showing improved performance in the 4-16 Hz range. [PERSON] et al.