## 3 Modal Analysis of the Bridge  
Modal analysis provides a fundamental approach to understanding the dynamic characteristics of structural systems, particularly for complex engineering structures like bridges. This investigation explores the intricate vibrational behavior of a bridge system through a comprehensive eigenvalue analysis, employing two distinct modeling strategies: a traditional fixed-base configuration and an advanced soil-structure interaction approach. The comparative analysis aims to elucidate the subtle yet significant variations in structural dynamic response induced by foundation flexibility.  
The methodology encompasses a systematic examination of the bridge's modal properties, capturing the inherent vibrational modes that characterize the structure's dynamic behavior. By meticulously analyzing the eigenperiods and associated mode shapes, researchers can gain profound insights into the structural system's fundamental mechanical characteristics. These modal properties serve as critical indicators of the bridge's potential responses to dynamic loading conditions, including seismic excitation and high-speed train traversal.  
The modal analysis provides a comprehensive understanding of the bridge's dynamic characteristics, demonstrating the profound influence of foundation modeling approaches on structural vibrational behavior. The detailed examination of five distinct modes provides a nuanced perspective of the structure's dynamic response mechanisms.  
Mode 1, representing the fundamental global vibration mode, exhibits a particularly notable transformation when transitioning from a fixed base to a soil-structure interaction model. The eigenperiod elongates from 4.245 s to 4.657 s, representing a significant 10% increase (Figure 7). This elongation suggests a more compliant structural response is obtained when foundation flexibility is explicitly considered, highlighting the critical importance of sophisticated modeling techniques in capturing the true dynamic characteristics of bridge systems.  
The subsequent modes (Modes 2-5) demonstrate a consistent pattern of period elongation, with each mode experiencing an increase ranging from 6% to 10.0%. This systematic variation underscores the comprehensive impact of soil-structure interactions on the bridge's modal properties. The progressive changes across multiple modes reveal the complex interactions between structural components and the supporting foundation system.  
The differential behavior observed across various modes provides critical insights into the bridge's dynamic performance. Higher-frequency modes exhibit more localized deformation patterns, capturing intricate structural responses that extend beyond global movement characteristics. These modes are particularly significant for understanding potential vulnerability points and dynamic stress distributions within the bridge structure.  
The analysis transcends traditional modeling approaches by demonstrating the limitations of fixed-base assumptions. The soil-structure interaction model introduces a more sophisticated representation of real-world structural behavior, accounting for the nuanced interactions between the bridge components and the surrounding soil environment.