## 1 Introduction  
Modern pedestrian bridges are evolving to be more innovative in their design, not only serving functional purposes but also acting as prominent urban landmarks. Advances in structural understanding and new materials have enabled longer spans and more slender structures [1, 2]. In engineering practice, particularly concerning pedestrian bridges, resonant vibrations due to pedestrian traffic are highly significant. This is particularly understandable because footbridges are increasingly designed with lower frequencies and lighter weights, which can cause their fundamental vibration frequency to align with the walking frequency of pedestrians [3, 4]. This alignment often leads to potential issues with vibration comfort, highlighting the importance of understanding how the natural frequencies of bridge structures change with different positions of additional masses. As a result of specific incidents related to footbridge vibration issues over the past several decades such as the Millennium Bridge in London [5, 6, 7] and the Solferino Bridge in Paris [8, 9], research has focused on enhancing design standards that consider pedestrian loads, traffic density, and comfort requirements [10]. Structures with natural frequencies outside the range of 1-3 Hz, which is typically induced by pedestrian loads [11], are not at risk of resonant vibrations according to many regulations and standards and, therefore, do not require further dynamic analysis. According to the Eurocode [12], dynamic analysis is necessary for pedestrian bridges if their natural frequencies fall within the critical range of 0.5 to 5.0 Hz. Meanwhile, the AASHTO LRFD Guide Specifications for the Design of Pedestrian Bridges [13] indicate that pedestrian-induced vertical vibrations are negligible if a bridge's fundamental vertical frequency exceeds 3.0 Hz, and transverse vibrations are negligible if the transverse frequency exceeds 1.3 Hz. If these conditions are not met, AASHTO recommends conducting a dynamic analysis according to European Standards, specifically following SETRA [14]for methods of analysis, loading cases, and criteria for acceleration acceptance. The serviceability assessment is generally based on a comparison between the pedestrian-induced acceleration and a suitably defined limit value [15]. In dynamic analysis, a significant aspect involves monitoring the shift in the fundamental frequency caused by additional masses, such as groups of pedestrians standing in specific sections of a bridge while others are in motion. It is essential to pinpoint the critical mass configuration capable of pushing a system into an undesirable frequency range, potentially inducing resonant vibrations. Such vibrations could compromise pedestrian comfort and raise concerns regarding a bridge's stability. The first and simplest way to simulate people's presence on a structure is by modeling human occupants as added masses [16; 17], which can effectively explain reductions in natural frequencies. However, to realistically explain a dynamic response, it is preferable to use models that consider human-structure interaction when simulating the presence of people on a footbridge [18; 19; 20; 21; 22; 23; 24; 25]. Analytical models for analyzing the vibrations of a beam under the action of a harmonic moving mass [26] and moving mass with constant [27] and variable speed [28], as well as for the evaluation of vertical vibrations of footbridges using response spectrum [29], have been proposed. Regarding the current codes of practice [14; 30], the influence of a static pedestrian mass, corresponding to up to 5% of a bridge mass, can be ignored since it results in a decrease in natural frequency of less than 2.5%. Several studies, including those by [PERSON] et al. [31] and [PERSON] et al. [32], some relying on full-scale measurements, have demonstrated that passive humans (sitting and standing) significantly impact the dynamic properties of a structure.