## 1 Introduction  
With the wide application of high-strength and lightweight materials and the improvement of structural analysis technology, the development of bridges tends to be large-span, slim, and light types. However, light, flexible, low damping large-span bridges are prone to vibration problems. Seismic loads and wind loads are typically taken into account in the dynamic analysis of bridge structures, while the crowd loads are designed as static loads whose dynamic impacts are frequently ignored. For large-span bridges of highways and railroads, the error brought by such treatment can be neglected in most cases due to the small proportion of crowd loads, however, the dynamic response caused by crowd loads is important for pedestrian bridges and non-motorized passenger bridges. Especially in the design of large-span pedestrian bridges, neglecting the potential crowd-induced vibration response during operation can lead to inaccurate prediction of structural response. Accidents attributed to vibration caused by crowd loads have been recurrent on pedestrian bridges, with notable instances being the Millennium Bridge in London, and the T-Bridge in Japan [1]. Furthermore, numerous other pedestrian bridges have encountered vibration issues under pedestrian loading, such as the Mape Valley Great Suspension Bridge [2], the Tudor Park Bridge [3], the Changi Mezzaine Bridge [4], and the Solferino Bridge. For bridges with vertical fundamental frequencies close to pedestrian step frequencies, there are two damping methods when considering the bridge structure itself: the frequency adjustment method and the vibration damping method. The purpose of the frequency adjustment method is to avoid the sensitive range of frequency in the structure by increasing the static stiffness, and different requirements of the fundamental frequency of pedestrian bridges have been proposed for different standards [5; 6; 7; 8; 9]. However, the structural resonance caused by the higher step frequency load component of pedestrian traffic is not considered in this method, and the structural dynamic response from a quantitative perspective not reflected. As for the vibration damping method, the structural resonance response is reduced by increasing the damping of the pedestrian bridge structure, which has been widely accepted as an effective method to reduce the dynamic response of structures. There are many types of classical control devices to increase the damping of structures such as viscous dampers [10; 11; 12], tuned mass dampers (TMDs) [13], restrained flexural corrugated plates [14], and actively controlled pre-stressing tendons [15; 16]. The comfort and the vibration-damping design of pedestrian bridges have been widely studied by experts and scholars [17; 18; 19; 20]. Considering the crowd-induced control, a generic optimization method was proposed by [PERSON] et al. [21] to solve the vibration problem of pedestrian bridges. The vibration response of a 142 m long three-span continuous pedestrian bridge under pedestrian loads was tested, and the results of the experimental dynamics tests and the numerical simulation were compared [22]. What is more, TMD has been widely studied and applied in many practical structures [23; 24]. The dynamic response of large-span bridges installed with TMDs was analyzed, and the results showed that the torsional and vertical response of the bridge under wind loads and traffic loads can be effectively controlled by a TMD system [25; 26]. The mechanical performance of steel pedestrian bridges with and without TMD were investigated using experimental and numerical simulation methods by [PERSON] et al. [27], and it was found that the structural response induced by pedestrians can be reduced by TMD; the degree of which depended on the type of human excitation. According to German pedestrian bridge design guidelines, [PERSON] [28] evaluated the comfort of a bridge with TMD for vibration damping treatment using the time history analysis method.