# A Study on a Mechanism of Lateral Pedestrian-Footbridge Interaction
## 1 Introduction  
The vertical vibrations of a pedestrian-footbridge coupling have attracted considerable and increasing attention [1; 2; 3; 4; 5]. A new single DOF (degree of freedom) model of a bipedal pedestrian is proposed to study the dynamic interaction between a footbridge and pedestrians [6]. [PERSON] et al. [7] simulated crowd-footbridge interactions based on a multi free mass-damping coupling system. In line with the work of [PERSON] [8], the probability assessment method is applied to study the interaction between the vertical vibrations of multiple pedestrians and a footbridge. In a study by [PERSON] [9], a three-degree-of-freedom human-structure interaction model was established to study the system resonance frequency. Comparing the existing experimental results, it was found that the tandem three-degree-of-freedom human model is closer to the experimental results. A new biodynamic synchronized coupled model is based on a measured footbridge on a comparative basis [10]. In a study by [PERSON] et al. [11], pedestrian excitation was considered to model the structure-human body model-damping of Phase Angle interactions based on biomechanics. [PERSON] et al. [12] established a model of pedestrians' movements based on biomechanics. [PERSON] et al. [13] studied the parametervibration responses while considering the time-lag effect by taking the Millennium Bridge in London as an example, which shows that the time-lag has no effect on the amplitude of the parameter vibration response. [PERSON], [PERSON] et al. [14] discussed parameter variation trends that were used to obtain coupled pedestrian-bridge oscillations, notably the influence of the mass ratio of pedestrians to the bridge mass in the mode considered, and the phase variation with the amplitude.  
However, there are few studies that have been conducted on lateral coupled pedestrian-footbridge vibrations. [PERSON], [PERSON], and [PERSON] considered pedestrians using a linear inverted pendulum model and studied the changes in parameters such as mass, damping, and force in the case of pedestrian-footbridge laterally coupled vibrations [15; 16; 17]. Among these studies, most researchers did not highlight the importance of pedestrians' vibration responses, but simply focused on the vibration responses of the footbridge. Some scholars did not analyze the decoupling work of pedestrian-footbridge coupling systems when studying pedestrian-footbridge coupling vibrations [18]. Instead, the vibration of the footbridge was regarded as a vibration table in their studies. Its vibration amplitude and frequency were assumed to be a constant value [18]. In this case, this assumption is not reasonable because the lateral vibration response of pedestrians and footbridges interacts mutually. However, if the coupling is weak, it is reasonable to assume that the amplitude and frequency of the footbridge are constant. When the lateral swaying frequency of pedestrians and lateral vibration of the footbridge are the same or similar, the coupled vibration between pedestrians and the footbridge is strongly coupled. Moreover, both have a major influence on each other. The pedestrian-footbridge system resonance effect cannot be presented because the vibration amplitude and frequency of the footbridge are simply considered as being constant. [PERSON] et al. [19; 20] adopted a modified hybrid Van der Pol/Rayleigh model to simulate the lateral forces of pedestrians on periodic moving floors and rigid floors, respectively.  
Therefore, based on a lateral force model of the hybrid Van der Pol/Rayleigh model proposed in their paper, the lateral interaction dynamic model of pedestrian-flexible footbridge coupling system was established. By decoupling the coupling equation, it derives the amplitude-frequency curve of pedestrians and pedestrian bridges. The paper also studies the lateral displacement and lateral force between pedestrians and footbridges. Finally, the study obtains a dynamic response of interactions between pedestrians and footbridges through an example.