# A Study on a Mechanism of Lateral Pedestrian-Footbridge Interaction
## 5 Study on Pedestrian-Footbridge Lateral Displacement and Lateral Interaction  
The Phase Angle between the Lateral Relative Displacement of Pedestrians and Lateral Displacement of the Footbridge  
In Equation (40), \(\theta_{p}\) is the Phase Angle between the lateral relative displacement of pedestrians and the lateral displacement of the footbridge. Assuming \(\theta_{p}<0\), the lateral absolute displacement of pedestrians lags behind the lateral displacement of the footbridge; if \(\theta_{p}>0\), the lateral absolute displacement of pedestrians is ahead of the lateral displacement of the footbridge; the amount of time lagging behind or in advance of is shown in the equation \(\tau=\left|\varphi\right|/\omega_{p}\). The relationship between Phase Angle \(\varphi\) and frequency tuning parameters \(\sigma\) can be derived as follows based on Equation (34):  
\[\cos\varphi=-\frac{2a_{p}\omega_{p}\sigma}{a_{s}\omega_{s}^{2}}. \tag{54}\]  
The phase angle is located at \([-\pi,\pi]\). Figure 2 presents the difference between pedestrians' lateral walking frequency and the footbridge fundamental frequency (frequency tuning parameters). It also shows the impact of the difference on the Phase Angle between pedestrians and the footbridge in the case of different amplitude ratios. According to Figure 3, when pedestrians' lateral walking frequency tends towards the fundamental frequency of the footbridge, the corresponding Phase Angle approaches \(\pi/2\). Taking [PERSON]'s study into account, it can be concluded that the range of Phase Angles between pedestrians and footbridge motion is limited to \([0,\pi/2]\) degree. In addition, the phase angle at resonance is \(\pi/2\). The results deduced in the paper are in accordance with the results based on the [PERSON] model [18]. In [PERSON] study of the Millennium Bridge, it is concluded that the Phase Angle of pedestrians with regard to the footbridge is \([-\pi,\pi]\), which is similar to the results found in this paper [21]. First, in the case where pedestrians' lateral walking frequency is consistent with the lateral fundamental frequency of the footbridge, that is, in the case of \(\sigma=0\), while considering Equation (54), it can be proven that \(\sigma=0\) and \(\cos\varphi=0\). Thus, \(\cos\varphi=0\) is inserted into Equation (35), after which we can obtain:  
\[\sin\varphi=-\frac{2M\zeta_{\mathrm{s}}\omega_{\mathrm{s}}^{2}a_{\mathrm{s}}}{ m\omega_{\mathrm{p}}^{2}a_{\mathrm{p}}}. \tag{55}\]  
Taking Equation (54) into consideration, it can be derived that \(\sin\varphi=-1\); thus, \(\varphi=-\frac{\pi}{2}+2k\pi\), \(k=0\), \(\pm 1\), \(\pm 2\), \(\pm 3\),..., which is consistent with the above results based on the relationship between the Phase Angle and frequency tuning parameters. Therefore, if \(\sigma=0\), then \(\theta_{p}=-\varphi\); if \(\theta_{p}>0\), then the pedestrians' vibration response is ahead of the footbridge response, with the amount of time in advance of the footbridge response shown as \(\tau=\theta_{p}/\omega_{\mathrm{s}}\). When the pedestrians walk along the footbridge, the amount of time before the footbridge's response is represented as \(\tau=(\pi-4k\pi)/2\omega_{\mathrm{s}}\). For \(\theta_{p}<0\), the pedestrians' vibration response lags behind the footbridge's response, with the lag time \(\tau=-\theta_{p}/\omega_{\mathrm{s}}\); when the pedestrians walk on the footbridge, the lag time becomes \(\tau=(-\pi+4k\pi)/2\omega_{\mathrm{s}}\). Figure 3: Amplitude frequency curve of pedestrians’ lateral vibration. Figure 2: Effect of frequency tuning parameters on the Phase Angle between pedestrians and footbridges under different amplitude ratios. Phase Angle between the Lateral Absolute Displacement of Pedestrians and the Lateral Displacement of the Footbridge  
With regard to resonance, the vibration frequency caused by pedestrians is the same as that of footbridges but differs in phase.