# A Study on a Mechanism of Lateral Pedestrian-Footbridge Interaction
## 5 Study on Pedestrian-Footbridge Lateral Displacement and Lateral Interaction
### Phase Angle between Lateral Interaction Force and Lateral Excitation Speed  
When the pedestrians' lateral walking frequency is not equal to the fundamental frequency of the footbridge, the interaction force can be presented as:  
\[\begin{array}{l}F=m(-\frac{d^{2}U_{y}}{dt^{2}}-\frac{d^{2}u_{y}}{dt^{2}})=m \omega_{s}^{2}a_{s}\cos(\omega_{s}t)+m\omega_{p}^{2}a_{p}\cos(\omega_{p}t+ \theta_{p})\\ =m\sqrt{\left(\omega_{s}^{2}a_{s}+\omega_{p}^{2}a_{p}\cos\varphi\right)^{2}+ \left(\omega_{p}^{2}a_{p}\sin\varphi\right)^{2}\cos(\omega_{s}t-\phi)}\end{array}. \tag{61}\]  
Because \(\cos\varphi=-\frac{2a_{p}\omega_{p}\sigma}{a_{s}\omega_{s}^{2}}\), in this case \(\sin\varphi=\pm\frac{\sqrt{\frac{\omega_{s}^{2}a_{s}^{4}-4\omega_{p}^{2}\sigma _{p}^{2}\sigma^{2}}{a_{s}\omega_{s}^{2}}}}{\text{; hence, Equation (61) can be written as:  
\[\begin{array}{l}F=m(-\frac{d^{2}U_{y}}{dt^{2}}-\frac{d^{2}u_{y}}{dt^{2}})\\ =m\sqrt{\left[\omega_{s}^{2}a_{s}-\omega_{p}^{2}a_{p}\frac{2a_{p}\omega_{p} \sigma}{a_{s}\omega_{s}^{2}}\right]^{2}+\left[\pm\omega_{p}^{2}a_{p}\frac{\sqrt {\frac{\omega_{s}^{2}a_{s}^{4}-4\omega_{p}^{2}\sigma_{p}^{2}\sigma^{2}}{a_{s} \omega_{s}^{2}}}}\right]^{2}}\cos(\omega_{s}t-\phi)\\ =m\sqrt{\omega_{s}^{4}a_{s}^{2}-4\omega_{p}^{3}a_{p}^{2}\sigma+\omega_{p}^{4}a_ {p}^{2}\cos(\omega_{s}t-\phi)}\end{array}. \tag{62}\]  
The lateral dynamic amplitude of pedestrians on the footbridge is defined as:  
\[F_{\max}=m\sqrt{\omega_{s}^{4}a_{s}^{2}-4\omega_{p}^{3}a_{p}^{2}\sigma+\omega_ {p}^{4}a_{p}^{2}}. \tag{63}\]  
Equation (37) is inserted in Equation (63) to easily obtain:  
\[\begin{array}{l}F_{\max}=m\omega_{p}a_{p}\sqrt{\left(\eta-\frac{1}{4}a_{p}^ {2}\varsigma\right)^{2}+4\sigma^{2}-4\omega_{p}\sigma+\omega_{p}^{2}}\\ =m\omega_{p}a_{p}\sqrt{\left(\eta-\frac{1}{4}\xi a_{p}^{2}\frac{1}{4}a_{p}^{2} \rho\omega_{p}^{2}\right)^{2}+4\sigma^{2}-4\omega_{p}\sigma+\omega_{p}^{2}} \end{array} \tag{64}\]  
As a result, the lateral dynamic coefficient is presented as:  
\[\alpha=\frac{\omega_{p}a_{p}}{g}\sqrt{\left(\eta-\frac{1}{4}\xi a_{p}^{2}- \frac{1}{4}\rho\omega_{p}^{2}a_{p}^{2}\right)^{2}+4\sigma^{2}-4\omega_{p} \sigma+\omega_{p}^{2}}. \tag{65}\]  
The phase angle between the interaction force and vibration displacement of the footbridge can be computed as:  
\[\tan\phi=\frac{m\omega_{p}^{2}a_{p}\sin\varphi}{m\omega_{s}^{2}a_{s}+m\omega _{p}^{2}a_{p}\cos\varphi}. \tag{66}\]  
Equations (37), (54), and (55) are substituted into Equation (66), and the following equation holds:  
\[\tan\phi=\frac{\omega_{p}(\eta-\frac{1}{4}\xi a_{p}^{2}-\frac{1}{4}\rho\omega _{p}^{2}a_{p}^{2})}{\left(\eta-\frac{1}{4}\xi a_{p}^{2}-\frac{1}{4}\rho\omega _{p}^{2}a_{p}^{2}\right)^{2}+4\sigma^{2}-2\omega_{p}\sigma}. \tag{67}\]  
Since the lateral vibration displacement of the footbridge is expressed as \(U_{s}=a_{s}(T_{1})\cos(\omega_{s}t)\), the lateral vibration speed of the footbridge becomes:  
\[\dot{U}_{s}=-a_{s}(T_{1})\omega_{s}\sin(\omega_{s}t)=a_{s}(T_{1})\omega_{s} \cos(\omega_{s}t+\frac{\pi}{2}). \tag{68}\]  
Hence, the Phase Angle between the pedestrian-footbridge interaction force and the lateral vibration speed of the footbridge is:  
\[\phi^{\prime}=\phi-\frac{\pi}{2}. \tag{69}\]Therefore, it can be proven that the lateral interaction force amplitude and the dynamic load coefficient are related to the relative lateral response amplitude of pedestrians, pedestrians' parameters, walking frequency, and frequency differences but have nothing to do with the mass of the footbridge. When these parameters are considered to be constant, in this case, the dynamic load coefficient is linear with respect to the amplitude of the pedestrian lateral response. Based on this, the larger the response amplitude is, the larger the dynamic load coefficient becomes, which is proven to be the main principle of pedestrian-footbridge interactions.  
It is assumed that when \(\phi<0\), the lateral interaction force lags behind the response of the footbridge; conversely, when \(\phi>0\), the lateral interaction force is ahead of the pedestrians' response.