# A Study on a Mechanism of Lateral Pedestrian-Footbridge Interaction
## 3 Study on Dynamic Response of Pedestrian-Footbridge Interactions
### Dynamic Coupling Equation Solution of the Lateral Interaction Pedestrian-Footbridge
\tag{24}\]  
Hence, Equations (21)-(24) can be rewritten as follows in the assumption of \(\varphi=\sigma T_{1}-\theta_{p}\):  
\[\omega_{p}D_{1}a_{p}=\frac{1}{2}\eta a_{p}\omega_{p}-\frac{3}{8}\rho a_{p}^{3 }a_{p}^{3}-\frac{1}{8}\xi\omega_{p}a_{p}^{3}+\frac{a_{s}}{2}\omega_{s}^{2}\sin\varphi \tag{25}\]  
\[a_{p}\omega_{p}D_{1}\varphi=a_{p}\omega_{p}\sigma+\frac{a_{s}}{2}\omega_{s}^{ 2}\cos\varphi \tag{26}\]  
\[ \begin{array}{l}2\omega_{s}D_{1}a_{s}=-\zeta_{s}a_{s}\omega_{s}^{2}-\frac{m} {2M}\omega_{p}^{2}a_{p}\sin\varphi-\eta\frac{m}{2M}a_{p}\omega_{p}\cos\varphi +\\ \frac{3m}{8M}\rho a_{p}^{3}\omega_{p}^{3}\cos\varphi+\frac{m}{8M}\xi a_{p}^{3} \omega_{p}\cos\varphi\\ \\ \frac{m}{2M}\omega_{p}^{2}a_{p}\cos\varphi-\eta\frac{m}{2M}a_{p}a_{p}\sin\varphi +\frac{3m}{8M}\rho a_{p}^{3}\omega_{p}^{3}\sin\varphi+\\ \frac{m}{8M}\xi a_{p}^{3}\omega_{p}\sin\varphi=0\\ \end{array} . \tag{27}\]  
To obtain the amplitude frequency equation of pedestrians and the footbridge structure as well as the Phase Angle of displacements for pedestrians and the footbridge, it is assumed that \(D_{1}a_{p}=0\), \(D_{1}\varphi=0\), and \(D_{1}a_{s}=0\) are the equations for which the amplitude and Phase Angle are satisfied. Subsequently, one obtains:  
\[\frac{1}{2}\eta a_{p}\omega_{p}-\frac{3}{8}\rho\omega_{p}^{3}a_{p}^{3}-\frac{ 1}{8}\xi\omega_{p}a_{p}^{3}=-\frac{a_{s}}{2}\omega_{s}^{2}\sin\varphi \tag{29}\]  
\[a_{p}\omega_{p}\sigma=-\frac{a_{s}}{2}\omega_{s}^{2}\cos\varphi \tag{30}\]  
\[\begin{array}{l}\zeta_{s}a_{s}\omega_{s}^{2}=-\frac{m}{2M}\omega_{Equations (29)-(32) can be rewritten as follows. If \(\varsigma=\rho\omega_{p}^{2}+\eta\), one obtains the following equations:  
\[\frac{1}{2}\eta a_{p}\omega_{p}-\frac{1}{8}\omega_{p}a_{p}^{3}\varsigma=-\frac{ a_{s}}{2}\omega_{s}^{2}\sin\varphi \tag{33}\]  
\[a_{p}\omega_{p}\sigma=-\frac{a_{s}}{2}\omega_{s}^{2}\cos\varphi \tag{34}\]  
\[\zeta_{s}a_{s}\omega_{s}^{2}=-\frac{m}{2M}\omega_{p}^{2}a_{p}\sin\varphi-\eta \frac{m}{2M}a_{p}\omega_{p}\cos\varphi+\frac{m}{8M}a_{p}^{3}\omega_{p}\varsigma \cos\varphi \tag{35}\]  
\[\frac{m}{2M}\omega_{p}^{2}a_{p}\cos\varphi-\eta\frac{m}{2M}a_{p}\omega_{p}\sin \varphi+\frac{m}{8M}a_{p}^{3}\omega_{p}\varsigma\sin\varphi=0. \tag{36}\]  
Multiplying Equations (33) and (34) and then adding them both together leads to:  
\[\left(\frac{1}{2}\eta a_{p}\omega_{p}-\frac{1}{8}\omega_{p}a_{p}^{3}\varsigma \right)^{2}+\omega_{p}^{2}a_{p}^{2}\sigma^{2}=\frac{a_{s}^{2}}{4}\omega_{s}^ {4}. \tag{37}\]  
Similarly, Equations (35) and (36) are multiplied, and then both of them are added together. The resulting equation can be defined as follows:  
\[\zeta_{s}^{2}\omega_{s}^{4}a_{s}^{2}=\frac{m^{2}}{4M^{2}}\omega_{p}^{4}a_{p}^ {2}+\frac{m^{2}}{4M^{2}}\eta^{2}\omega_{p}^{2}a_{p}^{2}-\frac{m^{2}}{8M^{2}} \gamma\varsigma\omega_{p}^{2}a_{p}^{4}+\frac{m^{2}}{64M^{2}}\omega_{p}^{2}a_{ p}^{6}\varsigma^{2}. \tag{38}\]  
Equations (37) and (38) are jointly connected to decouple the lateral amplitude of pedestrians and footbridges, for which the amplitude frequency equation of pedestrians' lateral vibration can be derived as:  
\[ \begin{array}{l}(\frac{1}{16}-\frac{m^{2}}{64M^{2}\zeta_{s}^{2}})\varsigma^ {2}a_{p}^{6}+(\frac{m^{2}}{8M^{2}\zeta_{s}^{2}}-\frac{1}{2})\eta\varsigma a_{ p}^{4}+(\eta^{2}+4\sigma^{2}-\\ \frac{m^{2}}{4M^{2}\zeta_{s}^{2}}\omega_{p}^{2}-\frac{m^{2}}{4M^{2}\zeta_{s}^{ 2}}\eta^{2})a_{p}^{2}=0\end{array} . \tag{39}\]  
With respect to Equations (33) and (34), the Phase Angle between the lateral relative pedestrians' displacement and lateral vibration displacement of the footbridge is derived as:  
\[\tan\varphi=\frac{4\eta-a_{p}^{2}\varsigma}{8\sigma}. \tag{40}\]  
It can be concluded from Equation (40) that when the pedestrians' lateral walking frequency is consistent with the vibration frequency of the footbridge, that is, when the resonance occurs, the Phase Angle between the pedestrians' relative displacement and the vibration displacement of the footbridge tends to \(\pm 90^{\circ}\).