## 2 Dynamic Excitation Mechanism  
The bipedal pedestrian structure system [13] is shown in Figure 1. The left end of the beam is defined as the origin of a planar coordinate system \(x\)-0-\(z\); \(L_{B}\) is span length; \(m^{(q)}\) is the lump mass of the bipedal model; the superscript \"\(q\)\" means the ordinal number of pedestrians; \(x^{(q)}\) and \(z^{(q)}\) are longitudinal and vertical displacements of the center of mass (CoM), respectively; \(x_{t}^{(q)}\) and \(x_{l}^{(q)}\) are the trialing and leading footholds positions, respectively; \(\left(x_{l}^{(q)},t\right)\) and \(\left(x_{t}^{(q)},t\right)\) are structural vertical displacements in the leading and trialing footholds, respectively. By only considering the first modal property, the dynamic Equation (8a) from the study [13] can be rewritten as:  
\[ \begin{split}& M_{1}\ddot{Y}_{1}(t)+2\zeta_{1}\omega_{1}M_{1} \dot{Y}_{1}(t)+\omega_{1}^{2}M_{1}Y_{1}+\sum\limits_{q=1}^{X}\left(c_{zz}^{(q )}\otimes\Phi_{1,1}\right)\dot{Y}_{1}(t)\\ &-\sum\limits_{q=1}^{X}\left(c_{zz}^{(q)}\otimes\phi_{1}\right)z ^{(q)}-\sum\limits_{q=1}^{X}\left(k_{z}^{(q)}\otimes\phi_{1}\right)z^{(q)}= \sum\limits_{q=1}^{X}\left(c_{xz}^{(q)}\otimes\phi_{1}\right)\dot{x}^{(q)}\end{split}  \tag{1}\]  
where \(M_{1}\) is modal mass of structure; \(\xi_{1}\) and \(\omega_{1}\) are the damping ratio and circular frequency of structure, respectively; \(Y_{1}\), \(Y_{1}\) and \(\dot{Y}_{1}\) are general displacement, velocity and acceleration, respectively;  
Figure 1: Crowd-structure interaction model. \(\phi_{1}(x)=\sin(\pi x/L_{B})\) and \(\Phi_{1,1}(x)=\phi_{1}^{2}(x)\) are the structural modal [13]. The tensor symbols \(c_{zz}^{(q)}\otimes\Phi_{1,1}\) and \(c_{xz}^{(q)}\otimes\phi_{1}\) are shown as  
\[c_{zz}^{(q)}\otimes\Phi_{1,1}=c_{zz}^{(q)}\Phi_{1,1}\Big{(}x_{l}^{(q)}\Big{)}+c _{tz}^{(q)}\Phi_{1,1}\Big{(}x_{t}^{(q)}\Big{)} \tag{2}\]  
\[c_{xz}^{(q)}\otimes\phi_{1}=c_{lxz}^{(q)}\phi_{1}\Big{(}x_{l}^{(q)}\Big{)}+c_{ lxz}^{(q)}\phi_{1}\Big{(}x_{t}^{(q)}\Big{)} \tag{3}\]  
The leading and trailing leg damping coefficients are shown as  
\[c_{lzz}^{(q)}=c_{l}^{(q)}\Big{(}L_{lz}^{(q)}/L_{l}^{(q)}\Big{)}^{2},c_{lxz}^{( q)}=c_{l}^{(q)}\Big{(}L_{lx}^{(q)}L_{lz}^{(q)}\Big{)}/\Big{(}L_{l}^{(q)}\Big{)}^{2} \tag{4}\]  
\[c_{lzz}^{(q)}=c_{t}^{(q)}\Big{(}L_{tz}^{(q)}/L_{t}^{(q)}\Big{)}^{2},c_{lxz}^{( q)}=c_{t}^{(q)}\Big{(}L_{lx}^{(q)}L_{lz}^{(q)}\Big{)}/\Big{(}L_{t}^{(q)} \Big{)}^{2} \tag{5}\]  
where \(c_{l}^{(q)}\) and \(c_{t}^{(q)}\) are the leading and trialing leg damping, respectively; \(L_{l}^{(q)}\) and \(L_{t}^{(q)}\) are the leading and trialing leg lengths, respectively. The expressions of \(L_{lx}^{(q)}\), \(L_{lz}^{(q)}\), \(L_{lx}^{(q)}\) and \(L_{tz}^{(q)}\) can be found in the study [13]. The tensor product \(k_{z}^{(q)}\otimes\phi_{1}\) is shown as:  
\[k_{z}^{(q)}\otimes\phi_{1}=k_{lz}^{(q)}\phi_{1}\Big{(}x_{l}^{(q)}\Big{)}+k_{lz }^{(q)}\phi_{1}\Big{(}x_{t}^{(q)}\Big{)} \tag{6}\]  
where \(k_{lz}^{(q)}\) and \(k_{lz}^{(q)}\) are given by  
\[k_{lz}^{(q)}=k_{l}^{(q)}\Big{(}1-L_{0}^{(q)}/L_{l}^{(q)}\Big{)}\Big{(}1-_{l}^ {(q)}/z^{(q)}\Big{)},k_{lz}^{(q)}=k_{l}^{(q)}\Big{(}1-L_{0}^{(q)}/L_{l}^{(q)} \Big{)}\Big{(}1-_{t}^{(q)}/z^{(q)}\Big{)} \tag{7}\]  
\(k_{l}^{(q)}\) and \(k_{t}^{(q)}\) are the stiffness of leading and trialing legs, respectively. \(L_{0}^{(q)}\) is the relaxed length of the leg. Substituting Equations (2)-(7) into Equation (1), the dynamic equation of a crowd-structure system can be expressed as:  
\[\ddot{Y}_{1}(t)+2\vec{\xi}\tilde{\omega}\dot{Y}_{1}(t)+\vec{\omega}^{2}Y_{1} =\widetilde{F} \tag{8}\]  
where \(\vec{\xi}\) and \(\vec{\omega}=2\pi\widetilde{f}\) are the damping ratio and circular frequency of the structure induced by humans; \(\widetilde{f}\) is the structure frequency induced by humans; \(\widetilde{F}\) is the excited force.