## 3 Excitation Assumption  
All pedestrians are assumed to be uniformly distributed on a footbridge deck, without an overlap, as shown in Figure 2. Additionally, pedestrians are assumed to be moving with consistent walking behavior and that they have the same dynamic properties. For simplicity, the superscript \"(\(q\))\" is omitted in the following contents. Each pedestrian is assumed to walk in a fixed position, as shown in Figure 3. To be more specific, each pedestrian's walking behavior is similar to the movement on a treadmill. The numbers of pedestrians in longitudinal and lateral directions are \(N_{x}\) and \(N_{y}\), respectively. The total number of pedestrians can be obtained with \(\chi=N_{x}N_{y}\). Considering the step length \(L_{s}\) is much less than the structure span \(L_{B}\), the models of two footholds are approximated as:  
\[\phi_{1}\Big{(}x^{(q)}\Big{)}\simeq\phi_{1}\Big{(}x_{l}^{(q)}\Big{)},\phi_{1} \Big{(}x^{(q)}\Big{)}\simeq\phi_{1}\Big{(}x_{t}^{(q)}\Big{)} \tag{15}\]  
Without considering leg compression, the lengths of two legs are kept as the original length. Based on the above assumption, the damping parameters \(\sum\limits_{q=1}^{X}\Big{(}c_{zz}^{(q)}\otimes\Phi_{1,1}\Big{)}\) can be approximated by:  
\[\sum\limits_{q=1}^{X}\Big{(}c_{zz}^{(q)}\otimes\Phi_{1,1}\Big{)}\simeq N_{y}c _{leg}\Big{(}\frac{z}{L_{0}}\Big{)}\sum\limits_{q=1}^{N_{x}}\Big{(}\sin^{2} \!\left(\frac{q\pi}{N_{x}\,+\,1}\right)\Big{)} \tag{16}\]  
Figure 3: Assumption of stepping in a fixed position. Figure 2: Diagram of a uniformly arranged crowd on a footbridge. where \(c_{leg}\) is the damping of leg. The \(\sum\limits_{q=1}^{\chi}\left(k_{\Delta}^{(q)}\otimes\Phi_{1,1}\right)\) in Equation (10) is calculated by  
\[ \begin{array}{ll}\sum\limits_{q=1}^{\chi}\left(k_{\Delta}^{(q)}\otimes\Phi_{1,1}\right)&\simeq N_{y}\sum\limits_{q=1}^{N_{x}}\left(k_{leg}(\Delta L_{l}/L_{ l}+\Delta L_{t}/L_{t})\Phi_{1,1}\left(x^{(q)}\right)\right)\\ &\simeq 2N_{y}k_{leg}\frac{\Delta z}{L_{0}}\sum\limits_{q=1}^{N_{x}}\sin^{2} \left(\frac{q\pi}{N_{x}+1}\right)\end{array}  \tag{17}\]  
where \(\Delta z\) is vertical vibration amplitude of CoM and \(k_{leg}\) is leg stiffness. \(\sum\limits_{q=1}^{\chi}\left(c_{zz}^{(q)}\otimes\phi_{1}\right)\dot{x}^{(q)},\quad\sum\limits_{q=1}^{\chi}\left(c_{zz}^{(q)}\otimes\phi_{1}\right)\dot{z}^ {(q)}\) and \(\quad\sum\limits_{q=1}^{\chi}\left(k_{\Delta}^{(q)}\otimes\phi_{1}\right)z^{(q)}\) are calculated by Equations (18)-(20), respectively. \[\sum\limits_{q=1}^{\chi}\left(c_{xz}^{(q)}\otimes\phi_{1}\right)\dot{x}^{(q)} \simeq\sum\limits_{q=1}^{\chi}\left(c_{tz}^{(q)}+c_{tz}^{(q)}\right)\phi_{1} \left(x^{(q)}\right)\dot{x}^{(q)}\simeq N_{y}\frac{c_{dz}\dot{x}^{z}}{L_{0}^{ 3}}(2x-x_{l}-x_{l})\sum\limits_{q=1}^{N_{x}}\sin\left(\frac{q\pi}{N_{x}+1}\right) \tag{18}\]  
\[\sum\limits_{q=1}^{\chi}\left(c_{zz}^{(q)}\otimes\phi_{1}\right)\dot{z}^{(q)} \simeq N_{y}c_{leg}\frac{zz}{L_{0}}\sum\limits_{q=1}^{N_{x}}\sin\left(\frac{q \pi}{N_{x}+1}\right) \tag{19}\]  
\[ \begin{array}{ll}\sum\limits_{q=1}^{\chi}\left(k_{\Delta}^{(q)}\otimes\phi_ {1}\right)z^{(q)}\simeq N_{y}k_{leg}\Delta z\left(\frac{1}{\sin\theta_{1}}+ \frac{1}{\sin\theta_{t}}\right)\sum\limits_{q=1}^{N_{x}}\sin\left(\frac{q\pi }{N_{x}+1}\right)\\ \simeq N_{y}k_{leg}\Delta z\sum\limits_{q=1}^{N_{x}}\sin\left(\frac{q\pi}{N_{x }+1}\right)\end{array}  \tag{20}\]  
The \(\sum\limits_{q=1}^{\chi}\sin^{2}\left(\frac{q\pi}{N_{x}+1}\right)\) is simplified by employing the Taylor expansion as:  
\[