## 3 Excitation Assumption
\begin{array}{ll}\sum\limits_{q=1}^{N_{x}}\sin^{2}\left(\frac{q\pi}{N_{x}+1} \right)\simeq\left(\frac{\pi}{N_{x}+1}\right)\sum\limits_{q=1}^{N_{x}}q^{2}- \frac{1}{3}\left(\frac{\pi}{N_{x}+1}\right)^{4}\sum\limits_{q=1}^{N_{x}}q^{4} +\frac{1}{36}\left(\frac{\pi}{N_{x}+1}\right)^{6}\sum\limits_{q=1}^{N_{x}}q^{6} \\ =\frac{\pi^{2}}{6}\frac{N_{x}(2N_{x}+1)}{N_{x}+1}-\frac{\pi^{4}}{90}\frac{N_{x} (2N_{x}+1)\left(3N_{x}^{2}+3N_{x}-1\right)}{\left(N_{x}+1\right)^{3}}+\frac{ \pi^{6}}{1512}\frac{N_{x}(2N_{x}+1)\left(3N_{x}^{4}+6N_{x}^{3}-3N_{x}+1\right)} {\left(N_{x}+1\right)^{5}}\\ \simeq\left(\frac{1}{3}-\frac{\pi^{2}}{15}+\frac{\pi^{4}}{252}\right)\pi^{2}N _{x}\end{array}  \tag{21}\]  
Similarly, the \(\sum\limits_{q=1}^{N_{x}}\sin\left(\frac{q\pi}{N_{x}+1}\right)\) is approximated as:  
\[\sum\limits_{q=1}^{N_{x}}\sin\left(\frac{q\pi}{N_{x}+1}\right)\simeq\frac{\pi} {2}\left(1-\frac{\pi^{2}}{12}\right)N_{x} \tag{22}\]  
Substituting Equations (16) and (21) into Equation (9), the system damping ratio induced by the crowd can be estimated as:  
\[\widetilde{\xi}\simeq\zeta_{1}\frac{f_{1}}{\widetilde{f}}+\left(\frac{1}{3}- \frac{\pi^{2}}{15}+\frac{\pi^{4}}{252}\right)\frac{\chi\pi c_{leg}}{4M_{1} \widetilde{f}}\frac{z}{L_{0}} \tag{23}\]  
Similarly, by substituting Equations (17) and (21) into Equation (10), the system frequency induced by crowd is:  
\[\widetilde{f}=\sqrt{f_{1}^{2}-\left(\frac{1}{3}-\frac{\pi^{2}}{15}+\frac{\pi^{ 4}}{252}\right)\frac{\chi k_{leg}\Delta z}{2M_{1}L_{0}}} \tag{24}\]By substituting Equations (18)-(20) and (22) into Equation (11), the system excitation force induced by the crowd is calculated by:  
\[\widetilde{F}=\left(1-\frac{\pi^{2}}{12}\right)\frac{\pi\chi}{2M_{1}}\Bigg{[} \frac{c_{leg}\dot{x}z}{L_{0}^{2}}(2x-x_{l}-x_{l})+\frac{c_{leg}\dot{z}z}{L_{0}}- k_{leg}\Delta z\Bigg{]} \tag{25}\]  
Longitudinal movement velocity of the CoM is approximated to follow cosine [14]. The velocity of the CoM is at its highest when the CoM is approximately located at the mid-position between two footholds. In this study, longitudinal velocity of the CoM is defined as:  
\[\dot{x}(t)=\frac{L_{s}}{T_{s}}-A_{vx}\cos\left(\frac{2\pi t}{T_{s}}\right),\ (0 \leq t<T_{s}) \tag{26}\]  
where \(T_{s}\) is the period of stepping; \(A_{vx}\) is the varying amplitude of the CoM velocity around mean velocity. Longitudinal displacement of the CoM can be obtained by Equation (27) after integration with Equation (26)  
\[x(t)=\frac{L_{s}}{T_{s}}t-\frac{A_{v}T_{s}}{2\pi}\sin\frac{2\pi t}{T_{s}}+ \frac{x_{l}+x_{t}}{2}-\frac{L_{s}}{2},\ (0\leq t<T_{s}) \tag{27}\]  
When the supporting leg is upright in the single stance phase, the CoM reaches the summit in the vertical direction. When the CoM is located at the mid-position between two footholds, it reaches the lowest place in the vertical direction [14]. The vertical movement trajectory of the CoM is approximately sinusoidal and its expression is defined as:  
\[z(t)=L_{0}-\Delta z_{g}-A_{z}\sin\frac{\pi t}{T_{s}},\ (0\leq t<T_{s}) \tag{28}\]  
where \(\Delta z_{g}\) is the compression deformation of the leg, induced by gravitational effect; \(A_{z}\) is the vertical change amplitude of the CoM.