## 2 Experimental Study of a Human Mass-Spring-Damping Model
### Theory of Mode Test
#### 2.3.2 Study Methodology  
In this study, the pedestrian was simplified to a single-degree-of-freedom mass-spring-damper (MSD) model, and a two-degree-of-freedom coupled pedestrian-structure model was established, as shown in Figure 3. The coupling effect between the vibration induced by the first mode of the slab and the pedestrian was considered, thus the slab could be assumed to be equivalent to a SDOF dynamic system. In this coupled system, the pedestrian is equated to a SDOF system for simplicity. In Figure 3, \(k_{S}\) denotes the equivalent stiffness of the first mode of the slab, \(m_{S}\) refers to its first modal mass and \(c_{S}\) denotes its first modal damping. \(m_{H}\) denotes the equivalent mass of the pedestrian and is taken to be the total mass of the pedestrian. \(k_{H}\) is the equivalent stiffness of the pedestrian and \(c_{H}\) is its equivalent damping. Once the mass and damping ratio of the pedestrian are derived from the test results, the equivalent stiffness \(k_{H}\) and \(c_{H}\) of the pedestrian can be easily calculated by dynamic methods, which are not presented in this paper. Drawing on the test method proposed by [PERSON] and [PERSON] [39], the dynamic characteristic parameters of the pedestrian MSD model were derived by identifying the modal parameters of the empty structure and identification of the coupled pedestrian-structure system, respectively. The difference is that the modal testing method used in this study is the ambient excitation method. The coupled two-degree-of-freedom pedestrian-structure model with damping is considered, with the dynamic equations expressed as  
\[ \begin{bmatrix}m_{S}&0\\ 0&m_{H}\end{bmatrix}\begin{bmatrix}\dot{x}_{S}(t)\\ \dot{x}_{H}(t)\end{bmatrix} + \begin{bmatrix}c_{S}+c_{H}&-c_{H}\\ -c_{H}&c_{H}\end{bmatrix}\begin{bmatrix}\dot{x}_{S}(t)\\ \dot{x}_{H}(t)\end{bmatrix} + \begin{bmatrix}k_{S}+k_{H}&-k_{H}\\ -k_{H}&k_{H}\end{bmatrix}\begin{bmatrix}x_{S}(t)\\ x_{H}(t)\end{bmatrix} = \begin{bmatrix}F_{S}(t)\\ F_{H}(t)\end{bmatrix} . \tag{15}\]  
With the floor slab and the human body both simulated with a single-degree-of-freedom mass-spring-damper (MSD), symbols \(x_{S}\) and \(x_{H}\) are the vibrational displacements of the structure and the human body as a single degree of freedom, respectively; \(\dot{x}_{S}\) and \(\dot{x}_{H}\) are the vibrational velocities; \(\dot{x}_{S}\) and \(\dot{x}_{H}\) are the vibrational accelerations; and \(F_{S}\) and \(F_{H}\) are the external forces acting on the structure and the human body. Random perturbations and excitation from the outside environment can be denoted by \(F_{S}(t)\). \(F_{H}(t)\) indicates the time history of the human dynamic force. Using the vibration test described above, the dynamic properties of the coupled system can be identified from the free vibration process of the slab structure. With the vibration vector \(\phi_{r}\), the eigenvector equation of the coupled system can be expressed as  
\[ \begin{pmatrix}\lambda_{r}^{2}\begin{bmatrix}m_{S}&0\\ 0&m_{H}\end{bmatrix}+\lambda_{r}\begin{bmatrix}c_{S}+c_{H}&-c_{H}\\ -c_{H}&c_{H}\end{bmatrix}+\begin{bmatrix}k_{S}+k_{H}&-k_{H}\\ -k_{H}&k_{H}\end{bmatrix}\end{pmatrix}\begin{bmatrix}\phi_{S}\\ \phi_{Hr}\end{bmatrix} = \begin{bmatrix}0\\ 0\end{bmatrix} . \tag{16}\]  
The eigenvalue equation for the pedestrian-structure coupled system is  
\[ \begin{bmatrix}m_{S}\lambda_{r}^{2}+(c_{S}+c_{H})\lambda_{r}+k_{s}+k_{H}&-c_{ H}\lambda_{r}-k_{H}\\ -c_{H}\lambda_{r}-k_{H}&m_{H}\lambda_{r}^{2}+c_{H}\lambda_{H}+k_{H}\end{bmatrix} =0. \tag{17}\]  
By simplifying Equation (17), we get  
\[ \begin{pmatrix}m_{S}\lambda_{r}^{2}+(c_{S}+c_{H})\lambda_{r}+k_{S}+k_{H}(m_{H }\lambda_{r}^{2}+c_{H}\lambda_{r}+k_{H})-(-c_{H}\lambda_{r}-k_{H})^{2}=0.\end{pmatrix}