## 2 Experimental Study of a Human Mass-Spring-Damping Model
### Theory of Mode Test  
The method for the identification of the structural dynamic properties is the ambient random vibration method. When modal tests are carried out on large structures such as bridges and architectural buildings, artificial excitation in a sinusoidal, random or impulsive form is not possible or permissible. Vibration environments exist for almost any large structure: wind, water impact, ground movement, vibrations caused by moving vehicles, etc. Under the excitation of these natural environments, structures are expected to produce tiny vibrations. Although we cannot precisely quantify the characteristics of these excitation signals, they can be estimated to some extent. It is a reasonable hypothesis that the ambient excitation can be treated as stochastic excitation signals with a continuous spectrum with a certain bandwidth where frequency bands of structural relevance are mainly covered. Thus, the vibration response of the structure induced by ambient excitation can reflect the dynamic properties of the structure. With the assumption that the structural vibration system with \(n\) degrees of freedom (\(n\)DOF) is stimulated by an identical white noise, the frequency response function of these \(n\) degrees of freedom to the input white noise can be exhibited as  
\[\{H_{1}(\omega),H_{2}(\omega),\dots,H_{n}(\omega)\}. \tag{1}\]  
One column or one row of the vibration shape matrix of the system is defined by Equation (1). The frequency response function for the \(i\)th degrees of freedom under a stochastic excitation can be expressed as  
\[H_{i}(\omega)=\frac{X_{i}\cdot\bar{F}}{F\cdot\bar{F}}=\frac{G_{x_{i}f}}{G_{ff}}, \tag{2}\]  
where the power spectrum of the external force is denoted by \(F\) and the conjugate value of \(F\) is denoted by \(\bar{F}\). \(G_{x_{i}f}\) denotes the cross-power spectrum of the vibration response \(x_{i}\) at point \(i\) with respect to the excitation \(f\). \(G_{ff}\) represents the self-power spectrum of the excitation \(f\). Figure 2: Test system and excitation system. With the signal of the input force \(F\) unmeasurable, the identification of the structural modes can be done by way of substituting the vibration quantity \(X_{i}\) of a reference degree of freedom \(i\) for F in Equation (2), and the transfer function of the \(j\)th DOF to the \(i\)th DOF is obtained as  
\[T_{ji}(\omega)=\frac{X_{j}\cdot\bar{X}_{i}}{X_{i}\cdot\bar{X}_{i}}=\frac{G_{ji} }{G_{ii}}, \tag{3}\]  
where the power spectrum of the vibration response of \(x_{i}\) is denoted by \(X_{j}\). The conjugate value of \(X_{i}\) is denoted by \(\bar{X}_{i}\). \(G_{ji}\) indicates the cross-power spectrum of the vibration response at point \(i\) and the vibration response at point \(j\). \(G_{ii}\) denotes the self-relative power spectrum of the vibration response at point \(i\). From Equations (2) and (3), we get  
\[T_{ji}(\omega)=H_{j}/H_{i}. \tag{4}\]  
From Equation (4), we can obtain its amplitude as  
\[\Phi_{j}=\left|T_{ji}(\omega_{k})\right|=\sqrt{G_{jj}(\omega_{k})/G_{ii}( \omega_{k})}\qquad(j=1,2,\ldots,n), \tag{5}\]  
where \(\Phi_{j}(j=1,2,\ldots,n)\) forms the vector of vibrational shapes. The subscript \(k\) of \(\omega_{k}\) denotes the \(k\)th order natural frequency of the structure. The \(k\)th order frequency of the structure is picked from the peak of the power spectral density function and its range is determined by the frequency range of the analysis. The phase of \(T_{ji}\) is the difference between the phase of \(X_{j}\) and the phase of \(X_{i}\) (in-phase or anti-phase), i.e., the positive and negative sign of the real part of the cross-spectrum \(G_{ij}\) between the \(j\)th measured DOF with the \(i\)th reference DOF. Each frequency of the structure can be identified from the frequency at the corresponding peak on the amplitude-frequency curve of the transfer function \(T_{jj}\). The approach adopted is to pick the natural frequency corresponding to the peak value of the total set average curve of the power spectrum at all measured points.