## 2 Experimental Study of a Human Mass-Spring-Damping Model
### Theory of Mode Test
#### 2.3.1 System Damping Identification Method
Combining Equations (8) and (9), we can obtain  
\[R_{ijk}(\tau)=\sum_{r=1}^{2N}\sum_{s=1}^{2N}\phi_{ir}\phi_{js}a_{kr}a_{ks}a_{ k}\int_{-\infty}^{t}e^{\lambda_{r}(t+s-p)}e^{\lambda_{s}(t-p)}dq. \tag{11}\]  
Integrating, we have  
\[R_{ijk}(\tau)=\sum_{r=1}^{2N}\sum_{s=1}^{2N}\phi_{ir}\phi_{js}a_{kr}a_{ks}a_{ k}(-\frac{e^{\lambda_{r}\tau}}{\lambda_{r}+\lambda_{s}}). \tag{12}\]  
Record \(b_{jr}\) as  
\[b_{jr}=\sum_{s=1}^{2N}\phi_{js}a_{kr}a_{ks}a_{k}(-\frac{e^{\lambda_{r}\tau}}{ \lambda_{r}+\lambda_{s}}). \tag{13}\]  
Notice that \(b_{jr}\) is a constant related exclusively to the reference point \(j\) and to the mode order \(r\). Combining Equations (12) and (13), we can obtain  
\[R_{ijk}(\tau)=\sum_{r=0}^{2N}\phi_{ir}b_{jr}e^{\lambda_{r}\tau}. \tag{14}\]Comparing the pulse response function \(h_{ijk}(t)\) with the cross-correlation function \(R_{ijk}(t)\) between points \(i\) and \(j\) of the system, it can be found that the mathematical expressions for the two are identical under white noise excitation. The specific constants \(a_{kr}\) and \(b_{jr}\) are equivalent to multiplying the same order modal shape function by the same constant for each measurement point and will not alter the characteristics of the mode shape function. It can be argued that it is feasible at a theoretical level to substitute the impulse response function of a system with a cross-correlation function between the responses of its two measurement points and to integrate it with conventional modal parameter identification methods for modal identification under ambient excitation. The specific process of the NExT method is:  
* Collect response data of measurement points using sensors. * Select a measurement point with a lower response as a reference point and calculate the correlation function between the other measurement point and the reference point. * Use the calculated cross-correlation function between the measurement points as the impulse response function and then combine it with other mode parameter identification methods to identify the dynamic parameters. The implementation of ambient excitation for mode identification of structure is as follows. * Select \(i\)th DOF on the structure as the parametric degrees of freedom. * Obtain the transfer function \(T_{ji}(j=1,2,\ldots,n)\) of \(j\)th DOF with respect to the \(i\)th reference degree of freedom from the measurements. That is, a two-channel fast Fourier transform (FFT) analysis of the vibration response of the \(j\)th and the \(i\)th points is performed to obtain the mutual power spectrum \(G_{ji}(\omega_{k})\) of the \(j\)th DOF with respect to the \(i\)th DOF, the self-power spectrum \(G_{ij}(\omega_{k})\) of the \(j\)th DOF and the self-power spectrum \(G_{ii}(\omega_{k})\) of the \(i\)th DOF, and the transfer function \(T_{ji}\) is calculated from Equation (4). * Choose the peak frequency on the curve of \(G_{jj}(\omega_{k})\) or \(G_{ii}(\omega_{k})\) as the natural frequency of the structure. * Estimate the structural frequency as \(\omega_{k}\) and obtain the value of \(\left|T_{ji}(\omega_{k})\right|\) from Equation (5) as the amplitude of the shape at the \(j\)th DOF in the shape function at the estimated frequency.