## 2 Experimental Study of a Human Mass-Spring-Damping Model
### Theory of Mode Test
#### 2.3.1 System Damping Identification Method  
Among the modal parameters of a structure, the accuracy of identifying the damping ratio is far lower than that of identifying the natural frequency and vibration shape, and it is even more difficult to guarantee the accuracy when the test results are disturbed by noise. Thus, improving the accuracy of the identification of structural damping ratio has been the pursued goal of structural dynamics researchers. In this study, two methods were used to identify the damping ratio of the system: (1) the natural excitation technique method (NExT) in the field of time-domain data pre-processing; and (2) free vibration decay method based on the free vibration decay curve of the structure. The fundamental principles and applications of NExT are described briefly below. The NExT method was first proposed by [PERSON] et al. [50] at SADIA National Laboratory in the USA. The modal parameters of a high-speed turbine blade in operation are identified with this method. The basic principle is that the correlation function calculated from the response signal between two points of the structure has shown many similarities to the impulse response function under ambient excitation. Thus, the impulse response function of a structure can be substituted by a correlation function to identify the modal parameters of the structure more conveniently and efficiently. The theoretical analysis of the NExT method is as follows. Assuming a linear system with degrees of freedom \(N\), when an external excitation \(f_{k}(t)\) is applied to point \(k\) of the system, the response \(x_{ik}(t)\) at point \(i\) of the system can be expressed as  
\[x_{ik}(t)=\sum_{r=1}^{2N}\phi_{ir}a_{kr}\int_{-\infty}^{t}e^{\lambda_{r}(t-p)} f_{k}(p)dp, \tag{6}\]  
where \(\phi_{ir}\) denotes the \(r\)th-order mode shape at the \(i\)th measurement point, and \(a_{kr}\) denotes the constant term associated with the excitation point \(k\) and the mode order \(r\). \(\lambda_{r}\) denotes the eigenvalue of the \(r\)th-order mode. \(p\) represents the time-integrated variable used to calculate the response value \(x_{ik}(t)\) at point \(i\). When the system is excited by a unit impulse force at point \(k\), the unit impulse response at point \(i\) of the system can be expressed as  
\[h_{ik}(t)=\sum_{r=1}^{2N}\phi_{ir}a_{kr}e^{\lambda_{r}t}. \tag{7}\]  
For a linear system with responses obtained by testing at points \(i\) and \(j\) that can be represented as \(x_{ik}(t)\) and \(y_{jk}(t)\), respectively, the cross-correlation function of these two columns of response data can be expressed as  
\[R_{ijk}(\tau)=E[x_{ik}(t+\tau)x_{jk}(t)]. \tag{8}\]  
It defines the cross-correlation function \(R_{ijk}(\tau)\) as the expected valued of the product of two responses evaluated at a time separation of \(\tau\). The operator symbol \(E\) indicates the process of taking the expected value. \(\tau\) means the time separation. Then, we get  
\[R_{ijk}(\tau)=\sum_{r=1}^{2N}\sum_{s=1}^{2N}\phi_{ir}\phi_{js}a_{kr}a_{ks}\int _{-\infty}^{t}\int_{-\infty}^{t+\tau}e^{\lambda_{r}(t+s-p)}e^{\lambda_{s}(t-q) }E[f_{k}(p)f_{k}(q)]dpdq. \tag{9}\]  
\(p\) and \(q\) are used to distinguish the time integration variables for the two integration stages, rather than using the traditional time variables \(\tau\). When \(f(t)\) is an ideal white noise excitation, by definition of the cross-correlation function, we have  
\[E[f_{k}(p)f_{k}(q)]=a_{k}\delta(p-q), \tag{10}\]  
where \(\delta(t)\) denotes the pulse function and \(a_{k}\) is the constant associated with the stimulation point \(k\).