## 2 Research Method  
The equation of motion of a \(n\) Degree Of Freedom (DOF) linear vibration system with nonstationary excitation \(f(t)\) can be expressed as  
\[M^{\underline{x}}_{\underline{\Sigma}}(t)+C\underline{\dot{x}}(t)+K_{ \underline{\Sigma}}(t)=\underline{f}(t) \tag{1}\]  
where \(M,\ C\), and \(K\) are the mass, damping, and stiffness matrices of the vibration system, respectively, \(f(t)=\Gamma(t)\underline{w}(t)\) is a nonstationary white noise with the product model, \(\underline{x}(t)=\Gamma(t)\underline{u}(\overline{t})\), \(\underline{u}(t)\) is the stationary displacement response when the system is excited by a stationary white noise \(\underline{w}(t)\), \(\Gamma(t)\) is a deterministic envelope function. If \(\Gamma(t)\) is a time-varying function with the secular change, meaning \(\dot{\Gamma}(t)\approx 0\) and \(\ddot{\Gamma}(t)\approx 0\), then \(\Gamma(t)\underline{\ddot{u}}(t)\approx\underline{\dot{x}}(t)\) and \(\Gamma(t)\underline{\dot{u}}(t)\approx\underline{\dot{x}}(t)\), where \(\underline{\dot{u}}(t)\) and \(\underline{\ddot{u}}(t)\) are the response signals of the system corresponding to stationary white noise \(w(t)\), the system response can be expressed as a nonstationary model coincident with envelope function \(\Gamma(t)\) of nonstationary excitation, and Equation (1) can be rewritten as:  
\[M\Gamma(t)\underline{\ddot{u}}(t)+C\Gamma(t)\underline{\dot{u}}(t)+K\Gamma(t )\underline{u}(t)=\Gamma(t)w(t) \tag{2}\]In practical application, the dynamic response data of structures are in discrete form, so the continuous state equation is transformed to discrete state space:  
\[\underline{x}_{k+1}=A\underline{x}_{k}+B\underline{f}_{k} \tag{3}\]  
\[\underline{y}_{k}=C\underline{x}_{k} \tag{4}\]  
where \(\underline{x}_{k}\), \(\underline{y}_{k}\), and \(\underline{f}_{k}\) are state, output, and input vectors, respectively. \(A\), \(B\), and \(C\) are the system, input, and output matrices, respectively. \(k\) is the sampling point sequence. As \(\underline{f}_{k}\) is a nonstationary white noise with zero mean, the Hankel matrix \(H(k)\) is formed of the system output vector \(\underline{y}_{k}\). The nonstationary correlation function matrix \(T_{\tau}\) between response data, in different time-delayed times, \(\tau\) is defined as:  
\[T_{\tau}\equiv T(\tau)=E\left[\underline{y}_{k+\tau}\underline{y}_{k}^{T}\right] \tag{5}\]  
According to Equations (3) and (4), \(T_{\tau}\) can be expressed as  
\[T_{\tau}=CA^{\tau-1}G+\sum_{i=1}^{\tau}CA^{i-1}B\overline{t}_{k}^{2}E\left[ \underline{w}_{k+\tau-i}\underline{u}_{k}^{T}\right]C^{T} \tag{6}\]  
where \(G=E[\underline{x}_{k+\tau}\underline{y}_{k}^{T}]\) is the covariance matrix. In \(E\left[\underline{w}_{k+\tau-i}\underline{u}_{k}^{T}\right]\), \(\underline{w}_{k+\tau-i}\) is the stationary white part of nonstationary white noise in the form of a product model, \(\underline{u}_{k}\) is the displacement response when the system is excited by a stationary white noise. As \(\underline{w}_{k+\tau-i}\) is uncorrelated with \(\underline{u}_{k}\), \(E\left[\underline{w}_{k+\tau-i}\underline{u}_{k}^{T}\right]=0\). If the excitation can be expressed as the product model of the nonstationary white noise, and the nonstationary response of the system can be expressed as a nonstationary process coincident with the envelope function of excitation, the nonstationary correlation function matrix \(T_{\tau}\) can be treated as an approximately stationary correlation function matrix. In the analysis process, if the stochastic process can be reasonably assumed to be the whole process, the statistical value of correlation can be derived from a single (delay is long enough) sample function, so for \(T_{\tau}\), the calculation can be approximated as  
\[