## 4 Simulation Verification
### The Evaluation Indexes  
In the simulation, the average model assurance criterion (MAC\({}_{i-avg}\)) will be used to identify the average accuracy of modal shapes. The specific definition of MAC\({}_{i-avg}\) is,  
\[\text{MAC}_{i-avg}=\frac{1}{m}{\sum\limits_{j}^{m}}\frac{{(\overset{\rightarrow} {\phi}_{ij}T^{\rightarrow}\overset{\rightarrow}{\psi}_{ij})}^{2}}{{(\overset{ \rightarrow}{\phi}_{ij}T^{\rightarrow}\overset{\rightarrow}{\phi}_{ij})( \overset{\rightarrow}{\psi}_{ij}T^{\rightarrow}\overset{\rightarrow}{\psi}_{ ij})} \tag{26}\]  
where \(\overset{\rightarrow}{\phi}_{ij}\) is the modal shape of \(j\)-th window of the \(i\)-th order identified by algorithm; \(\overset{\rightarrow}{\psi}_{ij}\) is real modal shape of \(j\)-th window of the \(i\)-th order identified; \(m\) is the number of windows. It can be seen from Equation (26) that the value range of MAC\({}_{i-avg}\) is \(0\leq\text{MAC}_{i-avg}\leq 1\). The closer the MAC\({}_{i-avg}\) value is to 1, the higher the average identification accuracy of the modal shape of \(i\)-th order. Average error rate (\(\delta_{i-avg}\)) will be used to evaluate the natural frequency identified by algorithm. The specific definition of \(\delta_{i-avg}\) is,  
\[\delta_{i-avg}=\frac{1}{m}{\sum\limits_{j}^{m}}\left|\frac{f_{ij}-f_{ij-real}} {f_{ij-real}}\right|\times 100\% \tag{27}\]  
where \(f_{ij}\) is the natural frequency of \(j\)-th window of the \(i\)-th order identified by algorithm; \(f_{ij-real}\) is real natural frequency of \(j\)-th window of the \(i\)-th order; \(m\) is the number of windows. It can be seen from Equation (27) that the value range of \(\delta_{i-avg}\) is \(\delta_{i-avg}\geq 0\). The closer the \(\delta_{i-avg}\) value is to 0, the higher the average identification accuracy of the natural frequency of \(i\)-th order. It is important to choose an appropriate limited memory length L of moving window. Frequency resolution \(\triangle f\) and average frequency variation of the \(i\)-th modal in a window \(\triangle f_{L}(i)\) was used to select window length. Window length \(L\) is proportional to sample frequency \(f_{s}\) and frequency resolution \(\triangle f\), \(\triangle f\) cannot be too small, because it cannot reflect the change of frequency, and \(\triangle f\) cannot be too large, because the change of \(\triangle f_{L}(i)\) cannot be identified, at the same time, \(\triangle f_{L}(i)\) cannot be larger too much than \(\triangle f\), otherwise, LTV structure cannot be regarded a LTI structure in a window. Therefore, different structures may choose different window lengths. The idea was proposed in [28; 30]. \[\triangle\,f=\frac{f_{s}}{L} \tag{28}\]  
\[\triangle\,f_{L}(i)= \frac{L}{f_{s}}\times\frac{f_{end}(i)-f_{begin}(i)}{t_{end}-t_ {begin}} \tag{29}\]  
where variables \(f_{s}\), \(f_{end}(i)\), \(f_{begin}(i)\), \(t_{end}\), \(t_{begin}\) are sample frequency, end-frequency of the \(i\)-th modal, begin-frequency of the \(i\)-th modal, end-time, begin-time of the whole data. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Real Natural Frequency (Hz)**} \\ \hline
**Order** & \(t\) = 0.5 s & \(t\) = 2 s & \(t\) = 3.795 s \\ \hline
1 & 16.70 & 17.80 & 19.46 \\
2 & 104.66 & 111.56 & 121.96 \\
3 & 293.03 & 312.38 & 341.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The real natural frequency of the density slow-time-varying cantilever beam structure when \(t\) = 0.5 s, \(t\) = 2 s and \(t\) = 3.795 s.  
The model assurance criterion (MAC) will be used to identify the accuracy of modal shapes. The specific definition of MAC is,  
\[\text{MAC}= \frac{\overset{\rightarrow}{(\phi\overset{\rightarrow}{\psi})}^{T \rightarrow}\overset{2}{\phi}}{(\overset{\rightarrow}{\phi\overset{ \rightarrow}{\psi})}(\overset{\rightarrow}{\psi\overset{\rightarrow}{\psi})}} \tag{30}\]  
where \(\overset{\rightarrow}{\phi}\) is the modal shape identified by algorithm; \(\overset{\rightarrow}{\psi}\) is real modal shape. The value range of MAC is \(0\leq\text{MAC}\leq 1\).