## 5 Simulations
### Analysis of Modal Identification Results  
(1) For the LTV three-DOF system considered in the simulations, Figure 7 clearly shows that different forgetting factors have an effect on the recognition of the modal shape for ERPCAWF-based OMA method. The current choice of forgetting factor is not a uniform standard, and is mostly based on \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Order** & **First Order** & **Second Order** & **Third Order** \\ \hline ARPCAWF/ERPCAWF MAC value (112.775 s) & 0.8481/0.9998 & 0.8410/0.9996 & 0.9806/0.9998 \\ ARPCA/ERPCA MAC value (112.775 s) & 0.8481/0.9998 & 0.8410/0.9996 & 0.9806/0.9998 \\ ARPCAWF/ERPCAWF MAC value (581.4 s) & 0.9145/0.9996 & 0.8170/0.9957 & 0.8579/0.9961 \\ ARPCA/ERPCA MAC value (581.4 s) & 0.9306/0.9999 & 0.8503/0.9946 & 0.9162/0.9947 \\ ARPCAWF/ERCAWF ARC value (1275.3 s) & 0.9151/1.0000 & 0.7696/0.6180 & 0.7626/0.6198 \\ ARPCA/ERPCA MAC value (1275.3 s) & 0.9367/0.9999 & 0.6044/0.8994 & 0.6521/0.8984 \\ ARPCAWF/ERPCAPMF MAC value (1987.2 s) & 0.9183/0.9999 & 0.8368/0.8582 & 0.8232/0.8603 \\ ARPCA/ERPCA MAC value (1987.2 s) & 0.9302/0.9999 & 0.5617/0.6381 & 0.5744/0.6352 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MAC value of identified modal shape in ARPCAWF-, ERPCAWF-, ARPCA-, and ERPCA-based OMA when \(\mu=0.9990\). Figure 12: Cumulative variance contribution rate of PCs extracted by ARPCAWF-based OMA at different moments when \(\mu=0.9990\): (**a**) 112.775 s; (**b**) 581.4 s; (**c**) 1275.3 s; and (**d**) 1987.2 s.  
empirical choice. In this study, the recognition results under different forgetting factors were compared. After 1250 s, the best forgetting factor that makes the experimental structure is 0.9990. (2) With \(\mu=0.9990\), Figure 8 shows that ERPCAWF-based OMA can effectively identify and track frequency changes in the time-varying system. This further validates the ERPCAWF-based OMA method. In Figures 8 and 9, we can see that the natural frequencies can be well identified by ERPCAWF-, ARPCAWF-, ARPCA-, and ERPCA-based OMA methods. The results seem to be exactly the same between the methods because the three-DOF system in Figure 6 is a slow time-varying system and the natural frequencies do not change much (first order is 0.0697 Hz, second order is 0.9460 Hz, and third order is 3.1748 Hz). In addition, the natural frequencies are calculated by picking the peaks of the Fourier transform of the latest 4096 modal coordinate responses, instead of picking the peaks of the Fourier transform of all the identified modal coordinate responses. (3) Figure 10 and Table 2 show that the ERPCAWF-based OMA method achieves a high MAC value, which means that this approach can effectively identify the modal shapes of LTV structures. ERPCAWF-based OMA offers higher precision than ARPCAWF-based OMA. The time-varying system changes quickly at some time steps and slowly at some time steps. In Figure 10, ERPCA looks more stable than ERPCAWF, because the the forgetting factor in ERPCAWF is fixed (\(\mu=0.9990\)). To make ERPCAWF-based OMA stable, a variable forgetting factor that can self-adaption change related to the time-varying speed characteristics of vibration signals is needed. (4) As shown in Figures 11 and 12, for LTV structures, the engineering parameters (mass, stiffness, and damping) change over time, thus the modal contribution rate of each order varies with time. The contribution order is not the same as the frequency order, and the correspondence between the two rankings may change over time. This phenomenon is called modal exchange. In Figure 7, each modal shape identified by ERPCAWF corresponds to the theoretical value. By avoiding the modal exchange phenomenon, the ERPCA- and ERPCAWF-based OMA methods offer greater stability than ARPCA- and ARPCAWF-based OMA in vibration control [34; 36]. (5) From the results in Table 3, we can confirm that ERPCAWF-based OMA has a shorter runtime and requires less memory than ARPCA-, ERPCA-, and ARPCAWF-based OMA method.