## 4 Simulation Verification
### Analysis of Simulation Results  
1. From Figures 12-21 and Tables 3-14, the MWLPP method can identify the operational modal parameters of the slow time-varying structure well. 2. Comparing Figures 12 and 16, and Tables 6 and 10, MWLPP has high accuracy in identifying modal shapes of linear slow-time-varying structures. In particular, MAC  
Figure 21: The MAC value of modal shapes identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise. Figure 20: The natural frequencies identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise. values of modal shapes are very close to 1 in complex density slow-time-varying cantilever beam structures. 3. Comparing Figures 13 and 17, the natural frequency of the linear slow-time-varying structure changing with time, and MWLPP can well track the change of the natural frequency. Combined with Tables 4 and 8, MWLPP has a high accuracy in identifying the natural frequency of the linear slow-time-varying structure. 4. Comparing Tables 5 and 9, the total number of un-identified windows in MWLPP method is lower than that in MWPCA whether the structure is a linear mass slow-time-varying 3-DOF structure or a complex linear density slow-time-varying cantilever beam structure. Comparing Tables 6 and 10, the MAC value of modal shape identified by MWLPP is higher than that MWPCA. Therefore, the MWLPP method is superior to the MWPCA method in identifying the operational modal parameters of linear slow time-varying structures. 5. From Figure 17 and Table 8, the first natural frequency seems contain fluctuations because the ordinate scale of the first mode natural frequency is small. According to Equation (29), in the cantilever beam structure, the first average frequency variation is \(\triangle f_{L}(1)=0.15\) Hz, the second average frequency variation is \(\triangle f_{L}(2)=1.09\) Hz, the third average frequency variation is \(\triangle f_{L}(3)=3.06\) Hz. The smaller the average frequency variation is, the smaller the variation range of the natural frequency value is. Due to the small variation range of the first order natural frequency value, the fluctuation phenomenon of the first order natural frequency is obvious in Figure 15[28]. In addition, according to Equation (27), the denominator of the first order is much smaller than that of the second and the third order when calculating the average error rate \(\delta_{i-avg}\) of natural frequency because the natural frequency value of the first order is smaller than that of the second and the third order. Therefore, in Table 8, the average error rate of natural frequency calculated in the first order is greater than that in the second and third order. In fact, the result of modal shape identification shows that the first order identification is indeed the best. 6. Figure 15 shows that the damping ratio identified by MWLPP algorithm fluctuates greatly because theoretical analysis and numerical simulations indicate that a decreasing or moving mass and density will generate additional damping in the LTV structures [41; 42]. In addition, compared with theoretical values, the identification accuracy of damping ratios has a certain error. It is generally predictable because the identification of the damping ratio itself is a difficult problem in the field of structural dynamics, and easily affected by the adopted identification algorithm. Therefore, the time-varying transient mode damping ratio identified by MWLPP is not suitable to compare with the mode damping ratio calculated by finite element methods. 7. From Figures 14, 18 and 21, the MAC varies very much and have low values. The time-domain method used in this article cannot use the average technique in the frequency domain to identify the time-domain modes from the non-stationary random vibration response signals. Therefore, the algorithm is unstable, and some moments cannot identify the modal parameters, or the identified modal natural frequency and modal shape are not good.