## 4 Numerical Simulation  
In this section, a simulated three DOF model incorporating TV stiffness and damping is constructed to testify the effectiveness of the proposed method, which is illustrated in Figure 3. The initial structural parameters of the TV model are given as follows, and the excitation force was random Gaussian white noise. The sampling frequency was set to 1200 Hz, and the sampling time was set to 10 s.  
\[\left\{ \begin{array}{l}m_{1}=10(\mathrm{kg}),m_{2}=15(\mathrm{kg}),m_{3}=20( \mathrm{kg})\\ k_{1}=50000(\mathrm{N}/\mathrm{m}),k_{2}=30000(\mathrm{N}/\mathrm{m}),k_{3}=200 00(\mathrm{N}/\mathrm{m})\\ c_{1}=8(\mathrm{Ns}/\mathrm{m}),c_{2}=6(\mathrm{Ns}/\mathrm{m}),c_{3}=4( \mathrm{Ns}/\mathrm{m})\end{array} \right. \tag{48}\]  
This paper considers the identification of stiffness and damping, as these parameters are more likely to change compared to the structural mass in practical engineering structures. Moreover, assessing stiffness and damping is more meaningful in terms of structural damage and vibration control. In order to simulate the dynamic characteristics of practical structures more accurately, three different cases of variation in time of stiffness and damping parameters are investigated, namely, linear variation, quadratic variation, and periodic variation. The detailed cases are given as follows:  
\[\left\{ \begin{array}{l}\Delta k_{1}=2000t\\ \Delta k_{2}=2000t^{2}-5000t\\ \Delta k_{3}=-2000\cdot\sin(0.5\pi t)\end{array} \right. \tag{49}\]  
\[\left\{ \begin{array}{l}\Delta c_{1}=-0.5t\\ \Delta c_{2}=0.5\cdot\sin(0.5\pi t)\\ \Delta c_{3}=0\end{array} \right. \tag{50}\]  
The raw vibration acceleration signal under a random excitation force was computed through the Newmark algorithm, and the corresponding time-domain waveform is shown in Figure 4a. In order to extract the modal component corresponding to each order DOF, the multi-level KRRD was then applied to the raw acceleration signal. The decomposed components are shown in Figure 4b-d, respectively. Subsequently, the polynomial frequency modulation integral algorithm was employed to each modal component. Figure 5 illustrates the signal reconstruction result of velocity and displacement signals (blue) obtained from the integral algorithm. Additionally, the velocity and displacement signals (red) obtained from numerical simulation are also included in Figure 5 to further verify the reconstruction performance of the integral algorithm. The time-domain waveforms in the first line of Figure 5 represent the displacement signals, while the second line represents the velocity signals, with each column corresponding to a different DOF. By comparing the blue and red waveforms in Figure 5, it can be seen that the proposed integral algorithm exhibits good signal reconstruction performance. Figure 3: The three DOF TV model. Finally, the physical parameters (stiffness, damping) can be identified according to Equation (47), and the corresponding identification results of linear variation, quadratic variation, and periodic variation are given in Figures 6 and 7, respectively. As shown in Figures 6 and 7, the identification values fluctuate around the theoretical values for the three  
Figure 4: The multi-level KRRD decomposition result: (**a**) raw acceleration signal, (**b**) third-order component, (**c**) second-order component, and (**d**) first-order component. Figure 5: The velocity and displacement signals obtained by integral algorithm (blue) and numerical simulation (red). different cases. Consequently, the proposed method can effectively capture the variation of physical parameters with time and track the trend of various variations. However, it is worth noting that the identification of damping is more volatile than that of stiffness, as seen in Figure 7. This is because the damping coefficient, due to its nature and usually small values in structural contexts, is more prone to error than the stiffness.