## 4 Simulation Verification
### An Introduction to Simulation Systems and Data Sets
#### 4.1.1 A Mass Slow-Time-Varying 3-DOF Structure  
The mass slow-time-varying 3-DOF structure is designed in Matlab/Simulink. The model of the structure is shown in Figure 5. The dynamic equation of the structure is shown in Equation (16).  
\[\left[\begin{array}{ccc}m_{1}(t)&0&0\\ 0&m_{2}(t)&0\\ 0&0&m_{3}(t)\end{array}\right]\left[\begin{array}{ccc}\ddot{x}_{1}(t)\\ \ddot{x}_{2}(t)\\ \ddot{x}_{3}(t)\end{array}\right]+\left[\begin{array}{ccc}c_{1}+c_{2}&-c_{ 2}&0\\ -c_{2}&c_{2}+c_{3}&-c_{3}\\ 0&-c_{3}&c_{3}\end{array}\right]\left[\begin{array}{ccc}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\\ \dot{x}_{3}(t)\end{array}\right]+\left[\begin{array}{ccc}\dot{x}_{1}+k_{2}& -k_{2}&0\\ -k_{2}&k_{2}+k_{3}&-k_{3}\\ 0&-k_{3}&k_{3}\end{array}\right]\left[\begin{array}{ccc}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\\ \dot{x}_{3}(t)\end{array}\right]=\left[\begin{array}{ccc}\dot{F}_{1}(t)\\ 0\\ 0\end{array}\right] \tag{16}\]  
The structural parameters are as follows: The damping ratio of the three objects is \(c_{1}=c_{2}=c_{3}=0.01(\mathrm{Ns/m})\); the stiffness is \(k_{1}=k_{2}=k_{3}=1000(\mathrm{N/m})\); the initial displacement is 0. The quality is \(m_{2}=m_{3}=1\mathrm{kg}\). The mass \(m_{1}\) are time-varying, and the change rule is shown in the following Equation (17).  
\[m_{1}(t)=\left\{\begin{array}{ll}1,\,t\leq 50\,\mathrm{s}\\ e^{-0.0005(t-50)},\,50\,\mathrm{s}<t\leq 2000\,\mathrm{s}\end{array}\right. \tag{17}\]  
\(F_{1}(t)\) is white Gaussian noise, and is applied to object \(m_{1}\). The simulation consists of two parts. In \(t\leq 50\,\mathrm{s}\), the mass of object \(m_{1}\) is constant value. In \(50\,\mathrm{s}<t\leq 2000\,\mathrm{s}\), the mass of object \(m_{1}\) changes with time, and the system becomes a linear time-varying structure. The 3-DOF simulation structure has only third order mode at most. From Table 1, the highest natural frequency of the third order mode is relatively small, much less than 20 Hz. In general, the higher the sampling frequency is set, the more accurate the modal parameter identification results will be. However, at the same time, the vibration response semaphore and calculation amount of sampling will also be larger. Therefore, according to Nyquist theorem, the sampling frequency is set to 40 Hz, and the sampling interval of simulation is 0.025 s. The displacement response signal data set \(\mathbf{X}(t)\in\mathbb{R}^{n\times T}\) of the structure is obtained by Runge-Kutta algorithm in Matlab/Simulink module. The data set \(\mathbf{X}(t)\in\mathbb{R}^{n\times T}\) of three degrees of freedom structure is shown in Figure 6, and the sampling duration is 2000 s. Therefore, the final data set is \(\mathbf{X}(t)\in\mathbb{R}^{3*80,000}\).  
Figure 5: The mass slow-time-varying 3-DOF (degree of freedom) structure with external force.  
The real natural frequency of the structure is shown in Figure 7. The real natural frequencies at \(t\) = 50.025 s, \(t\) = 1200 s and \(t\) = 1974.375 s are shown in Table 1 below. The natural frequencies identified by MWLPP at \(t\) = 50.025 s, \(t\) = 1200 s and \(t\) = 1974.375 s are shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Real Natural Frequency (Hz)**} \\ \hline
**Order** & \(t\) = 50.025 s & \(t\) = 1200 s & \(t\) = 1974.375 s \\ \hline
1 & 2.24 & 2.29 & 2.31 \\
2 & 6.28 & 7.02 & 7.27 \\
3 & 9.07 & 10.56 & 12.26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The real natural frequency of the mass slow-time-varying 3-DOF structure when \(t\) = 50.025 s, \(t\) = 1200 s and \(t\) = 1974.375 s.  
Figure 6: The Gaussian white noise and three object displacement response signal graphs. (**a**) Gaussian white noise; (**b**) The first object displacement response signal; (**c**) The second object displacement response signal; (**d**) The third object displacement response signal.  
Figure 7: The real natural frequency of the mass slow-time-varying 3-DOF structure.