# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 4 Numerical Examples
### High-Speed Pantograph Modal  
#### 4.2.1 Numerical Simulation Verification  
This section uses the finite element model of a high-speed train pantograph to numerically simulate the actual modal test. The simulation test adopted the hammer excitation method with a single-point excitation and multi-point pickup vibration. Fixed constraints were applied to the base of the pantograph's finite element model, and the pantograph head was a free boundary. The hammering point is shown in Figure 5, and the hammer excitation was simulated using a triangular wave pulse signal with an amplitude of 30 N and a pulse width of 0.01 s. Seven measurement channels were set up on the pantograph head strip to measure the Z-direction's acceleration response, with a sampling frequency of 1000 Hz, a data length of 1500 for each measurement channel, and a measurement \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Pamper**} & **Mode 1** & **Mode 2** & **Mode 3** \\ \hline \multirow{3}{*}{\(f\) (Hz)} & MPV & 3.854 & 6.237 & 7.338 \\  & Theory & 3.854 & 6.238 & 7.340 \\  & MPV & 1.00 & 0.96 & 0.96 \\  & Theory & 1.00 & 1.00 & 1.00 \\  & MPV & 1.65 & 0.48 & 0.68 \\  & Theory & 1.65 & 0.50 & 0.70 \\  & MAC & 0.9999 & 0.9992 & 0.9990 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modal identification results and their theoretical values, spring–mass system. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Modal Parameter** & **Mode 1** & **Mode 2** & **Mode 3** \\ \hline \(f\) (\%) & 0.003 & 0.022 & 0.020 \\ \(\zeta\) (\%) & 0.464 & 3.190 & 2.869 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Posterior COV of modal parameters, spring–mass system. two pairs of modes with a relatively close natural frequency, i.e., modes 2 and 3 and modes 4 and 5. From the mode shape of the finite element model shown in Figure 8, it can be seen that the mode shapes of the strip in modes 4 and 5 were very similar, with a MAC of 0.96, and modes 4 and 5 were closely spaced. Six modes will be identified simultaneously in the modal identification, which involves solving twelve modal parameters (six natural frequencies and six modal damping ratios) using an optimization algorithm. Table 3 shows the modal parameters of the pantograph identified using the method presented in this paper and gives the \"theoretical values\" of the natural frequencies calculated by the finite element software. For the natural frequencies, the identification results of the first five modes were in good agreement with the \"theoretical values\", and the maximum error was only 0.21%, while the identification result of the sixth mode was slightly worse compared to the first five, with an error of 1.03%. The identified modal damping ratios were also close to the accurate values, but the accuracy of the damping ratio was more easily affected by the measured noise, which can be verified from the uncertainty analysis results of the damping ratio. Since the MAC between the identified and the finite element mode shape was greater than 0.99, it indicated that the identified mode shape values have a high accuracy. It can also be seen from the results of the mode shape shown in Figure 8 that the identified mode shapes were relatively consistent with the results of the finite element analysis. It can also be seen from the identification results that the method proposed in this paper can also identify close modes. In addition, the MPV of the variance of the identified prediction error \(\sigma_{e}^{2}\) was 4.60 \(\times\) 10\({}^{-7}\) and its theoretical value was 4.99 \(\times\) 10\({}^{-7}\), with a relative error of 7.82%. Figure 6: Transient excitation and measured acceleration response of pantograph. (**a**) Triangular wave pulse excitation force; (**b**) The acceleration response of measured node 1. Figure 7: PSD of measured acceleration response. The posterior COV for the parameters are shown in Table 4. The COVs of natural frequency were less than 0.06%, indicating that the uncertainties of natural frequency were relatively small.