# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 5 Conclusions  
This study presents a Bayesian system identification method that uses the time domain response of a structure. This method applies to cases where a single excitation force or acceleration excitation is used as the system input. In the developed method, the modal bases were used to transform the theoretical predictive model based on the modal state space model into physical space. Addressing the challenge of the coupling between the dynamic spectral parameters and spatial parameters, which complicates the process of finding an optimal solution using optimization algorithms, firstly, the quadratic function for the spatial parameter vector was obtained through the vectorization of the spatial parameter matrix. Then, the optimization objective function for only the spectral parameters was established based on the function extremum, thereby decoupling the spectral and spatial parameters. In addition, in this paper, the posterior covariance matrix of the parameters to be identified was solved by the semi-analytic expression of the derivative function and the finite difference method. The effectiveness of the method was verified using a spring-mass system and a high-speed pantograph model. The identified natural frequencies and damping ratios were extremely close to their theoretical values, and the mode shapes showed the dynamic characteristics of a structure. This method can identify measured \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Modal** & **Mode 1** & **Mode 2** & **Mode 3** & **Mode 4** & **Mode 5** \\
**Parameter** & & & & & \\ \hline \(f\) (\%) & 0.005 & 0.006 & 0.020 & 0.005 & 0.010 \\ \(\zeta\) (\%) & 3.845 & 5.141 & 0.699 & 0.347 & 1.054 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Posterior COV of modal parameters, pantograph modal test. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Modal Parameter** & **Mode 1** & **Mode 2** & **Mode 3** & **Mode 4** & **Mode 5** \\ \hline Proposed & \(f\) (Hz) & 10.17 & 12.46 & 49.06 & 134.69 & 221.32 \\ methodology & \(\zeta\) (\%) & 0.46 & 0.37 & 3.92 & 2.54 & 1.54 \\ Ref. [32] & \(f\) (Hz) & 10.0 & 12.5 & 48.4 & 128.5 & 213.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Modal identification results and their theoretical values, pantograph modal test. errors consistent with the modeling assumption. Moreover, the proposed method possesses the ability to identify dense modes. The effects of the noise level on the identification results were discussed. As the measurement noise increases, the MPVs of the modal parameters gradually deviate from the theoretical value, and the posterior uncertainties increase. In addition, the method in this paper can be extended to the modal identification of multi-input linear time-invariant systems. According to the superposition principle, the modal response \(\hat{\eta}_{l}\) of the system under multiple excitations can be obtained by superimposing the modal responses \(\hat{\eta}_{l,l}\) under a single excitation (\(\hat{\eta}_{l}=\sum_{j=1}^{N_{f}}\hat{\eta}_{l,j}\), where \(l\) is the number of the modal order and \(N_{f}\) is the number of the system inputs). The physical response \(\mathbf{y}_{k}(\overline{\mathbf{\theta}})\) of the system can still be written in the form of the multiplication of the spatial parameter matrix \(\hat{\mathbf{\psi}}\) with the modal response vector \(\hat{\mathbf{\eta}}\) (\(\overline{\mathbf{\theta}})=\sum_{l=1}^{m}\sum_{j=1}^{N_{f}}\mathbf{\varphi}_{ l}\gamma_{l,j}\hat{\eta}_{l,j}=\hat{\mathbf{\psi}}\hat{\mathbf{\eta}}\)), and then a structural modal identification can be carried out based on the decoupling strategy in Section 3.2. Conceptualization, Y.P. and Y.Z. ; methodology, Y.P. and Y.Z. ; software, Y.P. ; validation, Y.P. ; investigation, Y.P. and Y.Z. ; resources, Y.Z. ; data curation, Y.P. ; writing--original draft preparation, Y.P. ; writing--review and editing, Y.Z. ; visualization, Y.P. ; supervision, Y.Z. ; and funding acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript.