## 4 Experimental Investigation
### Case 1: The Linear System  
A shaking table test of a 5-story steel frame model (with the model dimension shown in Figure 9) was carried out to investigate the applicability of the SCEM-UA algorithm. The \(15\times 180\times 1060\) mm sized columns were made of Q690 high-strength steel plates, and the \(2.0\) m \(\times\)\(2.0\) m sized floors were of Q345 steel plates. The structure can be simplified into a 5-story shear frame as shown in Figure 9, and the structural dynamic equation can be described as Equation (2). In this case, the structural dynamics behavior can be determined by the parameter set \(\theta=(m_{1},\,m_{2},\,\ldots,\,m_{5},\,k_{1},\,k_{2},\,\ldots,\,k_{5},\,\zeta _{1},\,\zeta_{2})\) with the mass matrices \(\mathbf{M}=(1032\) kg, \(1032\) kg, \(1032\) kg, \(1032\) kg, \(829.9\) kg) already known. The El Centro wave (1940, NS) was utilized in the shaking table test as the ground motion in this research, and the peak ground acceleration (PGA) was \(0.15\,g\) (\(g\) is the gravity acceleration). The response of the structure was measured by acceleration sensors on all floors, as shown in Figure 9. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Parameter** & **GA** & **PSO** & **BB-BC** & **SCE-UA** & **SCEM-UA** \\ \hline Avg. error—\(m\) (\%) & 2.98 & 2.74 & 2.61 & 1.04 & 0.68 \\ Max. error—\(m\) (\%) & 6.62 & 7.11 & 4.55 & 3.17 & 1.49 \\ Avg. error—\(k\) (\%) & 3.00 & 3.00 & 2.25 & 1.12 & 0.73 \\ Max. error—\(k\) (\%) & 6.81 & 6.10 & 3.33 & 2.62 & 1.52 \\ Avg. error—\(c\) (\%) & 8.41 & 7.96 & 2.79 & 2.94 & 2.71 \\ Max. error—\(c\) (\%) & 15.29 & 10.7 & 4.72 & 6.13 & 5.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average error (avg. error) and maximum error (max. error) between the parameter value found with the lowest posterior probability and the true value. Figure 9: Schematic diagram and the test model of the 5-DOF structure system. To demonstrate the reliability of the identified values, the identified results are used to re-simulate the structural response by the given excitation. The comparison results of the acceleration histories at the 3 rd and 5 th floor between the experiment data and the best simulation results, which have the largest likelihood value based on the principle of Bayesian statistics, are shown in Figure 13. After convergence was accomplished, the last 10% of the samples that adequately fit the model responses onto observations were exerted to recalculate model outputs. Figure 14 demonstrates the 95% acceleration histories prediction uncertainty depicted by computing 2.5% and 97.5% percentiles associated with the posterior distribution of the parameter estimates. In this linear system, the sources of uncertainty are considered as the measurement uncertainty and model uncertainty. The simplification, inadequacy, and ambiguity in description of realistic structural system through mathematical equation leads to model structural uncertainty. The results show the SCEM-UA algorithm has feasibility with good accuracy and is able to estimate the uncertainties associated with the posterior distribution of the unknown parameters simultaneously within a single optimization run. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **M.V.** & **S.D.** & **R.D.** \\ \hline \(k_{1}\) (kN/m) & \(4.434\times 10^{5}\) & \(2.52\times 10^{2}\) & 0.057\% \\ \(k_{2}\) (kN/m) & \(5.034\times 10^{5}\) & 96.8 & 0.019\% \\ \(k_{3}\) (kN/m) & \(4.755\times 10^{5}\) & \(3.65\times 10^{2}\) & 0.077\% \\ \(k_{4}\) (kN/m) & \(5.107\times 10^{5}\) & \(1.73\times 10^{2}\) & 0.034\% \\ \(k_{5}\) (kN/m) & \(5.980\times 10^{5}\) & \(3.75\times 10^{2}\) & 0.063\% \\ \(\zeta_{1}\) & \(2.5\times 10^{-3}\) & \(2.58\times 10^{-6}\) & 0.10\% \\ \(\zeta_{2}\) & \(8.59\times 10^{-4}\) & \(1.18\times 10^{-6}\) & 0.14\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Summarizing statistics of the posterior parameter distribution mean value (M.V., standard deviation (S.D. ), and relative deviation (R.E.)).