## 4 Experimental Investigation
### Case 2: The Nonlinear System
The prior information of estimation parameters consists of realistic lower and upper bounds on each of the parameters with a uniform prior distribution as tabulated in Table 8.  
The evolution of the Gelman-Rubin scale reduction factor for estimation parameters using the SCEM-UA algorithm is illustrated in Figure 16, indicating that the parallel sequences converged to the stationary distribution after approximately 400 iterations. The evolution of the generated samples converges rapidly as presented in Figure 17. Similar results were obtained for the other parameters. Figure 18 presents the marginal posterior probability distributions of the estimated parameters constructed after convergence. Note that none of the parameters in Figure 18 correspond to the respective prior uniform distributions. Table 9 presents summarizing statistics of the posterior parameter distributions. In general, the width of posterior distribution in a nonlinear system is relatively larger than in a linear system. Figure 17: Generated samples for the six estimation parameters (\(k_{1},k_{3},k_{5},\zeta_{1},\omega_{\rm p},\zeta_{\rm c}\)). \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **M.V.** & **S.D.** & **R.D.** \\ \hline k1 (kN/m) & \(5.011\times 10^{5}\) & \(7.73\times 10^{3}\) & \(1.54\%\) \\ k2 (kN/m) & \(5.619\times 10^{5}\) & \(4.45\times 10^{3}\) & \(0.79\%\) \\ k3 (kN/m) & \(4.879\times 10^{5}\) & \(6.98\times 10^{3}\) & \(1.43\%\) \\ k4 (kN/m) & \(4.721\times 10^{5}\) & \(5.61\times 10^{3}\) & \(1.19\%\) \\ k5 (kN/m) & \(5.190\times 10^{5}\) & \(4.93\times 10^{3}\) & \(0.95\%\) \\ \(\zeta\)1 & \(4.423\times 10^{-3}\) & \(3.11\times 10^{-5}\) & \(0.70\%\) \\ \(\zeta\)2 & \(6.601\times 10^{-4}\) & \(3.79\times 10^{-6}\) & \(0.57\%\) \\ \(\omega_{\rm c}\) (rad-s) & \(1.584\times 10^{2}\) & \(2.3\times 10^{-1}\) & \(1.45\%\) \\ \(\omega_{\rm p}\) (rad-s) & \(1.14\times 10^{2}\) & \(4.94\) & \(4.33\%\) \\ \(\zeta_{\rm c}\) & \(3.175\times 10^{-1}\) & \(2.38\times 10^{-3}\) & \(0.75\%\) \\ \(\zeta_{\rm p}\) & \(2.621\times 10^{-1}\) & \(1\times 10^{-2}\) & \(3.81\%\) \\ \hline \end{tabular}
\end{table} 
Table 9: Summarizing statistics of the posterior parameter distribution (Mean Value (M.V. ), Standard Deviation (S.D. ), and Relative Deviation (R.E.)). Figure 18: Marginal posterior probability distributions of the six estimation parameters (\(k_{1},k_{3},k_{5},\zeta_{1},\)\(\omega_{\rm p},\zeta_{\rm c}\)). To demonstrate the reliability of the identified values, the identified model parameters are input to operate the simulation of the structural response to the given excitation. The simulation results of the acceleration histories at the 3 rd and 5 th floor in this study are shown in Figure 19 as dashed curves, whereas the solid curves are the corresponding experimental acceleration histories for comparison. The PTMD performs nonlinearly under seismic excitation, but the results of the proposed identification algorithm generally matched. After the convergence was accomplished, the last 10% of the samples were employed to recalculate model outputs, and then by computing 2.5% and 97.5% percentiles, the authors obtained a 95% confidence interval. Figure 20 demonstrates the 95% acceleration histories prediction uncertainty associated with the posterior distribution of the parameter estimates. The predicted motion coincides with reasonably accurate estimates of the structural response showing the feasibility of the SCEM-UA algorithm for structural parameter identification and the ability to simultaneously identify the parameters of the primary structure and the PTMD. The prediction uncertainty bounds include almost all observed values during the calibration period. Note that compared with the linear structural system (Figure 14), the prediction uncertainty ranges are quite large, indicating considerable uncertainty given the current model structure.