## 3 Simulation
### Degree-of-Freedom (DOF) Lumped Model
The same trend is confirmed in results of other modes. In the previous section, it was assumed that if one increases the amplitude of the external force as z times, then the relative displacement will linearly increase by z times. As confirmed in Table 1, however, it is observed that the results are less than z times (in this study, 2 times). The results of the estimated slope of the curve shown in Figure 5 show lower values than the true value. Eventually, this results in a larger amplitude (0.52-0.53) of the estimated \(\chi\) than the true value. Next, \(R\) is estimated using the estimated \(\chi\) and Equation (37), as shown in Figure 5. Similar to \(\varphi_{max}\), \(R\) can be obtained with the defined parameters in Equation (39) by the following relation which is introduced in Reference [12]. \[R=\frac{F_{s}(\chi+1)}{\varphi_{\max}^{\chi+2}\Big{(}\beta+\frac{\chi+1}{\chi+ 2}\Big{)}} \tag{42}\]  
The obtained \(R\) by Equation (42) is \(1.69\times\mathrm{e}^{5}\). In this study, the constant \(\overline{B}\) in Equation (37) was obtained in two different ways. First, relative displacements that are necessary to estimate this constant are obtained by the same way that is introduced in the previous section. This indicates that the system matrices \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K_{j}}\) are obtained and estimate the relative displacements at nodes associated with the Iwan model by solving the inverse problem with these matrices. Then, \(\overline{R}\) can be estimated by using the estimated \(\overline{B}\) and Equation (37). The errors between the estimated \(\overline{R}\) and the obtained true value of \(R\) by Equation (42) are listed in Table 2 in percent format (%). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Order of Exps. **} & \multicolumn{2}{c}{**2\({}^{\text{nd}}\) Mode**} & \multicolumn{2}{c}{**4\({}^{\text{th}}\) Mode**} & \multicolumn{2}{c}{**6\({}^{\text{th}}\) Mode**} \\  & **Iwan1** & **Iwan2** & **Iwan1** & **Iwan2** & **Iwan1** & **Iwan2** \\ \hline
1 & 0.00005 & 0.00016 & 0.00001 & 0.00003 & 0.00001 & 0.00001 \\
2 & 0.00010 & 0.00030 & 0.00003 & 0.00006 & 0.00002 & 0.00002 \\
3 & 0.00018 & 0.00057 & 0.00005 & 0.00011 & 0.00004 & 0.00004 \\
4  
In Table 2, the inverse matrix shows the obtained results using the method proposed in this study with relative displacement, whereas the Newmark methods results are given in \[C=aM+bK_{j} \tag{40}\]0 If one examines the characteristics observed in this table, the same region corresponding to the linear region where the curve fitting process was performed in Figure 5 can confirm that the results in the table show more closed values than the true value of the defined R. In addition, the results, which were estimated using the results in Table 1 (Newmark method), are more accurate than the estimated results using the proposed method (Inverse matrix). Furthermore, as the mode number increases, the error of the estimated results of the proposed method also increases. This result is related to the mode shape because the estimated mode shape \(\varphi_{n}\) from the lumped model is different to the true mode shape in dynamic behavior. Based on the results in the table, the difference between these two mode shapes (i.e., the lumped model vs. practical behavior) increases as the mode number increases.