## 4 Experiments
This was because slip rarely occurs when the external force is small, and in this case, the estimated results by the proposed method were more sensitive to the noise in the experiment. This tendency was also confirmed in the simulation results. The objective \(\chi\) was estimated using the damping ratios calculated from the first seven experiments. Figure 8 represents the difference in energy dissipation in successive experiments; thus, \(\chi\) was estimated by curve fitting the first six results by a 1\({}^{\text{st}}\) order straight line (). The estimated result was \(-0.37366\). Next, the parameter R of the Iwan model was estimated using Equation (37), but first the mode shape of the partially connected double beam shown in Figure 6 had to be determined. The mode shape of a partially connected double beam was studied analytically and numerically in [20]. By applying Equation (37) to the mode shape vector, R was estimated as \(3.5431\times\text{e}^{+5}\). Utilizing the estimated \(\chi\) and R, the natural frequencies and damping ratios were estimated using the measurements in the 8\({}^{\text{th}}\), 9\({}^{\text{th}}\), and 10\({}^{\text{th}}\) experiments. The  
Figure 6: Experimental configuration. variation of natural frequency, \(\Delta\omega_{n}^{2}\), when increasing the external force is represented in Equation (30). It can also be represented as follows, similar to Equation (33):  
\[ \begin{split}\Delta\omega_{\textit{I0}}^{2}&=\kappa_{2 }\sum\limits_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})A_{m_{0}}^{\chi+1}=\kappa_{2}B\\ \Delta\omega_{\textit{I1}}^{2}&=\kappa_{2}\sum \limits_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})(zA_{m_{0}})^{\chi+1}=z^{(\chi+1)} \kappa_{2}B\\ \Delta\omega_{\textit{I2}}^{2}&=\kappa_{2}\sum \limits_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})\big{(}z^{2}A_{m_{0}}\big{)}^{\chi+1}= z^{2(\chi+1)}\kappa_{2}B\\ \Delta\omega_{\textit{I3}}^{2}&=\kappa_{2}\sum \limits_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})\big{(}z^{3}A_{m_{0}}\big{)}^{\chi+1}= z^{3(\chi+1)}\kappa_{2}B\\ &\hskip 142.26378 pt\vdots\\ \Delta\omega_{\textit{I0}}^{2}&=\kappa_{2}\sum \limits_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})(z^{v}A_{m_{0}})^{\chi+1}=z^{v(\chi+1) }\kappa_{2}B\\ \end{split}  \tag{43}\]  
where \(B=\sum_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})A_{m_{0}}^{\chi+1}\). Using the \(y\)-intercept in Figure 8 and Equation (35), \(|\kappa_{1}|B\) was found to be 29.431. By substituting the estimated \(\chi\) and R into Equation (A15), \(\kappa_{1}\) was obtained. Utilizing the estimated value and \(|\kappa_{1}|B\), the following result was determined: \(B=2.787\times\text{e}^{-4}\). Additionally, \(\kappa_{2}\) was found using Equations. (A12) and (A15). \(\Delta\omega_{n}^{2}\), which represents the change in natural frequency, could then be obtained for the 8 th through 10 th experiments. Based on obtained \(\Delta\omega_{n}^{2}\), the natural frequency \(\omega_{n}^{2}\) was estimated. Figure 7: Measured frequency response functions. The second column of Table 3 is the variation in the natural frequencies \(\Delta\alpha_{n}^{2}\) between successive experiments. The third column shows the variation of the natural frequencies estimated by applying the proposed method, Equation (43), to the experimental results from the first to the seventh trials. The fourth column shows the error between the experimental and estimated values. On average, the error was less than 10%. Table 4 shows the natural frequencies \(\omega_{n}^{2}\) for the individual experiments, as obtained through Table 3. The second column is the measured \(\omega_{n}^{2}\) in the experiment and the third column is the result estimated from Table 4. The fourth column shows the error between the experimental and the estimated values. Since the natural frequency is large, even if there was an error of 10% in the variation of the natural frequency listed in Table 3, the estimated natural frequency was very accurate at an error of less than 0.1%.