## 4 Experiments
\(\Delta\alpha_{n}^{2}\) in the denominator of Equation (31) was directly related to the variation in natural frequencies measured experimentally. However, the relation between \(\Delta c_{n}\) and the damping ratio \(\zeta_{J}\) measured through the experiment should be confirmed. The denominator of the FRF for the individual modes in the modal domain, as derived from the equations of motion for the linear structure is:  
\[\left(\omega_{J}^{2}-\omega^{2}\right)+j2\zeta_{J}\omega_{J}\omega \tag{44}\]  
In Equation (44), \(\omega_{J}\) and \(\zeta_{J}\) denote the natural frequencies and damping ratios measured, respectively, by applying the experimental modal analysis method to the experiments. Through the imaginary part of Equations (31) and (44), \(\Delta c_{n}\) and \(\zeta_{J}\) have the following relation. \[\zeta_{J}=\zeta_{n}\frac{\omega_{n}}{\omega_{J}}+\frac{\Delta c_{n}}{\omega_{ J}^{2}}\cong\zeta_{n}+\frac{\Delta c_{n}}{\omega_{J}^{2}} \tag{45}\]  
In Equation (45), \(\omega_{n}\) and \(\zeta_{n}\) are the natural frequency and damping ratio of the structure, respectively, when there is no Iwan model in the structure or when the structure vibrates with very small forces. Assuming that the ratio of \(\omega_{J}\) and \(\omega_{n}\) is close to one, the damping ratio measured from the experiment is expressed as the right-hand side of Equation (45). From Equation (33), \(\Delta c_{l0}=|\kappa_{1}|B\); thus, every \(\Delta c_{l0}\) in Equation (33) can be determined. \(\omega_{J}\) and \(\zeta_{J}\) were measured through experiments and \(\Delta c_{n}\) was predicted through the proposed method. \[\Delta\zeta_{J0}=\zeta_{J0}-\zeta_{J0-1}\cong\frac{\Delta c_{n0}}{\omega_{J0}^ {2}}-\frac{\Delta c_{n0-1}}{\omega_{J0-1}^{2}} \tag{46}\]  
Figure 8: Estimated y in Equation (35) (\(\Circle\)), results of curve fit with 1\({}^{\rm st}\) polynomial equation (\(\blackarrow\)) and estimated \(\chi\) from the curve fitting results. Equation (46) was derived from Equation (45) and shows the damping ratio change when the external force increases. Subscript \(J\) is the damping ratio measured in the experiment, and subscript \(v\) is the index of the experiment performed with increasing force level. As a result, the damping ratio measured from the 8 th through 10 th experiments could be estimated from Equation (46). The second column of Table 5 is the variation of damping ratio \(\Delta\zeta_{Jp}\) between consecutive experiments and the third column is the variation of the estimated damping ratio when applying the proposed method to the experimental results of the first seven experiments. The fourth column shows the error between the experimental and estimated values. The overall average error was less than 7%. Table 6 shows the damping ratios for individual experiments estimated based on Table 5. The second column was measured in the experiment and the third column is from Table 5. The fourth column shows the error between the experimental value and the estimated value. Since the damping ratio was larger than the variation in the damping ratio, the estimated damping ratio guaranteed a very accurate estimate with an error of less than 1%. This held even if the variation of the damping ratio estimated in Table 5 was 7%. Through the experiments performed, the estimation of variations in the natural frequency and the damping ratio had an error of 10% compared with the actual measurements. The natural frequencies and damping ratios; however, estimated based on the estimated variations were confirmed to be very accurate with less than 1% error. The reliability and effectiveness of the proposed method were, thus, verified.