## 3 Simulation
### Degree-of-Freedom (DOF) Lumped Model
It is one of the experimental modal parameter estimation methods and it guarantees stable identification of the system poles and participation factors. PolyMAX is applied to these three modes. Figure 5 shows the results of Equation (35) for the three modes (\(\square\)) obtained from the twelve experiments that were performed. In this figure, three plots are included as Figure 5a-c. These plots show the results for the selected modes. Among the results for each mode shown in this figure, a certain set is selected, which shows a constant increase of \(y\) values in each experiment. First degree polynomial curve fitting (--) has been applied on the selected data set in each mode, as shown in the figure. The parameter \(\chi\) is estimated from the slope obtained by the curve fitting. The estimated results of \(\chi\) are \(-0.528\), \(-0.526\) and \(-0.531\) for the 2 nd, 4 th and 6 th mode, respectively. These results show approximately 5.2-6.5% error compared to the value defined for the system in Figure 3 (\(-0.5\)). In Figure 5b,c, there are no values defined in the range of \(y\) (\(\square\)) from a value of zero (i.e., \(\
u-1=0\)) to the other end of the function domain. This is because the variation of damping, which is obtained from the first and second experiments, is too small to sense its value. Therefore, the variation cannot be represented by a logarithmic scale. In addition, in Figure 5c, this causes the similarity of values between 1 and 2 in the function domain. This is because the vibration magnitude at the higher modes (i.e., 4 th, 6 th modes) decreased compared to the lower mode (i.e., 2 nd mode). We performed other experiments to investigate the results in Figure 5 in more detail. PolyMAX was used to estimate the  
Figure 4: Numerically obtained twelve number of frequency response functions with respect to step sine excitations. modal parameters of the twelve performed experiments. We have harmonically excited the system to the estimated natural frequency in each experiment with the same amplitude of force that was used in the twelve performed experiments at the same positions. Relative displacements are confirmed with respect to these excitations at the nodes where the Iwan model is connected (i.e., the relative displacement between \(\mathrm{m}_{1}\) and \(\mathrm{m}_{6}\), and the relative displacement between \(\mathrm{m}_{2}\) and \(\mathrm{m}_{7}\)). These results are listed in Table 1.  
Figure 5: Estimated y in Equation (35) (\(\square\)), results of curve fit with 1\({}^{\mathrm{st}}\) polynomial equation (—) and estimated results of \(\chi\) from the curve fitting results: (**a**) for 2\({}^{\mathrm{nd}}\) mode, (**b**) for 4\({}^{\mathrm{th}}\) mode, (**c**) for 6\({}^{\mathrm{th}}\) mode. Table 1 lists half of the peak to peak values of the measurements, which are measured in the time domain one second after a long period of time (i.e., 9 seconds was used in this study). The results in the table can be associated with the parameter \(\varphi_{max}\) of the Iwan model. \(\varphi_{max}\) can be obtained by the following relation [12] with the defined parameters in Equation (39):  
\[\varphi_{\max}=\frac{F_{s}(1+\beta)}{K_{\mathrm{T}}\Big{(}\beta+\frac{\chi+1}{ \chi+2}\Big{)}} \tag{41}\]  
The obtained \(\varphi_{max}\) from Equation (41) for the defined Iwan model in Equation (39) is 0.0042. From the results in the table, it can be confirmed that for the second mode in the second Iwan model of the seventh experiment (i.e., the relative displacement between m\({}_{2}\) and m\({}_{7}\)), a larger value is obtained compared to the value of \(\varphi_{max}\) (0.0042) for the first experiment. This suggests that the macro-slip occurs in the seventh experiment for the first time. This also indicates that the system remained in the micro-slip region through the sixth experiment. The sixth experiment is associated with the 4 (v-1) in the function domain in Figure 5. Therefore, in Figure 5a, the constant increase continues until function domain 4, but it shows abrupt changes at function domain 5.