## 3 Simulation  
### Degree-of-Freedom (DOF) Lumped Model  
In this section, a simulation is conducted to verify the proposed method from the previous section. Figure 3 shows the configuration of the 10 DOF lumped model that is used in the simulation. Here, \(m\) and \(k\) represent the lumped mass and stiffness, respectively. In addition, \(u\) represents the displacement of each mass. As shown in the figure, two numbers of the Iwan model are assigned (one is connected between m\({}_{1}\) and m\({}_{6}\) and another is connected between m\({}_{2}\) and m\({}_{7}\)). These two models show their dynamic characteristics according to the relative displacement of the connected nodes in the \(u\) direction. The frequency response function for this system is obtained by applying a force \(f\) at the lumped mass m\({}_{5}\), and we define the step sine as the applied force. In addition, a total of twelve experiments were conducted in this simulation, and in each experiment, the magnitude of the applied force follows the step sine, which increases consistently. Detailed introductions and specifications on the system and the performed experiments are as follows:  
\[ \begin{array}{c}m_{1}=m_{2}=\cdots=m_{10}=1\\ k_{1}=k_{6}=1.5\times\ \mathrm{e}^{6},\ k_{2}=k_{3}=\cdots=k_{5}=k_{7}=\cdots=k_{10}= \ 1\ \times\ \mathrm{e}^{6}\\ k_{11}=0.3\ \mathrm{e}^{6},k_{12}=0.5\ \times\ \mathrm{e}^{6}\\ f=2\times\left[2^{0}2^{1}2^{2}2^{2}3\cdots 2^{11}\right]\end{array}  \tag{38}\]  
The four parameters on Iwan model include the following:  
\[K_{T}=1.1e5,F_{s}=400,\chi=-0.5,\beta=4 \tag{39}\]  
Moreover, damping is assigned to increase the accuracy of the measurements made by the simulation. Proportional damping of the mass and stiffness matrices is assumed. As mentioned in the previous section, the stiffness matrix involved in this proportional damping is related to the stiffness K\({}_{j}\) introduced in Equation (25):  
\[C=aM+bK_{j} \tag{40}\]  
In Equation (40), \(a\) and \(b\) are constants, and in this study, values of \(5\times\mathrm{e}^{-1}\) and \(7\times\mathrm{e}^{-5}\) are assigned as these constants, respectively. In addition, the frequency resolution in the step sine experiment is \(0.5\) Hz and the time integral in the simulation is performed by the Newmark method [18]. Figure 3: Description of the 10 DOF system with two numbers of the Iwan model. Figure 4 shows the obtained FRFs from performed simulations. As shown in this figure, it can be confirmed that each mode has a different effect with respect to the magnitude of the external force. For instance, some modes (i.e., 2 nd, 4 th, 6 th and 8 th modes) show that their natural frequencies are shifted and that their peak values are decreased according to the increased magnitude of the force, whereas the other modes (i.e., 1 st, 3 rd, 5 th, 7 th and 9 th modes) hardly change their natural frequencies or their peak values. This is because the effect of the Iwan model is activated when relative displacement occurs in the region of interest. However, the modes that show no variations in the FRF results with respect to the changes of the external force all have in-phase motion in the interface between the masses of the Iwan model. The results shown in this figure are consistent to the statements in Ref. [13]. In addition, this figure magnifies the results for the 2 nd, 4 th and 6 th modes where variations in the frequency results occur with respect to the magnitude of the external force. From these magnified plots, changes of damping and the natural frequency can be identified in each experiment. To identify the parameter \(\chi\) with the proposed method in this study, the results of Equation (35) for these three modes have been identified. In this process, the modal parameters for these modes have to be estimated. Recently, polyreference least-squares complex frequency-domain method which is so-called PolyMAX [19] is introduced as a modal parameter estimation method conducted in the frequency domain.