## 2 Theory
### Modal Formulation of the Full Structure  
In this section, the configuration of the variation force, which is generated in the constitutive model during dynamic behavior, is examined through a 2 degrees-of-freedom (DOF) system. In  
Figure 1: Description of the parallel-series Iwan model. addition, this result is extended to a multi-degree-of-freedom system, which involves the constitutive model. The configuration of the FRF for this system with respect to the harmonic excitation is obtained. The methodology, which is intended in this study to estimate certain parameters out of the parameters of the constitutive model, is introduced through these results. Figure 2 represents a 2 DOF system, which is connected by the stiffness \(\mathbf{k}^{ij}\) and the Iwan model. Here, \(\mathbf{u}^{i}\) and \(\mathbf{u}^{j}\) represent the displacement at each node. The relative displacement between these two nodes when an external force is applied can be represented as follows:  
\[u=u^{j}-u^{i}=A_{ij}sin(\omega t-\theta) \tag{19}\]  
In Equation (19), \(A_{ij}\) denotes the amplitude of the relative displacement between the \(i^{th}\) and \(j^{th}\) nodes. By using this equation, the restoring force in the stiffness and the generating force in the Iwan model can be represented in vector-matrix form using Equations (5) and (18):  
\[\dot{f}_{r}^{ij}=-\mathbf{K}^{ij}\mathbf{u}^{ij},\left\{ \begin{array}{c}f_{r}^{ i}\\ f_{r}^{j}\end{array} \right\}=-\left[ \begin{array}{cc}k^{ij}&-k^{ij}\\ -k^{ij}&k^{ij}\end{array} \right]\left\{ \begin{array}{c}u^{i}\\ u^{j}\end{array} \right\} \tag{20}\]  
\[\dot{f}_{J}^{ij}=-\mathbf{K}_{T}^{ij}\mathbf{u}^{ij}+\Delta f_{s}^{ij} \tag{21}\]  
where  
\[ \begin{array}{c}\mathbf{K}_{T}^{ij}=\begin{bmatrix}K_{T}^{ij}&-K_{T}^{ij}\\ -K_{T}^{ij}&K_{T}^{ij}\end{bmatrix},\mathbf{u}^{ij}=\begin{cases}\mathbf{u}^{i}\\ \mathbf{u}^{j}\end{cases}\\ \mathbf{and}\\ \Delta f_{s}^{ij}=-\frac{|\kappa_{1}|}{\omega}\mathbf{A}_{ij}\mathbf{u}^{ij}+\kappa_{ 2}\mathbf{A}_{ij}\mathbf{u}^{ij}\mathrm{or}\begin{pmatrix}\Delta f_{s}^{i}\\ \Delta f_{s}^{i}\end{pmatrix}=-\frac{|\kappa_{1}|}{\omega}\begin{bmatrix}\mathbf{ A}_{ij}^{i+1}&-\mathbf{A}_{ij}^{i+1}\\ -\mathbf{A}_{ij}^{i+1}&\mathbf{A}_{ij}^{i+1}\end{bmatrix}\begin{pmatrix}\mathbf{u}^{i}\\ \mathbf{u}^{j}\end{pmatrix}+\kappa_{2}\begin{bmatrix}\mathbf{A}_{ij}^{i+1}&-\mathbf{A}_{ij}^ {i+1}\\ -\mathbf{A}_{ij}^{i+1}&\mathbf{A}_{ij}^{i+1}\end{pmatrix}\begin{pmatrix}\mathbf{u}^{i}\\ \mathbf{u}^{j}\end{pmatrix}+\kappa_{2}\begin{bmatrix}\mathbf{A}_{ij}^{i+1}&-\mathbf{A}_{ij }^{i+1}\\ -\mathbf{A}_{ij}^{i+1}&\mathbf{A}_{ij}^{i+1}\end{pmatrix}\begin{pmatrix}\mathbf{u}^{i}\\ \mathbf{u}^{j}\end{pmatrix}\end{array}  \tag{22}\]  
Equations (20) and (21) represent the restoring force in the stiffness and the variation force in the constitutive model when an external force is applied, respectively. Detailed configurations on each element of Equation. (21) are introduced in Equation. (22). In this equation, the matrix \(\mathbf{K}_{T}^{ij}\) denotes a matrix that is composed of combinations of the added stiffness (\(K_{T}\)) by joints, which is one of the parameters defining the Iwan model. The matrix \(A_{ij}\) denotes a matrix that is composed of the combinations of relative displacement between nodes. Actually, the results in these equations can be extended without a loss of generality for general built-up structures. This means that the results of Equations (20)-(22) can be regarded as the local matrix for the interface between the jointed surfaces of general built-up structures. If one represents the equilibrium of force in general built-up structures by a vector summation form according to the [PERSON]'s second law, then,  
\[\mathbf{f}_{t}=\mathbf{f}_{e}+\mathbf{f}_{c}+\mathbf{f}_{r}+\mathbf{f}_{f} \tag{23}\]  
In Equation. (23), the subscripts I, e, c, r and J denote the force by inertia, the external force, the force by damping, the restoring force by stiffness and the force occurring in the joint component, respectively.