## 2 Theory
### Modal Formulation of the Full Structure
\begin{array}{l}\text{Eigenvalue}:\omega_{ln}\\ \text{Eigenvector}:\mathbf{\varphi}_{n}\end{array}  \tag{27}\]  
where \(\mathbf{\varphi}_{n}=\left[\varphi_{n}^{1}\varphi_{n}^{2}\cdots\varphi_{n}^{N} \right]^{T}\). Figure 2: Description of the 2 DOF system connected by the stiffness \(k\) and the Iwan model between the \(\mathrm{i}^{\mathrm{th}}\) and \(\mathrm{j}^{\mathrm{th}}\) nodes. Equation (27) shows the configuration of the n\({}^{\text{th}}\) eigenvalue and eigenvector. Here, the eigenvector represents the mass normalized eigenvector. The superscript \(T\) stands for the transpose. If assuming proportional damping (e.g., \(\mathbf{C}=a\mathbf{M}+b\mathbf{K}_{j}\); here, \(a\) and \(b\) are real constants), then the eigenvector will have the equivalent to the eigenvector in Equation (27), and the eigenvalue will have following form;  
\[\lambda_{Jn}=-\zeta_{n}\omega_{Jn}+j\omega_{Jn}\sqrt{1-\zeta_{n}^{2}}\text{ where }\zeta_{n}=\frac{a}{2\omega_{Jn}}+\frac{b}{2}\omega_{Jn} \tag{28}\]  
If the effect of stiffness in the static state is much larger than the effect of the variation stiffness by dynamic behavior, it is suggested that the following conditions are satisfied,  
\[\|\mathbf{K}_{l}\|>>\kappa_{2}\|\mathbf{A}\|\text{ and }|\kappa_{1}\|\|\mathbf{A}\| \tag{29}\]  
one can obtain the following results:  
\[\kappa_{2}\mathbf{\varphi}_{r}^{T}\mathbf{A}\mathbf{\varphi}_{l}=\Delta\omega_{r}^{2} \delta_{rl},|\kappa_{1}|\mathbf{\varphi}_{r}^{T}\mathbf{A}\mathbf{\varphi}_{l}=\Delta c_{ r}\delta_{rl} \tag{30}\]  
where, \(l=1,N\). From the results in Equation (30), Equation (26) can be represented by the model summation form. \[\mathbf{u}=\left[\sum_{n=1}^{N}\frac{\mathbf{\varphi}_{n}\mathbf{\varphi}_{n}^{T}}{(\omega _{Jn}^{2}-\Delta\omega_{n}^{2}-\omega^{2})+j(2\zeta_{n}\omega_{n}\omega+ \Delta\epsilon_{n})}\right]\mathbf{f}_{e} \tag{31}\]  
In Equation (31), the term in the bracket represents the frequency response function of a built-up structure that includes the joint component. In the term, the values of \(\Delta\omega_{n}^{2}\) and \(\Delta c_{n}\) in the denominator represent a real and imaginary part, respectively, and indicate the amount of the decreased natural frequency and increased damping by slip at the interface of jointed structures during dynamic behavior. The amount of the damping variation during dynamic behavior will be investigated in more detail. The amount of this quantity for a certain mode is  
\[\Delta c_{l}=|\kappa_{1}|\mathbf{\varphi}_{l}^{T}\mathbf{A}\mathbf{\varphi}_{l}=|\kappa_{ 1}|\sum_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})A_{m}^{\times+1} \tag{32}\]  
In Equation (32), \(P\) represents a quadratic polynomial with variables of the \(l^{\text{th}}\) eigenvector. \(M\) denotes the number of the Iwan model. \(A\) represents the magnitude of the relative displacement between the connected nodes by the constitutive model. If the magnitude of the initially applied force was assumed to be \(\mathbf{f}_{\alpha_{0}}\) and the experiment was performed by increasing the applied force \(z\) times, then the relative displacement \(A\) also increased \(z\) times. By using this assumption, the variation of damping in each experiment could be expressed as:  
\[ \begin{split}\Delta c_{0}&=|\kappa_{1}|\sum_{m=1}^{ M}P_{m}(\mathbf{\varphi}_{l})A_{m_{0}}^{\times+1}=|\kappa_{1}|B\\ \Delta c_{l1}&=|\kappa_{1}|\sum_{m=1}^{M}P_{m}(\mathbf{ \varphi}_{l})(zA_{m_{0}})^{\times+1}=z^{(\times+1)}|\kappa_{1}|B\\ \Delta c_{2}&=|\kappa_{1}|\sum_{m=1}^{M}P_{m}(\mathbf{ \varphi}_{l})\Big{(}z^{2}A_{m_{0}}\Big{)}^{\times+1}=z^{2(\times+1)}|\kappa_{1 }|B\\ \Delta c_{3}&=|\kappa_{1}|\sum_{m=1}^{M}P_{m}(\mathbf{ \varphi}_{l})\Big{(}z^{3}A_{m_{0}}\Big{)}^{\times+1}=z^{3(\times+1)}|\kappa_{1 }|B\\ \vdots\\ \Delta c_{lp}&=|\kappa_{1}|\sum_{m=1}^{M}P_{m}(\mathbf{ \varphi}_{l})(z^{p}A_{m_{0}})^{\times+1}=z^{p(\times+1)}|\kappa_{1}|B\end{split}  \tag{33}\]where \(B=\sum_{m=1}^{M}P_{m}(\mathbf{\varphi}_{l})A_{m_{0}}^{\chi+1}\).