## 2 Theory
### Modal Formulation of the Full Structure
Equation (33) shows the variation in the magnitude of damping values between each sequential experiment when the magnitude of the external force was increased \(z\) times based on \(f_{e_{0}}\). The total number of experiments shown in Equation (33) was \(v\)+1. However, the variation of damping values in each experiment shown in Equation (33) was difficult to identify practically. Therefore, the relative variation of damping values between sequential experiments was investigated. \[ \begin{split}\Delta d_{l1}&=\Delta c_{l1}-\Delta c _{l0}=(z^{(\chi+1)}-1)|\kappa_{1}|B\\ \Delta d_{l2}&=\Delta c_{l2}-\Delta c_{l1}=z^{(\chi+ 1)}(z^{(\chi+1)}-1)|\kappa_{1}|B\\ \Delta d_{l3}&=\Delta c_{l3}-\Delta c_{l2}=z^{( \chi+1)}(z^{(\chi+1)}-1)|\kappa_{1}|B\\ &\qquad\qquad\qquad\qquad\qquad\vdots\\ \Delta d_{lp}&=\Delta c_{lp}-\Delta c_{lp-1}=z^{( \
u-1)(\chi+1)}(z^{(\chi+1)}-1)|\kappa_{1}|B\end{split}  \tag{34}\]  
Taking the logarithm of the terms on both sides of Equation (34) and using \(z\) as the logarithm base, the following was obtained:  
\[ \begin{split} y_{1}&=a_{0}\\ y_{2}&=(\chi+1)+a_{0}\\ y_{3}&=(\chi+1)\cdot 2+a_{0}\\ &\qquad\qquad\qquad\qquad\vdots\\ y_{\
u}&=(\chi+1)\cdot(\
u-1)+a_{0}\end{split}  \tag{35}\]  
where \(y_{\
u}=log_{z}\Delta d_{l\
u}\), \(a_{0}=log_{z}\big{(}z^{(\chi+1)}-1\big{)}+log_{z}|\kappa_{1}|B\). Equation (35) shows the difference in the damping values of sequential experiments in which the external force was increased \(z\) times in each trial. The results of this equation were obtained experimentally. If the results are plotted after taking the logarithmic function, which has \(z\) as the logarithm base, then the slope of this plot is \(\chi+1\). Here, the abscissa indicated a negative one order for the performed experiments, and the ordinate indicated the measured values. This concept was similar to the conventional method used to estimate the value of \(\chi\) by converting the dissipated energy with respect to the external harmonic force to a logarithmic scale. To estimate \(\chi\) using Equation (35), linear curve fitting was performed on the measured damping values converted to a logarithmic scale. Here, the \(y\) intercept was regarded as \(a_{0}\). Since \(z\) could be measured through experiments or as the magnitude ratio of the applied force in successive experiments, \(|\kappa_{1}|B\) was estimated through \(a_{0}\) in Equation (35). This was the same as \(\Delta c_{l0}\) in Equation (33). As a result, every \(\Delta c_{l\
u}\) in Equation (33) could be estimated. By combining Equations. (33), (34), and (A15), the following relations were obtained:  
\[R=\frac{\pi}{4}\frac{(\chi+2)(\chi+3)}{\big{(}z^{(\chi+1)}-1\big{)}}\frac{ \Delta d_{l1}}{B}=\frac{\pi}{4}\frac{(\chi+2)(\chi+3)}{z^{(\chi+1)}\big{(}z^{( \chi+1)}-1\big{)}}\frac{\Delta d_{l2}}{B}=\dots=\frac{\pi}{4}\frac{(\chi+2)( \chi+3)}{z^{(\
u-1)(\chi+1)}\big{(}z^{(\chi+1)}-1\big{)}}\frac{\Delta d_{l\
u} }{B} \tag{36}\]  
In Equation (36), all of the variables except the constant \(B\) could be estimated by experiments. If a numerical model for the full system exists, then the stiffness \(\mathbf{K}\) in Equations (24) and (25) can be estimated, which was the case for the absence of joint components. If an experiment was performed by applying a very small force to the built-up structure, which includes the joint component, then the added stiffness \(\mathbf{K}_{T}\) at the interface could be estimated. Eventually, a static stiffness \(\mathbf{K}_{I}\) could be obtained. From this result, the mode shape for an undamped system could be estimated. In addition, if the external forces used in the experiments were applied to the numerical model to obtain the results in Equation (33), then the relative displacement at the contact surface could be estimated.