## 2 Theory
### Modal Formulation of the Full Structure
If the equation of motion for the full structure is represented with this equation, then the following is obtained:  
\[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{f}_{e}-\mathbf{K}_{T}\mathbf{u }-\frac{|\kappa_{1}|}{\omega}\mathbf{A}\dot{\mathbf{u}}+\kappa_{2}\mathbf{A}\mathbf{u} \tag{24}\]  
On the left side of Equation (24), \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) denote the mass, damping and stiffness matrices for the full system, respectively. Note that, in general, there are multiple numbers of joint components in the built-up structures. Therefore, as mentioned in [13], \"if one wants to estimate information on a certain constitutive model for a certain joint, the other joint components should be excepted from the full built-up structures apart from the joint component of interest\". Hence, the numerical model for the full structure, which includes the joint component shown in Equation (24), indicates that only the joint component of interest is connected in the full system. In this study, the full system stands for the same meaning of this statement. On the right side of Equation (24), every term apart from the external forcing vector \(f_{e}\) represents forces generated from the joint in a global coordinate system. In particular, matrices \(K_{T}\) and \(A\) are all of the symmetric matrices and are composed of real values. Re-arranging Equation (24) after transferring all of the terms in the right hand side of the equation apart from the term in the external forcing vector to the left side produces the following:  
\[\mathbf{M}\ddot{\mathbf{u}}+\Big{(}\mathbf{C}+\frac{|\kappa_{1}|}{\omega}\mathbf{A}\Big{)} \dot{\mathbf{u}}+\Big{(}\mathbf{K}_{J}-\kappa_{2}\mathbf{A}\Big{)}\mathbf{u}=f_{e} \tag{25}\]  
where \(\mathbf{K}_{J}=\mathbf{K}+\mathbf{K}_{T}\). \(\mathbf{K}_{J}\) denotes the stiffness matrix in a static state and both \(\frac{|\kappa_{1}|}{\omega}\mathbf{A}\) and \(\kappa_{2}\mathbf{A}\) denote the effects in the joint component by the dynamic behavior of the system. It can be confirmed through the signs of these two matrices that each term of these two effects of dynamic behavior will cause a decrease of stiffness and an increase of damping in the full system. These confirmations are consistent with previous research [13; 17]. In addition, assuming an external force with a harmonic component, then the velocity vector can be expressed by the multiplication of \(j\omega\) with the displacement vector. Here, \(j\) stands for the imaginary value. From this assumption,  
\[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\Big{(}\mathbf{K}_{J}-\kappa_{2}\mathbf{A}+j| \kappa_{1}|\mathbf{A}\Big{)}\mathbf{u}=\mathbf{f}_{e} \tag{26}\]  
From Equation (26), it is also confirmed that the increased damping due to dynamic behavior shows the form of structural damping (hysteric damping). It is a natural consequence because the hysteric behavior in elastoplastic elements is described in bi-linear form in Reference [9]. In conclusion, the physical meaning of Equations (25) and (26) are representative of the full system; if the joint component is connected, then the original stiffness \(\mathbf{K}\) is increased by \(\mathbf{K}_{J}\). In addition, if dynamic behavior is initiated by external forces, then slip occurs at the interface of the jointed structures and results in an increase in damping and decrease in stiffness. The type of damping variation during dynamic behavior shows the form of structural damping. Hereafter, the frequency response function will be obtained for the full system. To this end, the eigenvalue problem of Equation (26) is solved. For the case of no external force, there are no dynamic effects (\(\kappa_{1}\kappa_{2}\)), and therefore, this equation can be transformed into a simpler form. First, for the un-damped case, the \(\mathrm{n}^{\mathrm{th}}\) eigenvalue and eigenvector can be represented by \(\omega_{ln}\) and \(\mathbf{\varphi}_{n}\), respectively. These two results are composed of all real values. \[