## 2 Theory  
### Modeling of Joint Force  
Figure 1 shows the parallel-series Iwan model that has an equivalent stiffness \(k\). The mathematical description of this model is as follows:  
\[F(t)=\int_{0}^{\infty}\rho(\phi)[u(t)-x(t,\phi)]d\phi \tag{1}\]  
In Equation (1), F denotes the applied force. \(\phi\) denotes the sliding distance of each element and \(\rho(\phi)\) denotes the population density of Jenkins element of strength \(\phi\). \(u\) denotes the imposed distance and \(x\) denotes the slider displacement. Each quantity has the following dimensions. F has the dimension of force (N). \(\phi\), \(u\) and \(x\) have the dimension of length (m). The slider displacement \(x\)(t,\(\phi\)) evolves from the imposed displacement \(u\)(t). \[\dot{x}(t,\phi)=\left\{ \begin{array}{ll}\dot{u}&if\|u-x(t,\phi)\|=\phi\text{ and }\dot{u}[u-x(t,\phi)]>0\\ 0&\text{otherwise}\end{array} \right. \tag{2}\]  
In Equation (2), \(\dot{x}\) and \(\dot{u}\) represent the slider velocity and the imposed velocity, respectively. If \(u\) is subject to frequency \(\omega\) and amplitude A, it means  
\[u(t)=A\sin\omega t \tag{3}\]  
Then, Equation (1) can be represented as follows:  
\[F(t)=\int_{0}^{A}\rho(\phi)[u(t)-x(t,\phi)]d\phi+\int_{A}^{\infty}\rho(\phi)u (t)d\phi \tag{4}\]  
The second term on the right-hand side of Equation (4) can be obtained because \(x(t,\phi)=0\) for all \(\phi>\) A. If one re-arranges Equation (4),  
\[F(t)=u(t)K_{T}-\Delta F_{S}(t) \tag{5}\]  
where \(K_{T}=\int_{0}^{\infty}\rho(\phi)d\phi\), \(\Delta F_{S}(t)=\int_{0}^{A}\rho(\phi)x(t,\phi)d\phi\) and \(u(t)=A\sin(\omega t-\theta)\). In Equation (5), \(K_{T}\) represents the added stiffness by joints in built-up structures. \(\Delta F_{S}\) represents the variation force for the occurring slip. We adopt the representation of the density function \(\rho\) in Reference [12] to investigate in more detail the result in this equation; this representation is composed of the 4 parameters and is shown below:  
\[\rho(\phi)=R\phi^{\chi}[H(\phi)-H(\phi-\phi max_{max})] \tag{6}\]  
In Equation (6), \(R\), \(\chi\), \(\phi_{max}\) and \(S\) are four parameters that describe the density function. Here, \(R\) is the parameter related to the magnitude of the density function. \(\chi\) is a constant, which has a value between \(-1\) and \(0\). \(\phi_{max}\) is a constant representing the border line that determines the micro-slip and macro-slip in the contact surfaces of two objects. S is a constant to describe the dynamic characteristics when macro-slip occurs. In addition, H and \(\delta\) in this equation represent the Heaviside function and the Dirac delta function, respectively. These four parameters {\(R\), \(\chi\), \(\phi_{max}\) and \(S\)} can be transformed into the following four parameters {\(\chi\), \(K_{T}\), \(F_{S}\) and \(\beta\)}. In these newly introduced parameters, \(\chi\) is the existing  
the fundamental frequency component (i.e., the first harmonic component), then the following result can be obtained (a detailed description on how to obtain this result is in Appendix A):  
\[\Delta F_{s}(t)\cong a_{1}\cos\omega t+b_{1}\sin\omega t=A^{\chi+2}\{\kappa_{1} (\chi)cos\omega t+\kappa_{2}(\chi)sin\omega t\} \tag{14}\]  
Configurations of \(\kappa_{1}\) and \(\kappa_{2}\) in Equation (14) refer to Equation (A15). If one considers the more practical case of the imposed displacement compared to the case in Equation (3),  
\[u(t)=A\sin(\omega t-\theta) \tag{15}\]  
The variation force with respect to the case in Equation (15) can be obtained as follows if one follows a similar procedure to obtain the result in Equation (14).