# Displacement-Dependent Damping Inerter System for Seismic Response Control
## 2 Theoretical Analysis of Displacement-Dependent Damping Inerter System (DDIS)
### Stochastic Dynamic Analysis and Stochastic Linearization Method  
Considering the uncertain nature of the seismic excitation, the acceleration of the applied ground motion \(\ddot{u}_{g}\) in Equation (4) is modeled as a random process. The Kanai-Tajimi filtered white-noise process is adopted as excitation, as it has been widely employed in previous studies [35; 57]. Denoting the bedrock white noise process as \(W(t)\), the differential equation of the Kanai-Tajimi model can be established in time domain as follows:  
\[\ddot{u}_{g}(t)=-\omega_{g}^{2}u_{g}(t)-2\xi_{g}\omega_{g}\dot{u}_{g}(t)+W(t) \tag{5}\]  
where \(\omega_{g}(t)\) and \(\xi_{g}(t)\) are, respectively, the fundamental circular frequency and damping ratio related to the soil characteristics. The power spectral density function corresponding to the Kanai-Tajimi model is as follows:  
\[S_{\ddot{u}_{g}}(\omega)=\frac{\omega_{g}^{4}+4\xi_{g}^{2}\omega_{g}^{2}\omega ^{2}}{\left(\omega_{g}^{2}-\omega^{2}\right)^{2}+4\xi_{g}^{2}\omega_{g}^{2} \omega^{2}}S_{w} \tag{6}\]  
where \(S_{w}\) is the power spectral density of the white noise excitation [58]. A relationship between \(S_{w}\) and the peak ground acceleration was introduced in [59]. The solution of the governing equations of motion in Equation (4) is a non-linear problem owing to the non-linear restoring force of the bilinear model. Linear random vibration theory is thus not applicable. In this section, the stochastic linearization method (SLM) [60] is applied to solve Equation (4) using an equivalent linear-viscous damping \(c_{\alpha\beta}\) and a linear stiffness coefficient \(k_{eq}\) to represent the bilinear model. Equation (4) can thus be rewritten in terms of SLM as follows:  
\[\left\{ \begin{array}{l}\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u }}+\mathbf{Ku}+\mathbf{r}=\mathbf{E}\mathbf{M}\ddot{u}_{g}(t)\\ \dot{z}_{h}(t)+c_{\alpha\beta}\dot{u}_{d}(t)+k_{eq}z_{h}(t)=0\end{array} \right. \tag{7}\]  
Figure 4: Mechanical model of the single-degree-of-freedom (SDOF) frame structure (**a**) without and (**b**) with the DDIS. where,  
\[ \begin{array}{l}\mathbf{M}=\left[\begin{array}{cc}m&0\\ 0&m_{in}\\ \end{array}\right],\ \mathbf{C}=\left[\begin{array}{cc}c&0\\ 0&0\\ \end{array}\right],\ \mathbf{K}=\left[\begin{array}{cc}k+k_{t}&-k_{t}\\ -k_{t}&\alpha k_{d}+k_{t}\\ \end{array}\right],\\ \mathbf{r}=\left[\begin{array}{cc}0\\ (1-\alpha)k_{d}z_{h}\\ \end{array}\right],\ \mathbf{E}=\left[\begin{array}{cc}1\\ 0\\ \end{array}\right]\end{array}  \tag{8}\]  
and \(\mathbf{u}=\{u,u_{d}\}^{T}\) is the displacement vector. Here, \(c_{\alpha q}\) and \(k_{\alpha q}\) are determined by an equivalence principle [61; 62], minimizing the squared error between the linearized Equation (7) and non-linear Equation (4), which leads to:  
\[c_{\alpha q}=\mathrm{E}\left[\frac{\partial F_{d}(u_{d},z_{h})}{\partial\dot{u }_{d}}\right],\ k_{\alpha q}=\mathrm{E}\left[\frac{\partial F_{d}(u_{d},z_{h} )}{\partial z_{h}}\right] \tag{9}\]  
where \(\mathrm{E}\left[\,\right]\) denotes the expectation operator to obtain the mean value. The coefficients \(c_{\alpha q}\) and \(k_{\alpha q}\) of the bilinear model resulting from Equation (9) are given by [61; 62]  
\[\begin{array}{l}c_{\alpha q}=\left\{