# The Role of a Simple Inerter in Seismic Base Isolation
## 2 Base-Isolated Rigid Structure with TMDI and ADAS
### Supplemental TMDI Device  
The tuned mass damper inerter (TMDI) device comprises components including an auxiliary mass (\(m_{t}\)), stiffness (\(k_{l}\)), and damper (\(c_{t}\)), collectively known as TMD, along with an inertial device featuring an inertance coefficient denoted \(b\) (refer to Figure 1b). The two terminals of the inertial device are connected to the auxiliary mass and the ground. A reaction force is generated by the inertial device's motion that, roughly speaking, can be regarded as proportional to the relative acceleration developed between the two terminals. The inertance, also known as apparent mass, is a proportional coefficient that can be numerous times the inerter's actual mass. The energy is dissipated through damping when the resonance in the TMD is introduced together by the vibration of the base-isolated structure. The modeling aspects of the TMDI for practical usage had been confirmed through an experimental study of the dynamic response of an isolated structure system equipped with TMDI tested on a shaking table [16; 17]. Also, TMDI is observed to be robust when a change in the design parameters from their optimal values does not produce significant response variations [39]. The auxiliary and inertial masses coupled to the TMDI are represented by the mass and inertance ratio, defined as  
\[\mu=\frac{m_{t}}{m}\text{ and }\beta=\frac{b}{m} \tag{2}\]  
The TMDI's corresponding stiffness and damping are expressed as  
\[\xi_{t}=\frac{c_{t}}{2(m_{t}+b)\omega_{t}},\text{ }\omega_{t}=\sqrt{\frac{k_{t}}{m _{t}+b}}\text{ and }f=\frac{\omega_{t}}{\omega_{b}} \tag{3}\]  
where \(\xi_{t}\) and \(f\) represent the damping ratio of the TMDI and tuning frequency ratio, respectively. The equations governing the motion of the building model with a rigid superstructure, base isolation, and TMDI are represented by  
\[ \begin{bmatrix}m&0\\ 0&m_{t}+b\end{bmatrix}\begin{Bmatrix}\ddot{x}\\ \ddot{x}_{t}\end{Bmatrix} + \begin{bmatrix}c_{b}+c_{t}&-c_{t}\\ -c_{t}&c_{t}\end{bmatrix}\begin{Bmatrix}\dot{x}\\ \dot{x}_{t}\end{Bmatrix} + \begin{bmatrix}k_{b}+k_{t}&-k_{t}\\ -k_{t}&k_{t}\end{bmatrix}\begin{Bmatrix}x\\ x_{t}\end{Bmatrix} =- \begin{Bmatrix}m\\ m_{t}\end{Bmatrix} (\ddot{x}_{g}) \tag{4}\]  
Figure 1: Idealization of the rigid superstructure of the multi-story building with base isolation and supplemental devices. where \(x\) = relative displacement of the isolated structure; \(x_{t}\) = relative displacement of the auxiliary mass of the TMDI; and \(\tilde{x}_{g}\) = earthquake acceleration. Let the earthquake acceleration be harmonic as \(\tilde{x}_{g}=e^{j\omega t}\); where \(\omega\) is the circular frequency and \(j=\sqrt{-1}\), the steady-state displacement \(x\) is expressed as  
\[x=H_{x}(j\omega)\ e^{j\omega t} \tag{5}\]  
where \(H_{x}(j\omega)\) = frequency response function (FRF) of the displacement \(x\). The SDOF model of the base-isolated structure with TMDI is subjected to white-noise earthquake acceleration with the power spectral density function (PSDF) as \(S_{0}\). The mean square response of base displacement for the isolated structure is given by the following equation:  
\[\sigma_{x}^{2}=\int_{-\infty}^{\infty}\left|H_{x}(j\omega)\right|^{2}S_{0}d\omega \tag{6}\]  
By dividing the corresponding response from the base-isolated structure with no control, the mean square response obtained from Equation (6) is further normalized, i.e.,  
\[\widetilde{\sigma}_{x}^{2}=\frac{\sigma_{x}^{2}}{\sigma_{x,0}^{2}}\text{ and }\sigma_{x,0}^{2}=\frac{\pi S_{0}}{2\xi_{b}\omega_{b}^{3}} \tag{7}\]  
The TMDI system is effective at response mitigation if the normalized variance, \(\widetilde{\sigma}_{x}^{2}\) value, is less than unity. The parameters \(\mu\), \(\beta\), \(\xi_{t}\), and \(f\) provide a complete description of the supplemental TMDI for the base-isolated structure. For all numerical solutions in the current study, \(\mu\) is taken to be equal to 0.01 because the TMDI performs best for a smaller value of \(\mu\) and higher values of \(\beta\)[39].