# The Role of a Simple Inerter in Seismic Base Isolation
## 2 Base-Isolated Rigid Structure with TMDI and ADAS
### Supplemental TMDI Device
It is also worth mentioning that when \(\mu\) = 0, the TMDI is reduced to the TID. When \(\xi_{t}\) and \(f\) are set to their optimum values, the TMDI performs at its best. For a given base-isolated structure (i.e., specified by \(T_{b}\) and \(\xi_{b}\)) and white-noise earthquake excitation, the optimum parameters of the TMDI (i.e., optimum \(\xi_{t}\) and \(f\)) are the ones that minimize the \(\sigma_{x}^{2}\). This is referred to as 'optimal TMDI' in the present study and the corresponding optimums \(\xi_{t}\) and \(f\) are obtained using an algorithmic numerical search technique for values spanning their feasible regions with increments of 0.0001. Using the above-specified procedure with isolation damping, \(\xi_{b}\) not exceeding 10 percent, the optimum damping ratio (\(\xi_{t}^{opt}\)) and tuning frequency ratio (\(f^{opt}\)) for a selected value of \(\beta\) can be calculated using the following expressions from [40]. \[\xi_{t}^{opt}=\sqrt{\frac{\beta(4+3\beta)}{8(1+\beta)(2+\beta)}} \tag{8}\]  
\[f^{opt}=\frac{1}{1+\beta}\sqrt{\frac{(1+\frac{\beta}{2})}{(1+\xi_{b})}} \tag{9}\]  
The changes in FRF of base displacement for an isolated structure with optimal TMDI are plotted in Figure 2. The inertance ratio is considered to be 0.1, 0.2, and 0.4, and the FRF responses are compared with the corresponding base-isolated structure without TMDI (referred to as BIS). The FRF of isolated structures with optimal TMDI appears to be a two-peaked response. The optimization of parameters occurs when the response to the second peak is minimized. The maximum reduction in FRF occurs at the resonating frequency, and there is a reduction of 60.8, 68.1, and 75.1 percent for the inertance ratio of 0.1, 0.2, and 0.4, respectively. Thus, it can be concluded that the TMDI effectively reduces the FRF and, thereby, the mean square of the displacement response of the base-isolated structure. It was observed in Figure 2 that the optimal TMDI mitigates the FRF of the base-isolated structure under resonating conditions. However, it is interesting to know how much time or the number of cycles the TMDI needs to attain fully tuned conditions for maximum response reduction. To verify this, the displacement of the base-isolated structure with and without optimal TMDI subjected to earthquake acceleration of \(\ddot{x}_{g}=sin(2\pi t/3)\) m/s\({}^{2}\) (\(t\) denotes here the time in s) is calculated using the numerical integration procedure and plotted in Figure 3. It is observed from the figure that after one cycle of response, the reduction of the displacement is 12.4, 27.7, and 39.1 percent for the inertance ratio of 0.1, 0.2, and 0.4, respectively. To achieve the maximum reduction in the response under resonating conditions, the TMDI should undergo several cycles of responses (i.e., about 10 cycles as per Figure 3). Normally, such many cycles may not be possible during the strong motion duration of past recorded earthquakes. Despite that, there is a reduction in the response of the TMDI in the initial cycles, which is primarily attributed to the added heavy damping of its auxiliary damper. It is to be noted that the added damping by optimal TMDI is 0.16, 0.48, and 0.87 times the isolation damping for the inertance ratio of 0.1, 0.2, and 0.4, respectively. Figure 3: Time variation of displacement of base-isolated structure with optimal TMDI under harmonic excitation (\(T_{b}=3\) s and \(\xi_{b}=0.1\)). Figure 2: Changes in FRF of base displacement of isolated structure with optimal TMDI (\(T_{b}=3\) s and \(\xi_{b}=0.1\)).