# The Role of a Simple Inerter in Seismic Base Isolation
## 5 Flexible Isolated Structure under Real Earthquakes
The auxiliary stiffness and damping of the optimal TMDI (for the specified value of \(T_{b}\) and \(\xi_{b}\)) are selected corresponding to which minimizes the \(\sigma_{x}^{2}\) of a rigid-base isolated structure under white-noise earthquake excitation using Equations (8) and (9). The viscous damping of FID is defined by \(\xi_{f}\) using Equation (10) and considering the total mass of the isolated structure, and taken as \(\xi_{f}=\beta/2\). The analytical seismic response of the isolated structure with supplemental TMDI, ADAS, FID, and FD devices is evaluated numerically by solving the governing equations of motion using the step-by-step method [49]. Three real earthquake motions, namely the north-south component of El-Centro, 1940 earthquake, the N00E component of the 1989 Loma Prieta earthquake (measured at Los Gatos Presentation Centre), and the 270 component of the 1992 Landers earthquake (measured at Lucerne Valley), are selected. The peak ground acceleration (PGA) of El-Centro, Loma Prieta, and Landers earthquake motions are 0.34 g, 0.57 g, and 0.73 g, respectively. The response quantities of interest are the top floor absolute acceleration (\(x_{5}\)), relative base displacement (\(x_{b}\)), force in the isolation system (\(F_{is}\)), force in the supplemental device (\(F_{sup}\)), and total base shear (\(F\)). The selected force quantities are normalized with \(W\)  
Figure 8: Modeling of the flexible base-isolated building with TMDI, ADAS, FID, and FD and bi-linear characteristics of LRB. The time history of the top floor/story acceleration, relative base displacement, the force developed in the isolator, the force developed in the supplemental control system used, and total base shear are shown in Figure 9. The responses are shown for the Loma Prieta, 1989 earthquake, and they are compared with the corresponding uncontrolled response of isolated structures, i.e., without supplemental devices (referred to as BIS). The figure indicates that the peak value of the top floor acceleration for BIS + TMDL, BIS + ADAS, and BIS + FD is the same as that of BIS, suggesting that these supplemental control devices are not much altering the top floor acceleration. However, there is an almost 3.61 times increase in the peak top floor acceleration of the isolated structure with FID compared to the corresponding peak acceleration of BIS. This is unacceptable as the FID defeats the primary purpose of seismic isolation by increasing the structural acceleration. It was also noted that the base-isolated structure with FID's absolute acceleration contains high-frequency components, which could be harmful to equipment or a secondary system that is sensitive to high-frequency vibrations. There is a reduction in base displacement by all the supplemental devices. It was observed to decrease by 23.8 to 32.8 percent compared to the corresponding BIS. The base displacement of isolated structures with ADAS is less than the corresponding TMDL. This is interesting and primarily happens because the peak base displacement occurs in the large initial cycles of the response. By that time, the TMDL was not adequately tuned to dissipate the input seismic energy. However, the TMDL is more effective in reducing the subsequent peak response in comparison to the ADAS. The time variation in force within the isolation system mirrors the displacement of the isolator for all supplementary devices. Figure 9: Time history of response of a flexible base-isolated building under Loma Prieta, 1989 earthquake (\(T_{b}=3\) s and \(\beta=0.4\)). The effects of the \(\beta\) on the peak top floor absolute acceleration, relative base displacement, the force in the isolation system, the force in the supplemental device, and the total base shear of the base-isolated structure with different supplemental devices and earthquakes are shown in Figure 10. The figure indicates that the peak top floor acceleration is not very sensitive to \(\beta\) and quite comparable to the base-isolated structures with supplemental TMDI, ADAS, and FD.