# The Role of a Simple Inerter in Seismic Base Isolation
## 5 Flexible Isolated Structure under Real Earthquakes  
An earlier study attempted the seismic response of a rigid base-isolated structure with the optimum TMDI, ADAS, FID, and FD under a stationary earthquake ground excitation model. However, the behavior of these devices under real earthquake excitation and when the isolated superstructure is assumed to be flexible will be interesting to explore. This will provide insight into the tuning behavior and the frequency contents of the superstructure acceleration of the isolated structure with TMDI and FID. Figure 8a depicts a typical five-story linear shear-type building equipped with supplementary TMDI, ADAS, FID, and FD devices in addition to a base isolation system. The base mass and the \(i\)th floor mass are represented by \(m_{b}\) and \(m_{i}\), respectively. \(k_{i}\) stands for the \(i\)th floor's stiffness. The lead rubber bearing (LRB) system is selected as an isolation device. Hysteretic damping is additionally produced by the lead core of the LRB through yielding, which dissipates the seismic energy input. Figure 8b illustrates the simple bi-linear force-deformation behavior of the LRB. The bi-linear characteristics of the LRB (lead rubber bearing) are defined by three parameters: \(F_{y}\), (yield force), \(k_{b}\) (post-yield stiffness), and \(q\) (yield displacement). For the selected base-isolated structure with supplemental TMDI, ADAS, FID, and FD, the governing equations of motion are  
\[\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K}\mathbf{ x}(t)+\mathbf{D}_{i}F_{i}(t)+\mathbf{D}_{s}\mathbf{F}_{\mathbf{s}}(t)=-\mathbf{E} \ddot{x}_{g}(t) \tag{17}\]  
where \(\mathbf{M}\) = mass matrix; \(\mathbf{C}\) = damping matrix; \(\mathbf{K}\) = stiffness matrix; \(\mathbf{x}(t)\) = vector of horizontal displacements (relative to the ground) of each mass at time \(t\); \(\mathbf{D}_{\mathbf{i}}\) = location vector for the bi-linear restoring force of the LRB; \(F_{i}(t)\) = bi-linear restoring force of the LRB; \(\mathbf{D}_{\mathbf{s}}\) = location matrix for the vector of control forces \(\mathbf{F}_{\mathbf{s}}(t)\) produced by the supplemental devices TMDI, ADAS, FID, and FD; \(\mathbf{E}\) = vector containing the vibrating masses; and \(\ddot{x}_{g}(t)\) = earthquake acceleration specified by a time history. Figure 7: Comparison of normalized displacement variance of the base-isolated structure with optimal TMDI, ADAS, FID, and FD subjected to of Kanai–Tajimi model of earthquake excitation (\(\xi_{f}=\beta/2\) and \(\xi_{b}=0.1\)). The study focuses on a five-story building characterized by linear inter-story stiffness and viscous damping, with parameters sourced from [PERSON] [48]. The mass distribution is uniform across all floors and the base raft locations, implying equal masses (\(m_{b}=m_{i}\) for \(i=1\) to 5). The inter-story stiffness for each floor is denoted as \(k_{1},k_{2}\), \(k_{3},k_{4}\), and \(k_{5}\), corresponding to 15, 14, 12, 9, and 5 times the \(k\), respectively. Here, the value for \(k\) is chosen such that it gives a fundamental time period of 0.4 s for the fixed-base structure. The corresponding fixed base structure has five natural frequencies, quantified in radians per second: 15.71, 38.48, 60.84, 83.12, and 105.37. With the assumption that the modal damping ratio for all vibrational modes is 2 percent, the damping matrix for the superstructure is formulated. The parameters \(T_{b}\) and \(\xi_{b}\) (see Equation (1)) are used to calculate the post-yield stiffness and damping of the LRB while considering the flexible base-isolated structure's total mass, \(M\) (i.e., \(M=m_{b}+\sum_{i=1}^{5}m_{i}\)). The isolation damping ratio is chosen as 0.1 of the critical damping. The yield strength of the LRB is selected as 0.05\(W\) (i.e., \(W=Mg\) represents the total weight of the base-isolated structure) and the yield displacement as 0.025 m. The normalized inertance, \(\beta\) of the supplemental devices is calculated by considering the total mass, \(M\).