# The Role of a Simple Inerter in Seismic Base Isolation
## 3 Rigid Base-Isolated Structure with FID and FD  
Figure 5a depicts a schematic diagram of FID installed in a rigid model of the base-isolated structure. The two terminals, represented by 1 and 2 on the FID, are coupled with the ends depicted in Figure 1a of the base-isolated structure model. An FID typically consists of a piston and cylinder that force the fluid through a helical tube that is encircled  
Figure 4: The plot of normalized displacement variance of base-isolated structure with optimal TMDI and corresponding ADAS against the inertance ratio. by the cylinder. The hydraulic cylinder and the piston rod are the two terminals of the FID. The fluid flow through an external helical channel produces rotational inertia to compensate for the pressure loss [42; 43; 44]. The FID's resisting force depends on the friction, oil density, and viscosity of the fluid. The FID is modeled as a linear inerter in parallel with a linear [27; 28; 29; 32; 33] and a non-linear [30; 31] dashpot for studying the seismic response of the base-isolated structures. For simplicity, the FID is modeled in the present study as a linear inerter (i.e., with an inertance coefficient of \(b\)) in parallel with a linear viscous FD (i.e., with a damping coefficient of \(c\)). It is worth noting that this FID modeling is identical to the EIMD modeling described in the References [35; 36]. The FID is defined by two parameters, \(\beta\) and \(\xi_{f}\), which are both defined as follows:  
\[\beta=\frac{b}{m}\text{ and }\xi_{f}=\frac{c_{f}}{2m\omega_{b}} \tag{10}\]  
Consider the rigid superstructure model of the base-isolated structure (refer to Figure 1a) with supplemental FID. The governing equation of motion of a base-isolated structure with FID under earthquake excitation is expressed as  
\[(m+b)\ddot{x}+\left(c_{b}+c_{f}\right)\dot{x}+k_{b}x=-m\ddot{x}_{g} \tag{11}\]  
Dividing Equation (11) by \((m+b)\) and substituting \(\beta=b/m\), the equation of motion reduces to  
\[\ddot{x}+2\left(\frac{\xi_{b}+\xi_{f}}{\sqrt{1+\beta}}\right)\left(\frac{ \omega_{b}}{\sqrt{1+\beta}}\right)\dot{x}+\left(\frac{\omega_{b}}{\sqrt{1+ \beta}}\right)^{2}x=-\left(\frac{1}{1+\beta}\right)\ddot{x}_{g} \tag{12}\]  
The above indicates that FID elongates the vibration period and decreases the total damping ratio (i.e., \(\xi_{b}+\xi_{f}\)) of the base-isolated structure. It also suppresses the level of earthquake shaking (as the denominator on the right-hand side is always larger than unity). Let the rigid superstructure model of the base-isolated structure with FID be subjected to white-noise earthquake acceleration with PSDF as \(S_{0}\). The mean square displacement of the isolated structure with FID will be as follows [45]:  
\[\sigma_{x}^{2}=\frac{\pi S_{0}}{2\left(\xi_{b}+\xi_{f}\right)\omega_{b}^{3}} \tag{13}\]  
The above equation indicates that for broadband excitation, the displacement variance of the base-isolated structure is not influenced by the inertance of the FID. The reduction in displacement is primarily happening due to the added damping of the FID. Since the inerter of the FID does not influence the response of the isolated structures under broad band-excitation, it is therefore proposed that the corresponding FD only be added parallel to the isolation system (refer to Figure 5b). The absolute acceleration, \(\ddot{x}_{a}\), of the isolated structural mass with FID, is expressed as  
\[\ddot{x}_{a}=\left[\ddot{x}+\ddot{x}_{g}\right]=-\left[\omega_{b}^{2}x+2\left( \xi_{b}+\xi_{f}\right)\omega_{b}\dot{x}+\beta\ddot{x}\right] \tag{14}\]  
Figure 5: Schematic diagram of supplemental FID and FD devices. Interestingly, two terms, i.e., \(2\xi_{f}\omega_{b}\dot{x}\) and \(\beta\ddot{x}\), are added to the absolute acceleration of the mass compared to the conventional isolation system, and their effects are interesting.