## 3 Seismic Control Effectiveness of the TID for Base-Isolated Structures
### Optimal Design of Base-Isolated Structures Equipped with a TID  
Figure 9 illustrates the modeling of a shear model with \(N\) degrees of freedom and the installation of the TID in the multi-storey base-isolated structure.  
The dynamic equilibrium equation of the structure equipped with a TID under \(\ddot{x}_{g}\) is given by Equation (16):  
\[\left\{\begin{array}{l}\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X} }+\mathbf{K}\mathbf{X}+c_{d}\big{(}\dot{x}_{1}-\dot{x}_{d}\big{)}\mathbf{D}+k_{ d}(x_{1}-x_{d})\mathbf{D}=-\mathbf{M}\mathbf{I}_{N}\ddot{x}_{g}\\ m_{d}\ddot{x}_{d}+c_{d}\big{(}\dot{x}_{d}-\dot{x}_{1}\big{)}+k_{d}(x_{d}-x_{1}) =0\end{array}\right. \tag{16}\]  
In Equation (16), M, C, and K represent the mass, damping, and stiffness matrices of the base-isolated structure, respectively. Additionally, \(\mathbf{X}\), \(\dot{\mathbf{X}}\), and \(\dot{\mathbf{X}}\) represent the displacement, velocity, and acceleration vectors of the base-isolated structure, all referenced to the ground. \(\mathbf{I}_{N}\) is a unit vector, while \(x_{1}\) and \(\dot{x}_{1}\) refer to the displacement and acceleration of the first storey of the structure. Finally, \(\mathbf{D}=\left[1,0\cdots,0,0,0,\cdots,0\right]^{T}\) is the N-dimensional position vector of the TID.  
Figure 8: The relationship between displacement reduction ratio \(J\) and the mass ratio.  
Figure 9: Multi-storey building structures equipped with a tuned inerter damper.  
For base-isolated structures, the first-order vibration mode typically has the greatest influence on the seismic response. Therefore, this vibration mode is often selected as the primary control mode. Let \(M_{1}\), \(C_{1}\), and \(K_{1}\) represent the modal mass, modal damping, and modal stiffness associated with the first-order vibration mode, respectively. The generalized coordinates for the first-order vibration mode are denoted by \(q\), while \(\Phi_{1}\) represents the first mode shape vector, and \(\phi_{11}\) is the first element of this vector. For convenience, the mode shape vector is normalized such that its first element, corresponding to the base story, is set to one, i.e., \(\phi_{11}\). By performing mode decomposition of Equation (16), Equation (17) can be derived as follows:  
\[\left\{\begin{array}{l}M_{1}\tilde{q}_{1}+C_{1}\dot{q}_{1}+K_{1}q_{1}+c_{d} \big{(}\hat{q}_{1}-\dot{x}_{d}\big{)}+k_{d}(q_{1}-x_{d})=-\mathbf{\Phi}_{1}\bar{T} \mathbf{M}_{N}\ddot{x}_{g}\\ m_{d}\ddot{x}_{d}+c_{d}(\dot{x}_{d}-\dot{q}_{1})+k_{d}(x_{d}-q_{1})=0\end{array}\right. \tag{17}\]  
It can be observed that Equations (1) and (15) have the same form, so the above optimization results can be directly extended to the multi-storey base-isolated structure. Similarly, the system's mechanical parameters are represented by Equation (18) as follows:  
\[\begin{array}{l}\eta_{1}=\mathbf{\Phi}_{1}\bar{T}\mathbf{M}_{N}/M_{1},\ \omega_{1}=\sqrt{K_{1}/M_{1}},\ \tilde{\xi}_{1}=C_{1}/2M_{1}\omega_{1},\\ \omega_{d}=\sqrt{k_{d}/m_{d}},\ \tilde{\xi}_{d}=c_{d}/2m_{d}\omega_{d},\ \mu=m_{d}/M_{1}\end{array} \tag{18}\]  
in which \(\eta_{1}\), \(\omega_{1}\), and \(\tilde{\xi}_{1}\) are the first mode participation factor, structural modal frequency, and structural mode damping ratio, respectively; \(\omega_{d}\), \(\tilde{\xi}_{d}\), and \(\mu\) are still the frequency, damping ratio and mass ratio of the TID, respectively.