# Seismic Response Mitigation of Base-Isolated Buildings
## 2 Structural Model  
Schematic representation of different structures considered here is presented in Figure 1. An idealized \(N\)-story BI building is demonstrated in Figure 1a. The masses are lumped at the floor levels, and the floors are deemed to act as a rigid diaphragm. One horizontal translational degree of freedom (DOF) is assigned to each floor. BI structures with different control schemes are shown in Figure 1a-d. In Figure 1b, a TMD is placed at the \(n\)th floor, and this scheme is called as BI + STMD. In Figure 1c, MTMDs are placed at the \(n\)th floor, and the scheme is called as BI + MTMDs. When the multiple TMDs are placed on various floors, as displayed schematically in Figure 1d, the scheme is called BI + d-MTMDs. For the sake of simplicity, the superstructure is presumed to remain linearly elastic and soil structure interaction (SSI) is not considered.  
Figure 1: Cont.  
The governing equation of motion of the system under ground shaking can be written as  
\[[M]\Big{[}\ddot{x}(t)\Big{]}+[C]\Big{\{}\dot{x}(t)\Big{\}}+[K][x(t) ]=-[M][t]\Big{\{}\ddot{x}_{g}\Big{\}} \tag{1}\]  
where \([M]\) is the mass matrix; \([C]\) is the damping matrix and \([K]\) is the stiffness matrix of the structure; \(\{x\}=\{X_{1},X_{2},\cdots X_{N},X_{b},\cdots x_{T1},x_{T2},\cdots X_{Tn}\}^{T}\), \(\dot{x}\), and \(\ddot{x}\) are the unknown relative (floor, isolator, and TMD) displacement, velocity, and acceleration vectors, respectively; \(\Big{[}\ddot{x}_{g}\Big{]}\) is earthquake ground acceleration; and \(\{r\}\) is the vector of influence coefficients. Following [PERSON] et al. [28], the \([M]\), \([C]\), and \([K]\) matrices of the BI buildings installed with TMDs can be written as  
\[[M]=\left[\begin{array}{ccc}[m_{b}]_{1\times 1}&[M_{N}]_{1\times N}&[m_{n}]_{1 \times n}\\ [M_{N}]_{N\times 1}&[M_{N}]_{N\times N}&[0]_{N\times n}\\ [m_{n}]_{n\times 1}&[0]_{n\times N}&[m_{n}]_{n\times n}\end{array}\right] \tag{2}\]  
Figure 1: Schematic models of a \(N\)-story building with (**a**) base-isolation (BI), (**b**) BI + single tuned mass dampers (STMD) at top floor, (**c**) BI + multiple tuned mass dampers (MTMDs) at top floor, and (**d**) BI + distributed multiple tuned mass dampers (d-MTMDs).  
\[[C]=\left[\begin{array}{ccc}[c_{b}]_{1\times 1}&[0]_{1\times N}&[0]_{1\times n}\\ \left[0\right]_{N\times 1}&[C_{N}]_{N\times N}+[c_{n}]_{N\times N}&-[c_{n}]_{N \times n}\\ \left[0\right]_{n\times 1}&-[c_{n}]_{n\times N}&[c_{n}]_{n\times n}\end{array}\right] \tag{3}\]  
\[[K]=\left[\begin{array}{ccc}[k_{b}]_{1\times 1}&[0]_{1\times N}&[0]_{1\times n} \\ \left[0\right]_{N\times 1}&[K_{N}]_{N\times N}+[k_{n}]_{N\times N}&-[k_{n}]_{N \times n}\\ \left[0\right]_{n\times 1}&-[k_{n}]_{n\times N}&[k_{n}]_{n\times n}\end{array}\right] \tag{4}\]  
where \([m_{\rm b}]\) is isolator mass matrix, \([M_{\rm N}]\) is superstructure mass matrix, and \([m_{\rm n}]\) is the TMDs schemes mass matrix. \([c_{\rm b}]\), \([C_{\rm N}]\), and \([c_{n}]\) are the corresponding damping matrices. Similarly, \([k_{\rm b}]\) is isolator stiffness matrix, \([K_{\rm N}]\) is superstructure stiffness matrix, and \([k_{\rm n}]\) is the TMDs schemes stiffness matrix. The \(N\) story building is isolated by one DOF isolator and equipped by \(n\) number of TMDs, that results in the matrices of the order \((N+n+1)\times(N+n+1)\).