## 2 Nonlinear Dynamic Modeling of Controlled Building Structure
### Superstructure Modeling and SSI Effects  
The modeling of the multi-story residential concrete building structure is based on the following assumptions:  
* The effect of soil-structure-interaction (SSI) is taken into consideration with the use of nonlinear elastic springs;
* The slabs and grinders on the floors are rigid as compared to the columns;
* The concrete columns of the superstructure are inextensible and provide the lateral stiffness of the structure;
* The total superstructure mass is concentrated at the floor levels as the sum of the floor masses and half of the columns mass at either side of the slab;
* The superstructure is considered to remain within the elastic limit during the dynamic analysis. An \(N\)-story building structure is considered with \(N\) dynamic DoFs, that correspond to the relative to the ground floor displacements, and are collected in the array \(\mathbf{u}_{\text{str}}\). The equations of motion of the controlled structure with the proposed VCS are expressed in a matrix form as follows:  
\[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}(\dot{u})\dot{\mathbf{u}}+\mathbf{K}(u) \mathbf{u}=-\mathbf{M}\ddot{\mathbf{X}}g \tag{1}\]  
Figure 1: Implementation of an extended KDamper device as a supplement to a conventional base isolation system. The matrices and vectors entering Equation (1) are formed as:  
\[\mathbf{M}_{(N+3)\times(N+3)}=\left[ \begin{array}{cccc}\mathbf{M}_{N}&(N\times N )&\mathbf{0}_{N\times 1}&\mathbf{0}_{N\times 1}&\mathbf{0}_{N\times 1}\\ \mathbf{0}_{1\times N}&m_{B}&0&0\\ \mathbf{0}_{1\times N}&0&m_{D}&0\\ \mathbf{0}_{1\times N}&0&0&m_{F}\end{array} \right] \tag{2}\]  
\[\mathbf{K}(u)_{(N+3)\times(N+3)}=\left[ \begin{array}{cccc}\mathbf{K}_{N}& \mathbf{0}_{(N-1)\times 1}&\mathbf{0}_{(N-1)\times 1}&\mathbf{0}_{(N-1)\times 1}\\ \mathbf{0}_{1\times(N-1)}&-k_{1}&0&0\\ \mathbf{0}_{1\times(N-1)}&0&-k_{NS}(u_{NS})&k_{NS}(u_{NS})+k_{PS}&0\\ \mathbf{0}_{1\times(N-1)}&0&-k_{BI}&0&k_{BI}+k_{SSI}(u_{F})\end{array} \right] \tag{3}\]  
\[\mathbf{C}(\dot{u})_{(N+3)\times(N+3)}=\left[ \begin{array}{cccc}\mathbf{C} _{N}&(N\times N)&\mathbf{0}_{(N)\times 1}&\mathbf{0}_{(N)\times 1}&\mathbf{0}_{(N) \times 1}\\ \mathbf{0}_{1\times(N)}&c_{NS}+c_{BI}&c_{NS}&-c_{BI}\\ \mathbf{0}_{1\times(N)}&c_{NS}&c_{NS}+c_{PS}&0\\ \mathbf{0}_{1\times(N)}&-c_{BI}&0&c_{BI}+c_{SSI}(\dot{u}_{F})\end{array} \right] \tag{4}\]  
\[\mathbf{u}_{(N+3)\times 1}=\left[ \begin{array}{cccc}\mathbf{u}_{str}&(N\times 1 )&u_{B}&u_{VCS}&u_{F}\end{array} \right]^{T} \tag{5}\]  
where \(\mathbf{M}_{N}\), \(\mathbf{K}_{N}\), and \(\mathbf{C}_{N}\) are the superstructure mass, stiffness and damping matrices, assuming it is mounted on a fixed base, \(k_{1}\) is the lateral stiffness of the first floor, and \(\mathbf{I}\) is the \((\mathbf{N}+3)\times 1\) influence vector associated with the ground motion \(Xg\). The nonlinear NS spring \(k_{\mathrm{NS}}\) depends on the value of the NS stroke \(u_{NS}\) (relative displacement between the terminals of the negative stiffness element). The stiffness of the foundation system (\(k_{SSI}\)) that is derived due to the SSI effects depends on the foundation relative displacement \(u_{F}\). The controlled structure is mounted on a mat foundation, as in the case of a conventional BI structure. The effect of the SSI is taken into consideration with the use of nonlinear spring stiffnesses (\(k_{SSI}\)). The \(k_{SSI}\) is calculated assuming a rigid beam subjected to horizontal displacement at the top. The height of this element is measured from the bottom of the mat foundation to the bottom end of the implemented bearing (Figure 1). As a consequence, the calculated stiffness (\(k_{SSI}\)), is a resultant of both the rotational and horizontal displacement of the soil footing and is expressed at the top of the rigid beam element. The soil stratum is considered to be isotropic/homogeneous. The interface between foundation-soil is considered to be 'fully bonded' with the foundation elements considered attached to the ground.