## 2 Nonlinear Dynamic Modeling of Controlled Building Structure
### VCS Dynamic Properties and Nonlinear Behavior  
The fundamental advantage of the VCS is that in order to be effective, it does not require significant reduction in the fundamental frequency of the system, as in the case of BI systems, due to the increase of the effective damping that the KDamper configuration offers. Thus, the nominal system frequency\(f_{0}\) is a freely selectable parameter, the value of which can be selected with respect to the structural system to be controlled in its initial undeformed (static) state. \[(2\pi f_{0})^{2}\pi_{total}=k_{0}=k_{R}+\frac{k_{PS}k_{NS}(u_{NS}=0)}{k_{PS}+k_ {NS}(u_{NS}=0)} \tag{6a}\] \[k_{R}=\frac{k_{SSI}(u_{F}=0)k_{BI}}{k_{SSI}(u_{F}=0)+k_{BI}} \tag{6b}\]Although theoretically the values of \(k_{NS}\), \(k_{PS}\), and \(k_{R}\) are selected according to Equation (6a) to ensure the static stability, due to various reasons, such as temperature variations, manufacturing tolerances, or non-linear behavior, \(k_{NS}\), \(k_{PS}\), and \(k_{R}\) may present variations in practice, since almost all NS designs result from unstable nonlinear systems. An increase of the absolute value of \(k_{NS}\) and/or a decrease of the values of the positive stiffness elements \(k_{PS}\) and \(k_{R}\) by a factor \(\varepsilon_{NS}\), \(\varepsilon_{PS}\), and \(\varepsilon_{R}\), respectively, may result in the system being unstable. This happens when the determinant of the stiffness matrix (structure-VCS) equals zero:  
\[\text{det}\textbf{K}=|\textbf{K}|=0\Rightarrow(1-\varepsilon_{R})\times k_{ R}+\frac{(1-\varepsilon_{PS})\times k_{PS}\times(1+\varepsilon_{NS})\times k_{ NS}(u_{NS}=0)}{(1-\varepsilon_{PS})\times k_{PS}+(1+\varepsilon_{NS}) \times k_{NS}(u_{NS}=0)}=0 \tag{7}\]  
Contrary to the KDamper concept, which foresees variation only in the NS element \(k_{NS}\), in the proposed VCS, a variation in all stiffness elements is taken into account. Assuming that the nominal VCS frequency \(f_{0}\), and the value of the negative stiffness element \(k_{NS}\) are known, using Equations (6a) and (7), the rest of the stiffness elements result as:  
\[\frac{k_{R}}{k_{0}}=\frac{-b-\sqrt{b^{2}-4\times a\times c}}{2a} \tag{8}\]  
\[\frac{k_{PS}}{k_{0}}=\frac{(k_{NS}(u_{NS}=0)/k_{0})-(k_{R}/k_{0})(k_{NS}(u_{NS }=0)/k_{0})}{(k_{R}/k_{0})+(k_{NS}(u_{NS}=0)/k_{0})-1} \tag{9}\]  
where parameters \(a\), \(b\), and \(c\) are defined as:  
\[a=(1-\varepsilon_{R})[(1-\varepsilon_{PS})-(1+\varepsilon_{NS})] \tag{10}\]  
\[b=\frac{k_{NS}(u_{NS}=0)}{k_{0}}(1+\varepsilon_{NS})[(1-\varepsilon_{P})-(1 -\varepsilon_{R})]+(1-\varepsilon_{R})[(1+\varepsilon_{NS})-(1-\varepsilon_{ P})] \tag{11}\]  
\[c=-(1-\varepsilon_{P})(1+\varepsilon_{N})\frac{k_{NS}(u_{NS}=0)}{k_{0}} \tag{12}\]  
The NS element of the VCS configuration is realized with a displacement-dependent mechanism, proposed in [29] as illustrated in Figure 2. A vertical conventional positive stiffness element (coil spring) is attached to the additional mass \(m_{D}\), and is connected to the base floor with an articulated mechanism. The generated \(NS\) is calculated according to the following procedure. Initially, the potential energy due to the deformation of the vertical spring is given, and consequently, the value of the elastic nonlinear force and the equivalent generated \(NS\) are calculated:  
\[U_{NS}(u_{NS})=\frac{1}{2}k_{V}(l_{V}(u_{NS})-l_{VI})^{2} \tag{13}\]  
\[k_{R}=\frac{k_{SSI}(u_{F}=0)k_{BI}}{k_{SSI}(u_{F}=0)+k_{BI}} \tag{6b}\]0 Proposed mechanism for the realization of the NS element.