## 4 Numerical Results and Discussion  
The examined original structure (OS) is a three-story residential RC building (Figure 5a). The elastic modulus of reinforced concrete (assuming long-term cracked conditions) is equal to \(E\) = 26 GPa. The concentrated mass of each floor is \(mi\) = 80 tn. Rayleigh damping is assumed with \(\zeta i\) = 2%. A 15 \(\times\) 15 m floor plan-view (Figure 5b) has been considered with 0.3 \(\times\) 0.3 m square columns. The resulting natural periods of the OS are \(Ti\) [sec] = [0.495, 0.177, 0.122]. The mass of the base level and the mat foundation are equal to \(m_{B}\) = 60 tn and \(m_{F}\) = 80 tn, resulting in a total superstructure mass of \(m_{total}\) = 300 tn. The VCS mass is selected as 0.1% of the \(m_{total}\) and is \(m_{D}\) = 300 kg. The foundation system is designed based on the properties and approximate loading of a typical residential building. On the basis of empirical correlations, the initial, small  
Figure 4: Flowchart of the VCS design implemented in the bases of structures as a supplement to conventional base isolation approaches (BI). strain, elastic modulus (\(E_{0}\)) of the soil was considered equal to 1800 \(S_{u}\). Based on the provisions of the EC8, the undrained shear strength \(S_{u}\) varies with respect to the ground type. For ground type A (firm soil), the underlying soil is a rock-like formation and can be thus considered as a fixed base. For ground type C (medium soil), \(S_{u}\) varies in the range [70 250] kPa, and for the analysis, is selected as the average, \(S_{u}=160\) kPa. For ground type D (soft soil), \(S_{u}\) is equal or less than 70 kPa, and this value is adopted for this paper. The soil stratum is considered to be isotropic/homogeneous and the same is applied for the mat foundation which is, however, considered to be elastic (Reinforced Concrete). The boundaries of the model have been strategically selected sufficiently far to avoid spurious boundary-effects on the system response. Static pushover analyses are subsequently undertaken in order to derive the non-linear soil-foundation stiffness relationships (\(k_{SSI}\)). The mean PGAs of the artificial accelerograms of the database are 0.421, 0.545, and 0.614 (g) for the firm, medium, and soft soil, respectively. The acceleration filter (\(AF\)) is set to 80% of the respective mean PGA of the database, and the optimal VCS parameters are obtained following the procedure described in Section 3.2 and are presented in Table 1.  
In order to confirm the effectiveness of the proposed VCS, a comparison is made with a conventionally base isolated structure (BI) with a base nominal frequency of \(f_{B}=0.4\) Hz (\(T_{B}=2.5\) s) and a BI damping ratio of \(\zeta_{B}=5\%\). The system main dynamic responses, considering the max values of the dynamic responses for all the Artificial Accelerograms in the database, mean of 10 maxes for each soil category, of the original structure (OS), the conventionally base isolated one (BI), and the controlled structure with the proposed vibration control system (VCS) are collected and presented in the bar plots of Figures 6 and 7. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Soil** & \(f_{0}\) & \(m_{D}\) & \(k_{\text{NS}}\) & \(k_{PS}\) & \(k_{R}\) & \(c_{\text{NS}}\) & \(c_{PS}\) \\
**Type** & **(Hz)** & **(tn)** & **(txN/m)** & **(txN/m)** & **(txN/m)** & **(txN/m)** & **(txNs/m)** & **(txNs/m)** \\ \hline
**firm** & 0.838 & 0.3 & \(-14\),914 & 27,799 & 40,499 & 997 & 914 \\
**medium** & 0.548 & 0.3 & \(-12\),681 & 34,450 & 23,630 & 982 & 840 \\
**soft** & 0.527 & 0.3 & \(-13\),455 & 41,238 & 23,260 & 892.8 & 388.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimum VCS parameters for each soil type. Figure 5: Typical 3-story residential RC building examined in this paper: (**a**) sketch; (**b**) floor plan. Figures 6 and 7 present the structural top floor acceleration and first floor drift expressed as a percentage of the OS values.