## 3 Soil-Dependent Design of Vibration Control System
### Constrained Optimization for the Selection of VCS Parameters  
The fixed parameters of the VCS are the additional mass \(m_{D}\) and the stability factors \(\varepsilon_{NS}\), \(\varepsilon_{PS}\), and \(\varepsilon_{R}\). The independent design variables sought in the optimization problem are: (1) the nominal VCS frequency \(f_{0}\), (2) the value of the NS element \(k_{NS}\), and (3, 4) the value of the artificial dampers \(c_{NS}\) and \(c_{PS}\). The optimization problem is formed for the mean of the maximum dynamic responses for the 10 artificial records of the database of the respective soil type. The objective function and the geometrical and manufacturing constraints are selected from the geometrically non-linear time domain responses, as described below:  
1. Assign values to the fixed parameters. According to previous work of [PERSON] [31], an additional mass of 5% is efficient. In this paper, in an effort to drastically reduce the additional mass, \(m_{D}\) is selected as 0.1%. The stability factors of the positive stiffness elements \(k_{PS}\), \(k_{R}\) and of the NS element \(k_{NS}\) are selected as 5%;
2. Selection of soil type and thus the respective artificial accelerogram database;
3. Set the objective function (OF) as the minimization of the mean of the maximum relative to the ground base displacements of the structure; \[\min:mean[\max[abs(u_{B})]]\qquad(\text{OF})\] (21)
4. Set an acceleration filter (AF) as a constraint for the structure absolute acceleration, expressed as a percentage of the mean PGA of the artificial accelerograms of the database of the selected soil type; \[mean[\max[abs(a_{i})]]\leq AF\times mean(PGA)\qquad(i=1,\dots,10)\qquad(\text{ Constraint 1})\] (22)
5. The NS element value and stroke are set as constraints, indicated from the proposed design of the NS configuration presented in [29] (50% lower compared to [28,35]); \[|k_{NS}|\leq 50\ kN/m/tn\qquad(\text{Constraint 2})\] (23)
6. Set an upper limit for \(c_{NS}\), \(c_{PS}\) with respect to the superstructure mass. This constraint is based on previous works [29] as well as on manufacturing restrictions; \[c_{NS}\leq 1000\ kNs/m\qquad(\text{Constraint 4})\] (25)  
\[c_{PS}\leq 1000\ kNs/m\qquad(\text{Constraint 5}) \tag{26}\]  
Figure 3: Mean acceleration response spectrum of all the artificial accelerograms of the database for: (**a**) firm soil (ground type A); (**b**) medium soil (ground type C); (**c**) soft soil (ground type D).  
The nominal SBA frequency \(f_{0}\) varies in the range [0.15 1.5] (Hz) (Constraint 6);  
The VCS design process is schematically represented with a flowchart in Figure 4. For the optimization process, the harmony search algorithm (HS) [36] is adopted. After the optimal system parameters are obtained, the proposed NS mechanism is designed, and consequently, nonlinear dynamic analysis is performed with nonlinear NS, in order to verify the effectiveness and realistic design of the proposed VCS. The conversion criteria mentioned in the outer loop of the flowchart presented in Figure 4 are:  
* \(mean[\max[abs(a_{i})]]\leq AF\times mean(PGA)\) (i = 1,..., 10) (Considering nonlinear NS)
* \(mean[\max[abs(a_{B}-u_{D})]]\leq 0.10\) m (Considering nonlinear NS)  
The first one ensures the effectiveness of the VCS in mitigating the seismic effects on the superstructure at the desired level (AF). The second one safeguards the realistic design of the proposed NS mechanism with conventional structural elements.