## 5 MRD Activation Current and Evaluation of the Mitigated Seismic Response of the Structure
Equation (4) was solved by referring to the same methodology that was adopted for the evaluation of the seismic response of the structure without dampers: by relying upon the formulation reported in Appendix A, the response was first evaluated in the generalized coordinates domain, and then in terms of the acceleration and speed of the nodes with respect to the base of the structure. It is worth noting that, in the generalized coordinates domain, the equation that was used to evaluate the seismic response of the structure that was equipped with dampers is formally the same as that used in the case where no dampers were installed. The only difference is in the elements of the generalized damping matrix. In the case of dampers not being installed on the structure, the generalized damping matrix was built coherently to a modal damping equal to 0.015 for all the relevant modes involved in the seismic response. In the presence of dampers, this matrix is instead given by: \(\underline{\Phi}_{V}^{T}\underline{\underline{D}}\underline{\underline{\Phi}} _{V}\), where \(\underline{\underline{\Phi}}_{V}\), is the modal matrix. The damping matrix (\(\underline{\underline{D}}\)) was assembled on the basis of the equivalent damping that was exhibited by each MRD, and was calculated by means of the following iterative method (Figure 25):  
Step 1: The seismic response of the structure is evaluated in the case of no dampers installed (clean structure). For the generic MRD to be placed on the \(i\)-th bracing that connects the nodes (\(h\) and \(k\)), the equivalent damping (\(D_{\alpha i,l}\)) was found according to Equation (3) and considering:  
\(\dot{\delta}(t)=\Delta V_{i}(t)=V_{h}(t)-V_{k}(t)\), where \(V_{h}(t)\) and \(V_{k}(t)\) are the speed responses of the nodes, \(h\) and \(k\), respectively, along the \(h\)-\(k\) direction; the function (\(f\)) is coincident with the operative curve of the damper that corresponds to the selected activation current; T is equal to the duration of the seismic event. Step 2: The seismic response of the structure is evaluated in the case of an MRD that is installed and that exhibits the values of the equivalent damping that was found in the previous step. Step 3: The equivalent damping of the MRDs is recalculated on the basis of the speed responses (\(\Delta V_{i}(t)\)) that were evaluated in the previous step. Figure 24: Calculation loop for the individuation of the activation current compliant with the vibration-mitigation target (defined in Section 3). For each MRD, the new value of the equivalent damping, \(D^{\prime}_{eq,i}\), is compared to that assumed at the previous step by evaluating the error margin, \(\mu=\frac{D^{\prime}_{eq,i}-D_{eq,i}}{D_{eq,i}}\). If m is below a predefined threshold (\(t\)), the process stops, as well as the mitigated seismic response of the structure that is evaluated in the current step. If not, Steps 2 and 3 are repeated until m < t for each MRD. The above-described iteration was executed at the second step of the loop in Figure 24 in order to evaluate the mitigated seismic response of the structure and the equivalent damping of the MRDs that corresponded to the input value of the activation current. The mitigation target for the seismic-induced vibrations was reached in the case of all MRDs activated with a current of 0.5 A; the convergence on the equivalent damping was reached after one iteration only, with a precision threshold of t = 0.001. The values of the equivalent damping obtained at each step are reported in Table 4. The mitigated seismic response of the structure is plotted in Figures 26-40, for the time histories of the floor accelerations with respect to the base, the interstory accelerations, and the speeds. The unmitigated seismic response is also plotted for comparison. Figure 25: Iterative method for the calculation of MRDs’ equivalent damping. Figure 27: Seismic response, 2 nd-floor acceleration (Node 46) with respect to the ground.