## 5 MRD Activation Current and Evaluation of the Mitigated Seismic Response of the Structure  
As in the case of the oleopneumatic dampers, the energy that is dissipated by an MRD is due to the work that is performed by the forces that act on a piston that slides into a viscous fluid. These forces mainly depend on the piston speed, and the graphical representation of this dependence is generally referred to as the \"operative curve of the damper\". The substantial difference between oleopneumatic dampers and MRDs is not only in terms of the different trends of the operative curves but is also in terms of the fact that MRDs can operate along several curves. The operative curve of an oleopneumatic damper is unique and it depends on the device's geometry and the mechanical properties of the damping fluid. The same also applies to an MRD, but, in this case, the mechanical properties of the magnetorheological (MR) fluid can be modified by the magnetic field that is generated by an embedded electromagnet. Figure 22: LORD MRD-1005-3: main dimensions and sketch of the working principle. In more detail, during the motion of the piston, the MR fluid is forced to flow into tight gaps that are hosted by the piston's head (Figure 22). When the MRD is powered on, the electric current flows into a solenoid that is installed in the piston's head, and a magnetic field arises within the gaps. The ferromagnetic particles that are suspended in the MR fluid align themselves to the magnetic field and generate a system of microscopic structures (fibrils) that hamper the MR fluid's flow through the gaps [30]. At a macroscopic level, this increases the force that is exhibited by the damper, which depends on the intensity of the magnetic field and, consequently, on the intensity of the current into the solenoid. Therefore, different activation currents are associated with different working modes of the MRD and with different operative curves. The operative curves of the selected MRD are reported in Figure 23[31] and they correspond with three relevant intensities of the activation current: 0 A (no activation), 1 A (maximum intensity), and 0.5 A (medium intensity). The seismic response of the structure was evaluated in conjunction with different activation levels of the MRDs, starting from 0 A, and increasing by 0.25 A until the mitigation target of the VSS was reached (Figure 24). At each step, the activation current was equal for all the MRDs in order to preserve the structural symmetry with respect to the Y-Z plane. Each MRD was treated as a linear damper that was characterized by a specific (equivalent) damping coefficient. Under this assumption, the differential equation that governs the seismic response of the system is linear and takes the form (see Appendix A):  
\[\underline{\underline{M}}\,\underline{\Delta\dot{d}}+\underline{D}\,\underline {\Delta d}+\underline{K}\,\underline{\Delta d}=-\underline{\underline{M}}\, \underline{e}a(t), \tag{4}\]  
where:  
\(\underline{\underline{M}}\) is the mass matrix of the structure;  
\(\underline{\underline{K}}\) is the stiffness matrix of the structure that is constrained at its base;  
\(\underline{\underline{D}}\) is the damping matrix of the structure, the elements of which are related to the equivalent damping coefficients that are assumed for the MRDs;  
\(\underline{\Delta d}\) is the displacement vector of the nodes of the structure with respect to the base of the structure;  
\(a(t)\) is the seismic acceleration at the base of the structure;  
Figure 23: LORD MRD-1005-3, relevant operative curves. \(\underline{\varepsilon}\) is the vector that defines the direction of the seismic acceleration into the same reference frame that is used for the definition of \(\underline{\underline{M}}\), \(\underline{\underline{K}}\), \(\underline{\underline{D}}\), and \(\underline{\underline{\Delta}d}\).