## 3 TMD Design Method
In this method, the building structure is modeled through a single-degree-of-freedom (DOF) system of mass \(M\) and stiffness \(K\), connected to the TMD mass, \(m\), by a linear elastic spring of stiffness \(k\) and a linear viscous damper with viscous coefficient \(c\), disposed in parallel, as shown in Figure 9. Therefore, the TMD behavior is completely described by the three parameters \(m\), \(k\) and \(c\). The dynamics of the two-DOF system in Figure 9 when it is subjected to acceleration \(a(t)\) at the base are described by Equations (1) and (2). \[Ma_{OB}(t)+c[v_{OB}(t)-v_{TMD}(t)]+Kd_{OB}(t)+k[d_{OB}(t)-d_{TMD} (t)]=-Ma(t) \tag{1}\] \[ma_{TMD}(t)+c[v_{TMD}(t)-v_{OB}(t)]+k[d_{TMD}(t)-d_{OB}(t)]=-ma(t) \tag{2}\]  
where \(a_{OB}\), \(a_{OB}\), \(v_{OB}\), \(v_{TMD}\), \(d_{OB}\) and \(d_{TMD}\) are the accelerations, velocities and displacements of OB and TMD, respectively, relative to the base. The design method is based on the minimization of the building displacements caused by a random acceleration time history acting at its base. This criterion allows the limitation of the stress demand on structural members and the avoidance of damage to non-structural elements. The design is carried out independently for motion along the \(X\) and \(Y\) directions. The TMD parameters obtained for each direction may be different due to the different properties (periods and participating mass ratios) of the main vibration modes along the main directions. The periods and the participating mass ratios of the main translational modes of the building considered in the case study are similar. Hence, the TMD parameters are calculated on the basis of the mode with the higher participating mass ratio. Otherwise, a TMD solution providing different stiffness and damping along the main directions of the building would have to be considered. The determination of the TMD parameters \(m\), \(k\) and \(c\) is divided into the following steps:  
Figure 9: Schematization of the structure of the TMD system. Definition of the mass ratio \(\mu\) between the TMD mass, \(m\), and the effective mass of the main vibration mode of the building without the TMD, \(M_{i}\). In general, the choice of \(m\) has to be made taking into account the bearing capacity of the structural members. In this case study, the structure can easily support, at ULS, an additional vertical load up to about 500% of the total mass of the building. In order to optimize the choice of \(m\), the design procedure proposed by [PERSON] [44] is used herein. In this procedure, the mass ratio is determined as  
Figure 8.0  
where \(\xi_{i}\) is the viscous damping factor of the i-th vibration mode of period \(T_{i}\) and the participating mass ratio \(\alpha_{i}\), and \(\xi_{eq}\) is the damping factor of the acceleration and displacement spectra, \(S_{A}\) and \(S_{D}\), respectively. \(\xi_{eq}\) is determined so that the following inequalities are satisfied:  
Figure 8.1 Figure 8.2  
In (4) and (5), \(a_{max}\) and \(d_{max}\) are the target maximum acceleration and displacement at the top of the building, respectively, that can be borne without damage by the structural members. The values of \(a_{max}\) and \(d_{max}\) are determined by scaling the amplitude of the accelerograms. 2. Calculation of the frequency ratio, \(f\), between the frequencies of the TMD and of the structure with the following equation  
Figure 8.3  
and then the TMD stiffness, \(k\),  
Figure 8.4  
where \(\Omega_{i}=\sqrt{\frac{K_{i}}{M_{i}}}\) is the frequency of the i-th vibration mode of the building. 3. Calculation of the damping factor, \(\xi\), and the viscous damping coefficient \(c\) of the TMD with the following equations:  
Figure 8.5 Figure 8.6  
where \(\omega=\sqrt{\frac{k}{m}}\) is the TMD's natural frequency. The values of \(k\) and \(\xi\) are then used in the design of isolators and dampers. After the definition of these parameters, the design of the TMD is completed by the implementation of the FEM of the building and the performance of its analysis.