# Seismic Benefits from Coupling Frame Structures with a Hysteretic Mass Damper Inerter
## 4 Spectral Analysis
### Modal Mass Participating Coefficients of the Coupled System  
The values of the mass participating coefficients (MPCs) of the three modes of the structure with the HMDI provide insights into the dynamic effects of the coupling [40,41]. The modal participation factor is defined as \(\widehat{\Gamma}_{i}=\mathbf{\Phi}_{i}\mathbf{M}\mathbf{r}\), where \(\mathbf{\Phi}_{i}\) is the vector of the \(i\)-th mode \((i=1,2,3)\), \(\mathbf{M}\) is the mass matrix (see Equation (15)), and \(\mathbf{r}\) is, in this case,  
Figure 4: Spectral analysis of the uncoupled and coupled structures: (**a**) Periods and shapes of the modes of the uncoupled structure; (**b**) shapes of the modes of the coupled structure; (**c**) periods of the three modes of the coupled structure (\(\gamma=0.1\)). a unitary vector. The MPC of the \(i\)-th mode is the percentage of mass indicating the participation of the \(i\)-th mode to the total response and, in the general case, is defined as \(\text{MPC}_{i}=(\hat{\Gamma}_{i}^{2}M_{i}/\mathbf{r}^{T}\mathbf{M}\mathbf{r}) \times 100\). In the case under investigation, the normalization of the modes is performed assuming \(\mathbf{\Phi}_{i}\mathbf{M}\mathbf{\Phi}_{i}=1\) and \(\mathbf{\Phi}_{i}\mathbf{K}\mathbf{\Phi}_{i}=\omega_{i}\)2, and the values of the MPCs become \(\text{MPC}_{i}=(\hat{\Gamma}_{i}^{2}/\mathbf{r}^{T}\mathbf{M}\mathbf{r}) \times 100\) because \(M_{i}=1\). Footnote 2: The \(i\)-th mode is the percentage of mass indicating the participation of the \(i\)-th mode to the total response and, in the general case, is defined as \(\text{MPC}_{i}=(\hat{\Gamma}_{i}^{2}M_{i}/\mathbf{r}^{T}\mathbf{M}\mathbf{r}) \times 100\). In the case under investigation, the normalization of the modes is performed assuming \(\mathbf{\Phi}_{i}\mathbf{M}\mathbf{\Phi}_{i}=1\) and \(\mathbf{\Phi}_{i}\mathbf{K}\mathbf{\Phi}_{i}=\omega_{i}\)2, and the values of the MPCs become \(\text{MPC}_{i}=(\hat{\Gamma}_{i}^{2}/\mathbf{r}^{T}\mathbf{M}\mathbf{r}) \times 100\) because \(M_{i}=1\). Figure 5 shows the contour maps of the values of the MPCs in the \(\
u\)-\(T_{ext}\) parameter plane. The threshold value \(\hat{T}_{ext}=0.382\) s that corresponds to a change in the shape of Mode 2 is included in the contour maps of the MPCs of both Mode 1 and Mode 2 (left and middle graphs). It can be observed that the value of \(T_{ext}\) also corresponds to the minimum values of the MPCs of Mode 1 and Mode 2. Analyzing the values in Figure 5, the following remarks can be made:  
* The MPC of Mode 1 is higher than 90% (the dotted line refers exactly to 90%) in almost all of the parameter planes so that Mode 1 is largely the predominant oscillation mode. The dynamics associated with this mode do not reduce the displacements of the frame structure since the HMDI and the frame structure move in phase. However, the HMDI also has the effect of increasing the period of such a mode. In the case of seismic excitation, the increase of the prevailing period has a protective effect on the structures because earthquakes have lower spectral energy in the range of higher periods. The shifting of the period of the prevailing mode toward higher values is the same protection concept at the base of base isolation. * The MPC of Mode 2 is less than 1% (the dotted line refers to 1%) in almost all the parameter planes. Therefore, although the introduction of the HMDI has the effect of substituting Mode 1 of the uncoupled structure with Mode 2 (i.e., the two modes before and after the coupling with the protection device have close values to the period), it also reduces the participation mass of this mode. Consequently, it reduces the influence that this mode would have under seismic excitation, considering that the period of Mode 2 is usually in the range of values where earthquakes have higher spectral energy. The contour map of the MPC of Mode 2 also shows the shape of the mode obtained at the threshold value \(\hat{T}_{ext}\). The shape indicates that there is no displacement at the first level of the 2-DOF system.