# Seismic Benefits from Coupling Frame Structures with a Hysteretic Mass Damper Inerter
## 4 Spectral Analysis
### Frequencies and Modal Shapes of the Coupled System  
This section presents the analyses related to the four-story frame described in the second line of Table 1. The corresponding mechanical characteristics of the 2-DOF equivalent system are those in the second line of Table 2. Figure 4 shows how the periods and shapes of the oscillation modes of the frame structure change when it is coupled with the HMDI. Specifically, Figure 4a shows the periods and shapes of the two oscillating modes of the uncoupled structure, whereas Figure 4b shows the sketches of the three modes of the coupled system (provided by the 3-DOF equivalent system). For the coupled system, both periods and modal shapes are strictly related to the parameters of the mechanical system varied in the analyses. However, the modal shapes are less sensitive to such parameters. Specifically, the modal shapes of Mode 1 and Mode 3 remain qualitatively similar for all values of the system's parameters. On the contrary, the qualitative shape of Mode 2 changes, as explained below in detail.  
The spectral analysis presented in this section aims to clarify the dependence of the dynamics of the coupled system on the characteristics of the frame structure and protection device. Figure 4c shows the values of the periods of the three modes as contour plots in the parameter plane \(v\)-\(T_{ext}\). It is worth remarking that the choice of \(\
u\) (see Equation (7)) and \(T_{ext}\) (see Equation (10)) entails a combined variation of the linear elastic coupling stiffness \(k_{c}\) and virtual mass \(m_{I}\). As can be observed by the contour levels of the period of Mode 1, the values of such a period strongly depend on \(\
u\) (i.e., on the virtual mass \(m_{I}\)), since the contour levels are almost orthogonal to the \(\
u\)-axis. Qualitatively, the period of Mode 1 increases when \(\
u\) increases. The contour plots of the periods of Mode 2 and Mode 3 show that such periods are not significantly dependent on \(\
u\) as that of Mode 1, especially for high values of \(\
u\). The main results can be summarized as follows:  
* The period of Mode 1 can be increased using an appropriate virtual mass of the HMDI (i.e., increasing \(\
u\)) so that it is possible to reach values significantly larger than those of the corresponding Mode 1 of the uncoupled frame structure.
* The period of Mode 2 of the frame coupled with the HMDI is consistently close to the period of Mode 1 of the uncoupled structure. It can be asserted that Mode 2 of the structure with the HMDI substitutes Mode 1 of the structure without the HMDI. Depending on the values of \(T_{ext}\) (i.e., depending on the characteristics of the HMDI), the shape of Mode 2 changes. In one case (Mode 2a), the HMDI moves in counter-phase with respect to the whole frame. In the other case (Mode 2b), the HMDI moves in phase with the first level of the frame structure and in counter-phase with the part of the structure above the connection with the HMDI. The threshold value that separates the regions of the parameter plane characterized by the two different modal shapes corresponds to a specific value \(T_{ext}=\hat{T}_{ext}\) that exclusively depends on the characteristics of the uncoupled structure. In Figure 4c, such a value is \(\hat{T}_{ext}=0.382\) s and corresponds to the thick dashed line.  
* The period of Mode 3 of the structure coupled with the HMDI always remains close to that of Mode 2 of the uncoupled structure. It can be asserted that Mode 3 of the coupled structure substitutes Mode 2 of the uncoupled structure.