# Seismic Benefits from Coupling Frame Structures with a Hysteretic Mass Damper Inerter
## 6 Conclusions  
In this paper, the connection between a multi-story frame structure and an external oscillating mass is investigated with the aim to improve the seismic behavior of the frame structure. The frame structure is modeled with a two-degree-of-freedom system, whereas the external mass is described by a single-degree of-freedom-system. The external mass is connected at the first story of the frame structure through a hysteretic device and its inertial force is increased by means of an inerter device. The whole protection device, i.e., the external body together with the hysteretic connection and inerter device, is named \"hysteretic mass damper inerter\". The coupled structure, consisting of the frame structure and protection device, is modeled as a three-degree-of-freedom (3-DOF) mechanical system. The equations of motion are derived using a direct approach.  
A preliminary spectral analysis is conducted using the linearized mechanical system to study the dynamics of the coupled system. Results indicate that the HMDI increases the period of the main mode of the coupled system, which is similar to the effect of base isolation. Additionally, the modal mass participating coefficient (MPC) of the main mode increases up to 90% in most practical cases. It is also demonstrated that the period of the HMDI, evaluated as if it was directly connected to the ground (i.e., studying the relative one-degree-of-freedom model), is one of the most important parameters that determine the effectiveness of the HMDI. The value of such parameter is directly related to the shape of the second oscillating mode of the coupled system, which replaces the main mode of the stand-alone frame structure. By suitably choosing this parameter, it is possible to increase the effectiveness of the HMDI and reduce the displacement at the connection level and the inter-story drift.  
After the spectral analysis, a seismic analysis is conducted using three ground motion records or a set of seven earthquake records compatible with the spectrum of Los Angeles. Two gain coefficients are used to evaluate the effectiveness of the HMDI. Such gain coefficients provide the ratios between the corresponding displacements of the coupled and  
Figure 14: Spectral gain maps of the three- and five-story frame structures under spectrum-compatible seismic records with the Los Angeles design spectrum (\(\beta_{y}=1.5\); \(\gamma=0.1\); \(\psi=0.1\)).  
stand-alone frame structures. Gain coefficients smaller than unity indicate the effectiveness of the HMDI in reducing displacements and drifts. Results are presented in spectral gain maps, which are contour plots of the gain coefficients in specific parameter planes. It is found that there are large ranges of the parameters characterizing the HMDI where the protection device is effective in the reduction of the displacement of the first story and the drift of the upper stories of the frame structure. Moreover, it is found that the HMDI is particularly effective in low- and medium-rise frame structures.  
All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data presented in this study are available on request.  
The authors declare no conflict of interest.