## 5 Conclusions  
The aim of this paper was the improvement of the dynamic and seismic behaviour of rigid block-like structures. To reach this goal, a rigid block-like structure, modelled as a rigid block, was coupled to an external, auxiliary system that works as a hysteretic mass damper and is equipped with an inerter. The elasto-plastic behaviour of the coupling device was described by the Bouc-Wen model. Since it was assumed that the rigid block can only undergo a rocking motion, the mechanical system has two Degrees Of Freedom. The equations of motion and uplift conditions were written by following a direct approach. For the impact conditions, the classical assumption based on the conservation of the angular momentum was used. Two different analyses were performed. In the first analysis, coupled systems and corresponding stand-alone rigid blocks were excited by a harmonic base acceleration. The resulting frequency-response curves were compared to investigate the role of the parameters in the effectiveness of the proposed protection method. The results of this analysis show that the coupling of rigid blocks with auxiliary systems significantly improves the dynamic performance of the stand-alone rigid blocks. The relevant parameters on which the efficiency of the coupled system depends were found to be the apparent mass of the inerter and the dynamic characteristics of the auxiliary system. In the second analysis, different seismic records were used to excite the coupled systems. A gain coefficient was introduced to evaluate the ability of the auxiliary systems to reduce the maximum rocking angles of the coupled systems with respect to those of the corresponding stand-alone rigid blocks. A parametric analysis permitted to build gain  
Figure 14: Gain maps obtained for different dimensions of the rigid blocks and under the Christchurch seismic record (\(m=0.05M,u_{y}=1/3u_{cr},\psi=0.2\)). surfaces and maps by evaluating the gain coefficient inside a specific parameter plane. It was also found that the seismic performance of the coupled systems mostly depends on the apparent masses of the inerters and the dynamic characteristics of the auxiliary systems. The comparative analysis between the frequency-response curves of the coupled systems and the acceleration response spectra of the exciting seismic records provided good insights to explain the good seismic performance of the coupled systems. The results show that there are several possible choices of the parameters of the auxiliary system, the mass of the inerter included, that lead to a reduction in the maximum rocking angle in the rigid block and, consequently, to an increase in its safety against overturning. Therefore, the proposed protection method should be taken into consideration, and its effectiveness compared with those of similar protection methods. Obviously, especially in the case of seismic excitation, the best performance could be reached by tuning the characteristics of the protection system and, in particular, the apparent mass of the inerter. For this purpose, it would be ideal to design a smart structure that, together with the rigid block-like structure and the proposed auxiliary system equipped with an inerter with an adjustable virtual mass, includes the sensing equipment and an appropriate algorithm to analyse the seismic input and control the tuning of the auxiliary system. An interesting improvement of the study presented in this paper could analyse the use of a hysteretic loop with different shapes, as those described in [30; 31]. Although a precise assessment of the usability of the proposed protection method would require the choice of the best hysteretic connection between the rigid block and auxiliary system, provided the availability of a suitable space for the placement of the auxiliary system, the simplicity of the proposed protection method would definitely be an advantage in its choice for real applications. [PERSON] and A.C. contributed equally to this work.