# Elasto-Plastic Short Exoskeleton to Improve the Dynamic and Seismic Performance of Frame Structures
## 5 Harmonic Excitation
### The Role of the Elastic Stiffness and of the Yielding Force  
The role of \(k_{\text{exo}}\) and \(F_{y}\) in the dynamic response of the coupled system is investigated by plotting the gain indexes in the parameters' plane (\(\mu\)-\(\eta\)) (see Equations (5) and (6)). The harmonic analysis is carried out for the three-storey frame structure whose geometric and mechanical characteristics are reported in the first row of Tables 1 and 2. The main period of the coupled system depends on the mass of the exoskeleton, which is considered fixed (\(\gamma=0.1\)), and on the stiffness of the exoskeleton, which is accounted for by the parameter \(\mu\). Since the gain maps are represented in the parameter plane \(\mu\)-\(\eta\), the range of the main period of the coupled system depends only on \(\mu\), which is varied in the range 0-10. Hence, in the parametric analyses performed on the three-storey frame structure, the range of the main periods of the coupled system is approximately 0.32-0.39 s.  
As shown in Figure 3a, the yielding system is connected to the first storey of the structure. The gain maps in Figure 3b are contour plots of the gain indexes \(\alpha_{1}\) and \(\alpha_{2}\) in the parameters' plane (\(\mu\)-\(\eta\)). The light grey colour regions (named advantage regions) are the regions where the gain coefficient is lower than unity, i.e., the coupling with the yielding system reduces the displacements of the structure. In Figure 3b, the two maps of each row refer to the \(\alpha_{1}\) (left map) and \(\alpha_{2}\) (right map) indexes, which are evaluated for a different \(\Omega\)The maps in the first column of Figure 3b show that the coupling with the yielding system leads to a significant reduction in \(u_{1}\). In the \(\alpha_{1}\) maps, the advantage regions cover almost all the parameters plane, reaching values smaller than 0.1 (i.e., the coupling has the effect to reduce \(u_{1}\) up to 90% of its original value). The \(\alpha_{2}\) maps show that the reduction in \(\Delta u\) is significantly smaller that the reduction in \(u_{1}\). Although \(\alpha_{2}\) has higher values than \(\alpha_{1}\), up to \(\Omega=15\) rad/s, the advantage regions still cover almost the entire parameters plane. Figure 3: Gain maps: (**a**) geometric coupling scheme; (**b**) Gain maps \(\alpha_{1}\) and \(\alpha_{2}\) for different frequencies \(\Omega\) of the harmonic excitation (\(\psi=0.1\), \(\gamma=0.1\), \(A_{\rm s}=0.7g\) m/s\({}^{2}\)); Points \(A\)–\(D\) and \(P_{1}\)–\(P_{4}\) denote reference cases examined in the analyses. For frequencies higher than \(\Omega=15\) rad/s, the advantage region shrinks, and the value of \(\alpha_{2}\) increases. However, contrarily to \(\alpha_{1},\alpha_{2}\) has an absolute minimum inside the investigated parameters' range. Such minimum depends on \(\Omega\); the smallest value among the minimums (\(\alpha_{2}=0.42\)) is obtained for \(\Omega=15\) rad/s. Such points of minimum provide the optimal design characteristics of the yielding system that lead to the maximum reduction in the drift of the super-structure. To gain a better understanding of the changes in the dynamics of the coupled system, different mechanical characteristics of the coupling are considered. Specifically, points \(A\), \(B\), and \(C\) on the maps referring to \(\Omega=15\) rad/s of Figure 3b are investigated. The results are shown in Figure 4, which is arranged in three sub-figures (a, b, and c). Each sub-figure shows (i) ) the comparison among time-histories of \(u_{1}(t)\) and \(\Delta u(t)\) of the coupled system and stand-alone structure (upper ad lower-left graphs), (ii) the hysteretic cycle of the yielding system (upper right graph), and (iii) the comparison of the frequency-response curves of \(u_{1}(t)\) and \(\Delta u(t)\), referring to the coupled system and stand-alone structure (lower right graph). In the figure, a thick line is used for the coupled system, whereas a thin line is used for the stand-alone structure.