# 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
Figure 4a shows the behaviour of the system having mechanical characteristics given by point \(A\), which is the point that corresponds to the absolute minimum of \(\alpha_{2}\) in the map obtained for \(\Omega=15\) rad/s. The comparisons between the time-histories of the coupled system and those of the stand-alone structure reveal that the yielding system reduces both \(u_{1}\) and \(\Delta u\). The hysteretic cycle shows a significant energy dissipation due to the plastic deformation of the yielding system. The observation of the frequency-response curves reveals a slight increase in the main frequency of the coupled system with respect to the stand-alone structure and a reduction in the amplitude of the displacements, mostly for \(u_{1}\). The two vertical dashed lines reported in all the frequency-response curves correspond to \(\Omega=15\) and \(\Omega=17.5\) rad/s. Along the line that refers to \(\Omega=15\), both \(u_{1}\) and \(\Delta u\) of the coupled system have maximum amplitudes smaller than those of the stand-alone structure, since the frequency-response curves of the coupled system are below the curves of the stand-alone structure. In fact, point \(A\) is inside the advantage regions of both maps obtained for \(\Omega=15\) rad/s. On the contrary, along the vertical line passing through \(\Omega=17.5\) rad/s, the drift of the coupled system has a maximum value slightly higher than that of the stand-alone structure, and point A is outside the advantage region of the \(\alpha_{2}\) map obtained for \(\Omega=17.5\) rad/s (see Figure 3b). Figure 4b shows the behaviour of the system having mechanical characteristics represented by point \(B\). In this case, such a point is inside the advantage regions of both the \(\alpha_{1}\) and \(\alpha_{2}\) maps, although it is close to the boundaries of these regions, and the values of the gain indexes approach the unity. In this case, the difference between the maximum amplitudes of the coupled system and stand-alone structure is smaller than the corresponding difference for the structures identified by point \(A\). Instead, the hysteretic cycle shows a small energy dissipation, because the yielding system has very small plastic deformations. Compared to the case described by point \(A\), the frequency-response curves show a smaller reduction in the amplitudes of \(u_{1}\) and \(\Delta u\) due to the coupling. Along the lines that refer to \(\Omega=15\) and \(\Omega=17.5\) rad/s, both \(u_{1}\) and \(\Delta u\) of the coupled system have maximum amplitudes slightly smaller than those of the stand-alone structure. In fact, point \(B\) is inside the advantage regions in both maps obtained for \(\Omega=15\) and \(\Omega=17.5\) rad/s (see Figure 3b). Figure 4: Time-histories, hysteretic cycles and frequency-amplitude curves: (**a**) Point \(A\); (**b**) Point \(B_{s}\) (**c**) Point \(C\) (points labelled in Figure 3b, \(\psi=0.1\), \(\gamma=0.1\), \(A_{s}=0.7g\) m/s\({}^{2}\)). Figure 4c shows the behaviour of the coupled system with mechanical characteristics represented by point \(C\). Such a point is inside the advantage regions of both the \(\alpha_{1}\) and \(\alpha_{2}\) maps. The comparisons among time-histories show results coherent with the values of \(\alpha_{1}\) and \(\alpha_{2}\) at point \(C\). By referring to the hysteretic cycle, due to the small value of the parameter \(\eta\) at point \(C\) (i.e., small value of the yield displacement \(u_{y}\)), there are large plastic deformations of the yielding system that, nevertheless, provide limited energy dissipation. The frequency-response curves show that the coupling does not affect the maximum amplitude of \(\Delta u\) but provides only a shift towards higher values of the main frequency of the coupled system. Along the lines for \(\Omega=15\) rad/s, both \(u_{1}\) and \(\Delta u\) of the coupled system have maximum amplitudes smaller than those of the stand-alone structure and, coherently, point \(C\) is inside the advantage regions of both maps obtained for \(\Omega=15\).