# 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
On the contrary, along the lines for \(\Omega=17.5\) rad/s, only \(u_{1}\) slightly decreases after the coupling. Instead, the \(\Delta u\) of the coupled system is significantly higher than the one of the stand-alone structure. This occurrence is coherent with point \(C\) being inside the advantage region of the \(\alpha_{1}\) map and outside the advantage region of the \(\alpha_{2}\) map obtained for \(\Omega=17.5\) rad/s (see Figure 3b). Additionally, an almost resonance condition occurs at point \(C\) for the coupled system, since the main frequencies of the system and harmonic excitation are close to each other. During the harmonic excitation, after the transient dynamics, the coupled system reaches a dynamic motion characterised by stationary hysteretic cycles. To further explain the effects of the coupling, the equivalent damping ratio \(\xi_{\alpha\beta}\) of such stationary hysteretic cycles is evaluated. The value of \(\xi_{\alpha\beta}\) is obtained by equating the area of the hysteretic cycle \(E_{h}\) and the area of the cycle \(E_{v}\) of an equivalent linear viscous damped system under the same excitation. As in [41], the area of the equivalent viscous cycle is evaluated as \(E_{v}=\pi c_{eq}\omega_{eq}\tilde{u}_{1}\), where \(c_{eq}\) and \(\omega_{eq}\) are the damping coefficient and the frequency of the equivalent viscous system, respectively, and \(\tilde{u}_{1}\) is the maximum displacement of the hysteretic system that corresponds to the maximum strain of the equivalent system. For a linear viscous 1-DOF system \(c_{eq}=2_{eq}^{\kappa}\omega_{eq}m\), \(\omega_{eq}=\sqrt{k_{eq}/m}\), and \(k_{eq}\) can be related to the maximum restoring force reached in the hysteretic cycle \(\tilde{J}_{\text{eeeo}}\) and to \(\tilde{u}_{1}\) (\(k_{eq}=\tilde{J}_{\text{eeo}}/\tilde{u}_{1}\)); thus, the equivalent damping ratio reads  
\[\xi_{\alpha\beta}=\frac{E_{h}}{2\pi\tilde{u}_{1}\tilde{J}_{\text{eeeo}}} \tag{10}\]  
Figure 5 shows the stationary hysteretic cycles and the frequency-response curves of \(\Delta u\) that are related to points \(A\) and \(P_{i}(i=1,\dots,4)\). Such points are shown in the \(\alpha_{2}\) map obtained for \(\Omega=15\) rad/s of Figure 3b. Specifically, \(A\) is located at the point of absolute minimum, whereas the other points \(P_{i}\) are in the vicinity of \(A\). For each investigated point, \(\xi_{\alpha\beta}\) is reported near the plot of the limit cycle. For each plot, the frequency-response curves of the drift show (i) the main frequency of the coupled system and the maximum value of drift (thick curve), and (ii) the difference between the maximum drift of the coupled system and stand-alone structure at \(\Omega=15\) rad/s (the same frequency of the map on which points \(A\) and \(P_{i}\) are located). Along the horizontal direction (from point \(P_{1}\) to \(P_{3}\), passing through \(A\)), although numerically close to each other, the values of \(\xi_{\alpha\beta}\) slightly increase. Moreover, the increase in the pre-yielding stiffness of the yielding system, measured by the parameter \(\mu\), causes an increase in the main frequency of the coupled system and a minor increase in its maximum drift. These different effects combine in such a way that the largest difference between the drifts of the coupled system and stand-alone system along \(\Omega=15\) rad/s occurs at point \(A\). Along the vertical direction (from point \(P_{2}\) to \(P_{4}\), passing through \(A\)), the increase of the yielding displacement, which is measured by the parameter \(\eta\), causes a decrease in the equivalent damping of the hysteretic cycles. Additionally, the main frequency of the coupled system increases, and the maximum drift in correspondence of the main frequency decreases. Along the vertical direction, the different effects combine so that the coupling is more effective for the characteristics of the coupled system defined by point \(A\).