# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 3 Harmonic Analysis
### Frequency-Response Curves  
A first analysis compares the frequency-response curves of the MS and AS. Such curves represent the dimensionless maximum displacement of the substructure, \(\chi_{1}=\text{Max}(\left|u_{1}(t)\right|/u_{0})\), and the dimensionless drift of the superstructure, \(\chi_{2}=\text{Max}(\left|u_{2}(t)-u_{d}(t)\right|/u_{0})\), as functions of the frequency ratio \(\beta=\Omega/\omega_{1}\), where \(\omega_{1}=\sqrt{k_{1}/m_{1}}\). In the analyses where the value of \(\eta\) is fixed, such value is assumed as \(\eta=0.2\). Figure 3 compares the frequency-response curves of the MS obtained for \(\Psi=1.0\), and the frequency-response curves of the AS. Assuming \(\Psi=1.0\) entails a linear behaviour of the discontinuity, the comparison shows that the discontinuity introduces a second peak in the frequency-response curves of the MS. Such a peak corresponds to a second resonant frequency, whereas the AS is characterized by a single resonant frequency in the range of the observed frequencies. The existence of a second resonant frequency is a consequence of the presence of a TMD in the system, meaning that, due to the discontinuity, the superstructure works as a TMD for the substructure. The frequency-response curve of \(\chi_{2}\) shows a shift of the main frequency of the MS to lower values and highlights a behaviour as a base isolated structure. Thus, the superstructure acts as a TMD for the substructure and the superstructure behaves as if it was base-isolated. In both cases, it can be asserted that the distance between the two resonant frequencies of the MS is a qualitative measure of the effectiveness of the discontinuity. A second analysis investigates the sensitivity of the system to the values of the parameters characterizing the hysteretic behaviour of the disconnection devices. Figure 4 shows the frequency-response curves obtained for several values of the parameters characterizing  
Figure 3: Frequency-response curves: comparison between curves of AS and MS (\(L=3\) and \(\Psi=1.00\)). the Bouc-Wen constitutive law. Specifically, Figure 4a shows the frequency-response curves of the MS for \(L=3\), \(A=0.6\) g, \(\rho=0.2\), and several values of \(\Psi\) increasing from \(\Psi=0.10\) to \(\Psi=1.00\). The highest resonant frequency is not significantly sensitive to \(\Psi\); on the contrary, the smallest resonant frequency significantly depends on \(\Psi\). The values of both \(\chi_{1}\) and \(\chi_{2}\) in correspondence of the resonant peaks strongly depend on \(\Psi\). As \(\Psi\) decreases, the hysteresis cycles become larger, and the values of the dimensionless displacements in correspondence of the peak decrease. Moreover, the distance between the peaks increases as \(\Psi\) decreases. Figure 4b shows the frequency-response curves for \(L=5\), \(A=0.6\) g, \(\rho=0.2\), and the same values of \(\Psi\) of the case presented in Figure 4a. When the discontinuity is placed at higher levels, the values of \(\chi_{1}\) and \(\chi_{2}\) at the resonant peak that refers to the lower frequency reduce as \(\Psi\) decreases. The value of \(\mu\) obtained for \(L=5\) is smaller than the value obtained for \(L=3\), and the distance between the two resonant peaks decreases, thus reducing the frequency range where the discontinuity is effective. Figure 4c shows the frequency-response curves obtained for different values of \(\rho\) increasing from \(\rho=0.02\) to \(\rho=0.30\), \(L=3\), \(A=0.6\) g, and \(\Psi=0.25\). As \(\rho\) increases, both the first and second resonant frequencies shift toward higher values. Moreover, \(\chi_{1}\) manifests higher values at the resonant peaks for smaller values of \(\rho\). The same increase occurs for the values at the left peak of the \(\chi_{2}\) curves. Instead, the value of \(\chi_{2}\) at the higher resonant frequency decreases until it almost vanishes for decreasing values of \(\rho\), thus confirming that, for small values of \(\rho\), the discontinuity acts on the superstructure as a base isolation.