# Intermediate Isolation System with Nonlinear Lower Structure and Isolation System
## 4 Parametric Response Spectrum Analysis and IIS Design Spectra
For the case of Pozzuoli, the elastic acceleration response spectrum is plotted in Figure 3a with the following parameters: \(a_{g}\) = 0.162 g, site ground acceleration; \(F_{0}\) = 2.347, maximum value of the amplification factor for horizontal acceleration; \(T_{C}\) = 0.333 s, reference value to calculate the period corresponding to the starting point of the constant velocity branch; \(S\) = 1.472, factor depending on the soil and topographic categories; \(C_{C}\) = 1.509, factor depending on the soil category; \(\xi\) = 0.05, viscous damping ratio. \begin{table}
\begin{tabular}{l l} \hline
**Upper Structure** & **Isolation System** \\ \hline \(\widetilde{m}_{US}=0.6\cdot\widetilde{M}_{ISO}\) = 371 kNs\({}^{2}\)/m & \(\widetilde{m}_{ISO}=\widetilde{M}_{ISO}-\widetilde{m}_{US}=247\) kNs\({}^{2}\)/m \\ \(\widetilde{k}_{US}=K\cdot\widetilde{k}_{LS}\) = 799.854 kN/m & \(\widetilde{k}_{ISO}=4\pi^{2}\widetilde{M}_{ISO}/T_{ISO}\)\({}^{2}\) = [976, 9, 759, 065] kN/m \\ \(\widetilde{c}_{US}=2\xi_{US}\omega_{US}\widetilde{m}_{US}=689\) kNs/m & \(\widetilde{c}_{ISO}=2\xi_{ISO}\omega_{ISO}\widetilde{M}_{ISO}\) = [233, 23, 298] kNs/m \\ \(T_{US}=2\pi\cdot\Big{(}\widetilde{m}_{US}/\widetilde{k}_{US}\Big{)}^{0.5}\) = 0.135 s & \(T_{ISO}=2\pi\Big{(}\widetilde{M}_{ISO}/\widetilde{k}_{ISO}\Big{)}^{0.5}\) = [0.05, 5] s \\ \(\omega_{US}=\Big{(}\widetilde{k}_{US}/\widetilde{m}_{US}\Big{)}^{0.5}\) = 46.43 rad/s & \(\omega_{ISO}=\Big{(}\widetilde{k}_{ISO}/\widetilde{M}_{ISO}\Big{)}^{0.5}\) = [1.26, 126] rad/s \\ \hline \end{tabular}
\end{table} 
Table 1: Dynamic properties of the upper structure and isolation system in the 3 DOF IIS. The analysis results of the linear RSA can be plotted in terms of different global response parameters (absolute acceleration, a\({}_{s}\); base shear, \(V_{b}\); drift of the first DOF, \(d_{1,IIS}\)), and as a function of \(T_{ISO}/T_{LS}\), which is the major design parameter, once the size (and the mass) of the vertical extension has been set. The typical trend of the RSA results is shown in Figure 3b, which refers to the IIS made by the LS here chosen as an example and the steel vertical extension plus the HDRB isolation system. The graph represents the so called \"_nonconventional TMD design spectrum_\" or \"_IIS design spectrum_\", in terms of drift ratio \(\overline{d}=d_{1,IIS}/d_{LS}\), i.e., drift of LS, in the 3 DOF IIS model (design configuration) divided by the counterpart in the SDOF LS (as-is configuration). Three plus-one regions can be clearly distinguished in the graph, i.e.,: zone 1, characterised by displacement ratio values \(\overline{d}\) larger than 1; zone 2, characterised by displacement ratio values \(\overline{d}\) smaller than 1; zone 3, characterised by displacement ratio values \(\overline{d}\) almost equal to 1; and a transition zone, between zones 2 and 3, characterised by displacement ratio values \(\overline{d}\) gradually approaching to 1. Zone 1, where \(T_{ISO}/T_{LS}<1\) and \(\overline{d}>1\), exemplifies the case of existing buildings expanded with vertical additions rigidly connected to the substructure. The solutions that fall into the zone 3, where \(T_{ISO}/T_{LS}>5\) and \(\overline{d}\) is close to 1, correspond to a nearly perfect disconnection between vertical extensions and substructure (_isolation effect_). In this case, the isolated upper structure works as a structure isolated at its base (BIS), with values of \(T_{ISO}\) enough long to exclude any dynamic interaction with the substructure. For period ratios \(T_{ISO}/T_{LS}\) approximately in the range 1-5, the ratio \(\overline{d}<1\) and an advantageous dynamic interaction arises between the two structural parts. The isolated upper structure in fact acts as a TMD for the lower structure, thus reducing the displacement response of the system (_mass damping effect_).