# Intermediate Isolation System with Nonlinear Lower Structure and Isolation System
## 4 Parametric Response Spectrum Analysis and IIS Design Spectra  
The second step of the proposed procedure aims to check the viability of the approach, understand the dynamic response of the IIS, and choose the dynamic characteristics of the added structure and isolation system that contribute to minimizing the seismic demand on the entire structure. For these aims, parametric response spectrum analysis (RSA) was carried out on simplified three-degrees-of-freedom model of IIS (3 DOF IIS (Figure 2b)) by  
Figure 2: Simplified lumped-mass models: (**a**) SDOF LS, (**b**) 3 DOF IIS. varying the characteristics of the intermediate isolation system (ISO) and of the extension, considered as the upper structure (US). The mass, stiffness, and damping constants of the LS (i.e., \(\widetilde{m}_{LS}\), \(\widetilde{k}_{LS}\), and \(\widetilde{c}_{LS}\)) are the same dynamic properties of the SDOF LS (Figure 2a) described in the previous Section 3.2. The mass, stiffness, and damping constants of the ISO and US are termed as: \(\widetilde{m}_{ISO}\), \(\widetilde{k}_{ISO}\), \(\widetilde{c}_{ISO}\), and \(\widetilde{m}_{US}\), \(\widetilde{k}_{US}\), \(\widetilde{c}_{US}\), respectively. To define \(\widetilde{m}_{ISO}\), \(\widetilde{m}_{US}\), \(\widetilde{k}_{ISO}\), and \(\widetilde{k}_{US}\), the following parameters are introduced:  
* mass ratio \(\alpha=\widetilde{M}_{ISO}/\widetilde{m}_{LS}\) = 0.25, where: \(\widetilde{M}_{ISO}=\widetilde{m}_{US}+\widetilde{m}_{ISO}\) is the mass of the isolated vertical addition, and \(\widetilde{m}_{US}=1.5\cdot\widetilde{m}_{ISO}\);
* stiffness ratio K = \(\widetilde{k}_{US}/\widetilde{k}_{LS}\) = 1.25;
* period of the isolated extension \(T_{ISO}=2\pi\Big{(}\widetilde{M}_{ISO}/\widetilde{k}_{ISO}\Big{)}^{0.5}\) = [0.05-5] s.  
By selecting a steel braced frame structure for the addition, an equivalent viscous damping ratio \(\xi_{US}\) = 0.02 was adopted for the US, while for the isolation system made of High Damping Rubber Bearings (HDRB), an equivalent viscous damping ratio \(\xi_{ISO}\) = 0.15 was considered. With these assumptions, the values of the mass, stiffness, and damping constants of the isolated vertical addition are provided in Table 1; the values of the fundamental period and circular frequency of the standalone (not isolated) upper structure, \(T_{US}\) and \(\omega_{US}\), and of the isolation system, \(T_{ISO}\) and \(\omega_{ISO}\), are provided as well. Figure 2 also depicts the relative displacements of the DOFs with respect to the ground, i.e.,: \(d_{LS}\), relative displacement of the LS in the SDOF LS; \(d_{1,IIS}\), \(d_{2,IIS}\), and \(d_{3,IIS}\), relative displacements of the first, second, and third degree of freedom of the 3 DOF IIS. Since the three DOFs show very different damping ratios (i.e., \(\xi_{\text{LS}}\) = 0.05, \(\xi_{\text{ISO}}\) = 0.15, \(\xi_{\text{US}}\) = 0.02), the 3 DOF IIS model is characterized by _non-proportional_ (or _non-classical_) damping. The assumption of _proportional_ (or _classical_) damping, for which the off-diagonal terms of the damping matrix can be neglected, may result in nonconservative response prediction [55]. By adopting the fully populated damping matrix, the equations of motion are coupled and involve complex modal coordinates and complex eigenvectors [56]. Being a non-classically damped system, the RSA was developed by means of the method described by [PERSON] and [PERSON] [57]. The analysis takes as reference the horizontal elastic response spectrum in acceleration provided by the Italian seismic code [52] at the Life Safety level (10% probability of exceedance in 50 years).