# Intermediate Isolation System with Nonlinear Lower Structure and Isolation System
## 1 Introduction
[1] and [PERSON] and [PERSON] [26] have investigated the dynamic behaviour of nonconventional TMD by adopting both multi- and two-degrees-of-freedom models and by modelling the seismic input as a stationary Gaussian stochastic process with zero mean. The control criterion suggested in [1] is the minimization of the root-mean-square displacement response of the damped main structure; in [26], instead, the criterion is the maximization of an energy performance index, defined as the ratio between the energy dissipated in the isolation system and the input energy globally transferred to the model. An experimental campaign on a multi-story frame structure equipped with nonconventional TMD has also been reported in [27], with white noise, sine sweep, and natural earthquakes as input motion conditions. [PERSON] et al. [28,29] have considered IIS made by fluid viscous dampers and have derived the optimal parameters for minimizing the isolation drift and simultaneously controlling the superstructure performance. [PERSON] et al. [30] have proposed a multi-objective optimization approach applied to a 2 DOF model, representing an IIS working as a TMD. The objective functions were the minimization of the variance of the substructure drift and the variance of the isolation drift (or of the superstructure acceleration), with the seismic input modelled as a stationary Gaussian stochastic process with zero mean. This paper examines the interesting idea of utilizing the IIS to extend vertically existing buildings for addressing the increasing demand for real estate in contemporary cities with almost no land take [2,3,11,12,31,32,33,34,35,36,37,3]. In earthquake-prone zones, IIS can prevent any increase in seismic effects, despite the increase in mass due to the vertical extension. Furthermore, even a non-negligible reduction in the seismic response can be obtained if proper design procedures are followed for the new extension, based on the properties of the existing one. In fact, vertical extension of existing buildings is being adopted in several major European cities as a means of sustainable recovery and growth. Vertical extensions are also very useful in responding to the demand, born after the COVID-19 pandemic, for new living and working spaces that allow safe social distancing. In England, new laws were issued in 2020 to encourage the construction of new living space by adding up to two storeys to existing buildings without any application [38]. Upper extensions are arising in Geneva, Milan, Rotterdam, Vienna, London, and New York City; this strategy is worth spreading because, based on the concept of the circular economy, it is more sustainable than the common approach of demolition/reconstruction of existing buildings, which consumes more energy and materials [39]. Furthermore, within the current urban planning trend of Transit-Oriented Development (TOD), the realization of over-track buildings for exploiting the space above rail infrastructure is becoming quite popular in Japan. A very interesting example is the Dayia Gate Ikebukuro, in Tokyo, an over-track tall building with mid-story isolation, constructed over the Ikebukuro railway station [40]. The authors have deeply examined the dynamic problem of IIS application for building extensions, with the aim of promoting the diffusion of this strategy, which is both sustainable and effective in earthquake-prone zones. A first insight on this topic is provided by the authors in [32], in which a three-storey masonry building is selected as case study and extended by means of different two-storey steel structures, isolated at the base and erected on the top of the existing building. In [33], the authors developed a proof-of-concept study for the large-scale application of this strategy, with reference to the town of Pozzuoli (South Italy). Finally, a design procedure has been defined and illustrated in [31]. The key point of this procedure is the parametric response spectrum analyses performed on lumped mass models by varying the mechanical properties of the building extension.