# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 1 Introduction
A circuit based on magnetic reluctance is considered to model the negative stiffness characteristic of the isolator, and an active control is introduced to improve the system response. Paper [19] presents an electromagnetic spring consisting of coaxial permanent annular magnets and coils with a rectangular cross-section. To expand the region where the stiffness is negative and to increase its tuneable range in [20], a compact and contactless multi-layer electromagnetic spring is suggested. A horizontal electromagnetic vibration isolator is proposed in [21], where the stator is made of an array of coil and the mover coil is substituted by an array of permanent magnets. In [22], an electromagnetic actuator is proposed to isolate the equipment installed on ships and to prevent the noise radiated under water from disturbing the marine environment. A permanent magnet, with external radial magnetization, is used to achieve a high thrust-force-to-weight ratio. Hybrid vibration isolators consist of the coupling of a passive system with an active one; they have the same flexibility and adaptability as the active ones with the advantage of ensuring sufficient functioning even in the absence of an external energy source. Their flexibility is due to the presence of a smart element that allows tuning the stiffness or the damping of the isolator. This paper presents a hybrid isolator to be used in the case of vibrations coming from the ground with a predominant horizontal direction. Its main functions have been decoupled: in fact, the vertical load is sustained with spherical rolling supports, while the horizontal restoring force is provided by two components: (a) passive elastomeric pads, characterized by a positive stiffness; (b) an electromagnetic device able to provide a negative stiffness. This last component is constituted by three coils, two of which are fixed to the ground and the other one to the body to be isolated. In comparison to other electromagnetic devices adopted for vibration isolation purposes, it allows electromagnetic forces to be provided in a fairly wide range of displacements; in fact, the relative displacements, within which the electromagnetic spring can perform a significant action, are comparable with the maximum displacements allowed by the elastomeric pads. The electromagnetic active system acts only when the passive one does not allow it to achieve the isolation requirements. In such conditions, it acts only for short instants of time to add its action to that of the elastomeric passive system. Therefore, the system has the advantages of an active isolation system but requires a much lower energy contribution. Like the semi-active isolators, the device can be considered reliable since it can operate with less efficiency, even if the control system is not active, for example, due to power failure. The theoretical and experimental models developed in this paper have the aim of defining the isolator's restoring force components and the isolator's transmissibility; both models refer to a mono-directional isolator, but they are easily reconfigurable for a two-dimensional use that is able to act in all directions of the horizontal plane. The organization of the paper is as follows. In Section 2, the description of system isolation is provided; in Section 3, all the components of the restoring force are modeled and analyzed; in Section 4, the analytical solution of the model in steady-state and time-varying conditions is reported; in Section 5, the model electrical parameters are defined; and results and their discussions are reported in Section 6.