# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 1 Introduction  
Passive vibration isolators are a simple and effective way to protect structures and machines from vibrations if the forcing frequency is known a priori; they are also reliable as they do not require a power supply, which may not be available when it is needed. Linear vibration theory shows that an isolator is effective if the forcing frequency is greater than \(\sqrt{2}.f_{n}\), where \(f_{n}\) is the natural frequency of the isolated system; therefore, to increase the isolation frequency range, a less stiff isolator system should be adopted, but this choice could lead to excessive displacements. Therefore, an insulation system often has two conflicting requirements: in the case of vertical excitation, the isolation system must be sufficiently stiff to support the structure's weight, while it should be flexible enough to provide low transmissibility. In the case of horizontal excitation, it should be stiff enough to prevent any small actions from causing large displacements, but it should also be sufficiently compliant to isolate vibrations. Therefore, in many cases, the linear-elastic theory does not allow the problem of vibration isolation to be solved, and some solutions can be obtainedby adopting passive isolators with nonlinear behavior or by adopting active/semi-active isolation systems that allow the adjustment of the restoring force according to the state of the isolated body. Low natural frequencies can be achieved by adopting passive isolators such as air springs or wire rope springs. The stiffness of air springs [1,2] mainly depends on the air volume and pressure. Their behavior has been extensively studied in the field of vehicle suspensions, where they allow the vehicle attitude to be kept constant as the mass of the suspended body varies. They are often used in combination with a damper due to their low level of damping. Wire rope springs [3, 4, 5] have hysteretic nonlinear behavior depending on the length-diameter ratio of the ropes and on the spring configuration. The restoring force depends on the instantaneous and past deformation. They are characterized by a good level of damping due to the friction forces developed between the wires of the rope. These kinds of isolators combine a high level of isolation while taking up relatively little space; they are made of stainless steel and therefore do not suffer the attack of external factors (atmospheric agents, oils, etc.). Although both types of insulators can achieve sufficiently low natural frequencies, they may not achieve adequate isolation if the forcing frequency is not known in advance. If the forcing frequency is close to the resonance condition, the amplification of the transmissibility can be avoided only through an adequate increase in the damping of the isolation system, but this solution worsens the transmissibility for higher forcing frequencies. A significant nonlinear passive device was proposed by [PERSON] [6] in 1957 (Figure 1a) in the case of vertical excitation; it consists of two equally inclined springs (k\({}_{1}\)) combined with a vertical spring (k\({}_{2}\)). The vertical reaction of the two inclined springs decreases with the displacement until it becomes zero when their axes are aligned in the horizontal direction; beyond this configuration, the set of the two inclined springs provides a negative restoring force, as shown in Figure 1b, for k\({}_{2}\)/k\({}_{1}\) = 0. Thanks to the presence of the k\({}_{2}\) spring, the device provides a restoring force whose nonlinear trend depends on the ratio k\({}_{2}\)/k\({}_{1}\); with increasing k\({}_{2}\) stiffness, the restoring force becomes positive, and for further increments, its trend becomes monotonic (e.g., for k\({}_{2}\)/k\({}_{1}\) = 0.5 in Figure 1b) with a range of displacements characterized by a low equivalent vertical stiffness.