# Modelling and Experimental Testing of Passive Magnetic Bearings for Power Loss Reduction
## 1 Introduction  
Magnetic bearings have been the subject of scientific research in recent decades due to their advantages over traditional mechanical bearings, such as their lack of contact friction, lower maintenance, and no lubrication requirements [1, 2]. These advantages have led to a growing interest in rotating systems with magnetic supports [3, 4, 5]. Moreover, Permanent Magnetic Bearings (PMBs) are the only fully passive magnetic supports allowing for levitation without energy consumption [6], making them the ideal solution for environmentally sustainable energy storage and conversion applications such as Flywheel Energy Storage Systems (FESSs) [7]. Instead, in Active Magnetic Bearings (AMBs) [8], the stator conductive coils are supplied with current to allow for rotor position control, thus requiring continuous power supply and feedback control logic [9]. Electro-Dynamic Bearings (EDBs) [10] exploit the electromagnetic forces generated by eddy currents into a conductor moving in a magnetic field [11], causing power losses. Finally, Superconductive Magnetic Bearings (SMBs) [1] require a low temperature, with the drawback of the cooling system energy consumption.  
According to Earnshaw's Theorem [12], stable levitation cannot be achieved using only permanent magnets; additional bearings are needed to stabilise the system. Consequently, hybrid solutions, which exploit PMBs and other support typologies, are used to obtain stable levitation and the reduction in energy consumption. In [13], [PERSON] et al. analyse a hybrid AMB-PMB system whose radial behaviour is controlled by the active bearing,while the passive support provides the axial levitation. The passive support is composed of a thrust PMB and a passive damper with a Halbach magnet array, which provides both stiffness and damping. In [14], [PERSON] and [PERSON] describe a rotating system whose levitation is achieved by a radial EDB and a thrust PMB made of two coaxial disc-shaped permanent magnets to compensate for the rotor axial load. A further passive hybrid configuration is described in [15], where [PERSON] et al. exploit an SMB to provide the stable levitation of an FESS, compensating for its weight using a PMB as thrust bearing. In [16], a PMB is used to partially compensate the weight of a vertical axis wind turbine to limit the axial load and the resulting friction at the carbide tip sharp point base constraint. Further passive hybrid configurations in which PMBs are used both as thrust and as radial bearings are found in literature. In [17], [PERSON] and [PERSON] propose a hybrid EDB-PMB system and investigate the functioning of a rotating machine in which PMB supports are used as centring bearings with a partial compensation of the rotor weight. In [18], [PERSON] et al. analyse a PMB used as a radial and thrust bearing with a ball thrust bearing for the rotor axial stability; the axial force and the radial stiffness and damping are evaluated with tests performed at null angular speed.  
This paper introduces a configuration of a fully passive magnetic rotor that exploits the intrinsic radial and thrust effects of the PMBs.  
The proposed solution does not require any power supply or control system, thus being more cost-effective and easier to implement compared to the hybrid systems analysed in the state of the art.  
The PMB thrust-radial configuration is analysed, and a dedicated test rig is developed for static and dynamic experimental characterisation. The article is organised as follows. First, the model describing the PMB characterisation and the identification of the working range of the support in the thrust-radial configuration is presented. Then, the experimental setup consisting of rotor, bearings, actuation, and measurement systems is described. Finally, the experimental outcomes concerning static and dynamic tests, for the thrust force and power loss investigation, are discussed.