**1. Introduction**

Magnetic bearings can suspend rotating bodies without any mechanical contact. They have advantages such as being free of dust, noise, vibration and maintenance. Some magnetic bearings are already in commercial use in specific apparatuses such as high vacuum pumps or contamination free applications [1]. However, the high cost of the control apparatus for five degrees of freedom of the rotor prevents their wide application at present. It is thus necessary to develop a low-cost magnetic bearing system.

The authors have previously reported on the characteristics of the magnetic force acting between a couple of permanent magnets [2]. A magnetic top, consisting of a couple of ring-shaped permanent magnets, can be levitated without any control while maintaining rotation by itself. This fact suggests that the magnetic top may be a potential candidate for a passive magnetic suspension system. Several efforts have been made to explain the levitation mechanism of the magnetic top. San Miguel proposed noble analytical method with complex formulas showing that a magnetic top can maintain levitation if it rotates with slight precession [3].

In this chapter, an intuitive and easy analytical method based on the equivalent coil currents model for a ring-shaped permanent magnet is proposed.

A quasi-three-dimensional analysis, in which the three-dimensional shapes and layout of the ring-shaped permanent magnets are considered to estimate the magnetic forces acting on the levitating permanent magnet, is proposed. The principle of levitation of the magnetic top and the dimensions of the permanent magnets to realise levitation are discussed using the two-dimensional equations of motion for the magnetic top.

Furthermore, simulations based on the three-dimensional equations of motion are performed to investigate the dynamic behaviour of the magnetic top. The simulated results well predict the dynamic behaviour observed in the experiments. The simulations based on

the three-dimensional analysis are used to investigate the effects of the key parameters on the levitating characteristics, such as the sizes of both the ground and rotating permanent magnets, mass of the levitating top, tilt angle of the levitating top, rotation speed and initial position related to the restoring centre.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 137

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(2)

1 2 12 2 1 2 <sup>10</sup> sin

*r*

(1)

magnet. In an analysis based on the equivalent side currents approximation, magnetic forces acting on the magnetic top can be estimated by integrating magnetic forces acting between

The magnetic force *df* [N] acting between two current elements *dl*1 [m] and *dl*2 [m] and two transporting currents *I*1 [A] and *I*2 [A] is estimated by the following Biot-Savart's

> 12 1 2 7 <sup>2</sup> 10 sin *I I dl dl df r*

where *r* [m] is the distance between the two current elements and *φ* [rad] is the angle

Then, the magnetic force *f* [N] acting between two ring-shaped coil currents is estimated by integrating Equation (1) along the coil sides of the two coil currents *l*1 and *l*2 as follows:

*l l l l f df dl dl I I dl dl*

The magnetic forces acting on the rotor magnet are estimated by integrating Equation (2) for the equivalent side currents. The *x*, *y* and *z* components of the magnetic forces acting on the

1 2 1 2

rotor magnet *Fx*, *Fy* and *Fz* are estimated based on Equation (2).

equivalent coil currents in the rotor and stator magnets.

between the directions of the current elements.

**Figure 1.** Experimental magnetic top.

equation:

The ability and feasibility of the magnetic top as a magnetic bearing are also discussed.
