**3.5. Levitating area and tilt angle of the magnets**

As mentioned in the previous section, the tilting of a rotor magnet is essential in a magnetic top. In this section, relations between the shapes of the levitating area and the tilt angle of the rotor magnet are discussed.

Figure 10 shows the magnetic force map with levitating areas for different tilt angles of the rotor magnet. Figure 10(a) shows the magnetic force map when the tilt angle of the rotor magnet is zero. The magnetic forces acting on the rotor magnet are stable in the vertical direction but unstable in the radial direction at the upper singular point (0, 99.5). On the contrary, the magnetic forces acting on the rotor magnet are unstable in the vertical direction but stable in the radial direction at the lower singular point (0, 91.5). In this case, there is no levitating area because there is no restoring centre.

Figure 10(b) shows the magnetic force map when the tilt angle of the rotor magnet is *θ* = 0.4°, *i.e. θ* = 0.4 in the area *x* < 0 and *θ* = −0.4 in the area *x* > 0. Figures 10(c) and (d) show the magnetic force maps when the tilt angle of the rotor magnet is *θ* = 0.8° and *θ* = 1.2°, respectively. Figures 10 and 8, showing the case of *θ* = 1.0°, illustrate the fact that the levitating area becomes wider when the tilt angle becomes larger up to 1.2°, while the magnitude of restoring force around the restoring centre becomes saturated. We can intuitively observe considering the behaviour of a normal top that a magnetic top with a very large tilting angle will not levitate. Figure 10 also shows that the height of the restoring centre does not change by the tilt angle of the rotor magnet.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 149

148 Performance Evaluation of Bearings

magnet with a smaller outer diameter.

stator magnet with a larger outer diameter.

increases, the restoring point becomes higher and the levitating area becomes narrower in the radial direction and wider in the thrust direction. These results state that relatively well radial bearing characteristics can be obtained by a rotor magnet with a large outer diameter. On the contrary, relatively well thrust bearing characteristics can be obtained by a rotor

Figure 9(c) shows the levitating areas and the restoring points in case where the inner diameter of the stator magnet *dsi* changes. When the inner diameter of the stator magnet increases, the restoring point becomes lower and the levitating area becomes wider in the radial direction and narrower in the thrust direction. These results show that relatively well radial bearing characteristics can be obtained by a stator magnet with a smaller outer diameter. On the contrary, relatively well thrust bearing characteristics can be obtained by a

Figure 9(d) shows the levitating areas and the restoring points in the case where the outer diameter of the stator magnet *dso* changes. When the outer diameter of the stator magnet increases, the restoring point becomes higher and the levitating area becomes narrower in the radial direction. The outer diameter of the stator magnet hardly affects the axial height of the levitating area. These results indicate that relatively well radial bearing characteristics can be obtained by a stator magnet with a large outer diameter. The thrust bearing

As mentioned in the previous section, the tilting of a rotor magnet is essential in a magnetic top. In this section, relations between the shapes of the levitating area and the tilt angle of

Figure 10 shows the magnetic force map with levitating areas for different tilt angles of the rotor magnet. Figure 10(a) shows the magnetic force map when the tilt angle of the rotor magnet is zero. The magnetic forces acting on the rotor magnet are stable in the vertical direction but unstable in the radial direction at the upper singular point (0, 99.5). On the contrary, the magnetic forces acting on the rotor magnet are unstable in the vertical direction but stable in the radial direction at the lower singular point (0, 91.5). In this case,

Figure 10(b) shows the magnetic force map when the tilt angle of the rotor magnet is *θ* = 0.4°, *i.e. θ* = 0.4 in the area *x* < 0 and *θ* = −0.4 in the area *x* > 0. Figures 10(c) and (d) show the magnetic force maps when the tilt angle of the rotor magnet is *θ* = 0.8° and *θ* = 1.2°, respectively. Figures 10 and 8, showing the case of *θ* = 1.0°, illustrate the fact that the levitating area becomes wider when the tilt angle becomes larger up to 1.2°, while the magnitude of restoring force around the restoring centre becomes saturated. We can intuitively observe considering the behaviour of a normal top that a magnetic top with a very large tilting angle will not levitate. Figure 10 also shows that the height of the restoring

characteristics are not changed by the outer diameter of the stator magnet.

**3.5. Levitating area and tilt angle of the magnets** 

there is no levitating area because there is no restoring centre.

centre does not change by the tilt angle of the rotor magnet.

the rotor magnet are discussed.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 151

**4. Study of the dynamic behaviour of a magnetic top by three-**

whether the rotating magnetic top can maintain levitation.

We can obtain approximate guidelines for the size and shape of the levitating area by quasithree-dimensional analysis. Although the static analysis gives the 'levitating area', a magnetic top in this area may not always continue to levitate, considering the dynamic motion of the top. Furthermore, the static analysis mentioned in the previous section showed that the magnitude of the restoring force acting on the rotor magnet was small. Because the quasi-three-dimensional static analysis provides an approximate design of the magnetic top, the three-dimensional dynamic analysis should be performed to confirm

In this section, how the parameters such as rotating speed, mass of the top and initial position with regard to the restoring centre affect the behaviour of the levitating magnetic

To realise a successful rotation of a magnetic top, the rotation speed is one of the most important parameters. Simulated results show that the magnetic top (Table 1) can maintain levitating while it rotates in the range of 18–50 rps, *i.e.* 1,080–3,000 rpm, when the initial

Figures 11(a) and (b) show the trajectories of the centre of the magnetic top rotating at 1020 rpm and 3240 rpm, respectively. This characteristic is closely related to the tilt angle of the rotor magnet, that is, the rotor magnet with the shaft rotating at very low speed cannot maintain an adequate tilt angle because of the lack of mechanical inertia and the rotor magnet with the shaft rotating at a very high speed cannot maintain its tilt angle stable

Figure 12 shows the typical time dependency of the tilt angle of the rotor magnet. The tilt angle in this figure indicates the absolute values, *i.e.* the rotor magnet is tilting in a radial direction around *z*-axis. As shown in this figure, the tilt angle *θ* varies within 1.2° while the rotor magnet levitates with precession, as in this case. The maximum value of the tilt angle increases with increase in the rotation speed of the rotor magnet, as shown in Figure 13. In this analytical model, the gravity centre of the magnetic top is located at a little upper point along its shaft from the centre of the rotor magnet; therefore, the tilt angle becomes larger with an increase in the rotating speed. Then, the rotor magnet will be thrown in the radial direction, along the magnetic force vectors shown in Figure 10(a). If we design a magnetic top with the gravity centre located at the centre of the rotor magnet, the rotor magnet will rotate without tilting because of its mechanical inertia; however, such a rotor magnet cannot

Figure 14 shows the simulated trajectories of a levitating magnetic top rotating at 1,080 rpm for 60 s after starting from point (1, 0, 98.5), which is 1 mm apart from the restoring centre in

position is 1 mm apart in both radial and vertical directions from the restoring centre.

**dimensional analysis** 

top is discussed.

**4.1. Effects of rotating speed** 

because of the increasing centrifugal force.

realise levitation, according to previous discussions.

**Figure 10.** Relationship between the levitating area and the tilt angle of the rotor magnet.
