**4.1. Effects of rotating speed**

150 Performance Evaluation of Bearings

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

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

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 because of the increasing centrifugal force.

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 realise levitation, according to previous discussions.

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

both *x* and *z* directions. Figures 14(a) and (b) show the trajectories of the head of the 25 mm long shaft of the magnetic top and Figures 14(c) and (d) show the trajectories of the centre of the rotor magnet. Figures 14(a) and (b) show that the shaft head rotates with both smaller radius nutation and larger radius precession. On the other hand, Figures 14(c) and (d) show that the centre of the rotor magnet rotates with precession when the tilt angle varies periodically. Comparing these two figures, it is observed that a magnetic top, rotating at a low speed such as 1,080 rpm, is rotating in a complex motion with nutation mode in addition to precession mode [6].

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 153

Figure 15 shows the simulated trajectories of a levitating magnetic top rotating at 3,000 rpm for 60 s after starting at point (1, 0, 98.5). Figures 15(a) and (b) show the trajectories of the head of the 25 mm long shaft of the magnetic top and Figures 15(c) and (d) show the trajectories of the centre of the rotor magnet. From these figures, we can observe that both the trajectories of the shaft head and the centre of the rotor magnet are almost the same in shape. However, the shaft head rotates in a little wider range compared to the moving area of the centre of the rotor magnet. This means that a magnetic top rotating at a relatively higher speed, e.g. 3,000 rpm, maintains its levitation with precession mode. In this case,

Although it is difficult to repeat the experiments in the same conditions, these simulated

**Figure 13.** The maximum tilt angle *θ* vs. rotating speed.

results showed good accordance with the experiments [6].

nutation mode is hardly observed.

**Figure 11.** Trajectories of the magnetic top for 5 s.

**Figure 12.** Time dependency of the tilt angle of the rotor magnet.

**Figure 13.** The maximum tilt angle *θ* vs. rotating speed.

addition to precession mode [6].

**Figure 11.** Trajectories of the magnetic top for 5 s.

**Figure 12.** Time dependency of the tilt angle of the rotor magnet.

both *x* and *z* directions. Figures 14(a) and (b) show the trajectories of the head of the 25 mm long shaft of the magnetic top and Figures 14(c) and (d) show the trajectories of the centre of the rotor magnet. Figures 14(a) and (b) show that the shaft head rotates with both smaller radius nutation and larger radius precession. On the other hand, Figures 14(c) and (d) show that the centre of the rotor magnet rotates with precession when the tilt angle varies periodically. Comparing these two figures, it is observed that a magnetic top, rotating at a low speed such as 1,080 rpm, is rotating in a complex motion with nutation mode in

> Figure 15 shows the simulated trajectories of a levitating magnetic top rotating at 3,000 rpm for 60 s after starting at point (1, 0, 98.5). Figures 15(a) and (b) show the trajectories of the head of the 25 mm long shaft of the magnetic top and Figures 15(c) and (d) show the trajectories of the centre of the rotor magnet. From these figures, we can observe that both the trajectories of the shaft head and the centre of the rotor magnet are almost the same in shape. However, the shaft head rotates in a little wider range compared to the moving area of the centre of the rotor magnet. This means that a magnetic top rotating at a relatively higher speed, e.g. 3,000 rpm, maintains its levitation with precession mode. In this case, nutation mode is hardly observed.

> Although it is difficult to repeat the experiments in the same conditions, these simulated results showed good accordance with the experiments [6].

**Figure 14.** Simulated trajectories of the levitating magnetic top rotating at 1,080 rpm.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 155

**4.2. Effects of the mass of a magnetic top and thickness of the stator magnet** 

increase in the mass of the top and a decrease in the thickness of the stator magnet.

magnet to the height of the restoring centre are discussed.

may realise successful levitation for wider mass variations.

**Figure 16.** Height of the restoring centre vs. mass of levitating top.

**4.3. Effects of the initial position** 

restoring centre.

In this section, the effects of the mass of a levitating top and the thickness of the stator

Figure 16 shows the relationship between the height of the restoring centre *zr* [mm] and the mass of a levitating top *m* [g] when the thickness of the stator magnet is *h* = 60, 40 and 20 mm. Calculated results show that the height of the restoring centre *zr* decreases with an

Calculations were made for various values of the mass of the top in the analytical model described in Table 1. However, there is no restoring centre or levitation area for a heavier or a lighter top than those shown in Figure 16. According to these results, a thin stator magnet

Some experiments demonstrated that the initial position related to the restoring centre is one of the most important parameters. To realise successful rotation of a magnetic top, the initial position should be at least inside the levitating area defined in the previous section. The magnetic top shows various behaviours according to its initial point with regard to the

Figure 17 shows the simulated trajectory of the centre of the rotor magnet for 60 s starting from (1, 0, 98.5), which is 1 mm apart in both *x* and *z* directions from the restoring centre. The mass of the top is 20.37 g and rotation speed is 23 rps, *i.e.* 1380 rpm. These results show that the rotating top is levitated in the area of ±1.6 mm in both *x* and *y* directions and of ±1 mm in *z* direction, from the restoring centre. The maximum tilt angle is 1.26° with the *z-*axis.

**Figure 15.** Simulated trajectories of the levitating magnetic top rotating at 3,000 rpm.
