**3.3. Validity of the quasi-three-dimensional analysis**

142 Performance Evaluation of Bearings

it rotates with a slight precession.

The quasi-three-dimensional analysis shows that there is no restoring centre when the tilt angle of the rotor magnet is 0, but a slight tilt angle such as 1° brings the restoring centre into existence. These results suggest that a magnetic top equipped with a ring-shaped permanent magnet can levitate in the space above a stator ring-shaped permanent magnet if

**Figure 5.** Magnetic force map for different tilt angles *θ* of a levitating magnetic top.

To confirm the validity and effectiveness of quasi-three-dimensional analysis using the magnetic force map, dynamic behaviour of the rotor magnet is investigated by computer simulation based on the equations of motion introduced in the previous section. To make intuitive discussions, a dynamic simulation using two-dimensional equations of motion, Equations (3) and (4), is performed. In this simulation, the tilt angle of the rotor magnet is

(a) *θ* = 0°, rotor magnet is horizontal. (b) *θ* = 1° with respect to *z*-axis.

Figure 6 shows the simulated behaviour of the centre of the rotor magnet for 10 s starting from the point (1, 98.5), which is 1 mm apart in both *x* and *z* directions from the restoring centre (0. 99.5). The simulated time trajectory of the centre of the rotor magnet (Figure 6(a)) shows that the rotor magnet levitates in the area of ±1 mm in both vertical and horizontal directions from the restoring centre. The bottom left point of this rectangular space is the initial position of the rotor magnet. These results tell us that the magnetic top is swaying around the restoring centre and the range of swaying motion is determined by the initial position of the magnetic top with regard to the restoring centre. Figures 6 (b) and (c) show the time dependencies of radial and vertical motions of the centre of the rotor magnet. From these figures, we find that the frequencies of radial and vertical motions are 1.45 Hz and 1.13

**3.2. Simulation to investigate the behaviour a magnetic top** 

set to *θ* = 1° in the area *x* < 0 and to −1° in the area *x* > 0.

Hz, respectively.

To verify the validity of the above analytical results, experiments are performed using the test model. The dimensions of the rotor and stator magnets used in the test model are listed in Table 1. The weight of the top is adjusted to 20.37 g using a dummy weight. Behaviour of the levitating magnetic top is recorded using a video camera from the *y* direction. The levitation height of the centre of the rotor magnet is about 100 mm above the centre of the stator magnet. The digital image information is obtained using motion capture software 'Pv Studio 2D demo' and the software 'Graph Scan 1.8' are used to obtain Figure 7. The frame size and frame interval of the obtained video data are 640 × 480 pixels and 30 flames per second. However, finally obtained frame interval using the above software is 4 frames per second.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 145

Measured Calculated

the rotor magnet in this experiment were 3 mm and 2.3 mm apart from the restoring centres in *x* and *z* directions, respectively. Figures 7 (b) and (c) demonstrate the time dependence of the radial and vertical motions in 15 s. These figures show that the frequencies of swaying motion are about 0.75 Hz in radial direction and about 1.05 Hz in vertical direction. These

> Levitation height [mm] 100 99.5 Frequency of radial swaying [Hz] 0.75 1.45 Frequency of vertical swaying [Hz] 1.05 1.13

In spite of low accuracy of the measured data and difficulties in reenacting experiments in the same condition, the analysed levitation height and the frequency of vertical swaying are well in accordance with the experimental values. However, the analysed frequency of radial swaying is about twice the experimental value. This difference seem to be derived from assumptions in the two-dimensional analysis such as the constant tilt angle of the rotor magnet. Simulated results for various tilt angles showed that the magnitude of the tilt angle significantly affects the radial motion, but does not affect the vertical motion of the rotor magnet. Furthermore, the analysis is based on two-dimensional equations of motion, and three-dimensional behaviour of the magnetic top in the experiment is measured as two-

These results show that the fundamental parameters of a magnetic top, such as levitation height and dimensions of the permanent magnets, can be determined well using the quasi-

Figure 8 shows the magnetic force map for the test model shown in Table 1. The tilting angle of the rotor magnet is set as ±1°. This figure shows that a tilting magnetic top, located within the red dotted lines and named as the 'levitating area', will be guided by the magnetic force along the direction of vectors towards the restoring centre A (0, 0, 99.5). Although the levitating area is shown as a two-dimensional area in this figure, the real shape of the

The size and shape of the levitating area are closely related to the dimensions of the permanent magnets and the tilting angle of the rotor magnet. Figure 9 shows the relationship between the shape and size of the levitating area and the parameters of the permanent magnets. The effects of precession are considered to set the tilt angle *θ* to −1° in *x*  > 0 and to 1° in *x* < 0. The conical shape of the levitating area is approximated by the

rectangular area bounded by the green coloured dotted line in Figure 8 [5].

test results are compared to the calculated ones in Table 2.

**Table 2.** Comparison between analysis and experimental data

**3.4. Levitating area and parameters of the magnets** 

dimensional video information.

three-dimensional analysis.

levitating area is conic.

**Figure 7.** Measured behaviour of the magnetic top in the test model.

Experiment is performed according to the following steps : (1) place a non-magnetic plate on the pole surface of a stator magnet, (2) rotate a magnetic top on the plate at the centre of a stator magnet, (3) lift the plate with rotating top slowly until a magnetic top is pulled into the restoring centre. There are some hurdles to clear these steps. A magnetic top should be rotate at the exact centre of the stator magnet in a certain rotating speed range to clear step (2). Lift force should be less than vertical magnetic force acting on a top from the stator magnet to clear step (3). A magnetic top is rotated by fingers and the plate is lifted by hand in our experiment. Then, it is difficult to obtain experimental data of the same conditions.

Figure 7 shows the measured trajectory of the centre of the rotor magnet for 15 s. In Figure 7, the origin of *x* and *z* coordinates is the centre of the picture captured by the camera. In Figure 7(a), dots indicate the positions of the rotor magnet centre measured every 0.25 s, *i.e.* 4 frames per second, and a smoothing line connects these dots in sequential order. The smoothing line in Figure 7(a) does not show the swaying motion correctly; however, we can observe that the rotor magnet levitates and sways in the range of ±3 mm in radial direction and ±2.3 mm in vertical direction. In the experiment, it is difficult to start rotation of the magnetic top at the designated initial point. Figure 7(a) suggests that the initial positions of the rotor magnet in this experiment were 3 mm and 2.3 mm apart from the restoring centres in *x* and *z* directions, respectively. Figures 7 (b) and (c) demonstrate the time dependence of the radial and vertical motions in 15 s. These figures show that the frequencies of swaying motion are about 0.75 Hz in radial direction and about 1.05 Hz in vertical direction. These test results are compared to the calculated ones in Table 2.


**Table 2.** Comparison between analysis and experimental data

144 Performance Evaluation of Bearings

same conditions.

**Figure 7.** Measured behaviour of the magnetic top in the test model.

Experiment is performed according to the following steps : (1) place a non-magnetic plate on the pole surface of a stator magnet, (2) rotate a magnetic top on the plate at the centre of a stator magnet, (3) lift the plate with rotating top slowly until a magnetic top is pulled into the restoring centre. There are some hurdles to clear these steps. A magnetic top should be rotate at the exact centre of the stator magnet in a certain rotating speed range to clear step (2). Lift force should be less than vertical magnetic force acting on a top from the stator magnet to clear step (3). A magnetic top is rotated by fingers and the plate is lifted by hand in our experiment. Then, it is difficult to obtain experimental data of the

Figure 7 shows the measured trajectory of the centre of the rotor magnet for 15 s. In Figure 7, the origin of *x* and *z* coordinates is the centre of the picture captured by the camera. In Figure 7(a), dots indicate the positions of the rotor magnet centre measured every 0.25 s, *i.e.* 4 frames per second, and a smoothing line connects these dots in sequential order. The smoothing line in Figure 7(a) does not show the swaying motion correctly; however, we can observe that the rotor magnet levitates and sways in the range of ±3 mm in radial direction and ±2.3 mm in vertical direction. In the experiment, it is difficult to start rotation of the magnetic top at the designated initial point. Figure 7(a) suggests that the initial positions of In spite of low accuracy of the measured data and difficulties in reenacting experiments in the same condition, the analysed levitation height and the frequency of vertical swaying are well in accordance with the experimental values. However, the analysed frequency of radial swaying is about twice the experimental value. This difference seem to be derived from assumptions in the two-dimensional analysis such as the constant tilt angle of the rotor magnet. Simulated results for various tilt angles showed that the magnitude of the tilt angle significantly affects the radial motion, but does not affect the vertical motion of the rotor magnet. Furthermore, the analysis is based on two-dimensional equations of motion, and three-dimensional behaviour of the magnetic top in the experiment is measured as twodimensional video information.

These results show that the fundamental parameters of a magnetic top, such as levitation height and dimensions of the permanent magnets, can be determined well using the quasithree-dimensional analysis.
