**3. Levitating characteristics of a magnetic top based on quasi-threedimensional analysis**

#### **3.1. Principle of levitation of a magnetic top**

To investigate levitation characteristics intuitively, the authors have proposed a so-called 'magnetic force map' that shows the magnetic force acting on the rotor magnet at each mesh point in the magnetic field generated by the stator magnet. Magnetic forces at the mesh points above the stator magnet are shown in the vector diagram. Because the vertical component of the magnetic force is deducted by the weight of the levitating top, we can observe the net force acting on the top at a glance.

Table 1 shows the parameters of the analytical model used in this chapter. These parameters are for the experimental model introduced in Figure 1. The magnitude of current in each equivalent side current is determined to be equal to the magnetic field density at the surface of the permanent magnets and the measured values for the ferrite permanent magnets used in the experiments. Considering the thickness of the permanent magnets, the number of the equivalent current coils is set to be 2 for the rotor magnet and 24 for the stator magnet in the simulation. Each circular coil current is simulated as a set of 72 linear current elements. These parameters are determined considering the accuracy of calculated results and the required time for computation.


**Table 1.** Parameters used in simulation

140 Performance Evaluation of Bearings

**Figure 4.** Angular momentum of a top

the 4th order Runge-Kutta method:

neglected for easy calculation.

**dimensional analysis** 

**3.1. Principle of levitation of a magnetic top** 

observe the net force acting on the top at a glance.

The moment *N*

*dL dt N*

in Equation (7) corresponds to torque and can be estimated by the magnetic

force acting on the rotor magnet with a calculation based on the equivalent coil currents model. The motion of the magnetic top can be simulated using the following equations by

 *tn*+1 = *tn* + *h* (9)

 *k*1 = *h f*(*tn*, *vn*) (10)

 *k*2 = *h f*(*tn*+*h*/2, *vn*+*k*1/2) (11)

 *k*3 = *h f*(*tn*+*h*/2, *vn*+*k*2/2) (12)

 *k*4 = *h f*(*tn*+*h*, *vn*+*k*3) (13)

 *vn*+1 = *vn* + (*k*1 + 2 *k*2 + 2 *k*3 + *k*4)/6 (14)

where *h* is the incremental time. In this analysis, the aerodynamic damping effects are

To investigate levitation characteristics intuitively, the authors have proposed a so-called 'magnetic force map' that shows the magnetic force acting on the rotor magnet at each mesh point in the magnetic field generated by the stator magnet. Magnetic forces at the mesh points above the stator magnet are shown in the vector diagram. Because the vertical component of the magnetic force is deducted by the weight of the levitating top, we can

**3. Levitating characteristics of a magnetic top based on quasi-three-**

(7)

*n n* <sup>1</sup> *L L dL* (8)

Figure 5 shows the magnetic force map calculated for the parameters given in Table 1. The figure shows the distribution of the magnetic force acting on the rotor magnet at each mesh point in the vertical plane including the *z–x* plane. Although the magnetic force map displays the force distribution in a two-dimensional plane, the magnetic forces are calculated considering three-dimensional shapes and layout of the equivalent side currents.

Figure 5(a) shows the magnetic force map in case the tilt angle of the rotor magnet is zero, that is, the rotor magnet is laid out horizontally in the area above the stator magnet. This figure shows that the force distribution is not uniform in the space above the stator magnet. There are two singular points along the *z*-axis: points A (0, 99.5) and B (0, 91.5) (Figure 5(a)). At point A, the magnetic forces acting on the rotor magnet are stable in the vertical direction but unstable in the horizontal direction. On the contrary, at point B, the magnetic forces acting on the rotor magnet are unstable in the vertical direction but stable in the horizontal direction. These results show that the magnetic top cannot levitate when its axis is parallel to the vertical axis; this result accords with the Earnshaw's theorem.

Figure 5(b) shows the magnetic force map when the tilt angle of the rotor magnet *θ* is set to 1° in *x* < 0 to −1° in *x* > 0. This figure shows that there is a point where the magnetic forces acting on the rotor magnet are stable in the both horizontal and vertical directions, as shown by the point C (0, 99.5) in Figure 5(b). In other words, the magnetic forces will guide the rotor magnet to the equilibrium point C, named as the 'restoring centre' in this chapter.

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 it rotates with a slight precession.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 143

**Figure 6.** Simulated behaviour of the centre of the rotor magnet based on two-dimensional analysis.

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

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

second.

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

## **3.2. Simulation to investigate the behaviour a 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 set to *θ* = 1° in the area *x* < 0 and to −1° in the area *x* > 0.

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 Hz, respectively.

**Figure 6.** Simulated behaviour of the centre of the rotor magnet based on two-dimensional analysis.
