**2.2. Magnetic force acting on the rotor magnet**

Magnetic force acting on a magnetic top can be estimated by the interaction between the magnetic field generated by the stator magnet and the equivalent coil currents of the rotor 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 equivalent coil currents in the rotor and stator magnets.

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 equation:

$$df = \frac{I\_1 I\_2 dl\_1 dl\_2}{r^2} \times 10^{-7} \sin \phi \tag{1}$$

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

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:

$$f = \int\_{l\_1} \int\_{l\_2} df \, dl\_1 dl\_2 = I\_1 I\_2 \int\_{l\_1} \int\_{l\_2} \frac{10^{-7}}{r^2} \sin \varphi \, dl\_1 dl\_2 \tag{2}$$

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

**Figure 1.** Experimental magnetic top.

136 Performance Evaluation of Bearings

**2. Analytical methods** 

assumed equivalent coils.

position related to the restoring centre.

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

The magnetic top is composed of a couple of ring-shaped permanent magnets magnetised in the axial direction, as shown in Figure 1. The magnetic top, equipped with a smaller ring-shaped permanent magnet (a rotor magnet), can be levitated in the magnetic field generated by the larger ring-shaped permanent magnet (a stator magnet) situated at its base, if it can maintain its rotation within a certain speed range. The levitating height is determined by the shapes and magneto-motive forces of the permanent magnets. The authors propose two types of analytical methods: (1) a quasi-three-dimensional analysis to investigate the principle of levitation and the design parameters of the permanent magnets and (2) a three-dimensional dynamic analysis to simulate the behaviour of the levitating magnetic top. The ring-shaped permanent magnet is

In the equivalent coil currents approximation, a ring-shaped permanent magnet, magnetised in the axial direction, is assumed to exist by the set of circular coil currents located at the outer and inner side surfaces of the ring-shaped permanent magnet [4]. The directions of currents in the outer and inner equivalent coils are inversed with each other, describing the axial magnetization of the permanent magnet, as shown in Figure 2. The magnitude of these equivalent side currents is determined so as to coincide with the measured magnetic field density at the pole surface of the permanent magnet in relation to the number of the

Figure 3 shows the analytical model based on the equivalent side currents model. The outer and inner diameters and the height of the ring-shaped permanent magnets are represented as *dso*, *dsi* and *hs* for the stator magnet and *dro*, *dri* and *hr* for the rotor magnet, respectively. The angle *θ* is the tilt angle of the rotor magnet. The origin is set at the centre of the stator magnet. The stator magnet is located in the horizontal *x*–*y* plane and *z*-axis is set as the vertical direction along the axis of the stator magnet. The numbers of the equivalent side currents in both the rotor and stator magnets, indicated as one and three in Figure 3, are decided considering both the accuracy of the calculated results and the required time for computation.

Magnetic force acting on a magnetic top can be estimated by the interaction between the magnetic field generated by the stator magnet and the equivalent coil currents of the rotor

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

approximated to the equivalent coil currents model in both the analytical methods.

**2.1. The equivalent coil currents approximation** 

**2.2. Magnetic force acting on the rotor magnet** 

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 139

*dt* (3)

(4)

estimated using Equations (1) and (2), considering the circular shapes and layout of the equivalent coil currents. The behaviour of the levitating magnetic top in *z*–*x* plane is estimated by the following two-dimensional equations of motion for the rotor magnet:

> 2 *x* 2 *d x F m*

> > 2

where *Fx* and *Fz* [N] are the magnetic forces acting on the rotor magnet in *x* and z directions, respectively, *m* [kg] is the mass of the magnetic top, *g* [m/s2] is the acceleration due to gravity and (*x*, *z*) are the coordinates of the centre of the rotor magnet. Here, Equation (4) indicates that the vertical acceleration is derived from the difference between the vertical component of the magnetic force due to the stator magnet and the gravity force acting on

The quasi-three-dimensional analysis is used to investigate the principle of levitation of the magnetic top and determine with a short computing time the parameters of the magnetic

Because the quasi-three-dimensional analysis provides the design parameters of a magnetic top, behaviour of the magnetic top is investigated by a simulation based on threedimensional dynamic analysis considering rotation of the magnetic top. Behaviour of the magnetic top can also be estimated by the equations of motion on the angular moment of the rotor magnet, considering three-dimensional layout of the stator and rotor magnets, the tilt

When the magnetic top is rotating, the angular momentum force *ITop* will act around the axis of the rotating magnetic top. The momentum force *ITop* can be expressed as Equation (5), where *m* is the mass of the levitating magnetic top, *rro* and *rri* are the outer and inner radius of the ring-shaped rotor magnet. Angular momentum vector around the axis of the rotating magnetic top *Ln* at a time *tn* can be expressed as Equation (6), where *ω* is the angular velocity

*n Top L I*

is the moment caused by magnetic force acting on the

2 2 <sup>2</sup> *Top ro ri I mr r* (5)

at time *tn* + 1=*tn* + *dt* is expressed as

(6)

in an infinitesimal time *dt* is

top such as the sizes of the stator and rotor magnets and the levitation height.

angle of the rotor magnet and the mechanical inertia of the rotor magnet.

of the magnetic top. The incremental angular momentum *dL*

rotor magnet. The angular momentum vector *<sup>n</sup>* <sup>1</sup> *L*

expressed as Equation (7), where *N*

Equation (8):

**2.4. Three-dimensional dynamic analysis** 

*z* 2 *d z Fm g dt* 

the rotor magnet.

**Figure 2.** Equivalent side currents.

**Figure 3.** Analytical model.

## **2.3. Quasi-three dimensional analysis**

Because an ideal magnetic top is considered to levitate and rotate around the *z*-axis, basic information can be obtained by a simple discussion on the two-dimensional motion of the magnetic top in the vertical plane including the *z*-axis. Hence, the authors propose the quasi-three-dimensional analysis in which the magnetic force acting on the rotor magnet is estimated using Equations (1) and (2), considering the circular shapes and layout of the equivalent coil currents. The behaviour of the levitating magnetic top in *z*–*x* plane is estimated by the following two-dimensional equations of motion for the rotor magnet:

$$F\_x = m \frac{d^2 \chi}{dt^2} \tag{3}$$

$$F\_z = m \left(\frac{d^2 z}{dt^2} + g\right) \tag{4}$$

where *Fx* and *Fz* [N] are the magnetic forces acting on the rotor magnet in *x* and z directions, respectively, *m* [kg] is the mass of the magnetic top, *g* [m/s2] is the acceleration due to gravity and (*x*, *z*) are the coordinates of the centre of the rotor magnet. Here, Equation (4) indicates that the vertical acceleration is derived from the difference between the vertical component of the magnetic force due to the stator magnet and the gravity force acting on the rotor magnet.

The quasi-three-dimensional analysis is used to investigate the principle of levitation of the magnetic top and determine with a short computing time the parameters of the magnetic top such as the sizes of the stator and rotor magnets and the levitation height.

#### **2.4. Three-dimensional dynamic analysis**

138 Performance Evaluation of Bearings

**Figure 2.** Equivalent side currents.

**Figure 3.** Analytical model.

**2.3. Quasi-three dimensional analysis** 

Because an ideal magnetic top is considered to levitate and rotate around the *z*-axis, basic information can be obtained by a simple discussion on the two-dimensional motion of the magnetic top in the vertical plane including the *z*-axis. Hence, the authors propose the quasi-three-dimensional analysis in which the magnetic force acting on the rotor magnet is Because the quasi-three-dimensional analysis provides the design parameters of a magnetic top, behaviour of the magnetic top is investigated by a simulation based on threedimensional dynamic analysis considering rotation of the magnetic top. Behaviour of the magnetic top can also be estimated by the equations of motion on the angular moment of the rotor magnet, considering three-dimensional layout of the stator and rotor magnets, the tilt angle of the rotor magnet and the mechanical inertia of the rotor magnet.

When the magnetic top is rotating, the angular momentum force *ITop* will act around the axis of the rotating magnetic top. The momentum force *ITop* can be expressed as Equation (5), where *m* is the mass of the levitating magnetic top, *rro* and *rri* are the outer and inner radius of the ring-shaped rotor magnet. Angular momentum vector around the axis of the rotating magnetic top *Ln* at a time *tn* can be expressed as Equation (6), where *ω* is the angular velocity of the magnetic top. The incremental angular momentum *dL* in an infinitesimal time *dt* is expressed as Equation (7), where *N* is the moment caused by magnetic force acting on the rotor magnet. The angular momentum vector *<sup>n</sup>* <sup>1</sup> *L* at time *tn* + 1=*tn* + *dt* is expressed as Equation (8):

$$I\_{Top} = m \left(r\_{ro}^{-2} + r\_{ri}^{-2}\right) \Big/ 2 \tag{5}$$

$$L\_n = I\_{Top} \times \alpha \tag{6}$$

$$d\vec{L} = dt \times \vec{N} \tag{7}$$

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 141

Rotor magnet Stator magnet

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

> Outer diameter *do* [mm] 30 134 Inner diameter *di* [mm] 12 75 Thickness *h* [mm] 5 60

> Mass *m* [g] 20.37 - Tilt angle *θ* [deg] - 1 No. of equivalent coils 2 24

Magnitude of equivalent current *Ieq* [A/mm] 286 286

No. of current elements in an equivalent coil 72 72

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

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

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 vertical axis; this result accords with the Earnshaw's theorem.

required time for computation.

**Table 1.** Parameters used in simulation

currents.

$$
\vec{L}\_{n+1} = \vec{L}\_n + d\vec{L} \tag{8}
$$

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

The moment *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 the 4th order Runge-Kutta method:

$$t\_{n+1} = t\_n + lr\tag{9}$$

$$k\mathbf{u} = \ln f(t\_{\mathbf{u}\_{\prime}}, \mathbf{u}\_{\prime}) \tag{10}$$

$$kz = h \left\{ \mathbf{f}(\mathbf{t}\_{n} + h/\mathfrak{Z}, \,\,\mathrm{v}\_{n} + \mathrm{k}z/\mathfrak{Z}) \right\} \tag{11}$$

$$k\mathbf{z} = h\ f(\mathbf{t}\_{n} + h/\mathbf{\color{red}{2}}, \mathbf{v}\_{n} + k\mathbf{z}/\mathbf{\color{red}{2}})\tag{12}$$

$$k\omega = h\, f(t\_{\nu} + l\mathbf{r}, \,\mathbf{v}\_{\nu} + l\mathbf{x})\tag{13}$$

$$
\varpi\_{n+1} = \varpi\_n + (k\_1 + 2\text{ k}z + 2\text{ k}s + \text{k}z)/6 \tag{14}
$$

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