**4. Discussion and conclusions**

Finally, let's conclude this chapter by discussing a real online example where it is required to bring the pH from 3 to 7. This example was chosen to show how the fuzzy controller has successfully solved oscillation due to titration curve mentioned earlier. Using all the expertise to set the membership functions as well as the rules, plus a fine tuning process, the membership function is shown in figure (27), and the pH response is shown in figure (28). It is clear that the pH has been brought from 3 to 7 in a very smooth manner. This value has been confirmed through the pH meter reading.

As a general conclusion, due to non linearity inherent in the chemical waist, the study presented here has shown that using fuzzy logic control is probably best suited to control the pH of industrial waist, despite the expertise required to fine tune the controller as well as the time required. It has also been shown that no matter how the linear controllers are tuned especially PID, including the many restrictions, plus the delays and settling time for each disturbance, their action was not satisfactory. Finally, the study has also shown that implementing the fuzzy control technique, by careful selection of memberships and setting the right rules, it is possible to bring any waist pH to neutrality smoothly in an acceptable time, regardless if the waist is strong base nature or strong acidic or not.

**Figure 27.** First membership function

266 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

This section performs the task of converting the output of inference mechanism, the rules that are fired, and the DoS given by the fuzzification module into a signal to the control valve. For this, it uses "height defuzzification" which is computationally simple and

Finally, let's conclude this chapter by discussing a real online example where it is required to bring the pH from 3 to 7. This example was chosen to show how the fuzzy controller has successfully solved oscillation due to titration curve mentioned earlier. Using all the expertise to set the membership functions as well as the rules, plus a fine tuning process, the membership function is shown in figure (27), and the pH response is shown in figure (28). It is clear that the pH has been brought from 3 to 7 in a very smooth manner. This value has

As a general conclusion, due to non linearity inherent in the chemical waist, the study presented here has shown that using fuzzy logic control is probably best suited to control the pH of industrial waist, despite the expertise required to fine tune the controller as well

**3.5. Defuzzification**

**4. Discussion and conclusions**

been confirmed through the pH meter reading.

fast.

**Figure 28.** First test online at set point 7


**4.3. Effects of the initial position**

154 Performance Evaluation of Bearings

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

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

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 restoring centre.

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.

To investigate the effects of the initial point with regard to the restoring centre (0, 0, 99.5), simulations were performed for the case of the typical initial point of (1, 0, 99.5), *i.e.* 1 mm apart in *x* direction from the restoring centre, and (0, 0, 98.5), *i.e.* 1 mm apart in *z* direction from the restoring centre. Figures 18(a) and (b) show the simulated trajectories of the centre of the rotor magnet, rotating at 1,380 rpm for 60 s starting from (1, 0, 99.5) and (0, 0, 98.5), respectively.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 157

**Figure 19.** Trajectories of the rotor magnet centre for 60 s starting from the restoring centre (0, 0, 99.5)

0.001165°.

In contrast, the magnetic top, starting at the point 1 mm apart in *z* direction from the restoring centre, levitates in the range of ±0.002 mm in both *x* and *y* directions and ±1 mm in *z* direction around the restoring centre, as shown in Figure 18(b). The maximum tilt angle is

Figure 19 shows the simulated trajectory of the centre of the rotor magnet, rotating at 1,380 rpm for 60 s starting from the restoring centre (0, 0, 99.5). In this case, a magnetic top levitates in the area of ±0.0003 mm in *x*–*y* plane and +0.016 mm/−0.003 mm in *z* direction. These results show that a rotating magnetic top can maintain levitation within several

Thus, the magnetic top has the ability to function as an entirely passive magnetic bearing [7].

micrometres displacements in both radial and vertical directions.

The magnetic top, starting from the point 1 mm apart in *x* direction from the restoring centre, levitates in the range of ±1.07 mm in both *x* and *y* directions and from +0.12 mm/ to 0.06 mm in *z* direction around the restoring centre, as shown in Figure 18(a). The maximum tilt angle is 1.018°.

**Figure 17.** Simulated trajectories of a magnetic top in 60 s starting from (1, 0, 98.5), 1380 rpm.

**Figure 18.** Trajectories of the centre of the rotor magnet in *z*–*x* plane.

respectively.

tilt angle is 1.018°.

To investigate the effects of the initial point with regard to the restoring centre (0, 0, 99.5), simulations were performed for the case of the typical initial point of (1, 0, 99.5), *i.e.* 1 mm apart in *x* direction from the restoring centre, and (0, 0, 98.5), *i.e.* 1 mm apart in *z* direction from the restoring centre. Figures 18(a) and (b) show the simulated trajectories of the centre of the rotor magnet, rotating at 1,380 rpm for 60 s starting from (1, 0, 99.5) and (0, 0, 98.5),

The magnetic top, starting from the point 1 mm apart in *x* direction from the restoring centre, levitates in the range of ±1.07 mm in both *x* and *y* directions and from +0.12 mm/ to 0.06 mm in *z* direction around the restoring centre, as shown in Figure 18(a). The maximum

**Figure 17.** Simulated trajectories of a magnetic top in 60 s starting from (1, 0, 98.5), 1380 rpm.

**Figure 18.** Trajectories of the centre of the rotor magnet in *z*–*x* plane.

**Figure 19.** Trajectories of the rotor magnet centre for 60 s starting from the restoring centre (0, 0, 99.5)

In contrast, the magnetic top, starting at the point 1 mm apart in *z* direction from the restoring centre, levitates in the range of ±0.002 mm in both *x* and *y* directions and ±1 mm in *z* direction around the restoring centre, as shown in Figure 18(b). The maximum tilt angle is 0.001165°.

Figure 19 shows the simulated trajectory of the centre of the rotor magnet, rotating at 1,380 rpm for 60 s starting from the restoring centre (0, 0, 99.5). In this case, a magnetic top levitates in the area of ±0.0003 mm in *x*–*y* plane and +0.016 mm/−0.003 mm in *z* direction. These results show that a rotating magnetic top can maintain levitation within several micrometres displacements in both radial and vertical directions.

Thus, the magnetic top has the ability to function as an entirely passive magnetic bearing [7].

#### **4.4. Effects of the air drag force**

In the previous analysis, the aerodynamic effects were neglected to simplify the discussion. If a magnetic top is rotating in air, rotating speed of the top will decay because of the pneumatic resistance acting on the surfaces of the top. In actual, the experiments showed that the rotating speed of the magnetic top decreases as time passes and the attitude of the top changes to a larger precession that leads it to fall down in a few minutes. Because there are no conducting materials in the magnetic top, there is no electrodynamic drag force caused by eddy currents. Hence, the aerodynamic drag force can be considered as the main reason for the decreasing rotation speed. In this section, some simulations are performed based on the equations of motion considering the aerodynamic drag force.

The aerodynamic effects to the behaviour of a rotating magnetic top is estimated as the pneumatic resistance acting on the outer side surface of the magnetic top. Here, the aerodynamic drag effects caused by the pole surfaces of the magnetic top are neglected. The following expressions are added to estimate the aerodynamic effects to the rotating speed of the magnetic top:

$$
\rho o\_{n+1} = o\_n - \frac{F\_d}{I\_{Top}} dt \tag{15}
$$

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 159

**Figure 21.** Time dependence of rotating speed and tilt angle of a levitating magnetic top.

**Figure 22.** Trajectories of a levitating magnetic top started at 1380 rpm and fall down at 275 s after start.

Figure 20 shows the simulated trajectory of the centre of the levitating magnet starting from 1 mm apart in both *x* and *z* directions from the restoring centre. Initial rotation speed is set to be 1,380 rpm. The coefficient of pneumatic resistance *CD* is set to be 0.5 for Figure 20(a) and 5.0 for Figure 20(b). Figure 20 shows that the magnetic top can levitate for 275 s or 28.4 s, if the coefficient of pneumatic resistance *CD* is 0.5 or 5.0, respectively. Experiments showed that the magnetic top can be levitated for 3–4 min. Hence, in this study, the coefficient of

Figure 21 shows the time dependence of the rotation speed and the tilt angle of the levitating magnetic top. Figure 21(a) shows that the magnetic top started at 1,380 rpm and

pneumatic resitance *CD* is assumed to be 0.5.

maintained levitation till 166 rpm at 275 s.

$$F\_d = \frac{1}{2}\rho \mathcal{C}\_d A\_r v\_r^2 r\_{ro} \tag{16}$$

where *ρ* = 1.225 kg/m3 is the density of air, *CD* is the coefficient of pneumatic resistance, *Ar* = 2*πrroh* is the area of the outer side surface of the top and *vr* = *rroω* is the velocity of the outer side surface of the rotating top [8].

**Figure 20.** Simulated trajectory of the levitating magnet centre, initial rotating speed is 1380 rpm.

the magnetic top:

side surface of the rotating top [8].

**4.4. Effects of the air drag force**

In the previous analysis, the aerodynamic effects were neglected to simplify the discussion. If a magnetic top is rotating in air, rotating speed of the top will decay because of the pneumatic resistance acting on the surfaces of the top. In actual, the experiments showed that the rotating speed of the magnetic top decreases as time passes and the attitude of the top changes to a larger precession that leads it to fall down in a few minutes. Because there are no conducting materials in the magnetic top, there is no electrodynamic drag force caused by eddy currents. Hence, the aerodynamic drag force can be considered as the main reason for the decreasing rotation speed. In this section, some simulations are performed

The aerodynamic effects to the behaviour of a rotating magnetic top is estimated as the pneumatic resistance acting on the outer side surface of the magnetic top. Here, the aerodynamic drag effects caused by the pole surfaces of the magnetic top are neglected. The following expressions are added to estimate the aerodynamic effects to the rotating speed of

*d*

(15)

(16)

*Top F dt I*

1 2 2 *d d r r ro F C Avr*

where *ρ* = 1.225 kg/m3 is the density of air, *CD* is the coefficient of pneumatic resistance, *Ar* = 2*πrroh* is the area of the outer side surface of the top and *vr* = *rroω* is the velocity of the outer

**Figure 20.** Simulated trajectory of the levitating magnet centre, initial rotating speed is 1380 rpm.

based on the equations of motion considering the aerodynamic drag force.

1

*n n*

**Figure 21.** Time dependence of rotating speed and tilt angle of a levitating magnetic top.

**Figure 22.** Trajectories of a levitating magnetic top started at 1380 rpm and fall down at 275 s after start.

Figure 20 shows the simulated trajectory of the centre of the levitating magnet starting from 1 mm apart in both *x* and *z* directions from the restoring centre. Initial rotation speed is set to be 1,380 rpm. The coefficient of pneumatic resistance *CD* is set to be 0.5 for Figure 20(a) and 5.0 for Figure 20(b). Figure 20 shows that the magnetic top can levitate for 275 s or 28.4 s, if the coefficient of pneumatic resistance *CD* is 0.5 or 5.0, respectively. Experiments showed that the magnetic top can be levitated for 3–4 min. Hence, in this study, the coefficient of pneumatic resitance *CD* is assumed to be 0.5.

Figure 21 shows the time dependence of the rotation speed and the tilt angle of the levitating magnetic top. Figure 21(a) shows that the magnetic top started at 1,380 rpm and maintained levitation till 166 rpm at 275 s.

Figure 21(b) shows the time dependence of the tilt angle of the rotor magnet with respect to *z*-axis. This figure shows that the tilt angle varies within 1.4° while the top is levitating and indicates that precession is needed to maintain levitation for a magnetic top.

Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets 161

rotor weight and rotation speed. A rotor shaft should be designed considering mechanical

In the experimantal model, because ferrite magnets are used for the rotor and stator magnets, the restoring forces are very small for commercial applications. However, if rare earth permanent magnets and rigid suspension devices are used, sufficient restoring forces may be expected to be generated for use as a commercial passive magnetic

The authors thank Mr. Makoto Matsumoto, former student of the Graduate school of Natural Science and Technology, Kobe University, for his efforts in establishing simulation

[1] The Magnetic Levitation Technical Committee of the IEEJ (1993) Magnetic Suspension Technology - Magnetic Levitation Systems and Magnetic Bearings. Corona Publishing

[2] Matsumoto M, Azukizawa T (2004) Characteristics of Magnetic Guidance Force Between Coaxial Ring Shaped Permanent Magnets. IEEJ Technical Meetings on Linear Drives,

[3] Miguel A.S. et al. (2005) Numerical integration for the dynamics of the heavy magnetic

[4] Ebihara D , Suzuki T (1988) The Repulsive Characteristics of the PM Type Magnetic

[5] Azukizawa T, Matsuo N (2007) Feasibility Study of a Magnetic Top As a Magnetic Bearing. Proc. of the 6th International Symposium on Linear Drives for Industrial

[6] Azukizawa T, Yamamoto S, Matsuo N (2008) Feasibility Study of a Passive Magnetic Bearing Using the Ring Shaped Permanent Magnets. IEEE Trans. on Mag., Vol.44,

[7] Azukizawa T, Yamamoto S, Makino H (2008) Effects of the System Parameters to the

Levitation Devices. Trans. IEEJ, Vol.108-D, No.5, 455-461, in Japanese.

requirements such as torque.

bearing.

**Author details**

Teruo Azukuzawa

Shigehiro Yamamoto

**Acknowledgement**

Co. Ltd., in Japanese.

LD-04-93, in Japanese.

No.11, 4277-4280.

top. Physics Letters A 335 235–244.

Applications, LDIA 2007(CD), 138.

Behavior of a Magnetic Top. MAGLEV08, No. 63.

**6. References**

*Japan Transport Safety Board, Tokyo, Japan*

*Graduate School of Maritime Sciences, Kobe University, Kobe, Japan*

tools for analysing the dynamic behaviour of a magnetic top.

Figure 22 shows the trajectories of the magnetic top at the last 15 s of its levitation. Figures 22(a) and (b) demonstrate the trajectories of the shaft head and the centre of the levitating rotor magnet in its final 15 s levitation. These figures show that the precession quickly becomes larger once the magnetic top exits the levitating area.