versão impressa ISSN 0100-7386
J. Braz. Soc. Mech. Sci. v.24 n.4 Rio de Janeiro nov. 2002
Single axis controlled attraction type magnetic bearing
O. HorikawaI; I. da SilvaII
IIDepartment of Mechatronics and Mechanical System Engineering, Escola Politécnica of University of São Paulo, Av. Prof. Mello Moraes, 2231, 05508-900 São Paulo, SP. Brazil
This paper presents a new type of magnetic bearing with active control only in axial direction. The bearing uses two pairs of permanent magnets working in attraction mode to restrict the radial motion and a control system composed of two electromagnets, a gap sensor and a controller to keep the axis in a fixed axial position. The principle, the dynamic model for axial motion and the control system for this bearing are presented. Finally, by experiments conducted in a prototype, the effectiveness of the presented concept is shown.
Keywords: Magnetic bearing, mechatronics, controlled bearing
Nowadays, magnetic bearings are utilized in many field of applications: machine tools, vacuum pumps, gyroscopes, robot arms, etc.. Also, in the microelectronics industry, magnetic bearing is considered a promising alternative as a machine element capable of assuring a high accuracy positioning of masks containing pattern for IC fabrication and that can operate in vacuum, where IC's of the latest generation are produced.
There are many researchers concerning magnetic bearings but most of than aimed application such as vacuum pumps and by other hand, referred to bearings with active control in at least 5 d.o.f (degrees of freedom) (e.g., Schweitzer, 1991). Since the control of each d.o.f., requires a sensor, an actuator and a controller, the entire system becomes complex in terms of the design of its mechanical /electrical part and the control system. Considering this, this paper presents a new concept of magnetic bearing, in which only 1 d.o.f. of an axis, namely the axial position, is actively controlled. The axis motion in other directions is restricted only by the action of permanent magnets. Different from similar type of bearing proposed by other authors (for example, Ohji et al. 1996), here permanent magnets will work in attraction mode, in order to avoid the problem of demagnetization (Campbell, 1994). Such configuration, with active control in only one d.o.f., is the most simple as possible. This is what Ernshaw principle (Earnshaw, 1939) states. There are studies showing possibilities of obtaining magnetic bearings without any active control by using superconductive materials (for example, Marion-Péra et al, 1994). However, until the moment, superconductivity is obtained at temperatures not higher than 77K, imposing serious difficulties in any practical application.
Principle of the New Bearing
Fig. 1 shows the schematics of the proposed bearing. A permanent magnet is fixed to the extremities of a rotary axis that passes through a pair of annular electromagnetic actuators attached to the base. Each actuator consists of an electromagnet combined with a permanent magnet axially polarized. The magnets are mounted as indicated in Fig.1 so as to have an attraction force between each pair of facing magnets at each side of the bearing. As it will be shown in the next section, assuring a minimum length to the rotor, the magnets assure the stability of the rotor in terms of radial and angular movements of the rotor. The stability in the remaining axial direction is obtained by an active control loop composed by a non - contact type sensor (eddy current type), a controller and electromagnetic actuators.
Magnetic Force and Stiffness
Fig.2 illustrates the pair of magnets at one side of the bearing. When one magnet is radially displaced (Fig.2(a)), the axial force fa, the radial force fr, the axial stiffness ka and the radial stiffness kr can be represented by Eqs.(1)~(4) (Yonnet, 1981). When one magnet is inclined with respect to the other (Fig.2(b)), the stiffness in terms of this inclination kg, can be represented by Eq.(5) (Delamare, 1994).
In Eqs.(1) ~ (4):
In Eqs.(1)~(7), J, S, p, and a represents respectively the magnetization, the cross section area, the perimeter and the thickness of the magnet.
Eqs.(1)~(4) shows that high radial stiffness can be achieved by the passive part of the bearing by using magnets with small thickness, large cross section area and large perimeter.
Minimum Length of the Rotor
As indicated by Eq. (3), the radial stiffness is positive. Therefore, the rotor is stable in terms of radial displacements. In the axial direction, although the stiffness is negative (Eq.(4)), the stability is obtained by the active control. However, the stability in terms of inclining motion of the rotor around an axis orthogonal to the axis of rotation has to be analyzed as follows.
Consider the configuration shown in Fig.3, in which the distance between magnets is represented by l. In front of each magnet, there is an another one operating in attraction mode, keeping a gap d. The position of magnets fixed to the base is different of that presented in Fig.1 in order to simplify the figure. Consider the pair of magnets in the left side. When the rotor is inclined clockwise, the gap between magnets increases at the upper side and decreases in the lower side. Thus, the attraction force at the upper side becomes smaller than that at the lower side. The opposite occurs with respect to the pair of magnets at the right side. These forces produce a momentum (SMt) which tends to increase the inclination of the rotor. However, the inclination of the rotor produces a radial displacement of the magnets fixed to the rotor with respect to that fixed to the base. Such displacement produces a radial force in opposition to the displacement, resulting in a momentum (SMr) which forces the rotor back to its original horizontal position. Therefore, the stability in terms of inclination is assured if the following condition is satisfied.
In Fig.3, an inclination was indicated around a generic point P. However it was already shown that the rotor is stable in terms of translation in the radial direction. So, the analysis of Eq.(8) requires only the study of the case in which P is at the middle of the rotor (x=l/2). Considering this situation and considering only small inclinations of the rotor (g @ 2Dr/l), following equations are obtained.
Also, Eqs.(4) and (5) results in:
By substituting Eqs.(9), (10) and (11) in Eq.(8) following relation is obtained.
The rotor will be stable in terms of inclination if the distance l between magnets at extremities of the rotor (length of the rotor) is larger than the diameter of magnets. As larger the distance l, the larger will be the momentum SMr and a larger stiffness in terms of inclination is obtained.
System Modelling and Control
To model the system for controlling the axial position of the rotor, some simplifying assumptions are adopted: (a) the rotor keeps the symmetry relative to the axis of rotation and (b) displacements are small and occur around the operating position. Under these assumptions, the dynamic model becomes as shown in Fig.4. The magnetic attraction force of permanent magnets, fm(t) and the electromagnetic attraction force of electromagnets, fem(t) were linearized with respect to the displacement x(t) (axial position of the rotor relative to an arbitrary point) and the current i(t) at the coils, as follow.
Where, kh and kt are magnetic and electromagnetic constants and ka is the gain of the voltage amplifier used to drive the electromagnets. Assuming the use of electromagnets with an inductance L and resistance R, the relation of applied voltage v(t) and the current i(t) that passes through the electromagnets is represented as follows:
Using Eqs(13) ~ (15), and being M the mass of the rotor including the magnets, following open loop transfer function G(s) of the system is obtained.
In this system, only one displacement sensor is utilized. The measured variable is the axial position x of the rotor relative to the operating position. This system is stabilized by a PID type controller given by Eq.(17).
Here, k, Ti , Td and t are respectively the gain of the controller, the time constant of integrator, the gain of differentiator and the time constant of the first order filter attached to the differentiator. The block diagram of the control system is presented in Fig.5.
As the first step in the study of this bearing, FeBa annular magnets (J = 0.25T, axial magnetization, S = 63mm2, p = 84.8mm and a = 7mm) was selected for constructive conveniences. Using such magnets, axial and radial forces were determined experimentally and simulated by using Eqs.(1)~(6) and also by finite elements method. Results are presented in Figs.6(a) and 6(b).
By the PID controller mentioned above, a series of experiments were executed in the prototype shown in Fig.7. Parameters of the bearing were measured and the parameters of the controller were defined by simulations based on Eqs.(16) and (17). All parameters and respective values are listed in Tab.1.
Fig.8 shows the axial position x of the rotor at 0rpm without applying it any intentional disturbance. A continuous vibration of 2mm amplitude is observed. This was mainly due to electrical noises of the sensor and of the electronic circuit of the controller.
Fig.9 shows the response to an impulse force applied to the rotor. Due to the impulse, the rotor displaces from its nominal position but restores this position rapidly showing its capability to attenuate vibrations.
The controller was also equipped with a entrance for the reference signal (reference position xr). Fig.10 shows the response to a stepwise input of 0.08mm. In less than 0.2s, the rotor reaches the commanded position and keeps this new position an accuracy of 2mm.
This result shows the capability of the bearing to execute fast and precise positioning of its rotor, opening possibilities, for example, to compensate systematic motion errors of the rotor in the axial direction.
Fig.11 shows results of measurement of the stiffness of the bearing in the radial direction. The radial stiffness is non-linear and increases as far the rotor is from the center.
Besides the sensor to measure the axial displacement of the rotor, the prototype was also equipped with two other displacement sensors (sensors 2 and 3 shown in Fig.7) to measure radial displacements and inclining motions of the rotor. Fig.12 shows readings of the sensors in the axial and radial directions (sensors 1, 2 and 3) with the rotor at 50rpm. The rotor keeps a fixed position with an error of less than 2mm. In addition to the error caused by electrical noises (Fig.8) an error synchronized with the rotation is observed. There are two possible causes for this error. The first is the geometrical error of the target measured by the sensor. The second is the non-uniformity in the magnetization of permanent magnets that produces cyclic forces in the axial direction. Similarly factors would caused errors also in the radial direction. Readings of sensors 2 and 3 contains cyclic errors synchronized with the rotation with amplitude of about 0.3mm.
Similar measurements were also executed with the rotor at 500rpm. Results are shown in Fig.13. In the axial direction, the error amplitude increased to approximately 4µm indicating that the vibration damping capability of the control system decreases as the frequency of the elements mentioned above, that originated the vibration, increases. However, in the radial direction, the amplitude of errors kept amplitude of about 0.3mm despite the rotation speed of the rotor was increased 10 times. This suggests that oscillations of the readings of the sensors 2 and 3 do not represent vibration of the rotor but effects of the form error of the target measured by the sensors.
Increasing the Radial Stiffness
Aiming the improvement of the radial stiffness of the bearing, two more permanent magnets were added to each side of the bearing as indicated in Fig.14. In this new configuration, each magnet of the rotor has at both sides a magnet fixed to the base. In this new configuration, for a same radial displacement of the rotor, the restoring force given by Eq.(3) simply becomes the twice, i.e., the radial stiffness is increased twice.
Even in this new configuration, the criterion presented in item 4 to assure the rotor stability is still valid. If the momentum that tends to incline the rotor (SMt, Eq.(9)) increases twice because of the two additional magnets, the restoring momentum (SMr, Eq.(10)) also increases twice for the same reason. Thus, the same conclusion given by Eq.(12) is obtained.
Without modifying the rotor and the electromagnets, two magnets were added to the prototype, as illustrated in Fig.14. Rigorously, the control systems suffer some changes. The constant kh of the dynamic model (Fig.4) and of the block diagram (Fig.5), becomes 2kh. However, simulations of the prototype showed that the effect of this change on the performance of the bearing is small. Thus, the same controller of the Tab.1 was used. Under these conditions, the same experiments executed with respect to the configuration using four magnets were repeated.
Similar results were obtained, excepting that, as expected, a two times larger radial stiffness was obtained (Fig.15).
Fig.16 shows readings of sensors 1, 2 and 3, with the rotor at 500rpm. Vibrations are observed on readings of sensors 2 and 3. Although differences in terms of shape, their amplitude is of approximately 0.3mm, very close to that of vibrations in the case of four magnets (Fig.13). A larger difference is observed in the amplitude of vibrations in axial direction. The amplitude increased almost twice compared to the case of Fig.13. When additional magnets were installed, all sensors and their holders had to be removed and mounted again. It is supposed that, this caused slight changes in the portion of targets where the sensors made the measurements. A conclusive study on these problems will be object of future works. Despite these problems, results showed that the bearing with increased stiffness is still capable of suspend the rotor with stability even at a speed of 500rpm.
A new concept of magnetic bearing with active control in only axial direction was presented. Other degrees of freedom are restricted only by the action of permanent magnets that operate in attraction mode. The principle of the bearing was presented and the most relevant points to be considered in its design were described. Finally, by experiments, it was shown that the bearing is capable of: (a) suspended its rotor in a stable way, (b) keep the rotor in a fixed axial position with an accuracy of 2mm at 50rpm and (c) execute fast and precise positionings of the rotor in axial direction. Although the presented prototype does not show an enough radial stiffness for some practical application, this can be improved by optimizing characteristics of permanent magnets. This is the theme for future works.
The authors register their gratitude to the "Fundação de Amparo à Pesquisa do Estado de São Paulo" (FAPESP) and the Electro Mechanical Technology Advancing Foundation (EMTAF, Japan) for the financial support and, to Prof. Akira Shimokohbe of Tokyo Institute of Technology for the valuable comments and discussions.
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Presented at COBEM 99 15th Brazilian Congress of Mechanical Engineering, 22-26 November 1999, São Paulo, SP. Brazil.
Technical Editor: José Roberto F. Arruda.