Magnetic Force Transmission of a Reciprocating Motion

Journal of Bionic Engineering 5 (2008) 143í147

Magnetic Force Transmission of a Reciprocating Motion Qing-chang Tan, Fu-sheng Zheng, Jian-gang Li Institute of Mechanical Science and Engineering, Jilin University, Changchun 130022, P. R. China

Abstract Magnetic force transmission of a reciprocating motion is studied by theoretical analysis and experiment. A mathematical model for calculating the magnetic force is derived using the theory of equivalent magnetic charges. An experimental rig is constructed to test the transmission and the model is verified by experiment. Effect of the transmission parameters on the magnetic force is analyzed theoretically from the model, and characteristic of the transmission is studied experimentally. Since the transmission is without direct contact between two elements, it is suitable for application in an organism. Keywords: reciprocating motion, magnetic force; transmission Copyright © 2008, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved.

1 Introduction Creatures are adaptable. Their senses, musculature and morphology seem more elaborate and perfect than artificial machinery, such as fish motion in water is efficient, with low noise, high speed and good flexibility. Therefore, bionic underwater robots have become one of the focuses of robotics in recent years[1]. Such a robot can be classified into one of three classes: fish robot[2], walking underwater robot[3,4] and wriggling underwater robot[5]. Some underwater robots are actuated using a reciprocating motion produced by electromagnetic or piezoelectric actuator. Bionic reciprocating motion also appears in medical and biological applications such as microsyringes, micropumps and ventricular assisting devices. Several different principles have been applied such as piezoelectric, electrostatic, magnetic and electromagnetic to produce reciprocating motion[6]. Evans et al[7] studied a high-performance linear device for the artificial heart. Fukunaga et al[8] developed a linear oscillatory actuator for a left ventricular assisting device. The developed linear devices provide reciprocating motion without any movement converter such as a roller screw or a hydraulic system. They consist of stators with coils and movers with permanent magnets powered electrically. Battery life is important when the devices Corresponding author: Qing-chang Tan E-mail: [email protected]

work in the organism. Present work studies a linear magnetic consists of two movers made of rare-earth Permanent Magnetic (PM) material. The two movers are separated by non-ferromagnetic structures (such as pipe or plate). When an active mover moves, the magnetic force between the two movers drives the passive one, transmitting the reciprocating motion. Thus, the passive mover does not directly use electricity. This allows the passive mover to work within the organism without a source electric of electricity. The two movers are made of neodymium-iron-boron (Nb-Fe-B) PM. A mathematic model is established by the theory of equivalent magnetic charges to calculate the magnetic force of transmitting the linear motion. An experimental rig of the transmission is constructed and the model is verified by experiment. The characteristic of the transmission is analyzed theoretically and experimentally.

2 Model of the magnetic force of transmitting the reciprocating motion The principle by which magnetic force transmits the reciprocating motion between two movers is shown in Fig. 1. The magnetic force drives the inner magnetic mover in the same direction when the outer magnetic mover moves and the outer and inner movers do not

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144

contact directly. From the theory of equivalent magnetic charges[9], the magnetic force comes from the interaction of magnetic charges distributed on the two mover surfaces. The surfaces are perpendicular to the magnetization direction of the PM material. The magnetic force is produced by magnetic charges on outer mover surfaces 1, 2 acting with those on inner mover surfaces 3, 4. The magnetic force F13 produced by magnetic charges on the surfaces 1 and 3 is calculated first. Magnetic charges at any point P on surface 1 are expressed using the face density of the magnetic charge distribution as

From Ref. [10], the magnetic charge face density of the PM material is related to the intensity of the residual magnetic induction: ı1 = Br1 , ı2 = Br2 ,

where Br1, Br2 are the residual magnetic-induction intensities of PM material. Component of dF13 in axial direction of the movers is dF13

V 1V 2 r1r2 dr1dr2 dD dE 4ʌP0 r13

3

˜ r13 ,

(3)

where ȝ0 is magnetic conduct rate of the vacuum, ȝ0 = 4ʌ × 10í7 (H·mí1); r13 is position vector from point P to point Q, r13 = r + z0 = r2 í (r1 + e) + z0.

Br1 Br 2 r1r2 dr1dr2 dD dE ˜ ˜ Z0 , 3 4ʌP0 r13

ª r13 ˜ i 2   r13 ˜ j 2   r13 ˜ k 2 º ¬ ¼

3/2

2

 r2 cos E  r1 cos D  e

2

(6)

 Z 0 2 ]3/2 ,

where i, j, k are unit vectors of three coordinate directions; e is eccentricity between the two movers; Z0 is difference between axial positions of the inner and outer movers. Eq. (5) is a differential form of the magnetic force F13. Similarly, dF24, is a differential form of the magnetic force produced by the magnetic charges on surfaces 2 and 4, is equal to dF13 since the position of surface 2 is symmetrical to surface 4. In the same way, differential forms of the magnetic forces between surfaces 1 and 4, surfaces 2 and 3, are expressed respectively as dF14

A

dF23

A

where A r14

3

r1r2 dr1dr2 dD dE r14

r1r2 dr1dr2 dD dE r23

˜  h1  Z 0 ,

(7)

˜  h2  Z 0 ,

(8)

3

3

Br1 Br 2 ; 4ʌP0 2

ª¬ r2 sin E  r1 sin D 

 r2 cos E  r1 cos D  e r23

3

2

2   h1  Z 0 º ¼

3/2

,

(9)

2

ª¬ r2 sin E  r1 sin D 

 r2 cos E  r1 cos D  e Fig. 1 Sketch of calculating the magnetic force.

(5)

[ r2 sin E  r1 sin D 

(2)

where ı1 and ı2 are the face densities of magnetic charges on surfaces 1 and 3 respectively; r1 is a vector from outer mover center to point P, r1=|r1|; r2 is a vector from inner mover center to point Q, r2=|r2|. As shown in Fig. 1, the magnetic force produced by the magnetic charges at points P and Q is expressed using Eq. (1) and Eq. (2) as[9] dF13

dF13 ˜ k

3

r13

(1) MP = ı1r1dĮdr1 , similarly, magnetic charges at point Q on surface 3 are MQ = ı2r2dȕdr2 ,

(4)

2

2   h2  Z 0 º ¼

3/2

, (10)

where h1, h2 are axial widths of the inner and outer movers.

Tan et al.: Magnetic Force Transmission of a Reciprocating Motion

Directions of the above forces are determined by the magnetic polarities of the movers. Thus, a differential form of the total magnetic force in the axial direction is obtained using Eqs. (5), (7) and (8) as

dFZ

dF13 ˜ 2  dF14  dF23 ª 2Z h  Z h  Z º A « 03  1 3 0  2 3 0 » ˜ r1r2 dr1dr2 dD dE r14 r23 «¬ r13 »¼ (11)

Let A1

§ 2Z h Z h Z · ¨ 03  1 3 0  2 3 0 ¸ ; ¨r r14 r23 ¸¹ © 13

 r2 sin E  r1 sin D

A2

3/ 2

2

2

  r2 cos E  r1 cos D  e .

Thus r13

3

ª¬ A2  Z 0 2 º¼

r23

3

ª A2   h1  Z 0 2 º ¬ ¼

; r14

ª A2   h1  Z 0 2 º ¬ ¼

3

3/ 2

3/ 2

;

.

145

ignored. The displacement sensor (9) of the resistance type is connected to the strain meter (DH3818) through a half bridge circuit, is used to measure the axial position difference Z0 between the two movers. The inner mover (5) is loaded gradually by the mass (1) until it separates from the outer one. Each mass increment and the difference Z0 with respect to the mass are measured. The relation between the measured magnetic force and the difference Z0 is shown in Fig. 3. The figure also includes results predicted by Eq. (12) and data in Table 1. The curves from three time measures verify that Eq. (12) is correct for predicting the magnetic force transmitting the reciprocating motion. The difference between the calculated magnetic force and the measured results is due mainly to two factors: the friction of the rig (such as that between the inner mover (5) and the sleeve (4)) and the assumption that the magnetic charge distribution is uniform as in Eq. (12).

Total magnetic force is obtained by integrating Eq. (11) along surfaces 1, 2, 3 and 4 of the movers: 2ʌ 2ʌ R2 R4

FZ

A³

³ ³ ³ A r r dr dr dD dE 1 1 2

1

2

.

(12)

0 0 R1 R3

Eq. (12) expresses the magnetic force of transmitting the reciprocating motion. The magnetic force depends on the axial position difference between the two movers as other conditions are given.

3 Experiment for verifying the magnetic force model To verify Eq. (12), the magnetic force between the two movers is measured using an experimental rig (Fig. 2) which consists of the two magnetic movers, a slider-crank linkage, the load unit, and a measuring device. The PM material of the movers is Nb-Fe-B magnet[11]. The magnetic parameters and dimensions of the movers are listed in Table 1. The slider-crank linkage makes the outer mover reciprocate. The motion velocity of the slider (7) is adjusted by the motor through a frequency converter and the outer mover (6) is attached to the slider. The magnetic force makes the inner mover (5) reciprocates with the mover (6). The inner mover is loaded by a mass (1) which equals the magnetic force if the friction in the rig is

Fig. 2 Experimental rig. 1-Salver and poises, 2-Pulley, 3-Force sensor, 4-Sleeve, 5-Inner magnetic mover, 6-Outer magnetic mover, 7-Slider-crank linkage, 8-Motor and frequency converter, 9-Displacement sensor, 10-Bridge boxes.

Table 1 PM material parameters and dimensions of the magnetic movers PM material parameters

Geometry dimensions of the magnetic movers

Remanence Br =1.298 (T)

Inner movers Thickness h2 = 0.0340 m Inner radius R1 = 0.0150 m Outer radius R2 =0.0394 m

Coercive force Hc = 900 (kA·m) Energy product (BH)max = 305 (kJ·mí3)

Outer movers Thickness h1 = 0.0340 m Inner radius R1 = 0.0454 m Outer radius R2 = 0.0810 m

Eccentricity e = 0.0001 m

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the transmission is not suitable for the reciprocating motion with high frequency when the driven load is large.

Fig. 3 Comparison of the calculated magnetic force with the measured results.

4 Discussion First, the effects of the mover parameters (see Fig. 1) on the magnetic force are analyzed by Eq. (12), as shown in Figs.4 and 5. The curves in Fig. 4 indicate that the magnetic force increases with the width of the inner mover. From Fig. 5, one can see that the magnetic force reduces with increase of the radial distance Lg between the two movers. The magnetic force is 0.3 N when the radial distance is 50 mm, the inner mover diameter is 10 mm and the width of the inner mover is 2 mm. Thus it is possible for the inner mover in the organism to be driven magnetically at a distance up to 50 mm, without direct contact between the two movers. Therefore, the inner mover can work in the organism without a power supply or cables. Since the magnetic force transmission is without contact between the two movers and has magnetic hysteresis, the inner mover may separate from the outer when the outer starts or accelerates quickly, causing the transmission to fail. This also affects the frequency of reciprocation of the inner mover. Experimentally, a load is exerted on the inner mover, and the outer one is driven by the slider-crank linkage. The frequency of oscillation of the outer mover is increased by the motor through the frequency converter (see Fig. 3) until it separates from the inner mover. The highest frequency with respect to the load is recorded. Then, the load is changed and the above step is repeated. Curve in Fig. 6 shows relationship between the highest frequency of the outer mover and the load borne by the inner mover. The maximum frequency decreases as the load increases, showing that

Fig. 4 Effect of axial width of the magnetic movers on the magnetic force at Z0 = (h1+h2)/2, h1 = 20, R1 = 0 and t1 = 5.

Fig. 5 Effect of radial distance between two magnetic movers on the magnetic force at Z0 = (h1+h2)/2, h1 = 20, t2 = 32, R1 = 0 and t1 = 5.

Fig. 6 Relation of the magnetic force with the highest reciprocating frequency.

Tan et al.: Magnetic Force Transmission of a Reciprocating Motion

5 Conclusions From the above work, the following results are obtained: (1) The reciprocating motion can be transmitted by the magnetic force with two movers separated (up to a certain distance) so that a mover can work in the organism without ant power supply. This characteristic can be used to develop some bionic equipment, such as underwater robots, micropumps and ventricular assist devices in which the work is driven by reciprocating motion without using cable for power supply. (2) A mathematic model for predicting the magnetic force has been developed by the theory of equivalent magnetic charges and verified by experiment. The reciprocating frequency decreases as the transmitted load increases.

Acknowledgement This work is financially supported by the Natural Fund of Jilin Province, P. R. China (Grant No. 20030525).

References [1]

Sfakiotakis M, Lane D M, Davies J B C. Review of fish swimming modes for aquatic locomotion. IEEE Journal of Oceanic Engineering, 1999, 24, 237–252.

[2]

Fukuda T, Kawamoto A, Arai F, Matsuura H. Steering mechanism of underwater micro mobile robot. Proceedings

147

of the 1995 IEEE International Conference on Robotic and Automation, Nagoya, Japan, 1995, 363–368. [3]

Safak K K, Adams G G. Modeling and simulation of an artificial muscle and its application to biomimetic robot posture control. Robotics and Autonomous System, 2002, 41, 225–243.

[4]

Chen H, Zhu C A, Yin X Z, Xing X Z, Cheng G. Hydrodynamic analysis and simulation of a swimming bionic robot tuna. Journal of Hydrodynamics, 2007, 19, 412–420.

[5]

Vaidyanathan R, Chiel H J, Quinn R D. A hydrostatic robot for marine applications. Robotics and Autonomous Systems, 2000, 30, 103–113.

[6]

Lemoff A V, Lee A P. An AC magnetohydrodynamic micropump. Sensors and Actuators B: Chemical, 2000, 63, 178–185.

[7]

Evans S A, Smith I R, Kettleborough J G. Permanent-magnet linear actuator for static and reciprocating short-stroke electromechanical systems. IEEE/ASME Transactions on Mechatronics, 2001, 6, 36–42.

[8]

Fukunaga K, Funakubo A, Fukui Y. Newly developed ventricular assist device with linear oscillatory actuator. ASAIO Journal, 2003, 49, 333–339.

[9]

Wazed Miah M A. Fundamentals of Electromagnetics, Tata Mcgraw-Hill Publishing Company Limited, New Delhi, 1982.

[10] Zahn M. Electromagnetic Field Theory: A Problem Solving Approach, John Wiley and Sons Inc, New Delhi, 1979. [11] Nb-Fe-B Magnet, [2008-5-7], http://www.rknerc.com/cp_yc.htm