Design of Electromagnetic Pre-Pressure Device for 3-DOF Spherical Ultrasonic Motor

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Available online at www.sciencedirect.com Available online at www.sciencedirect.com Procedia Engineering Procedia Engineering 00 (2011) 000–000 Procedia Engineering 29 (2012) 2393 – 2398 www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE) Design of Electromagnetic Pre-Pressure Device for 3-DOF Spherical Ultrasonic Motor

Bin Lia*, Junjie Shia, Guidan Lia, Hongfeng Lia

aSchool of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

Abstract

This paper presents a new type of pre-pressure device for the three degree-of-freedom (3-DOF) spherical ultrasonic motors (USM). The device is composed of stator coils and spherical rotor with permanent magnet (PM) poles, and the electromagnetic forces between the coils and the PM poles are employed to generate the pre-pressure on the vibrator of the spherical USM. The pre-pressure balance equation of the device is obtained based on the theorem of force translation and the symmetry of the coils arrangement. To control the pre-pressure, the electromagnetic force of a coil with a current of 1A in the rotor spherical coordinate is studied by the finite element analysis (FEA), and the force matrix of the device is built by the coordinate rotation transformation, and then the control currents of the coils are determined. The new type of pre-pressure device can actively vary the preload of spherical USM and provide more flexibility for the control system.

Keywords:spherical motor; ultrasonic motor; permanent magnet; electromagnetic force analysis; pre-pressure generation

1. Introduction

Spherical ultrasonic motor (USM) is a new type of motor, which utilizes the friction force to drive the rotor to do 3-DOF motion[1]. In order to improve the output torque, the large pressure between the vibrator and the rotor is necessary. More importantly, the pressure is an important parameter of USM, which can directly impact on the contact state and energy transfer between the stator and the rotor, and

a * Corresponding author. Tel.: +86-022-27400983 ; fax: +86-022-27400983 . E-mail address: [email protected]

1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.321 2394 Bin Li et al. / Procedia Engineering 29 (2012) 2393 – 2398 2 Li Bin/ Procedia Engineering 00 (2011) 000–000 then affects the resonant frequency of the vibrator, operating efficiency, output power, and torque-speed characteristic of USM[2, 3]. To meet the requirements of the spherical USM, some pre-pressure devices employing the elastic force of springs have been designed [4, 5]. Besides, in Ref.[6], the permanent magnet plate was attached to the top of the vibrator, and the magnetic force attracted the steel rotor to generate the pre-pressure. Compared to the force generated by the springs or the permanent magnets, the electromagnetic force has better controllability. Therefore, the combination of the electromagnetic force and the USM can be taken advantages to design more sophisticated actuators. Yang Zhigan. has designed the electromagnetic clamping device for a precise piezoelectric actuator[7]. E. Purwanto has adopted the Hall sensor to measure the magnetic field generated by the magnetic rotor, and then the rotor position of spherical USM has been obtained [8]. A. Ferreira has designed an electromagnetic device in nano-positioning actuator, and adopted the variable structure control to adjust the pre-pressure [9]. However, the abovementioned devices are fit primarily for linear or plane USM, but can not be directly employed in spherical USM. Spherical permanent magnet (PM) motor is another choice for 3-DOF actuation [10-11]. Adopting the stator coils and PM rotor like that in the spherical PM motor, the paper has designed a new type of pre- pressure device for the spherical USM. The device utilizes the electromagnetic forces between the coils and the PM poles to generate the pre-pressure on the ultrasonic vibrator. By varying currents of stator coils, the pre-pressure can be regulated linearly.

2. The Principle of Electromagnetic Pre-Pressure Device

2.1. The Structure of 3-DOF Spherical USM with Electromagnetic Pre-Pressure Device

Fig.1(a) shows the structure of a 3-DOF spherical USM with electromagnetic pre-pressure device, which mainly includes cylindrical ultrasonic vibrator, stator coils and spherical rotor. The air-core stator coil is a cylindrical structure. The stator coils of three layers, eight coils per layer, are distributed uniformly around the equator of the spherical stator wall. The spherical rotor is placed on the ultrasonic vibrator, and two layers PM poles are distributed uniformly around the rotor equator. There are six poles with alternate polarity in each layer. As shown in Fig.1 (b), the rotor magnetic pole has the shape of a dihedral cone. The parallel magnetization is applied in permanent magnets, with the direction of the normal line at the center of a single rotor pole.

Rotor output shaft

Coils

X Y

Ultrasonic Vibrator

Fig.1. The motor structure (a) Overall structure (b) A single magnet pole

2.1. The Generation of Electromagnetic Pre-Pressure

The electromagnetic pre-pressure device consists of the stator coils and the spherical rotor. The Lorentz force act on the current-carrying stator coil, and the reactive force will act on the rotor. By controlling the coil currents, the vector sum of reactive forces can be pointed to the ultrasonic vibrator, Bin Li et al. / Procedia Engineering 29 (2012) 2393 – 2398 2395 3 Li Bin/ Procedia Engineering 00 (2011) 000–000 and then the pre-pressure is generated. Assuming the Lorentz force acting on a coil is denoted as F, its reactive force is denoted as F'. The F' is supposed to act on the point A of the rotor surface, as shown in Fig.2. Based on the theorem of force translation, the F' can be translated from the point A to the rotor center O. In order to maintain the same effect on the rotor, a pair of forces F'' and F''', which are equal and parallel with F', are added at the point O. The force system (F', F'', F''') is equivalent to the original force F. Since F' and F''' form a moment M, F' is equivalent to a force and a moment exerted on the rotor center, as shown in Fig.2. F ''=F ' M = rF × ' (1)

Where, r indicates the radius vector from point O to point A. After translation, all equivalent forces and moments pass through the rotor center, and it is very easy to solve the sum of forces or moments. In the stator Cartesian coordinate system ΣXYZ, in which Z-axis passes through the ultrasonic vibrator, as shown in Fig.1(a), the balance conditions of force and moment can be written as

∑∑FFiZ=−= M i0 (2)

Where, Fz is the pre-pressure required for the spherical USM.

Fig.2. The equivalent force translation

3. Current Control of The Electromagnetic Pre-Pressure

3.1. Electromagnetic Force Analysis

In the pre-pressure device, the air gap magnetic field is established mainly by the rotor poles. It is difficult to analyze directly the electromagnetic force of a current-carrying coil in the stator coordinate, which depends on three Euler angles. Therefore, the electromagnetic force in the rotor coordinates should be studied first.

Table 1. Parameters of structure and material

Symbol Quantity Value Rin Rotor radius 45mm Ri Inner radius of the stator windings 1.5mm Ro Outer radius of the stator windings 8mm H Height of stator windings 30mm N Turns 1000 δ Air gap length 1.5mm φr Latitude angle of PM 30° θr Latitude angle of PM 60° dr Thickness of permanent magnet 5mm 2396 Bin Li et al. / Procedia Engineering 29 (2012) 2393 – 2398 4 Li Bin/ Procedia Engineering 00 (2011) 000–000 The FEA model of the pre-pressure device is established in ANSOFT. The structure parameters are given in Table 1. Fig.3 shows the relationship between the electromagnetic force and the position of a coil when the current is 1A.

0.5 0.5 1

0.5 0 0 (N) (N) 0 (N) r φ θ F -0.5 F -0.5 F -0.5

-1 -1 -1 90 90 90 80 60 80 60 80 60 40 40 40 70 20 70 20 70 20 φ(deg) 60 0 60 0 φ(deg) 60 0 θ(deg) φ(deg) θ(deg) θ(deg) (a) (b) (c)

Fig.3. Electromagnetic force in ∑rθφ (a) fr component (b) fφ component (c) fθ component

Since the polarization direction of each magnetic pole is centerline, the magnetic field is symmetric about the magnetic pole centerline. Correspondingly, three components of magnetic force are symmetric about θ=30°, as shown in Fig.3, whose direction can be determined by right-hand rule.

3.2. Coordinate transformation of the electromagnetic force

When the spherical USM runs, the Eq.(2) must be satisfied. The electromagnetic forces of the stator coils should be transformed to the stator Cartesian coordinates. In the stator coordinate, given the coordinate (Xi, Yi, Zi) of the coil labeled ‘1’ in the ith phase winding and Euler angles (α, β, γ), the coil position in the rotor Cartesian coordinate system can be obtained as (,xyz , )= ( XYZ ,, )*(,,) Rα βλ i=A…L (3) i i i ii i Where R(α,β,γ) is the rotation transformation matrix between stator and rotor Cartesian coordinate systems The coil position in the rotor spherical coordinate can be expressed as

⎧ x i ⎪ arccos yi > 0 xy22+ z i ⎪ ii ϕθii= arccos = ⎨ (4) 2 22 x xyzi++ ii ⎪ i 2π − arccos yi < 0 ⎪ 22 ⎩ xyii+ The corresponding coordinates of the coil in magnetic model, as shown in Fig.3, can be expressed as ⎧ϕϕ 0≤≤ 90 ϕ''=ii θθθ=−⋅[ / 60] 60 i⎨ iii (5) ⎩180−ϕϕii 90 ≤≤ 180

Where, [x] is the rounding function. The effects of three components of the electromagnetic force acting on the rotor are different. The radial component fr passes through the rotor center and don’t produce the moments, but the tangential Bin Li et al. / Procedia Engineering 29 (2012) 2393 – 2398 2397 5 Li Bin/ Procedia Engineering 00 (2011) 000–000 components fφ and fθ do. It can be known according to the pre-pressure generation principle that fφ and fθ of the symmetric coils will cancel each other out, and only fr will generate the required pre-pressure. The electromagnetic force in the rotor spherical coordinate can be obtained from Fig. 5.

⎧F (θϕ'' , ) [ θ / 60]= 0, 2, 4 60 ≤≤ϕ 90 ⎪ ri i i i ⎪ [θϕ / 60]= 1,3,5 90 <≤ 120 F (,θϕ )= ii (6) rii ⎨ '' ⎪−Fri(θϕ , i ) [ θ i / 60]= 1,3,5 60 ≤≤ϕi 90 ⎪ ⎩ [θϕii / 60]= 0, 2, 4 90 <≤ 120

Specially, the stator coil will be beyond the coverage of a pole as φi>120° or φi<60°, then let f(θ’,φ’)=0. In the rotor coordinate ∑xyz, the components of the electromagnetic force can be expressed as

⎡⎤sinϕii cosθ cos ϕθ ii cos− sin θ i T =⋅=T ⎢⎥ϕ θ ϕθ θ FFFxyz T F r0 0 T ⎢⎥sini sin i cos i sin i cos i (7) ⎢⎥ ⎣⎦cosϕϕii− sin 0 From Eq.(3) and (7), the electromagnetic force of the ith phase winding with a current of 1A in the stator Cartesian coordinate can be written as

T −1 Fi(,,) XYZ= 2(,)**(,,) Fiiθϕ i T Rα β λ i=A…L (8)

3.3. Current control of the pre-pressure

The magnetic circuit in the stator coils is not saturated for the air-core coil, so the electromagnetic force is linear with the current. Let F denote the force matrix composed by the electromagnetic forces of all the 12-phase windings, I denote the current vector. The relationship between the matrix F and the vector I can be expressed as

T [0 0FZ ]= F ( XYZ , , ) • I (9) It is known from (9) that the control currents of the stator windings can be obtained by solving the generalized inverse of the force matrix.

4. Results

For simplification, let Fz=1N, and the range of Euler angles is limited to α=0°, 0°≤β≤30° and 0°≤γ≤120°. To solve the generalized inverse, the software Matlab is adopted, and the currents of the phase windings, IA~ID, are shown in Fig.4. A,B,C,D express 4 phases coils in X-Y plane in stator Cartesian coordinate system. It is useful for increasing the pre-pressure output by larger currents.

0.5 0.5 0.5 0.5

0.25 0.25 0.25 0.25 (A)

0 (A)

0 (A) 0 (A) 0 A B C D I I I I -0.25 -0.25 -0.25 -0.25

-0.5 -0.5 -0.5 -0.5 30 30 30 30 120 120 120 120 20 100 20 100 20 100 20 100 80 80 80 80 10 60 60 60 60 40 10 40 10 40 10 40 20 20 20 20 0 0 0 0 0 β(deg) γ（deg） β(deg) 0 γ（deg） β(deg) 0 γ（deg） β(deg) 0 γ（deg）

Fig.4. Current surface 2398 Bin Li et al. / Procedia Engineering 29 (2012) 2393 – 2398 6 Li Bin/ Procedia Engineering 00 (2011) 000–000 If the stator coils with iron core are employed, the electromagnetic force generated by the current would be improved. But it will become more complex to analyze. In the future, we will study the influence of iron core to improve the pre-pressure output.

5. Conclusion

Due to the 3-DOF motion of the spherical USM, the traditional device, which adopts springs to generate the pre-pressure, has deficiencies in design, controllability and so on. Referencing to the spherical PM motor, a new type of pre-pressure device composed of stator coils and spherical rotor with PM poles was designed. The electromagnetic forces were employed to generate the pre-pressure on the vibrator of the spherical USM. The relationship between pre-pressure, force matrix and the coil currents was analyzed. This device has the following characteristics: 1. The pre-pressure can be regulated linearly by controlling the phase currents. 2. A phase winding is composed of two stator coils, which located symmetrically about the rotor center. The size of power circuit is smaller.

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant 50907043 and 51007061, Tianjin Research Program of Application Foundation and Advanced Technology under Grant 10JCYBJC07900.

References

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