Original contributions of this dissertation

The original contributions of this dissertation are summarized as follows:

A very compact and simple MB structure for miniature motors is proposed;

moreover, the developed mathematical model can successfully describe the dynamic behaviour of the MB motor. An entirely passive MB structure consisting of two sets of very compact sintered NdFeB rings is successfully achieved to make the contactless rotor of the prototyped motor stably levitated in the radial direction, and low friction torque is experimentally verified. A gyroscopic effect is observed, and it effectively suppresses the radial vibration due to the mass imbalance of the rotor. The consistent dynamic results are evaluated by adopting a proposed mathematical model numerically solved with RK4.

An innovative damping structure is presented. An innovative damping induced by magnetic force was designed successfully for the totally passive magnetic bearing motor.

A magnetic ring of high permeability and an annular-shaped rubber pad were mounted on the stator 0.55mm below the permanent magnet of the rotor. Computer simulations were compared with experimental measurements for deciding material and configuration of the critical components. The natural frequencies for lateral and rotational modes of the rotor are around 22 Hz measured by impulse method. Both the magnetic bearing motor without and with magnetic damping are rotated at rated speed 3840 rpm, which is far above the first critical speed of 1305 rpm. Without magnetic damping, the natural damping ratio in radial direction of the rotor is 0.0655. After damping, it increases to 0.1401. We have demonstrated by both experimental measurement and theoretical calculation that the anti-shock performance is significantly improved by the innovative damping technology in a passive magnetic bearing motor.

The use of bias-magnetic force shows that the radial vibration of the MB motor is significantly reduced. A compact method using a novel bias magnet to induce an axial bias-magnetic force in order to reduce the radial vibration of the MB motor is introduced. After applying the bias-magnetic force, the optimal radial vibration of the developed MB motor is only around 0.197 G when the natural frequency of the bias magnet reaches the resonant frequency of 62 Hz. In this optimal condition the ratio in

the radial vibration of the MB motor with the bias-magnetic force to the original one is around 89 percent, and the driving power of 0.24 W is saved.

A MMB motor intends to apply in a portable optical disk drive was developed. The novel MMB motor was developed with an inner and outer magnetic-ring stack structure of high permeability. These rings were arranged on the shaft above the permanent magnet of the rotor and the stator. The magnetic force between the MMB, the permanent magnet and the stator induced the magnetic coupling effect that generated the demon friction torque loss. However, the loss was controlled well under 1.93x10^-4 Nm by the proposed approach, so that the good stable system was achieved. After suppressing the loss caused by the magnetic coupling effect, the shaft can be rotated without any frictional contacts in radial direction, the running current was around 0.18 A, and the maximum speed was around 1850 rpm. It shows that the MMB demonstrates the lower friction torque loss in comparison with the conventional MBB. Moreover, the radial vibration of our device is 21 % lower than the conventional MBB type. To verify the performance of the provided system, both the computer simulation and experiment were performed.

Chapter 2

Mathematical model for the MB

2.1 Theoretical model

The prototype of the developed magnetic bearing motor, in which the rotor is stably levitated, is equipped with two MBs, upper and lower, as shown in Fig. 2- 1. Each of the MBs consists of two permanent magnet rings made of NdFeB with an energy

product of 45 MGOe, and each is axially magnetized. The inner diameter, outer diameter, and height for the inner ring are 310^-3, 610^-3, and 410^-3 m, respectively, and the corresponding dimensions for the outer one are 710^-3, 1010^-3, and 410^-3 m,

respectively. These two MBs are properly located along the axial direction by employing two cylindrical washers. Only a single pivot point of the rotor in conjunction with the stator is introduced, and it is supported by a thrust plate fixed above the base.

In addition to the inside part of the two MBs, a rotor part composed of ferrite permanent

magnet with a yoke attached to it is connected to the shaft via a yoke bush, and the diameter of the rotor is 4.110^-2 m. The power can be input through the printed circuit

board (PCB) to generate the repulsive and attractive magnetic torque between rotor and stator to drive the spindle.

Frames of reference are defined as shown inFig. 2- 2. The coordinate (fixed frame)

origin of which is O', is located at the center of mass of the rotor (CM), and the orientation of this coordinate is supposed not to be altered. The origin o of the body coordinate with xyz axes is attached to the same position as O', and the z axis is along the symmetry axis of the rotor. Then, the Eulerian angles , , and , as well as the corresponding angular velocities ^, , and ^, can clearly be defined and associated with a vector on which an infinitesimal rotation or time derivative is imposed.

Supposing that the xyz axes are not to be rotated about the z axis, the angular velocities for the rotor and axes of the body can be represented as:

k j

ω^sini^(^^cos) (1)

k j

Ωsinicos (2)

where ω is the angular velocity of the rotor, Ω is the angular velocity for the axes of the body, and i, j, and k are the unit vectors in the x, y, and z directions, respectively.

Referring to the Newton-Euler equations of motion for a rigid body, which the desired system is assumed to be, the mathematical model for the proposed system can be expressed in the following.

CM

X mX

F  



₍₃₎

CM

Y mY

F  



₍₄₎

CM

Z mZ

F  



₍₅₎



^{sin2cos}

 

^{ cos}







^Tx  ^{Iy Iz} ₍₆₎



^{2sincos}



^{ sin}



^{cos}





^Ty ^{Iy Iz} ₍₇₎

cos

^Tz^Iz_dt^d (8)

where ^XCM, ^YCM, and ^ZCM are accelerations of the center of mass of the rotor along X-axis, Y-axis, and Z-axis, respectively. ^Iy and ^Iz are the principle moments of inertia about y and z axes, respectively. It is obvious that total forces and torques acting on the rigid body are expressed in fixed frame body coordinate system, respectively.

The external forces considered in the MB motor are a mass unbalance force due to the mass imbalance of the rotor, a MB force along the radial direction, and friction and axial preload forces due to the pivot thrust plate. Then, the nonlinear equations (3)-(8) can be numerically solved by applying the RK4, combined with an iteration procedure

[27]which Wang et al. derived for modeling the porous bearing of a spindle motor.

For solving differential system equations described above, on the left hand side of the equations (3)-(8) are needed to be connected to the parameters of the MB motor which are shown inTable 2- 1. The detail derivation of these external forces and torques are illustrated in the following.

To rewrite equations (3)-(8) as the expected form of equation,



^{F ,X}



^{F ,Y}



^{F ,Z}



^{T ,x}



^{T , andy}



T have to be represented in terms of X^z CM, YCM, ZCM,

Eulerian angles, and their derivatives. Considering the external forces acting on the rotor which are pivot force, radial magnetic bearing force, and unbalance force, the derivation is carried out as follows:

(1) Pivot force

As shown in Fig. 2- 3, a point 3 designates the pivot point. The l₁, l₂, and l₃ designate the axial distances from CM to MB 1, MB 2, and the bottom of the shaft, respectively. Suppose that the rotor is unbalanced, and the unbalance can be represented by a massm, which is located at a distance eand l_efrom the CM along radial and axial directions, respectively.

The external forces acting at the mentioned point 3 are friction force and normal contact force. The weight of the rotor and the friction coefficient between the shaft and the thrust plate are mg and μ, respectively. Supposing the velocity of the pivot point is

known, the total force acting at this point can be calculated. In the fixed coordinate system XYZ the velocity of the pivot point can be represented as

' 3 '

3 VO VO

V   (9)

The relative velocity V3Ocan be calculated as follows:

' 3 '

3O ω RO

V   (10)

where R3O(seeFig. 2- 3) is the position vector of point 3 relative to Oand can be expressed as follows.

k i

R_3O'rsin (l₃(rrcos)) (11) Substituting equation (1) and (I-3) into (I-2), so that V3Ocan be expressed in the

moving coordinate system xyz and defined as follows:

k j

i

V_3O'(V_3O')_x (V_3O')_y (V_3O')_z (12)

In the fixed coordinate system, these vectors of V3Oare obtained by performing the matrix multiplication of inverse of transformation matrix [DCM]^-1 and V3O, i.e., the [DCM]^-1matrix implements the coordinate transformation of a vector in moving frame

)

where DCM is the well known coordinate transformation matrix, and is expressed as



Substituting equation(14)into equation(13), the velocity of the pivot point can be represented as,

Supposing v is the unit vector of V₃, and then v can be represented as V K

The magnitude of the velocity vector V₃is

2

Because an axial force of 1.887 N is applied along the axial direction, it is believed that the shaft is always in contact with the thrust plate. Therefore, the unit vector of the velocity of the contact point should be as

J I

vv_X v_Y (18)

The friction force can then be represented as ) )(

( I J

F_f mgF_a v_X v_Y (19)

where μ, m, and F_aare friction coefficient between thrust plate and shaft, mass of the

rotor, and axial preload force, respectively.

Thus, the total force acting at the contact point is

K J

I

F₃ (mgF_a)v_X (mgF_a)v_Y (mgF_a) (20) By operating the transformation matrix [DCM] to F₃, so that F₃can be expressed in the xyz coordinate system to be as

 

Finally, the torque about Oresulted from the contact force is as

z

l₃= distance from CM to shaft bottom in z-axis r = radius of the fillet of shaft bottom

It is assumed that the radial stiffness of MB 1 (upper) and MB 2 (lower) are identical.

The components of radial MB force along X-axis are

 

F_k1 _X k_X



X_CM X1



(23)

 

F_{k2 X} k_X



X_CM X₂



(24) The components of radial MB force along Y-axis are

 

F_k1 _Y k_Y



Y_CM Y1



(25)

 

F_{k2 Y} k_Y



Y_CM Y₂



(26) Thus, in the fixed coordinate system, the force due to the stiffness of radial MB can be represented as

By operating the transformation matrix [DCM] to the force vectors

 

Fk1 _XYZ and

 

Fk 2 _XYZ, respectively. Then, in the moving frame, the corresponding force vectors can be represented as

The torque about Odue to the stiffness of radial MB can be expressed as

   

  ⁱ  ^{   }  ^j

T

_k

 F

_{k1 y}

l

₁

 F

_{k2 y}

l

₂

 F

_{k1 x}

l

₁

 F

_{k2 x}

l

₂ (30)

where

△X1, △X₂= upper and lower MB displacement relative to the CM in the X direction (these two variables can be further expressed in terms of the unknown variables Eulerian angles)

△Y1, △Y₂= upper and lower MB displacement relative to the CM in the Y direction (these two variables can be further expressed in terms of the unknown variables Eulerian angles)

k_X= radial MB stiffness in X direction

k_Y= radial MB stiffness in Y direction

 

Fk_{1 X},  Fk_{2 X} = upper and lower MB forces in X direction

 

Fk_1Y,

 

Fk_{2 Y} = upper and lower MB forces in Y direction

 

Fk_{1 Z},

 

Fk_{2 Z} = upper and lower MB forces in Z direction

 

Fk_{1 x},

 

F_{k2 x} = upper and lower MB forces in x direction

 

Fk_{1 y},

 

F_{k2 y} = upper and lower MB forces in y direction

 

F_{k1 z},

 

Fk_{2 z} = upper and lower MB forces in z direction

l₁(l₂) = axial distance between upper (lower) MB and CM of rotor (seeFig. 2- 3)

Tk = torque about Odue to radial MB stiffness F = radial MB forcek

(3) Unbalance force

The unbalance force relative to the moving frame can be expressed as follows:

 

By applying the inverse transformation [DCM]^-1to

 

Fe _xyz, in the XYZ coordinate system,

 

Fe _XYZ can be expressed as

The torque about Odue to unbalance force is then as

j

m = mass unbalance of rotor

l_e= radial distance from the location of mass unbalance of rotor to CM (seeFig. 2- 3).

x

T = torque about Odue to mass unbalancee

F = unbalance forcee

After performing the derivation of the external forces



^{F ,X}



^{F , andY}



^{FZ in}

equations(3)-(5)and torques



^{T ,x}



^{T , andy}



T in equations^z (6)-(8), the total reaction force and torque can be determined by

     

F_{3 XYZ} F_{e XYZ} F_{k XYZ}

F   



⁽³⁴⁾

     

T3 _xyz Te _xyz Tk _xyz

T   



⁽³⁵⁾

Thus, equations(3)-(8)can be rearranged as the following equation(37)because



^{F and}



T are all in terms of XCM, YCM, ZCM, Eulerian angles, and their derivatives.

The demon friction force generated at the contact pivot point leads the system of equations to be highly nonlinear. Particularly, each of the two parameters will be known, only if the other one of them is given. Thus, an iteration procedure was developed by Wang et al.[27]in order to determine these two parameters.

It is not difficult to reduce the above second-order differential equations to a set of coupled first-order differential equations by defining a set of new variables, q₁, q₂,…, and q₁₂, and they are defined as follows.



Referring to the fourth-order Runge-Kutta formula, the RK4 approximation of {Q(t_i1)}

can be given by the following equations:



1

           

1 2 2 2 3 4



t

where subscript i and t are the time at i-th instant and step size of time increment, respectively. Thus, the next value {Q(t_i1)} is determined by the current value {Q(t_i)}

plus the product of the interval t and an calculated slope

       



1 2 2 2 3 4



6

1 K  K  K K .

To investigate the dynamic characteristics of the MB motor, a physical model of the MB is used. Each of the MBs is represented in terms of stiffness and damping coefficients k and c. In order to simplify the model, the cross-coupled stiffness coefficients are ignored. The modeling parameters for solving the ordinary differential equations mentioned above are listed inTable 2- 1.

Fig. 2- 1 Schematic structure of the magnetic bearing motor.

Fig. 2- 2 Frames of reference.



X_CM





X_CM

Fig. 2- 3 Geometry of the rotor

Table 2- 1 Parameters of the MB motor

Rotor mass, (kg) 0.071

Polar moment inertia, (kgm²) 2.09210^-5 Transverse moment of inertia, (kgm²) 1.25410^-5

Rotational speed of rotor, (Hz) 60.5

Friction coefficient thrust plate, (-) 0.01

Axial preload force, (N) 1.887

Clearance, (m) 510^-4

Radial stiffness of MB, (N/m) 1548

Damping coefficient of MB, (Ns/m) 0 Damping ratio of the system, (-) 0.0655 Mass unbalance of rotor, (kgm) 0.1010^-5

2.2 Procedures for prototyping the MB

As described in section 2.1, the mentioned MB prototype has been carried out using a developed procedure which will be thoroughly addressed in the following.

Finite element analysis was used to analyze the magnetic force of the magnetic bearing motor. To design a stable magnetic bearing motor system, there are three terms should be considered.

First, in the static state, the system should be stabilized in all axes when the rotor is tilted at a critical angle defined as Өalong the x-axis. Due to the geometry constraints of theradialairgap is500 μm and thedesired radial run-outshould besmallerthan 75 μm (peak). The force analysis is considered, in the critical condition when the Өis equal to 1° shown asFig. 2- 4(b).

Fig. 2- 4 (a) Rotor is tilted at a critical angle 0° about positive x-axis. (b) Rotor is tilted at a critical angle 1° about positive x-axis.

The restoring torque about y-axis was defined as Ty and the axial force was defined as F_z. The 3D full model of the finite element analysis was constructed to compute these parameters, as shown in Fig. 2- 4. For each variation of the axial displacement defined as Dzz (When the surface A of the inner annular magnet is lower than the surface B of the outer one along the positive z-axis, it was specified with D_zz> 0) of inner and outer magnetic annular rings, the rotor was rotated 1° about the y-axis. Then Ty and Fz were calculated and shown asTable 2- 2. Itwasclearthatthestate ofthe system with 0 ≤ D_zz

≤ 310^-4m was stable.

x z y

Θ=0° Θ=1°

(a) (b)

Surface A Surface B

Yoke

Magnet Stator

Magnetic bearing

Yoke

Magnet Stator

Magnetic bearing

Fig. 2- 5 3D solid model mesh of the magnetic bearing motor.

Table 2- 2 T_yand F_zversus various D_zz.

Secondly, in the dynamic state, the system should be stabilized in all axes when the rotor is loading the unbalance 9.810^-6Nm, rotated in the rated speed 3840 rpm and the

centrifugal force 0.1617N is generated. For supporting the dynamic load (0.1617N), the Dzz(10^-5m) Ty(Nm) Fz(N) State

﹣10 ﹣0.01658 2.420 Unstable

0 ﹣0.01525 ﹣1.997 Stable

10 ﹣0.01219 ﹣4.086 Stable

20 ﹣0.00788 ﹣5.719 Stable

30 ﹣0.00487 ﹣7.070 Stable

radial magnetic force is considered two times as it. That is equivalent to 0.3234N. The target of the radial run-out was supposed to be lower than 150 μm (peak to peak)in this prototype. To achieve the stable zone of D_zz, whenever the radial displacement was greaterthan 75 μm,theradialforcemustbegreaterthan 0.3234 N.Approach to this

computation of radial force of the development system, the stator was fixed and the

rotor was shifted along the positive x-axis as the following five different positions, 0, 110^-4 m, 210^-4 m, 310^-4 m, 410^-4 m and 4.910^-4 m for each Dzz. Three relative

positions of magnetic inner and outer ring were shown asFig. 2- 5, in which Fig. 2- 5 (a-1)showed the radial displacement of the rotor along positive x-axis was 0. Fig. 2- 5 (a-2)showed the displacement of the rotor along positive x-axis was 310^-4m. AndFig.

2- 5 (a-3)showed the radial displacement of the rotor along positive x-axis was 4.910^-4 m.

From these locations, arrangement of the rotor and stator, the radial force could be obtained as a function of the radial displacement during the force calculation. In order to analyze the radial force as a function of radial displacement at various D_zz values, the variation of Dzz is chosen as follows 0, 110^-4 m, 210^-4 m, 310^-4 m, and 410^-4 m.

And its 3D structure distribution at D_zz= 0 was shown asFig. 2- 6, in which, , ,  and  showed upper magnetic outer ring, lower magnetic outer ring, upper magnetic inner ring, lower magnetic inner ring, respectively. The rotor was shifted along negative

axis. Both in Fig. 2- 5 and Fig. 2- 6 only the magnetic inner and outer rings were demonstrated.

Both radial force and axial force were known, and then the stiffness of the magnetic bearing is perfectly determined. Because of the axial symmetry and the magnetic bearings made exclusively of permanent magnets, the axial stiffness is twice as radial stiffnessaccording to Earnshaw’stheorem [7].

After computations were performed, we found the value of radial force for each radial displacement. And the radial force as a function of D_zz could be obtained. Also, use the same process for F_z as a function of radial displacement was calculated. The final results were presented in Fig. 2- 8 (a) and Fig. 2- 8 (b). Fig. 2- 8 (a) shows the radial force as a function of radial displacement of the rotor. The radial force is linearly

proportional to radial displacement at Dzz = 0. Also as the Dzz = 110^-4 m, 210^-4 m, 310^-4 m, and 410^-4 m, the radial force versus radial displacement have the same characteristic, except the radial displacement is greater than 410^-4 m. Graphically the

equation Fr = ﹣0.3234 N and radialdisplacement= 75 μm givetwo linesA and B,as shown inFig. 2- 8 (a). It is manifest that in the fourth quadrant of line A crosses line B, the Fr is greater than the rated load whenever the radial displacement is greater than 75 μm.From thedataanalysisofradialforceand theresults ofTable 2- 2, it indicates that

the stable zone of Dzzis between 0 and 310^-4m. The rotor can be levitated in the radial

direction without any frictional contact when the radial run-out (peak to peak) is less than 150 μm.Fig. 2- 8 (b) shows the axial force as a function of radial displacement of

rotor. The number of axial force at D_zz= 0 was around zero from radial displacement = 0 to 4.910^-4m. For a fixed value of Dzz, the axial force rose slowly in the same radial

displacement scope. For a fixed value of radial displacement of rotor, the axial force

increased with increasing D_zz. The axial force of the magnetic bearing motor with radial displacement = 4.910^-4m increase from 0 N for Dzz= 0 to ﹣6.543 N for Dzz= 410^-4

m.

Thirdly, the axial force of the development system should be controlled under a proper value. The upper limit of this scope could be decided by two constraints. One is the friction torque produced by the MB system in the dynamic state has to smaller than the ball bearing one. The other is the capacity of the thrust plate, with loading capacity defined as P, which must be smaller than 10000 kg/m², the maximum speed defined as V, which must be smaller than 7 m/sec, and the capacity of PV value must be smaller than 1000 kg/(msec) (The coefficient of kinetic friction 0.08). Referring to the friction

Thirdly, the axial force of the development system should be controlled under a proper value. The upper limit of this scope could be decided by two constraints. One is the friction torque produced by the MB system in the dynamic state has to smaller than the ball bearing one. The other is the capacity of the thrust plate, with loading capacity defined as P, which must be smaller than 10000 kg/m², the maximum speed defined as V, which must be smaller than 7 m/sec, and the capacity of PV value must be smaller than 1000 kg/(msec) (The coefficient of kinetic friction 0.08). Referring to the friction

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