Modelling of AMB Force for Residual and Additional Trial Misalignments

3.2 System Configuration and Mathematical Modelling of Rotor-AMB System

3.2.2 Modelling of AMB Force for Residual and Additional Trial Misalignments

The force modelling for perfectly aligned AMB with the rotor is already discussed in Section 2.3.2. The same section also presented the innovative derivation for the linearized form of AMB force for the case of residual misalignment and additional trial misalignment. In the Chapter 2, the force-displacement (ks) and force-current (ki) stiffness of both AMBs were assumed to be same as well as the parallel misalignment between rotor and AMBs was considered. However, this is not always the case in a real rotor-AMB system, the stiffness constants (force-displacement and force-current) may be different for both AMBs and there

can be a combination of parallel and angular misalignments. In these conditions, the expression for AMBs force for perfect alignment state and for the residual and additional trial misalignment cases will be modified.

Rotor

Actuator

N

S

S N

N

N S

S

x

y

O

Figure 3.3 Side view of a hetero-polar eight pole actuator in x-y plane when the rotor is perfectly aligned with both actuators.

In the perfect alignment state as shown in Figure 3.3, the force due to AMB1 on the rigid rotor in the x and y directions can be written as

1 1 1 1 1; 1 1 1 1 1

x s x i x y s y i y

f k u k i f k u k i (3.3)

with

2

1 0 1 0 2

1 3 1 2 1 0 1 1

0 0

4 4 1

; ; cos

4 2

s i a

k i k i

k k k N A

s s

 

   (3.4)

Similarly, the force due to AMB2 on the rigid rotor in x and y directions can be expressed as

2 2 2 2 2; 2 2 2 2 2

x s x i x y s y i y

f k u k i f k u k i (3.5)

with

2

2 0 2 0 2

2 3 2 2 2 0 2 2

0 0

4 4 1

; ; cos

4 2

s i a

k i k i

k k k N A

s s

 

   (3.6)

where N1 and N2 are the number of coils in AMB1 and AMB2, respectively; Aa1 and Aa2 are the areas of the magnetic pole of AMB1 and AMB2, respectively; ks1 and ki1 are, respectively, the force-displacement and force-current constants of AMB1, and ks2 and ki2 are, respectively, the force-displacement and force-current constants of AMB2. The controlling current in the x and y directions at AMB1 and AMB2 locations are represented by { ( ), ( )}i t i t_{x1 y1} and

2 2

{ ( ), i_x t i_y ( )}t , respectively. Following Equation (2.7), the expressions for controlling current in the x and y directions at the AMB1 and AMB2 locations can be expressed as

   

   

1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2

;

;

x P x I x D x y P y I y D y

x P x I x D x y P y I y D y

i k u k u dt k u i k u k u dt k u

i k u k u dt k u i k u k u dt k u

       

       

 

 

^(3.7)

Rotor

Actuator

N

S

S N

N

N S

S

x

y

O

δy1

δ_x1

C

(a)

Rotor

Actuator

N

S

S N

N

N S

S

x

y

O

δ_y2

δx2

C

(b)

Figure 3.4 Misalignment of the rotor in the x-y plane with (a) AMB1 by δx1 and δy1 amounts (b) AMB2 by δx2 and δy2 amounts.

Similarly, for the residual misalignment case of AMBs as shown in Figure 3.4(a) (with the concept discussed in Section 2.3.2.1), the linearized force equations at AMB1 location in the x and y directions can be expressed as

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 ; 1 2 1 2 1 2

m m m m m m m m m m m m

x s x i x y s y i y

f k u k i  f f k u k i  f (3.8)

with

   

   

   

   

2 2

1 1

1 1 1 1 1 1 1 0 1

1 2 2 1 2 2 1 2 2 1 2 1

0 0

1 1 1

2

1 2 1

1 1 1 1 1 2

2 2 2 2 2 2 2 2 2 2

2 2 2 0

1 4

; ; ; ;

1 1 1

; 1 ; ;

1 1 1

m a m i m a x

s i a

i y

m a m m a

s i

k k f k i

k k f f

s s

k k f

k k f

s

   

  

   

  

     

  

    

  

(3.9)

where (u_x^m₁¹,u^m_y1¹) and (i_x^m₁¹,i^m_y1¹) are, respectively, the x- and y-directional displacement of the rotor and the controlling current at AMB1 location, and (k_s^m₁¹,k_i^m₁¹) and (k_s^m₂¹,k_i^m₂¹) is the modified force-displacement and force-current stiffness constants of AMB1 in the x and y directions, respectively. The new constant force terms arise due to AMB1 misalignment in the x and y directions are represented by f₁^m1and f₂^m1, respectively. Similarly, the AMB2 force on the rotor, in the x andy directions, when there is a misalignment of δx2 and δy2 amounts as shown in Figure 3.4(b), can be written, respectively, as

1 1 1 1 1 1 2 2 2 2 2 2

2 3 2 3 2 3 ; 2 4 2 4 2 4

m m m m m m m m m m m m

x s x i x y s y i y

f k u k i  f f k u k i  f (3.10) with

   

   

   

   

2 2

2 3

1 2 1 1 2 3 2 0 2

3 2 2 3 2 2 3 2 2 2 2 3

0 0

3 3 3

2

2 4 2

1 2 1 1 2 4

4 2 2 4 2 2 4 2 2 4

4 4 4 0

1 4

; ; ; ;

1 1 1

; 1 ; ;

1 1 1

m a m i m a x

s i a

i y

m a m m a

s i

k k f k i

k k f f

s s

k k f

k k f

s

   

  

   

  

     

  

    

  

(3.11)

where (u_x^m₂¹,u^m_y2¹) and (i_x^m₂¹,i^m_y2¹) are, respectively, the displacements of the rotor and the controlling currents in the x and y directions at AMB2 location, and (k_s^m₃¹,k_i^m₃¹) and (k_s^m₄¹,k_i^m₄¹ ) are the modified force-displacement and force-current stiffness constants of AMB2 in the x

and y directions, respectively. The constant force terms appearing due to AMB2 misalignment in the x and y directions are denoted by f₃^m1and f₄^m1, respectively.

One of the prime objectives of this chapter is to obtain the four unknown residual misalignments, i.e. δx1, δy1, δx2 and δy2 from the proposed model-based identification algorithm derived from novel trial misalignment approach. For the estimation of these parameters, the known trial misalignments in addition to the unknown residual misalignments are provided at the AMB1 and AMB2 locations. These trial misalignments between the rotor and supported AMBs in the vertical and horizontal directions can be created by using the physical trial misalignment (PTM) concept, as discussed in Section 2.3.2.2. The combination of trial and residual misalignments can be termed as additional trial misalignment. Figure 3.5 represents for additional trial misalignment (residual plus trial) between the axes of the rotor and AMBs.

Rotor

Actuator

N

S

S N

N

N S

S

x

y

O

δy1 +∆ _y1

δx1 +∆x1

C

(a)

Rotor

Actuator

N

S

S N

N

N S

S

x

y

O

δy2 +∆ _y2

δx2 +∆x2

C

(b)

Figure 3.5Misalignment of the rotor in x-y plane with (a) AMB1 by (δx1 + ∆ x1) and (δy1 + ∆ y1) amounts (b) AMB2 by (δx2 + ∆ x2) and (δy2 + ∆ y2) amounts.

Here, C is the rotor center, ∆x1 and ∆y1 are the trial misalignments in the vertical (x) and horizontal (y) directions at the AMB1 position. Trial misalignments in the vertical and horizontal directions at AMB2 location are represented by ∆x2 and ∆y2, respectively. This case will be beneficial to determine the unknown residual misalignment. After the application of

trial misalignment, ∆x1, the new misalignment between the rotor and AMB1 axis is (δx1+∆x1) in the x-direction. For the present case, the force on the rigid rotor due to AMB1 misaligned by (δx1 + ∆ x1) and (δy1 + ∆ y1), in the x and y directions, respectively, can be expressed as

f_x^m₁² k u_s^m₁² _x^m₁²k i_i^m₁² _x^m₁²  f₁^m2; f_y^m₁² k u_s^m₂^{2 m}_y1²k i_i^m₂² _y^m₁² f₂^m2 (3.12) with

   

   

   

   

2

1 1

2 1 2 2 1 1 1

1 2 2 1 2 2 1 2 2 1 1 1 1

1 1 1 0

2

1 2 1

2 1 2 2 1 2

2 2 2 2 2 2 2 2 2 2

2 2 2

2 2 2 0

; 1 ; ; ;

1 1 1

; 1 ; ; ;

1 1 1

m s m i m a x

s i

i y

m s m m a

s i

k p

k f p

k k f p

p p p s

k p

k f p

k k f p

p p p s





 

       

  

 

       

  

(3.13)

Similarly, the force on the rotor due to AMB2 misaligned by (δx2 + ∆ x2) and (δy2 + ∆ y2), in the x and y directions, respectively, can be written as

2 2 2 2 2 2 2 2 2 2 2 2

2 3 2 3 2 3 ; 2 4 2 4 2 4

m m m m m m m m m m m m

x s x i x y s y i y

f k u k i  f f k u k i  f (3.14)

with

   

   

   

   

2

2 3

2 2 2 2 2 3 2

3 2 2 3 2 2 3 2 2 3 3 3 3

3 3 3 0

2

2 4 2

2 2 2 2 2 4

4 2 2 4 2 2 4 2 2 4 4 4 4

4 4 4 0

; 1 ; ; ;

1 1 1

; 1 ; ; ;

1 1 1

m s m i m a x

s i

i y

m s m m a

s i

k p

k f p

k k f p

p p p s

k p

k f p

k k f p

p p p s





 

       

  

 

       

  

(3.15)

where (u^m_x1²,u^m_y1²,u_x^m₂²and u^m_y2²) and (i^m_x1²,i^m_y1²,i_x^m₂²and i^m_y2²) are the displacements of the rigid rotor and the controlling current in the x and y directions, respectively, at AMB1 and AMB2 locations; (k_s^m₁²,k_i^m₁²), (k_s^m₂²,k_i^m₂²), (k_s^m₃²,k_i^m₃²) and (k_s^m₄²,k_i^m₄²) are the modified force- displacement and force-current stiffness constants of AMB1 and AMB2 in the x and y directionsfor additional trial misalignment, respectively. The constant force terms appearing

due to AMB1 and AMB2 misalignment in the x and y directions for the present misalignment case are denoted by f₁^m2, f₂^m2, f₃^m2and f₄^m2, respectively.

3.3 Equations of Motion of the Misaligned Rotor System considering Gyroscopic