Identification Scheme for Estimation of the Unbalance, Misalignment and AMB

4.4 Identification Scheme for Estimation of the Unbalance, Misalignment and

Following the concept of Section 2.4.2, the equation of motion (i.e., Equation (4.15)) can be written in frequency domain as

     



^{ i 2Md j i Cd Gd Kd}



^{Nmism FunbFAMBmis (4.17)}

where, the force due to misaligned AMB, F_AMB^mis can be written in terms of the summation of force due to AMB modified stiffness (which includes forces from both the displacement and current stiffness coefficients), F_{AMB s}^mis_, and constant AMB force, F_c^mis (follow Equation (4.9)), as

,

mis mis mis

AMB  AMB s c

F F F (4.18)

For (i = 0) and (i = 1) respectively, Equations (4.17) and (4.18) for both the misalignment cases can be expressed as

,

d mis mis mis

m  AMB s  c

K N F F (4.19)

 



^{2Md j CdGd Kd}



^{Nmism FunbFAMB smis , (4.20)}

It can be noticed that the left-hand side (LHS) of Equations (4.19) and (4.20) contains all known terms (i.e., mass, stiffness, damping and gyroscopic matrices) explicitly while right-hand side (RHS) has unknown terms (i.e., unbalance fault parameters, the modified AMB force- displacement and force-current stiffness parameters as well as constant AMB forces due to misalignment) in the form of unbalance force and misaligned AMB force. Hence, for estimation purpose, the above equations can be suitably manipulated such that all the unknown terms are in a vector in the left-hand side. Thus, by taking the terms relating to the unbalance and modified AMB parameters and AMB constant forces on LHS, Equations (4.19) and (4.20) can be given as

,

mis mis d mis

AMB s  c  m

F F K N (4.21)

 



²



,

j

mis d d d d mis

unb



AMB s

       

m

F F M C G K N

(4.22)

Further, the force vectors (includes both the unbalance and misaligned AMB forces) on the LHS can be transformed in such a way that all the known terms (i.e. rotor design parameters, generated rotor displacement and AMB current information) are in the regression matrix in the left hand side. Here, the right-hand side vector itself includes all known quantities. Thus, the transformation for the residual and additional trial misalignments for (i = 0 and 1) can be expressed as

1,0 2,0 1,0

mis mis mis mis mis mis

s  i  c 

A k A k F B (4.23)

3,1 1,1 2,1 1,1

mis mis mis mis mis

s i

  

A e A k A k B (4.24)

with

1 1 11

1 1 1 21

2 2 2 12

2 2 22

1,

; ; ;

mis mis mis

sx ix

mis mis mis

sy iy

mis mis mis mis mis mis

s sx i ix c

mis mis mis

sy iy

k k f

e k k f

e k k f

k k f

     

     

       

       

       

       

       

     

     

e k k F

B^{misi  }

  

^{i 2Md j}

 

^i



^{Cd Gd}



^Kd



^{Nmism ; for}

^

^i0,1

^

(4.25)

Further, Equations (4.23) and (4.24) for both the cases of misalignment can be rearranged in matrix form as

1 1 1

mis mis  mis

A x B (4.26)

with

1,0 2,0 1,0

1 1 1

3,1 1,1 2,1 1,1

0 ; ;

0

mis

mis mis mis

mis mis s mis

mis mis mis mis

i mis c

 

 

     

     

 

     

 

 

e k

A A I B

A x B

k

A A A B

F

(4.27)

Equation (4.26) is observed to have the common vector of unbalance parameters e, in the unknown vector x₁^misof both cases of misalignment. Hence, Equation (4.26) for the residual and additional trial misalignments, i.e., the superscript mis replaced with m1 and m2, respectively, can be combined together to estimate unique value of parameters associated with the unbalance fault in the rotor system. Hence, the above equation can be written in combined and condensed matrix form as

 

_{   }

 

_{ }

2  a b_ 2b_1  2  a_1

A x B (4.28)

with

 

1 1

1,0 2,0

1 1

3,1 1,1 2,1

2 2 2

1,0 2,0

2 2

3,1 1,1 2,1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

m m

m m

m m

m m



 

 

 

  

 

 

 

A A I

A A A

A A A I

A A A

(4.29)



^{1 1 1 2 2 2}



2

m m m m m m T

s i c s i c



x e k k F k k F (4.30)

 



^{1 1 2 2}



2 1,0 1,1 1,0 1,1

m m m m T

 

B B B B B (4.31)

Subscripts in Equation (4.28) denote the matrix or vector size, which depend upon the number of elements taken in the finite element method formulation. The total number of equations associated with (i = 0) and (i = 1) as well as the number of unknowns to be determined are shown by ‘a’ and ‘b’, respectively. The subscript ‘b’ in the combined unknown vector x2

depends on the total number of disc unbalances and the number of misaligned AMB parameters with constant forces for both misalignment cases. On substituting the unknown vectors of Equation (4.25), Equation (4.30) can be written in expanded form, as





1 1 1 1 1 1

2 1 2 1 1 1 1 11 21

2 2 2 2 2 2

1 1 1 1 11 21

m m m m m m

sx sy ix iy

m m m m m m T

sx sy ix iy

e e k k k k f f

k k k k f f

 x

(4.32)

Equation (4.28) is in complex form, hence it can be segregated into the real and imaginary components and written as

   



A^Re2  jA^Im2 



x^Re2 jx^Im2







B^Re2

^{ }

 jB^Im2

^{ }





(4.33)

where the subscripts Re and Im represent the real and imaginary components, respectively.

Further, the real and imaginary components of this equation can be written separately to obtain all real quantities, as

   

^Re

 

Re Im 2 Re

2 2 Im 2

2

 

^{ }



   

  

A A x B

x ^{and Im}2

 

^Re2

 

^Re2Im ^Im2

 

2

 

^{ }



  

  

A A x B

x ^(4.34)

Equation (4.34) can be expressed in the combined form, to get estimation equation for the overall unbalance and misaligned AMB parameters, as

 

 _2a c_{  }c_1 

 

 _2a_1_

A x B (4.35)

with

     

       

 

Re Im Re

2 2 2

Im Re Im

2 2 2

;

  

 

  

    

    

 

   

A A B

A B

A A B ^(4.36)





Re Im Re Im 1 1 1 1 1 1

1 1 2 2 1 1 1 1 11 21

2 2 2 2 2 2

1 1 1 1 11 21

m m m m m m

sx sy ix iy

m m m m m m T

sx sy ix iy

e e e e k k k k f f

k k k k f f

 x

(4.37)

where the subscript ‘c’ in the unknown vector x contain the real and imaginary parts of disc eccentricities and AMB misaligned stiffness parameters along with AMB constant forces for both misalignment cases. Here, A(ω) is the regression matrix and B(ω) is the vector containing known quantities, respectively. The real and imaginary parts of k^th disc eccentricity (with k = 1, 2, …, p) are

   

Re Im

cos ; sin

k k k k k k

e e



e e



(4.38)

To solve simultaneous linear system of equations and obtain an exact or optimal solution, the number of unknowns must be either equal to or less than the number of equations. If the number of equations (i.e., number of rows in matrix A of Equation (4.35)) are more than unknown terms (x), then it is called overdetermined system and solved by the Moore-Penrose inverse technique. In case, Equation (4.35) is underdetermined (i.e., 2a < c), the rotor-AMB system can be operated with at least two different speeds such that the system is transformed into overdetermined type. Increase in the number of rows of matrix A from more sets of displacement and current response measurements due to these different speeds, will result in Equation (4.35) as over-determinate. Moreover, to identify all the unknowns utilizing the Moore-Penrose inverse approach, Equation (4.35) can be further given in matrix form as

   



^{ T }



^1

^{   }

^{ T }



x A A A B (4.39)

Equation (4.39) can be used to determine the unknown parameters (i.e., unbalance eccentricities and its phases from real and imaginary parts (i.e., using Equation (4.38)), modified AMB stiffness coefficients, AMB constant forces for both the residual and additional trial misalignment cases) with the knowledge of measured rotor displacement and AMB current responses in the frequency domain for (i = 0) and (i = 1) and the known rotor model parameters.

Better estimation of the unknown faults and AMBs parameters can be achieved by taking the system responses for the range of multiple ‘n’ number of rotor spin speeds using the below equation.

 

 

 

 

 

 

1 1

2 2

n n

 

 

 

   

   

   

   

   

   

   

A B

A B

x

A B

(4.40)

Further, the identified AMBs force-displacement constants (i.e., k_sx^m1₁, k_sy^m1₁, k_sx^m1₂, k_sy^m1₂, ) and force-current constants (i.e., k_ix^m₁¹, k_iy^m₁¹, k_ix^m1₂, k_iy^m1₂, ) and AMBs force constants (i.e., f₁₁^m1,

1 21

f m , f₁₂^m1, f₂₂^m1, ) for the first misalignment level (residual misalignment) as well as AMBs force-displacement constants (i.e.,k_sx^m₁², k_sy^m₁², k_sx^m2₂, k_sy^m2₂, ) and force-current constants (i.e.,

2 1 m

kix , k_iy^m₁², k_ix^m₂², k_iy^m₂², ) and AMBs constant forces (i.e., f₁₁^m2, f₂₁^m2, f₁₂^m2, f₂₂^m2, ) for the second level of misalignment (additional known trial misalignment) accommodated in the x vector, can be utilized in estimating the amounts of misalignment (i.e., _x1, _y1, _x2, _y2, ) between the rotor and AMBs axes. Subsequently, the identified AMB coefficients (i.e, k_sx^m₁¹,

1 1 m

ksy , k_sx^m1₂, k_sy^m1₂, , k_ix^m₁¹, k_iy^m₁¹, k_ix^m₂¹, k_iy^m1₂, , k_sx^m₁², k_sy^m₁², k_sx^m2₂, k_sy^m2₂, and k_ix^m₁², k_iy^m₁², k_ix^m₂²,

2 2 m

kiy , ) for both the misalignment cases and the estimated misalignment amounts (i.e., _x1,

1

y , _x2, _y2, ) can be manipulated to identify the actual force-displacement stiffness (ksx1, ksy1, ksx2, ksy2, ) and force-current stiffness (kix1, kiy1, kix2, kiy2, ) coefficients (i.e., for perfectly aligned case) for ‘r’ number of AMB, presented in Figure 4.1. The ratio of k_sxq^m1 and

2 m

ksxq of Equations (4.5) and (4.8) will be useful in estimating the misalignment amount_xq, with the known values of trial misalignment,_xqand AMB air gap, s0. Their ratio can be given as

 

 

 

 

 

2 2 2 2

1 1 1 1

1 1

2 2

2 2 2

0 0

1 1

1 1

; ;

1 1

m sxq q q q

sxq xq xq

q q

m

sxq sxq q q

k p

k

k k s s

 

  

  

 

    

  ^(4.41)

Similarly, for the known values of ∆yq and s0, the same Equation (4.41) would be followed to estimate the residual misalignment δyq by replacing x with y. Thereafter, Equation (4.5) can be utilized for estimating the actual AMB force-displacement and force-current coefficients (ksxq,

ksyq, kixq and kiyq) for q^th AMB, with the identified values of modified AMB constants and estimated x- and y-directional misalignment amounts. Thus, the trial misalignment approach can be utilized in estimating the disc unbalances, AMBs stiffness parameters and their residual misalignments, employing the developed mathematical model based identification algorithm.

In the next section, the numerical experiments have been presented to establish the procedures proposed in this section. This will be useful in checking the accuracy and effectiveness of the methodology in a flexible rotor-AMBs system.