Modal analysis of rotor system using modulated coordinates for asymmetric rotor system with anisotropic stator

(1)

Modal analysis of rotor system using modulated coordinates for asymmetric rotor system with anisotropic stator

Chong-Won Lee

Center for Noise and Vibration Control, Dept. of Mech. Eng., KAIST, Daejeon, Korea

Jeong-Hwan Suh

Digital Appliance Research Laboratory, LG Electronics Inc., Seoul, Korea

Seong-Wook Hong

School of Mech. Eng., Kumoh National Institute of Technology, Kyungbuk, Korea

ABSTRACT

A new modal analysis method for general rotor systems with time-varying parameters is proposed.

The essence of method is to introduce the modulated coordinates to derive the equivalent infinite order time-invariant matrix equation from the finite order time-varying matrix equation. Timevarying parameter equation is inevitable for general rotors, of which rotating and stationary parts both possess asymmetric properties. An extensive modal analysis method is rigorously developed for the infinite order time-invariant system and an approximate, yet effective modal analysis technique is introduced with the reduced order system, in order to derive four types of directional frequency response functions. It is shown theoretically that they can be effectively used for detection of the presence of system anisotropy and asymmetry. A numerical example with a simple rotor model and a flexible asymmetric rotor are also provided to demonstrate the theoretical findings and practicality of the proposed method.

INTRODUCTION

The present study proposes a new modal analysis method, employing the modulated coordinates, for general rotor systems, of which rotating and stationary parts are asymmetric. For asymmetrical rotors with isotropic stators, the periodically time-varying linear differential equation expressed in stationary coordinates can be transformed to the time-invariant linear differential equations expressed in the rotating coordinates. Then the modal analysis becomes essentially the same as the ordinary complex modal analysis method developed for anisotropic rotors, which possess asymmetric properties only in the stator part. However, the asymmetric rotor system with anisotropic stator cannot be transformed to a finite order equation of motion with the time-invariant parameters. It is well known that the concept of directional frequency response functions (dFRFs) is very useful for the vibration analysis of rotating machinery with asymmetric and/or anisotropic properties. In particular, the reverse dFRF has been found to be a sensitive indicator to presence of anisotropic or asymmetric properties. However, the usefulness of dFRFs has not been fully proven in the presence of both the asymmetric and anisotropic properties in the rotor system.

Unlike the asymmetric rotor system with isotropic stator, the general rotor cannot be transformed to a finite order equation of motion with the time-invariant parameters, regardless of the coordinate system

(2)

method has been introduced to perform a modal analysis of rotor systems with periodically time- varying parameters by employing time-varying eigenvectors but confined to real coordinate systems.

This paper introduces a generalized theory of complex modal analysis method using modulated complex stationary coordinates for asymmetric rotor systems with anisotropic stators. The method introduces modulated complex stationary coordinates to derive an equivalent, infinite-order time- invariant equation of motion. Then, the characteristics of eigenvalues and eigenvectors are theoretically investigated thoroughly by using the equivalent time-invariant equation of motion. Two analytical and numerical examples are treated to demonstrate the effectiveness of the proposed method.

1. ANALYTICAL MODAL ANALYSIS

1.1 Equation of motion in the modulated coordinates

The equation of motion of an asymmetric rotor system with anisotropic stator part can be written, using the complex stationary coordinates, as [1]

2 2

2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ),

j t j t

j t

t t t e t t t e

t t t e t

Ω Ω

Ω

+ + + + +

+ + + =

f b r f b r

f b r

M p M p M p C p C p C p

K p K p K p g

(1) where M_i, C_i and K_i denote the complex valued N × N generalized mass, damping and stiffness matrices, respectively; the subscripts f, b and r refer to the mean value, and, the deviatoric values for anisotropy (stationary asymmetry) and asymmetry (rotating asymmetry), respectively;

( )t = ( )t +j t( )

p y z and g( )t =f_y( )t + jf_z( )t are the N × 1 complex response and input vectors, respectively; _y_{( )}_t and _z_{( )}_t are the real valued response vectors, and, _f_y_{( )}_t and _f_z_{( )}_t are the real valued input vectors, in the direction of Y and Z in the stationary coordinates, respectively; Ω is the rotational speed of the shaft; N is the dimension of the complex coordinate vector; j is the imaginary number; the bar indicates the complex conjugate. Note here that the time-varying parameters associated with the harmonic frequency of twice the rotational speed appear due to the asymmetry in the rotor part.

Equation (1) can be transformed to an infinite order matrix equation, introducing the modulated complex coordinate and force vectors, p_;n and g_;n, where the modulation index, n, is an arbitrary integer, defined as

2 2

;_n( )t ≡ ( )t e^{j n t}^Ω, ;_n( )t ≡ ( )t e^{j n t}^Ω

p p g g

(2)

2 2 2

;_n( )t ≡ ( )t e^{j n t}^Ω = ( )t e⁻^{j n t}^Ω, ;_n( )t ≡ ( )t e⁻^{j n t}^Ω

p p p g g

. (3) We can rewrite equation (1) as

( )t + ( )t + ( )t = ( )t

Mp Cp Kp g (4)

where

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

f b

b f r

r f b

b f r

r f b

b f

M M 0 0 0 0

M M M 0 0 0

0 M M M 0 0

M 0 0 M M M 0

0 0 0 M M M

0 0 0 0 M M

…

;1 ;1

; 1 ; 1 ;0

;1 ;0 ;0

;0 ;0 ;1

;0 ; 1 ; 1

;1 ;1

,

− −

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

f b

b f r

r f b

b f r

r f b

b f

C C 0 0 0 0

C C C 0 0 0

0 C C C 0 0

C 0 0 C C C 0

0 0 0 C C C

0 0 0 0 C C

…

(3)

;1 ;1

; 1 ; 1 ;0

;1 ;0 ;0

;0 ;0 ;1

;0 ; 1 ; 1

;1 ;1

− −

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

f b

b f r

r f b

b f r

r f b

b f

K K 0 0 0 0

K K K 0 0 0

0 K K K 0 0

K 0 0 K K K 0

0 0 0 K K K

0 0 0 0 K K

where, ^;

2 2

;

4

2 4

i n i i

i n i i i

j n

j n n

= − Ω

= − Ω − Ω

C C M

K K C M

( ) { }

;1 ; 1 ;0 ;0 ; 1 ;1

( ) ( ) ( ) ( ) ( ) ( ) ,

t t t t t t t

− −

=

p p p p p p p

g g g g g g g

, ,

i=r b f , n= ± ±0, 1, 2,....

Note that the differential equation (2) with periodically time-varying parameters is transformed into equation (4) with time-invariant parameters, at the expense of introducing the coordinate vector of infinite dimension.

1.2 Modal analysis

The equation of motion can be written, in the state space, as ( )t = ( )t + ( )t

Aw Bw F (5)

where,

⎡ ⎤

= ⎢⎣ ⎥⎦

0 M

A M C

, _{= ⎢}^⎡ ^⎤_⎥

⎣ − ⎦

M 0

B 0 K

, _{( )} ^{( )} ( ) t t

t

⎧ ⎫

= ⎨ ⎬

⎩ ⎭ w p

p

and _{( )} t ( )

t

⎧ ⎫

= ⎨ ⎬

⎩ ⎭ F 0

g .

Here, 0 represents the zero vector and matrix of infinite dimension. Assuming the solution form of e^λt

=

w r , we can obtain the eigenvalue and its adjoint problems associated with equation (5) as

( ) ( ) ( )

i i i

r m r m r m

λ Ar =Br and λ_{r m}ⁱ_{( )}⋅A^Tl_{r m}ⁱ_{( )}=B^Tl_{r m}ⁱ_{( )}, (6a) or the equivalent latent value problem as

( ) ( )

(λ_{r m}ⁱ ) ⁱ_{cr m} =

D u 0 and v^iT_{cr m}_{( )}D(λ_{r m}ⁱ_{( )})=0^T (6b)

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

r= ± ± N m= ± ± , i=B,F, where the lambda matrix of degree two is given by

( )λ =λ2 +λ +

D M C K

and the right and left eigenvectors, and the latent vectors take the form of

^c ^c

c c

, ,

λ λ

⎧ ⎫ ⎧ ⎫

=⎨ ⎬ =⎨ ⎬

⎩ ⎭ ⎩ ⎭

u v

r u l v

{ }

c ;1 ; 1 ;0 ;0 ; 1 ;1

, .

T T T T T T T

− −

=

u u u u u u u

v v v v v v v

Here, the pair of eigenvalues, equal in subscript value but different in sign of subscript, are dependent upon each other; they can be complex conjugate pairs with or without a shift parameter ±j n2 Ω, as will be shown later. The superscripts B and F implicitly refer to the backward and forward modes, respectively, and the subscript r(m) refers to the r-th eigen (latent) solution in the m-th cluster, as will be shown later (refer to equation (14a)). The eigenvalues and eigenvectors, obtained from equation (6), are normalized so as to satisfy the bi-orthonormality conditions given by

( ) ( ) ( ) ( )

iT k ki

r mArs =δs r m

l and l_{r m}^iT_{( )}Br_s^k_{( )}=λ δ_{r m}ⁱ_{( )} _s^ki_{( ) ( )}_{r m} , (7)

(4)

( ); ( );

s n= r m n

u u , λ_{r m}_{( )} =λ_s_{( )}. (8b)

Recall that the subscript n denotes the order of the modulated coordinates. Since it holds

, ,

i=k s= −r =m, from the above relations, we obtain, for every pair of latent vectors ₍_u_;^T_n_,_u_;^T_n₎^T associated with the eigenvalue λ_{r m}ⁱ_{( )},

( ); ( );

i i

r m n r m n

− =

u u , λ_{r m}ⁱ_{( )}=λ₋ⁱ_{r m}_{( )}. (9) On the other hand, since it also holds

( ) ( )

2 2

;_n( )t = ;0( )t e^{j n t}^Ω = ⁱ_s( );0e^λ^sⁱ ^te^{j n t}^Ω = ⁱ_{r m n}( ); e^λ^{r m}ⁱ ^t

p p u u , (10a)

we obtain the circulation formula between the eigenvalues associated with the latent vectors u_;n and u;0 as

( ) ( ) 2 , ( ) ( ) ( ) 2 ( ) 2

i i i i i i

r m s j n r m r m s j n s j n

λ =λ − Ω λ₋ =λ ₋ =λ₋ + Ω =λ ₋ − Ω (10b)

which means that, unless A is a singular matrix (M_f and thus A seldom become singular), the shifted eigenvalue by −j n2 Ω and its complex conjugate also become eigenvalues for any integer value of n.

Note that such relation holds only between the same directional modes, forward or backward. For example, if λ₀^F is an eigenvalue, λ λ₀^F, ₀^F − j n2 Ω, λ₀^F − j n2 Ω also become eigenvalues.

In this case, since λ₀^'^F =λ₀^F− j n2 Ω is an eigenvalue, we can derive the relations such as

'

0^F 0^F j n2

λ =λ − Ω, λ₀^'^F− j n2 Ωλ₀^F− j n2 Ω −j n2 Ω =λ₀^F and λ₀^'^F =λ₀^F− j n2 Ω +j n2 Ω =λ₀^F. In other words, there exists a circulation relation between the eigenvalues with the shift of −j n2 Ω. The structure of the eigenvalues and latent vectors can be summarized as:

eigenvalues:

( )

(0) (0) 1(0) 1(0)

1(0) 1(0) (0) (0)

{ } {...,

(..., 2 , 2 ,..., 2 , 2 ,

2 , 2 ,..., 2 , 2 ,...),

...

i r m

F B F B

N N

F B F B

N N

j n j n j n j n

λ

λ λ λ λ

=

− Ω − Ω − Ω − Ω

(0) (0) 1(0) 1(0) 1(0) 1(0) (0) (0)

...,

(..., , ,..., , , , ,... , ,...),

...

F B F B F B F B

N N N N

λ λ λ λ λ λ λ λ

(0) (0) 1(0) 1(0)

1(0) 1(0) (0) (0)

...,

(..., 2 , 2 ,..., 2 , 2 ,

2 , 2 ,..., 2 , 2 ,...),

...

F B F B

N N

F B F B

N N

j n j n j n j n

λ λ λ λ

+ Ω + Ω + Ω + Ω

..}

(11a)

latent vectors:

{ }

c ( );1 ( ); 1 ( );0 ( );0 ( ); 1 ( );1

, .

iT iT iT iT iT iT T

r m r m r m r m r m r m

iT iT iT iT iT iT T

r m r m r m r m r m r m

− − − − −

=

u u u u u u u

v v v v v v v

(11b)

1.3 Directional frequency response functions

Directional frequency response matrices (dFRMs) can be represented, in the form of modal summation using the eigensolutions, as

( )

^c^{( )} ^c^{( )}

, ( )

'

i i T

N

r m r m

i

m i B F r N r m

jω j

ω λ

∞

=−∞ = =−

⎡ ⎤

⎢ ⎥

=

∑ ∑ ∑

⎢⎣^u −^v ⎥⎦ H

(5)

( )

;1 ;1 ;1 ; 1 ;1 ;0 ;1 ;0 ;1 ; 1 ;1 ;1

; 1 ;1 ; 1 ; 1 ; 1 ;0 ; 1 ;0 ; 1 ; 1 ; 1 ;1

;0 ;1 ;0 ; 1 ;0 ;0 ;0 ;0 ;0 ; 1 ;

, ( )

' 1

T T T T T T

T T T T T

N

i

k i B F r N r m

jω j

ω λ

− −

− − − − − − − −

∞ − −

=−∞ = =−

=

∑ ∑ ∑

−

u v u v u v u v u v u v

u v u v u v u v u v u

H ⁰ ^;1

;0 ;1 ;0 ; 1 ;0 ;0 ;0 ;0 ;0 ; 1 ;0 ;1

; 1 ;1 ; 1 ; 1 ; 1 ;0 ; 1 ;0 ; 1 ; 1 ; 1 ;1

;1 ;1 ;1 ; 1 ;1 ;0 ;1 ;0 ;1 ; 1 ;1 ;1

T

T T T T T T

r

− −

− − − − − − − −

− −

⎡ ⎤

⎢ ⎥

⎣ ⎦

v

u v u v u v u v u v u v

( ) i

m

(12)

where the prime notation in the summation implies the exclusion of r=0. Using equation (12), we introduce, among others, four dFRMs, that are important in characterizing the system asymmetry and anisotropy, as

( )

_{;0 ;0}

( )

^{( );0} ^{( );0}

, ( )

'

i iT

N

r m r m

i

i B F m r N r m

j j

ω ω j

ω λ

∞

= =−∞ =−

⎡ ⎤

= =

∑ ∑ ∑

⎢⎢⎣ − ⎥⎥⎦

gp g p

u v

H H

( ) ( )

;0 ;0

; 1 ;0

( );0 ( );0 ˆ

, ( )

( );0 ( ); 1

, ( )

( );0 ( ); 1 ( )

'

i iT

N

r m r m

i

i B F m r N r m

i iT

N

r m r m

i

i B F m r N r m

i iT

r m r m

i

r r m

j j

j

j j

j

j j

j

ω ω

ω λ

ω ω

ω λ

ω ω

ω λ

−

∞ −

= =−∞ =−

∞ − −

= =−∞ =−

−

=

⎡ ⎤

= = ⎢⎢⎣ − ⎥⎥⎦

⎡ ⎤

= = ⎢⎢⎣ − ⎥⎥⎦

⎡ ⎤

= = ⎢⎢⎣ − ⎥⎥⎦

∑ ∑ ∑

g p gp

gp g p

u v

H H

u v

H H

u v

H H

,

N

i B F m N

∞

=

∑ ∑ ∑

=−∞ −

(13)

where

{ }

;_n(jω)= j(ω− Ω2n )

G G , Gˆ (_;_n jω)⁼^G^ˆ

{

^j⁽^ω^{+ Ω}²ⁿ ⁾

}

^.

Here G(jω) and Gˆ (jω)are the Fourier transforms of g( )t and g( )t , respectively. In equation (13),

; ;0( )

n jω

Hg p and

; ;0( )

n jω

Hg p are referred to as the normal and reverse directional frequency response matrix of modulation index n, respectively, which are associated with the shifted input by 2nΩ in the frequency domain.

3. MODAL ANALYSIS FOR SIMPLE ASYMMETRIC ROTOR SYSTEM 3.1 Equation of motion

This section illustrates the modal analysis procedure using modulated coordinates with a simple general rotor system shown in Fig. 1. The equation of motion for the system can be written as

( ) ( ) ( ) ( )

^{j2 t}

( ) ( ) ( )

p t + 2 - jζ αΩ p t +p t +δe ^Ωp t + ∆p t =g t (14)

where ∆ and δ represent the degree of anisotropy and asymmetry, respectively. The reduced equation of motion, including only the coordinates of modulation indices 0, -1 and +1, may be written as

( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)

( )t ( )t ( )t ( )t

+ + + + + + + + = +

M p C p K p g (15)

where

{ }

( 1)

; 1 ; 1 ;0 ;0 ; 1 ; 1

p ( )^T t p ( )^T t p ( )^T t p ( )^T t p ( )^T t p ( )^T t ^T

+ = + − − +

p

(6)

1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1(0)

1(0) 1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1(0) ; 1 ; 1 ;0

eigenvalue left latent vector right latent vector

( , , , , , ) ( , , , , , )

( , , , , , ) ( , , ,

i iT iT

i i iT

v v v v v v u u u u u u

v v v v v v u u u u

λ

λ λ

+ − − + + − − +

− = + − − + − + − _;0 _{; 1} _{; 1} ₁₍₀₎

1( 1) 1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1( 1) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1( 1)

1( 1) 1( 1) 1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1( 1) ; 1 ; 1 ;0

, , )

2 ( , , , , , ) ( , , , , , )

2 ( , , , , , ) ( , ,

iT

i i iT iT

i i i iT

u u

j v v v v v v u u u u u u

j v v v v v v u u u

λ λ

λ λ λ

− + −

− + − − + − + − − + −

− − − + − − + − + + −

= − Ω

= = + Ω _;0 _{; 1} _{; 1} _{1( 1)}

1( 1) 1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1( 1) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1( 1)

1( 1) 1( 1) 1(0) ; 1 ; 1 ;0 ;0 ; 1 ; 1 1( 1) ; 1 ; 1

, , , )

2 ( , , , , , ) ( , , , , , )

2 ( , , , , , ) ( , ,

iT

i i iT iT

i i i iT

u u u

j v v v v v v u u u u u u

j v v v v v v u u

λ λ

λ λ λ

− + − +

+ + − − + + + − − + −

− − + + − − + − − + −

= + Ω

= = − Ω u u u_;0, _;0, _{; 1}₋,u_{; 1}₊)^iT_{− −}_{1( 1)}

⎧⎪

⎪⎪⎪

⎨⎪

⎪⎪

⎪⎩

(16)

3.2 Directional frequency response functions

The directional frequency responses can be expressed as

( ) ( ) ( ) ( ) ( ) ( )

( )

(0);0 (0);0 (0);0 (0);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0

gp

1 0 0 1 1 1 1 1

(0);0 (0);0 (

0

u v u v u v u v u v u v

H ( )

u v u

F F B B F F B B F F B B

N r r r r r r r r r r r r

F B F B F B

r r r r r r

F F

r r r

F r

j

j j j j j j

j

ω ω λ ω λ ω λ ω λ ω λ ω λ

ω λ

− − − − + + + +

= − − + +

− − −

−

≤ + + + + +

− − − − − −

+ +

−

∑

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

0);0 (0);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 1( 1);0 1( 1);0

0 1 1 1 1 1

4 4

1 0 0 1

v u v u v u v u v

(1) (1) ( , ) ( , )

~

B B F F B B F F B B

r r r r r r r

B F B F B

r r r r

N

F B F

r r r r r

j j j j j

O O O O

j j j j

ω λ ω λ ω λ ω λ ω λ

δ δ

ω λ ω λ ω λ ω λ

− − − − − − − − − − + − + − + − +

− − − − − − + − +

= −

+ + + +

− − − − −

∆ ∆

+ + +

− − − −

∑

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

4 4

1 1 1

2 2 6 6 2 2

0 0 1 1 1 1

( , ) ( , )

( , ) ( , ) ( , ) ( , ) ( , ) ( , )

B F B

r r

F B F B F B

r r r r r r

O O

j j

O O O O O O

j j j j j j

δ δ

ω λ ω λ

δ δ δ δ δ δ

ω λ ω λ ω λ ω λ ω λ ω λ

− + +

− − − − − − − + − +

∆ ∆

+ +

− −

∆ ∆ ∆ ∆ ∆ ∆

+ + + + + +

− − − − − −

(17a)

( ) ( ) ( ) ( ) ( ) ( )

(0);0 (0);0 (0);0 (0);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0

ˆgp

1 0 0 1 1 1 1

(0);0 (0);0

H ( )

u v

F B F B F B

r r r r r r r

F F

r r

j

j j j j j j

j

ω λ

− − − − + + + +

= − − + +

− −

−

≤ + + + + +

− − − − − −

+ −

∑

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

(0);0 (0);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0

0 0 1 1 1 1

2

1 0 0

u v u v u v u v u v

( ) ( ) ( ( , ) )

~

B B F F B B F F B B

r r r r r r r r r r

F B F B F B

r r r r r r

N

F B

r r r r

j j j j j

O O O

j j j

δ

ω λ ω λ ω λ

− − − − − − − − − − − + − + − + − +

− − − − − − + − +

=

+ + + + +

− − − − −

∆ + ∆ + ∆ ∆

− − −

∑

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 4 4

1 1 1 1

4 4 2 2

0 0 1 1 1 1

( ( , ) ) ( ( , ) ) ( ( , ) )

( ) ( ) ( ( , ) ) ( ( , ) ) ( ( , ) ) ( ( , ) )

F B F B

r r r

F B F B F B

r r r r r r

O O O

j j j

O O O O O O

j j j j j j

δ δ δ

ω λ ω λ ω λ

δ δ δ δ

− − + +

− − − − − − − + − +

∆ ∆ ∆ ∆ ∆ ∆

+ + +

− − −

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+ + + + + +

− − − − − −

(17b)

( ) ( ) ( ) ( ) ( ) ( )

( )

(0);0 (0); 1 (0);0 (0); 1 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1);0 ( 1); 1 ( 1);0 ( 1); 1

gp

1 0 0 1 1 1 1 1

(0);0 (0); 1 0

H ( )

u v

F B F B F B

r r r r r r

F F

r r

r

j

j j j j j j

j

ω λ

− − − − − − + + − + + −

= − − + +

− − −

−

≤ + + + + +

− − − − − −

+ −

∑

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

(0);0 (0); 1 ( 1);0 ( 1); 1 ( 1);0 ( 1); 1 ( 1);0 ( 1); 1 1( 1);0 1( 1); 1

0 1 1 1 1 1

1 0 0 1

u v u v u v u v u v

( ) ( ) ( )

~

B B F F B B F F B B

r r r r r r r r

F B F B F B

r r r r

N

F B F

r r r r

j j j j j

O O O O

j j j

δ δ δ

ω λ ω λ ω λ

− − − − − − − − − − − − − − + − + − − + − + −

− − − − − − + − +

= −

+ + + + +

− − − − −

∆ + ∆ + ∆ +

− − −

∑

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

4 4

1 1 1

2 2 2 2

0 0 1 1 1 1

( ) ( ( , ) ) ( ( , ) )

( ) ( ) ( ( , ) ) ( ( , ) ) ( ( , ) ) ( ( , ) )

B F B

r r r

F B F B F B

r r r r r r

O O

j j j

O O O O O O

j j j j j j

δ δ δ δ δ

ω λ ω λ ω λ

δ δ δ δ δ δ δ δ δ δ

− + +

− − − − − − − + − +

∆ + ∆ ∆ + ∆ ∆

− − −

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+ + + + + +

− − − − − −

(17c)

( ) ( ) ( ) ( ) ( ) ( )

(0);0 (0); 1 (0);0 (0); 1 ( 1);0 ( 1); 1 ( 1);0 ( 1); 1 ( 1);0 ( 1); 1 ( 1);0 ( 1); 1

gp

1 0 0 1 1 1 1

H ( )

F B F B F B

r r r r r r r

j

j j j j j j

− − − − − − − − + + − + + −

= − − + +

≤ + + + + +

− − − − − −