magnetohydrodynamical element in the problems of rc and sc stabilization

Magnetohydrodynamical element in the problems

of RC and SC stabilization

Boris I. Rabinovich

Electronic version

Victoria Prokhorenko and Aleksey Grishin

Magnetohydrodynamical element in the problems of RC and SC

stabilization The use of the control system including the MHD

elements for stabilization of the dynamically unstable objects has been considered. The mathematical model of MHD element is transformed to the model of equivalent oscillator.

The possibilities of the control system with MHD elements are presented: RC unstable in the longitudinal direction (POGO) and the unstable rotating SC with the flexible spike antenna located along the rotation axis (like Auroral probe of INTERBALL project).

2

MHD element

• The main constants

•

.V

aAl;Sh;lRe

;UV;V

VSh;

VlRe

2

2

MM

2

MM

maxF

.Ha;l;1 0

0 MMM

.1Al;1Sh;1Re;1~Sh;1ReMM

3

• Reynolds, Strouhal, and Alfven`s numbers

• The criterion of applicability of the mathematical model

• General view

The mathematical model of the MHD element

• Equivalent oscillator

Vortex Processes and Solid Body Dynamics

Spacecraft and Magnetic Levitation Systems Dynamic Problems

byBoris I. RabinovichMoscow Institute for Control Devices Design, RussiaValeriy G. LebedevResearch and Design Institute, Moscow, RussiaAlexander I. MytarevResearch and Design Institute, Moscow, Russiatranslated byA.S.. Leviant

FLUID MECHANICS AND ITS APPLICATIONS 25

Translated from the Russian

October 1994, 308 pp. Kluwer Academic Publishers Group

δ(t).IRαUJL*IL

;0t

τt

dτ)(τJγαUIJL*

;0τt

dτUβI)α(JUm*

t

t

0

2rr

ti0

δ(t)dt.kr)rr(

;rU;eδδ;0R;0γ

U – liquid velocity; I – external current; J – eddy current;

4• General equations

Methodical example

• G - Gravity center• M – MHD element;• О0 –Accelerometer;• Y – Non conservative

force

5

The maintenance of dynamical stability

• Mathematical model

• Characteristic equation and stability condition

.1

~a;~a;~a;~a

;0)x(aaass

;0saa

;0saaa

s2

s2

s

0sss

2s

s2s

s2s

0

;0)](

[)]()[(

aaaaxaa

pxaaappaa

aaxaappaap

ss0

ss

40ss

22s

2

ss0

ss2s

2422s

2

6

POGO – problem

• RC body strains during its longitudinal oscillations

The eigen frequencies of the longitudinal oscillations of the RC body (f q j ) and of the LOX in the oxidizer line (f s 2 ) of Saturn 5 RC ( ___ AS-501, AS-502 ; __ . __ AS – 503)

7

The mathematical model of POGO for the RC with MHD element and accelerometer

, q, s, r – the generalized coordinates of RC as a solid body, and as a elastic bar, of the liquid in the propellant line and inside the MHD element

).q)x(();ss(as

)];(exp[)(A)(L

;,r,;s,s,;(p)L

;δraqaξarωrβr

;0qaξasωsβs

;δsarasaqωqβq

;δsarasaξ

02s

0s

2s

0s

0

00

rrqrξ2ss

sqsξ2ss

qqrqs2qq

ξrξs

ii

8

Approximate solution of the characteristic equation

• The non-dimensional parameters: , , .• The subscripts: q -RC body; r - liquid in MHD element;

s - liquid propellant in the line;

;)(1)(12

1

;)()(2

1

srqssss

qrqsqqq

BB

BB

.)(

;1;

).(sin)()(

);(sin)()(

0

2

2

0

axa

aaaa

AaaBAaaaB

r

qr

s

q

s

qsq

rrrrr

sssss

99

The designation of the stability and instability regions

1010

0pLs )( ; 0pLr )( or 0 q s

Stability

Sign < 0 < 0

Designation

Instability

Sign > 0 < 0

Designation

Sign < 0 > 0

Designation

Sign > 0 > 0

Designation

- -

+ -

- +

+ +

• The initial propellant line (instability at the frequency ~ q)

• The improved propellant line with

hydro-accumulator (stability)

The stability and instability regions. LPM with the phase retarding

- -

+ -

- +

1

0β <

α

0

α

β >

- +- +- -

+ -

1

11

• The initial propellant line ( instability at the frequency ~ q)

• The propellant line with hydro-

accumulator (instability at two

frequencies: ~ s and ~ q )

The stability and unstability regions. LPM with the phase outstripping

- -

+ -

- +

1

0β <

α

0

α

β >

- +- +- -

+ -

1

12

- +

+ -

+ +

1

0

1

0

+ -

- +

+ +

The control law for the MHD element

• The real parts of the characteristic equations roots

• The conjugate control law

13

.0)(,0)(

);()(),()(

srqr

sssrqsqr

BB

BBBB

.10;t

t;

;)(2

1

;)(2

1

k

*

sr

*

sss

qr

*

qqq

B

B

• The propellant line with hydro-accumulator and damping device (instability at the frequency ~ q)

• The use of the additional control

loop with MHD element and

accelerometer (stability)

The stability and instability regions. LPM with the phase outstripping

+ -

0 β >

α

0

α

β >

- +- +- -

+ -

1

14

- -

- +

+ -1

Auroral Probe (AP) spacecraft of the INTERBALL project

• The flexible spike antenna located along the rotation axis

15

The samples of unstable nutation of AP, 0 = 3/s, TMI from Sun sensor

• а) 23.10.96, 17 : 38 MT;

• b) 24.10.96, 05 : 10 MT;

• c) 02. 08.97, 07 : 20 MT

а)

b) c)

16

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8

Evolutionary rife unstable nutation of the AP, the

attitude control system is switched on (а, b),

TMI from Sun sensor • a) 23.10.96, 05 : 58 MT, 0= 3/s ;

• b) 24.10.96, 11 : 47 MT, 0= 3/s;

• c) 03.09.96, 12 : 04 MT, 0= 4/s

17

a)

b) c)

βо

αо

The main designations

j (j = 2, 3) – the angles characterizing the attitude of SC relative to the inertial frame;

j (j = 2, 3) – the angular velocity components in the frame connected with SC;

• p j, q j (j = 1, 2) – the transversal shifts of the attached masses of the flexible and MHD elements relative to the SC;

• m, l – the attached mass and the length of the flexible element;

• а – the distance from the connection point of the flexible element to the center of masses of SC;

0 – the angular velocity of SC rotation around the longitudinal axis;

c – the eigen frequency of the flexible elements oscillations.

18

The mathematical model of SC of AP type with MHD elements and accelerometers (k=2) and

non-controlled SC (k=1, a0=0, a1=0) • The equations of disturbed motion

• The generalized coordinates

19

.)(22(

;0)2(kDI)1I(

01

aiaii

ii

)

.t;d

d;

d

dθθθ;θ

;2

ppp;

2

qqq;

z

pq;θθθ

032

2121

032

ii

ii

l.az;J

zmD

;σ;JJJ;1σΔ;J

JJI

0

20

0

c32

21

• The main parameters

Stability and instability regions for the rotating SC of AP type

- - stability

+ - instability, one root

+ + instability, two roots

20

0,04

0,0530,03

0,05432

0,055

0,06 0,065 0,07

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

-1 -0,5 0 0,5 1 1,5

- -

+ -++

Root locuses for variable parameter (solid line -exact, thin line _ approximate)

21

0.070.06

0.040.03

0.05432

0.07 0.06 0.05432

0.04 0.03

0.03

0.07

0.06

0.04

0.05432

-2,5

-2

-1,5

-1

-0,5

0

0,5

1

1,5

-0,46 -0,36 -0,26 -0,16 -0,06 0,04

Re

Im

Approximate value of IM 2

4 root

locus

2 root locus

root locus

0.07

0.06

0.04

0.03

=0.07

0.03

0.04

0.054320.06

-0,5

-0,3

-0,1

0,1

0,3

-0,08 -0,06 -0,04 -0,02 0 0,02 0,04 0,06

Re

Im

Root locuses for variable parameters and I

22

23 4 5

6

7

8

9

1011

24

23

22

21

20

1918

17 16

15

12 - 1413

1

20

19

18

13

12

11

109

876

5

4

14

17 15

16

231 2 - 24 3

22

21

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03

Re

Im

рис.3

12345

6789

10111213 14 15 16 17

18192021222324

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.3 -0.1 0.1 0.3 0.5

3 root locus

2 root locus

The analytical solution for SC of AP type

vector

vectors locus

23

.)1([expB])1(exp[A

;);)1

1(2

;0)2(;e

;t;0)2(

0

с1

20

0

ii

i(

i

i

].2

)1(exp[CB];

2

)1(exp[CA

);exp(C

C

BA

AB;)1()](arg[

);2cos1)(2

1(B)(

12

1

2

22

2

The variable parameters of mathematical model of AP and the initial values of

coordinates and velocities(c = const = 0.0465 s-1)

24

The variable

parameters

The initial values

and the time of integrationFig. №

0

/с

0 θ20

20 /с

30 /с t с

25а 3.0 0.68 0.0033 0 0 5500

25б 4.0 0.68 0.0033 0 0 1500

26а 3.0 0.165 4.0 -0.05 -0.05 540

26б 3.0 0.165 2.5 -0.02 -0.02 540

26в 3.0 0.165 2.0 -0.28 0.28 450

The initial stage of the unstable nutation of the AP (0 = 3/ s and 0 = 4/s),

mathematical simulation • a) See table 24

• b) See table 24

25

-0.30 -0.20 -0.10 0.00 0.10 0.20a,grad

-0.30

-0.20

-0.10

0.00

0.10

0.20

b,g

rad

-0.20 -0.10 0.00 0.10 0.20

-0.20

-0.10

0.00

0.10

0.20

Unstable nutation of AP (0 = 3° / s), mathematical simulation

• а), b), c) See table 24

a)

b) c)

26

-4.00 -2.00 0.00 2.00 4.00

-4.00

-2.00

0.00

2.00

4.00

-8.00 -4.00 0.00 4.00 8.00

-8.00

-4.00

0.00

4.00

8.00

-8.00 -4.00 0.00 4.00 8.00

-8.00

-4.00

0.00

4.00

8.00

Stability and instability regions for variable parameters а0 , а1 (0 = 0.06 s -1)

27

-2

-1

0

1

2

3

4

5

6

-20 -15 -10 -5 0 5 10 15a0

a1

instability corresponding to 2 root


instability corresponding to two roots

stability

а0=2, а1=3а0=3, а1=2.5

а0=0.5, а1=2

Stability and instability regions for variable parameters а0 , а1 (0 = 0.03 s -1)

28

-2

-1

0

1

2

3

4

5

-25 -20 -15 -10 -5 0 5 10a0

a1

а 0 =0.5, а 1 =2а0=3, а1=2,5

а0=2, а1=3


instability corresponding to3 root

instability corresponding to two roots

stability

Root locuses for SC AP type with MHD elements and accelerometers in the control loop

(а0=2, а1=3) for variable parameter (solid line - exact, thin line _ approximate)

29

0.07

0.06

0.05

432

0.05

0,03

0,03

5

0,04

0,04

5

0.07

0.06

0.05

432

0.05

0,03

5

0,04

0,03

0,04

5

-1

0

1

2

3

4

5

6

7

8

-0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0 0,1

Re

Im

4 root locus

0.045

0.04

0.035

0.03

0.05432

0.05

0.06

0.07

0.045

0.04

0.035

0.03

0.050.05432

0.06

0.07

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

-0,12 -0,1 -0,08 -0,06 -0,04 -0,02 0 0,02

Re

Im

3 root locus

0.07

0.05

432

0.05

0.06

0,04

0,04

5

0,03

0,03

5

0.07 0.

06

0.05

432

0.05

0,03

5

0,04

0,030,04

5

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

-0,07 -0,06 -0,05 -0,04 -0,03 -0,02 -0,01 0 0,01

Re

Im

2 root locus

4 root locus

3 root locus

Mathematical simulation of the nutation of gyro stable SC of the AP type

(c = 0.06 c -1)

30

S vector locus corresponding to the mass m displacement by the strains of the flexible element

vector locus corresponding to the angular velocity of the rotating SC

-0,0015

-0,001

-0,0005

0

0,0005

0,001

0,0015

0,002

-0,002 -0,0015 -0,001 -0,0005 0 0,0005 0,001

-0,006

-0,004

-0,002

0

0,002

0,004

0,006

0,008

0,01

-0,01 -0,005 0 0,005 0,01 0,015

q

- p

Stabilization of the gyro stable SC of AP type with MHD elements and accelerometers, mathematical simulation

(с = 0.06 s -1, a0 = 2, a1= 3)

31


vector locus corresponding to the angular velocity of the rotating SC-1,2

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

-1 -0,5 0 0,5 1

q

-p

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0,01

0,02

0,03

0,04

-0,06 -0,04 -0,02 0 0,02

Mathematical simulation of the nutation of gyro unstable SC of the AP type

(c = 0.03 c-1)

32

vector locus corresponding to the angular velocity of the rotating SC


-4

-3

-2

-1

0

1

2

3

-6 -4 -2 0 2 4q

- p

-0,1

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

-0,06 -0,04 -0,02 0 0,02 0,04 0,06 0,08

Stabilization of the gyro unstable SC of AP type with MHD elements and accelerometers, mathematical simulation

(с = 0.03 s-1, а0 = 2, а1 = 3) 33


vector locus corresponding to the angular velocity of the rotating SC-4

-3

-2

-1

0

1

2

3

4

5

-4 -2 0 2 4 6

q

-p

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

-0,06 -0,04 -0,02 0 0,02 0,04 0,06

Liquid hyroscope as MHD element

Mathematical model

r0 , h - mean radus and thickness of the liquid sheet

The roots of the characteristic equation

The stability borders

Stability regions Instability regions

.)1(~

);

~

1)(2

(

;04/~

)~

)~

( 222

h2

hi

h(1hhi

.2/~1;2/~2,12,1 hh

.~

;/;/

;0]4/~

)1~

[~

(

000

0

2

/rp;qsrs

()

hh i

hhhi 2

34

• The RPM of new generation having the open loop response from pump inlet pressure to the combustion chamber pressure with the phase outstripping on the low frequencies make the POGO probability much higher.

• The use of the flexible elements with relative low eigen frequencies located along the rotation axis of the gyro-stabilized SC may lead to non-stability of the steady-state rotation around the axis with maximum moment of inertia. The logarithmic increment of nutation oscillations is proportional to the oscillations decrement of the flexible element and the difference between the SC angular velocity and the eigen frequency of the flexible element.

• One of the possible approaches to solve the stability problems is the use of the additional control system with MHD elements, accelerometers, and (or) angular velocity sensors, accelerometers, and (or) angular velocity sensors.

35Magnetohydrodynamical element in the problems of RC and SC stabilization

Summary

magnetohydrodynamical element in the problems of rc and sc stabilization

Documents

mhd elements

q rc body r liquid

improved propellant

rc body f q j

problemsof rc

problems of rc

oxidizer line f s

mhd elementthe real