Magnetohydrodynamical element in the problems of RC and SC stabilization

1
Magnetohydrodynamical element in the problemsof
RC and SC stabilization

• Boris I. Rabinovich
• Electronic version
• Victoria Prokhorenko and Aleksey Grishin

2
Magnetohydrodynamical element in the problems of
RC and SC stabilization
2

• 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).

3
MHD element
3

• General view

• Reynolds, Strouhal, and Alfvens numbers

• The main constants

• The criterion of applicability of the
  mathematical model

4
The mathematical model of the MHD element
Vortex Processes and Solid Body
Dynamics Spacecraft and Magnetic Levitation
Systems Dynamic Problems by Boris I.
Rabinovich Moscow Institute for Control Devices
Design, Russia Valeriy G. Lebedev Research and
Design Institute, Moscow, Russia Alexander I.
Mytarev Research and Design Institute, Moscow,
Russia translated by A.S.. Leviant FLUID
MECHANICS AND ITS APPLICATIONS 25 Translated
from the Russian October 1994, 308 pp.
4

• General equations

• Equivalent oscillator

Kluwer Academic Publishers Group
U liquid velocity I external current J
eddy current
5
Methodical example
5

• G - Gravity center
• M MHD element
• ?0 Accelerometer
• Y Non conservative force

6
The maintenance of dynamical stability
6

• Mathematical model
• Characteristic equation and stability condition

7
POGO problem
7
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)

• RC body strains during its longitudinal
  oscillations

8
The mathematical model of POGO for the RC with
MHD element and accelerometer
8

• ?, 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

9
Approximate solution of the characteristic
equation
9
9

• The non-dimensional parameters ?, ?, ?.
• The subscripts q -RC body r - liquid in MHD
  element s - liquid propellant in the line

10
The designation of the stability and instability
regions
10
10
- -
-
-

11
The stability and instability regions. LPM with
the phase retarding
11
a
a

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

-
-
-
- -
1
ß lt 0
ß gt 0
0
0
1

• The improved propellant line with
  hydro-accumulator (stability)

- -
-
-
12
The stability and unstability regions. LPM with
the phase outstripping
12
a
a

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

-
-
-
-
-
- -
1
1
1
ß lt 0
0
0
ß gt 0

1
0

• The propellant line with hydro-accumulator
  (instability at two frequencies ?s
  and ?q )

- -

0
-
-
-
-
13
The control law for the MHD element
13

• The conjugate control law

• The real parts of the characteristic equations
  roots

14
The stability and instability regions. LPM with
the phase outstripping
14
a
a?

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

-
-
-
-
1
- -
1
0
ß gt 0
0
ß gt 0
- -

• The use of the additional control loop with MHD
  element and accelerometer (stability)

-
-
15
Auroral Probe (AP) spacecraft of the INTERBALL
project
15

• The flexible spike antenna located along the
  rotation axis

16
The samples of unstable nutation of AP, ?0
3?/s, TMI from Sun sensor
16

• ?) 23.10.96, 17 38 MT
• b) 24.10.96, 05 10 MT
• c) 02. 08.97, 07 20 MT

?)
b)
c)
17
Evolutionary rife unstable nutation of the AP,
the attitude control system is switched on (?,
b), TMI from Sun sensor
17
a)

• 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

??
b)
c)
??
18
The main designations
18

• ?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
• ? 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.

19
The mathematical model of SC of AP type with MHD
elements and accelerometers (k2) and
non-controlled SC (k1, a00, a10)
19

• The equations of disturbed motion

• The generalized coordinates

• The main parameters

20
Stability and instability regions for the
rotating SC of AP type
20

• - - stability
• - instability, one root
• instability, two roots

21
Root locuses for variable parameter?? (solid
line -exact, thin line _ approximate)
21
22
Root locuses for variable parameters??? and ? I
22
?2 root locus
?3 root locus
23
The analytical solution for SC of AP type
23

• ? vector

• ? vectors locus

24
The variable parameters of mathematical model of
AP and the initial values of coordinates and
velocities(??c const 0.0465 s-1)
24
25
The initial stage of the unstable nutation of
the AP (?0 3?/ s and ?0 4?/s), mathematical
simulation
25

• a) See table 24

• b) See table 24

26
Unstable nutation of AP (?0 3 / s),
mathematical simulation
26

• a)

• ?), b), c) See table 24

b)
c)
27
Stability and instability regions for variable
parameters ?0 , ?1 (?0 0.06 s -1)
27
28
Stability and instability regions for variable
parameters ?0 , ?1 (?0 0.03 s -1)
28
29
Root locuses for SC AP type with MHD elements and
accelerometers in the control loop (?02, ?13)
for variable parameter ?? (solid line - exact,
thin line _ approximate)
29
?2 root locus
?3 root locus
?4 root locus
30
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
31
Stabilization of the gyro stable SC of AP type
with MHD elements and accelerometers,
mathematical simulation(?? 0.06 s -1, a0 2,
a1 3)
31
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
32
Mathematical simulation of the nutation of gyro
unstable SC of the AP type (?c 0.03 c-1)
32
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
33
Stabilization of the gyro unstable SC of AP type
with MHD elements and accelerometers,
mathematical simulation(?? 0.03 s-1, ?0 2,
?1 3)
33
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
34
Liquid hyroscope as MHD element
Stability regions Instability regions
34

• Mathematical model

r0 , h - mean radus and thickness of the liquid
sheet
The roots of the characteristic equation
The stability borders
35
Magnetohydrodynamical element in the problems of
RC and SC stabilization Summary
35

• 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.