MHD element for Attitude Control and Stabilization of the Rotating

Магнитогидродинамические
эффекты в задачах
ориентации вращающихся
космических аппаратов
Борис Рабинович
Институт Космических Исследований РАН
Январь 2003
Аннотация
• Излагаются основные результаты теоретических
и экспериментальных исследований, связанных с
использованием магнитогидродинамических
(МГД) эффектов в задачах стабилизации и
ориентации вращающихся КА с
деформируемыми элементами .
• Рассматривается новый принцип использования
этих эффектов, основанный на идее «жидкого
гироскопа» (вращающаяся тороидальная полость,
частично заполненная электропроводной
замагниченной жидкостью).
2
• Предлагается использовать такого рода МГДэлементы для создания не требующих затрат
рабочего тела, бесшарнирных систем
ориентации и стабилизации вращающихся КА.
• МГД элементы новой конфигурации
позволяют, в отличие от их первоначальной
версии, реализовать постоянные и медленно
меняющиеся управляющие моменты.
• При этом открываются широкие возможности
включения в состав измерений не только
интегрирующих акселерометров, но и
солнечных датчиков
3
• Автор благодарен ктн Алексею Гришину за
большую работу по математическому
моделированию и построению корневых
годографов и кф-мн Виктории Прохоренко за
подготовку электронной версии этого доклада
4
Stability of a rotating
SC with a flexible
element located along
its rotation axis
[1, 2]
5
Auroral Probe
(INTERBALL project)
6
Mathematical model
θ  i( ΔI  1)θ  ΔIθ  D(ζ  2iζ  ζ )  0;
ζ  (2i  γ)ζ  Δωζ  θ  2iθ  θ  0,
where
θ  θ 2  iθ 3 ;
s
ζ  ;
z0
  ω2  iω3  θ  iθ ;
dζ

ζ 
; τ  ω0 t
dτ
s  q  ip;
dθ

θ
;
dτ
7
Stability condition
I D
det
 I    D  0,
 1 
where
Δω   2  1;
J1  J
ΔI 
;
J
J  J2  J3;
2
0
mz
D
;
J
z0  a  l;
ωc
σ
ω0
8

-1
-0.5
0
0.5
1
1.5
1
0.03
0.053
0.04
0.05432
0.055
0.8
0.06
0.065
0.07
0.6
--
0.4
0.2
0
-0.2
++
+-
-0.4
-0.6
-0.8
I
-1
I
Figure 2. Stability and instability domains for the rotating SC of the AP type:
- - stability; + - instability (one unstable root); + + instability (two unstable roots)
Stability and instability domains for the rotating SC of the AP type:
- - stability; + - instability (one unstable root);
+ + instability (two unstable roots)
9
0.9
Im
0.5
5
0.8
the locus of the
second root l 2
1
0.7
-0.5
2
3
4
5
6
23
17
14
24
7
0.2
23
8
9
0.1
22
0
21
10
-0.2
-0.1
-0.1
0
0.1
19
12
-0.3
18
-0.4
14
15
13
16
12
10
-0.03
-0.02
-0.01
20
0.1
11
1
23
2 - 24
0
120- 14
0.4
0.5
9
8
9
10
0.3
the locus of the
third root l 3
11
19
15

17
16
-0.5
0.2
17
20
-0.2
18
8
18
0.2
11
0.3
20
1
2
3
0.3
15
7
21
I
0.4
6
13
16
0.4
22
-0.04
-0.3
0.5
24
19
-0.4
0.6
4
5
21
22
7
6
4
13
3
Re
0.01
0.02
0.03
-0.1
Figure 4. Root loci of the second(0,0) and third(0,0) roots for the variable parameters  and  I.
Root loci of the second and third roots for the variable parameters  and I
10

s
 
0.01 0.01
-p
0.008 0.008
-p
0.002 0.002
3 3
(a) (a)
(b)(b)
0.00150.0015
0.006 0.006
q
0.004 0.004
0.001 0.001
q
0.00050.0005
0.002 0.002
-0.01 -0.01 -0.005 -0.005
0
0
0
0
-0.002 -0.002
-0.004
-0.006
2 2
-0.004
-0.006
0.005 0.005 0.01 0.01
0
0
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.015 0.015
-0.0005
-0.0005
-0.001
-0.001
-0.0015
-0.0015
Figure 5. Mathematical simulation of the nutation of the gyro-unstable SC of the AP type ( 0 = const =
Figure 5. Mathematical simulation of the nutation of the gyro-unstable SC of the AP type ( 0 = const
=
-1
-1
0.0523
s and c = 0.06
s ): (a) s is a vector locus corresponding to the mass m displacement by the
-1
-1
0.0523 s and c = 0.06 s ): (a) s is a vector locus corresponding to the mass m displacement by the
strains of the flexible element; (b)  is a vector locus corresponding to the angular components of the SC.
strains of the flexible element; (b)  is a vector locus corresponding to the angular components of the SC.
Mathematical simulation of the nutation of the gyro-stable AP- type
SC (0 = const = 0.0523 s-1 and c = 0.06 s-1): (a) s is a vector
locus corresponding to the mass m displacement by the strains of
the flexible element; (b)  is a vector locus corresponding to the
angular components of the SC
11

s
3
0.08
3
-p -p
2
2
1
1
0
0
3
(a)
(a)
qq
-4
-4
-3
-3 -2
-2 -1
-1
0 0
-1 -1
11
2
2
33
0.08
4 4
0.06
0.06
0.04
0.04
0.02
0.02
0
-0.06-0.06
-0.04 -0.04
-0.02
3
0
-0.02
0
0.02
0
-0.02
-0.02
-0.04
-0.04
-3 -3
-0.06
-0.06
-4 -4
-0.08
-0.08
-2 -2
(b)
(b)
2
2
0.04
0.02 0.060.040.08 0.06
0.08
Figure
6. Mathematicalsimulation
simulation of
of the
gyro-unstable
SC of the
const( 
= = const =
0 =type
Figure
6. Mathematical
ofthe
thenutation
nutation
of the
gyro-unstable
SCAP
oftype
the (AP
0
-1
-1
-1 s and c = 0.03 s
-1 ): (a) s is a vector locus corresponding to the mass m displacement by the
0.0523
0.0523 s and c = 0.03 s ): (a) s is a vector locus corresponding to the mass m displacement by the
strains
of flexible
the flexible
element;(b)
(b) 
locus
corresponding
to the angular
components
of the SC. of the SC.
 is
strains
of the
element;
isaavector
vector
locus
corresponding
to the angular
components
Mathematical simulation of the nuta tion of the gyro-unstable SC
of the AP type (0 = const = 0.0523 s-1 and c = 0.03 s-1): (a) s is a
vector locus corresponding to the mass m displacement by the
strains of the flexible element; (b)  is a vector locus
corresponding to the angular components of the SC
12
MHD-element
Theory and experiment
[3, 4]
13
The MHDelement of the
torus shape
completely filled
with an
electroconductive
magnetized
liquid
14
Stability of a rotating SC
with MHD-element
in the control loop
[5, 6]
15
Mathematical model
of a rotating SC with MHD control
  i(I  1)  I  D(  2i   )  0;
  (2i  γ)  Δ    2i    a0   a1 ( - i ).
The root of the characteristic
equations responsible for the stability
~
~
l    i ;
~  
~
~
Da    2 (I  a )
1
1
I  D(1  a0 )
;
~

I  D(1  a0 )
(1  I )  2I  2 D(2  a0 )
16

s
-p
-p
0.6
0.6
0.03
0.03
(a)
(a)
0.4
0.4
0.02
0.02
0.2
0.2
0.01
0.01
q
-0.6
-0.6
-0.4
-0.4
-0.2
-0.2
(b)
(b)
22
00
00
-0.8
-0.8

33
00
0.2
0.2
0.4
0.4
0.6
0.6
-0.05
-0.05
-0.04
-0.04
-0.03
-0.03
-0.02
-0.02
-0.01
-0.01
-0.2
-0.2
-0.01
-0.01
-0.4
-0.4
-0.02
-0.02
-0.6
-0.6
-0.03
-0.03
-0.8
-0.8
-0.04
-0.04
00
0.01
0.01
0.02
0.02
Figure 10.
10.Stabilization
Stabilization of
of the
the gyro-unstable
gyro-unstable SC
SC of
of the
the AP
AP type
type with
with MHD
Figure
MHD elements
elements and
and accelerometers.
accelerometers. The
The
-1
-1
-1
-1
mathematical
simulation
for
=
const
=
0.0523
s
,
=
0.06
s
(a
=
2,
a
=
3):
(a)
s
is
a
vector
locus
the
c
Stabilization
offorthe
SC
AP- type
elements
andlocus
mathematical
simulation
= const = 0.0523
s of
, 
(a00 =with
2, a11MHD
= 3): (a)
s is a vector
00gyro-stable
c = 0.06 s
corresponding to the
mass
m displacement simulation
by the strainsfor
of the
flexible
element;
(b)  is
locus saccelerometers.
mathematical
0 flexible
= const
= 0.0523

corresponding
to theThe
mass
m
displacement by
the strains of
the
element;
(b)  s
is-1aa, vector
vector
locus
c = 0.06
corresponding
to
the
angular
components
of
the
SC.
1corresponding
angular
of thelocus
SC. corresponding to the mass m displacement
(a = 2, a to=the3):
(a) scomponents
is a vector
0
1
by the strains of the flexible element; (b)  is a vector locus corresponding to the
angular components of the SC
17
0.03
0.03

s
5
-p
-4 -2
-2
3
3
2
2
1
1
-2
0.06
a)
a)
0
0
-1
2
2
4
4
6
-0.06
6
0.04
0.04
0.02
0.02
-4
3
b)
b)
2
2
-0.06
-0.04
-0.04
-0.02
0
-0.02 0
-0.02
0
0 0.02
0.020.04
0.040.06
-0.02
-2
-0.04
-3
0.06
3
q
q
0
0
-1
-p
4
4
-4
5
-0.04
-3
-4
-0.06
-0.06
Figure 11. Stabilization of the gyro-unstable SC of the AP type with MHD elements and accelerometers.
Figure 11. Stabilization of the gyro-unstable SC of the AP type
with MHD-1elements and accelerometers.
-1
The mathematical simulation for 0 = const = 0.0523s
, c -1
= 0.03s (a = 2, a1 = 3): (a) s is a vector locus
-1
The mathematical simulation for 0 = const = 0.0523s , c = 0.03s (a0 = 2, a10= 3): (a)
s is a vector locus
corresponding to the mass m displacement by the strains of the flexible element; (b)  is a vector locus
corresponding to the mass m displacement by the strains of the flexible element; (b)  is a vector locus
corresponding to the angular components of the SC.
corresponding to the angular components of the SC.
Stabilization of the gyro-unstable SC of the AP- type with MHD elements
and accelerometers. The mathematical simulation for 0 = const = 0.0523
s-1, c = 0.03 s-1 (a0 = 2, a1 = 3): (a) s is a vector locus corresponding to
the mass m displacement by the strains of the flexible element; (b)  is a
vector locus corresponding to the angular components of the SC
18
0.06
Facility for the experimental studying
of the MHD-phenomena
1, 2, 3 – the rotating MHD-element
19
Experimental results
Amplitude and phase
responses of I/V control loop
, 
- Theory
- Experiment, A(f)
- Experiment, (f)
The hydrodynamical moment
acting on the torus during the
slow braking of its rotation
- Theory
- Experiment without magnetic field
- Experiment with magnetic field
20
MHD-element for
the attitude control
and stabilization of a
rotating spacecraft
Some new ideas [7, 8, 9, 10]
21
Reminiscences concerning some
problems of Rocket Carriers
dynamics and stability
The launches of N-1 heavy Rocket Carrier (RC)
in the years 1969 – 1972 discovered the disturbing
moment in the roll plane, caused by the twist of
the Liquid Propellant Engines (LPE) jets
combination around the longitudinal axis of the
RC.
22
The heavy RC N-1
The view from the tail on the 30 LPE of the N-1 RC
23
The equilibrium forms of 8
interacting LPE jets
• a – The form with
the regular
symmetry
• b – The form with
two planes of
symmetry
• c - The form with
screw symmetry
24
Mechanical models of the LPE jets
forms presented in the previous slide
• a – The form with the
regular symmetry
• b – The form with two
planes of symmetry
• c - The form with the
screw symmetry
25
The launch of N-1 RC 3-L
26
General comment to the slide 22
• Analyzing the situation described above we see the arising
in particular cases of the roll moment caused by a gas
dynamical eccentricity of LPE jets. The moment is acting
on a non rotating object (RC).
• We are looking forward to use the analogous phenomena
for generating the pitch and yaw moments for the attitude
control of the rotating SC. These moments must be in the
contrary to the previous case under strict control. The point
is that we can use for this purpose a hydro dynamical
eccentricity with MHD control.
Let us consider this problem more closely.
27
MHD effects in
the Nature
Force lines of the Jovian magnetic
field in the vicinity of the Io orbit
Eccentric Jovian plasma torus
including the Io moon’s orbit
The forces acting on the elements
of a rotating plasma torus
28
Table 1. Parameters of Jupiter
1
Radius R0 [km]
71 950
2
Period of self rotation T [hr]
9. 9
3
Gravitational acceleration on the planets
equator g0 [g]
2. 64
4
Strength of magnetic field on the planets
equator μ0H0 [Gauss]
4. 28
5
Eccentricity of dipole ε0 [R0]
0. 14
6
Inclination of dipole to the planets
axis γ [deg]
9. 6
29
Table 2. Parameters of Jovian torus
1
Radius r [R0]
2
Mean eccentricity ε [r]
3
Thickness h [r]
4
Mean strength of magnetic field
μ0H [Gauss]
5
Strength of magnetic field [μ0H]
Mean
Min
Max
6.19
5.64
6.73
0. 223
Mean
Min
Max
0.176
0.167
0. 191
0. 00242
Mean
Min
Max
1
0.385
1.72
30
New
MHD-element
realizing the
attitude control of
a spinning SC
Ferromagnetic
magnet guide
Electro
conductive liquid
Winding
31
Rotating SC with a new MHD-element
Mathematical model
2
L1 , s   M ;
2    0   k I I 0 2 ;
L 2 , s   i0 s  2 s  2 I 0 I  J   0.
LI  RI  L J  m I 0 s  V ;
t
Jd

0


L I  J   m  I s  
 0.
 t  
Steady-state regime
.
I  I 2  iI 3 ; J 2  iJ 3 ; V  V2  iV3 ; V  L;

V  V0 1 e lt

2 I 0 V0
iz 0 ms0 I 0V0
 V ; s0 
; 0 
.
2
 R
CA
32
MHD control of the three
surface of the liquid
33
• The fact of vital importance is that the
system being under consideration has no
hinges and does not need any special fuel
expenses
• To confirm the new conception and to make
the next step for its practical application
we must fulfill a good deal of theoretical
and experimental investigation.
34
References
1.
2.
3.
4.
5.
Dokuchaev, L.V., Rabinovich, B.I. Analisis of Perturbed Motion near the
Stability Boundary of a Rotating Spacecraft of the INTERBALL Auroral
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Dokuchaev, L.V, Nazirov, R.R., Rabinovich, B.I., Ulyashin, A.I., On the
Concordance of the Mathematical Model of Nutation of the Interball-2
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35
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7.
Rabinovich, B.I., Prokhorenko., V.I. Concerning the rolling
disturbance caused by the joint work of a Rocket Carriers LPR
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Scientific and technical journal «Polyot» («Flight»), No 5, 1999,
pp. 9 – 16 (In Russian).
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the Magnetohydrodynamic Phenomena. The Application to the
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Astronautics, Ottawa, Canada, November 2000, p. 240a.
36