Magnetohydrodynamical element in the problems of RC and SC stabilization
description
Transcript of 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 3 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 to 2 root
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
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-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
S vector locus corresponding to the mass m displacement by the strains of the flexible element
-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
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-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