Transcript SPIN STABILILIZATION

SPIN STABILILIZATION
1. INTRODUCTION
•Dynamics, Astrodynamics
•Orbital Dynamics, Attitude Dynamics
•Basic terminology
•Attitude
Z
x
y
Y
z
X
Z
z
x
X
Y
y
•Spin stabilization

H
H

Z
X
Y
Single Spinners
Dual Spinners
2. The Euler’s Moment Equations
•Rigidy body dynamics: rotational motion in space
•Torque-free motion
•Reference systems:
•geometrical
•Angular momentum axis
•instantaneous rotation axis
•principal axes

z H p
x
H, ,p
H

y
Pure rotation
Conning
Nutation
z
H   r    r dm
m
I   y  z dm
I   xydm
I    x  z dm
I   xzdm
2
2
x
m
2
2
y
m
xy
m
xz
m
I    x  y dm I   yzdm
2
z
2
yz
m
M
O
dm
m
dH
dt
Torque-free motion
dH
 0  H  const
dt
x
y
Spin stabilization with passive/active control
Major Axis Rule for Spin Stabilization
•Stability of rotation about principal axes
•Consider the the perturbed the steady motion given
by the Euler’s moment equation for torque-free motion:
I   I  I  
x
y
z
y
I
  


I
  


z
  I  I    
I
y
y
y
z
x
0
z
z
z
x
y
0
y
  I  I    
I
y
Differentating w.r.t. time
and eliminating 


 

I 
 
I 
I
I
x
0
z
0
y
y
x
y
z
 I  I I  I  
  

 
II


 I  I I  I  
  

 
II


x
z
z
y
x
z
y
2
0
y
x
y
x
z
z
2
0
y
y
z
z
z
 I  I I  I  
  

 
II


 I  I I  I  
  

 
II


x
Differentating w.r.t. time
and eliminating 

y
x
z
y
y
z
2
0
x
z
y
x
z
z
2
0
y
z
Both of these equations represent simple harmonic oscillator
with general solution:    e   e (j  1, 2)
it
j
Where

j1
 it
j2
( I  I )( I  I )
II
x
y
x
y
y
3
z
If  is imaginary j will diverge andis unstable.  must be real for
stability. This is satisfied when (Ix-Iy)(Ix-Iz) > 0 . Motion is stable
when Ix>Iy e Ix>Iz or when Ix<Iy e Ix<Iz
Conclusion: motion is stable about major or minor axis but motion
about intermediate axis is unstable.
z
Internal Energy Dissipation Effects
All real spacecraft have, at least, some nonrigid properties.
These include: elastic structural deflection and sloshing.
Some lessons learned from the past:
• Explorer I (1958)
Energy dissipation
H
2T 
I
2
H
2T 
I
2
at the major axis
max
at the minor axis
min
Since for torque-free motion the angular momentum must be
conserved motion about the major axis corresponds to the
minimum energy state. Conclusion: a semirigid body is stable
only when spinning about the major axis, bringing about the major
axis rule for spin stabilization.
ATS-5 Satellite - 1969
Examples of Flexibility and/or Dissipation Effects
Year
Satellite
1958
Explorer I
1952
Alouette
1964
Explorer XX
1969
ATS-5
Control
System
Spin
Stabilized
Adverse
Effect
Spin
Stabilized
Rapid Spin
Decay
Spin
Stabilized
Spin Stabilized
with active
Nutation Control
Unstable
Probable
Cause
Internal Energy
dissipation
Solar Torque on
Thermally Deformed
Satellite
Rapid Spin Solar Torque on
Thermally Deformed
Decay
Satellite
Unstable
Internal Energy
Dissipation
Momentum precession and spin thrusters locations
F
R
SACI-1: Spin Stabilized with Geomagnetic Control
Nutation Damper
Torque
coil
SCD-1: Spin Stabilized
Partially Filled Ring Nutation Damper
Torque Coil
SACI-2
Spin stabilized
with geomagnetic
control
Partially filled ring
Nutation Damper
Nutation damper
z
a=15 mm
R=95 mm
h=180 mm
y
x
Spin plane coils
Mathematical model: Satellite With a Partially Filled Ring
Nutation Damper to Prevent Nutation Motion
( I x  I xf )x  [( Iz  Iy )  ( Izf  Iyf )] y z 
 I xy y  I xzf ( z   )  Iaz   I x x  I xyf y 
 I xzf ( z   )  Iyz (  z
2
 2y
)  I xy x z 
 I xz x y  Iyz z  ( Izf  Iax )y   N x
( Iz  Izf )z  [( I x  Iy )  ( I xf  Iyf )] x y 
 ( Iz  Iax )  I xzf x  Iyzf y  Izf z  ( Iz  Iax ) 
 I xzf x  Iyzf y  I xyf ( 2y
 2x
)
 [ I xzf y  Iyzf x ]( z   )  Iaz y   N z
( Iy  Iyf )y  [( I x  Iz )  ( I xf  Izf )] x z 
 I xyf x  Iyzf ( z   )  Iyf y  I xyf x  Iyzf ( z   ) 
 I xzf ( 2x  2z )  Iyzf x y  I xyf y z 
 ( Iaz  I xzf )z  ( Izf  Iax )x  N y
mf R 2  ( Izf  Iax )z  ( I xzf  Iaz )x  I yzf y 
 ( Izf  Iax )z  I xzf x  I yzf y 
1 
( I xf   Izf z ) 
2 

Izf  Iax z  Ixzf x  Iyzf y  




 I yzf y z  I xzf x z  I xyf x y  Q






Computer Simulation
Hz
x  x ( y ,z ,, )
z
y  x ( x ,z ,, )

z  x ( x ,y ,, )
H
Hy
  ( x ,y ,z ,x ,y ,z ,, )
Hx
HT
x
y
H x  AT z  I xy y  I xz ( z   )  Iaz 
  a tan
H x2  H y2
Hz
H y  BT y  I xy x  Iyz ( z   )  Iaz 
H z  CT z  I xz x  I yz y  ( Iz  Iax )
10.00
12.00
Nutation Angle (Deg.)
Nutation Angle (Deg)
8.00
C
 1.1
A
10.00
8.00
6.00
C
 1.2
A
6.00
4.00
2.00
4.00
0.00
2.00
0.00
0
100
200
300
400
500
600
700
800
1000.00
900
3000.00
12.00
60.00
C
 1.05
A
A  B C
Nutation Angle (Deg.)
50.00
Nutation Angle (Deg)
2000.00
Time (Sec)
Time (Sec,)
40.00
30.00
8.00
4.00
20.00
0.00
10.00
0.00
200.00
400.00
600.00
Time (Sec)
800.00
1000.00
0.00
1000.00
2000.00
Time (Sec.)
3000.00
Conclusion
Directional Stability: inertial pointing
Gyroscopic properties of rotating bodies
Major axis rule: rigid body are only idealizations
Single and Dual Spinners
Nutation Dampers: passive and active
Spin stabilization combined with active control