Vibrations
And their effect on electro-optical imaging spacecraft
Steve Hearon
Applied Math Seminar
December 14, 2009
(to fullfill requirements of an optomechanics course)
Outline
•
•
•
•
•
•
Material Properties, Stress/Strain and Bending
Vibration Transfer Basics (from J. Burge, Univeristy of Arizona)
Electro-optical Spacecraft Examples
Types of Disturbances that cause vibration in optical path
FEA Models
Effects of Vibration on Imaging
– Boresight
– Primary Mirror Distortion
• Conclusions
Material Properties
• Materials will deform when subjected to a load
• Stress-strain relationship is
E  
– where ε is strain (normalized change in length); σ is stress (force/area);
and E is Young’s modulus
– E = 70 x 10^9 N/m^2 for Aluminum
•
For an aluminum bar, 1cm x 1cm x 10 cm, 1 Newton force,
change of length is :
L  (1N  0.1m) /( 70e9 Pa 104 m2 )  14nm
– Stiffness k = F/ΔL = 70,000 N/mm
Bending example: Bar with force applied at end
F
Max deflection occurs at end. Max deflection for the Aluminum bar
As before 1N force, 1cm x 1 cm x 10 cm:
3FL3
3(1N )(0.1m)3


 2.1m
4
4
2 EW
2(70e9 Pa)(0.01m)
Stiffness
2 EW 4
k
 467 N / mm
3
3L
Natural Frequency of Vibration
• Natural Frequency of Vibration of a mode is
fN 
1
2
k
m
– where k is stiffness and m is mass of the moving structure
• For previous examples, assuming a 1-pound mass at
the end of the aluminum bar:
f AXIAL 
1
2
7 e7 N / m
 1880 Hz
0.5kg
f BENDING 
1
2
4.7e5 N / m
 154 Hz
0.5kg
• Bending direction is much more compliant than axial
tension
Analysis of Vibrations (J. Burge, University of Arizona)
• Each degree of freedom can be
represented as a simple mode that
has mass, stiffness, and damping
• This can be modeled using a simple
2nd-order differential equation
• (Charts from J. Burge, University of
Arizona)
Disturbance
Natural Frequency of Vibration
Analysis of Vibrations (J. Burge, University of Arizona)
•
Critical Damping Ratio is
Critical Damping Ratio:
CR  C /( 2mN )
(From J. Burge, University of Arizona)
(From J. Burge, University of Arizona)
Electro-optical Spacecraft Example Hubble
•2.4m Primary Mirror
•Richey-Chretien (2-mirror)
•SA and Coma corrected
•Staring Arrays
•24 arcminutes Full FOV for
widest sensor
•Reaction Wheels for attitude
control
•Gyroscopes for stabilization
Electro-optical Spacecraft Example -IKONOS
•0.7m primary mirror
•10m focal length
•Three-Mirror Anastigmat (TMA)
•Controls SA, Coma, Astig, FC
•Attitude controlled by Reaction Wheels
•Pushbroom detector technology
•6500 lines/sec, 12 um pixels PAN
•48 um pixels MSI
•Fully steerable, swath 12 km
•Resolution 0.8m PAN
Types of Disturbances That Can Affect Optical Path (1)
• Reaction Wheel
–
–
–
–
Electric motor attached to a flywheel
Upon spinup, causes spacecraft to turn other way
Work around a nominal zero rotation rate
IKONOS and Hubble
• Control Moment Gyroscope (CMG)
–
–
–
–
–
Gyroscope that is always spinning during operation ~ 6000 rpm
Spacecraft attitude controlled by turning axes of CMG’s
S/C rotates in reaction to CMG rotation (ang. mom. conserved)
More energy efficient than Reaction Wheels
Used in WorldView Satellite, International Space Station
• Thrusters
– Used for orbit adjusts
Types of Disturbances That Can Affect Optical Path (2)
• Slewing to acquire a target
– s/c decelerates after slewing to begin imaging
– Vibration damps out before imaging can begin
• Motors
– Solar array
– Communications antenna
– Cryocoolers
• Sudden temperature change; thermal snap
– Spacecraft enters or exits umbra
Finite Element Analysis
•Describes material in terms of
small elements connected by
nodes
• Each element models the
compliance of the material
(J. Burge, University of Arizona)
Using FEA Tools in Vibration Analysis (1)
• FEA Tools allow the natural frequencies of a complex structure
to be obtained
• Additionally, the shape of the natural mode is obtained,
• For a complete dynamical description, the damping ratio is
also needed
–
–
–
–
Damping ratio is difficult to model for space structures
Often a low value is assumed e.g. 0.005
It is possible to perform a factory test (“modal test”) to obtain this
However, difficult to transform measurements at std pressure and 1 g
to vacuum and no gravity
Using FEA Tools in Vibration Analysis (2)
• With natural frequencies, mode shapes, damping, and the
input disturbance, the time-varying motion of the structure
can be obtained
• Effect on imaging obtained by importing structure into ASAP
or another optical modeling tool
• Optionally, can perform an analysis based on analytical
calculations
FEA Results from a Study for an 8-Meter space telescope
• Shown on next slide is a table of natural modes
from following report:
– “HST Optics Enhancement, preliminary feasibility
study summary report”, 2000
• Note that mode shapes and frequencies have
been calculated, and damping ratios have all
been set to 0.005
FEA Results from a Study for an 8-Meter space telescope
Trussed Mirror Concept
• Following two slides show NASTRAN results
for first natural frequency for a trussed-mirror
concept
Strutted Mirror - 1st Natural Mode at 73.7 Hz
Lockheed Martin
Baseline Petal - 1st Natural Mode at 63.2 Hz
Lockheed Martin
Effects of Vibration on Optical Path –
Boresight Variation -- Jitter
•
•
Vibrational Modes that affect telescope as a whole
cause jitter
Jitter is often spec’d as a PSD in units of microrad^2/Hz
–
•
•
On a per-axis basis
Frequencies greater than line rate cause image blur
Frequencies less than line rate cause frame-to-frame
jitter (Line-to-line jitter)
PSD (urad^2/Hz)
Frequency (Hz)
Lockheed Martin
Jitter realization With programmed
Motion added
(pushbroom)
Effects of Vibration on Optical Path –
Primary Mirror distortions
• Distortions of the Mirrors (e.g. primary or secondary mirror)
cause spreading of point-spread-function
– Results in a decrease of Strehl Ratio
• Movie showing time-dependent PM distortions, and resulting
variation in Point-Spread Function:
– X:\7\7CNO\Hearon\PSFvsTime_2kHz_04.avi
Lockheed Martin
Conclusions
• Vibrations are an important consideration for
EO spacecraft, across their entire lifecycle
– Development, build, operation
• Vibration analysis tools help designers to
understand and mitigate potential vibration
issues early in the design process
Lockheed Martin