analysis and design of a thin shell active mirror · analysis and design of a thin shell active...
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Analysis and design of a thin shell active mirror
Item Type text; Thesis-Reproduction (electronic)
Authors Radau, Rudolph Emile, 1948-
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/348341
ANALYSIS AND DESIGN OF A
THIN SHELL ACTIVE MIRROR
by
Rudolph Emile Radau, Jr .
A Thesis Submitted to the Faculty of the
COMMITTEE ON OPTICAL SCIENCES (GRADUATE)
In P a r t ia l F u l f i l lm e n t o f the Requirements For the Degree of
MASTER OF SCIENCE
In The Graduate College
THE UNIVERSITY OF ARIZONA
1 9 7 8
STATEMENT BY AUTHOR
This thesis has been submitted in p a r t ia l f u l f i l l m e n t of requirements fo r an advanced degree at The Univers ity o f Arizona and is deposited in the Univers i ty L ibrary to be made a v a i la b le to borrowers under rules o f the Library.
B r ie f quotations from th is thesis are a l lowable without special permission, provided that accurate acknowledgment o f source is made. Requests fo r permission fo r extended quotation from or reproduction of th is manuscript in whole or in par t may be granted by the head o f the major department or the Dean of the Graduate College when in his judgment the proposed use of the mater ial is in the in terests o f scholarship. In a l l other instances, however, permission must be obtained from the author.
SIG N ED: ^
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
7 / DateR. R. SHANNON ..........Professor Of Optical Sciences
ACKNOWLEDGMENTS
I wish to extend my appreciation to my thesis d i re c to r
Professor Robert R. Shannon fo r providing the impetus to conduct th is
study, and to Dr. W. Scott Smith fo r his enl ightening discussions
during a l l phases of th is work. A special thanks is due to Dr. Ralph
M. Richard in f e r t i l i z i n g my motivation to pursue th is area of
research with his support and enthusiasm.
Addit ional thanks must be forwarded to those persons who
provided the m ater ia ls , f a b r ic a t io n , and adv ice : . Charles Burkhart,
Johannes Appels, Wi l l iam P r a t t , S te r l ing Kopke, and Charles N, Brown,
Last but not least I must express my g ra t i tu d e to Norma Emptage for
preparing th is manuscript.
This work was supported by the United States A i r Force Space
and M iss i le Systems Organization under Contract F04701-75~C-0106.
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS. . . . . . . vi
LIST OF TABLES . . . . . . . . . . . . . . . . . x i i
ABSTRACT . . . . . -. . x i i !
1. INTRODUCTION.......................................................... 1
2. CONCEPTUAL DESIGN. . ... . . . . . . . . . . . . . . . . . . . . 12
The Seal loping E f fe c t . . . . . ............................ 12Relation between Actuator Control and Posit ion . . . . . . 13Previous Studies ............................... . 21
The Origional Concept. . . . . . . . . . 21Single Actuator Model. .......................... 23Nine-Actuator Model. - 30Nine-Actuator Prototype and Experimental Results . . . 42Experimental V e r i f i c a t io n of Nine-Actuator
Using Holographi c Interferometr.y .. . . . . . . . . . 40Experimental Technique ...................... 50Results. . . . . . . . . . . .................. . 52
The 33 Actuator System . . . . . . . . . ........................... 55The 41 Actuator S y s te m ...................... 57
3. STRUCTURAL ANALYSIS. . ........................... .... . . . . .60
The Mathematical Model . . . . . . . . . . . . . .................. . 60Results. ...................... 70
4. MECHANICAL DESIGN OF THE PROTOTYPE ................................... 105
Thin Shell M i r r o r ........................... 106Reference P l a t e . ............................... I l lInact ive Actuator ...................... 115Act ive Actuator. . . . . . . . . . . . . . . . . . . . . . 120
1 v
V
TABLE OF CONTENTS*"~-Cpnt ? nued
Page
SUMMARY AND CONCLUSIONS, . . . ...................... 126
SELECTED BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . 130
LIST OF ILLUSTRATIONS
Figure
1.1. Astronomical M ir ror Support S y s t e m . ................... .... ................................
1.2. Active Mir ror Configurat ion . . . . . . . . . . . . . . . . . .
1.3. Pre-tensioned Truss ....................................
1.4. Off Axis Ray Bundles and the P u p i l s . .
1.5. Ray Aberrations . . . . . . . . . . . . . . . . . . . . . . . .
1.6. Fourth-order Spherical Aberration . . . . . . . . .
1.7. Astigmatic Ray Aberrat ions. ....................................
1.8. Comatic Ray Aberrations ............................
2 .1 . ft versus .8/8 and y / r fo r Posit ion Contro l . . . . . . . . . . .
2 .2 . 3 0 - inch Robertson M ir ror . . . . . . . . . . . . . .
2.3« H versus 8/8 and y / r fo r Posit ion and Slope Control. . . . .o
2 .4 . Orig inal Concept Using S t i f f Outer Rings. .................. ....
2 .5 . Model Constructed to Demonstrate Active Rigid Structure . . . .
2 .6 . Scale Model with One Active Control ...........................................................
2 .7 . Perspective o f Computer Simulation of Design with OneActive Control. ................................ ................................
2 .8 . Plan and Section Views o f Computer Simulation of Designwith One Active Control . . . . . . . . . . . . . . r . . .
2 .9 . Computer P lot fo r f i r s t Analysis of Single Actuator Model . . .
2 .10. Computer Plot fo r Second Analysis of Single Actuator Model. . .
2 .11. S im pl i f ied Three-dimensional Truss . . . . . . . . . . . . .
LIST OF ULUSTRATIONS— Continued
F igu re Page
2.12. Plane View of Support Structure fo r Computer Modelwith Nine Active Controls. .. . . . . . . . . . . . . . . 32
2.13. Sect ion Vi ew of Support Structure fo r Computer Modelwith Nine Active Controls. . . . . ....................... . . . . 32
2 .14. Plan and Cross Section o f Mir ror Structure fo r ComputerModel with Nine Act ive Controls. . . . . . . . . . . . . 33
2 .15. Computer Plot fo r Mir ror with Astigmatism . . . . . . . . . 35
2 .16. Computer Plot o f Thin M irror fo r Astigmatism underActive Control, with Nine Actuators .................. 36
2 . 1 7 . Computer Plot of M ir ror with Focus S h i f t . . . . . . . . . . 37
2.18 . Computer Plot of Thin Mir ror fo r Focus S h i f t underActive Control with Nine Actuators . ...................... 37
2.19. Cross Sections of Thin Mir ror with Focus S h i f t ShowingScalloped E f fect . . . . . . . . . . . . . . . . . . . . . 3 8
2.20. Computer Plot o f Thick Mir ror fo r Focus S h i f t underActive Control with Nine A c t u a t o r s ........................... 39
2.21. Comparison of Reactions fo r Sandwich Type M irror and Thin 1Shell Mir ror . . . . . . . . . . . . . . . . . . . . . . 40
2.22. Computer Plot of Sandwich M irror fo r Focus S h i f t underActive Control with Nine Actuators . . . . . . . . . . . 41
2.23. Photograph of Nine-Actuator M o d e l ....................... 43
2.24. Perspective of Support System of Scale Model with NineActive Supports. ........................... 44
2.25. Plan and Section Views of Support System of Scale Modelwith Nine Active Controls. . . . . . . . . . . . . . . . 44
2.26. Detail ' of Actuators and Wire Assembly . . . . . . . . . . . 46
2.21. Detai l of Typical Thumbscrew Attachments. . ............................. 46,
LIST OF. ILLUSTRATIONS--Continued
F igu re Page
2.28. Detai l of Actuator Movement fo r F i r s t HolographicExperiment. . . . . . i . . . . . ............................................. 47
2.29. Copy o f Holographic Interferogram . . . . . . . . . 48
2 .30. Geometry Used to Form In te r f e r o g r a m s .......................................... 51
2 . 3 1. Slope Change Mode. .................. . 53
2.32. Defocus Mode— F i r s t Attempt............................ 53
2.33. Defocus Mode a f t e r M irror Assembly . . ................... . . . . . . 53
2.34. Defocus Mode— Smal1 Force Applied to Central Actuator Pin. . 53
2.35. Elevation View of the 33 Actuator System ........................ 55
2 . 36 . Plane View of the 33 Actuator System ....................... 56
2.37. T r im e tr ic View of the 33 Actuator System . . . . . . . . . . 56 .
2 .38. Modified Center Actuator Configuration . ................... 58
2.39. Section View of Truss Configuration along Meridonal Plane. . 59V - ‘ •
2.40. Top View of Truss Conf igurat ion................................ 59
3.1 . Top View of the F in i t e Element Model . . . . . . . . . . . . . 6 1
3.2 . Side View of the F i n i t e Element Model . ........................ 62
3.3* 30° Oblique View of the F in i t e Element Model . . . ................... 62
3.4 . Section View of the F in i t e Element Model Along the X-axis . . 63
3.5. In-plane Slope Control . . . 64
3 .6 . Ideal Normal Posit ion Control Truss Model. . . . . . . . . . 65
3.7* Normal Posit ion Control Truss Model that was Used. . . . . . 66
3 .8 . F in i t e Element Model of the M ir ror . ................................. 67
3 .9 . Tr iangular Edge Element. . . . . . . . . . . . . . . . . . . 68
I X
LIST OF I LLUSTRATIQMS— Continued ,
F igu re Page
3.10 The Composition of the Plate-bending Quadr i la tera lElement. . . . . . ...................... 69
3.11. Truss Deflect ions fo r In-plane Slope Control . . . . . . . 71
3.12. Normalized Mir ror and Reference P late Deflect ions fo rNormal Posit ion Control. . . . . . . . 72
3 . 1 3 . Cubic Spline Contour Plot of Central Actuator 's SlopeControl, Load Case #1 73
3.14. Cubic Spline Contour Plot o f Central Actuator 's NormalPosit ion Control, Load Case #2 . . . . . . . . . . . . . 74
3.15. Cubic Spline Contour Plot of 4 . 8 " r Actuator 'sTangential Slope Control, Load Case #3 . . . . . . . . 75
3.16. Cubic Spline Contour Plot of 4 . 8"r Actuator 's RadialSlope Control, Load Case #4.......................... 7&
3.17. Cubic Spline. Contour Plot of 4 . 8"r Actuator 's NormalPosit ion Control , Load Case #5 . . . . . . . . . . . . 77
3.18. Cubic Spline Contour Plot of 9 . 6 " f Actuator 's TangentialSlope Control, Load Case # 6 , ............................... 78
3.19. Cubic Spline Contour Plot of 9 . 6"r Actuator 's RadialSlope Contro l , Load Case #7. . . . . . . . . . . . . . 79
3.20. Cubic Spline Contour Plot o f 9 - 6"r Actuator's NormalPosit ion Control, Load Case #8 ...................... .... . . . . 80
3.21. Cubic Spline Contour Plot of Edge Actuator 's TangentialSlope Control, Load Case #9* • ............................................ 81
3.22. Cubic Spline Contour Plot o f Edge Actuator 's RadialSlope Contro l , Load Case #10 . . . . . . . . . . . . 82
3.23. Cubic Spline Contour Plot o f Edge Actuator 's NormalPosit ion Control, Load Case # 11 i .......................... .... 83
3.24. Zernike Polynomial Contour Plot o f Central Slope Control,Load Case #1 . . . . .................................... . . . . . . 85
X
LIST OF I [LUSTRATIONS— Continued
Figure Page
3.25. Zernike Polynomial Contour Plot o f Central Posit ionControl, Load Case #2. . ...................... 86
3.26. Zernike Polynomial Plot of Tangential Slope Controlof 4 .8 " r Actuator, Load Case #3- • • . . . . . . . . . 87
3.27. Zernike Polynomial Plot o f 4 . 8 " r Actuator Radial SlopeContro l , Load Case #k. . . . . ............................... 88
3.28. Zernike Polynomial Contour Plot of 4 . 8"r Actuator 'sNormal Posit ion Control, Load Case #5.................. .... . . . 89
3.29. Zernike Polynomial Plot of 9 . 6"r Actuator 's TangentialSlope Control , Load Case #6. ...................... 90
3.30. Zernike Polynomial Contour Plot o f 9 . 6"r Actuator 'sRadial Slope Contro l , Load Case # 7 .................. . . . . . 91
3.31. Zernike Polynomial Contour Plot of 9 . 6"r Actuator 'sNormal Posit ion Control, Load Case #8. . . . . . . . . 92
3.32. Zernike Polynomial Contour Plot o f Edge A c tua to r ' sTangential Slope Control, Load Case #9 • . . . . . . . 93
3.33. Zernike Polynomial Contour Plot o f Edge Actuator 'sRadial Slope Contro l , Load Case #10. . . . . . . . . . . 94
3.34: Zernike Polynomial Contour Plot o f Edge Actuator 'sNormal Posit ion Contro l , Load Case #11 ...................... 95
4 .1 . Sca t te rp la te Interferograms of M irror before I t WasMounted to the Assembled Support S t r u c t u r e .................. ... 108
4.2 . Foucault Test of M ir ror a f t e r I t Was Mounted to theAssembled Support Structure. .................. 108
4 .3 . Cracks in Pyrex Mir ror . . ............................ 110
4 .4 . Final Grinding of Aluminum M ir ror . . . : . . . . . . . . . . 112
4 .5 . Removable Spoke of Reference Plate . . . . . . . . . . . . 113
LIST OF ILLUSTRATIONS--Continued
F ig u re Page
4 .6 . The Removable Spoke of the 4 .8 - inch Active Actuator .Mounted to the Reference P la te . . . . . . . . . . . . . .114
4 .7 . The Assembled 2 4 - inch Prototype Mounted in the Test Mount . 116
4 .8 . Inact ive Actuator Configuration . . .................... . . . . . . 117
4 .9 . Mir ror End Cap. . ....................... 117
4.10. Preload End Cap . . . . . . . . . . . .................... . . . . . 119
4.11. Slope Control o f Inact ive Actuators , . ........................................... 119
4.12. Active Actuator Configuration . . . . . . . . . ................... 120
4.13- Normal Posit ion Control S l ide and the Actuator Post . . . . 121
4.14 . Methods of Actuation in the Servomechanism. . . . . . . . . 123
4.15. Worm Gear Drives in the Servomechanism....................... . . . . . . 124
LIST OF TABLES
Table Page
1. Modified C i rc le Polynomials Q™(r). . . . . . . . . . . . . 97
2. Zernike Polynomials Used by FRINGE.......................... 98
3. Zernike Polynomial Coeff ic ients in Wavelengths fo r LoadCase 1 through 11. . . . . . . . . . . . . . . . . . . . 100
4. Root-Mean-Square Error of the Best F i t Polynomial to theNodal Displacements of the M irror in Terms ofWavelength fo r Load Case 1 through 11............................ 102
ABSTRACT
This thesis is part o f a continued study on the app l ica t ion of
t e n s i le structures to f ie x ib ie -m ? r ro r ac t iv e opt ics , conducted j o i n t l y
by members of the Optical Sciences Center and the C iv i l Engineering
Department of The Univers i ty of Arizona. Such a system is very advanta
geous fo r l ightweight systems ap p l ica t ion s , since the shell of the mirror
is an integra l p a r t of the t e n s i 1 e-membrane s t ructure .
In the course of th is study a 2 4 - inch 41 actuator prototype was
designed and fab r ic a te d , with one d i g i t a l l y control led actuator at every
unique actuat ing pos i t ion , ac t ive in three degrees-of-freedom.
Tests yet to be conducted w i 11 demonstrate the f e a s i b i l i t y of
such a system. The results of these tests should ex h ib i t a certa in
degree of c o r re la t io n with those of a f i n i t e element ana lys is , and
demonstrate the. loca l ized posit ion and slope c o n t r o l .
This thesis includes a l l work completed to date.
CHAPTER 1
INTRODUCTION
An ac t ive op t ica l support system d i f f e r s from a passive system
in the obvious respect that rea l - t im e control of the shape o f the mir ror
is obtained. Since the ac t ive mirror does not depend e n t i r e ly upon i ts
s t i f fn e s s to re ta in the f ig u re of the sur face , i t is inherent ly l ig h t e r
than the completely passive system.
A large astronomical mirror and i ts ce l l is an example of an
ac t ive support system that is su b s ta n t ia l ly passive, since the mirror
surface and support are not mechanically coupled. Because the mirror is
sub s ta n t ia l ly s t i f f , only rea l - t im e feedback o f g rav i ty in format ion, to
keep the mirror in a s t a t i c s t a t e , is used in the supporting technique
of the c e l l , i l l u s t r a t e d in F ig . 1.1.
Here the load along the edge o f the mirror w i l l be a sinusoidal
d is t r ib u t io n as determined by the o r ie n ta t io n of the 1 ever arms. In
equaliz ing the pressure over the back of the m irror , the airbags are
interconnected with pressure l ines .
In the transformation to an a c t ?ve-surface contro l led mirror
support system, the in te r face between the astronomical m ir ror and c e l l
is replaced by a set of e i th e r force or posit ion actuators referenced to
a r ig id w e l l -de f ined backing s t ruc tu re , i l l u s t r a t e d in F i g . 1.2.
1
2
Mirror
Pressure nesLever Arm
Ai r Pads
Fig. 1.1. Astronomical M ir ror Support System
Mirror
ActuatorsReference Structure
F ig . 1.2. A c t iv e M i r r o r C o n f ig u ra t io n
The reference s t ructure is not absolute ly r ig id but serves as a good
reference fo r the mirror which becomes e i th e r a thin or moderately th ick
s h e l l . Sensors are appropr ia te ly placed so that re a l - t im e o n - l in e
measurement o f the surface f ig u re and feedback to the actuators can
then be made.
The integrated ac t iv e mirror has the inherent l ightweightness of
the ac t iv e mir ror enhanced by the app l ica t ion of t e n s i 1 e-membrane
structures to the design of the support system. The e f f ic ie n c y of these
structures is due to the loads being carr ied in the d i re c t io n of maximum
structure 1 s t I f f n e s s , i . e . , in tens i 1 e-membrane act ion . Lightweightness
is maximized with the membrane of the .she l l becomes an integral part o f
the s t ructure , i . e . , the integrated ac t ive m ir ror . This ac t iv e mir ror
would then correct for the coupl ing due to the integrated mi rror-mount
system. The amount of coupling can be control led by a v a r ia t io n of the
s t i f fn e s s in the s t ruc tu re . in order fo r the actuators to be e s s e n t ia l ly
independent, the def lect ions of the m ir ro r 's surface must be lo c a l iz e d ,
at the indiv idual actuator points. This would correspond to a diagonal
influence matrix having dominate diagonal terms as defined below:
where: [ I ] { f } = {d}
[ I ] = influence matr ix
{ f } = actuator force vector
{d} - mirror d e f lec t io n vector ' '
An example of a t e n s i le s t ructure that has become a basic
component o f the systems studied in th is thesis is the pre-tens I oned
truss i l l u s t r a t e d in Fig. 1.3.
4
T russ Element
ActuatorPost
Fig. 1.3. Pre-tensioned Truss
Now i f the tension of the bottom truss elements of this simple
truss is increased, the actuator post w i l l move upward. S im i la r ly the
actuator post w i l l move downward i f the tension of the bottom truss
elements is decreased. I f the top end of the actuator post is attached
to the surface of the mirror so that i t is normal to the surface, a
change in tension of the bottom truss elements w i l l re s u l t in a force
applied normal to the mirror surface.
I f the bottom truss element is adjusted so that the r ight h a l f
of the bottom element is shortened and the l e f t h a l f lengthened, the
actuator post w i l l ro ta te about i ts top end. This movement w i l l produce
a change in slope of the attached mirror surface.
Throughout this thes is , the funct ional form of opt ica l wavefront
aberrations in pupil coordinates (F ie ld dependence excluded) are used to
character ize the m ir ror 's surface d e f lec t ion s . For the reader to whom
> '5
the terms wavefront and ray aberrat ions have no meaning, a b r i e f o u t l in e
of the f i r s t order imaging properties and the higher order imaging
properties (aberrat ions) of an opt ica l system w i l l be presented.
The f i r s t - o r d e r propert ies o f an op t ica l system are i l l u s t r a t e d
in Fig. 1.4. By convention, the object and image points l i e along the Y
axis . The aperture stop is a p h y s ic a l . r e s t r ic t io n on the s ize of the ray
bundles fo r a l l object points , causing the central ray of the o f f axis
bundles, i . e . , the ch ie f ray, to cross the opt ica l axis at i ts axia l
posi t ion . The image o f the aperture stop by the opt lca l components to
the 1 e f t of the stop is the entrance pupi1. Such object and image planes
are said to be conjugate. The e x i t pupi1 is a 1 so conjugate to the
aperture stop, causing the ray bundles to appear to be projected from i t .
Geometric o p t ic s , in ignoring the e f fec ts of d i f f r a c t i o n , pre-
dicts the ideal opt ica l system w i l l image a point into a point . This
occurs when the surface of constant phase o f the l ig h t bundle in the e x i t
pupi1, i . e . , the wavefront, is spher ica1. The amount the actual wave-
f ront deviates from th is ideal surface, i . e . , the reference sphere, is
termed a wavefront aberra t ion . The e f fe c ts of these deviat ions may be
eas i ly v isua l ized by the "ray" aberrations i l l u s t r a t e d in Fig. 1.5. In
th is case the aberrated wavefront was also spherical but had a smaller
radius o f curvature causing i t to focus 1n front of the foca1 plane.
This second-order wavefront aberration is obviously ca l le d defocus and2 2
is represented, by a second-order term, where W(x,y) = C(x +y ) , in
smal1 aberration theory (C is an a r b i t r a r y constant) . This aberration
may be elirriinated by redef ining the reference sphere about the wave-
f ro n t ' s focus, i . e . , focal s h i f t .
ImageP1 ane
ChiefRayEntrance V Aperture Stop
Optica1
J Exit Pupi 1
Ob je c t Plane Pup i
System
H = Object Height
H' = Image Height
Fig. 1.4. Off Axis Ray Bundles and the Pupils
7
W(x,y)
WavefrontExit Pupi1
Reference Sphere
where: W(x,y) = wavefront aberration
n = index of re f rac t ion
e = transverse ray aberra tion ( in y -d i re c t io n ) Y
-R BW n 3y
6 = log i tudina i ray aberra t ion ( in y -z plane)
= _ iitan<j)
- — = 2 £ f /no0 y
F ig . 1.5. Ray A b e r ra t io ns
Higher order r a d ia l ly symmetric aberrat ions such as fourth-orde
spherical aberra t ion , where W(x,y) = C(x2+y2) 2 , w i l l have the ray
aberrations o f the e n t i re aperture minimized but not el iminated by a
proper choice of a reference sphere. Since this aberra t ion does not
include a second-order term, the focal point of the portion of the wave
f ront that is In f i n i t e s im a l ly close to the axis , i . e . , Gaussian reg ion,
w i l l coincide with the center of curvature of the reference sphere, i . e
Gaussian focus, as i l l u s t r a t e d in Fig. 1.6.
AberratedWavefront
Minimum spot size focal plane
GaussianFocalPlane
Marg i na 1 Ray
ExitPup I 1
Fig . 1.6. Fou r th -o rd e r Spher ica l A b e r ra t io n
9
Fourth-order astigmatism, where W(x,y) = C (y2) , causes the wave-
f ront curvature to have extreme values in the x-y coordinate d irect ions
due to the second-order term of one coordinate in i ts d e f i n i t io n . This
creates two d is t in c t focal posit ions that correspond to the coordinate
d irec t io n s , as i l l u s t r a t e d in Fig. 1.7. At these posit ions the image of
a point becomes a l in e . The image becomes c i r c u la r at the medial focal
plane which l ies h a l f way between the tangentia l and s a g i t ta l focal
planes. Normally expressed as a second-order c y l in d r ic a l function
p a ra l le l to the x -a x is , the aberration is rotated 45° about the z-axis
and defocus, with minus one-ha lf the peak value of the astigmatism,
is superimposed (focal s h i f t to medial focus) so that the aberration may
be expressed in the form: W(x,y) = C(xy).
Sag i t t a 1Focus
Tangent ia 1 Focus
Chief Ray
Fig. 1.7. Astigmatic Ray Aberrations
10
Fourth-order coma, where W(x,y) = C(x2+y2) y , and astigmatism are
f i r s t and second-order functions of image height (H1) respect ive ly .
Since coma is a cubic function of pupil height ( y ) , the transverse ray
aberration (e^) of the e n t i re aperture w i l l have the same sign at the
Gaussian image plane. There fore , the center o f the image w i l l not
coincide with the ch ie f ray as i l l u s t r a t e d in Fig. 1.8. Because of th is
c h a ra c te r is t ic , coma is of ten combined with the f i r s t - o r d e r aberration
t i l t , where W(x,y) = C ( y ) , for the minimization of the ray aberra tions.
In actual opt ica l systems, decentering and t i l t i n g the opt ica l components
w i l l produce th is e f f e c t .
Ch i e f Ray
Gaussian 1 Focal Point
Fig . 1.8. Comat i c Ray A b e r ra t io n s
Assuming that in a s t re s s - f re e s ta te the m i r r o r 1s-surface f igu re
is o p t i c a l l y "pe r fec t" , aberrations w i l l be introduced into the wavefront
upon r e f l e c t i o n , when the mirror is stressed. The aberrations w i l l be
of the same functional form as the m ir ro r 's d e f lec t io n . Therefore, i t
is logical that the def lect ions may be expressed in terms o f aberra t ion .
CHAPTER 2
CONCEPTUAL DESIGN
This chapter describes how the basic t e n s i 1 e-membrane s t ructura l
concepts were combined with higher order shell concepts and f igure
control analysis (Optical Sciences Center, 1974, p p .11-19) for a basis
of design. E a r l i e r studies are reviewed (Optical Sciences Center, 1974,
pp. 20-55) in the descr ip t ion of the evolution of the 41 actuator system..
The Scalloping Ef fect
I t is obvious t h a t , for the same maximum values, fourth and
s ix th -o rd er spherical aberra tion produce larger slopes near the edge of
the aperture than does a second-order e r ro r such as defocus. However
any radius change introduces a fundamental problem in . th e fo ld ing or
"sca l lop ing11 o f the e-ge o f the m irror . This e f fe c t is a non - loca l?zed
displacement c h a ra c te r is t ic o f the shall under the displacements of an
actuator , espec ia l ly the centra l actuator under posit ion c o n t r o l .
A pure radius change of the shell o f the mirror would require a
sta te o f stress that had, for i ts boundary condit ions, a uniform radial
membrane stress reacted by a uniform pressure applied to the surface of
the s h e l l . Bending and shear stresses along the boundary would not e x is t .
In attempting to produce th is change through the bending of the shell by
a d iscre te number of actuator points , def lect ions that are functions of
12
13
angular posit ion are obtained. Since the membrane s t i f fn e s s (the s t i f f
ness associated with middle surface stre tch ing) fo r the mirror is several
orders of magnitude larger than the bending s t i f fn e s s (the s t i f fn ess
associated with the formation of a "developable" shape), and these are
coupled in shell a c t io n , the membrane s t i f fn e s s e f fe c t dominates,
forcing considerable bending to occur in order to accommodate the
enforced displacements at the actuator points. Since the membrane
s t i f fn e s s varies with t , the thickness o f the s h e l l , and the bending
s t i f fn e s s with ^ 3, i t is apparent that by increasing the mirror thickness,
the scalloping e f fe c t may be reduced.
This e f fe c t is considerably smaller for an ast igmatic and comatic
type o f surface e rror as the edge zone is warped (bending action) rather
than becoming compressed or extended (membrane a c t io n ) . Thus ast igmatic
and comatic errors are both eas ier to correc t , and more l i k e l y to occur,
for the same reason.
Relation between Actuator Control and Posit ion
A method o f maintaining the surface f igu re of the mirror requires
a set of actuated support points d is t r ibu ted across the mir ror . The
optimizat ion of the number and posit ion of the actuators becomes a major
design problem. The number of ac t ive support points required for
operation of. a th in mirror is dependent upon a combination of mirror
s t i f fn ess and scale o f the errors to be corrected. Obviously, the
smaller the e r ro r scale with respect to the mirror diameter, the smaller
w i l l have to be the spacing of the actuators in the m ir ro r . . .
14
I f the f igu re e r ro r is known a t each actuator locat ion , i t is
possible to move the actuators so the e r ro r is reduced to zero a t these
locat ions. I f the actuator e f fec ts are e s s e n t ia l ly independent, the
process is rap id ly convergent. However, even though the e r ro r is reduced
to zero at the actuator locat ions, there remains a f ig u re e rror between
those locat ions.
This problem is analogous to the problem of f i t t i n g a prescribed
curve with a sequence o f spl ine funct ions. In th is case the prescribed
curve is the f igu re e rror which we want to remove, and the sp l ine
functions are the s t ructura l deformations of the mirror between the
actuator loca t ions . These local deformations may be approximated by
spl ine functions to simulate the s t ructura l behavior. In p a r t ic u la r ,
fo r th is study one-dimensional cubic splines were chosen with the
propert ies that d e f le c t io n , slope, and curvature are continuous at the
actuator locat ions, and that curvature in the radial d i rec t io n at the
outer edge of the mirror is zero , The second property fo r the outer
edge is imposed because the actuators as described here would not produce
a change of curvature at the outer edge. A sinusoidal e r ro r function was
assumed, and the maximum in te rpo la t ion e r ro r with the cubic splines was
determined for various actuator spacings. The results o f th is numerical
experiment may be summarized with th is empirical equation:
in which 3 is the amplitude of the i n i t i a l e r ror funct ion , g is the0
amplitude of the residual e rror a f t e r correction has been applied,
15
s is the ac tuator spacing, and y is the "sp a t ia l frequency" or the
generalized s ize of the i n i t i a l e r r o r .
In applying th is re s u l t in to the required number n o f actuators
fo r a two dimensional system, i t is assumed that the regions of
influence fo r each actuator are equal in area. This resu l ts in the
r e l a t i o n :
ns 2 = irr2
in which r is the m irror radius, and then
6
[6 XK j
r
This r e la t io n is p lo t ted in Fig. 2 .1 .
f 61r
10
-s-
5
Fig. 2 .1 . n versus 6 /6 and y / r for Posit ion Control
16
The greatest residual e rror may be expected near the outer edge
of the mirror where the curvature of the error function is not changed.
This e rror may be fu r the r reduced by closer spacing of actuators in the
radial d i rec t ion near the edge. More c losely spaced (s^) pairs would be
s u f f i c i e n t . Then the greatest residual er ror is reduced by the factor 2
(s^/s) . I t would not be advisable for s^ to be too smal l , say s ̂ < s /4 ,
because the actuat ing forces could be large and have detr imental e f fec ts
on the mirror materia l and on convergence of the error reduction scheme
that would be employed.
To i l l u s t r a t e the use of the chart shown in Fig. 2 .1 , consider
for example the 3 0 - inch Robertson Mirror (Optical Sciences, 1974, p. 15).
In this example, the maximum f igu re e r ro r has a spat ia l frequency along
the l in e AA of approximately twice the mirror radius, and has been
reduced by a fac tor of 25 (Fig. 2 . 2 ) . Also, there is a f igu re error
along the l in e BB with a spat ia l frequency that is approximately equal to
the mirror radius. The er ror has been reduced about 12 t imes. Thus we
have two points marked on the chart marked A for B/8o = 0 .04 , y / r = 2; and
B for B/8o = 0 .08 , y / r = 1. The number of actuators required for these
corrections is 35 and 70, respect ive ly .
The number of actuators a c tu a l ly used was 61. The greatest r e s i
dual error is at the outer edge. By placing addit ional actuators near
the edge with h a l f the spacing in the radial d i r e c t io n , and about 15
a round the m ir ro r , the residual error there would be reduced about 4
times. There would be a smaller reduction elsewhere.
17
Before A f te r
A
Contour In te rva l = yA Contour In te rva l = 7- X40
Fig. 2 .2 . 3 0 - inch Robertson Mir ror
For a second example, consider a th in 1ST m ir ro r . I t
has been proposed to f ig u re such a m irror to a smooth sphere and perform
the aspheric correc t ion with a c t ive controls while the instrument is
in o r b i t . I f the aspheric f ig u re is to be a parabola, the maximum
c o r re c t io n , 6o , is given by
v
2 0 0 0 ( f /n o )4
in which R is the radius of c u rv a tu re , of the m ir ror .
For a 250 cm F/5 m ir ro r , 3q is 0.002 cm with a spa t ia l
frequency four times the radius, or y / r = 4. I f we want the
residual f ig u re e r ro r to be 2 x 10 ̂ cm or 1/20 wavelength (X = .6238 pm),- 3
8 / 3 q = 10 . The required number of ac tua to rs , from Fig. 2 .1 , is about
400. I f the expected thermal deformations in o r b i t are e s s e n t ia l l y
s i m i l a r , th is manner o f actuators should be s u f f i c i e n t to control the
18
f igu re . However, i f the spat ia l frequency of the e r ro r is expected to be,
say, y / r = 0.5 with amplitude 0.002 cm, then about 25,000 actuators would
be required, which is get t ing well beyond the range of f e a s i b i l i t y .
The number of actuators may be reduced by decreasing the radia l
spacing at the outer edge to achieve slope control th e r e . I f the radia l
spacing is 1/4 the usual spacing over the m irror , the maximum error is
reduced by a factor of 16. To use the chart , 8/g may be m ul t ip l ied
by 16 to get 0.016 for the parabolic correct ion . The number of actuators
indicated is then only 25. However, the actual number required would be
about 40, because at least 15 should be added at the outer edge to
provide the reduced spacing in the radia l d i rec t ion without scalloping.
In general , i f the chart indicates actuators, the actual number, n,
required may be estimated by
n = 30 + 0 .4 n 0 < n < 50.c c
In conclusion, the formulas and chart given here provide an order of
magnitude estimate of the number of ac t ive control elements required
to reduce a given f igure error by a specif ied amount. These results
should be helpful in ac t ive mirror studies.
Now, consider ac t ive supports with slope and displacement
correction c a p a b i l i ty . As before, one-dimensional cubic splines are
used to model the mirror behavior. The splines have the properties of
prescribed def lec t ion and slope at each end. Assuming a sinusoidal
19
error funct ion, the maximum in te rpo la t ion error was determined for various
actuator spacings. Results of th is invest igat ion may be summarized with
th is empirical equation:
Each location has three decoupled a c tu a to rs , one for displacement
correction and two for slope component correct ion . Then i t is found that
gives the required number of locations.
To i l l u s t r a t e the use of the chart in Fig. 2 . 3 . , the Robertson
Mirror (Optical Sciences Center, 1974, p. 15) may be used for an example.
Points A and B indicate the number of actuator locations to be 2 and 5,
respect ive ly , with the number of actuators three times as much. However,
at least 20 locations may be required to prevent scalloping of the
mirror . Thus for th is example there would be no advantage to employing
slope correc t ion . In f a c t , the addit ional actuator complexity would be
a decided disadvantage.
o
As before, ns2 = irr2 , where n is now the number of actuator locations.
1.4n
20
Considering a second example of a 300 cm F /1 .5 mirror with
y/4 = 4 and B/B0 = 15 % turns out to be about 30, compared to 4000 for
simple displacement actuators with close radia l spacing near the outer
edge. A decided advantage of slope correction is apparent here.
A *
Xr
.01
10" '
6
Fig. 2 .3 . M. versus 3 /6 q and y / r for Posit ion and Slope Control
n is the number of actuator locat ions. Eachlocation has three decoupled actuators.
I t should be noted that th is s im p l i f ie d , one-dimensional analysis
does not take into account the higher order non-localized shell e f fec ts
such as scalloping. For a more accurate f igure ana lys is , a parametric
study u t i l i z i n g the f i n i t e element method in modeling the shell would
have to be made. Empirical equations in terms of the parameters of
general shell response, such as Gaussian curvature, could then be formu
lated .
21
Previous Studies
This section w i l l present a review of the previous syterns
studies and how that study a f fected the evolution of the system that is
the subject of th is thesis .
The Original Concept
Fig. 2.4 shows an o r ig in a l concept in which a s t i f f outer r ing
was used to d is t r ib u t e in compression the tension forces in a number of
cables supporting the mirror . A photograph of a model b u i l t during th is
concept is shown in Fig. 2 .5 . This approach shows some promise, but i t
does not appear to be eas i ly accessible to simple actuator arrangements.
The actuator approach consists of tension changes in the wire support,
plus a posit ional orthogonal to the wire from each of the connection
points.
Fig . 2 .4 . O r ig in a l Concept Using S t i f f Outer Rings
Fig. 2 .5 . Model Constructed to Demonstrate Active Rigid Structure
23
Single Actuator Model
An i n i t i a l design of a s im p l i f ied mirror support system with one
act ive control was undertaken to determine the f e a s i b i l i t y o f a pre-
tens ioned cable truss support system. The simplest pre-tensioned cables
spread apart at t h e i r center with an actuator post is shown in Fig. 2 .6 .
As previously described, with a few Simple adjustments in the bottom
, cable, a pre-tensioned cable truss attached to a mirror can produce slope
control in the mirror in the plane of the truss as well as v e r t ic a l
d ef lec t ion contro l . Complete three-dimensional slope control can be
obtained by using a three-dimensional truss composed o f two trusses at
r ig h t angles to each other with a common actuator post a t the center.
Adjustments in the two bottom cables can produce a change in slope o f an
attached mir ror in any d i rec t ion of the actuator post.
A scale model was constructed as shown in Fig. 2 .6 . The supports
for the trusses are mounted on a 1 2 - inch square aluminum base p la te . Two
thumbscrews are attached to each support. These thumbscrews control the
ends of the two truss wires terminat ing a t each support. The trusses are
constructed with 0 .014- inch diameter sta in less steel w i re , and the s ingle
post in the center is a 1 . 8 8 - inch long s ta in less steel rod, 1 /2- inch In
diameter, with an enlarged upper end which is cemented to the mirror
surface. The "mirror" is a 12-inch square single Strength glass p la te .
In addit ion to being supported by the movable actuator post in the center ,
the mirror is f ixed to supports at e ight places; i . e . , to four f ixed
posts located on a radius from the center o f 3.53 inches, and to the
four truss supports located on a f iv e - in c h radius as shown.
mo
The computer simulation o f the scale model consisted o f a
structure containing a to ta l o f 42 nodes and 45 elements as shown in
perspective in Fig. 2 .7 , and in plan and section in F ig . 2 .8 . The truss
system contains e ight bar elements (no bending s t i f fn e s s ) simulating the
truss wires and one beam element simulating the actuator post. The
mirror contains 20 rectangular elements and 16 t r ia n g u la r elements. The
four corner sections of the mirror were neglected since they were out
side o f the support points and would have very l i t t l e a f f e c t on deforma
t ions produced at the center. The e ight f ixed support points for the
mirror are at nodes 1, 5, 9, 18, 26, 34, 38, and 42. The four f ixed
support points fo r the truss system are a t nodes 2, 17, 25, and 41.
Al l degrees o f freedom are suppressed at these nodes. Five degrees of
freedom, consisting of three displacements (x, y, arid z d i rec t ions) and
two rotations (about x -ax is and about y - a x i s ) , are permitted at every
other node including node 21 where an add it ional ro ta t ion about the
z -ax is is also permitted.
For the analysis of th is computer model, a temperature
d is t r ib u t io n method of loading was used. By this method two of the
bottom truss wires are given a thermal c o e f f ic ie n t of expansion of
un i ty , and a l l other elements are given a zero value. Then, by placing
d i f f e r e n t temperatures at the end nodes o f the two truss elements, the
elements w i l l undergo a change in length.
The f i r s t analysis was performed as fol lows. The two bottom
truss elements between nodes 17 and 21 and nodes 21 ad 25 were given un i t
values fo r t h e i r thermal co e f f ic ie n ts o f expansion, and a l l other elements
were maintained at zero values. A temperature of -2 degrees was placed
Fig. 2 .7 . Perspect ive of Computer Simulation of Design with One Act ive Control
39 -40
28 30
2018 22 23 26
10 14
T l j r i ' l 1 111111 Y r n - r m 11 j n 2 6
Fig. 2 .3 . Plan and Section Views of ComputerSim u la t io n o f Design w i th One A c t iv e Control
27
at node 17 and also at node 25. All other nodes were given a temperature
of zero degrees. With this temperature d is t r ib u t io n the two bottom truss
elements should decrease in length and tend to produce an upward movement
of the actuator and corresponding pos it ive displacement at the center of
the m i r r o r .
A p lot of the computer resul ts for this f i r s t analysis is shown
in Fig. 2 .9- As expected, a pos it ive displacement occurred at the center
of the m irror . The contour in terval is 0.0703 inches, thus the maximum
posit ive displacement at the center is 0.703 inches. This , of course,
is based on a thermal c o e f f ic ie n t of expansion of one for the two bottom
truss elements instead of the actual value of 6.6 x 10 ̂ i n / i n degrees F.
By using the actual value, the resu l t ing maximum displacement would be
scaled down to a value of 4.64 x 10 ̂ inches, or approximately 1/4-wave
length. The computer resul ts also give a stress in the two bottom truss
elements of 28,259,700 ps i , which scaled down would be 186.5 psi . This
would correspond to a tension of 0.029 lbs. in the bottom wires. This,
of course, neglects any pre-tension since the computer model assumes
s t i f f elements p r io r to loading.
28
Z Deflect ions
Datum = 0 .0 , Contour Interval = 0.0703"
F i r s t Ana lysis
Node 17 = ~2 degrees
Node 25 = -2 degrees
Fig. 2 .9 . Computer Plot for F i r s t Analysis of Single Actuator Model
A second analysis was performed to determine the ef fectiveness
of the truss support system in producing slope con tro l . For this analy
sis the temperature d is t r ib u t io n s and thermal c o e f f ic ie n ts of expansion
were ident ical to those of the f i r s t analysis except tha t a temperature
of +2 degrees was placed a t node 17, and -2 degrees was placed at node 25.
With this temperature d i s t r i b u t io n , the bottom truss element on the r ig h t
should decrease in length and the one on the l e f t should increase in
length. This should tend to s h i f t the bottom of the actuator to the
r igh t producing a change in slope of the mirror at the center.
29
Z Deflect ions
Datum = 0 .0 , Contour Interval = 0.119"
Second Analysis
Node 17 = +2 degrees, Node 25 = -2 degrees
Fig. 2 .10. Computer Plot for Second Analysis of Single Actuator Model
A p lot of the computer resul ts for this second analysis is shown
in Fig. 2 .10. A slope change has occurred. I f the resul ts are scaled
down as in the f i r s t ana lys is , the maximum pos i t ive displacement occurs
at node 23 and has a value of 3•96 x 10 ̂ inches, or approximately
1/5-wavelength. The maximum negative displacement occurs a t node 20 and
has the same magnitude as a t node 23. The computer results indicate
a stress in the bottom r ig h t truss element of +58.5 ps i , and -58 .5 ps i
for the bottom l e f t truss element. This corresponds to a change in
tension of 0.009 lbs for these elements.
3 - V
30
These computer results can be projected to the actual scale
mode 1. These analyses demonstrate the ef fectiveness of the truss support
system in producing posit ion and slope control in the m irror .
Nine-Actuator Model
Based on the promising results o f the analyses fo r the s im p l i f ied
design with one ac t ive c o n t r o l , computer studies were done with a more
sophisticated design. These studies were done to determine the e f fe c - .
t iveness of a truss support system with a number of actuators in
deforming the mirror to a predetermined shape. .
For these computer studies a computer model w i th .n ine act ive
controls was designed. The support system for the computer model was
designed on the basis o f supporting a 12-?nch th in spherical mirror with
a radius o f curvature of 36 inches. ' A s im p l i f ie d truss design was also
used. Instead o f forming a three-dimensional truss by placing two
trusses at r igh t angles to each other with a common actuator post in the
center , a three-dimensional truss was formed from a plane truss by
placing a single wire through the center o f the actuator post at r ig h t
angles to the truss as shown in Fig. 2 .11 .
31
Z
Fig. 2 .11. S impl i f ied Three-dimensional Truss
The computer model consisted of a s tructure containing a tota l
of 60 nodes and 136 elements. The support system, shown in plan in Fig.
2.12 and in section in Fig. 2 .13 , contains e ight bar elements (no
bending s t i f fn e s s ) simulating the outer r ing , eight bar elements
simulating the eight spokes, and s ix bar elements and two beam elements
simulating the truss wires and center actuator for each of nine trusses.
32
59
50 X 52
4 y < ^ 5 .4 6
28.2Q 3 i \
4 Z V ’ 4"
/ 3 6 30 313 7 / \ J 8
I p X s , H1z v 6 , 17 /
8 >Z X 5. .6,7 y X \ MO
Fig. 2 .12. Plane View of Support Structure for Computer Model with Nine Active Controls
29
23 28 37
27 54 47 30
Fig. 2.13- Section View of Support Structure for Computer Model with Nine Active Controls
33
The mirror s t ruc ture , shown in plane and cross section in Fig.
2.14 , contains 48 t r ia n g u la r elements in both bending and membrane states
of stress. The common nodes between the mirror and support structures
are nodes 7, 14, 17, 29, 32, 38, 46, 49, and 56, which, of course, are
also the nine points of ac t ive contro l . Node 35 which is located at the
same point in the center as node 37, is a hub point to which a l l the
spokes are jo ined. This is a f ixed support point at which a l l degrees
of freedom are suppressed. Six degrees of freedom (x, y, and z displace
ments, and rotat ions about x, y , and z-axes) are permitted at a l l other
nodes except at e ight f ixed support points along the outer r ing , i . e . ,
at nodes 1, 8, 10, 23, 35, 50, 52, and 59, where three rotations and
the z displacement are suppressed at each node.
60
------
\ jX 5 3
/ / \ \4?r / \ . / ^ 49 \ \ 43
l \ / \ / \
yp v.! v
\ / \ \ //\ ''■’iK* \ / — /
/ z ' 1
--------- - ----- - 4
Fig. 2 .14. Plan and Cross Section of Mirror Structure for Computer Model with Nine Active Controls
34
The computer model has an overa l l diameter o f 12 inches. The
center actuator is 6 inches long, and the other e ight actuators are
6.5 inches long to account fo r the spherical shape of the m irror . The
mirror has a thickness of 0.08 inches and a radius o f curvature o f J6
inches.
Numerous studies were performed u t i l i z i n g th is computer model.
The general procedure involved the fo l lowing: f i r s t of a l l , a given
mirror deformation p a t t e r n , or shape, was chosen; secondly, the v e r t ic a l
displacements and slopes a t the nine control points were computed for
th is given shape; t h i r d l y , these were used as I n i t i a l displacements and
rotations at the nine control points fo r the computer model; and, f i n a l l y ,
the computer analysis was run and the deformation pattern for the mirror
of the computer model was p lo t ted to compare i ts shape with that o f the
predetermined shape. ,
In some of the studies the predetermined shapes were based on
the second-order aberration defocus and on the four th -order aberra tions,
astigmatism, and coma. In other studies a deformation pat tern
based on the shape o f the mirror used in the Robertson report (1970) was
used. Also, d i f f e r e n t analyses were done with various mir ror thicknesses
and with addit ional i n i t i a l displacements.
35
Z Deflect ions-# o --------------------- • o #-
Datum = 0 .0Contour interval = 3.60 W(x,y) = (1) (xy)
Fig. 2 .15. Computer Plot for M ir ror with Astigmatism
A typical computer study was performed u t i l i z i n g an ast igmatic
aberration as the given f igure error of the mirror . A contour plot for
th is shape, using W(x,y) = ( 1 ) ( xy) which resul ts in an exaggerated
v e r t ic a l scale is shown in Fig. 2 .15. Note that the contour inte rval is
3.60 units . The v e r t ic a l displacements and slopes in the x and y
direct ions were computed and used as i n i t i a l displacements a t the nine
act ive control points; i . e . , nodes 7, 14, 17, 29, 38 , 32, 46, 49, and
56 in the computer model. The computer analysis was then performed and
the resul t ing mirror deformations were p lo t ted . The results are shown
in Fig. 2 .16. Note that the contour in terval is 3.18 un i ts . A
comparison of the two plots demonstrates a f a i r l y good f igu re match with
nine control points. The maximum erro r at any point is less than 12%.
36
-2
-3
Z Def lect ions
Datum = 0.0 Contour interval Mi rro r thickness 0.08 in.
= 3 . 18
Fig. 2 .16 . Computer Plot of Thin Mir ror for Astigmatism under Active Control with Nine Actuators
Another typical analysis was performed in more d e ta i l u t i l i z i n g
2 2defocus. A contour plot for th is shape, using W(x,y) = ( 1 ) (x +y ) is
shown in F ig . 2 .17. The contour in te rva l is 3.60 un i ts . Again, the
v e r t ic a l displacements and slopes were computed and used as i n i t i a l
displacements at the nine control points. The computer analysis was
performed and the resu l t ing deformations were plotted as shown in Fig.
2.18. The contour in te rval is 1.92 un i ts . A comparison of the two plots
in th is case demonstrates a very poor match. The e r ro r at the outer
edge of the mirror approaches 58%. In add it ion , there is a scalloped
e f fe c t at the outer edge of the m irror .
37
Z Deflect ions
Datum = 0.0Contour In terva l = 3.60 W(x,y) = ( 1 ) (x2+y2)
Fig. 2 .17. Computer Plot of M ir ror with Focus S h i f t
Z Deflect ions
Datum = 0 .0 Contour Interval Mirror Thickness
1 .92 0 . 08"
F ig . 2 .18 . Computer P lo t o f Thin M i r r o r f o r Focus S h i f tunder A c t i v e Contro l w i t h Nine Ac tua to rs
33
Plots of the cross sections of the thin mirror under a focus
s h i f t are shown in Fig. 2 .19. The cross sections are taken through the
high point of the scallop (through the control point) and through the
low point o f the scallop (between control p o in t s ) .
I t would appear that several things can be done to reduce or
remove the sca lloping, such as placing addit ional control points closer
to the outer edge of the mirror or disp lacing the control points l a t e r a l l y
when required, or using a th icker mir ror .
AZ (X)
Crosa Section for Ideal Focus Shift
Crosa Section through
Control Point
Actuator Control Point
Cross Section between
Control Points-
F ig . 2 .19 . Cross Sections of Thin Mirror with Focus Shif t Showing Scalloped Ef fect
A second analysis was done with a focus s h i f t using a thicker
mirror to check i ts e f fectiveness in reducing the scalloped e f fe c t .
Instead of a mirror thickness of 0.08 inches which was used in the
preceding computer studies, a mirror thickness of 0 .8 inches was used.
All other input data remained the same as in the f i r s t focus s h i f t
analysis . The computer results for th is th icker mirror are plot ted in
Fig. 2.20. The contour in terval is 3•06 un i ts , and the maximum error has
been reduced to 15% at the outer edge. In add it ion , the scalloping has
been d r a s t i c a l l y reduced. Thus, a th icker mirror does represent one
solution to the problem. However, the solution is not very good when
one is attempting to design a l ightweight s tructure since a th icker
mirror is a heavier mir ror .
Z Deflect ions
Datum = 0.0 Contour Interval Mirror Thickness
3 . 0 60 . 8"
Fig. 2.20. Computer P lo t o f Th ick M i r r o r f o r Focus S h i f tunder A c t i v e Contro l w i t h Nine Ac tua to rs
40
An a l te r n a t i v e to a thick mirror would be a sandwich type mirror
with a l ightweight core. With th is type of mirror an added advantage
should resu l t i f the actuator is f ixed to the core at the middle plane
of the mir ror . Then, when forces are applied to the ac tua tor , the
actuator should t rans fe r them to the upper and lower shells of the mirror
pr im ar i ly as membrane type stresses instead of as bending stresses which
occur with the single thin shell m ir ror . This should fu r th e r reduce the
scalloping without adding to the weight of the mir ror . The d i f f e r e n t
reactions with the two types of mirrors are shown in Fig. 2.21.
Fig. 2 .21 . Comparison of Reactions for Sandwich Type Mirror and Thin Shell Mirror
41
A th ird analysis was done with a focus s h i f t using a sandwich
type mirror . This involved a much more complicated computer model for
the mir ror . The en t i re model required a to ta l of 369 nodes. The to ta l
thickness of the mirror was 0 .76 inches with a top shell thickness of
0.20 inches and a bottom shell thickness o f 0.13 inches. Without going
into fu r the r d e t a i l , a p lot of the computer results for th is analysis is
shown in F ig . 2 .22. The contour in terval is 2.80 units and the maximum
error at the outer edge when compared to the ideal focus s h i f t is 22%.
This is not as good as the th ick m irror , but the scalloped e f fe c t has
almost disappeared.
Z Deflect ions
Datum = 0.0Contour Interval = 2.80 Sandwich Mirror
F ig . 2 .22. Computer P lo t o f Sandwich M i r r o r f o r Focus S h i f tunder A c t i v e Contro l w i t h Nine Ac tua to rs
42
Many other computer studies were performed in add it ion to the
ones described above. In general , the studies have shown that the shape
of a mirror can be qu i te e f f e c t i v e l y contro l led with a system of actua
tors providing the slope control in add it ion to v e r t ic a l displacement
contro l . With only nine control points a surpr is ing ly good match can be
achieved. The studies have also shown that i f a very th in mirror Is to
be used, control points should be placed close to the outer edge to
reduce the warping occurring under cer ta in loading condit ions. In
addit ion the studies have demonstrated the e f fect iveness o f a sandwich
type mirror in adjust ing i t s shape with a minimum of warping under ac t ive
slope and de f lec t ion contro l .. ' * . •
Nine-Actuator Prototype and Experimental Results
During the time that the preceding computer studies were being
performed, an actual scale model, shown in Fig. 2 .23 , with nine .active
controls was being constructed. The design for th is model was s im i la r
to that o f the computer model.
The actual mirror was a 12-inch diameter thin spherical ground
down from a 1-inch blank to a thickness o f 0 .08 inches and a radius of
curvature o f 36 inches. The main support s tructure consisted of two
concentric rings cut from 1-inch aluminum p la te stock and held together
by four bar type spokes. These rings support the nine w ire trusses and
actuators and also support the 54 thumbscrews required to control the
truss wires. This support system is shown in perspective in Fig. 2 .24 ,
and in plan and cross section in Fig. 2 .25 . The outer r ing has
Fig. 2 . 2 3 . Photograph of Mine Actuator Model
F ig . 2.2k. Perspective of Support System of Scale Model with Nine Active Supports
Fig. 2 .25. Plan and Section Views of Support System of Scale Model with Nine Active Controls
an inside diameter o f 12 inches and a channel cross s e c t io n . The inner
ring has the shape of an I -s e c t io n with the v e r t ic a l portion o f
the I having a diameter o f f i v e inches. The remainder o f the structure
is dimensional according to the scale drawings. The actuators were cut
from 1 / 2 - inch diameter sta in less steel rods and were connected to the
truss wires as shown in Fig. 2 .26. Spring steel piano wire with a
diameter of 0.010 inches was used fo r the wire trusses. The wires a t the
ends of the trusses were f i t t e d through small holes in the rings and
attached to the thumbscrews. This is shown in Fig. 2.27 fo r f i v e of the
thumbscrews on the outer r ing. The thumbscrews on the bottom of the
rings control the horizontal and bottom truss wires, whereas the thumb
screws on the top o f the rings control the top truss w ires.
Once the support s tructure was assembled and the trusses and
actuators a l i g n e d , the mirror was attached to the actuators by means of
a heat sens it ive wax. This was used so tha t the mirror could be eas i ly
removed by the app l ica t ion of h e a t . i f fu r th e r adjustments in the trusses
or actuators were required.
I t was thought that th is completed model could be best evaluated
by the use of holographic inter ferometry . By th is method the exact
deformation pattern of the 12-inch spherical mirror could be immediately
determined before, dur ing , and a f t e r a c t iv e control experiments.
Experiments s im i la r to those done in the various computer studies could
be performed, and the results corre la ted .
46
V7
Front View Side View
Wi re Assemb1y (Top)
Wire Assembly (Kiddle)
Fig. 2 .26. Detai l of Actuators and Wire Assembly
Fig . 2 .2 1 . D e ta i l o f Typ ica l Thumbscrew Attachments
A considerable amount of time was spent in obtaining and set t ing
up equipment required for the holographic in te r fe rom etry . In addit ion to
constructing a stable mount for the mirror support system, numerous
adjustments and refinements had to be made with the laser set -up. Due to
the time involved, only one experiment was completed as of th is w r i t in g .
The f i r s t experiment was performed to determine the ef fectiveness
of one actuator in producing a change of slope in the m irror . To
accomplish t h is , the hor izontal truss wire between the center of the
actuator and the outer ring was lengthened by loosening the thumbscrew.
This should cause the bottom of the actuator to move toward the center
of the support system and produce a change of slope in the mirror at the
top of the actuator . This is shown to an axaggerated scale in Fig. 2 .28.
1engthened
Fig. 2 .28 . Detai l of Actuator Movement for F i r s t Holographic Experiment
48
The holograph i c i nterferogram of this experiment was obtained on
a 2" x 2" photographic glass p la te . Although the q u a l i ty o f the image
on the developed p la te was quite poor, the fr inges could be readi ly seen
and counted. A f a i r l y accurate scale drawing of this holographic in te r -
ferogram is presented in Fig. 2.29.
Fig. 2.29. Copy of Holographic Interferogram
The actuator at the top of the mirror in Fig. 2.29 is the one
that was a c t iv e ly con tro l led . The f r inge pattern was pe r fe c t ly symmet
r ica l about an axis from the top to the bottom of the mirror and
corroborates the predicted behavior. The change in slope of the mirror
at the actuator is very obvious.
The f r inge pattern also demonstrates that the actuators can act
r e l a t i v e l y independently of each other . Except for the scalloped e f f e c t ,
49
the deformations are local ized about the one actuator that was a c t iv e ly
contro l led . The scalloped e f fe c t in the hologram also corroborates the
computer resu l ts .
This f i r s t experiment was considered highly successful since i t
did corroborate the predicted behavior o f the actuator and has shown that
a truss support system is .very e f f e c t i v e in producing deformations in
the surface of a m ir ror .
■ Experimental V e r i f i c a t io n o f Mine-Actuator Model Using Holographic Interferometry
The 12-inch diameter deformable mirror shell was tested by
double exposure holographic ?nter ferometry. Of the two modes tested, the
slope change mode performed nearly as expected, whereas the defocus mode
performed less well than expected. The cause may have been due to
inaccurate assembly of the shell in i ts support s t ruc ture .
A computer model for a l ightweight deformable mirror had been
developed and a scale model was constructed in order to v e r i f y the
predicted model. Because the r e l a t i v e deformations of the mirror and
not the actual f ig u re were of in te re s t , holographic in terferometry would
be the ideal test ing method.
Two modes of deformation were to be evaluated— slope change, in
which a radial force would be applied to an actuator pin located at the
0.7 zone; and focal s h i f t , in which the central actuator pin is given an
axia l force. A double exposure hologram, with one exposure made before
and the other made a f t e r the d e f le c t io n , would reveal in terference
f r inges on the Surface o f the mirror which correspond to contours of
surface displacement. '
50
Experimental Technique. The layout of the apparatus is shown: ' 0
in Fig. 2 .30 . An expanded Argon laser beam U=5145A) was.divided by
means of a beam s p l i t t e r cube. One beam would serve as the referenced
beam. The other beam was diverged using a 50 mm camera lens whose focus
approximate ly coincided w i th the Center o f curva ture o f the deformable
m i r r o r . At the conjugate image Of t h is focus was located a 200 mm f i e l d
lens that would form an image o f the mirror surface. Between the f i e l d
lens and the image of the mirror surface was located the holographic
pla te upon which the reference beam also f e l l . Upon exposure and
processing the p la te was returned to the o r ig in a l posit ion and i l luminated
with the reference beam only to form the reconstruction of the mirror
surface image.
The holographic mater ial used was Agfa 10E56 photographic
p la tes , which required a to ta l exposure o f approximately 10 to 20 e r g s /c m ^
Since each exposure was to be 1/125 sec. , the requi red i rradiance was
to be approximately 0.15 mW/cm . The reference beam/subject beam r a t io
was estimated v is u a l l y to be about 10:1. Plates were developed in D-19
for f i v e minutes, or un t i l the f i lm density became about 0 .6 as estimated
v is u a l ly . Reconstructions of the image were recorded on Polaroid Type
57 f i lm with an irrad iance of ,35 mW on the hologram surface and an
exposure o f 1/125 sec.
51
s p a t l u l c h u t t e r f . » . m i r r o rfilter
L a s e r 5 1 6 5 A
s u r f a c eu n d e rt e s t 50=a F.L.
c a m e ra o b j e c t i v e beam s p l i t t e r c u b e
f . s . m i r r o r
r e e li s a g e c f t e s t s u r f a c e
ZOOmn F . L f i e l d le n s
Fig. 2 .30. Geometry Used to Form Interferograms
. 5 2
Results . Figure 2.31 shows the e f fe c t of the slope change mode.
Note that the f r inge pattern is near ly , but not exac t ly , b i l a t e r a l l y
symmetric as predicted by the computer ana lys is . Figure 2.32 shows the
f i r s t attempt at performing the defocus operation. Because the design of
the shell support s t ructure did not al low easy access to the defocus
controls , the defocus was accomplished by t ightening a screw against the
central actuator pin. The symmetrical sca lloping o f the edge is not ~
seen in the subsequent f r inge p a t t e r n , whi le concentric f r inges are seen
almost to the 0 .7 zone as predicted. There does not appear to be
scal lop ing, but there is no obvious symmetry about the pattern as would
have been expected.
I t was subsequently believed that the screw was not making contact
with the dead center of the central actuator post. The screw was thus
replaced with a micrometer head on a movable support in order to enable
b et te r posit ioning o f the contact point on the actuator post.
At th is point in the experiment, i t became necessary to dismantle
and real 1gn the mlrror assembly because some of the actuator posts had
separated from the mirror she l1. Subsequently, the defocus patterns
were d i f f e r e n t from that o f Figure 2 .32 (See Figure 2 .3 3 ) .
The p o s s ib i l i t y that the screw was not pushing exact ly on the
center of the actuator pin was ruled out by moving the micrometer screw
o f f ax is . No matter where on the end of the post the micrometer screw
made contact, the resu l t ing holograms were v i r t u a l l y the same as Figure
2.33.
53
Fig. 2.31. Slope Change Mode Fig. 2.32. Defocus Mode--FirstAttempt
Fig. 2.33. Defocus Mode a f te r Fig. 2.34. Defocus Mode--SmallMirror Assembly Force Applied to
Central Actuator Pin
Torquing of the actuator post by the micrometer screw was. also
considered. To el im inate the torque, a ba l l bearing was suspended between
the micrometer and the post. This , too, resulted in a s im i la r hologram
as in F ig . 2 .33 .
To prove that the actual mi r ror f igu re was not a f fe c t in g the
r e l a t i v e change in the m ir ror , the f ig u re was grossly a l te re d by moving
. a rb i t ra ry actuators and then making the double exposure hologram. Again
the same basic pattern of Fig. 2.33 was observed, demonstrating that the
r e l a t i v e f igure.change is independent o f the actual f r in g e .
One remaining hologram. Fig . 2 .34 shows the mir ror when a very
s l ig h t displacement is applied to the central actuator . In th is case,
the same general f r inge contours are presented as before, although
there are fewer fr inges in the hologram. Thus i t would seem that the
displacement o f the mirror over the e n t i re surface is a l in e a r function
of the displacement o f the central actuator .
In th is experiment nearly a l l the p o s s ib i l i t i e s fo r the f a i l u r e
of the mirror to produce symmetrical defocus contours have been examined .
and ruled out . The one remaining outstanding p o s s ib i l i t y involves the
precision with which the mirror and i ts support s tructure was assembled.
During the reassembly o f the mirror an e f f o r t was made to keep a l l o f the
actuators evenly spaced about the mirror s h e l l . The posit ions of the
actuator pins were w ith in two m il l im eters of the nominal locat ions, but
th is to lerance may not be good enough. Furthermore, since each actuator
pin has a f l a t surface that is waxed onto the mirror s h e l l , the actual
55
contact point might not be prec ise ly located. Thus i t is recommended
that a more pos it ive alignment method be developed i f successful results
are to be expected.
The 33 Actuator System
As a resu l t o f the study of the 9 actuator system, more advanced
configurat ions were designed. The 33 and the 41 actuator systems were
studied in depth. The 33 actuator system is i l l u s t r a t e d in Figs. 2 .35 ,
2.36, and 2.37. This design was abandoned in favor of the 41 actuator
system because of the elongated truss configurat ion used and the f l e x i
b i l i t y of the reference s tructure .
Fig. 2 .35 . Elevation View of the 33 Actuator System
56
Fig. 2 .36. Plane View of the 33 Actuator System
Fig . 2.37* T r im e t r i c View o f the 33 A c tua to r System
/ ' • ■
57
The 41 Actuator System
A deviat ion was made from the previous truss and system
configurat ions studied in order to r a d ic a l ly increase the s t i f fn e s s and
structura l e f f i c ie n c y of the support s t ruc tu re . The truss and the
reference s tructure become in tegrated, as i l l u s t r a t e d by the truss
configurat ion in Fig. 2 .38 . Here the truss is modified to include the
addit ional hor izontal reference s t ructure element. The angle of the truss
elements is also changed to 45 degrees fo r the central t russ , to
maximize the truss s t i f fn e s s . This al lows fo r the e l im inat ion of pre
tension in the truss fo r large systems where the truss elements are
s t r u c t u r a l l y s tab le .
Radial al ignment o f the trusses is i n t r i n s i c to the system being
a component of a symmetrical opt ica l system. Radial slope control of
the m ir ro r 's surface can be achieved by the in-plane slope control of
the truss. Simultaneously the s t i f fn e s s of the reference s t ructure is
increased by the spoke configurat ion created by th is rad ia l al ignment.
Since the spat ia l frequency o f the e r ro r can be adjusted by a proper
modif icat ion o f the s t i f fn e s s of the s t ruc ture , a minimum spat ia l f r e
quency o f D/5 is assumed. The width of the trusses is reduced to
D/5 to increase t h e i r s t i f fn e s s and al low for th e i r rad ia l alignment
and loca1ized f ig u re c o n t r o l .
53
TrussElements
ActuatorPost
ReferenceStructure
Fig. 2 .38 . Modif ied Center Actuator Configurat ion
A top view o f the support s t ruc tu re is shown in Fig. 2 .39 . In
th is f ig u r e , 16 ta n g e n t ia l ly al igned trusses have been added at the
points where sca llop ing would have maximized in t h e i r absence. The
posit ions of the actuators were made four-way symmetric in order to
make the width of the edge actuator truss approximately D/5. The centra l
actuator is a three-dimensional t russ. F ig . 2.40 is a meridonal section
view of the s t ruc tu re that i l l u s t r a t e s the change in the actuator truss
height with rad ia l pos i t ion and the symmetry o f the truss about the
reference s t ru c tu re .
59
> Reference Structure
D/5------5 places
Fig. 2.39* Section View o f Truss Conf igurat ion along • Heridonal Plane
22.5
= D/5
F ig . 2 .40 . Top View o f Truss C o n f i g u r a t io n
CHAPTER 3
STRUCTURAL ANALYSIS
This chapter describes in d e ta i l the f i n i t e element analysis
performed upon the mathematical model o f the 24-inch 41 actuator proto
type using the s t ructura l analysis program SAP IV (Bathe, Wilson,
Peterson 1973).
The Mathematica1 Model
The mathematical simulation of the prototype consists o f a
244 node f i n i t e element model i l l u s t r a t e d in a top view in F ig . 3 .1 ,
a side view in Fig. 3 .2 , and a 30° obl ique view in Fig. 3 .3 - The
geometric configurat ion o f the s tructure was described in the previous
chapter. 168 bar elements (no bending s t i f fn e s s ) represent the 0.031-
inch diameter steel wire truss elements. 198 beam elements are used to
model both the 0 . 375-inch diameter aluminum actuator posts and the
0 .75“ inch square aluminum components o f the reference p la te . The model
of the 2 4 - inch diameter pyrex m ir ror , having a thickness of 0 . 125-inch
and a radius of curvature of 72 inches, is composed of 112 plate-bending
elements. Since the s tructure is four-way symmetric, the boundary
condit ions are the three suppressed t ra n s la t io n a l degrees of freedom of
the reference p la t e 's four outside nodes along the X and Y axes, as
i l l u s t r a t e d in F ig , 3.1 and 3 .2 . A section view of the model along the
60
Fig. 3 .1 . Top View o f the F i n i t e Element Model
62
Fig. 3 .2 . Side View of the F in i te Element Model
Fig. 3 .3 . 30° Oblique View of the F in i te Element Model
63
Y axis is shown in Fig. 3 .4 . The nodes in the proximity of the in t e r
section between the actuator post and the reference place in a c t u a l i t y
coincide.
F ig . 3*4. Section View of the F in i t e Element Model Along the X-axis
Slope control in the d i rec t ion p a ra l le l to the plane of the truss
is i l l u s t r a t e d in Fig. 3-5 . For mechanical s im p l ic i ty of the prototype,
the actuator is bent to produce the angular d e f lec t ion . The angular
def lec t ion of the mirror surface depends upon how much the actuator post
is bent and how much the lower truss elements are stretched since move
ment of the top of the truss is restrained due to the membrane s t i f fn ess
of the shell coupled with the truss s t i f fn e s s of the rest of the struc
ture.
Fig. 3 .5 . In-plane Slope Control
Slope control in the d i rec t ion perpendicular to the plane of the
truss is done in the same manner ( i . e . , force and reaction is between
the coincident nodes). The angular displacement for th is case w i l l be
much la rger , fo r the same load, than fo r the in-plane slope control ,
since the truss has no s t i f fn e s s in th is d i rec t ion and the moment
created by the forces must be reacted e n t i r e ly by the bending s t i f fness
of the sh e l1.
Normal posit ion control of the prototype is most accurately
represented by the modified truss model i l l u s t r a t e d in Fig. 3 .6 , using
the "slave node" option in SAP IV. This option forces a specif ied d is
placement of a beam element to be equal to that of the so-cal led "Master
Node".
65
Node "A" is the master node of node "B" in a l l degrees-of- freedom except Z-1rans1 at ion
Sl id ing beam element
Mode "C" is the master node of node "D" in a l l degrees-of- freedom except Z-1rans1 at ion
- ?
Fig. 3-6. Ideal Normal Posit ion Control Truss Model
Besides requiring these add it ional nodes, which generally increase the
bandwidth of the s t i f fn e s s matr ix , th is method a l te rs the s t i f fn e s s of
the truss at the mirror (no s t i f fn e s s in Z - t ra n s 1 at ion at node "A").
In other words, th is method requires a modificat ion of the model's s t i f f
ness matrix before the equation solution routine begins in the program.
Since only the actuator upon which the normal posit ion loads are applied
can be modified, only one load case of posit ion control can be treated
at a time. This g reat ly increases the cost of the ana lys is , since the
equation solut ion routine in the program is the most expensive routine
in a s t a t ic analysis . All slope control load cases can be handled
66
in a single computer run since the s t i f fn e s s matrix is not a l te red .
Normal posit ion control using the unmodified truss is i l l u s t r a t e d in
Fig. 3 .7 .
F ig . 3 . 7 . Normal Posit ion Control Truss Model That Was Used
Here the loads are applied to the ends of the truss, d i r e c t l y loading
not only the s h e l l , but also the truss elements and the actuator post.
The results w i l l be the same as the other method but are scaled, since
most of the load is reacted by the actuator post.
67
The d iscre t ized shell of the mirror is i l l u s t r a t e d in Fig.
3.8.
Fig. 3 .8 . F i n i t e Element Model o f the M i r r o r
68
The large nodal points in the f igure are the locations of the actuator
posts. Addit ional nodal points were necessary in order to proper 1y
model the shell ( i . e . , geometr ical ly isotropic model) and to provide
more data points for in te rpo la t ion of the data. Close inspection of
the f igu re w i l l reveal the fact that the model is composed o f , in par t ,
t r ian g u la r elements having edges along the circumference of the model
that are not r a d ia l ly symmetric. This problem could have been avoided
i f the two elements at the edge were replaced by a s ingle t r iang u la r
element, i l l u s t r a t e d in Fig. 3-9.
Fig. 3 .9 . T r ia n g u la r Edge Element
69
Such representation would not have produced any information about the
scalloping e f fe c t that occurs between the edge actuators. Another
representation that would re ta in the radial symmetry and also provide a
data point between the actuators would be the q u a d r i la te ra l plate-bending
element from SAP IV i l l u s t r a t e d in Fig. 3.10.
\
Subnode
I
F ig . 3.10. The Composition of the Plate-bending Quadri la tera l Element
This element is in a c t u a l i t y Clough's compatible plate-bending element.
In making the element compatible, i t is subdivided into four smaller
70
t r i a n g u la r elements with a common "sub-node1! at the centroid of the
element. Since the element is very skewed, almost to the point of
becoming a t r i a n g le , some of the t r ia n g u la r sub-elements become very
narrow leading to an overly s t i f f element. Thus, in the f in a l
representat ion , the qu a d r i la te ra l element is replaced by two w e l l -
proportioned t r ia n g u la r elements. The use of two elements to represent
th is region of the mirror also al lows fo r a reduction in the bandwidth
of the global s t i f fn e s s matrix.
The membrane and bending s t i f fn e s s o f the actual shell are
coupled. The convergence to shell act ion o f the model which is composed
of t r ia n g u la r and f l a t q u a d r i la te ra l p la te elements (where the membrane
and bending s t i f fnesses are uncoupled) has been demonstrated when the
mesh size becomes increasingly small (Zienkiewicz 1971, p. 238 ) . The
bending solut ion w i l l converge non-monotonically while the membrane
solut ion is monotonic in i ts convergence, y ie ld ing accurate resul ts with
a r e l a t i v e l y coarse mesh.
Results
The de f lec t ion of the truss fo r in-plane slope con tro l , shown
(exaggerated) in Fig. 3 .11 , is a good i l l u s t r a t i o n of the e f f ic ie n c y
of t e n s i le s t ru c tu res . Even though the actuator post is 0 .375“ Inch in
diameter and the wire only 0 . 031“ inch in diameter, most o f the angular
def lec t ion o f the shell is produced not by the movement of the lower
end of the post, as would be i n t u i t i v e l y expected, but by the bending
d ef lec t ion of the post.
71
Fig. 3.11. Truss Def lect ions for In-plane Slope Control
Figures 3.12a, 3.12b, and 3.12c i l l u s t r a t e the p r in c ip le of the act ive
mirror where the s t i f fnesses of the s tructure were chosen to make the
def lect ions loca l ized . These f igures correspond to the normal posit ion
control of actuators A, B, and C of Figure 3.4 respect ive ly . Here the
normalized def lect ions of the shell (dashed l ine) and the reference
pla te (dotted l ine ) along the Y axis are superimposed. The mir ror 's
def lec t ion in a l l three cases is loca l ized in a region of influence
roughly o n e - f i f t h the m ir ror 's diameter in s ize . The d e f lec t ion of
the reference p late in turn is more broad having less than 10% the
magnitude of the s h e l l 's de f lec t ion for a l l load cases.
1 . 072
Load Case ,f2
Centra
1.0
Load Case #5
4 .8 " r rad ia l actuate
1 .0
Load Case #8
9 . 6"r radia l ac tuator
F i q . 3 .12 . Normalized M ir ro r and Reference Plate Deflect ions fo r Normal Posit ion Control
Figures 3.13 through 3.23 are lo ca l ized cubic sp l ine f i t contour
plots from the normal and angular d e f l e c t io n data o f the model of the
s h e l l . Normal pos it ion con tro l , and tangent ia l and rad ia l slope c o n t r o l ,
fo r each unique actuator pos it ion were analyzed using u n i t loads. A
small amount o f sca l lop ing can be seen in F i g . 3.14 where the contour
curve near the edge i s n ' t c i r c u l a r . The amount of sca l lop ing has been
Z Deflect ions
Contour Interval 5.18 x 10-6
Datum = 0 .0
g. 3.13- Cubic Spline Contour Plot of Central Actuator 's Slope Contro l , Load Case #1
74
Contour Interval 0.160 x 10-&
Datum = 0.0
Fig. 3.14. Cubic Spline Contour Plot of Central Actuator 's Normal Posit ion Control, Load Case #2
75
Z Def lect ions
Contour Interval = 0.167 x 10-4
Datum = 0.0
Z Deflections
Contour Interval 0.566 x 10-5
Datum = 0.0
Fig. 3*16. Cubic Spline Contour Plot of 4 .8" r Actuator 's Radial Slope Control, Load Case #4
Z D e f l e c t i o n s
Contour Interval = 0.566 x 10~5
Datum = 0.0
Fig. 3.17. Cubic Spline Contour Plot of 4 .8 " r Actuator's Normal Posit ion Control , Load Case #5
Z D e f l e c t i o n s
Contour Inte rval 0.133 x 10-4
Datum = 0 .0
F ig . 3 .18 . Cubic S p l in e Contour P lo t o f 9 . 6 " r A c t u a t o r ' sTan gen t ia l Slope C o n t r o l , Load Case #6
Z D e f le c t io n s
Contour Interval 0.953 x 10"5
Datum = 0 . 0
Fig. 3.19. Cubic Sp l ine Contour P lo t o f 3 . 6 " r A c t u a t o r ' sRadial Slope C o n t ro l , Load Case #7
80
Z Deflect ions
Contour In terva l = 0.228 x 10"°
Datum = 0.0
F ig . 3 .20 . Cubic S p l in e Contour P lo t o f 9 . 6 “ r A c t u a t o r ' sNormal P o s i t i o n C o n t r o l , Load Case #8
81
Z Deflect ions
Contour Interval = 0.333 x 10-5
Datum = 0.0
Fin . 3-21. Cubic Sp l ine Contour P lo t o f Edge A c t u a t o r ' sTangent ia l Slope C o n t ro l , Load Case #9
Z D e f le c t io n s
Contour Interval = 0.173 x 10-4
Datum = 0.0
Fig. 3-22. Cubic Spline Contour Plot of Edge Actuator's Radial Slope Control, Load Case #10
Z D e f l e c t i o n s
Contour In terva l = 0.285 x 10-&
Datum = 0.0
Fig . 3.23. Cubic Sp l ine Contour P lo t o f Edge A c t u a t o r ' sNormal P o s i t io n C o n t ro l , Load Case #11
34
great ly reduced from that of the 9 -actuator system described in Chapter
2, thus i l l u s t r a t i n g the ef fectiveness of the edge actuators in reducing
th is e f fe c t .
Figures 3.24 through 3.34 are contour plots of the same load
cases using the program f r inge (Loomis 1976). This program, o r d in a r i ly
used to determine opt ica l wavefront aberrations from interferometer d a ta ,
makes a global Zernike polynomial f i t using only the normal de f lec t ion
d a ta .
The Zernike polynomials are a complete set o f polynomials in the
two var iab les , r , 9, which are orthogonal over the i n t e r i o r of the unit
c i r c l e . Their simple ro ta t iona l symmetry properties lead to a polynomial
product of the form
R(r) G (0 ) ,
where G(6) is a continuous function that repeats i t s e l f every 2w radians
and s a t i s f i e s the requirement that ro ta t ing the coordinate system by
an angle cj> does not change the form of the polynomial, that is:
G(9+<f)) = G (9) G (<j>) .
The tr igonometr ic functions
G(9) = e+im9,
where m is any pos i t ive integer or z e rg , have the required propert ies.
The radial function must be a polynomial in r of degree n and contain
no power of r less than m. R(r) must also be even i f m is even and
odd i f m is odd.
Z D e f l e c t i o n s
g. 3 .24. Zernike Polynomial Contro l , Load Case
Contour Interval = 5.18 x 10-&
Datum = 0
Contour Plot of Central Slope #1
NOTE: Figures 3.24 through 3-34 contain p lo t te rerrors in FRINGE.
86
Z Deflections
Contour Interval = 0.160 x 10-6
Datum = 0
Fig. 3-25. Zern ike Polynomial Contour P lo t o f Centra lP o s i t i o n C o n t ro l , Load Case #2
87
Z Deflect ions
Contour Interval = 0.167 x 10-4
Datum = 0
Fig. 3-26. Zern ike Polynomial P lo t o f Tangent ia l SlopeContro l o f 4 . 8 " r A c t u a t o r , Load Case #3
88
Z Deflections
Contour Interval = 0.566 x 10'5
Datum = 0
Fig . 3.27. Zern ike Polynomial P lo t o f 4 . 8 " r A c tu a to rRadial Slope C o n t r o l , Load Case #4
89
Z Deflections
Contour Interval = 0.566 x 10-5
Datum = 0
Fig. 3.28. Zern ike Polynomial Contour P lo t o f 4 . 8 " rA c t u a t o r ' s Normal P o s i t i o n C o n t r o l , Load Case #5
90
Z Def lect ions
Contour Interval = 0.133 x 1 0 '4
Datum = 0
F i g . 3.29. Zern ike Polynomial P lo t o f 9 . 6 " r A c t u a t o r ' sTangent ia l Slope C o n t r o l , Load Case #6
91
— L
Z Deflections
Contour Interval = 0.958 x 10-5
Datum = 0
Fig. 3-30. Zern ike Polynomial Contour P lo t o f 9 . 6 " r A c t u a t o r ' sRadial Slope C o n t ro l , Load Case #7
Z D e f l e c t i o n s
Contour 0.228 x
Datum =
Fig. 2 .31. Zernike Polynomial Contour Plot of 9 . 6"r Normal Posit ion Control, Load Case #8
Interval = 10"6
0
Actuator 1s
93
Z Def lect ions
Contour Interval =0.333 x 10“5
Datum = 0
Fig . 3.32. Zern ike Polynomial Contour P lo t o f Edge A c t u a t o r ' sTangent ia l Slope C o n t r o l , Load Case #9
94
• p
Z Deflections
Contour Interval = 0 . 1 7 3 x 10-4
Datum = 0
F ig . 3 .33. Zern ike Polynomial Contour P lo t o f Edge A c t u a t o r ' sRadial Slope C o n t ro l , Load Case #10
95
Z Deflect ions
Contour Interval
Datum = 0
Fig. 3.34. Zern ike Polynomial Contour P lo t o f Edge A c t u a t o r ' sNormal P o s i t i o n C o n t r o l , Load Case #11
96
The radia l polynomials can be derived as a special case of Jacobi
or hypergeometric polynomials and tabulated as R™(r). Their or thogonali ty
and normalization propert ies are given by
1
where <5nni is the Kronecker d e l ta , and R^( 1) = 1.
To s im pl i fy the computation of the Zernike polynomials, we fac tor
the radia l polynomial into
C n / r ) = * > ) r " ,
where Q^(r) is a polynomial of order 2 (n-m). This polynomial can be
generally w r i t te n as
s (2n-m -s ) ! 2 (n-m-s)O r > = l o r
The f in a l Zernike polynomial series may be w r i t ten
o o nAZ = AZ + Z [ A nQ°(r ) + ^ Q^(r) rm(Bnm cos m6
n=l m=l
+ C sin m6)I nm -I
Where AZ is the mean de f lec t ion of the mirror surface (or the mean
wavefront opt ica l path d i f fe rence) and A , B , and C are individualn nm nm
polynomial c o e f f ic ie n ts .
Table 1 l i s t s the modified c i r c u la r polynomials in a Pascal 's
t r i a n g le manner and Table 2 l i s t s the Zernike polynomials used by
FRINGE. I t is seen from Table 2 that the highest order angular terms
Table 1. M od if ied C i r c l e Polynomials Q^(r)
n 0 1 2 3 4 5
0 1
1
CMCM 1
2 Sr1* - 6 r2 + 1 3r2 - 2 1
3 20r6 - 30r4 + I 2 r 2 - 1 10r4 - 12r2 + 3 4 r2 - 3 1
4 70r8 - 140r6 + 90r4 - 20r2 + 1 35r6 - 60r4 + 30r2 - 4 I5 r 4 - 20r2 + 6 5 r2 - 4 1
5 25210 - 630r8 + 560r6 - 21 Or4 126r8 - 280r6 + 210r4 56r6 - 105r4 21r4 - 3 0 r 2 + 10 6r2 - 5 1+ 3 0 r 2 - 1 - 60r2 + 5 + 60r2 - 10
vo
Table 2. Zernike Polynomials Used by FRINGE
98
No Polynomial
01
1r cosG
2 r sinG3 2 r 2 - 14 r 2 cos205 r 2 sin206 (3 r 2 - 2) r cos07 (3 r2 - 2) r sinG8 6 r 4 - 6 r 2 + 19 r 3 cos3B
10 r 3 sin3011 (4 r 2 - 3) r 2 cos2012 (4 r 2 - 3) r 2 sin2013 (10 r 4 - 12 r 2 + 3) r cosG14 (10 r 4 - 12 r 2 + 3) r sinG15 20 r 6 - 30 r4 + 12 r 2 - 116 r 4 c o s 4 g
17 r 4 sin4G18 (5 r 2 - 4) r 3 cos3Q19 (5 r 2 - 4) r 3 sin3020 (15 r 4 - 20 r 2 + 6) r 2 cos2021 (15 r 4 - 20 r 2 + 6) r2 sin2022 (35 r 6 - 60 r 4 + 30 r 2 - 4) r cosG23 (35 r 6 - 60 r 4 + 30 r 2 - 4) r sinG24 70 r 8 - 140 r 6 + 90 r 4 - 20 r 2 + 125 r 5 cos5G26 r 5 sin5G27 (6 r 2 - 5) r 4 cos4028 (6 r 2 - 5) r 4 sin4029 (21 r 4 - 30 r 2 + 10) r 3 cos3630 (21 r 4 - 30 r 2 + 10) r 3 sin30
. 31 (56 r 6 - 105 r 4 + 60 r 2 - 10) r 2 cos2032 (56 r 6 - 105 r 4 + 60 r 2 - 10) r 2 sin2G33 (126 r 8 - 280 r 6 + 210 r 4 - 60 r 2 + 5 )34 (126 r 8 - 280 r 6 + 210 r 4 - 60 r 2 + 5)35 252 r 10 - 630 r 8 + 560 r 6 - 210 r 4 + 3036 924 r 12 - 2772 r 10 + 3150 r 8 - 1680 r 6
cos0 sinG
- 1+ 420 r 4 - 42 r 2 + 1
99
(terms 25 and 26) have fo r an argument 56. Since there are sixteen edge
actuators, the angular terms would need to have arguments up to and
including 89 in order to properly represent ju s t the r a d i a l l y symmetric
scalloping e f fe c t near the edge of the mir ror .
Table 3 l i s t s the Zernike polynomial c o e f f ic ie n ts fo r a l l
eleven load cases. A descr ip t ion of each load case is given in the
t i t l e s of the contour p lo ts . The r a d i a l l y symmetric load case No. 2
has f i v e dominate c o e f f ic ie n ts o f pure radial polynomials and two
r e l a t i v e l y small c o e f f ic ie n ts of polynomials with angular dependence.
Since both angular polynomials have arguments of 48, a small amount of
scalloping exists in the region o f the e ight actuators that correspond
to actuator "B" in Fig. 3 .4 . A b e t te r representation o f the scalloping
fo r th is load case would not require the use of a complete set Of higher
order polynomials since the symmetry and angular dependence properties
of the def lect ions are already known.
Table 4 i l l u s t r a t e s the convergence o f the polynomial approxi
mations by l i s t i n g the root-mean-square ( i . e . , RMS) o f the error between
the polynomial and the data points fo r various orders of complete sets
of the polynomials used. The def lect ions of load case No. 2 are
represented very accurately by a 36 term set while that o f the r a d i a l ly
asymmetric def lect ions of load case No. 1 has i ts RMS error reduced
only by a fac to r of three fo r the same number of terms. The cubic
spl ine contours fo r these load cases (Figs. 3.13 and 3.14) show that
both are loca l ized w ith in regions of comparable s iz e , whi le those of
Table 3. Zern ike Polynomial C o e f f i c i e n t s in Wavelengths f o r Load Case 1 through 11
TermNo, 1 2 3 k 5 6 7 8 9 10 1 1
i + .1 1 4 67 0 .0 + .36439 +.0 0028 - .0 0 0 0 2 +.4 0415 .00000 + .00002 . + .1 6387 - .0 2 8 3 8 - .0 0 1 4 02 +.00031 0 .0 + .00336 + .07287 + .00524 - .0 7 2 8 7 - .3 3 8 8 4 + .0 0 8 89 - .2 0 2 6 4 - .3 0 5 0 7 +-,00608
3 0 .0 - .0 0 1 3 0 + .00067 +.17352 - .0 0 4 0 5 - .0 0 6 1 7 + .15346 +.00052 - .0 0031 +.24201 + .00222
k 0 .0 0 .0 - .0 0 1 6 8 - .0 0 8 2 7 - .00451 + .01289 + .07547 - . 0 0 0 0 0 +.03732 +.24871 - .0 0 7 2 4
5 0 .0 0 .0 + .6 9620 +.00055 +.00001 +1.10146 + .00137 .00000 +.4 5533 ■ - .0 7 5 2 3 - .0 0 3 2 0
6 - .3 6 2 2 0 0 .0 - .4 6 3 1 2 - .6 0 0 1 4 + .00006 +.01710 +.00114 - .0 0 0 0 5 - .0 1 6 13 - .0 7 6 8 6 - .0 0 1 2 0
7 - .0 0 0 7 3 0 .0 +.0 0336 - . 10.235 - .0 1 0 8 3 +.03279 + .25903 - .00371 - .0 0 5 4 8 + .39060 +.00612
8 0 .0 + .01486 + .00437 - .1 2 6 0 4 +.03924 +.1 1239 - .0 0 4 1 5 +.24634 - .0 0 0 6 2 +.24634 +.00291
9 +.00582 d.o - .2 2 1 3 2 7 .00069 - .0 0 0 0 2 - .5 6 0 2 6 - .0 0 1 1 4 - .00001 - .1 8 4 4 5 - .1 0 5 24 +.00500
10 0 .0 0 .0 + .00336 - .0 9 1 1 9 +.00095 - .0 2 5 5 0 - .1 3 5 6 7 - .0 1 1 0 2 - .1 2 1 1 7 +.14741 - .0 0 7 5 5
11 0 .0 0 .0 + .00235 +.23335 + .00879 - .0 8 8 0 0 - .3 4 4 7 7 +*00696 - .0 0 2 8 9 - .4 0 2 03 - .0 0 5 3 712 0 .0 0 .0 - .7 2 3 7 4 +.00152 + ,00010 + .14826 + .00160 - .0 0 0 0 7 ' - .0 4 0 1 2 - . 1 4386 - .0 0 2 1 5
13 +.34587 0 .0 + .20218 +.0 0096 +.00002 +.00813 - .0 0 2 51 +.00001 +.0 0476 - .1 2 5 7 3 - .0 0 0 8 0
14 - .0 0 0 1 0 0 .0 + .00134 - .2 0 8 4 2 “ +.01030 - .0 2 2 4 2 +.04560 . - .0 0 7 6 0 + .00165 • +.63931 +.00403
15 0 .0 - .01601 - .00301 + .09505 +.00296 - .0 5 8 8 6 - .07251 - .0 0 0 9 3 +.00093 +.26290 + .00048
16 0 .0 . - . 0 0 0 0 4 - .0 0 2 3 5 +.08678 +.00197 +.06278 + .05085 + .01115 +.07692 - .0 8 5 14 + ,00699
17 0 .0 0 .0 - .5 0 5 7 3 - .00041 - .00001 - .1 5 5 2 7 + .00182 +.00002 - .0 7 6 9 2 - .1 1 3 12 +.0 0694
18 - . 0 4 1 5 8 0 .0 + .92860 - .0 0 1 6 5 - .0 0 0 0 7 - .18021 + .00046 + .00006 +.03133 • + .27788 +.00238
19 0 .0 0 .0 + .00739 + .17852 + .00828 - .0 8 8 8 5 - .2 8 8 9 0 +.00791 +.02461 - .44421 - .0 0 3 7 5
20 0 .0 0 .0 - .0 0 0 6 7 + .00317 - .0 1 2 4 4 +.0 1710 + .07183 ' + .00429 - .0 0 1 5 5 - .4 7 6 53 - .0 0 1 6 2
21 0 .0 0 .0 + .53600 ■ + .00028 - ,0 0 0 0 2 - .1 5 4 9 9 - .0 0 5 2 4 +.00002 +,00393 - .1 8 8 4 0 - ,0 0 0 6 9
22 - .4 4 3 4 9 0 .0 +.00537 - .0 0 0 5 5 - .0 0 0 0 5 - .1 6 9 5 6 - .0 0 2 0 5 +.00002 - .0 0 9 6 2 - .0 0 6 7 0 . +.OOO32
23 +.00042 0 .0 - .0 0 4 3 7 +.40760 . - .DOB? - .0 3 0 5 5 - .1 3 3 8 5 +,0 0266 - .0 0 1 1 4 +.06937 - .0 0 1 5 6
24 0 .0 - .0 0 2 2 7 - .0 0 4 7 0 . - .1 7 8 9 4 - .0 0 2 1 5 - .0 4 2 3 0 - .0 4 1 2 7 +.00233 +.OOO3 I - .0 1 7 34 - ,0 0 1 0 3
25 0 .0 0 .0 + .04030 + .00041, +.00001 +.09165 - .0 0 4 7 9 - ,0 0 0 0 5 +.02523 - .2 6 9 99 - .0 0 9 4 726 0 .0 0 .0 - .0 0 4 0 3 + .00758 +.00001 +.1 7657 +.2 0545 + .00555 +.03577 +.22821 + .00649
27 0 .0 +.00003 - .0 0 3 6 9 - .0 9 2 9 8 - .0 0 2 6 5 +.1 0230 +.2 0887 - .0 0 8 5 6 - .0 1 9 9 5 +.50451 + .00225
Table 3* Ze rn ike Polynomial C o e f f i c i e n t s in Wavelengths— (Cont inued)
TermNo. 1 2 3 4 5 6 7 8 . 9 1 0 11
28 0 .0 0 .0 + .59947 - .0 0 0 5 5 - .0 0 0 0 2 +.02831 +.00388 + .00005 + .01737 +.47495 + .00212
29 +.00229 0 .0 - .8 7 3 8 6 +.00124 + .00006 +.44479 +.00684 - .00001 - .0 0 2 5 8 +.16712 + .00006
30 0 .0 0 .0 - .0 3 3 5 8 - .1 4 1 0 6 - .0 1 0 0 5 - .0 1 5 7 0 + . I 3932 +.00112 - .0 0 5 1 7 - . 28063 -.0 0001
31 0 .0 0 .0 + .00067 - .2 9 2 8 6 +.00917 - .0 3 5 5 9 + .05906 - .0 0 4 3 9 +.00289 +.14111 +.00224
32 0 .0 0 .0 - .0 5 6 0 9 - .0 0 1 2 4 - .0 0 0 0 5 - .2 8 8 9 6 - .00251 - .00001 - .0 0 7 5 5 +.07962 + .00095
33 +.42201 0 .0 +.01981 0 .0 + .00003 + .02775 +.0 0068 - .0 0 0 0 5 - .0 0 4 5 5 + .12770 +.00059
34 0 .0 0 .0 +.00134 - .24561 - .0 0 1 1 0 +.20067 +.1 6007 +.00354 - .0 0 2 2 7 - .55851 - .0 0 2 9 4
35 0 .0 - .0 1 3 4 4 + .00202 - .1 5 7 7 2 + .00182 +.08100 +.0 7707 +.0 0074 - .00041 - .2 0 1 0 2 - .0 0 0 8 2
36 0 .0 + .01559 +.0 0202 +.19891 - .0 0 1 1 7 +.06774 +.01482 - .0 0 0 2 7 - .00041 - .1 2 2 5 8 - .0 0 0 2 7 -
Table 4. Root-Mean-Square Error of the Best F i t Polynomial to the Nodal Displacements of the Mirror in Terms of Wavelength fo r Load Case 1 through 11
No of T e rms 1 2 3 4 5 6 7 8 9 10 11
0 .2412 .0099 -.7086 .2369 .0083 . 7595 . 3945 .0111 .2771 .7607 .01452 .2339 . 0099 .6952 .2369 »0080 . 7343 . 3808 .0105 .2388 .7607 .0136
3
<T
\
CM .0088 .6952 . 2259 . 0078 .7343 . 3580 .0105 .2388 .7213 .01358 .2100 .0064 .611 2 . 2025 .0069 .5718 , 3352 ,0097 .1055 . 6661 .0120
15 .1767 .0043 .5676 .1763 .0057 .5409 .2964 .0088 . 0468 .5794 .0105
24 . 1227 . 0028 .5105 . 1295 .0044 . 5269 .2668 .0079 .0238 . 4966 . OO89
36 .0884 ,.0000 .4332 .0937 ,0032 .5017 .2371 .0071 .0124 . 3902 .0072
102
103 '
the global polynomial approximation show the de f lec t ion of load case
No. 1 (Fig. 3.25) to be more broad than that o f load case No. 2 (Fig.
3 .26 ) . The i n a b i l i t y o f the 36 term polynomial approximation to
represent th is asymmetric d e f lec t ion in a local ized manner is due to
the r e l a t i v e l y low order of the radial components of the dominate
polynomials (maximum is e ighth-order) with respect to the order of the
dominate pure radial polynomials o f the 2nd load case (maximum is
t w e l f t h - o r d e r ) .
The representation of the unsymmetric def lect ions o f lead cases
No. 3 through No. 11 with the Zernike polynomials show an even greater
diversion from the cubic spl ine f i t results than does the asymmetric
d e f lec t ion of load case No. 1. Al l 36 polynomial c o e f f ic ie n ts are used
fo r these approximations. Although no set o f terms c l e a r ly dominate,
the ones that have the la rgest magnitude usual ly have angular terms with
9 for an argument. Fig. 3.28 i l l u s t r a t e s th is problem with a looped
FRINGE ju s t below the center of the p lo t . The reason fo r th is as
i l l u s t r a t e d in Table 3, is that the r a d i a l l y symmetric propert ies of
the polynomials are not in t r i n s i c to the representation of these
unsymmetric d e f lec t ion s . Complete sets o f very high order polynomials
would be required. A s h i f t of coordinate axes to the actuat ing point
w i l l not a l l e v i a t e the problem since the Zernike polynomials would no
longer be orthogonal w i th in the uni t c i r c l e .
The f i n i t e element analysis also provided slope information
with the normal de f lec t ion data. The complete set of Zernike polynomials
104
could be d i f f e r e n t i a t e d to obtain another complete set o f orthogonal
polynomials so that the co e f f ic ien ts of an even higher order set could
be more accurate ly solved f o r . The cubic spl ine f i t is the most accurate
representation since both normal d e f lec t ion and slope information are
used in a local in te rpo la t ion w i th in the f i n i t e elements. There fore , due
to the "smoothing" ch a ra c te r is t ics of a global polynomial f i t coupled
with the i n t r i n s i c propert ies of the Zernike polynomial approximations
described above are not accurate, in representing the loca l ized de f lec
tions of the mirror .
CHAPTER 4
MECHANICAL DESIGN OF THE PROTOTYPE
This chapter describes the mechanical design o f a 24-inch
diameter 41 actuator ac t ive mirror prototype to be used in conducting
studies on the f igu re control e f f i c ie n c y o f force actuators u t i l i z i n g
both a loca l ized posit ion control and local ized slope contro ls . I t was
not designed to i l l u s t r a t e other advantages of th is system, such as i ts
l ightweightness and structura l , e f f i c ie n c y , since the cost of making
an accurate s t ructura l representation.would make such a study impract ica l .
I ts geometric configurat ion is the Same as that o f the f i n i t e element
model described in Chapter 3.
A single actuator , ac t ive in posit ion control and in radia l and
tangentia l slope- con tro l , Is located at each unique radia l actuator
posit ion outside the central actuator . Each degree-of-freedom is con
t r o l l e d independently by a single servomotor. The rest of the actuator
posts are inact ive with a manual control a t each degree-of-freedom for
"tuning11- the f igure o f the mirror a f t e r the mirror has been bonded to
the support s t ructure . The reference p la te is symmetrically supported
at four points.
A prac t ica l ac t ive mirror system may u t i l i z e hybrid ac tuators ,
i . e . , actuators having both a coarse and f in e control mode. The three
105
106
act ive actuator posts o f th is prototype use only a coarse control mode,
since both modes are not necessary to perform the studies described
above.
Th.in She! 1 Mi r ro r .
The fa b r ic a t io n o f the 24" Inch diameter mirror was a de l ica te
process since i ts thickness is only 0 . 1 2 5 - in c h .1 The convex side was
generated f i r s t from a 1 .5 - in c h - th ic k f ine-annealed pyrex blank. This
surface was then ground into contact with a 1 .5~ inch-th ick tool with a
radius of curvature of 7 2 .1 2 5 - inch, which would la te r be the support
fo r the mirror when i ts f ro n t surface was processed. Successively f in e r
g r i ts of grinding compound were used to remove any local s tra ins
introduced by the surface generator and coarse gr inding. With the convex
side of the mirror blank bonded with beeswax to the t o o l , the concave
surface was then generated, ground, and polished to a 7 2 - Inch radius o f
curvature. The amount of warping -due to released stress In the mirror
was expected to be smal l , since the materia l was r e l a t i v e l y stress f re e .
A considerable amount of "pr in t - th rough" was found in the regions of the
mirror that were posit ioned over the grooves in the tool a f t e r i t was
removed, even though the grooves were f i l l e d with beeswax. A small dr ive
button for a polishing machine was then attached d i r e c t l y to the mirror
so that i t could be polished over a f u l l s ize lap to smooth out these
i r r e g u l a r i t i e s .
1. Al l of the opt ica l components were fabr icated by J . Apples of the Optical Sciences Center, Univers ity of Arizona.
107
. Two sca t te rp la te interferograms of the mirror are depicted in
Fig. 4.1 where the mirror is supported a t two points along i ts edge,
i . e . , in a stressed s ta te . The surface q u a l i ty was s t i l l good enough
to y ie ld an ?nterferogram over the e n t i r e surface of the mir ror .
Figure 4 .2 shows a Foucault Test being displayed on a
t e le v is ion screen. The tes t was made a f t e r the mirror had been bonded
to the assembled support s t ructure . Since the kn i fe edge of the Foucault
t e s te r could not be posit ioned close to the 1ens of the te le v is ion camera,
the surface is not f u l l y i 11uminated (the te le v is io n camera lens had a
larger f/number than the mir ror a t i ts rad 1 us of c u rv a tu re ) . Low
shrinkage aluminum-f i11ed epoxy cement was used to bond the glass to the
aluminum actuator posts. The presence of "craters" around the actuators
in th is f igu re indicates that the shrinkage o f the epoxy is s ig n i f i c a n t .
This e f fe c t may be due to the discrepancy between the rad?us o f curvature
. of the convex surface o f the mirror and the f l a t in te r face surface of
the actuator posts. Any residual epoxy outside the actuator posts could
also have contributed to t h 1s e f f e c t .
The shrinkage of the epoxy a 1 so caused broader deformations in
th is very f l e x i b l e mirror to the to va r ia t ion s in the thickness o f
the bonds between the various ac tua tors . This problem cou1d be
eliminated with a th icker mirror which, in t u r n , would grea t ly reduce
the local 1 zed nature o f the d e f le c t io n s . Such a mlr ror would exclude the
p o s s ib i l i t y o f i l l u s t r a t i n g the e f f i c ie n c y of f igu re control by loca l ized
posit ion and slope c o n t r o l . The slope errors that are seen in th is
108
Fig. 4 .1 . Scatterplate Interferograms of Mirror before I t Was Mounted to the Assembled Support Structure
Fig. 4 .2 . Foucault Test of Mirror a f te r I t Was Mounted to the Assembled Support Structure
f igu re correspond to def lect ions having magnitudes o f hundreds of
wavelengths (A = .6328 pm), making normal interferometry impossible.
The e l im inat ion of the e rror in the f igure o f the surface would
not be a p rob lem . i f a l l o f the actuators were ac t iv e . The ac t ive mirror
system with an image e r ro r sensor and a feedback control system wou1d
establ ish the necessary corrections fo r each actuator and physically
move the actuators to these posit ions to e l im inate th is e r r o r . Al l that
can be done with the prototype is to attempt to eradicate the error
with the manual controls of the ?nactive posts. Even 1f a systematic
method o f ad just ing the manual controls of the 1nac t1ve posts were
devised, the convergence to a spherica1 surface would be d i f f i c u l t .
This is because the method would s t i l l be one of t r i a l and e r ro r . There
would also be a problem of seal loping i f the surface f ig u re converged
to a second-order surface where the s h e l l ' s s t ra in energy is not
minimized. This of course assumes that the opt ica l fa b r ic a t io n process
can produce a mirror with an accurate f igu re in a s t r e s s - f re e s ta te .
The app l ica t ion of rep l ica surfaces to the system would e l im inate the
problem of f igu r ing the surface of th is f l e x i b l e s h e l l . An act iveI
m ir r o r 1s image e r ro r sensor would have to be able to detect the sea l lop-
ing in the mir ror and correct for i t with a defocus adjustment through
the bending of the shell rather than a r ig id body motion of the system
in order to remove the stresses causing th is e f f e c t .
The use of epoxy to bond pyrex mir ror to the support structure
yie lded some undesi rable results as is seen in Fig. 4 .3 . In these cases
the bond between the epoxy and the actuator post s tar ted to separate ’
but to stop before the the components completely debonded. This created
1 1 0
Fig. 4 .3 . Cracks in Pyrex Mirror
stress concentrations in the glass, which has a smal1 te n s i le s t reng th ,
and produced a chip from the back side as is seen at actuator "A" and
a to ta l f ra c tu re at actuator "B". The cause fo r th is separation is
probably due to the incomplete cleansing of the surfaces. An opt ica l
cement may be a b e t te r choice fo r a bonding agent . The use of beeswax
or pi tch would not be p r a c t i c a l , owing to the creep deformations
that would Occur in these m ater ia ls . Even with a good bond, shear
stresses w i l l be introduced into the glass with thermal deformations
because o f the d i f fe rence in the thermal co e f f ic ie n ts o f expansion be
tween the aluminum actuator post and the pyrex mir ror .
An aluminum mirror with a nickel coating w i l l replace the
damaged pyrex s h e l l . Having approximately equal compressive and te n s i le
strengths, the 0 . 1 2 5 - inch- th ick mir ror can be bonded with epoxy without
any p ro b a b i l i ty o f f ra c tu r e , although an epoxy cement with an extremely
small amount of shrinkage would have to be used. Figure 4 .4 shows the
aluminum mirror being ground (before the nickel coating was applied) on
the same lap used to polish the pyrex m ir ror , but with emery paper placed
on each facet o f the lap.
Reference Plate
The reference p la te consists of a 0 . 7 5 - Inch-th ick aluminum p la te
with the appropr iate cutouts being made in order to dupl icate the con
f ig u ra t io n described in Chapter 2. I t was machined from a solid p la te
in order to guarantee a simple homogeneous structure . The cutouts were
machined so that each spoke in the reference p late had a 0 . 75~i nch-square
1 1 2
Fig. 4 .4 . Final Grinding of Aluminum Mirror
113
section. For an actual l ightweight system the s t i f fn e s s and weight
of the p la te is much too large. Since i t w i l l not be used in a study of
the i n t r i n s i c propert ies of the 41 actuator system's geometric con
f ig u ra t io n , i t is an adequate representat ion. Three spokes corresponding
to the horizontal element o f the three ac t ive actuator trusses were cut
out so that removable spokes, with the servomechanisms attached to them
could be fastened to the reference p late as i l l u s t r a t e d in Fig. 4 .5 .
Reference Plate
Removab1e Spoke
Fig. 4 .5 . Removable Spoke of Reference Plate
The e f fe c t o f the removable spokes upon the s t i f fness of the plate is
minimal, since the ac t ive actuator posts are not w ith in proximity of
each other. Fig. 4.6 shows one ac t ive actuator , located 4 .8 - inch
from the center , with the removable spoke barely v i s i b le inside the
servomechan i sm.
m
For mechanical s im p l ic i ty , the 0 .031-inch steel wire (music wire)
that represents the truss elements in the support s tructure , is not
attached d i r e c t ly to the reference p la te . Instead, i t is sharply bent
by the pre-tension in the truss when passing through holes in the p la te ,
as can be seen in Fig. 4 .6 . This sharp bend, combined with the close
proximity of the other wires that pass through the same hole, provides
enough resistance to slippage for the loads that w i l l occur.
Fig. 4 .6 . The Removable Spoke of the 4 .8- inch Active Actuator Mounted to the Reference Plate
115
In order to dupl icate the boundary conditions enforced upon the
f i n i t e element model, four countersunk holes were d r i l l e d in exact 90°
i n t e r v a ls , in those corners of the reference p la te corresponding to the
four simply supported nodes of the model. The p la te is then supported
by four cone-tipped screws, also located in exact 90° in te rva ls in the
te s t mount. The completely assembled prototype mounted in the test
mount is shown in Fig. 4 .7 .
inact ive Actuator
The inact ive ac tuator , as i l l u s t r a t e d in Fig. 4 .8 , is composed
of four basic components: the 0 . 3 7 5 “ i n c h diameter aluminum actuator
post, the O.625- inch diameter aluminum mirror end lap, the 0 . 5 0 - inch
diameter reload end cap, and the 0 . 0 3 1 - i n c h diameter s teel wire (music
wire) that represents the four truss elements in the truss system.
The mirror end cap provides a rough adjustment in the height
of the pre-tensioned actuator to insure contact with the m ir ro r 's
convex surface before boning. Fig. 4.9 displays a cross-sectional view
o f th is end cap with the actuator post and steel w ire . I t can be seen
in th is f igu re that a s ingle wire is used fo r each actuator with the
ends fastened to the preload end cap.
116
Fig. 4 .7 . The Assembled 24-inch Prototype Mounted in the Test Mount
M i r r o r
ReferencePlate
Mirror End Cap
l____
Actuator Post
Preload End CapSteel Wi re
Fig. 4 .8 . Inact ive Actuator Configuration
Mirror End Cap
Actuator Post
Fig . 4 .9 . M i r r o r End Cap
1 1 8
The basic problem with th is configurat ion is that i t was designed to be
assembled only once. When the set screws in the end cap have been
t ightened into the actuator post, any small readjustment in height is
very d i f f i c u l t , due to the o r ig ina l indentations made by these set screws.
In order to make these adjustments, the truss had to be detensioned and
the wires allowed to s l ip through the holes in the reference p la te .
The preload end cap stresses the truss by elongating i t . Set
screws in the cap protrude into a keyway in the actuator post to prevent
the Cap from rota t ing with respect to the post . These same set screws
also lock the cap to the post once the truss is property preloaded.
As i l l u s t r a t e d in Fig. 4 .10 , the wire is bent 45° and clamped in order
to prevent sl ippage. The 0.003- inch clearance between the wire and the
holes in the preload end cap, combined with the 45° bend in the wire
produced by the f i r s t clamping end cap, was not s i f f i c i e n t to prevent
wire sl ippage in some of the actuator posts. For these posts, the
aluminum around the end caps' holes near the bend in the wire yielded
under the pre load, thus al lowing the wire to s l ip . A second clamping
end cap was added to prevent sl ippage.
Posit ion control of the m ir ro r 's surface is made by changing the
length of the actuator post with a small va r ia t ion in the preload o f the
t russ. Tangential and radial slope control is achieved by con tro l l ing
the posit ion of the actuator post's midpoint. Fig. 4.11 shows two
horseshoe-shaped plates w i t h .set screws clamped onto the reference p la te
to provide th is contro l .
ActuatorPost
- Preload End Cap
C1 amp ing End Caps
Fig. 4.10. Preload End Cap
Reference Plate
Actuator Post
Fig. 4.11. Slope Control o f I n a c t i v e Ac tua to rs
120
Active Actuator
The ac t ive actuator , as i l l u s t r a t e d in Fig. 4 .12 , is composed
of f iv e basic components: the 0 . 3 7 5 - inch diameter aluminum actuator
post, the 0 . 2 0 - inch diameter aluminum normal posit ion control s l ide with
a 0 . 6 2 5 - inch diameter head, the servomechanism, the 0 .031- inch steel
w ire , and the preload end cap which is the same as those o f the
inact ive actuators.
Mir ror
Actuator Post
ReferencePlate
Stee1 Wire
Servo- mechanism
Fig. 4 .12. A c t i v e A c tu a to r C o n f ig u ra t io n
121
Since the ends o f the wire are bent 45° and clamped at the
m irro r 's end of the truss, two steel wires are used. Fig. 4.13
i l l u s t r a t e s this and the normal posit ion control s l id e in place w ith in
the actuator post. A mirror end cap s im i la r to that of the inact ive
actuator post was not used on the s l id e , since the three ac t ive actuator
posts are kinematic reference points fo r the m irror 's pos it ion . Normal
posit ion control of the m ir ro r 's de f lec t ion is obtained by insert ing
an eccentr ic arm from an output shaft of the servomechanism into the
groove at the bottom of the s l id e . A spring is located between the
actuator post and the s l id e to e l im inate backlash between the s l ide and
the eccentr ic arm. The preload in the spring is several times larger
y— Normal Posit ion Control Slide
C1 amping End Cap
ActuatorPost
Ant i -backlashS p r i n g
4.13. Normal Pos i 1 1 Control Sl ide and theA c tu a to r Post
1 2 2
than the force necessary to produce the magnitude of de f lec t ions desired
fo r the studies described e a r l i e r , i . e . , 20 wavelengths (X = 0.6328 pm).
A stack of p ie z o e le c t r ic c rys ta ls were not used for actuat ion , since the
length of the stack would become nearly as long as the actuators them
selves in producing these deformations.
The servomechanism is a movable gear box having there servo
motors, each independently c o n t r o l l in g one o f the degrees-of-freedom of
the actuator through a s ingle output shaft . The two I l l u s t r a t i o n s of
Fig. 4.14 show how the ac tua tor 's three degrees-of-freedom are indepen
dently control led. Slope control in general is obtained by posit ion ing
the midpoint of the actuator post. The motion of the transverse s l id e ,
which make contact with the post at i ts midpoint and passes through a
clearance s lo t in the removable spoke, produces the ou t -o f -p lane slope
control by means of a screw d r ive . The motion o f the e n t i r e
servomechanism along the spoke controls the in-plane slope of the mir ror
and is produced by an eccentr ic arm that engages a s lo t in the spoke.
Teflon pads provide bearing surfaces in the servomechanism for th is
motion. The normal posit ion control output shaft passes through a c l e a r
ance hole in the transverse s l id e . Fig. 4.15 shows the 50:1 worm gear
drives between the servomotor and the output shafts fo r normal posit ion
control on the l e f t and in-plane slope control on the r ig h t . These two
worm gears were machined into sectors because of the reference p late
space l im i ta t io n s . A f u l l gear is used in the transverse s l ides dr ive
since there are no space l im i ta t io n s on th is portion o f the servomechanism
and the l i n e a r i t y of the motion is not dependent upon the angular
posit ion o f the output shaf t .
123
Te f lon Pads
Norma 1 Posi t ion Control
x
\
Out-of-Plane Slope Control
Transverse Slide
• In-Plane S1 ope Control
Reference Plate
F ig . 4.14. Methods o f A c tu a t io n in the Servomechanism
124
F i g . 4 .15. Worm Gear Drives in the Servomechanism
, 125
A f u l l y ac t ive operational system may use actuat ing devices of
a completely d i f f e r e n t configurat ion than the ones used fo r th is model.
In-plane slope control and normal posit ion control may a c tu a l ly be
produced by the v a r ia t io n of the tension o f the lower two truss elements.
The two actuators that control th is tension would be coupled. I f the
in -p lane slope of the mir ror is control led by the ac tua to r 's midpoint,
a bending j o i n t would be used at th is point so to remove the bending
s t i f fn e s s o f the actuator in th is d i rec t io n and thus minimize the
actuat ing fo rc e . The s t a b i l i t y o f the post would be maintained through
the posit ion control o f th is point .
CHAPTER 5
SUMMARY AND CONCLUSIONS
In t h is . th e s is a study of the conceptual f e a s i b i l i t y o f the 41
actuator ac t iv e mirror fo r l ightweight system applicat ions is presented.
This includes an eva luation of the app l icat ion of t e n s i 1 e-membrane
structures to the system design and the. in te ract ion of the mirror into
the support s t ructure to minimize weight and maximize s t ructura l
e f f i c ie n c y . Figure control e f f i c ie n c y and l ightweightedness are shown
to be fu r th e r enhanced by the app l ica t ion of actuators having both
local ized posit ion and slope contro l , e l im inat ing the requirements
fo r a s t i f f reference p la te . The reference p la te , in t u r n , is s t i f fened
in place by i ts "spoke" conf igurat ion which results from i ts in te rac t ion
with the r a d i a l l y and tan g e n t ia l ly al igned local truss systems.
Scalloping of the m ir ro r 's f ig u re is a de f lec t ion c h a ra c te r is t ic
of thin shell ac t ive mir rors . I t occurs when def lect ions of the shell
that are produced by membrane stresses are enforced a t d iscre te actuat ing
points through the bending of the she l1. This e f fe c t may be reduced by
increasing the thickness of the s h e l l , which in turn reduces the
Ideal ized nature o f the m ir ro r 's d e f lec t ion and thus reduces the f ig u re
control e f f ic ie n c y of the actuator .
1 2 6
127
Actuator spacing depends upon the spat ia l frequency of the
f igu re e r ro r . The one-dimensional e r ro r analysis presented in th is
thesis does not take into account the higher order shell s t i f fn e s s
c h a ra c te r is t ic s , such as the radia l dependence of the normal s t i f fness
o f the s h e l l . A parametric study of the s h e l l ' s response using a f i n i t e
element analysis would have to be made in order to der ive empirical
equations fo r the ac tuator 's spacing. For a f ixed number of ac tuators ,
the magnitude o f the scalloping can be minimized by e f f i c i e n t l y posi
t ioning the actuators in a manner s im i la r to that of the 41 actuator
system.
The design of a l ightweight ac t ive mirror consists of a number
o f t radeoffs between the various s t ructura l ch a ra c te r is t ics of the
system in minimizing sca lloping. Disregarding thermal stresses and
grav i ty loads in the m irror , the number o f t radeoffs would be d r a s t i c a l ly
reduced i f a very accurate surface f ig u re could be produced upon the
mirror while being in a s t r e s s - f re e s ta te . The actuators would then,
move to the posit ions that e l im inate the internal stress and thus the
f ig u re e rror in th e ,m ir ro r , including scalloping. I t is outside the
scope of th is thesis to determine whether the use of fused s i l i c a or
Cervi t would reduce the thermal stresses in the thin shell to a level
having a r e l a t i v e l y in s ig n i f ic a n t e f f e c t upon the surface f igu re .
An image e r ro r sensor in a th in shell ac t ive mirror system wo.uld
have to be able to detect sca lloping. Without the a b i l i t y to detect
th is e r r o r , the surface f igu re may converge to any one o f an i n f i n i t e
128
number of second-order surfaces , whi le only one corresponds to the
s t re s s - f re e s ta te of the mirror . The magnitude o f the coe f f ic ie n ts
corresponding to the Zernike polynomials having the same angular
dependence as the scalloping is a good measure of th is e r r o r . Because
of the r a d ia l ly symmetric propert ies of these polynomials, o f f - a x is
deformations are not e f f i c i e n t l y represented. Even so, the polynomials
may be used to cad u la te the lower order (broader) orthogonal aberrations
of the d e f le c t io n . The ac t ive mirror system can then independently
el im inate each one of these aberrations with the proper set of motions
of the ac tua to rs . The e rror which remains w i l l be more local ized in
nature than that o f the o r ig in a l d e f le c t io n . This remaining error can
be corrected independent1y by the local ized posit ion and slope control
of the actuators.
A possible f igu re control alogrithm may i n i t i a l l y e l im inate
the aberrations in t r i n s i c to sca l lop ing, such as defocus and spherical
aberra t ion . Aberrations c h a ra c te r is t ic of a "developable" surface, ( i . e . ,
a surface produced by bending stresses) such as coma and astigmatism,
would then be el iminated along with the local surface i r r e g u la r i t i e s
without a s ig n i f i c a n t amount of scalloping being reintroduced. Even so,
scalloping must be monitored throughout both phases of th is f igure
correction procedure since i t is a non-local ized d e f lec t io n character
i s t i c of an ac tuator 's displacement. I ts e l im inat ion over the en t i re
surface o f th e .m ir ro r , not ju s t the edge, w i l l ensure the minimization
of the s h e l l ' s s t ra in energy.
129
I t is ant ic ipated that the f e a s i b i l i t y w i l l be demonstrated in
the near fu ture using the 2 4 - inch prototype that was designed and
fabr icated fo r th is study. As a resu l t o f these te s ts , fu r the r studies
w i l l be made into the refinement o f the system.
SELECTED BIBLIOGRAPHY
Bathe, K. J . , E. L. Wilson, and F. E. Peterson, 1973, "A Structural Analysis Program for S ta t ic and Dynamic Response of Linear Systems, EERC-73- 11." College of Engineering, Berkeley, CA.
Born, M . , and E. Wolf, Pr incip les of Opt ics , Pergamon Press, F i f th Edit ion , 1975.
Koterwas, D. J . , 1974, A Pre-Tensioned Truss System for Active Control of Mi rrors, Department of C iv i l Engineering and Engineering Mechanics, Thesis, Univers ity of Arizona.
Koterwas, D. J . , R. M. Richard, and R. R. Shannon, 1975, Mi r ror Slope and Posit ion Controlled by Prestressed Tr iangular Truss, Optical Sciences Newsletter, September 1975, Optical Sciences Center, Un ivers i ty of Arizona.
Loomis, John S. , FRINGE User's Manual, Version 2 , November 1976, Optical Sciences Center, Un ivers i ty of Arizona.
Optical Sciences Center, S ta f f o f , October 1974, Final Report onLarge Diameter Active M ir ror with Holograph i c Figure Sensing, SAMSO TR 75-17, The Univers ity o f Arizona, 106 pp.
Robertson, H. J . , 1970, Development of an Active Optics Concept Using A Thin Deformable M i r r o r , NASA CR-1593• Norwalk: The Perk i n- Elmer Corp.
Wei ford, W. T . , Aberrations o f the Symmetrical Optical System,Academic Press, 1974.
Zienkiewicz, 0. C. , The F in i te Element Method in Engineering Science McGraw Hi l l Publishing Co., 1971.
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