Download Telescope mirror supports: plate deflections on point supports

Telescopemirror supports: plate deflectionson point supports
Jerry E. Nelson
Lawrence BerkeleyLaboratory,Space Sciences Laboratory,University of California,Berkeley,California 94720
Jacob Lubliner
Departmentof Civil Engineering, University of California, Berkeley,Calitornia94720
Terry S, Mast
Lawrence BerkeleyLaboratory,Space Sciences Laboratory,University of California,Berkeley,CaliÍornia94720
Abstract
'ùy'edescribethe deflectionsof uniform-thickness
plates
supportedby discretepoints and by continuousrings. The
calculationsare basedon the theory of deflectionsof thin
plates. In somecasesthe effectof shearon the deflectionsis
also included. The optimum locationsof the support points
for a wide variety of simple geometriesare given. The
deflectionsand methodsfor estimatingthe deflectionsfor the
limiting caseof a large number of supportpoints are also
desc¡ibed.Sincethe slopeerrorsinducedin a mirror by its
supportmay be relevantto opticalimagequality,we describe
a method of relating the surfacedeflectionsto the surface
slopes,and hencethe geometricopticsimageblur that results
from a support system. Theseresultsare particularly useful
for the supportof very thin mirrors,wherethe optimum support is needed.
l. Introduction
problemsin the designand
One of the most challenging
constructionof largetelescopes
is that of properlysupporting
the primarymirror so that the largeforcesdue to gravity do
alter its shape.High qualitytelescopes
not objectionably
typically requirethat the mirror deformarions
be limited to some
small fractionof the wavelengthof light that the telescopeis
designedto use. For optical telescopes
in particular, the
deflectiontolerancesare sufficientlysmall that great caremust
be taken in the mirror supportif thesetolerancesare to be
met.
As a telescopepointsto differentzenithanglesthe direction of gravity relativeto the primarymirror changes.As a
result of this, a mirror supportsystemmust be able to carry
this force both in the axial direction(normal ro the mirror
surface)and in a radialdirection(parallelto the mirror surface). Thesetwo components
typicallyhavedifferentsupport
systems,and it is often the casetha! the axial supportis the
more delicateand difficultsupportproblem.For largemirrors
this axial supportis usuallyprovidedby some distributionof
forcesappliedalong the backside of the mirror, or at some
points locatedalong the centerof gravityof the mirror. The
radialsupportsystemis similarin form, typicallybeing a system of forces applied either around the periphery of the
mirror, or at discretepointslocatedalongthe midplaneof the
mirror. In this paperwe will not dealexplicitlywith the problems of radial support, but will only addressthe nature of
axial supports. This generalproblem has been of great
interestin the past,and a detailedsrudyof axial supportsystems in theory and practicewas made by Couder (1931).
More recently,the entire field of the supportand testingof
astronomical
mirrors wasreviewedin a symposium(Crawford
et al., 1968). Several specificmirror supports are also
describedby Pearson(1980).
The largesttelescopes
have useda discreteset of point
supportsfor their axialsupport.The Soviet6m telescopehas
60 backsupports,while the Hale200" (5m) has 36 backsupports, and smaller telescopes
such as the Lick 3m have 18
back strpportpoints. Thesepointsof supportare points at
2l 2 / SPIE Vol. 332 Advanced Technology Optical Telescopes (1982)
which specifredforcesare applied,often with counterweight
systems.Other methodsof applyingtheseforcesincludeair
pads and whifrletreesystems.In other cases,back support
has been achievedby usingring supportswith, for example,
fluid filled tubes to achievea uniform distributionof force
alongthe circumferential
lengthof the tubes.
Althoughaxialsupportsfor telescope
mirrors havebeen
dealt with in some detail by previousauthorsand telescope
designers,the currentdesireto build extremelythin and large
mirrors makesthe subjectof great interest. Generaldesign
guidelinesfor possiblesupportsof thesevery flexiblemirrors
will be most useful.
In addition,the recentinterestin telescopemirrors that
are segmentedor composedof multiple telescopeprimaries,
makesthe uniformityof focallengtha new and criticalissue.
With the supportof only a singlemirror concernover the
accuratecontrol of the radiusof curvatureof the mirror is
minimal. In this case,changesin the focallengthof the mirror due to imperfectionsin the support only require a
refocusingof the telescope,
somethingwhich is often required
as a resultof thermaleffectsin any case. For telescopes
that
have primarieswith multiple mirror components,the focal
lengthsof the mirrors shouldeither not changeat all, or at
least all changetogether. Particularlyfor segmentedpriboth obsermaries,whereone may attemptto accommodate
vationsin the visible(seeingtimited)and infrared (diffraction
limited) regimes,all segmentsmust have the same focal
length to very high accuracy.Of coursefor the supportof
flatsthe propercontrolover the curvatureis essentialas well.
In section2 we will developsomescalinglawsand other
designof axial
semiquantitativerulesto aid in the conceptual
support systems. In keepingwith our conceÍn over focal
length,we will considerany platedeformationsthat alter the
focal lengthas detrimental.A varietyof examplesof optimized supportsystemswill also be given as further aid in the
designof supports.Because
the goalsof this work are rather
general, simple idealizedcasesare discussed,rather than
specificrealisticcasesthat must be dealtwith in the detailed
designof a specificsupportsystem.In this spirit we will typicallyignorethe effectsof centralholes,variablemirror thickness,shelleffects,and the effectsof shearin mirror deformageneralscalinglawsthat applyto the
tion. Section2 describes
the deflection
deformationof thin plates.Section3 discusses
of plates that are infinite in extent. These have no edge
effectsand thus representthe limiting caseof a mirror that is
supportedat a great many points. Section 4 gives for a
varietyof casesthe optimizedsupportconfigurationsfor circular plates.Thesecoverboth point supportsand continuous
ring supports.In the final sectionwe brieflyaddressthe issue
of supporttolerances
and sensitivity
In keepingwith the opticalmotivationfor thesestudies,
we evaluatethe effect on the image quality of imperfections
in the mirror support. There are two common ways of
evaluatingmirror distortions. l) Using the geometricoptics
approximationthe imageblur is evaluatedfrom slopeerrors
the
in the mirror surface.2) The other extremecharacterizes
mirror surfaceby the root mean squaredeviation (rms) of
the
the mirror from its desiredshape,and thus characterizes
imageby an rms wavefronterror.
This approach assumes that the wavefront error can be
characterizedby a singlenumber and that this determinesthe
optical effects. Although this is an excellentapproximation
when the wavefront errors are small compared to the
wavelengthof light, and the geometriclimit is a Soodapproximation when the wavefront errors âre largecomparedto the
wavelengthof light, the intermediateøse requires the full
and complexapplicationof the physicaltheory of light. Some
tools for understandingthis intermediateregionare contained
in our report to this conference"The Effectsof Mirror Segmentationon TelescopeImage Quality (Mast et al.)." Conceivablythe "optimumnsupportsupportwill dependon which
approachone choosesfor analyzingimageblur, but as we
relationcan be
show in the next section,a fairly quantitative
made betweenrms surfaceerror and maximum slope errors,
identicalresults.
should give essentially
so both approaches
rrVetypicallywill optimize on the rms surfacedeviation from
the mean deflection rathe¡ than the slopeerrors. The peakto-valleyerror is also closelyrelatedto the rms surfaceerror.
error is about
It is commonlythe casethat the peak-to-valley
five times the rms, and hencethey are essentiallyequivalent.
However, since the peak-to-valleyonly gives information
from two extrelne points, whereasthe rms is averagedover
the entire surface.the rms is normallythe more useful surfacecharacterization.
2. ScaiingLaws
In the theo¡y of thin plateswhereonly flexuralbending
contributes to the plate deflection and shear effects are
ignored,deflectionsof circularplatestakethe form:
(1)
^-rms: k "q a 4
D
Ìvnere
q - the appliedforce per unit area
a - the plateradius.
O - nnjll2(1-vz), the flexuralrigidity,
E : the materialelasticmodulus.
v - the materialPoissonsratio,
h - the platethickness,and
k is a parameterthat dependson the natureof the
supportconfiguration.
For self-weightdeflectionsthis gives the well known scaling
law that the deflectionsvary as:
ô-4
h¿
e)
form of this relationis
For non-circularplatesa generalized
more useful.
ô,,":t4,*
4,
'
D
h2'
(3)
where A is the plate area and (, the supportefficiency,is a
dimensionlessparameter that depends on the support
configuration. This form is useful for estimatingfor example
the deflectionsof hexagonalplates,a shapedesired in segmentedmirrors.
When one considersthe deflectionwith N support
points,it is evidentthat a pertinentparameteris the areaper
supportpoint, ratherthan the areaitself. Thus one cancouch
the rms deflectionof a platesupportedby N points in the followingform:
ô.^:tNfitftlz
(4)
In this form, once 7¡ is known, the deflection for a mirror
of any size can be found.
The virtue of this form of the deflection law is that the
parameter y¡¡ becomesa fixed constant for an optimized support configuration, at least in the limit of large N. For small
values of N there will be noticeable edge effects that will
make 7¡ a function of the number of support points. As the
number of support points increases/¡y will decrease to an
asymptotic limit y-. Improperly optimized support systems
will of course have a larger y than otherwise. Recognizing
that 7 is a measure of the efficiency of a given support
coofiguration, we call y the support point efficiency.
We will typically ignore the effects of shear. This eases
the calculations and allows us to ignore the effects of plate
thickness, except in D. However, a rough estimate of the
additional deflectionsthat will result from the inclusion of the
effects of shear can be found by use of the formula
r
+
6¡oat: Earn¿¡nell
"(!)21
[
" l
t
(5)
where u is an effective length between support points, and a
is a constant. If one deflnesu from the relation
Nru;¿: A
(6)
then for a variety of casesnumericallyevaluatedone obtains
an approximatevaluefor a of o : 2. Note that this rough
formula only appliesto platesthat have optimizedsupports.
A non-optimizedsupportmay have a much larger bending
deflection but essentiallythe same shear deflection, and the
inclusionof sheareffectsthenresultsin a smallerchangethan
indicated. We emphasizethat the effects of shear scale
spanratio, rather
accordingto the thickness-to-support-point
ratio.
than accordingto the thickness-to-plate-radius
As mentionedearlier,the optic¿leffectsdue to mirror
deflectionsare often estimatedusing surfaceslopesrather
than the ¡ms deviationof the surfacefrom its desiredshape.
The slope,the rms deflection,and the effectivelength must
haveat leasta roughrelationship,
(7)
Sn õr^r/ u
This gives the scalingrelation of
l=14=yn
ùc - 8- Nat 1\ I4.¡uz
:/
t?rñ,'
(8)
One can test these relationsin a variety of casesto flnd a
value for the parameterg. We have found for a surprising
variety of cases that, for systems optimized to give the
minimum rms deviationfrom the desiredsurface,the maximum slopecanbe found (within-20V0)by using
(9)
g9:9v ¡,t
The maximum imagediameteris four times the maximum
relaof thiswell definedslopevs displacement
slope. Because
tion, and becausewe are particularlyinterestedin deflections
that are sufficientlysmall to allow diffraction limited perforthe surfaceby its rms deviation
mance,we will characterize
from the idealone for the rest of the paper. This meansthat
the averagedeflectionon a given supportsystemhas been
removed. We includeall other effectsincludingchangesin
the overallradiusof curvature(focus).
3. InflnitePlates
Real finite plates have a variety of edge effects that tend
to increase the plate deflections and appreciably complicate
the analysis of their deflection. If a plate is simply supported
SPIE Vot. 332 Advanced Technotogy Optical Telescopes (1982) / 213
,i;
neara givensupportpoint will
at many points,the deflections
be dominatedby the locationsof the neighboringsupport
points and will be only weaklyeffectedby the locationand
shape of the free edge of the plate. For these reasons'
infinite platesare worth studying. We also expectthat no
frnite plate can be supportedas efficientlyas an infinite plate,
thus, the efficiencycoeñcient for an infinite plate will set a
lower bound on the emciencycoefficientof any point support
systemfor a frnite plate. Finally,the studyof infinite plates
cån serveas a useful guidefor the optimaltopologyof support point locationsfor frniteplates.
If we restrict ourselvesto point supportgrids that are
are posidenticalfor all supportpoints,,only threetopologies
sible. These are shown in Figure I and are the ones with
three neighboringpoints (hexagonal),with four neighboring
points (triangupoints (square),and one with six neighboring
iar). Becausethese support topologiesare different, we
expectthat their efficiencycoefficientswill differ as well.
The deflectionsof an infinite plate supportedby rectangularrows of supportpostsis a classicproblemin plate
theory. A discussionof this problem and its solution are
given by Timoshenko(1959). This solutioncan be readily
appliedto the squaregrid case.
The deflectionsof the plate on triangularand hexagonal
supportpoint grids can also be readilyfound' Followingan
of a
ideaby Medwadowski(1981),we considerthe deflections
plate supportedby a rectangulargrid of supportpoints and
to flnd the triangular
use the principleof linearsuperposition
and hexagonalcasedeflections.Detailsof this derivation,and
the resultsare given in AppendixA. The resultsfor these
grids can be summarized by determining 7 as defined in
Equation4. We find
^l
: 2'36 x 10{
hexasonal
- l'33 x 10-r
I square
^f
triansutar:1'19 x 10-r
ratio v.
of Poissons
Note that theseresultsareindependent
Not surprisingly,the triangulargrid is the most efficient,
and the hexagonalgrid is decidedlythe worst' It followsfrom
this that the triangulargrid is the basictopologicalpatternthat
shouldbe usedas a startingpoint in the optimizationof support point forcesand locationsfor finite plates.
4. CircularPlates
We now turn to the most common shapefor primary
mirrors, the circle. As mentionedbefore,we will not explicitly discussannular platesor variablethicknessplates,or
include the effects of shear or the curvature of the plate.
These effectsshould be includedin any detailedanalysisof a
specifrcsystem. We will first discuss the deflections that
result from point supports. We then evaluatedeflectionsdue
to ring supportswhich for some configurationsform a limiting caseof point supports.
In analyzingand optimizingsupportsystemsin general,
we strongly emphasizethat the supportsprovide specified
forces,but do not by themselvesdefinethe positionof the
mirror. In the caseof N supportpoints,threeof thesepoints
can be specifiedto definethe rigid body motion of the mirror,
but the other supportpointsshouldin practicebe allowedto
float in position and only provide the specifiedforce. It fol'
lows from this that the deflectionsat the support points are
not required to be zero for the overall optimizedsupport.
for these
This is importantin developinga physical-intuition
systemsand differs markedlyfrom the intuition one has of
conventionalsupportsof plateson fixedpoints.
214 / SPIE Vot. 332 Advanced Technology Optical Telescopes (1982)
To find the deflectionsof a circularplate supportedby a
and
seriesof pointswe will use the principleof superposition
divide thè supportpointsup into groups,eachgroupcontaining k pointsúnifortirtvspacèdarounda circleof a fixed radius
as-shownin Figure2. The derivationfor the deflectionsdue
tõ a singlesucli groupis givenin AppendixB. The deflection
has the form
w(p,o): iw^$)cos(kmg)
(lo)
m-0
where the formulae for w^(p) aregivenin AppendixB' To
find the deflectionsfrom ihe full supportsystem'we simply
udd th. deflectionsfrom eachgroup,weightingthe deflections
by the fractionof the loadbeingcarriedby that group' Thus
w(p,Ð : f e¡w¡(n¡,ß¡,p,9-þ),
(12)
i-I
where
w(p.0) - Plate deflection
n - number of groups or rings
e, : the fractional load carried by i't ring
)rrj :
¿, :
B¡ :
é¡ :
Et': I
platedeflectionswith a singlering of point support:
numberof supportpointson the irn ring
radiusof the l¡n ring
azimuttraloffsetof pointson ifi ring
A given supportsystemis optimizedby varying the forces,
radii, and rotationanglesof eachgroupto minimizethe rms
defleition. For the õasesdescribedbelow the optimization
was made by numericallyevaluatingthe rms deflectionwith a
computer,and using a standardoptimizingpackagefor non'
lineaì functions. In addition,someof theseresultshavebeen
checkedwith a finiteelementanalysisprogram'
independently
(Tayìor,Asfura,and Budiansky,1982)
A singlegroupof supportpointsgivesan rms deflection
that variesïitñ ttré raOiusof the ring and with the numberof
supportpointson the ring. The rms deflection(or ratherthe
tupport èffñciency)as a functionof thesetwo parametersis
shoin in Figure 3. Note that as the radius of the ring
one, and the numberof support,pointsbecomes
approaches
tárge, ttrat the deflectionsapproachthose of a plate simply
suõportedalong its rim' It is interestingthat six points is
to a continuousring' As
atreä¿ya uery ðtoseapproximation
zero,the deflectionsbecome
the raãiusof the ring approaches
thoseof a platesupforié¿Uy a singlepoint at its center' The
deflectionsof the piatesupportedby a ring at the outer edge
define a kind of "àatural"scalefor the deflections. The rms
deflectionnearthe optimumis shownin Figure4 and is a factor of about 25 smallerthan the naturalscale. This dramatic
improvementis a reflectionof the fact that deflectionsvary as
a4'. This also leads to great sensitivityto support errors'
Exceptvery near the minimum itself' the rms variesessen'
tially linearlywith the radiusof the supportring.
When one analyzessupportswith multiple rings of
becomepossible'[n other
pointsa varietyof point topologies
words,specifyingthe numberof supportpointsdoesnot pro'
vide enoughinformationto readilyoptimizethe supportsyscharaaterof the rms
tem. If oñe considersthe mathematical
numberof points,the funcdeflectionfunctionfor a specified
tion will have multiple extremain general. These extrema
and thus the topologies
typicallyexhibit differentsymmetries,
fôì eaitr extrema should be found and then each used as a
startingpoint for the optimization.
The problemof optimizingthe positionsand forcesof N
:
oointshas 3N variablesand 3 constraints( net force l, net
-o-ents : 0). In additionrotationof the coordinategrid
about the plate center has no effect. Thus we
seek
minimum value of a functionin a 3N-4 dimensional the
Sincethis can be a very largespace,and thË-ïunction space.
is non_
Iinear in the variables,the matLematical
difficutties... errãi_
mous. We will reduce..the
difñcultybV addingsymmeÍy con_
ditions^whenever
possible.For most casesoi int"r.st,
t9q of support with reflectionsymmetryãUout ¡ " lvï_
ä*"r'i.
desired. This reducesthe problemto Zñtl-_ 1 variaUlei.
We now divide.rhis spacein_ioregionswiitr'¿istinct
propertiesrelaliveto the reflectioìaxes. Theie ,yãããö
can be exhibited on one sixth of rhe circularpl"te. Àñìxumple
'ftreof this
decomposirion
for lS.pointsis shown¡n ¡igure S.
noìã_
tion usesthe superscripr
for the numbei oi"point, on the first
symmetryaxis, the subscriptfor the second;ymmetry
axis.
_ Some topologies are obviously not candidatesfor
efficientsuppoit syitems. otlers a¡å, .nã-trr.r" have
been
optimizedwith the computer.The number'of local
mrnrrnu
for eachtopologyis belþvedto Ueveiv smaä,
-*" probablyone
in many cases.When multipleminimueilsi,
u." nor cer_
tain thar the lowestminimumhasbeenfounã, Uutexpect
that
this is the case.
We have found. thar the topologiesthat produce
the
lowest extrema (smallesrrms deflèctiö)-.iä-ìr,òr. .ppr*i_
matlng a triangula¡grid, as we wouldexpectfrom the
discussion on- infinite plates. As the number óf-support points
grows, the number of possibletopologiesgroru,
., well,
for largenumbersof supporrpointss-electäitt. U"rt and
täpology may be a difficult task. We have foünà in fact
ihat
seve¡al topologiesmay- have essentiallyidentical
,uppãii
efficiencies.Particularlyfor circularp-lãiËi'"nJlo-. values
of
N, the triangulargrid may not ahvaysproviãe an
obvious
naturalropology. For example,.fo¡
N : 9, thelriangulãr
is unsuitableand the supportis quite inemc¡ent.iVe LriO
tñãw
llrat n-o support sy-stemcàn have'an .m¿i"r;t ractor uéiier
than that of the infinite triangulargriO. ftrui,'in practice
the
effciency of a given optimiz;d topäloiy ä-i,;
comparedro
the triangular grid and .n uppé, bã"rd";
the possible
improvementby choosinganoth;; topoloÀicarr'Ue
assessed.
A variety of supportshave been optimized. For
these
caseswe have assumed v - 0.25. Thè simplest
cases
shown in Figure 6, indicatingttre numuei anã iåcat¡on aÍe
or trre
supportpoints,and the alrernatetopologies
considered.Cón_
tour plo_ts
of the plaredeflections
foi t¡õse ..s.j.re shownin
actualoptimumparameters
are given in Table
.rhe
frS,I9./.
r rvntcngtvesthe numberof supportpoints,the
number of
oprimized, the.ring radii, the forces for e.ò¡ ,lng
!1t:T:t.*
oI
supports,the azimuthalrotationfor eachring of suppoiisl
th_eachievedefficiency from which ttre rms åËflectio;'for;
gl:: 1*.? can readityþe fouqd using Equationq), unO-i¡ð
rauo or tne peak-to-valley
deflectionto the rms deflection.
The efficiencycoefficientsî*;¿;;;;jr;pl'oti.ä'in
Figures I 5
1n^d.16 along with the efficiencyfunciions for the three
geometries.
supporr
We now give brief descrip:l^r1T,:"ll"r.
uons
ol the supportsystems.
. The-singlesupportpointsystemhasno parameters
to be
optimized,and has rms defleðtion,,orn. zii-imes greatãi
than rhe limit suggested
by the innn¡æiiiangulargrid. The
two.point sysremhas a radiusthat can Ue uãiËã,
ãnOwhãn
optimized,..this
systemgivesan effic¡ency
iunciiä tnat is 7.2
trmesthe limit. The threepointsupportis the most common
of
and its optimum'oc."*äiti,å well known
.supportsystems,
radius of ß : 0.645. Ir is iomewhats"rp¡ri"s (anà'ålõ':
pointing) rhat the best rhreepoint ãmcienli,'li:'?.A
ti..r-trî"
limit Continuingon the singiering d;;;;
see that a six
':,_op_ti-ir:9.for B : 0.ó83and has an efficiency
Î".'i::lT:T
z.r
umes rne ltmrt. Addingmore pointsto a singlering now
becomesless efficient than creatingn"* iingî of
süppãrt
points. The simplestadditionis a neú ring withîero
radius.a
point at the center.
forln of sevenpoint supportsystem
lhi¡
is notice3blymore efficienr
than the ,if iäììrt system,as is
seenin Figure 16. This naturallyraisestnl-lúestion
whether
six point systemwith one poini .t tt"
surroundedby
I
nve
points can be more efficientthan tire
"óntJi
Cónventionats¡nglä
ring six point sysrem.
.Optimilatiãni.r..l, ti"t the single6
point ring is betterby about
130/0.
ring systemsare neededwhen we move
ro
-,_^
pornt supports.Here^sym_metry
nrnelll_jhr.e
' allowsthe rwo iãpãlå_
gies shown in Ficure g.- Both
'po¡nL,r,år,
nãu"'"uirost identical
efficiencies.Note tñat t¡e
nine
have much
lower efficiencythan rne slx or sevenpoint
supports.
be explainedas due to.rhefact that ,¿;m;i;;;'riungutar It can
grids
cannotbe madewith nine points.
A completeriangutargrid can be made with
and the resulrsfor the besi topologi aiä-gluen 12 points
Table l.
The topotosies and locarions ôf qf", ãntiüirço inpoints
are
shownin Figure9. The best t*o, 12¡l ^írä'tT¿,
-it
i,ãu. èrsrn_
tially the same efficiency.As expecteJ,
grid
_
givessuperiorperformance.,
"-t.¡.ngular
with an em"iLnåi.uout
worse rhan the infinite.rriangulargriO. OpiimizeO1.6times
systemsare shownin Figure.10. TopotogieåtS,tån¿15 poini
isolîiË
about equal,but sinceno triangular
iri¿îãnngurution
is pos_
sible,the efûciencyis poor.
ìVfovingup to the,next completetriangulargrid
. -18.point
we find
the
system (we omit it. .rnifai-pìlnt).
topotogiesgive equivalentefrciencies(t-gg; 1E";;d Several
lgrr');
indicatingthat strict triangulargrl¿ topofoiìe.-"r.
no, .rr"ntial, and rhe circularnature oftiri
;úËË;;r.
importanr.
optimizedtopologies,r"-r¡J*i-in Éiäìr. lr.
l-1_v_erat
Contour mapsof the deflections..are
shownin Fijui. lZ. À.tuãilv
the.grid byjd¿ing the lgrh polni'rt the center
!?-Tll:t¡ne
gves
an even higher efficiencyas is seenin Figures
15 and
Iô.
As we increasethe numberof supportpoints, probably
it
becomeslessimportanrto choose
points
consisrenr_with
triangulargrids. The
";ñ'b;;;'J';upporr
n.if iriunöunr grid sys_
27.supportpoints,.with an
1"T.þ.r
L¿l times that of the infinite triangular
"ptir*rn-åfr.t"nõy-ãuãut
Severaloptim_
-euite
ized topoloeiesare shorvq in-Ed*"'iã:Àii¿.
similar
efficiencies
are found for t t6 ana¿tf .
. We completeour examplesof disc¡etesupportswith a 36
point
supportsysrem. The'200" t.l.ììopãïr'iiì.ry has
a 36
point support system, and the propoi.o
r,r'pïãit system for
the indivídualmirror segment,oïtli" uni"-;r-r'irü
of California
T^enÌvleter Telescopetr.s ttris numue, ãirrîiäí,r.
There are
28 different.topologies
that satisfy-ìhe
-beenrt i.O'!v-*.try condi_
t¡ons. The bestcandidates
have
ôpti.¡iËä,.n0 rhe support point^locations
areshownin Figure,la. Ãiìsual, several
systems( 36d, 36i, 36j) areequa[] efficieni.-"fne
-*årrî'ìr,*
optimized
ppporr efficiency'is on-lyauoui +õã¡o
rhat of the
infinitetriangulargrid.
When so manysupportpointsare used,
the
is so
complexthar a varietyof similarsupport;o;ii;irssystem
givessimi_
lar results. For example,_ifone assumeìthat
all 36 points
must lie on three rings of 6,.12, and iS-pó-i";,
one greaily
reducesthe number of variablei,U"t tr,Jï"åi'optimum
is
onty about 150/o
worsethan the fú,ty ;pi;;i;;d
-itris-'iÃ-"ãr"ti.uf
system. This
does nor mean that one can actr¡eíe
-*
perfor_
mancewithout great care.
.On the .ontrrry, Lne goes to
more and more supports,the
carean¿ oàtäilln tr¡e ãn"rysii
requiredro achievethe optimumgo., up
áiá_iiicaily. euite
minor changesin the parameters
èanlea'dto largè
the deflecrions.Consideredi" tf,"ärìrr-.ilävïàver,changesin
it ooes
seem to be the caserhat.the multiple ãi..^ion.f
spacein
which one searchesfor the minimïm
does ffi; rather flat
"valleys".where more than one
of p"rumete.swill give
_set
rather similar rms deflecrions.eìen-soi-itr'îãlr,
of rhese
SPIE Vol. 332 Advanced Technology Optical Telescopes
lt9B2),r
215
I
I
t
t\
l
i
I
I
I
I
I
¡
I
I
"valleys"are exceedinglysteep,and greatcareis neededto
achievethe performanceindicatedhere.
Based on our experiencewith the supports analyzed
here, it appearslikely that if one goesto even more support
points, concentricrings of point supportsare an adequate
approximationto the ideal, but more complex optimum
geometry. The most efficientuse of supportpoints in this
situationis likely to be one with the numberof supportsper
ring increasingoutwardsin the series6, 12, 18, 24, etc. As
noted before,the degreeto which one hasoptimizedthe use
of the support poínts can be checkedby comparingthe
efficiencycoefÊcientagainstthat of an infinite triangulargrid.
system,as the numberof supportpoints
For a well-designed
becomeslarge, the edge effectsbecomenegligible,and the
efficiencycoefficientshould equalthat of the infinite plate.
Another way to evaluatethe qualityof the point support
systemsis to comparethem with specialcaseshaving an
infinitenumberof supportpointson eachring. Thus.weturn
to the analysisof continuousring supports.Theseare noticeably simplerto analyzeand optimizethan discretepoint supports, since the number of parametersfor optimizationis
for the deflections
imaller and the actualanalyticexpressions
are substantiallysimpler. One again uses the principle of
superposition,and uses the formula for a ring support as
given in AppendixB. Sincethe numberof supportpointsis
not defined, \r'e return to the notation of Equation 3 and
study the parameterf as a function of the number and radii
of the ring supports.As with points,we do not insistthat the
deflectionsvanish at the rings but rather let the deflectionsat
the rings be determinedfrom the elasticequationsgoverning
the platedeflections.
We have optimizedthe radiusand force for eachring,
and the results are shown in Figure 15. We have also
included the theoreticallimits for the three infinite plate
cases,and some of the optimizedpoint supportsystemsin
this figure.
It is interestingto use this graph to comparea ring
system's efficiency against the point support system
efficiencies. For example, notice that the deflectionachievable from three continuousrings is only slightlybetter than
that achievedwith three "¡ings"of discretesupportpointsas
seen in the 36 point support system. This provides
confirmationfor the idea that addingmore pointsto a given
ring is not productivebeyonda certainpoint, and that one
benefitsmuch more by addingnew ringsof supportpoints.
The supportpoint efficiencyis shownin Figure 16. As
expected,as the number of support points grows, the
efficiencyapproachesthat of a triangulargrid.
5. Sensitivityof Deflectionsto SupportPerturbations
In the precedingdiscussion,we havegiven the theoretical limits for plate deflectionsthat can be achieved. In pracand fabrication
tice, various approximations,simplifrcations'
of the supor alignmenterrors will degradethe performance
port system. Particularlywhen the number of supportsis
associated
of the deflections
superposition
large,the necessary
with eachsupportpoint that will allowalmostperfectcancellationsof thesedeflectionsbecomesa very delicateprocedure.
When one is supportinga singlecurved mirror, maintainingthe exactradiusof curvatureis not alwayscritical,and
is often possiin this case,some relaxationof the tolerances
ble, sincethe flrst order effectintroducedby analysiserrors,
etc.,is typicallya changein the curvatureof the plate. One is
fortunatewhen this can be focusedout. For caseswherean
opticalflat is beingsupported,,or when a seriesof segments
216 / SPIE Vol. 332 Advanced Technology Optical lelescopes (1982)
are being supportedthat must haveidenticalradii of curvacritical.
ture, any error in the mirror shapebecomes
If we assumethat the optimumsupportis known' the
effect of displacementsof the supportsfin the plane o[ the
mirror) from their optimum locationcan be understoodby
of a point is equivalentto
the fotlowing. The displacement
the supportforce beingin the correctpositionplus a moment
beingappliedat the supportpointwith a sizeequalto the proof the
duct-of ih" fot." at that point timesthe displacement
of the resultingdeflections
supportpoint. The superposition
from eachof the supportpoint errorsgivesthe additionalsurfaceerror introducedby the supportdeflections.
Perturbationsof support positionsand forces must
satisfythe constraintsof overallequilibrium;the sum of the
forcesequalsthe weightof the mir¡or, and the net moments
about the mirror centermust vanish'
The true optimization of a support system is an
extremelypainstakingprocedure.However,we conjectureon
the basis òf severalnumericalèxperiments,that the exact
supportpositionscan be adjustedover a small rangewitlout
gròisty increasingthe deflections,as long as one carefully
õptimizesthe fories appliedat thosepoints. Fortunately,the
mathematicaloptimizationof the forcesis a linear problem,
hencea uniqueoptimumexists. It canbe easilyfound if the
deflectionfoi eachgroup of supportpoints (evaluatedas sole
supports) is known. As indicatedearlier, the deflectionsat
vanish,but will follow
thè'supportpointswill not necessarily
from tiù análysis. An exampleshowingtwo configurationsof
the sametopologyis shownin Figure17. Thesehaveessentially identicalefficiencies.
6. Conclusions
We have describedthe generalconcepts,generalscaling
laws, and given some specificexamples0nfinite plates and
circular pÈtes) for supportingthin plates on a number
canserv-eas the basisfor
discretepoints. Thesedescriptions
beginning
-Inthe designof supportsystemsfor thin telescope
additionthey providea basisfor evaluatingthe
miirors.
of a givensystem.For any real mirror suprelativeeffÊciency
port system, an analysis that includesthe detailed shape
inon-circulargeometry,centralholes,etc.)and the curvature
is essential. A careful examinationof tolerances,although
difficult,is alsoinvaluable.
Acknowledgments
rùy'ethank Michael Budiansky,Alex Asfura, Robert Taylor, Karl Pister,and Enzo Ciampifor their excellentcounsel
the work
and aid with the computing.We gratefullyappreciate
of BarbaraSchaeferon the illust¡ations'
References
of Large Mirro-rs
Couder, A. Researchon the Deformations
Bulletin AstronomiqueVII
Observations
UsedFor Asrronomical
(1931) MemoiresEt Varietesp. 201 ff (translatedby E' T'
Pearson)
Crawford,D. L., Meinel, A. B' , and Stockton,M. W' Supoort and Testiw of Large AstonomicalMirrors. Symposium
Center
(Tucsón,Arizona,1968)OpticalSciences
Þroceedings
TechnicalReport# 30
S. , (1981)privatecommunication
Medwadowski,
AÞPendixA
for-.the1990'sKitt
Pearson,8., Opticaland InfraredTelescopes
(1980)
Piocèedings
Observatory"òõJ"ttnt"
National
Peak
M'' (1982)private
Taylor, R. ,. Asfura, 4., and Budiansky'
communlcatlon
S'' Theoryof Plaæs
Timoshenko, S. and Woinow-sþ-Krieger'
(1959)
edition
S"cond
;';; lì';ß.
Deflectionsof Semi-InfinitePlates
Points
Supportedon ManYSuPPort
plate can
The questionof efficiencyof supportof a large
deviation
square
mean
root
the
u" uO¿iãtt-"0W àetermining
since the plates
;i tdJ;fÈ¿tiãn ftorn the ñreundeflection'
points'edge
support
with
manv
u".v
large,
;";;;;lõ-b.
dis'
best^support
the
õoísequentlv,
;ffeJiJä; ilìãgtecte¿.
will be amóng the class of geometries
;,Ñil-;¡;óin-is
where all supportpointslook alike'
If we conThere are three geometriesthat are possible'
-Ñi itian have either 3, 4' o¡ 6 nearestneighsiOerïiiùn point,
ottrei ãistribution of points will have the
Ë.iil ñi"tt.
points
took atike' These three topologies
all
Jimriöli;t
are shownin Figurel.
plate supnow wish to calculatethe deflectionsof a
we aÍe
Since
types'
support
three
thtt.
portediv- each-ãf
'ioi"."rt"O
the.sup'
arrange
must
we
in the supportefficiency,
int-.iea Canie¿bv eachsupportpoint is
ö;îö;il;tð1ñ"f
cases' ln this way, we cån directly comthree
ltté r"'*" ii all
parethe deflections.
The deflectionsof a rectangulargrid of.support-points
and
tp..itgt ïi- ä .n¿ å are dJscriueãbv Timoshenko of
-":n;i;ü:ir;åeãt
"itft
tlssõl' to the deflections.forthe case
can be readilyfound'
supportpoints
four neigh-boring
'fVe
:
Drectorstr(a,b;x,y)
#r_'
t
+)2
+ Ao
-'
'1-¡mlz"orWt
a
oa3b S
2n3D ^-?1,t.. n3sinhc.tanha.
.---.
mTJ-(c'*tanhc
m¡ t rin¡
I t rnno -)
l \ u
where
t
t
"
I
) cosn@l
*
m
'
a
a
"
'
a
l
a'*tanha
.
oåå
s
- |7 \ a/ ^ - - = - - '
As--:T
"
L
tann'dm
"
Zt" D ^_T¡.t... r¡l
and
mrb
a^- T
Setting4 - ä givesthe squaregrid case'
T h e d e f l e c t i o n s f o r t h e s i x p o i n t c a s . e ( a t r i a n g uAn
largrid)
can Uï'iounO by using the principleof superposition'tiis pròblem ìvas..obtainedbv-S' J'
ìoiuìion'for
;Ëñ
difrerent
i,r"?¡"îoãirii ü sb1) utïng irt. superpositionof two
we- prèsent,here a somewhat
;;ú;t.ti";t'
;ffi"ilü'
the
JJ"rion b-asedon the sãme idea: Consider
íiäil*
geometrysnown ln {eure A1'^ We can *:rr1"-th" deflections
grid of supports
ã. ir,. ,ú* of the deflótions of a rectangular
grid of supports.
recrangular
a
displaced
of
äirJtr,Jãånã.tiðns
and
If we assumethe spacingof the triangr¡largrid is ä'
ö' we can
assumethe rectangular-lridhãsa spacingof a and
write the rectangulargrid deflectionsas
Drr*ongt"(a,b;x,Y)
with
a - Jîu,
of
The triangular grid will then give deflections
Telescopes(1 982) / 21 7
SPIEVol. 332 Advanced TechnologvOpticat
D,rions!"(
b, x, Ð :
AppendixB
or"rr*t"( a, bix,t)
I
Defìectionsof a Circularplateon a Ring of point Supports
I
* lD,"r,oner"(a,b;x
i,,
*r.
The t!1ee noin! supporrsystemcan be handledin a simi_
.
rar way. ttgure AZ shows the hexagonalgrid of supports.
Assuming the point separationis ã, onã ãn- write the
deflections as the sum. of fgur ãirpf"..¿ rectangular
deflectionsas is indicatedin the figure. 'iV" ifr.n obtain for
the hexagonal
grid, deflections
of
( b;x,Ð :
Dhuoson
y)
IIo,,,,*¡"(a,B; x,
+ Dr"r,onglr(o,A;*-ll $)
+ Dr"rto
nsr,(a,B;x- y,y) * Drrr**¡" (a,A,.!,,
lll
The deflectionsof thin, constantthicknessplatesare
of
great,interestin opticals.upportsystems.Many problems
can
oe solvedby the appropriate
superpositions
of soiutionsof the
caseot_a crrcularplateaxiallysupportedby a singleconcentric
point
(as
supports
shoùà ¡n F¡dure-ã)l- we ¿erivì
Sins of
here the solutionto this problem
__ Assume a uniformly loaded plate of radius ¿ and
stiffnessD, with total load
suipoitgã-ul-i point -...7.
sup_
porrs locatedat r - b (- a) ana''0-2 jt¡l
ß",,is
tc,j : ¡;
The deflectionw(r,0) is governedby
- 4* 4f a(':¿r
Dyaw:
,[r_+1.
ta2 kíi, á "t" kl
TheFourier
series
for É tfr- +l isoftheform
/cJ
i-t t
Q
where
a:3b,
Ê:ß8,
and
q)
n
æ
++E
^t: -b.
a^coskm9,
with
+É,'['þ-Tl**^,*
"^-
Note that for thesecases,the areasper point are:
At iorgle :
f,,
k
17
hencethe loadingis
. ¿ : 62
" square
,
../3
âhexagon: tT
- \*$atr - al*
7ra"
,7
t-,
P ô( r :á)
û
b
i- c os * m e.
.ar-
Assumethedeflection
hastheform
_
æ
w\r,0): f, 4(r) coskmï,
We write the solution in the form
m:0
where rir¡ is governedby
.
.l
t
t
ìlì
¡- : -, !D .t lNt z'
where _ø - the appliedpressureon the plate,
D" - the modulusof rigidiry otitré pUie,
- the plateareaper supportpoint.
l:
We_can define ô as either the peak deflection, the
mean
deflection,or the root meansquaredeviationoi nb ¿en..tion
from the mean. Fo¡ each of ihese we ..n õo*pãr, the rela'[*
fti:l.ies
üãïu", õl ,
lXt,
^of the. three confi_gurationsLv') aÍe grven fbr
eachof the definitionsand configurations
in the followingtable.
triangular
square
hexagonal
¿ltD
peak mean
4.95 3.71
e)
L dt rtr dt rþlll__ P=+å¡(r_¿),
; d , l ' d , l ; d , l ' d , J J-J- l ¿ r ç ¡ t
i.e., w¡.is ju-stthe deflectionof a platesupportedon a ring
of
radius á. We can obtain w6 by superposingtne ãeflectioni
ãi
thãt are uniformli loadãdand ring:lTf]I :gp"fed.ptates
loaoeo,
wtth totalloadsequaland opposite,to obtain
(3)
rms
p>p.
For m2 1, we have
l¿
I d
È,,?12
p
Iof" ,-ã-Tl "': hõG-Ð'
In other words,
[æ,r
l..-+
ldl
d
rdr
tt^212
¿ |
If we define w-(o) ang
of wry^$.) ^, the resrrictions
ø (b,al and [0, å),'iespectivelíl
tn.o *ã i,"u. irrLcontinuiry
equations
21 8 / SPIE Vot. 332 Advanced Technotogy Optical
felescopes (t 982)
\
'.1
w m{ a" ) ( 6 . l ) : w ^ ( Ð ( b - ) ,
d
- - r o-G) 16+¡
ar
+(1-v)
#lr#-Ëll,:.
(5)
: fiw^$)
Ø-)'
so that
& w^(ù{b+):
(b-),
#w^(b)
¿P
ê w^(ò(b+)-ã
tt-l : -1.
*n^r,
ar
Thesolutiontakestheform (with n : km)
n^G) (r) : A^rì + B^r'+2+ cman + D^r-n+2,
w ^ ( Ð( ò : A ', r ' + B ' ^ r n + z ,
l+l-
!"r"'l
¿
n ^ ( ò +r l 4 n ^ ' " ' - 4 n ^ < o t
dr"^
¿P
(6)
and the continuity equations become
(A^- A' ,) bn+ (Bn- B'^) 6n+z
0)
+ C ^ b - ' + D ^ b - n + 2: 0
n ( A . - A ' ^ ) b n+ ( n + Ð ( B ^ - B ' , ) 6 n +-2 n C ^ b - ,
- (n-2) D^b-n+2- g
n b - l ) ( A ^ - A '^ ) b n+ ( n + Ð G + l ) G ^ B '^ ) 6 n + z
* n ( n * 7 ) C ^ b - n+ ( n - 2 ) ( n - l ) D ^ b - ' + z: g
n ( n - I ) ( n - 2 ) ( A , - A ' ^ )b '
+ G + Ð ( ¡ t+ 1 )n ( B ^ - B ' ^ ) 6 n + z
- n ( n 1 1 )( z + 2 )C ^ b - '
- n(n-Ð (n-2)D^6-n+2:
nOL
Thesolutions
are
(13)
*(1-,'
#ll r,t
[å#n^,",-
Substituting(6) into (10) and (13), we see that 1.,
Bn, Cm, D. must satisfy
(14)
(1-z) n(n-1)A^a
(
(
* n + D l,n + 2 v n 2)l B, a'+2
+(1-z) nGll)C^a-n
+G-
1)[¿- 2-vG+2)]D^o-n+2: g,
( l - v ) n z ( n - l ) A ^ a n * n ( n * I ) ( n - 4 - v n )B ^ a n + z
- ( 1- v ) , ? G + l ) C ^ a - n
- n(n-I) (n+4-vòQ^a-'+z - 0'
Since C- ar'd D^ are known alreadyfrom (8) we can
solvefor A^ and B^:
t ^ a ^P_a 2f3fßþ
in t . - , f t ïolt *ì ,f f i8](,1 + y ) ì
[ r r _t ," -r ' _
Pa2 B' /1 .f
¡ - + ) -;;õfr(r-,,[;B^an*'-
I
ß2 I
ffi|.
(15)
(8)
We rewrite the remainingsolutionsas
(B^-B'^)6n+2:
#{*,
a
- m "h - n : -
|
n(n*l)
1
"m'
Pb2
8¡ D'
Pb2
Paz ßn+2
ïnD n(n*I)'
g n Dßnf f i ,
un r -e"-+i- : -Pa2
D^6-'--:ffiffi
A ' . a ' - A ^ a n * â " _3 f f i ,
In addition, ur(o) must satisfy the conditions of zero
moment and shear at r - a. Now, using standardexpressions.
P - n, .
B , - a n + z :B - a , + 2 - y
8¡D n(n*l)'
[^r
(.
lðl
I
06)
o^2
-1
ì'l
: o.
La.:
l4*,1!9s*+.È+ll
D
r ör I ð0¿
(e)
p-n+2
Thus the solutionto (1) has been obtained. It is given
by usins(15)and (16)in (6), and (6) in (2).
ll,-o
so
æ
I
I
, \ *,1:
. l t . a
:
(ro)
w^G)
ftw^G)
fr
I "^{a)lo;
furthermore
*r,,:(r-,,+
#[*-+ì,
0l)
-f,o,:'fi'[#.+#.
,#1,
and
-f,v,It-*n,*+f¡ll,-a (,2)
:[*[#.+#.
i#l
SPIE Vot. 332 Advanced Technology Optical Telescopes (l 982) / 21 9
=. -È-!+i-- -+rL+
llvar
Yr(xlO3¡
2.60
3.8
8.56
3.9
3
), t6
6\
2.93
2.36
ni, ßí, eÍ, Oí
P-p
rms
ß =
.3s38
ß =
.64s
.t.J
ß =
.681
4.9
n =
I
I
7r
o
ß =
0
E = .1183
o
0 =
6
. 73 7
.8817
0
n =
3
ß = .282s
e = .2309
,Þ=
o
3
.7936
.3637
0
n =
3
ß = .3lsl
e = ,2783
3
.7662
.2843
r .0472
3
.8257
.2187
.3497
3
.BZs1
.2187
_.3497
3
.7765
.2046
.7833
3
.8412
.1538
.2615
3
.7165
.2046
_.7833
2
3.76
t' - 2l
5.0
t
1.94
0 =
I
l5o
2.32
s.4
t
l8t
r.89
5.5
n =
ß =
e =
3
,3192
.2833
0 =
o
n =
ß =
e =
A =
2
.ut1 L
r. 65
9
I . OJ
3
.4741
.1625
3
.3195
s
.817r
3
.8536
3
.84r2
, l53B
-.zol)
o
.2071
1.0472
,.t731
.7820
.1421
.2663
3
.Bl7l
. I 73r
-.7820
3
.8536
.1421
-.2663
n
ß
e
A
=
=
=
=
3
.2044
.1152
0
3
.8740
. LO49
o
3
.4777
. ß47
1.0472
3
.s55s
. rt62
.33gg
3
,8787
.o95s
.4280
3
.8574
. tl10
.8383
n
ß
e
9
=
=
=
=
6
.2s6g
.1671
0
6
.5771
. l8l2
.2649
6
.5771
.1812
.7823
6
.8830
.t54g
.1703
6
.8834
.1607
.5236
6
.8830
.1549
.8769
I
361
o
3
.7700
.4054
1.0472
TABLE I
3
.5555
. Ir62
-.3399
J
.ötó/
.0955
-.4280
'ì
. 8 5 74
.tlt0
-.8383
: -..:tr:È*æi.:i.@
h
T
T
a
a
a
TRIANGULARGRID
,'
Ztr-
i -<-r
,
a
\
t
a
I
t
SQUAREGRID
\
\
b
n= Eh3
l2(l-vz)
a
a
HEXAGONAL
GRID
x8L 823-8449
x B L8 2 3 - 8 4 4 6
2. The geometry and notation for supportinga circular plate
on a_single ring of support points. P is the appliedforce. In
the figuresix supportpointsare used.
l. The three kinds of point symmetric infinite grids. These
grids are shown with equal areasper point. As describedin
the text, the triangulargrid is the most efficient.
SPIEVol.332 Advanced TechnologyOpticatTetescopes
(t982) / 221
iif
lii
arr.=Ç-.9f
€rrrxlO4
€rr.xtO4
4
N=4
N=6
N=æ
02
o.4 o.6
p
o.8
t.o
o
060
xBL en-8445
3. The support efüciencyas a function
.
of the ring radius for
po_ints
on a single ring. Caseswitn 2, 3, ãnó?
lippo.t po¡nt,
are shown. More than 6 points i,
i"dirìi;;ùJhiule rrom e
points. Note the dramatic improvement
*îth-increasing N,
and the great sensitivitvof theïfficiånl;';,il1#
normalized
radiusB.
0.70
xBL 823-8448
a,
Jhe region of Figure 3 nearJhe minimum is expandedfor
clarity.
Note againthe small differen* üñ;eliñ
: 6 and N
: infinitv.
222 / SplE Vot. 332 Advanced lechnology
Optical Telescopes (l 9g2)
.
-
t83
xsl 823-8447
:1,,,,_**
5. The possiblesupporrtopologies-foran
-ìuppãr,
lg point supporr.
Three fold ¡eflectiôn s¿mmetri of tire
,yrt"* i,
6. The optimal locationsof supportpoints
ass¡med.inestablishing
given fgr the
are
-iuai'i¡-ethe posiibletopolog¡åi.Tf,e nom"nsupport configurationsihown. note
ctarure.gtvesln the superscriptthe numbe¡ of points
¿j' ;i;
i:i"]"l
on one
force appti^e$
axts or symmetry and the subscript gives thê number
to the central point.
--' T¡e" plãts
l-"1
..
l"g"j1ue
of
are orderedin the senseof increasing
pointson the other axis.
"mr¡"n'.v.
SPIEVol.332 Advanced Technotogy
Optical Tetescopes
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7.
^Contour maps of
conngurationsof Figure ^ tfr-o^"li.lions
for the
support
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heavy
tine is the mean deflection. ll";
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224 / SplE Vot. 332
Advanced Technotogy
Opticat Tetescopes
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9. The optimal locations of support points for the general
supportconfigurations
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8. The optimal locationsof supportpoints for the general
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10. The optimal locationsof supportpointsfor the general
supportconfigurationsshown.
SPIE Vol. 332 Advanced Technotogy Optical Tetescopeslt 982) /
225
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11. The optimal locationsof supportpoints for the general
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12. _ Contour maps of the deflections for the support
configurationsof Figure I l. The contourinterval for eachþlot
is one half times îhe rms deflectionof the surfacefrom the
mean deflections. The heavyline is the meandeflection.
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13. The optimal locationso[ supportpoints for the general
supportconfigurationsshown.
226 / SPIE Vol. 332 Advanced Technology Optical Tetescopes (19g2)
14. The op-timallocationsof supportpointsfor the general
supportconfigurationsshown.
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15. The support efficiencyas a function of the number of
supportpoints. For eachnumber the bestresult is given, and
the-optimum for other topologiesis also occasionallyindicated. The limiting value for each of the three infrnite g4ds
(triangular, square, and hexagonal) is also given. The
efficienciesfor optimizedcontinuousrings are shown--bythe
dashedhorizontal lines . As N becomeslarge the efficiency
that of a triangulargrid.
of the best topologyapproaches
16. The efficiency per support point as a function of the
number of supporipoints' The vertical scaleis normalizedto
the limiting álcienóy, that of an infrnite triangulargrid' As
triangulargrid as the
expected,Ihe efficiencyapproaches-the
the
best op-timumfor
pointsgrows.
Only
nu'.u"iäf iupport
each number-ii shown. The limits set by each of the three
infinite grid systemsare alsoindicated.
SPIE Vol.332 Advanced TechnotogY Optical Tetescopes (1982) / 227
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A.l: T.ry infinite triangular.gridjs shown. points
are labeled
with different symbots-to.strã* irã*
tnË;r:irilil
be decom_
rectansutargrids,mî'äi'Ë.-inatyzed
L:::_d_l11q.two.
superposed
to give the deflectionsof tfre tiiã¡u-uräigria. and
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17. Two different ootimizations
for the same topologyare
shown. one wasrestiictedt"^h*.i'h.
io¡nts tte
'rwo
s,ame radius- The rms deflections ilnää
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of t¡eiè
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rather
ornerentoptimizations
are.essentially
i¿"ìt¡..1.iti, example
itrdicatesthe difficuttiesin finoini
ìïã Ëöhil.
A2' The infinite hexaronar.grid-is
shown. pointsare labered
with different symbotito,shó*
h";-'rhË"sr¡¿
-äî'-uJînalyzea
äi u. decom_
four recrangutargrios thaf
L:::-d_lll".
an¿
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superposed
to give the deflectionsroiiir"iå*"goial
gri¿.
229 ,/ SplE Vot. 532 Advanced
Technotogy Opticat Telescopes
fl9g21
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