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The concept of passive surface error control via tension is explored for flexible square panels and for a rectangular array of such panels suspended with four catenaries. The panels are 1 mm thick 1 m square graphite-epoxy composite plates with a

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Error Suppression via Tension for Flexible SquareAntenna Panels and Panel Arrays
Gyula Greschik
∗
TentGuild Engineering Company, Inc., Boulder, Colorado 80305
Juan M. Mejia-Ariza
†
L
’
Garde, Inc., Tustin, California 92780
Thomas M. Murphey
‡
U.S. Air Force Research Laboratory, Kirtland Air Force Base, Albuquerque, New Mexico 87116
andSungeun K. Jeon
§
Moog CSA Engineering, Albuquerque, New Mexico 87123
DOI: 10.2514/1.J052691
Theconceptofpassivesurfaceerrorcontrolviatensionisexploredforflexiblesquarepanelsandforarectangulararrayofsuchpanelssuspendedwithfourcatenaries.Thepanelsare1mmthick1msquaregraphite-epoxycompositeplates with a four-ply symmetric layup. Individual panel response to a uniform through-thickness temperaturegradientfieldbothaloneandcombinedwithin-planetension(stretch)areexamined.Whenintegratedintoa
2
×
5
grid(a simple phased array configuration), the effects of a transient slew are considered. Characteristic responses areidentifiedandstudiedsymbolicallyaswellasnumerically.Theresultsdemonstratethefeasibilityofcombinedglobalandcomponent-levelerrorsuppressionfortheconsideredstructure,exercisedsingularlybyglobalprestressthatalsomaintainstheintegrityofthetensionstructure.Limitedattentionispaidtomission-specificissuessuchasstowageanddeployment. Assuming an X-band radar context, surface errors are related to a 1 mm limit. Out-of-flatness isevaluated with three metrics relevant to different aspects of signal processing: the maximum depth of the deformedsurface, its maximum lateral deviation from the best-fit plane, and the phased-array radiometric rms surface error.
Nomenclature
A
= area; panel area; aperture area
a
= panel square edge length
a
0
= gap between antenna panels in array
a
q
= traction footprint of link force on panel mesh
b
= location of link hinge inbound from panel edge
C
= preloading compliance (displacement per force)
c
= distance of link from parallel panel edge
c
0
,
c
= orbital Earth-view cone base circles
D
=
Et
3
∕
12
1
−
ν
2
, plate flexural rigidity
E
,
ν
= Young
’
s modulus, Poisson
’
s ratio
EI
= beam cross section flexural stiffness
e
= cross section load eccentricity
F
= force (positive when tension)
F
c
,
F
s
= center- and side-link loads on panel edge
F
E
= full tension exerted on a panel edge
G
= shear modulus
g
r
= auxiliary function (axisymmetric analysis)
h
ρ
= nondimensional auxiliary function (axisymmetricanalysis)
h
= orbital altitude
k
=
EI
∕
F
, beam stiffness to tension coefficient
L
= member length
M
= bending moment
N
= membrane force (force per length)
p
b
=
L
∕
2
k
, beam loading parameter
Q
in
= irradiation absorbed by panel face
q
s
= solar irradiation
q
T
= through-panel heat flow (flux)
R
= radius of curvature
r
,
r
e
= radial position, disk radius (axisymmetric model)
T
= temperature; torque
T
mean
= panel temperature through-thickness mean
T
offs
= hot side temperature offset (
“
”
offset) from theaverage
t
= plate structural thickness or beam depth; time
u
,
v
= radial, lateral displacement (axisymmetric model)
W
= parameter for cable grid rotation model
x
,
y
= coordinate axes in the array long and short directions
z
= out-of-plane coordinate axis
α
T
= coefficient of thermal expansion
α
s
= solar absorptance of sun-facing panel side
β
= slope angle of disk contour (axisymmetric model)
γ
= panel face to nadir angle (thermal analysis)
γ
c
=
F
c
∕
F
E
, panel edge center load ratio
Δ
l
= optical path-length error (for antenna accuracy);constant-force spring preloading length change
Δ
T
=
T
−
T
−
, through-thickness temperature gradient
ε
= direct strain (stress analysis); gray body emissivity(thermal analysis)
ϵ
= surface out-of-flatness (geometric error analysis)
ϑ
= angular orientation (transient slew)
κ
=
1
∕
R
, curvature
κ
=
r
∕
R
, normalized radial curvature
λ
= wavelength
ξ
=
u
∕
r
e
, normalized radial displacement
ρ
=
r
∕
r
e
, dimensionless radius
ρ
,
ρ
A
= material volume density; panel areal density
ρ
E
= Earth albedo (reflectance of solar radiation)
σ
= direct stress
σ
=
σ
1
−
ν
2
∕
E
, dimensionless stress
Thispaperisbasedonan earlierpublicationpresentedas Paper2010-2748at the 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,andMaterialsConference,Orlando,FL,12
–
15April2010;received20March2013; revision received 16 November 2013; accepted for publication 11January 2014; published online 8 January 2015. Copyright © 2013 byGreschik. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 1533-385X/15 and $10.00 in correspondencewith the CCC.*Structural Engineer; greschik@teguec.com. Senior Member AIAA.
†
Senior Mechanical/Aerospace Engineer. Member AIAA.
‡
Engineer. Associate Fellow AIAA.
§
Project Engineer. Member AIAA.513
AIAA J
OURNAL
Vol. 53, No. 3, March 2015
σ
SB
=
5.6704
×
10
−
8
W
·
m
−
2
·
K
−
4
, Stefan
–
Boltzmannconstant
ω
,
β
= rotational velocity; rotational acceleration
ω
0
,
ω
= Earth-view cone half-angles (thermal analysis)
Subscripts
b
= beam; beam solutionCF = constant force spring
dz
= surface depth as measure of out-of-flatness
e
= edge (end, perimeter) region
F
= association with tensionoffs = maximum offset from mean surface
r
,
c
= radial and circumferential directionrms = radiometric root-mean-square surface error
s
,
b
= panel sun-facing and back side (thermal analysis)
T
= association with thermal load0 = association with reference state or quantity1, 2 = responsestepsincompressionringanalysis;plyin-plane principal stress or strain directions3 = ply through-thickness principal direction0, 1, 2 = beginning, midpoint, and end of slew acceleration
= beam or plate top (
z >
0
) fiber
−
= beam or plate bottom (
z
<
0
) fiber
I. Introduction
T
ENSION structures
—
cable networks, membranes, tether systems, etc.
—
that are inherently lightweight and relyon prestress for mechanical and global geometric integrity areincreasingly common in space. However, the idea to use globalprestress to suppress errors also in flexible components such as shell-membraneantennapanelsisnew(earlierdiscussedonlyinaprecursor paper [1]). In fact, coupling local (component) and global (system-level) error control may prove practicable only in coincidentalconditionsbecausethecharacteristicsofatensionstructurearefarfrom those of its components. The present work takes a first step towardunderstandingsuchconditionsbyexploring,inthecontextofasimpletensioned phased array structure, some component and system-levelerrormechanismsandtheeffectivenessoftheircombinedsuppressionbysystempretension.Thestructureconsideredisa
2
×
5
arrayof1mm thick 1 m square flexible panels, connected with tension links andsuspended with a catenary on each side (Fig. 1).Thepaperisorganizedasfollows.Theproblemdefinitionisrefinedin Secs. II
–
VI, where the loads (thermal gradient and transient slew),antenna surface precision requirements, and panel specifications arediscussed. Sections VII
–
IX cover component (panel) analysis: panelresponsetoathermalgradientandtheeffectivenessoftensiontoreducethe resulting out-of-flatness.The tension structure is considered in Secs. X
–
XIII in its entirety,except its external support that is here replaced with a simpleidealization. Catenary system design is also considered. Surfaceerrors from transient slew are evaluated, and the effectiveness of preloadtoreducethemisassessed.Finally,theobservationsfromthepanel- and array-level analyses are summarized in Sec. XIV.
II. Problem Specification
To enable the assessment of the practical merits of the arrayconceptinFig.1andofthefeasibilityofemployingitsglobaltensionto suppress both array and panel errors, it has to be placed in a re-presentative quantitative framework. This is done in the followingby defining performance metrics, loads, and geometric and materialproperties.The performance of a mechanical aperture system is the accuracywith which its actual shape, in mission conditions, approachesthe ideal geometry. For a phased array, the ideal shape is planar,the proximity to which can be quantified with various measures of out-of-flatness, discussed in Sec. III.Perturbations to out-of-flatness can results from kinematic, thermal,and mechanical loads. These include imperfections, even or unevenchangesoftemperatureoverthesurface,through-thicknesstemperaturegradients, and transient responses to slew and to orbit adjustments.Imperfectionsareignoredherebecausetheydependprimarilynotonthe structural design but on fabrication technology, and their controlis best done and optimized for in the workshop. Simple thermalextension (near-uniform areal change of the mean through-thicknesspanel temperature) is not considered either because it merely
“
scales
”
the array, without out-of-flatness. Although this scaling may effect aminorwavefrontdirectionalerrorsimilarinmagnitudetothe
relative
change of the array size, wavefront coherencewill not be affected andRF (radio frequency) performance will not suffer as long as the gapsbetweentheelectronicunitsonthesurfacedonotincreasebeyondtheir operational limit. On the other hand, a nonuniform temperature fieldcould,intheory,buckleapanel.Suchfields,however,arenotincludedin the scope of this paper. One type of transient load, perturbation byorbit adjustments, is also ignored.Remaining within the scope of this study are through-thicknessthermalgradientloadsandtransientsleweffects.Thesearediscussedin Secs. IV, V.
III. Surface Error Metric and a Limit Error Value
Twotypesoferroraresignificanttomostcommunicationorimagingantenna systems. The first, the so-called rms figure, is a measure of overall surface accuracy and is directly linked to the antenna gain [2].Theother,localerrors,mayinterferewiththeeffectivecontinuityofthesurface and may thus constitute a less-negotiable design metric.Asalsodoneelsewhere[3],thermserror
ϵ
rms
inthecurrentcontext of a phased array antenna is calculated from
half
the offset of theantenna surface from its ideal geometry according to
ϵ
rms
Z
z
−
z
0
∕
2
2
d
A
∕
A
s
(1)
in which
z
is the position in the surface normal direction,
A
is theaperture,andthereferencesurface
z
0
itselfisdeterminedbyminimizing
ϵ
rms
. If a definition other than Eq. (1) were used, the rule to relate for a desired gain a limit on
ϵ
rms
to the wavelength
λ
would need to differ from that for a reflector dish.The compatibility between Eq. (1) for a phased array aperture andthe
“
classic
”
rms error [4] for a reflector is due to the definition of antenna errors, which entails [2] the application of the generic
“
root-mean-square
”
formula to half the optical path-length error
Δ
l
ϵ
rms
Z
Δ
l
∕
2
2
d
A
∕
A
s
(2)
Forareflector,
Δ
l
istwicethesurfaceoffsetfromthereference,whileit is the offset itself for a phased array [2,5].
The allowable rms error for a device depends on the gain requiredand usually lies in the
ϵ
rms
;
max
∈
λ
∕
15
;
λ
∕
50
range. Thus, for thepresent generic study the
ϵ
rms
<
λ
∕
30
midrange limit is adopted.Further, assuming an X-band, medium-class, radar application, a 1mmerrorlimit(about1/30ofthe3.75
–
2.50cmwavelength)results:
ϵ
rms
<
ϵ
rms
;
max
λ
∕
30
1 mm (3)
x
catenary
x yF
ly
F
lx
y
catenary
y
link
x
link
Fig. 1 Tension array design paradigm.
514
GRESCHIK ETAL.
More stringent than the rms surface error is the maximum offset
ϵ
offs
of the surface from its best-fit plane, and a yet less-forgivingmetric is the surface depth
ϵ
dz
(the absolute difference between thelateralextremesofthesurface).Thesetwomeasures,whilenotdirectlylinked to antenna gain, are still considered herein. Deformations aredeemed acceptable if all three errors are below the 1 mm limit:
ϵ
rms
;
ϵ
offs
;
ϵ
dz
<
ϵ
err
;
max
1 mm (4)
IV. Thermal Gradient Loads
The thermal gradient across a phased array structural panel aswell as the panel mechanical response to this gradient are hard toreliably assess because of the complex constraints imposed by thesurface-mounted components. The latter have radiation proper-ties different from the panel
’
s, they generate as well as buffer heat,and their internal structure (
“
thickness
”
) both flexurally stiffens thepanel underneath and can reduce its share of the gross thermalgradient.The authors are unaware of a simple method to assess the re-presentative panel properties and thermal loads in such complexcases. Therefore, and considering the mission-specific nature of panel configurations in contrast to the generic scope of this work, noattempt is made here to model a full assembly. Rather, the structuralpanel is considered by itself, and the effects of mounted electronicsare ignored for the flexural stiffness and accounted for in thecalculation of a representative thermal gradient implicitly, via thecautious selection of panel reflectance properties. The gradient, inturn, is assessed from a parametric study of generic panel thermalproperties in the context of selected orbital configurations: for the1 mm thick panels, it is selected as
Δ
T
3.5
°
C
.
A. Definition of Parametric Study
Consider three orbital position-attitude combinations for a panel(setofpanels).Inthefirst(baseline)state,thepaneldirectlyfacesthesunat1astronomical unit(AU)in
“
deepspace
”
(outsidethe rangeof influence of other celestial objects); see Fig. 2a . In the second case(Fig. 2b), the same surface is in an
h
0
250 km
altitude low Earthorbit (LEO) over the sunrise line on the planet
’
s surface. At thislocation, the panel is also irradiated by the Earth (that blocks nearlyhalf the view to space), more intensely on its hot side that is alsoexposed to Earth-reflected solar heat (albedo effect). The thirdconfiguration is an attitude perturbation to the second, with thepanel turned away from directly facing the sun, with it no longer being perpendicular to the Earth horizontal plane,
γ
≠
90 deg
in Fig. 2c.In the search for a representative through-thickness thermalgradient, the first two configurations serve as reference states; thethird permits some calibration for a worst-case scenario. For eachstate, thepanel hot (
“
sunny
”
)and cold (
“
back
”
)side emissivities andthesunny-sidesolarabsorptivitiesconsideredare
ε
s
;
ε
b
0.25
,0.55,0.85, as well as
α
s
0.2
, 0.5, respectively. These data ranges, basedon published emittance and solar absorptance data [6
–
8], areintended to bracket the likely options for an actual design.The emissivity lower bound
ε
0.25
is moderately low, in theballpark of some common polished, galvanized, or otherwise treatedmetals. The upper limit
ε
0.85
befits a variety of paints and somemetallic and nonmetallic surface types. Although there are viableoptions below the lower and even more so above the upper bound,such values are not considered here, recognizing a likely bias ina real-life design toward symmetric thermal performance betweenthe panel faces, a bias that will
“
pull
”
the designed property valuestoward the midrange. Symmetry is preferred because the panelshould perform equally well, regardless of which side is exposed toirradiation, the dominant thermal load. A preference for midrangeradiation properties follows because, if either orientation were con-sidered alone, the thermal gradient would be minimized with lowemissivityonthecoldandhighonthehotside
—
oppositetothebest design for irradiation on the other side.The lower solar absorptance value considered,
α
s
0.2
, iscomfortably matched with a number of white paints, some of whichcan actually lower
α
s
even under 0.1 (not considered here for thereasonsof symmetryandpreferredmidrangeperformance,mentionedabove). The higher value,
α
s
0.6
, is a compromise within reach of some metallic and ceramic surfaces, chosen to account for theuncertainties of mission-specific electronics. These bounds are rather conservative, given that one direct means to minimize panel thermalgradient isvia lowering solar absorptance, and that several options for this,hereignored,maybeavailableduringdetailingevenforthepanelface where the electronics are mounted.Two through-thickness thermal conductance values are consid-ered:
k
0.20
and
0.50 W
∕
m
·
K
, the boundaries of the rangegiven in [9] for epoxies
—
values near the
k
0.35 W
∕
m
·
K
midpoint of which are often cited elsewhere as a single
“
epoxy
”
values (for example,
k
0.33 W
∕
m
·
K
, in Table 12 on page 390in [10]). Although epoxy conductivities as low as
k
0.15 W
∕
m
·
K
can also be found in the literature, the lower value of
k
0.20 W
∕
m
·
K
is still deemed conservative here because, one,epoxies can be engineered for high thermal conductance and, two,our interest is in the composite with substantial graphite-fiber contents, not in the matrix alone. Usually much higher values
—
k >
1 W
∕
m
·
K
in [9]
—
are cited for carbon-fiber-reinforcedcomposites. (Such values are not adopted here because their relevance to the present study is uncertain: sources often do not spellout whether the values there refer to through-thickness or in-planeconductance.)
B. Calculation of Thermal Gradients
The through-thickness thermal gradient has been evaluated for allcombinations of the parameters and orbital configurations describedabove with a spreadsheet. In this program, the irradiation of (energyinflux on) the two panel sides by the sun, the Earth
’
s bright and dark sides as applicable, and deep space are evaluated first.The view coefficients are obtained with geometric formulas com-bined with numerical integration. In particular, the satellite
’
s Earthviewconewithaxis
OBA
inFig.3isdefinedwithits
c
0
basecircleand
ω
0
apexhalf-angle,andthesolidviewanglecontributionsofthebright and dark Earth regions are numerically summed ring by ring on theEarth
’
s visible cap (circles
c
with view angles
ω
in Fig. 3).The solar irradiation used is
q
s
1419 W
∕
m
2
, a worst-casenumber accounting for both seasonal and 11-year solar cyclicvariations [11]. The Earth albedo
ρ
E
0.4
is extrapolated from statistical values in Figs. 7 and 14 of [12] with a conservative bias,needed for the worst-case scenario of interest. The Earth long wavesurface radiation flux was set similarly in Fig. 14 of [12] to
Q
s
240 W
∕
m
2
. From these data, the view coefficients, attitudeangles (where necessary), and values for the panel surface radiationproperties (cf. Sec. IV.A), and the radiation energy fluxes
Q
in
;s
and
Q
in
;b
absorbed by the sunny and back faces are obtained.The thermal balance is solved next, formulated in terms of themean and offset temperatures
T
mean
and
T
offs
:
solar radiation
γ
(a)(b)(c)
Earth day ... night Earth day ... night
Fig.2 Orbitalpositionsconsideredinstudytobracketthermalgradient loads.
GRESCHIK ETAL.
515
T
s
T
mean
T
offs
(5)
T
b
T
mean
−
T
offs
(6)
T
mean
T
s
T
b
∕
2 (7)
T
offs
T
s
−
T
b
∕
2
Δ
T
∕
2 (8)
The equations for through-thickness conduction and for the hot andcold face radiation-flux balances thus become
q
T
2
T
offs
k
∕
t
(9)
q
T
Q
in
;s
−
T
mean
T
offs
4
σ
SB
ε
s
(10)
q
T
T
mean
−
T
offs
4
σ
SB
ε
b
−
Q
in
;b
(11)
where
q
T
is the through-panel heat flux, and
σ
SB
is the Stefan
–
Boltzmann constant. Bringing the parenthesized terms in Eqs. (10)and (11) to one side, subtracting one equation from the other, andeliminating
T
offs
via Eq. (9) then yields a single equation, which issolved for
q
T
via Newton iteration. The gradient
Δ
T
is triviallyobtained next[cf.Eq. (8)] and ismaximized in terms of the
γ
attitudeangle when applicable. The software also shows the panel positionand attitude graphically as presented in Fig. 4.
C. Thermal Study Results and the Design Thermal Gradient
The results of the parametric study are presented in Tables 1
–
4.ShowninTables1
–
3arethethermalgradientsforallcombinationsof the parameters discussed in Sec. IV.A, for the satellite positions inFigs.2a
–
2c,respectively.Table4displaysthe
γ
≠
90
°
attitudeanglesassociated with the gradients in Table 3, which were maximized byvarying
γ
(cf.Fig.2c).Ineachtable,resultsareshownforthelowandthe high solar absorptance values considered,
α
s
0.2
and 0.6;results for variations of the remaining parameters (thermal con-ductivity
k
and panel surface emissivities
ε
s
and
ε
b
) are presented.The data are organized for the gradient values to generally increaseacross the tables to the right. As expected, the greater the solar absorptance, the greater the gradients.Comparing the gradients when far from Earth to those in LEOreveals that the immediate vicinity of Earth significantly affects thethermalloads,insomecasesincreasingtheworst-casegradientswithmorethan60%,e.g.,
α
s
0.2
,
k
0.2 W
∕
m
·
K
,
ε
s
0.85
,
ε
b
0.55
or
α
s
0.2
,
k
0.5 W
∕
m
·
K
,
ε
s
0.85
,
ε
b
0.25
. Theadditional step of relaxing the panel orientation has a lesser effect;although in a few cases it further increased thegradient with a nearly7% [see
α
s
0.2
,
k
0.2 W
∕
m
·
K
,
ε
s
0.85
,
ε
b
0.55
], thegain is often realized only in digits less significant than those shownin the tables. However, the attitude change to maximize the gradient (Table4)canbelarge;itapproaches30deg(i.e.,
γ
≈
60 deg
)forhighgraybodyemissivity
ε
s
andlowsolarabsorptance
α
s
values.Astudyof these optima has revealed that actually the
Δ
T
γ
function oftenhas two local maxima in the
γ
60
:::
90 deg
range, sometimesmore than a dozen degrees apart.Probably the gradient could be further increased by also allowingsatellite locations not precisely over the sunset line. However, theassociated gain is not expected to be significant and is thus hereinignored.The absolute maximum gradient obtained is
Δ
T
3.46 K
: therightmost value in the first row of Table 3, for the variable panelorientationat250kmaltitude,withshortwaveabsorptance
α
s
0.6
,composite through-thickness conductivity
k
0.2 W
∕
m
·
K
, andsunny and back face emissivities
ε
s
0.25
and
ε
s
0.85
. Accord-ingly, the thermal gradient used in the rest of this study is chosen as
Δ
T
3.5 K
, rounding the calculated maximum up.
V. Transient Slew
The major system-level perturbation to array flatness is transient dynamics:responsetoslew.Thiswasmodeledintheprecursorstudy[1] with a symmetric bilinear rotational acceleration history
β
t
applied to the external support,to reach from a stationary initial statea
1 deg
∕
s
rotational velocity in 10 s. With the variables used inFig. 5a , this meant
ω
t
0
0 deg
∕
s
and
ω
t
2
1 deg
∕
s
, with
t
0
0 s
and
t
2
10 s
. The maximum rotational acceleration,at half-time
t
1
t
2
∕
2
5 s
, was
β
t
1
0.2 deg
∕
s
2
β
max
;
ramp
.
βγ
h
o
r
c
r
c
' solar rad. R
E
AOEarthBC
ω
'
ω β
circles c', c
Fig. 3 Variables for the calculation of irradiation by the Earth.Fig. 4
γ
61
.
5 deg
attitude direction, maximizing
Δ
T
for
α
s
0
.
2
,
k
0
.
2 W
∕
m
·
K
,
ε
s
0
.
85
,
ε
b
0
.
25
, position Fig. 2c. The Earthhorizon (view cone base) from the satellite is in red.Table 1 Thermal gradients
Δ
T
in degrees of Kelvin for panel far from Earth,facing the sun (Fig. 2a)
k
0.5 W
∕
m
·
K
k
0.2 W
∕
m
·
K
ε
b
0.25
ε
b
0.55
ε
b
0.85
ε
b
0.25
ε
b
0.55
ε
b
0.85
Low solar absorptivit
y
,
α
s
0.2
ε
s
0.25
0.28 0.39 0.44 0.71 0.97 1.09
ε
s
0.55
0.18 0.28 0.34 0.44 0.71 0.86
ε
s
0.85
0.13 0.22 0.34 0.32 0.55 0.71
High solar absorptivity
,
α
s
0.6
ε
s
0.25
0.85 1.17 1.31 2.11 2.90 3.26
ε
s
0.55
0.53 0.85 1.03 1.32 2.10 2.55
ε
s
0.85
0.39 0.67 0.85 0.96 1.65 2.10516
GRESCHIK ETAL.
However, for small to medium spacecraft (increasingly favoredoverlargeplatforms),muchgreateragilityhasalreadybeenachieved.Forexample,DigitalGlobe
’
s2800kgWorldView-2imagingsatellitefeatures [13]
β
1.5 deg
∕
s
2
acceleration and an
ω
3.5 deg
∕
s
characteristic slew rate. A similar rate,
ω
3.0 deg
∕
s
, for a 130 kgplatform is used as a working number elsewhere [14].Consequently, acceleration time
t
2
2
t
1
5 s
and achievedrate
ω
t
2
2.0 deg
∕
s
are herein assumed, from which a peak accelerationof
β
max
;
ramp
β
t
1
0.8 deg
∕
s
2
followsfortherampslew profile used earlier (Fig.5a ). These numbers are belowthe onescited from the literature, for two reasons. One, the references citedpertain to state of the art attitude actuation, ahead of the mainstream.Two, less agility may be required of a phased array than thereferencedoptical
–
RFsystems,becauseelectronicbeamsteeringcanrelieve some demand on slewing.Analternativeslewprofile,achievingthesamevelocityandorientationin the same time, is also explored. This is constant
“
step
”
acceleration
β
t
1
β
max
;
ramp
∕
2
0.4 deg
∕
s
2
≈
0.006981 rad
·
s
−
2
, with half the earlier maximum but abrupt (step-function) beginning andend (Fig. 5b).
VI. Panel and Array Specifications
For relevance to cost-efficient near-future missions, the system herein considered (cf. Fig. 1) is defined to rely on mainstream technology both in architecture and some key details. Manageable1 m side four-ply
0
∕
45
s
composite square structural panels of
t
40 mil
1.016 mm
total thickness are used. Each lamina is
10 mil
0.254 mm
thick and is of common T300 [15] plain weavefabric in a generic epoxy matrix with effective in-plane
E
1
E
2
62 GPa
Young
’
s modulus and
G
12
4.14 GPa
shear modulus,
ν
12
ν
21
0.05
Poisson
’
s ratios, and coefficients of thermalexpansion (CTE)
α
T
1
α
T
2
2.5 ppm
∕
°
C
. For the antenna, theelectronics are mounted on the composite plates, yielding a grosssurface density herein assumed as
ρ
A
5 kg
∕
m
2
.
Table 2 Thermal gradients
Δ
T
in degrees of Kelvin for sun-facing panelat 250 km altitude (Fig. 2b)
k
0.5 W
∕
m
·
K
k
0.2 W
∕
m
·
K
ε
b
0.25
ε
b
0.55
ε
b
0.85
ε
b
0.25
ε
b
0.55
ε
b
0.85
Low solar absorptivit
y
,
α
s
0.2
ε
s
0.25
0.33 0.46 0.52 0.83 1.15 1.29
ε
s
0.55
0.25 0.40 0.48 0.62 0.99 1.20
ε
s
0.85
0.21 0.36 0.46 0.52 0.89 1.14
High solar absorptivity
,
α
s
0.6
ε
s
0.25
0.90 1.24 1.39 2.23 3.07 3.45
ε
s
0.55
0.60 0.96 1.16 1.49 2.38 2.89
ε
s
0.85
0.46 0.80 1.02 1.15 1.99 2.52
Table 3 Thermal gradients
Δ
T
in degrees of Kelvin for panel at 250 kmaltitude, worst-case attitude (Fig. 2c)
k
0.5 W
∕
m
·
K
k
0.2 W
∕
m
·
K
ε
b
0.25
ε
b
0.55
ε
b
0.85
ε
b
0.25
ε
b
0.55
ε
b
0.85
Low solar absorptivity
,
α
s
0.2
ε
s
0.25
0.34 0.46 0.52 0.84 1.15 1.30
ε
s
0.55
0.26 0.41 0.50 0.64 1.01 1.23
ε
s
0.85
0.22 0.38 0.49 0.55 0.95 1.21
High solar absorptivity
,
α
s
0.6
ε
s
0.25
0.90 1.24 1.39 2.23 3.07 3.46
ε
s
0.55
0.60 0.96 1.17 1.50 2.39 2.90
ε
s
0.85
0.47 0.81 1.03 1.16 2.00 2.55
Table 4 Attitude angles
γ
in degrees (Fig. 2d), associated with the thermalgradients in Table 3
k
0.5 W
∕
m
·
K
k
0.2 W
∕
m
·
K
ε
b
0.25
ε
b
0.55
ε
b
0.85
ε
b
0.25
ε
b
0.55
ε
b
0.85
Low solar absorptivity
,
α
s
0.2
ε
s
0.25
83.4 83.4 83.4 83.4 83.4 83.4
ε
s
0.55
76.8 76.8 76.8 76.8 76.8 76.8
ε
s
0.85
61.5 61.5 61.5 61.5 61.5 61.5
High solar absorptivity
,
α
s
0.6
ε
s
0.25
87.4 87.4 87.4 87.4 87.4 87.4
ε
s
0.55
85.3 85.3 85.3 85.3 85.3 85.3
ε
s
0.85
82.4 82.4 82.4 82.4 82.3 82.3
rot.accel.rot.velocityangular orientn.
a) b)
t t
1
t
0
=
0t
2
=
2t
1
t
2
t
0
β β
1
=
β
max
β
1
t
1
ω
1
= 2
ω
2
=
β
1
t
1
β
1
=
β
2
=
β
max
/2
ϑ
2
=
β
1
t
12
ω
1
ϑ
1
ω
2
ϑ
2
ω ϑ
t t
1
t
0
t
2
βϑ
1
ω
2
ϑ
2
ω ϑω
12
β
1
t
1
ϑ
1
= 6
β
1
t
1
ω
1
= 2
ω
2
=
β
1
t
1
ϑ
2
=
β
1
t
122
β
1
t
1
ϑ
1
= 4
Fig. 5 Ramp and step slew profiles.
GRESCHIK ETAL.
517

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