Analysis of Gossamer Structures - NASA€¦ · Wallops Island, VA 23337,...
Transcript of Analysis of Gossamer Structures - NASA€¦ · Wallops Island, VA 23337,...
An Optimization-Based Approach to Membrane Problems Arising in the
Analysis of Gossamer Structures
Frank Baginski, Michael Barg & Stacey Vezina, Department of Mathematics,
The George Washington University, Washington, DC 20052, [email protected]
Laura Cadonati, MIT, LIGO Project, NW17-161, 175 Albany St.,
Cambridge, MA 02139, [email protected]
William Collier, Flight Software Maintenance Group, Computer Sciences Corporation,
7700 Hubble Drive, Lanham-Seabrook, Md. 20706, [email protected]
Willi W. Schur, Physical Sciences Laboratory, NMSU, NASA-GSFC-WFF,
Wallops Island, VA 23337, [email protected]
Supported with funding from NASA’s Balloon Program Office, a mathematical model is being developed for theanalysis of large scientific balloons. Initially focusing on fully inflated and partially inflated zero-pressure natural-shape balloons, our approach has proven to be adaptable to a variety of problems that lend themselves to avariational formulation with an optimization-based solution process. Our model is implemented into a programcalled EMsolver.
We present a collection of membrane problems for gossamer structures, including natural-shape balloons,pumpkin balloons, a solar neutrino balloon, and an NGST-like sunshield. In each of these applications, thestructure consists of membrane elements and reinforcing tendons. Wrinkling in the membrane is modeled viaA. C. Pipkin’s energy relaxation approach.
Nomenclature
S - complete balloon shape.Ω - complete reference configuration.FI - discretized configuration in IR3
FI - flat unstrained reference configuration in IR2.T - a triangle in FI
T - a triangle in FI.ng - number of gores in a complete shape.rB - bulge radius for the ideal pumpkin gore.ET - total energy of the balloon system.EP
R
Ω
12bz2 p0zk dS - hydrostatic pressure potential energy.
E f R
S wf zdA - gravitational potential energy of the film.
Et ngwt
R Lt
0 τs k ds - gravitational potential energy of the load tendons.Etop wtopztop - gravitational potential of top fitting.St ng
R
Γ W
t εds - relaxed tendon strain energy.Sf
R
ΩW
f dA - relaxed film strain energy.
dS NdS, N is the unit outward normal to S .dS is surface area measure on S .dA - area measure in the reference configuration,ds - arc length measure along Γ,τs IR3 is the position of the load tendon.
Variational Formulation and Optimization-Based SolutionSee Baginski & Collier (2001)
Total potential energy:
E
T S EPE f Et EtopSt Sf
EP - hydrostatic pressure potential (lifting gas)E f - gravitational potential energy of film Sf - relaxed strain energy of film
Et - gravitational potential energy of load tendons St - relaxed strain energy of load tendonsEtop - gravitational potential energy of top fitting S - membrane
Optimization Problem
For S C ,minimize: E
T S
subject to: GS 0
EMsolver uses Matlab’s fmincon to solve a discretization of Problem .
C - class of shapes; G - constraints
Design parameters and related constants for a ULDB Phase IV balloon. Comparing a
standard gore design and a supergore design.
Description ULDBTop fitting weight (N) wtop 831
Cap weight density (N/m2) wc 0.18387Cap thickness (µm) hc 10.8
Film weight density (N/m2) wf 0.3440Film weight with caps (N) 16,562
Film thickness (µm) hf 38.1Youngs modulus (MPa) E 404.2
Poisson ratio ν 0.825Tendon weight density (N/m2) wt 0.094
Tendon weight (N) 4,144Tendon stiffness (MN) Kt 0.651
Payload (N) Lbot 20393Specific buoyancy (N/m3) b 0.0763
Number of gores ng 290Cap length (m) lc 49.45
Constant pressure (Pa) p0 135Target volume (mcm) 0.515
Curved edge length (m) Ld 152.03Center gore length (m) Lc 152.02
Pumpkin generator length (m) Lp 150.02Length of tendon (m) Lt 147.57
Wrinkling and film strain energy relaxation
F. Baginski & W. CollierSee Collier (2000) & Baginski & Collier (2001)
Relaxed membrane strain energy density:
W
f T
0 δ1 0 and δ2 0 (slack)
12tEδ2
2 µ1 0 and δ2 0 (wrinkled)
12tEδ2
1 µ2 0 and δ1 0 ( wrinkled)
tE21ν2
δ21δ2
22νδ1δ2
µ1 0 and µ2 0 (taut)
(1)
Relaxed membrane energy:
Sf ilm Z
ΩW
f dA
Λ1-taut, Λ2-slack, other - wrinkled
δ1δ2-plane W
f - quasi-convexification of Wf
•
•
•• •
↑δ
2=−0.5
δ1=−0.5 →
← δ2 = −δ
1/νδ
2 = −ν δ
1 →
Λ2
Λ1
−2
−1
0
1
2
−2
−1
0
1
20
5
10
15
20
δ1
δ2
Wf*
Reference Configuration Natural state of single gore
.LcL
d . . . . .
Ω1
Ω2
Ωn
g
Lc
Ld
The Pumpkin BalloonF. Baginski & W. Schur
Section of pumpkin gore Pumpkin Balloon
t(s)
b(s)
α(s)
j
← z = 0, −P(0) = p0
←z = ztop
, −P(ztop
) = bztop
+ p0
Averaged principal stresses of natural and pumpkin shapes
(a) σ1s meridional stress (b) σ1s
0 50 100 150
0
10
20
30
40
50
60
70
80
s
MP
a
σ1 − short tendon
σ1 − slack tendon
σ1 − film/tendon match
0 50 100 150
0
10
20
30
40
50
60
70
80
s
MP
a
σ1 − short tendon
σ1 − slack tendon
σ1 − film/tendon match
(c) σ2s hoop stress (d) σ2s
0 50 100 150
0
10
20
30
40
50
60
70
80
s
MP
a
σ2 − short tendon
σ2 − slack tendon
σ2 − film/tendon match
0 50 100 150
0
10
20
30
40
50
60
70
80
s
MP
a
σ2 − short tendon
σ2 − slack tendon
σ2 − film/tendon match
(a)-(b) slack tendons; (c)-(d) short tendons (structural lack-of-fit)
Max. principal stress µ2 vs. bulge radius rB for ng-gores.
F. Baginski & M. Barg
0.7 0.8 0.9 1 1.1 1.2 1.3 1.44
5
6
7
8
9
10
11
12
13
14
rB
MP
an
g=200
ng=220
ng=255
ng=290
ng=350
ng=375
ng=400
“Molded” Pumpkin Supergore
F. Baginski & W. Schur
Molded Pumpkin Supergore Flat unassembled configuration
Containment vessel for a solar neutrino detectorL. Cadonati & F. Baginski
Stainless Steel Water Tank18m ∅
Stainless SteelSphere 13.7m ∅
2200 8" Thorn EMI PMTs
WaterBuffer
100 ton fiducial volume
Borexino Design
PseudocumeneBuffer
Steel Shielding Plates8m x 8m x 10cm and 4m x 4m x 4cm
Scintillator
Nylon Sphere8.5m ∅
Holding Strings
200 outward-pointing PMTs
Muon veto:
Nylon filmRn barrier
Ø13700 mmØ10500 mm
OUTER VESSEL SSS
SUPPORTSTRAP
POLARSUPPORTROD
PIPES
Ø8500 mm
INNER VESSEL
Tendon/gore configuration in Borexino
Top mooring rope tobottom endcap
Bottom mooring rope totop endcap
Top endcap
Bottomendcap
Gore seam
−6 −4 −2 0 2 4 6
−1
0
1
2
3
4
5
6
7
8
9
10
vbot,ec
vbot,m
vtop,m
vi*
vn−i
*
vtop,ec
Mooring ropeto top endcap
Mooring ropeto bottom endcap
Polar support
Pipes
EMsolver and ABACUS Results Comparison
Constant Pressure Buoyant Force + Constant Pressure
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1minimum principal stress sp1 (MPa) ABA at seam
ABA midgoreEM at seamEM midgore
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1maximum principal stress sp2 (MPa)
arc length (m)
0 2 4 6 8 10 120
1
2
3minimum principal stress sp1 (MPa)
ABA at seamABA midgoreEM at seam EM midgore
0 2 4 6 8 10 120
2
4
6
8
10maximum principal stress sp2 (MPa)
arc length (m)
NGST-like sunshieldF. Baginski & S. Vezina
Principal strains
−0.5 0 0.5 1 1.5 2 2.5
−1.5
−1
−0.5
0
0.5
1
(ST 0.2028) (S
F 4.488)
(EG
0)
(E
F −
0.06
3565
) (E
T
0
)
(f−cnt: 1886) (ETOT
0.01596)strn2/1 L:(m),F:(N),E:(J)
−2
−1
0
1
2
3
x 10−3
Initially flat membrane. Corners displaced 2.5 cm out-of-plane; Fixed center box.
Wrinkle Pattern
−0.5 0 0.5
−1.5
−1
−0.5
0
0.5
1
(ST 0.2028) (S
F 4.488)
(EG
0)
(E
F −
0.06
3565
) (E
T
0
)
(f−cnt: 1886) (ETOT
0.01596)Tension Field
Tension Field − blueTaut −red
Corners: 2.5 cm out−of−plane 0.2 mm in−plane Center square fixed
Nonuniqueness of Ascent ShapesBaginski (2002)
References F. Baginski and W. Collier, “Modeling the shapes of constrained partially inflated high altitude balloons,”
AIAA J., Vol. 39, No. 9, September 2001, pp. 1662-1672. Errata: AIAA J., Vol. 40, No. 9, September2002, pp. 1253.
F. Baginski and W. Schur, “Structural Analysis of Pneumatic Envelopes: A Variational Formulation andOptimization-Based Solution Process,” inflated high altitude balloons,” AIAA J., Vol. 41, No. 2, February2003, pp. 304-311.
F. Baginski, “Nonuniqueness of strained ascent shapes of high altitude balloons,” COSPAR02-A-01591, TheSecond World Space Congress/34th COSPAR Scientific Assembly, Houston, Texas 10-19 October 2002.
F. Baginski and J. Winker, “The natural shape balloon and related models,” 34th COSPAR Scientific As-sembly, PSB1-0009-02, Houston, Texas, 10-19 October 2003.
W. Collier, Applications of Variational Principles to Modeling a Partially Inflated Scientific Research Balloon,Ph.D. Thesis, Department of Mathematics, The George Washington University, January 2000.
W. W. Schur, “Analysis of load tape constrained pneumatic envelopes”, 40th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, AIAA-99-1526, St. Louis, MO, April 1999.
A. C. Pipkin, “Relaxed energy densities for large deformations of membranes,” IMA Journal of AppliedMathematics, Vol. 52, 1994, pp. 297-308.
Acknowledgement
F. Baginski, M. Barg, W. Collier and S. Vezina were supported in part by
NASA Awards NAG5-5292 and NAG5-5353.