Abstract

Several applications, both on the ground and in space, are envisaged for large, light parabolic reflectors which are compact when folded. In this paper the aim is to calculate by a differential method the deformed meridian line and the creased zone of an elastic inflatable revolution membrane. Conversely, one can calculate the original meridian line which, under a given pressure, transforms into a selected meridian line, in this case parabolic. The theory is confirmed by the deformation measurements of a cap in 36-μm thick polyester film cut out of a sphere 4 m in diameter.

© 1981 Optical Society of America

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References

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  1. B. Authier, L. Hill, C. Fayard, “Optical Feasibility of Thin Film Spherical Solar Collectors of Material Science Experiments in Space,” (79-202), 30th International Astronautical Congress (IAF), Munich (1979).
  2. S. Timoshenko, Résistance des matériaux (Librairie Polytechnique, Paris, 1954).
  3. J. Courbon, Résistance des matériaux, tome 1 2 ìeme id. (Dunod, Paris1964), Chap. “Des voiles minces sans flexion.”
  4. B. Carnakan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  5. C. Fayard, “Etude de membranes réfléchissantes gonflables,” these de Docteur-ingénieur, Université de droit, d'économie et des sciences d'Aix-Marseille, Faculté des sciences et techniques de Saint Jérôme, Marseille (Nov.1980).

Authier, B.

B. Authier, L. Hill, C. Fayard, “Optical Feasibility of Thin Film Spherical Solar Collectors of Material Science Experiments in Space,” (79-202), 30th International Astronautical Congress (IAF), Munich (1979).

Carnakan, B.

B. Carnakan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Courbon, J.

J. Courbon, Résistance des matériaux, tome 1 2 ìeme id. (Dunod, Paris1964), Chap. “Des voiles minces sans flexion.”

Fayard, C.

C. Fayard, “Etude de membranes réfléchissantes gonflables,” these de Docteur-ingénieur, Université de droit, d'économie et des sciences d'Aix-Marseille, Faculté des sciences et techniques de Saint Jérôme, Marseille (Nov.1980).

B. Authier, L. Hill, C. Fayard, “Optical Feasibility of Thin Film Spherical Solar Collectors of Material Science Experiments in Space,” (79-202), 30th International Astronautical Congress (IAF), Munich (1979).

Hill, L.

B. Authier, L. Hill, C. Fayard, “Optical Feasibility of Thin Film Spherical Solar Collectors of Material Science Experiments in Space,” (79-202), 30th International Astronautical Congress (IAF), Munich (1979).

Luther, H. A.

B. Carnakan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Timoshenko, S.

S. Timoshenko, Résistance des matériaux (Librairie Polytechnique, Paris, 1954).

Wilkes, J. O.

B. Carnakan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Other (5)

B. Authier, L. Hill, C. Fayard, “Optical Feasibility of Thin Film Spherical Solar Collectors of Material Science Experiments in Space,” (79-202), 30th International Astronautical Congress (IAF), Munich (1979).

S. Timoshenko, Résistance des matériaux (Librairie Polytechnique, Paris, 1954).

J. Courbon, Résistance des matériaux, tome 1 2 ìeme id. (Dunod, Paris1964), Chap. “Des voiles minces sans flexion.”

B. Carnakan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

C. Fayard, “Etude de membranes réfléchissantes gonflables,” these de Docteur-ingénieur, Université de droit, d'économie et des sciences d'Aix-Marseille, Faculté des sciences et techniques de Saint Jérôme, Marseille (Nov.1980).

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Figures (9)

Fig. 1
Fig. 1

Stratospheric balloon during inflation at the CNES center in Toulouse, France.

Fig. 2
Fig. 2

Geometrical definitions: O, apex; A, connecting point on the supporting circle; M, a current point; M′, projection of M on Oz (OM′ = z); OA, meridian curve; r, radius of the parallel passing through M; a, radius of the parallel passing through A; s, OM curve; ϕ, angle of the tangent in M with MM′; α, angle of the tangent in A with AA′ (O < ϕ < α); ρ, radius of curvature in M (ρ = ds/dϕ); N1: component, on the tangent at the parallel of point M, of the tension at this point; N2: component, on the tangent at the meridian in M, of the tension in this point.

Fig. 3
Fig. 3

Example of a pleated zone: At rest the cap S0 is fixed on a ring of fixed radius a0. From a certain pressure, a creased zone, of revolution, appears in the neighborhood of the ring.

Fig. 4
Fig. 4

Alignment chart of the profile of an initially circular meridian line for a value of b equal to 1.2 × 10−2 (for the membrane studied in the experimental application, this value corresponds to 10.8 mbar). Curves 1–6 are plotted for a parameter (R/R0), respectively equal to 0.8, 0.9, 0.95, 0.97, 1, and 1.05. Curve 7 represents the limit between the creased (broken line) and stretched (continuous line) zones. The points on the curves are 5° graduations in ϕ0.

Fig. 5
Fig. 5

The network of curves represents the meridian lines of the revolution membranes which transform into a paraboloid whose axis is that of the ordinates. The unit of length is equal to 2f, and the parameter b = (2pf/Eh) is successively equal to 0, 0.02, 0.04, 0.06, 0.08, 0.10, 0.5, and 1.0 for the curves marked 1 to 8. The curves are graduated as a function of the angle ϕ with a step of 5°.

Fig. 6
Fig. 6

The connecting strips of the gores (four white lines) and the loops with their attachment springs at the circumference of the cylinder can be seen in this photograph of the spherical cap (1.88-m aperture diameter). The broken lines A and B represent the meridian lines whose profiles have been measured.

Fig. 7
Fig. 7

The meridian line studied, of which the portion ij is observed, is situated in the plane defined by the axis OZ of the mirror and the laser scanning direction Fx (direction of optical bench). The semitransparent plate L is oriented by rotation around the axis Fy until the image of the reflected beam NMS is centered on the laser source S.

Fig. 8
Fig. 8

Angular values ϕ of the tangents along a meridian line as a function of the distance at the axis. Curves 1, 2, and 3 represent the values calculated for initial aperture semiangles of 25°, 27°, and 29°, respectively, whereas curves 2A and 2B represent the values measured on meridian lines A and B of the experimental cap with initial aperture semiangle of 27°. Pressure is 17.5 mbar. To estimate the displacements better, we have subtracted from angular values ϕA, ϕB, and ϕth the value which concerns the tangent to the reflected point on the meridian circle of the original cap.

Fig. 9
Fig. 9

Theoretical and measured meridian profiles. From the ordinate z of a current point we have deducted the value z0 of the corresponding point of the meridian circle of the original cap of 27° aperture semiangle. r is the distance at the axis and the pressure is 17.5 mbar. Curves 2A and 2B relate to the meridian lines A and B, whereas curves 1, 2, and 3 are the theoretical profiles which concern the deformed meridian lines of the cap, respectively, of 25°, 27°, and 29° aperture semiangle.

Equations (45)

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d r = ρ cos ϕ d ϕ ,
d z = ρ sin ϕ d ϕ .
e 1 = r r 0 1 ,
e 2 = d s d s 0 1 or e 2 = ρ ρ 0 ϕ 1 ,
N 1 sin ϕ r + N 2 ρ = P , N 2 = P r 2 sin ϕ .
e 1 = 1 E h ( N 1 ν N 2 ) , e 2 = 1 E h ( N 2 ν N 1 ) ,
e 1 ν e 2 , e 2 = N 2 E h , and N 1 = 0 .
N 1 = N 1 E h N 2 = N 2 E h b = P R 0 E h .
d r = ρ cos ϕ d ϕ ,
d z = ρ sin ϕ d ϕ ,
e 1 = r r 0 1 ,
e 2 = ρ ρ 0 ϕ 1 ,
N 1 sin ϕ r + N 2 ρ = b ,
N 2 = r b 2 sin ϕ .
e 1 = N 1 ν N 2
e 1 ν e 2
e 2 = N 2 ν N 1
e 2 = N 2
N 1 > 0
N 1 = 0.
( E ) { ϕ = F ( r , ϕ , r 0 , ρ 0 , b ) r = G ( r , ϕ , r 0 , ρ 0 , b ) 0 < ϕ 0 < α 0 .
N 2 = r b 2 sin ϕ , e 1 = r r 0 1 , N 1 = e 1 + ν N 2 , e 2 = N 2 ν N 1 , 1 ρ = b N 2 ( 1 N 1 2 N 2 ) ,
ϕ = ρ 0 ρ ( e 2 + 1 ) , r = ρ 0 cos ϕ ( e 2 + 1 ) ,
z = ρ sin ϕ ϕ = ρ 0 sin ϕ ( e 2 + 1 ) .
r sin ϕ ϕ 0 0 R and r r 0 ϕ 0 0 R ϕ .
N 1 = N 2 = b R 2 and e 1 = e 2 = R ϕ 1 = b R ( 1 ν ) / 2 .
{ ϕ ( 0 ) = b R ( 1 ν ) / 2 + 1 R r ( 0 ) = R ϕ ( 0 ) .
z = z 1 + C 1 2 ϕ B ϕ A ( sin ϕ ) 1 / 2 d ϕ .
ϕ = ρ 0 [ b + 2 ( sin ϕ ) 1 / 2 C 1 ] ,
C 1 ( sin ϕ ) 1 / 2 r 0 1 ν C 1 b 2 1 ( sin ϕ ) 1 / 2 .
R S = 1 1 b [ ( 1 ν ) / 2 ] ,
N 1 = N 1 E h , N 2 = N 2 E h , b = P R E h
N 2 = r b 2 sin ϕ ,
N 1 = r b 2 sin ϕ ( 2 r ρ sin ϕ ) ,
e 1 = r b 2 sin ϕ ( 2 ν r ρ sin ϕ ) ,
e 2 = r b 2 sin ϕ ( 1 2 ν + ν r ρ sin ϕ ) .
r 0 = r e 1 + 1 ,
ρ 0 d ϕ 0 = ρ d ϕ e 2 + 1 .
r = sin ϕ cos ϕ , ρ = 1 cos 3 ϕ ,
r ρ sin ϕ = cos 2 ϕ .
N 2 = b 2 cos ϕ , N 1 = b 2 cos 2 ϕ 2 cos ϕ .
e 1 = b 2 ν cos 2 ϕ 2 cos ϕ , e 2 = b 1 2 ν + ν cos 2 ϕ 2 cos ϕ .
e 1 e 2 = b ( 1 + ν ) sin 2 ϕ cos ϕ 0 ,
r 0 = sin ϕ cos ϕ + b 2 ν cos 2 ϕ 2 ,
ρ 0 d ϕ 0 = d ϕ cos 2 ϕ ( cos ϕ + b 1 2 ν + ν cos 2 ϕ 2 ) , cos ϕ 0 = d r 0 ρ 0 d ϕ 0 = cos 2 ϕ ( cos ϕ + b 1 2 ν + ν cos 2 ϕ 2 ) ( 1 + b cos ϕ cos 2 ϕ ν 2 ) ( cos ϕ + b 2 ν cos 2 ϕ 2 ) 2

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