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Surface precision of optical membranes with curvature

D. K. Marker and C. H. Jenkins

Open Access Open Access

Abstract

Space-based inflatable technology is of current interest to NASA and DOD, and in particular to the Air Force and Phillips Laboratory. Potentially large gains in lowering launch costs, through reductions in structure mass and volume, are driving this activity. Diverse groups are researching and developing this technology for radio and radar antennae, optical telescopes, and solar power and propulsion applications. Regardless of the use, one common requirement for successful application is the accuracy of the inflated surface shape. The work reported here concerns the shape control of an inflated thin circular disk through use of a nonlinear finite element analysis. First, a review of the important associated Hencky problem is given. Then we discuss a shape modification, achieved through enforced boundary displacements, which resulted in moving the inflated shape towards a desired parabolic profile. Minimization of the figure error is discussed and conclusions are drawn.

©1997 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. Definition Sketch

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Fig.2.
Fig.2. Inflated Membrane (3-D view)

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Fig. 3.
Fig. 3. Comparison Measurements Fig.

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Fig. 4.
Fig. 4. Radial Displacements

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Fig. 5.
Fig. 5. Model Vs Parabola

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Equations (19)

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ε r = du dr + 1 2 ( dw dr ) 2
ε θ = u r
D ( d 2 d r 2 + 1 r ) 2 w − h r d dr ( d Φ dr dw dr ) = p
r d dr ( d 2 Φ d r 2 + 1 r d Φ dr ) + E 2 ( dw dr ) 2 = 0
σ r = 1 r d Φ dr , σ θ = d 2 Φ d r 2
k = ( 2 r a p Eh ) 1 3
w = w 0 ( 1 + α 2 r 2 a 2 + α 4 r 4 a 4 )
w 0 = 0.626 a ( pa Eh ) 1 3
M r = D ( d 2 w d r 2 + ν r dw dr )
ε θ = 0 ⇒ σ θ − ν σ r = 0 ⇒ d 2 Φ d r 2 − ν r d Φ dr
α 2 = − 6 + 2 ν 5 + ν
α 4 = 1 + ν 5 + ν
w b = w 0 ( 1 − 1.259 r 2 a 2 + 0.245 r 4 a 4 )
w m = w 0 ( 1 − 0.899 r 2 a 2 − 0.101 r 4 a 4 )
w p = w 0 ( 1 − r 2 a 2 )
k m = Eh 1 − ν 2
ε θ = 0 ⇒ σ θ = ν σ r
ν ( r = a ) = σ θ σ r
k m ( r = 0 ) k m ( r = a ) = 1 − [ v ( a ) ] 2 1 − [ ν ( 0 ) ] 2 > 1
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