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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  1. C.H. Jenkins and J.W. Leonard, “Nonlinear Dynamic Response of Membranes: State of the Art,” Appl. Mech. Rev. 44, 319–328 (1991).
    [CrossRef]
  2. C.H. Jenkins, “Nonlinear Dynamic Response of Membranes: State of the Art – Update,” Appl. Mech. Rev. 49 (10), S41–S48 (1996).
    [CrossRef]
  3. H. Hencky, “Uber den Spannungszustand in kreisrunden Platten,” Z. Math. Phys. 63, 311–317 (1915).
  4. A. Föppl, “Vorlesungen uber technische Mechanik,” B.G. Teubner, Bd. 5., p. 132, Leipzig, Germany (1907).
  5. T. von Kármán, “Festigkeitsproblem im Naschinenbau,” Encyk. D. Math. Wiss. IV, 311–385 (1910).
  6. H.H. Stevens, “Behavior of circular membranes stretched above the elastic limit by air pressure,” Exp. Stress Anal. 2, 139–146 (1944).
  7. W.Z. Chien, “Asymptotic behavior of a thin clamped plate under uniform normal pressure at very large deflection,” Sci. Rep. Natn. Tsing Hua Univ. A5, 71–94 (1948).
  8. R. Kao and N. Perrone, “Large deflections of axisymmetric circular membranes,” Int. J. Solids Struct. 7, 1601–1612 (1971).
    [CrossRef]
  9. J.D. Cambell, “On the theory of initially tensioned circular membranes subjected to uniform pressure,” Q. J. Mech. Appl. Math. 9, 84–93 (1956).
    [CrossRef]
  10. R.W. Dickey, “The plane circular elastic surface under normal pressure,” Arch. Ration. Mech. Anal. 26, 219–236 (1967).
    [CrossRef]
  11. N.A. Weil and N.M. Newmark, “Large plastic deformations of circular membranes,” J. Appl. Mech. 22, 533–538 (1955).
  12. R. Kao and N. Perrone, “Large deflections of flat arbitrary membranes,” Comput. Struct. 2, 535–546 (1972).
    [CrossRef]
  13. F.S. Shaw and N. Perrone, “A numerical solution for the non-linear deflection of membranes,” J. Appl. Mech. 21, 117–128 (1954).
  14. R. Schmidt and D.A. DaDeppo, “A new approach to the analysis of shells, plates, and membranes with finite deflections,” Int. J. Non-Linear Mech. 9, 409–419 (1974).
    [CrossRef]
  15. R. Schmidt, , “On Berger’s method in the non-linear theory of plates,” J. Appl. Mech. 41, 521–523 (1974).
    [CrossRef]
  16. B. Storakers, “Small deflections of linear elastic circular membranes under lateral pressure,” J. Appl. Mech. 50, 735–739 (1983).
    [CrossRef]
  17. H.J. Weinitschke, “On axisymmetric deformations of nonlinear elastic membranes,” Mech. Today 5, 523–542, Pergamon Press, Oxford (1980).
  18. H.J. Weinitschke, “On finite displacements of circular elastic membranes,” Math. Method Appl. Sci. 9, 76–98 (1987).
    [CrossRef]
  19. H.J. Weinitschke, “Stable and unstable axisymmetric solutions for membranes of revolution,” Appl. Mech. Rev. 42, S289–S294 (1989).
    [CrossRef]
  20. P.G. Ciarlet, “A justification of the von Kármán equations,” Arch. Ration. Mech. Anal. 73, 349–389 (1980).
    [CrossRef]
  21. Pujara and T.J. Lardner, “Deformations of elastic membranes--effect of different constitutive relations,” Z. Angew. Math. Phys. 29, 315–327 (1978).
    [CrossRef]
  22. M. Thomas and G. Veal, (1984), “Highly accurate inflatable reflectors,” AFRPL TR-84-021.
  23. L.M. Murphy, “Stretched-membrane heliostat technology,” J. Solar Energy Eng. 108, 230–238 (1986).
    [CrossRef]
  24. L.M. Murphy, “Moderate axisymmetric deformations of optical membrane surfaces,” J. Solar Energy Eng. 109, 111–120 (1987).
    [CrossRef]
  25. A. Palisoc and M. Thomas, “A comparison of the performance of seamed and unseamed inflatable concentrators,” Solar Engineering 1995, Proc. 1995 ASME/JSME/JSES Int. Solar Energy Conf.,  2, 855–864 (1995).
  26. J. P. Basart, S.A. Mandayam, and J.O. Burns, “An inflatable antenna for space-based low-frequency radio astronomy,” Proc. Space ’94: Engineering, Construction, and Operations in Space IV - Vol. 2, Albuquerque, NM (1994).
  27. C. Cassapakis and M. Thomas, “Inflatable structures technology development overview,” AIAA95–3738 (1995).
  28. P.Y. Chow, “Construction of pressurized, self-supporting membrane structure on the moon,” J. Aerospace Eng. 5, 274–281 (1992).
    [CrossRef]
  29. G. Grossman and G. Williams, “Inflatable concentrators for solar propulsion and dynamic space power,” J. Solar Energy Eng. 112, 229–236 (1990).
    [CrossRef]
  30. P.K. Malone and G.T. Williams, “A lightweight inflatable solar array,” Proc. 9th Annual AIAA/USU Conference on Small Satellites, Logan, UT (1995).
  31. P.S. Nowak, W.Z. Sadeh, and J. Janakus, “Feasibility study of inflatable structures for lunar base,” J. Spacecraft Rockets 31, 453–457 (1994).
    [CrossRef]
  32. C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
    [CrossRef]
  33. W.Z. Sadeh and M.E. Criswell, “A generic inflatable structure for a lunar/Martian base,” Space IV, Proc. Space ’94, Albuquerque, ASCE, 1146–1156 (1994).
  34. J.M. Hedgepeth, “Accuracy potentials for large space antenna reflectors with passive structures,” J. Spacecraft 19(3), 211–217 (1982).
    [CrossRef]
  35. H. Vaughn, “Pressurizing a prestretched membrane to form a paraboloid,” Int. J. Eng. Sci. 18, 99–107 (1980).
    [CrossRef]
  36. L.J. Hart-Smith and J.D.C. Crisp, “Large elastic deformations of thin rubber membranes,” Int. J. Eng. Sci. 5, 1–24 (1967).
    [CrossRef]
  37. M. Natori, Y. Shibayama, and K. Sekine, “Active accuracy adjustment of reflectors through the change of element boundary,” AIAA89–1332 (1989).
  38. C.H. Jenkins, D.K. Marker, and J.M. Wilkes, “Improved surface accuracy of precision membrane reflectors through adaptive rim control,” AIAA Adaptive Structures Forum, Long Beach, CA (to appear) (1998a).
  39. C.H. Jenkins, J.M. Wilkes, and D.K. Marker, “Surface accuracy of precision membrane reflectors,” Space 98, Albuquerque, NM (to appear) (1998b).
    [CrossRef]

1996 (1)

C.H. Jenkins, “Nonlinear Dynamic Response of Membranes: State of the Art – Update,” Appl. Mech. Rev. 49 (10), S41–S48 (1996).
[CrossRef]

1995 (3)

A. Palisoc and M. Thomas, “A comparison of the performance of seamed and unseamed inflatable concentrators,” Solar Engineering 1995, Proc. 1995 ASME/JSME/JSES Int. Solar Energy Conf.,  2, 855–864 (1995).

C. Cassapakis and M. Thomas, “Inflatable structures technology development overview,” AIAA95–3738 (1995).

P.K. Malone and G.T. Williams, “A lightweight inflatable solar array,” Proc. 9th Annual AIAA/USU Conference on Small Satellites, Logan, UT (1995).

1994 (2)

P.S. Nowak, W.Z. Sadeh, and J. Janakus, “Feasibility study of inflatable structures for lunar base,” J. Spacecraft Rockets 31, 453–457 (1994).
[CrossRef]

W.Z. Sadeh and M.E. Criswell, “A generic inflatable structure for a lunar/Martian base,” Space IV, Proc. Space ’94, Albuquerque, ASCE, 1146–1156 (1994).

1993 (1)

C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
[CrossRef]

1992 (1)

P.Y. Chow, “Construction of pressurized, self-supporting membrane structure on the moon,” J. Aerospace Eng. 5, 274–281 (1992).
[CrossRef]

1991 (1)

C.H. Jenkins and J.W. Leonard, “Nonlinear Dynamic Response of Membranes: State of the Art,” Appl. Mech. Rev. 44, 319–328 (1991).
[CrossRef]

1990 (1)

G. Grossman and G. Williams, “Inflatable concentrators for solar propulsion and dynamic space power,” J. Solar Energy Eng. 112, 229–236 (1990).
[CrossRef]

1989 (2)

H.J. Weinitschke, “Stable and unstable axisymmetric solutions for membranes of revolution,” Appl. Mech. Rev. 42, S289–S294 (1989).
[CrossRef]

M. Natori, Y. Shibayama, and K. Sekine, “Active accuracy adjustment of reflectors through the change of element boundary,” AIAA89–1332 (1989).

1987 (2)

L.M. Murphy, “Moderate axisymmetric deformations of optical membrane surfaces,” J. Solar Energy Eng. 109, 111–120 (1987).
[CrossRef]

H.J. Weinitschke, “On finite displacements of circular elastic membranes,” Math. Method Appl. Sci. 9, 76–98 (1987).
[CrossRef]

1986 (1)

L.M. Murphy, “Stretched-membrane heliostat technology,” J. Solar Energy Eng. 108, 230–238 (1986).
[CrossRef]

1983 (1)

B. Storakers, “Small deflections of linear elastic circular membranes under lateral pressure,” J. Appl. Mech. 50, 735–739 (1983).
[CrossRef]

1982 (1)

J.M. Hedgepeth, “Accuracy potentials for large space antenna reflectors with passive structures,” J. Spacecraft 19(3), 211–217 (1982).
[CrossRef]

1980 (3)

H. Vaughn, “Pressurizing a prestretched membrane to form a paraboloid,” Int. J. Eng. Sci. 18, 99–107 (1980).
[CrossRef]

P.G. Ciarlet, “A justification of the von Kármán equations,” Arch. Ration. Mech. Anal. 73, 349–389 (1980).
[CrossRef]

H.J. Weinitschke, “On axisymmetric deformations of nonlinear elastic membranes,” Mech. Today 5, 523–542, Pergamon Press, Oxford (1980).

1978 (1)

Pujara and T.J. Lardner, “Deformations of elastic membranes--effect of different constitutive relations,” Z. Angew. Math. Phys. 29, 315–327 (1978).
[CrossRef]

1974 (2)

R. Schmidt and D.A. DaDeppo, “A new approach to the analysis of shells, plates, and membranes with finite deflections,” Int. J. Non-Linear Mech. 9, 409–419 (1974).
[CrossRef]

R. Schmidt, , “On Berger’s method in the non-linear theory of plates,” J. Appl. Mech. 41, 521–523 (1974).
[CrossRef]

1972 (1)

R. Kao and N. Perrone, “Large deflections of flat arbitrary membranes,” Comput. Struct. 2, 535–546 (1972).
[CrossRef]

1971 (1)

R. Kao and N. Perrone, “Large deflections of axisymmetric circular membranes,” Int. J. Solids Struct. 7, 1601–1612 (1971).
[CrossRef]

1967 (2)

R.W. Dickey, “The plane circular elastic surface under normal pressure,” Arch. Ration. Mech. Anal. 26, 219–236 (1967).
[CrossRef]

L.J. Hart-Smith and J.D.C. Crisp, “Large elastic deformations of thin rubber membranes,” Int. J. Eng. Sci. 5, 1–24 (1967).
[CrossRef]

1956 (1)

J.D. Cambell, “On the theory of initially tensioned circular membranes subjected to uniform pressure,” Q. J. Mech. Appl. Math. 9, 84–93 (1956).
[CrossRef]

1955 (1)

N.A. Weil and N.M. Newmark, “Large plastic deformations of circular membranes,” J. Appl. Mech. 22, 533–538 (1955).

1954 (1)

F.S. Shaw and N. Perrone, “A numerical solution for the non-linear deflection of membranes,” J. Appl. Mech. 21, 117–128 (1954).

1948 (1)

W.Z. Chien, “Asymptotic behavior of a thin clamped plate under uniform normal pressure at very large deflection,” Sci. Rep. Natn. Tsing Hua Univ. A5, 71–94 (1948).

1944 (1)

H.H. Stevens, “Behavior of circular membranes stretched above the elastic limit by air pressure,” Exp. Stress Anal. 2, 139–146 (1944).

1915 (1)

H. Hencky, “Uber den Spannungszustand in kreisrunden Platten,” Z. Math. Phys. 63, 311–317 (1915).

1910 (1)

T. von Kármán, “Festigkeitsproblem im Naschinenbau,” Encyk. D. Math. Wiss. IV, 311–385 (1910).

Basart, J. P.

J. P. Basart, S.A. Mandayam, and J.O. Burns, “An inflatable antenna for space-based low-frequency radio astronomy,” Proc. Space ’94: Engineering, Construction, and Operations in Space IV - Vol. 2, Albuquerque, NM (1994).

Burns, J.O.

J. P. Basart, S.A. Mandayam, and J.O. Burns, “An inflatable antenna for space-based low-frequency radio astronomy,” Proc. Space ’94: Engineering, Construction, and Operations in Space IV - Vol. 2, Albuquerque, NM (1994).

Cambell, J.D.

J.D. Cambell, “On the theory of initially tensioned circular membranes subjected to uniform pressure,” Q. J. Mech. Appl. Math. 9, 84–93 (1956).
[CrossRef]

Campbell, T.G.

C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
[CrossRef]

Cassapakis, C.

C. Cassapakis and M. Thomas, “Inflatable structures technology development overview,” AIAA95–3738 (1995).

Chien, W.Z.

W.Z. Chien, “Asymptotic behavior of a thin clamped plate under uniform normal pressure at very large deflection,” Sci. Rep. Natn. Tsing Hua Univ. A5, 71–94 (1948).

Chow, P.Y.

P.Y. Chow, “Construction of pressurized, self-supporting membrane structure on the moon,” J. Aerospace Eng. 5, 274–281 (1992).
[CrossRef]

Ciarlet, P.G.

P.G. Ciarlet, “A justification of the von Kármán equations,” Arch. Ration. Mech. Anal. 73, 349–389 (1980).
[CrossRef]

Crisp, J.D.C.

L.J. Hart-Smith and J.D.C. Crisp, “Large elastic deformations of thin rubber membranes,” Int. J. Eng. Sci. 5, 1–24 (1967).
[CrossRef]

Criswell, M.E.

W.Z. Sadeh and M.E. Criswell, “A generic inflatable structure for a lunar/Martian base,” Space IV, Proc. Space ’94, Albuquerque, ASCE, 1146–1156 (1994).

DaDeppo, D.A.

R. Schmidt and D.A. DaDeppo, “A new approach to the analysis of shells, plates, and membranes with finite deflections,” Int. J. Non-Linear Mech. 9, 409–419 (1974).
[CrossRef]

Dickey, R.W.

R.W. Dickey, “The plane circular elastic surface under normal pressure,” Arch. Ration. Mech. Anal. 26, 219–236 (1967).
[CrossRef]

Föppl, A.

A. Föppl, “Vorlesungen uber technische Mechanik,” B.G. Teubner, Bd. 5., p. 132, Leipzig, Germany (1907).

Grossman, G.

G. Grossman and G. Williams, “Inflatable concentrators for solar propulsion and dynamic space power,” J. Solar Energy Eng. 112, 229–236 (1990).
[CrossRef]

Hart-Smith, L.J.

L.J. Hart-Smith and J.D.C. Crisp, “Large elastic deformations of thin rubber membranes,” Int. J. Eng. Sci. 5, 1–24 (1967).
[CrossRef]

Hedgepeth, J.M.

C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
[CrossRef]

J.M. Hedgepeth, “Accuracy potentials for large space antenna reflectors with passive structures,” J. Spacecraft 19(3), 211–217 (1982).
[CrossRef]

Hencky, H.

H. Hencky, “Uber den Spannungszustand in kreisrunden Platten,” Z. Math. Phys. 63, 311–317 (1915).

Janakus, J.

P.S. Nowak, W.Z. Sadeh, and J. Janakus, “Feasibility study of inflatable structures for lunar base,” J. Spacecraft Rockets 31, 453–457 (1994).
[CrossRef]

Jenkins, C.H.

C.H. Jenkins, “Nonlinear Dynamic Response of Membranes: State of the Art – Update,” Appl. Mech. Rev. 49 (10), S41–S48 (1996).
[CrossRef]

C.H. Jenkins and J.W. Leonard, “Nonlinear Dynamic Response of Membranes: State of the Art,” Appl. Mech. Rev. 44, 319–328 (1991).
[CrossRef]

C.H. Jenkins, D.K. Marker, and J.M. Wilkes, “Improved surface accuracy of precision membrane reflectors through adaptive rim control,” AIAA Adaptive Structures Forum, Long Beach, CA (to appear) (1998a).

C.H. Jenkins, J.M. Wilkes, and D.K. Marker, “Surface accuracy of precision membrane reflectors,” Space 98, Albuquerque, NM (to appear) (1998b).
[CrossRef]

Kao, R.

R. Kao and N. Perrone, “Large deflections of flat arbitrary membranes,” Comput. Struct. 2, 535–546 (1972).
[CrossRef]

R. Kao and N. Perrone, “Large deflections of axisymmetric circular membranes,” Int. J. Solids Struct. 7, 1601–1612 (1971).
[CrossRef]

Lardner, T.J.

Pujara and T.J. Lardner, “Deformations of elastic membranes--effect of different constitutive relations,” Z. Angew. Math. Phys. 29, 315–327 (1978).
[CrossRef]

Leonard, J.W.

C.H. Jenkins and J.W. Leonard, “Nonlinear Dynamic Response of Membranes: State of the Art,” Appl. Mech. Rev. 44, 319–328 (1991).
[CrossRef]

Malone, P.K.

P.K. Malone and G.T. Williams, “A lightweight inflatable solar array,” Proc. 9th Annual AIAA/USU Conference on Small Satellites, Logan, UT (1995).

Mandayam, S.A.

J. P. Basart, S.A. Mandayam, and J.O. Burns, “An inflatable antenna for space-based low-frequency radio astronomy,” Proc. Space ’94: Engineering, Construction, and Operations in Space IV - Vol. 2, Albuquerque, NM (1994).

Marker, D.K.

C.H. Jenkins, D.K. Marker, and J.M. Wilkes, “Improved surface accuracy of precision membrane reflectors through adaptive rim control,” AIAA Adaptive Structures Forum, Long Beach, CA (to appear) (1998a).

C.H. Jenkins, J.M. Wilkes, and D.K. Marker, “Surface accuracy of precision membrane reflectors,” Space 98, Albuquerque, NM (to appear) (1998b).
[CrossRef]

Murphy, L.M.

L.M. Murphy, “Moderate axisymmetric deformations of optical membrane surfaces,” J. Solar Energy Eng. 109, 111–120 (1987).
[CrossRef]

L.M. Murphy, “Stretched-membrane heliostat technology,” J. Solar Energy Eng. 108, 230–238 (1986).
[CrossRef]

Natori, M.

M. Natori, Y. Shibayama, and K. Sekine, “Active accuracy adjustment of reflectors through the change of element boundary,” AIAA89–1332 (1989).

Newmark, N.M.

N.A. Weil and N.M. Newmark, “Large plastic deformations of circular membranes,” J. Appl. Mech. 22, 533–538 (1955).

Nowak, P.S.

P.S. Nowak, W.Z. Sadeh, and J. Janakus, “Feasibility study of inflatable structures for lunar base,” J. Spacecraft Rockets 31, 453–457 (1994).
[CrossRef]

Palisoc, A.

A. Palisoc and M. Thomas, “A comparison of the performance of seamed and unseamed inflatable concentrators,” Solar Engineering 1995, Proc. 1995 ASME/JSME/JSES Int. Solar Energy Conf.,  2, 855–864 (1995).

Perrone, N.

R. Kao and N. Perrone, “Large deflections of flat arbitrary membranes,” Comput. Struct. 2, 535–546 (1972).
[CrossRef]

R. Kao and N. Perrone, “Large deflections of axisymmetric circular membranes,” Int. J. Solids Struct. 7, 1601–1612 (1971).
[CrossRef]

F.S. Shaw and N. Perrone, “A numerical solution for the non-linear deflection of membranes,” J. Appl. Mech. 21, 117–128 (1954).

Pujara,

Pujara and T.J. Lardner, “Deformations of elastic membranes--effect of different constitutive relations,” Z. Angew. Math. Phys. 29, 315–327 (1978).
[CrossRef]

Rogers, C.A.

C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
[CrossRef]

Sadeh, W.Z.

P.S. Nowak, W.Z. Sadeh, and J. Janakus, “Feasibility study of inflatable structures for lunar base,” J. Spacecraft Rockets 31, 453–457 (1994).
[CrossRef]

W.Z. Sadeh and M.E. Criswell, “A generic inflatable structure for a lunar/Martian base,” Space IV, Proc. Space ’94, Albuquerque, ASCE, 1146–1156 (1994).

Schmidt, R.

R. Schmidt, , “On Berger’s method in the non-linear theory of plates,” J. Appl. Mech. 41, 521–523 (1974).
[CrossRef]

R. Schmidt and D.A. DaDeppo, “A new approach to the analysis of shells, plates, and membranes with finite deflections,” Int. J. Non-Linear Mech. 9, 409–419 (1974).
[CrossRef]

Sekine, K.

M. Natori, Y. Shibayama, and K. Sekine, “Active accuracy adjustment of reflectors through the change of element boundary,” AIAA89–1332 (1989).

Shaw, F.S.

F.S. Shaw and N. Perrone, “A numerical solution for the non-linear deflection of membranes,” J. Appl. Mech. 21, 117–128 (1954).

Shibayama, Y.

M. Natori, Y. Shibayama, and K. Sekine, “Active accuracy adjustment of reflectors through the change of element boundary,” AIAA89–1332 (1989).

Stevens, H.H.

H.H. Stevens, “Behavior of circular membranes stretched above the elastic limit by air pressure,” Exp. Stress Anal. 2, 139–146 (1944).

Storakers, B.

B. Storakers, “Small deflections of linear elastic circular membranes under lateral pressure,” J. Appl. Mech. 50, 735–739 (1983).
[CrossRef]

Stultzman, W.L.

C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
[CrossRef]

Thomas, M.

A. Palisoc and M. Thomas, “A comparison of the performance of seamed and unseamed inflatable concentrators,” Solar Engineering 1995, Proc. 1995 ASME/JSME/JSES Int. Solar Energy Conf.,  2, 855–864 (1995).

C. Cassapakis and M. Thomas, “Inflatable structures technology development overview,” AIAA95–3738 (1995).

M. Thomas and G. Veal, (1984), “Highly accurate inflatable reflectors,” AFRPL TR-84-021.

Vaughn, H.

H. Vaughn, “Pressurizing a prestretched membrane to form a paraboloid,” Int. J. Eng. Sci. 18, 99–107 (1980).
[CrossRef]

Veal, G.

M. Thomas and G. Veal, (1984), “Highly accurate inflatable reflectors,” AFRPL TR-84-021.

von Kármán, T.

T. von Kármán, “Festigkeitsproblem im Naschinenbau,” Encyk. D. Math. Wiss. IV, 311–385 (1910).

Weil, N.A.

N.A. Weil and N.M. Newmark, “Large plastic deformations of circular membranes,” J. Appl. Mech. 22, 533–538 (1955).

Weinitschke, H.J.

H.J. Weinitschke, “Stable and unstable axisymmetric solutions for membranes of revolution,” Appl. Mech. Rev. 42, S289–S294 (1989).
[CrossRef]

H.J. Weinitschke, “On finite displacements of circular elastic membranes,” Math. Method Appl. Sci. 9, 76–98 (1987).
[CrossRef]

H.J. Weinitschke, “On axisymmetric deformations of nonlinear elastic membranes,” Mech. Today 5, 523–542, Pergamon Press, Oxford (1980).

Wilkes, J.M.

C.H. Jenkins, D.K. Marker, and J.M. Wilkes, “Improved surface accuracy of precision membrane reflectors through adaptive rim control,” AIAA Adaptive Structures Forum, Long Beach, CA (to appear) (1998a).

C.H. Jenkins, J.M. Wilkes, and D.K. Marker, “Surface accuracy of precision membrane reflectors,” Space 98, Albuquerque, NM (to appear) (1998b).
[CrossRef]

Williams, G.

G. Grossman and G. Williams, “Inflatable concentrators for solar propulsion and dynamic space power,” J. Solar Energy Eng. 112, 229–236 (1990).
[CrossRef]

Williams, G.T.

P.K. Malone and G.T. Williams, “A lightweight inflatable solar array,” Proc. 9th Annual AIAA/USU Conference on Small Satellites, Logan, UT (1995).

AIAA (2)

C. Cassapakis and M. Thomas, “Inflatable structures technology development overview,” AIAA95–3738 (1995).

M. Natori, Y. Shibayama, and K. Sekine, “Active accuracy adjustment of reflectors through the change of element boundary,” AIAA89–1332 (1989).

Appl. Mech. Rev. (3)

H.J. Weinitschke, “Stable and unstable axisymmetric solutions for membranes of revolution,” Appl. Mech. Rev. 42, S289–S294 (1989).
[CrossRef]

C.H. Jenkins and J.W. Leonard, “Nonlinear Dynamic Response of Membranes: State of the Art,” Appl. Mech. Rev. 44, 319–328 (1991).
[CrossRef]

C.H. Jenkins, “Nonlinear Dynamic Response of Membranes: State of the Art – Update,” Appl. Mech. Rev. 49 (10), S41–S48 (1996).
[CrossRef]

Arch. Ration. Mech. Anal. (2)

R.W. Dickey, “The plane circular elastic surface under normal pressure,” Arch. Ration. Mech. Anal. 26, 219–236 (1967).
[CrossRef]

P.G. Ciarlet, “A justification of the von Kármán equations,” Arch. Ration. Mech. Anal. 73, 349–389 (1980).
[CrossRef]

Comput. Struct. (1)

R. Kao and N. Perrone, “Large deflections of flat arbitrary membranes,” Comput. Struct. 2, 535–546 (1972).
[CrossRef]

Encyk. D. Math. Wiss. (1)

T. von Kármán, “Festigkeitsproblem im Naschinenbau,” Encyk. D. Math. Wiss. IV, 311–385 (1910).

Exp. Stress Anal. (1)

H.H. Stevens, “Behavior of circular membranes stretched above the elastic limit by air pressure,” Exp. Stress Anal. 2, 139–146 (1944).

Int. J. Eng. Sci. (2)

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[CrossRef]

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[CrossRef]

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[CrossRef]

Int. J. Solids Struct. (1)

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[CrossRef]

J. Aerospace Eng. (1)

P.Y. Chow, “Construction of pressurized, self-supporting membrane structure on the moon,” J. Aerospace Eng. 5, 274–281 (1992).
[CrossRef]

J. Aerospace Engineering (1)

C.A. Rogers, W.L. Stultzman, T.G. Campbell, and J.M. Hedgepeth, “Technology assessment and development of large deployment antennas,” J. Aerospace Engineering 6(1), 34–54 (1993).
[CrossRef]

J. Appl. Mech. (4)

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Z. Math. Phys. (1)

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C.H. Jenkins, J.M. Wilkes, and D.K. Marker, “Surface accuracy of precision membrane reflectors,” Space 98, Albuquerque, NM (to appear) (1998b).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Definition Sketch

Fig.2.
Fig.2.

Inflated Membrane (3-D view)

Fig. 3.
Fig. 3.

Comparison Measurements Fig.

Fig. 4.
Fig. 4.

Radial Displacements

Fig. 5.
Fig. 5.

Model Vs Parabola

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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