Abstract

A method is proposed whereby the spatial speckles created in front of a plate with an optically rough surface when illuminated by a coherent laser beam is used to generate slope contour fringes. This is done by photographing the speckles contained in a parallel plane in front of the plate before and after deformation via double exposure. The resulting speckle interferogram is then optically Fourier transformed to yield the fringe pattern of slope contours. It is shown that the method is analogous to the Ligtenberg reflection moiré method with a grating of continuously variable pitch and orientation. The method can be applied to plates made of almost any material.

© 1976 Optical Society of America

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References

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  1. P. S. Theocaris, “Moiré Method in Plates,” in Proc. Int. Assoc. Shell Struct. Symposium, Warsaw (North Holland, Amsterdam, 1963), pp. 877–889.
  2. F. P. Chiang, B. Ranganayakamma, Exp. Mech. No. 7, 296 (July1971).
    [CrossRef]
  3. J. Havanesian, J. Varner, “Method for Determining the Bending Moments in Normal Loaded Thin Plates by Holographic Interferometry,” in Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge U.P, London, 1970), pp. 173–185.
  4. F. K. Lightenberg, Proc. Soc. Exp. Stress. Anal. 12, No. 2, 83 (1955).
  5. J. P. Duncan, C. J. E. Brown, “Slope Contours in Flexed Elastic Plates by Salet-Ikeda Technique,” in Proceedings First International Congress Experimental Mechanics (Pergamon, London, 1963), pp. 149–176.
  6. F. P. Chiang, T. Y. Kao, “An Optical Method for Generating Slope and Curvature Contours of Bent Plates,” in Proceedings of the Joint Meeting of German Optical Society (DGaO) and Belgium Optical Committee, Brugge, Belgium, 4–8 June 1975, p. 58 (abstract).
  7. J. P. Duncan, P. G. Sabin, Exp. Mech. 5, No. 1, 22 (Jan.1965).
    [CrossRef]
  8. J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, No. 2, 101 (1968).
  9. E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
    [CrossRef]
  10. E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
    [CrossRef]
  11. D. E. Duffy, Exp. Mech. 14, 378 (1974).
    [CrossRef]
  12. Y. Y. Hung, C. P. Hu, C. E. Taylor, “Speckle-Moiré Interferometry—A Tool for Complete Measurement of In-plane Surface Displacements,” in Proc. 7th Southeastern Conference on Theoretical and Applied Mechanics, 1974, pp. 497–505.
  13. F. P. Chiang, “Multi-aperture Speckle Interferometry,” in Proc. Conference on Speckle Phenomena and Their Applications, Loughborough University of Technology, 27–28 March 1974, pp. 22–24.
  14. R. P. Khetan, F. P. Chiang, “Strain Analysis by One Beam Laser Speckle Interferometry, Part I, Single Aperture Method,” to appear in Appl. Opt.
  15. H. J. Tiziani, Opt. Commun. 5, 271 (1972).
    [CrossRef]
  16. Y. Y. Hung, I. M. Daniel, R. E. Rolands, “A New Optical Technique for Determining Derivatives of Surface Displacements,” paper presented at the 1975 Spring Meeting of SESA, 11–16 May, Chicago.
  17. D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
    [CrossRef]
  18. G. Rieder, R. Ritter, Forsch. Ingenieurwes. 31, No. 2. (1965).
    [CrossRef]
  19. F. P. Chiang, R. M. Juang, J. Opt. Soc. Am. 65, 1196 (1975).

1975 (1)

F. P. Chiang, R. M. Juang, J. Opt. Soc. Am. 65, 1196 (1975).

1974 (1)

D. E. Duffy, Exp. Mech. 14, 378 (1974).
[CrossRef]

1972 (2)

H. J. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

1971 (1)

F. P. Chiang, B. Ranganayakamma, Exp. Mech. No. 7, 296 (July1971).
[CrossRef]

1970 (2)

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

1968 (1)

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, No. 2, 101 (1968).

1965 (2)

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, No. 1, 22 (Jan.1965).
[CrossRef]

G. Rieder, R. Ritter, Forsch. Ingenieurwes. 31, No. 2. (1965).
[CrossRef]

1955 (1)

F. K. Lightenberg, Proc. Soc. Exp. Stress. Anal. 12, No. 2, 83 (1955).

Archbold, E.

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

Brown, C. J. E.

J. P. Duncan, C. J. E. Brown, “Slope Contours in Flexed Elastic Plates by Salet-Ikeda Technique,” in Proceedings First International Congress Experimental Mechanics (Pergamon, London, 1963), pp. 149–176.

Burch, J. M.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, No. 2, 101 (1968).

Chiang, F. P.

F. P. Chiang, R. M. Juang, J. Opt. Soc. Am. 65, 1196 (1975).

F. P. Chiang, B. Ranganayakamma, Exp. Mech. No. 7, 296 (July1971).
[CrossRef]

F. P. Chiang, T. Y. Kao, “An Optical Method for Generating Slope and Curvature Contours of Bent Plates,” in Proceedings of the Joint Meeting of German Optical Society (DGaO) and Belgium Optical Committee, Brugge, Belgium, 4–8 June 1975, p. 58 (abstract).

F. P. Chiang, “Multi-aperture Speckle Interferometry,” in Proc. Conference on Speckle Phenomena and Their Applications, Loughborough University of Technology, 27–28 March 1974, pp. 22–24.

R. P. Khetan, F. P. Chiang, “Strain Analysis by One Beam Laser Speckle Interferometry, Part I, Single Aperture Method,” to appear in Appl. Opt.

Daniel, I. M.

Y. Y. Hung, I. M. Daniel, R. E. Rolands, “A New Optical Technique for Determining Derivatives of Surface Displacements,” paper presented at the 1975 Spring Meeting of SESA, 11–16 May, Chicago.

Duffy, D. E.

D. E. Duffy, Exp. Mech. 14, 378 (1974).
[CrossRef]

Duncan, J. P.

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, No. 1, 22 (Jan.1965).
[CrossRef]

J. P. Duncan, C. J. E. Brown, “Slope Contours in Flexed Elastic Plates by Salet-Ikeda Technique,” in Proceedings First International Congress Experimental Mechanics (Pergamon, London, 1963), pp. 149–176.

Ennos, A. E.

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

Gabor, D.

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

Havanesian, J.

J. Havanesian, J. Varner, “Method for Determining the Bending Moments in Normal Loaded Thin Plates by Holographic Interferometry,” in Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge U.P, London, 1970), pp. 173–185.

Hu, C. P.

Y. Y. Hung, C. P. Hu, C. E. Taylor, “Speckle-Moiré Interferometry—A Tool for Complete Measurement of In-plane Surface Displacements,” in Proc. 7th Southeastern Conference on Theoretical and Applied Mechanics, 1974, pp. 497–505.

Hung, Y. Y.

Y. Y. Hung, C. P. Hu, C. E. Taylor, “Speckle-Moiré Interferometry—A Tool for Complete Measurement of In-plane Surface Displacements,” in Proc. 7th Southeastern Conference on Theoretical and Applied Mechanics, 1974, pp. 497–505.

Y. Y. Hung, I. M. Daniel, R. E. Rolands, “A New Optical Technique for Determining Derivatives of Surface Displacements,” paper presented at the 1975 Spring Meeting of SESA, 11–16 May, Chicago.

Juang, R. M.

F. P. Chiang, R. M. Juang, J. Opt. Soc. Am. 65, 1196 (1975).

Kao, T. Y.

F. P. Chiang, T. Y. Kao, “An Optical Method for Generating Slope and Curvature Contours of Bent Plates,” in Proceedings of the Joint Meeting of German Optical Society (DGaO) and Belgium Optical Committee, Brugge, Belgium, 4–8 June 1975, p. 58 (abstract).

Khetan, R. P.

R. P. Khetan, F. P. Chiang, “Strain Analysis by One Beam Laser Speckle Interferometry, Part I, Single Aperture Method,” to appear in Appl. Opt.

Lightenberg, F. K.

F. K. Lightenberg, Proc. Soc. Exp. Stress. Anal. 12, No. 2, 83 (1955).

Ranganayakamma, B.

F. P. Chiang, B. Ranganayakamma, Exp. Mech. No. 7, 296 (July1971).
[CrossRef]

Rieder, G.

G. Rieder, R. Ritter, Forsch. Ingenieurwes. 31, No. 2. (1965).
[CrossRef]

Ritter, R.

G. Rieder, R. Ritter, Forsch. Ingenieurwes. 31, No. 2. (1965).
[CrossRef]

Rolands, R. E.

Y. Y. Hung, I. M. Daniel, R. E. Rolands, “A New Optical Technique for Determining Derivatives of Surface Displacements,” paper presented at the 1975 Spring Meeting of SESA, 11–16 May, Chicago.

Sabin, P. G.

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, No. 1, 22 (Jan.1965).
[CrossRef]

Taylor, C. E.

Y. Y. Hung, C. P. Hu, C. E. Taylor, “Speckle-Moiré Interferometry—A Tool for Complete Measurement of In-plane Surface Displacements,” in Proc. 7th Southeastern Conference on Theoretical and Applied Mechanics, 1974, pp. 497–505.

Theocaris, P. S.

P. S. Theocaris, “Moiré Method in Plates,” in Proc. Int. Assoc. Shell Struct. Symposium, Warsaw (North Holland, Amsterdam, 1963), pp. 877–889.

Tiziani, H. J.

H. J. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

Tokarski, J. M. J.

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, No. 2, 101 (1968).

Varner, J.

J. Havanesian, J. Varner, “Method for Determining the Bending Moments in Normal Loaded Thin Plates by Holographic Interferometry,” in Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge U.P, London, 1970), pp. 173–185.

Exp. Mech. (2)

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, No. 1, 22 (Jan.1965).
[CrossRef]

D. E. Duffy, Exp. Mech. 14, 378 (1974).
[CrossRef]

Exp. Mech. No. 7 (1)

F. P. Chiang, B. Ranganayakamma, Exp. Mech. No. 7, 296 (July1971).
[CrossRef]

Forsch. Ingenieurwes. (1)

G. Rieder, R. Ritter, Forsch. Ingenieurwes. 31, No. 2. (1965).
[CrossRef]

IBM J. Res. Dev. (1)

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

F. P. Chiang, R. M. Juang, J. Opt. Soc. Am. 65, 1196 (1975).

Opt. Acta (3)

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, No. 2, 101 (1968).

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

Opt. Commun. (1)

H. J. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

Proc. Soc. Exp. Stress. Anal. (1)

F. K. Lightenberg, Proc. Soc. Exp. Stress. Anal. 12, No. 2, 83 (1955).

Other (8)

J. P. Duncan, C. J. E. Brown, “Slope Contours in Flexed Elastic Plates by Salet-Ikeda Technique,” in Proceedings First International Congress Experimental Mechanics (Pergamon, London, 1963), pp. 149–176.

F. P. Chiang, T. Y. Kao, “An Optical Method for Generating Slope and Curvature Contours of Bent Plates,” in Proceedings of the Joint Meeting of German Optical Society (DGaO) and Belgium Optical Committee, Brugge, Belgium, 4–8 June 1975, p. 58 (abstract).

J. Havanesian, J. Varner, “Method for Determining the Bending Moments in Normal Loaded Thin Plates by Holographic Interferometry,” in Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge U.P, London, 1970), pp. 173–185.

Y. Y. Hung, I. M. Daniel, R. E. Rolands, “A New Optical Technique for Determining Derivatives of Surface Displacements,” paper presented at the 1975 Spring Meeting of SESA, 11–16 May, Chicago.

P. S. Theocaris, “Moiré Method in Plates,” in Proc. Int. Assoc. Shell Struct. Symposium, Warsaw (North Holland, Amsterdam, 1963), pp. 877–889.

Y. Y. Hung, C. P. Hu, C. E. Taylor, “Speckle-Moiré Interferometry—A Tool for Complete Measurement of In-plane Surface Displacements,” in Proc. 7th Southeastern Conference on Theoretical and Applied Mechanics, 1974, pp. 497–505.

F. P. Chiang, “Multi-aperture Speckle Interferometry,” in Proc. Conference on Speckle Phenomena and Their Applications, Loughborough University of Technology, 27–28 March 1974, pp. 22–24.

R. P. Khetan, F. P. Chiang, “Strain Analysis by One Beam Laser Speckle Interferometry, Part I, Single Aperture Method,” to appear in Appl. Opt.

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Figures (7)

Fig. 1
Fig. 1

Arrangement for recording spatial speckles for plate bending studies. The camera is focused on a plane either in front or behind the plate surface.

Fig. 2
Fig. 2

Arrangement for optical Fourier filtering of processed speckle interferogram.

Fig. 3
Fig. 3

Fringe patterns of partial slope contours with different sensitivity of a clamped circular plate under concentrated central load. Circles at top of figure depict the positions of filtering aperture at transform plane.

Fig. 4
Fig. 4

Comparison of theoretical and experimental results.

Fig. 5
Fig. 5

Partial slope contours of a clamped centrally loaded triangular plate. Different patterns are obtained by placing the filtering aperture at different angles at the transform plane as shown at top of figure.

Fig. 6
Fig. 6

Patterns of partial slope contours showing sensitivity dependence on position of focused plane.

Fig. 7
Fig. 7

Partial slope contours obtained from a model made of cardboard.

Equations (9)

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s = 1.2 ( λ q D )
d = A [ tan ( 2 ϕ + α ) - tan α ] = tan 2 ϕ ( 1 + tan 2 α ) 1 - tan α tan 2 ϕ .
d = 2 ϕ A ( 1 + tan 2 α )
d = 2 ϕ A [ 1 + ( B 2 A 2 ) ] ,
I ( u ) = 4 cos 2 k ( u · d L ) I 1 ( u ) ,
d u cos θ = n λ L for bright fringes = ( n + 1 2 ) λ L for dark fringes ,
ϕ u cos θ = n λ L 2 A [ 1 + ( B 2 A 2 ) ] , n = 0 , ± 1 , ± 2 , , ϕ = ϕ .
λ L 2 A ϕ [ 1 + ( B 2 A 2 ) ]
w x = ϕ x = n λ L 2 A u x , w y = ϕ y = n λ L 2 A u y ,

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