Abstract

We analyze the sensitivity of a video-based moiré technique used to measure vibrations. By use of modulated projected fringes, full-field distributions are mapped for both the vibration amplitude and the vibration phase. We also show methods for visual interpretation of the vibration fringe pattern, and we describe phase-stepping algorithms for full-field calculations of the distributions by use of digital image-processing equipment. Implementation of these methods makes it possible to measure vibration amplitude and phase over a large range of amplitudes, from < 1 μm to several millimeters. We show that the method bridges the traditional gap between holographic interferometry and moiré methods.

© 1994 Optical Society of America

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References

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  1. C. M. Vest, D. W. Sweeney, “Measurement of vibrational amplitude by modulation of projected fringes,” Appl. Opt. 11, 449–454 (1972).
    [CrossRef] [PubMed]
  2. K. G. Harding, J. S. Harris, “Projection moiré interferometer for vibration analysis,” Appl. Opt. 22, 856–861 (1983).
    [CrossRef] [PubMed]
  3. A. Asundi, M. T. Cheung, “Moiré interferometry for vibration analysis of plates,” Exp. Mech. 27, 338–341 (1987).
    [CrossRef]
  4. Y. Y. Hung, C. Y. Liang, J. D. Hovanesian, A. J. Durelli, “Time-averaged shadow-moiré method for studying vibrations,” Appl. Opt. 16, 1717–1719 (1977).
    [CrossRef] [PubMed]
  5. F. P. Chiang, C. J. Lin, “Time-average reflection moiré method for vibration analysis of plates,” Appl. Opt. 18, 1424–1427 (1979).
    [CrossRef] [PubMed]
  6. P. Hopstone, A. Katz, J. Politch, “Infrastructure of time-averaged projection moiré fringes in vibration analysis,” Appl. Opt. 28, 5305–5311 (1989).
    [CrossRef] [PubMed]
  7. K. J. Gåsvik, “Moiré technique by means of digital image processing,” Appl. Opt. 22, 3543–3548 (1983).
    [CrossRef] [PubMed]
  8. E. Vikhagen, “Vibration measurement using phase-shifting TV-holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
    [CrossRef]
  9. J. H. Bruning, D. R. Elliot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wave-front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  10. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 351–393.
    [CrossRef]
  11. C. Koliopoulus, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).
  12. O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, J. C. Wyant, R. Shannon, eds. (Academic, New York, 1987), Vol. 10, pp. 455–504.
  13. O. J. Løkberg, G. Å. Slettemoen, “Improved fringe definition by speckle averaging in ESPI,” in ICO-13 Proceedings (ICO, Sapporo, Japan, 1984), pp. 116–117.
  14. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1981), Formula (8.511.4).
  15. Ref. 14, Formula (8.53 1.1).
  16. O. J. Løkberg, K. Høgmoen, “Use of modulated reference wave in electronic speckle pattern interferometry,” J. Phys. E 9, 847–851 (1976).
    [CrossRef]
  17. O. J. Løkberg, K. Høgmoen, “Vibration phase mapping. using electronic speckle pattern interferometry,” Appl. Opt. 15, 2701–2704 (1976).
    [CrossRef] [PubMed]
  18. O. J. Løkberg, K. Høgmoen, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
    [CrossRef] [PubMed]
  19. S. Ellingsrud, G. O. Rosvold, “Analysis of a data-based TV-holography system used to measure small vibration amplitudes,” J. Opt. Soc. Am. A 9, 237–251 (1992).
    [CrossRef]
  20. M. Abramowitz, I. A. Stegun, Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1984), Formula (9.1.7).
  21. G. O. Rosvold, “Fast measurement of phase using a PC-based frame grabber and phase stepping technique,” Appl. Opt. 29, 237–241 (1990).
    [CrossRef] [PubMed]
  22. V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24, 185–188 (1985).
    [CrossRef] [PubMed]

1992 (1)

1990 (1)

1989 (2)

P. Hopstone, A. Katz, J. Politch, “Infrastructure of time-averaged projection moiré fringes in vibration analysis,” Appl. Opt. 28, 5305–5311 (1989).
[CrossRef] [PubMed]

E. Vikhagen, “Vibration measurement using phase-shifting TV-holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
[CrossRef]

1987 (1)

A. Asundi, M. T. Cheung, “Moiré interferometry for vibration analysis of plates,” Exp. Mech. 27, 338–341 (1987).
[CrossRef]

1985 (1)

1983 (2)

1979 (1)

1977 (2)

1976 (2)

O. J. Løkberg, K. Høgmoen, “Use of modulated reference wave in electronic speckle pattern interferometry,” J. Phys. E 9, 847–851 (1976).
[CrossRef]

O. J. Løkberg, K. Høgmoen, “Vibration phase mapping. using electronic speckle pattern interferometry,” Appl. Opt. 15, 2701–2704 (1976).
[CrossRef] [PubMed]

1974 (1)

1972 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1984), Formula (9.1.7).

Asundi, A.

A. Asundi, M. T. Cheung, “Moiré interferometry for vibration analysis of plates,” Exp. Mech. 27, 338–341 (1987).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Cheung, M. T.

A. Asundi, M. T. Cheung, “Moiré interferometry for vibration analysis of plates,” Exp. Mech. 27, 338–341 (1987).
[CrossRef]

Chiang, F. P.

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 351–393.
[CrossRef]

Durelli, A. J.

Ellingsrud, S.

Elliot, D. R.

Gallagher, J. E.

Gåsvik, K. J.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1981), Formula (8.511.4).

Halioua, M.

Harding, K. G.

Harris, J. S.

Høgmoen, K.

Hopstone, P.

Hovanesian, J. D.

Hung, Y. Y.

Katz, A.

Koliopoulus, C.

C. Koliopoulus, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

Liang, C. Y.

Lin, C. J.

Liu, H. C.

Løkberg, O. J.

O. J. Løkberg, K. Høgmoen, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
[CrossRef] [PubMed]

O. J. Løkberg, K. Høgmoen, “Vibration phase mapping. using electronic speckle pattern interferometry,” Appl. Opt. 15, 2701–2704 (1976).
[CrossRef] [PubMed]

O. J. Løkberg, K. Høgmoen, “Use of modulated reference wave in electronic speckle pattern interferometry,” J. Phys. E 9, 847–851 (1976).
[CrossRef]

O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, J. C. Wyant, R. Shannon, eds. (Academic, New York, 1987), Vol. 10, pp. 455–504.

O. J. Løkberg, G. Å. Slettemoen, “Improved fringe definition by speckle averaging in ESPI,” in ICO-13 Proceedings (ICO, Sapporo, Japan, 1984), pp. 116–117.

Politch, J.

Rosenfeld, D. P.

Rosvold, G. O.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1981), Formula (8.511.4).

Slettemoen, G. Å.

O. J. Løkberg, G. Å. Slettemoen, “Improved fringe definition by speckle averaging in ESPI,” in ICO-13 Proceedings (ICO, Sapporo, Japan, 1984), pp. 116–117.

O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, J. C. Wyant, R. Shannon, eds. (Academic, New York, 1987), Vol. 10, pp. 455–504.

Srinivasan, V.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1984), Formula (9.1.7).

Sweeney, D. W.

Vest, C. M.

Vikhagen, E.

E. Vikhagen, “Vibration measurement using phase-shifting TV-holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
[CrossRef]

White, A. D.

Appl. Opt. (11)

C. M. Vest, D. W. Sweeney, “Measurement of vibrational amplitude by modulation of projected fringes,” Appl. Opt. 11, 449–454 (1972).
[CrossRef] [PubMed]

J. H. Bruning, D. R. Elliot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wave-front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

O. J. Løkberg, K. Høgmoen, “Vibration phase mapping. using electronic speckle pattern interferometry,” Appl. Opt. 15, 2701–2704 (1976).
[CrossRef] [PubMed]

Y. Y. Hung, C. Y. Liang, J. D. Hovanesian, A. J. Durelli, “Time-averaged shadow-moiré method for studying vibrations,” Appl. Opt. 16, 1717–1719 (1977).
[CrossRef] [PubMed]

O. J. Løkberg, K. Høgmoen, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
[CrossRef] [PubMed]

F. P. Chiang, C. J. Lin, “Time-average reflection moiré method for vibration analysis of plates,” Appl. Opt. 18, 1424–1427 (1979).
[CrossRef] [PubMed]

K. G. Harding, J. S. Harris, “Projection moiré interferometer for vibration analysis,” Appl. Opt. 22, 856–861 (1983).
[CrossRef] [PubMed]

K. J. Gåsvik, “Moiré technique by means of digital image processing,” Appl. Opt. 22, 3543–3548 (1983).
[CrossRef] [PubMed]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24, 185–188 (1985).
[CrossRef] [PubMed]

P. Hopstone, A. Katz, J. Politch, “Infrastructure of time-averaged projection moiré fringes in vibration analysis,” Appl. Opt. 28, 5305–5311 (1989).
[CrossRef] [PubMed]

G. O. Rosvold, “Fast measurement of phase using a PC-based frame grabber and phase stepping technique,” Appl. Opt. 29, 237–241 (1990).
[CrossRef] [PubMed]

Exp. Mech. (1)

A. Asundi, M. T. Cheung, “Moiré interferometry for vibration analysis of plates,” Exp. Mech. 27, 338–341 (1987).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. E (1)

O. J. Løkberg, K. Høgmoen, “Use of modulated reference wave in electronic speckle pattern interferometry,” J. Phys. E 9, 847–851 (1976).
[CrossRef]

Opt. Commun. (1)

E. Vikhagen, “Vibration measurement using phase-shifting TV-holography and digital image processing,” Opt. Commun. 69, 214–218 (1989).
[CrossRef]

Other (7)

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 351–393.
[CrossRef]

C. Koliopoulus, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

O. J. Løkberg, G. Å. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, J. C. Wyant, R. Shannon, eds. (Academic, New York, 1987), Vol. 10, pp. 455–504.

O. J. Løkberg, G. Å. Slettemoen, “Improved fringe definition by speckle averaging in ESPI,” in ICO-13 Proceedings (ICO, Sapporo, Japan, 1984), pp. 116–117.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1981), Formula (8.511.4).

Ref. 14, Formula (8.53 1.1).

M. Abramowitz, I. A. Stegun, Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1984), Formula (9.1.7).

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Figures (8)

Fig. 1
Fig. 1

Fringe projection and observation; a plane object surface is illuminated by a collimated pattern and is observed by a camera. The angle of incidence is θ0, and the observation angle is θ1. The period of the projected fringe pattern is Λ0, and the observed period is Λ1. When the object is displaced by dΛ, a 2π phase shift is obtained.

Fig. 2
Fig. 2

J02 function, or the amplitude response given by the squared zero-order Bessel function of the first kind. The most linear region of the function is indicated and used in Section 5.

Fig. 3
Fig. 3

Measurement area and fringe period versus accuracy. The size of the measurement area and the necessary fringe period Λ are functions of minimum detectable amplitude amin.

Fig. 4
Fig. 4

Experimental setups: (a) setup used to obtain vibration fringes as described in Section 3, (b) setup used to measure the vibration amplitude and phase measurements by use of phase modulation, as described in Sections 4 and 5.

Fig. 5
Fig. 5

Suppression of the carrier fringes. The results of averaging of multiple frames in which the carrier fringes are shifted a small distance between acquisition of each frame are shown. The serrated edge of a handsaw is shown at the bottom of each picture. (a) example of instantaneous fringes; the closely spaced vertical carrier fringes are clearly visible. (b) Result of averaging of several frames; the carrier fringes are now greatly attenuated.

Fig. 6
Fig. 6

Vibration fringes in different phase states: (a) without phase modulation; (b)–(e) phase differences of 0, π/2, π, and 3π/2 rad, respectively.

Fig. 7
Fig. 7

Vibration amplitude and phase distributions. The method described in Section 5 is used to calculate the vibration amplitude and phase of the same vibrating object as shown in Fig. 6. The vibration frequency is 77 Hz. (a) Same as Fig. 6(a); amplitude and phase distributions are calculated for parts of the object within the marked rectangular area. (b) Plot of the vibration amplitude distribution. (c) Plot of the vibration phase distribution.

Fig. 8
Fig. 8

Vibration amplitude measured by two techniques. The vibration frequency is 25 Hz. (a) Vibration amplitude distribution measured by use of TV holography; (b) plot of vibration amplitude distribution measured by use of projected fringes.

Equations (52)

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Λ 1 = Λ 0 cos θ 1 / cos θ 0 ,
I ( x , y ) = a ( x , y ) + b ( x , y ) cos ϕ s .
Λ = d Λ ( tan θ 0 + tan θ 1 ) = Λ 0 / cos θ 0 ,
Λ 0 = d Λ ( sin θ 0 + tan θ 1 cos θ 0 ) = d Λ ( sin θ 0 cos θ 1 + cos θ 0 sin θ 1 ) / cos θ 1 = d Λ [ sin ( θ 0 + θ 1 ) / cos θ 1 ] ,
d Λ = Λ 0 / g .
g = sin ( θ 0 + θ 1 ) / cos θ 1 .
ϕ d = d 2 π / Λ Λ = d g 2 π / Λ 0 .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ s + ϕ d ) ,
Λ 1 = Λ 0 cos θ 1 / cos θ 0 Λ c ,
Λ 0 cos θ 1 Λ c cos θ 0 ,
Λ 0 cos θ 1 Λ c cos ( π / 2 - θ 1 ) ,
Λ 0 cos θ 1 Λ c sin θ 1 ,
Λ 0 / Λ c tan θ 1 ,
θ 1 arctan ( Λ 0 / Λ c ) .
g max = 1 / cos [ arctan ( Λ 0 / Λ c ) ] .
d ( x , y , t ) = a 0 ( x , y ) sin [ ω t + ϕ ( x , y ) ] ,
I ( x , y , t ) = a ( x , y ) + b ( x , y ) Re { exp [ i ( ϕ s + ϕ 0 ) ] } ,
E ( x , y ) a ( x , y ) + b ( x , y ) ( cos ϕ s ) J 0 [ a 0 ( x , y ) 2 π g / Λ 0 ] ,
I ( x , y ) ( cos 2 ϕ s ) J 0 2 [ a 0 ( x , y ) 2 π g / Λ 0 ] .
ϕ r ( d r ) = d r 4 π / λ .
I ( x , y , t ) = a ( x , y ) + b ( x , y ) Re { exp [ i ( ϕ s + ϕ 0 - ϕ r ) ] } = a ( x , y ) + b ( x , y ) Re { exp ( i ϕ s ) exp [ i ( ϕ 0 - ϕ r ) ] } .
ϕ s = x 2 π cos θ 0 / ( Λ 0 cos θ 1 ) ,
ϕ 0 = a 0 ( x , y ) sin [ ω t + ϕ ( x , y ) ] 2 π g / Λ 0 = A 0 ( x , y ) sin [ ω t + ϕ ( x , y ) ] ,
ϕ r = a r sin ( ω t ) 4 π / λ = A r sin ω t ,
A 0 ( x , y ) = a 0 ( x , y ) 2 π g / Λ 0
A r ( x , y ) = a r 4 π / λ
exp ( i a sin x ) = n = - J n ( a ) exp ( i n x ) .
exp [ i ( ϕ 0 - ϕ r ) ] = exp ( i { A 0 ( x , y ) sin [ ω t + ϕ ( x , y ) ] - A r sin ω t } ) = n = - m = - J n [ A 0 ( x , y ) ] J m ( A r ) × exp { i n [ ω t + ϕ ( x , y ) ] } exp ( - i m ω t )
= n = - m = - J n [ A 0 ( x , y ) ] J m ( A r ) × exp [ i n ϕ ( x , y ) ] exp [ i ( n - m ) ω t ] .
( 1 / T ) 0 T exp [ i ( ϕ 0 - ϕ r ) ] d t = n = - J n [ A 0 ( x , y ) ] J n ( A r ) exp [ i n ϕ ( x , y ) ] = J 0 [ A 0 ( x , y ) ] J 0 ( A r ) + 2 n = 1 J n [ A 0 ( x , y ) ] J n ( A r ) cos [ n ϕ ( x , y ) ] .
( 1 / T ) 0 T exp [ i ( ϕ 0 - ϕ r ) ] d t = J 0 [ A 0 ( x , y ) 2 + A r 2 - 2 A 0 ( x , y ) A r cos ϕ ( x , y ) ] 1 / 2 .
E ( x , y ) a ( x , y ) + b ( x , y ) cos ϕ s × Re { ( 1 / T ) 0 T exp [ i ( ϕ 0 - ϕ r ) ] d t } .
I ( x , y ) ( cos 2 ϕ s ) J 0 2 [ A 0 ( x , y ) 2 + A r 2 - 2 A 0 ( x , y ) A r cos ϕ ( x , y ) ] 1 / 2 ,
J 0 [ A 0 ( x , y ) 2 + A r 2 - 2 A 0 ( x , y ) A r cos ϕ ( x , y ) ] 1 / 2 = J 0 ( A r ) - ( d / d A r ) [ J 0 ( A r ) ] A 0 ( x , y ) cos ϕ ( x , y ) + ,
J 0 2 [ A 0 ( x , y ) 2 + A r 2 - 2 A 0 ( x , y ) A r cos ϕ ( x , y ) ] 1 / 2 = J 0 2 ( A r ) - ( d / d A r ) [ J 0 ( A r ) ] A 0 ( x , y ) cos ϕ ( x , y ) + .
I ( x , y ) cos 2 ϕ s { J 0 2 ( A r ) - ( d / d A r ) [ J 0 2 ( A r ) A 0 ( x , y ) cos ϕ ( x , y ) ] } .
I ( x , y ) = I b ( x , y ) - k ( x , y ) A 0 ( x , y ) cos ϕ ( x , y ) ,
I 000 = I b ( x , y ) - k ( x , y ) A 0 ( x , y ) cos ϕ ( x , y ) ,
I 090 = I b ( x , y ) + k ( x , y ) A 0 ( x , y ) sin ϕ ( x , y ) ,
I 180 = I b ( x , y ) + k ( x , y ) A 0 ( x , y ) cos ϕ ( x , y ) ,
I 270 = I b ( x , y ) - k ( x , y ) A 0 ( x , y ) sin ϕ ( x , y ) ,
ϕ ( x , y ) = arctan [ ( I 090 - I 270 ) / ( I 180 - I 000 ) ] .
I + A r = I b ( x , y ) - k ( x , y ) Δ A r .
I - A r = I b ( x , y ) + k ( x , y ) Δ A r .
A 0 ( x , y ) = Δ A r [ ( I 180 - I 000 ) 2 + ( I 090 - I 270 ) ] 1 / 2 / ( I - A r - I + A r ) .
2 π g / Λ 0 = A 0 ( x 0 , y 0 ) / a 0 ( x 0 , y 0 ) ,
I ( x , y ) J 0 2 [ a 0 ( x , y ) 2 π g / Λ 0 ] ,
I ( x , y ) J 0 2 [ a 0 ( x , y ) 2 π g / ( Λ cos θ 0 ) ] .
a 1 drk 2 π g / ( Λ cos θ 0 ) = 2.41.
a 1 drk = 2.41 Λ cos θ 0 / 2 π g .
Λ = a min ( 100 ) 2 π g / ( 2.41 cos θ 0 ) ,
I ( x , y ) J 0 2 [ a ( x , y ) 4 π / λ ] ,

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