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Theory and Application of Sampling Moiré Method 239

Fig. 8. Experimental setup: (a) Side view; (b) Top view

Fig. 9. Photograph of experimental setup

240 Y. Morimoto and M. Fujigaki

Fig. 10. Experimental results: images of the specimen before (a) and after (a’) loading; (b)

phase-shifted images obtained by sampling moiré method; wrapped phase distribution be-

fore (c) and after (c’) loading at 98N; (d) deflection distribution

6.2 Experimental Results of Deflection Measurement

Figure 10 shows the captured and analyzed images. Figure 10(a) shows the captured

image of the specimen before loading. Figure 10(b) shows the phase-shifted images

obtained when the thinning-out index

N is equal to 6. Then the π/3 phase-shifted

moiré patterns were obtained. Figures 10(c) and (c’) show the wrapped phase distri-

butions before loading and after 98 N loading, respectively. Figure 10(d) shows the

displacement distribution obtained from Eq. (13). Figure 11 shows the deflection

distributions under loading at 9.8 to 127.4 N.

Fig. 11. Deflection distributions under loading at 9.8 to 127.4 N

Theory and Application of Sampling Moiré Method 241

Figure 12 shows the results of displacement along the line AA’ in Figure 10(d).

Figure 13 shows the comparison of the measured displacement values by the sam-

pling moiré method and with the values measured by the laser displacement sensor

at the position B in Fig. 10(d) in different loading condition. The results show that

the displacement values of the steel beam agree well with the data obtained by the

sensor. The average error was 3.8

μm and the standard deviation was 3.15 μm.

This means that the average error of the detected displacement was less than 1/500

of the grating pitch 2.0 mm.

Fig. 12. Displacement distribution along line AA’ in Figure 10(d)

Fig. 13. Accuracy check by comparing measured displacements by sampling moiré method

with those by laser sensor and those obtained from beam theory

242 Y. Morimoto and M. Fujigaki

7 Shape and Displacement Measurement by Sampling Moiré

Method

Sampling moire method was applied to measurement of shape, displacement dis-

tributions and strain distributions on the surface of a rotating rubber object running

at 80 km/h speed using a two-camera stereoscopic system [37].

Fig. 14. Arrangement of cameras, reference plane and object

7.1 Process of Measurement

Let us explain the process of the measurement using Fig. 14. Figure 14 shows a

schema of the arrangement of two cameras, two reference planes and an object. At

first, in order to calibrate, two reference planes R

and R

with gratings are re-

corded and the phase distributions of the gratings are obtained. Secondary, each

corresponding point of the gratings on the reference planes of the images recorded

by the two cameras is determined each other on the whole area of the planes. Then

the view lines of each pixel point of the cameras are obtained. Thirdly, an object

with a grating on the surface is recorded by the two cameras. Corresponding

points at the same point on the object are obtained from the two images. The coor-

dinates of each point of the object is determined from the intersection of the view

lines from the two cameras. From the coordinates, the shape is determined. From

the position of each point before and after deformation, strain distributions are ob-

tained. The concrete measurement is as follows.

Theory and Application of Sampling Moiré Method 243

(a) Photograph

(b) Diagram

Fig. 15. Experimental setup to measure moving object

244 Y. Morimoto and M. Fujigaki

7.2 Calibration with Two Reference Planes

Both left and right CCD cameras capture a reference planes. A liquid crystal dis-

play (LCD) is used as the reference plane. The reference plane is moved at two

positions, R

and R

. The reference plane can display grating patterns for the x-

and

y- directions with phase-shifting. Initial phase distributions of the grating pat-

terns can be obtained accurately using phase-shifting method. The initial phase

distribution is obtained as unwrapped phases for the

x- and y- directions (

)

on the two reference planes R

(n = 0, 1).

The 3-D coordinates (

x, y, z) at a point on the reference plane is calculated from

the

x-directional grating pitch p

, the y-directional grating pitch p

and the z posi-

tion of the reference plane

(n = 0, 1) as shown in Eq. (15).

x =

+ x

, y =

+ y

, z = z

, (15)

where

and p

are the grating pitches for the x- and y-directions, respectively,

and (

, y

) is the coordinates at the point where (

) = (0, 0).

7.3 Shape and Strain Distribution Measurement

An object is placed between the two reference planes. A 2-D grating attached on

the object is captured by two sets of CCD cameras as shown in Fig. 14.

The view line

from a pixel of CCD camera 1 passes through the point S on

the object and the view line

from a pixel of CCD camera 2 also passes through

the point S on the object. The points C

and C

are intersections of the view line l

with the reference planes R

and R

, respectively. The points C

and C

are inter-

sections of the view line

with the reference planes R

and R

, respectively. The

position of the reference plane R

and R

are z = 0 and z = Δz, respectively. An ex-

pression of the view lines

and l

can be obtained from the four points C

, C

and C

. The 3-D coordinates of the point S are obtained as the intersection of the

two view lines

and l

Strain can be calculated from the 3-D coordinates of the two points at each cor-

responding point on the object before and after deformation.

7.4 Experimental Results of Shape and Strain Distributions of Rotating

Object

Figure 15 shows the photograph of the experimental setup. A rotating object was

set on a flat belt cornering machine. This test machine can control the rotating of

the object. The rotating object had a 2-D grating with 2.0 mm pitch on the surface.

The 2-D grating was made of a thin rubber sheet. The 2-D grating was printed on

the rubber sheet.

Theory and Application of Sampling Moiré Method 245

(a) 2-D -grating (b) Phase distribution of moiré

Fig. 16. Phase analysis (in x-direction)

Fig. 17. Analyzed shape

(a) x-directional strain (b) y-directional strain (c) Shear strain

Fig. 18. Analyzed strain distributions

-0.05 0.05 -0.03 0.03 -0.08 0.08

246 Y. Morimoto and M. Fujigaki

In this experiment, the object was rotating at 80km/h in circumferential veloc-

ity. The images of the grating are captured by two CCD cameras. The trigger sig-

nal is produced by a trigger sensor placed behind the object. The sensor seeks a

white mark attached on the object. The CCD cameras 1 and 2 can capture images

simultaneously with the trigger signal when the rotating object comes at the same

position.

The grating image recorded by the left camera is shown in Fig. 16(a). The

phase distribution of the moiré fringe pattern analyzed by sampling moiré method

is shown in Fig. 16(b). The phase distribution of the grating shown in Fig. 16(c)

was obtained by using Eq. (9). By unwrapping the phase distribution, the un-

wrapped phase distribution was obtained as shown in Fig. 16(d). The image re-

corded by the right camera was also analyzed in the same process.

Figure 17 shows the measurement result of the 3-D shape. The measured shape

was the initial shape when the axis load was 0. Figures 18(a), (b) and (c) show the

distributions for the

x- and y-directional and shear strains, respectively.

8 Conclusions

Sampling moiré method using accurate phase analysis of a specimen grating was

proposed. Deflection of a beam and shape and strain distribution of a moving ob-

ject were measured by sampling moiré method. The main advantage of the sam-

pling moiré method is non-contact displacement distribution measurement with

simple, easy, high accuracy, high speed, and low cost. By comparing the conven-

tional phase shifting methods using multiple phase-shifted images, the sampling

moiré method requires only one image for phase-analysis. Therefore, this method

can be easily applied to dynamic phenomenon using a high-speed camera. More-

over, this method also can analyze from small to large deformation by changing

the grating pitch. It will be a new tool for experimental mechanics and health

monitoring of structures.

Acknowledgements

The authors appreciate for the contributions to this study by Dr. Shien Ri in To-

hoku University and Mr. Akihiro Masaya and Mr. Kosuke Shimo in Wakayama

University.

References

1. Takahashi, S. (ed.): Photomechanics. Sankaido Publishing, Tokyo (1997) (in Japanese)

2. Rastogi, P.K. (ed.): Photomechanics (Topics in Applied Physics). Springer, New York

(2000)

3. Japanese Society for Experimental Mechanics (ed.), Handbook on Experimental Me-

chanics, Asakura Publishing, Tokyo (2008) (in Japanese)

Theory and Application of Sampling Moiré Method 247

4. Sharpe, W.N.J. (ed.): Handbook of Experimental Solid Mechanics, Soc. for Exp.

Mech. Springer, New York (2008)

5. Theocaris, P.S.: Moiré Fringes in Strain Analysis. Pergamon Press, Oxford (1969)

6. Durelli, A., Parks, V.: Moiré Analysis of Strain. Prentice-Hall, Englewood Cliffs

(1970)

7. Japanese Society for Moiré Contourography (ed.), Moiré Contourography and Its Ap-

plication (1984) (in Japanese)

8. Post, D., Han, B., Ifju, P.: High Sensitivity Moiré. Experimental Analysis for Mechan-

ics and Materials. Springer, New York (1994)

9. Amidror, I.: The Theory of the Moiré Phenomenon. Kluwer Academic Publishers,

Dordrecht (2000)

10. Walker, C.A. (ed.): Handbook of Moiré Measurement. Series in Optics and Optoelec-

tronics. Inst. of Physics Publishing (2004)

11. Yoshizawa, T.: Handbook of Optical Metrology, Principles and Applications. CRC

Press, Boca Raton (2009)

12. Sciammarella, C.A., Sturgeon, D.L.: Digital-filtering Techniques Applied to the Inter-

polation of Moiré-Fringes Data. Experimental Mechanics 7(11), 468–475 (1967)

13. Brunning, J.H., Herriott, D.R., Gallagher, J.E., Rosenfeld, D.P., White, A.D., Brangac-

cio, D.J.: Digital Wave Front Measuring Interferometer for Testing Optical Surface

and Lenses. Appl. Opt. 13, 2693–2703 (1974)

14. Creath, K.: Phase Measurement Interferometry Techniques. Prog. Opt. 26, 349–393

(1988)

15. Takeda, M., Ina, H., Kobayashi, S.: Fourier Transform Method of Fringe Pattern

Analysis for Computer-Based Topography and Interferometry. J. Opt. Soc. Am. 72(1),

156–160 (1982)

16. Morimoto, Y., Seguchi, Y., Higashi, T.: Moiré Analysis of Strain by Fourier Trans-

form. Trans. of the Japan Society of Mechanical Engineers 54(504 A), 1546–1552

(1988) (in Japanese)

17. Morimoto, Y., Seguchi, Y., Higashi, T.: Two-dimensional Moiré Method and Grid

Method Using Fourier Transform. Exp. Mech. 29, 399–404 (1989)

18. Morimoto, Y.: Digital Image Processing. In: Kobayashi, A.S. (ed.) Handbook on Ex-

perimental Mechanics, ch. 21, pp. 969–1029. Prentice-Hall, Inc., Englewood Cliffs

(1993)

19. Morimoto, Y., Fujigaki, M.: Techniques for Stress and Strain Measurements using

Various Transformations. Journal of JSNDI 46(7), 473–480 (1997) (in Japanese)

20. Morimoto, Y., Fujigaki, M., Yoneyama, S.: Recent Studies on Shape and Deformation

Measurements by Moiré Method and Grid Method. Journal of JSNDI 52(3), 116–121

(2003) (in Japanese)

21. Morimoto, Y., Matui, T., Fujigaki, M., Yamamoto, Y., Ri, S.: Applications of Accu-

rate Phase Analysis to Optical Measurement. Journal of JSNDI 54(3), 120–126 (2005)

(in Japanese)

22. Idesawa, M., Yatagai, T., Soma, T.: Scanning Moiré Method and Automatic Measure-

ment of 3-D Shapes. Applied Optics 16(8), 2152–2162 (1977)

23. Morimoto, Y., Hayashi, T.: Deformation Measurement During Powder Compaction by

a Scanning-Moiré Method. Exp. Mech. 24(2), 112–116 (1984)

24. Morimoto, Y., Hayashi, T., Yamaguchi, N.: Strain Measurement by Scanning Moiré

Method. Transaction of JSME 50(451A), 489–494 (1984) (in Japanese)

25. Arai, Y., Kurata, T.: On the Representative of Scanning Moiré Fringe. Journal of the

Japan Society of Precision Engineering 51(3), 6–100 (1985) (in Japanese)

248 Y. Morimoto and M. Fujigaki

26. Yoshizawa, T., Tashiro, H., Nakagawa, S.: High-sensitive Moiré Method by Phase De-

tection. Journal of the Japan Society of Precision Engineering 51(4), 154–159 (1985)

(in Japanese)

27. Arai, Y., Kurata, T.: High Speed- and High Resolutive-Moiré Topography by the

Method of Fringe Scanning Interferometry. Japanese Journal of Optics 15(5), 402–406

(1986) (in Japanese)

28. Morimoto, Y., Yang, I.H., Gu, C.G.: Scanning Moiré Method for Obtaining Smooth

Fringe Patterns. Optics and Lasers in Engineering 24, 3–17 (1996)

29. Arai, Y., Yokozeki, S., Shiraki, K., Yamada, T.: High Precision Two-dimensional Spa-

tial Fringe Analysis Method. Journal of Modern Optics 44(4), 739–751 (1997)

30. Kato, J., Yamaguchi, I., Nakamura, T., Kuwashima, S.: Video-rate Fringe Analyzer

Based on Phase-shifting Electronic Moiré. Patterns. Appl. Opt. 36(32), 8403–8412

(1997)

31. Fujigaki, M., Kim, W.G., Morimoto, Y.: Shape Measurement by Phase Shifting Scan-

ning Moiré Method. In: Proc. of Int. Conf. on Advanced Technology in Exp. Mech.

(ATEM 1997), pp. 161–164 (1997)

32. Ri, S., Fujigaki, M., Morimoto, Y., Matui, T.: Pixel-to-pixel Correspondence Adjust-

ment Method Using Moiré Pattern for DMD Reflection-type CCD Camera. Journal of

JSEM 5(4), 361–366 (2005) (in Japanese)

33. Ri, S., Fujigaki, M., Matui, T., Morimoto, Y.: Accurate Pixel-to-pixel Correspondence

Adjustment in a Digital Micromirror Device Camera by Using the Phase-shifting

Moiré Method. Appl. Opt. 45(27), 6940–6946 (2006)

34. Fujigaki, M., Iwai, K., Morimoto, Y.: Shape Measurement with Grating Projection Us-

ing Sampling Moiré Method. In: 3rd International Symposium on Advanced

Fluid/Solid Science and Technology in Experimental Mechanics (ISEM2008),

#145(CD-ROM) (2008)

35. Morimoto, Y., Fujigaki, M.: Accuracy of Sampling Moiré Method. Mechanics of Ma-

terials 35, 511–522 (2008)

36. Ri, S., Morimoto, M., Fujigaki, M.: Non-contact Displacement Distribution Measure-

ment by Sampling Moiré Method. Inspection Engineering 14(5), 1–6 (2009) (in Japa-

nese)

37. Shimo, K., Fujigaki, M., Masaya, A., Morimoto, Y.: Development of Dynamic Shape

and Strain Measurement System by Sampling Moiré Method. In: Proc. SPIE 7522,

75224M (2009), doi:10.1117/12.854780 online Publication (2010)

38. Ri, S., Fujigaki, M., Morimoto, Y.: Sampling Moiré Method for Accurate Small De-

formation Distribution Measurement. Exp. Mech. 50(4), 501–508 (2010)