Abstract
Two techniques of moiré-pattern differentiation are described and implemented. One of the techniques is extremely simple since the grid printed in the model provides both the displacement information and the shearing mechanism required to display the displacement derivatives. This technique can only be applied to the derivatives in the direction parallel to the projected displacement. To find the derivativs in the direction perpendicular to the projected displacement, a double-exposure technique is utilized.
Experimental and theoretical values of the derivatives show a good agreement. In view of the speed with which the differentiation can be performed as compared to any other methods, the proposed techniques seem to be valuable practical tools for moiré work.
The optical differentiation is very attractive, but to be useful in the case of small deformations, it must produce patterns with large density of fringes. The techniques described in this paper offer a simple solution to this problem.
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Abbreviations
- A(x) :
-
grid trasmission function
- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} (x,z)\) :
-
amplitude vector of light wavefronts
- A n :
-
coefficients of the Fourier expansion of the grid transmission function
- \(\bar A_1 ,\bar A_2 \) :
-
summatories of the amplitudes contributing to the first- and second-order spatial frequencies, respectively
- B n :
-
amplitudes of light wavefronts
- \(\bar B_2 \) :
-
summatory of the amplitudes, contributing to the second-order spatial frequency
- I n :
-
light-intensity amplitude. Subscript 1, 2, ...n, indicate the corresponding spatial frequency
- T(x) :
-
plate-transmission function
- P :
-
grid pitch
- u :
-
displacement in thex direction
- x, y, z :
-
coordinates
- 2πα n :
-
phase angles of the diffracted wavefronts function ofz. n=0, 1, 2, 3...
- γ:
-
slope of the straight-line portion of the curve of density vs. the logarithm of the exposure
- Δx :
-
shear in thex direction corresponding to the first-order image
- Δx n :
-
shear in thex direction corresponding to thenth-order image
- \(\overline {\Delta x} \) :
-
shear introduced in thex direction equal to 2 Δx
- Δy :
-
shear introduced in they direction
- \(\overline {\Delta y} \) :
-
shear introduced in they direction equal to 2 Δy
- ε:
-
strain
- θ:
-
quantity between zero and one
- θ n :
-
diffraction angle of thenth-order diffraction wavefront
- λ:
-
wavelength of light
- ν:
-
spatial frequency equal to 1/p
- ξ(x):
-
dimensionless coordinate equal tox/p(x)
Bibliography
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Sciammarella, C. A. andDavis, D., “Gap Effect in Moiré Fringes Observed With Coherent Monochromatic Collimated Light,”Experimental Mechanics,8 (10),459–466 (Oc. 1968).
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Sciammarella, C.A., Chang, T.Y. Optical differentiation of the displacement patterns using shearing interferometry by wavefront reconstruction. Experimental Mechanics 11, 97–104 (1971). https://doi.org/10.1007/BF02328643
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DOI: https://doi.org/10.1007/BF02328643