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Experimental Mechanics

, Volume 5, Issue 5, pp 154–160 | Cite as

Basic optical law in the interpretation of moiré patterns applied to the analysis of strains—part 1

The moiré-displacement-field optical law is generalized. The light intensity at a point is related to its relative displacement. By applying the generalized law, the precision of the moiré method is increased far beyond the present limits
  • Cesar A. Sciammarella
Article

Abstract

The objective of this paper is to generalize the optical law that relates the displacement field to the fringes of moiré patterns. Until now, a discontinuous relationship has been applied. Displacements that are equal to an integral number of times the master grid pitch or half the master grid pitch are related to the points of maximum and minimum light intensity, respectively.

The generalized optical law gives a continuous relation-ship between displacements and light intensities. By applying this law it is possible to increase the precision of moiré far beyond the actual limits.

Keywords

Mechanical Engineer Light Intensity Fluid Dynamics Displacement Field Actual Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a

radius of the pupil of the optical system equal torR

A

area of the exit pupil of the optical system

b

width of the light-intensity rectangular pulse

\(\bar b\)

average value ofb

cn

coefficient of the Fourier expansion of the grid transmission function n=0,1, 2, 3...

E

amplitude of the light-intensity rectangular pulse

f(x, y)

grid transmission function

fT(x,y)

transmission function resulting from the superposition of two grids

\(\bar fr(x'y')\)

cross-correlation function off A (x,y) andf B (x,y)

F(g, h)

Fourier transform off(x, y)

\(\bar F_r (g, h)\)

Fourier transform of\(\bar f_r (x, y)\)

g

spacial frequency

G(g, h)

pupil function of the image-forming system

h

spacial frequency

i

imaginary unit

I0

image average background light intensity

I1

light-intensity amplitude of the image first harmonic

I(x)

light intensity at a point of coordinatex in the image plane

\(\Im (g, h)\)

Fourier transform of the light-intensity distribution function in the image plane

k

integer 1, 2, 3...

K

coefficient smaller than one

l

integer 1, 2, 3....

L

range of integration for the computation of the Fourier transform

\(\mathfrak{L}(g, h)\)

frequency-response function of the image-forming system

n

integer 1, 2, 3....

m

magnification of the optical system

p

grid pitch

p1,p2

pitches of the gridsA andB, respectively

\(\bar p\)

average pitch

px

grid pitch in thex-axis direction

p(x)

local pitch

\(\bar p_x\)

average value ofp x

pm

grid pitch of the image of the grid

R

radius of the Gaussian sphere

r

radius of the pupil of the optical system

S

grid area

s

integer 1, 2, 3...

u

displacement in the direction of thex axis

Uo*(x,y)U0(x,y)

disturbances arriving at and leaving the grid, respectively

x

coordinate

x′

relative displacement in thex-axis direction independent ofx

y

coordinate

y′

relative displacement in they-axis direction independent ofy

β

angle whose cosine ish/2a

δ

fringe spacing in thex-axis direction

\(\delta (\xi - \xi _1 )\)

Dirac delta function

Δp(x)

local change of pitch

Δθ(x)

local change of the angle θ

Δπ(x)

local change of π

θ

rotation of gridB with respect to thex axis

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References

  1. 1.
    Aciammarella, C. A., andDurelli, A. J., “Moiré Fringes as a Means of Analyzing Strains,”Transactions of the ASCE Paper No. 3329, 127, pt I, 582–603 (1962).Google Scholar
  2. 2.
    Guild, G., “The Interference System of Crossed Diffraction Gratings. Theory of Moiré Fringes,” Oxford at the Clarendon Press (1956).Google Scholar
  3. 3.
    Born, A., and Wolf, E., “Principle of Optics,” Pergamon Press, 479 (1959).Google Scholar
  4. 4.
    Encyclopedia of Physics. Edited by S. Flugge, XXIV, Fundamentals of Optics. “Interferences, diffraction et polarization,” 331 (1956).Google Scholar

Copyright information

© Society for Experimental Mechanics, Inc. 1965

Authors and Affiliations

  • Cesar A. Sciammarella
    • 1
  1. 1.Advanced Mechanics Research SectionUniversity of FloridaGainesville

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