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Strains from holographic data

Strains in a cantilever plate obtained from second derivatives of holographically measured mode shapes

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Abstract

In this paper, reliable derivatives up to second order over a surface are obtained from holographic interferograms. This is done on a large scale, handling an amount of data that would be prohibitive if attempted by hand smoothing. This opens the way for bending and shear-strain determinations in a noncontacting way over large fields of view. The surface need be treated in no special way, in contrast to the preparation required with brittle lacquers and photoelastic coatings. Strains in the microstrain range are readily measured. Holographic interferometry in conjunction with the numerical methods mentioned herein constitutes a noncontacting optical strain gage. Furthermore, it makes possible experimental application of sophisticated modal analysis because the normal modes upon which these techniques are based are directly measured. For example, if the vibratory behavior is known, through a sufficiently large sample of the normal modes, then the static behavior may be calculated. Similarly, if the normal modes of a nonrotating body are known, its behavior in a rotating system can be calculated. Thus, the centrifugal stiffening of a turbine rotor may be predicted from a knowledge of the spectrum of its normal modes.

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Abbreviations

D :

modulus of rigidity

g in (α):

\(\begin{array}{*{20}c} {\left( n \right)\alpha ^i \left( {1 - \alpha } \right)^{n - i} } \\ i \\ \end{array}\), weighting factors

K x :

stiffness inx direction

K y :

stiffness iny direction

M y :

bending moment about they axis

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \left( {\alpha ,\beta } \right)\) :

two-dimension vector defined in the parameter space α, β which approximates the true surface

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} _{ij}\) :

coefficients of the two-dimensional Bezier vector polynomial

t :

plate thickness

u o :

refers to specific displacements at a pointx, y, z, u o parallel tox

v o :

refers to specific displacements at a pointx, y, z, w o parallel toy

w o :

refers to specific displacements at a pointx, y, z, w o parallel toz

x :

distance measured along the first axis in a right-handed orthcnormal coordinate system

y :

distance measured along the second axis in a right-handed orthonormal coordinate system

z :

distance measured along the third axis in a right-handed orthonormal coordinate system

z −1 :

plate amplitude at a particular point,x −1 , y −1

z o :

plate amplitude at a particular pointx o , y o

z +1 :

plate amplitude at a particular pointx +1 , Y +1

z(x r , y s ):

a discrete set of measured amplitudes at the pointsx r , y s which constitute the measured mode shape

\(\mathop {(n)}\limits_i \) :

\(\frac{{n!}}{{(n - i) ! i!}} !\) implies the factorial function

γ xy :

shear strain

Δx :

spacing between fringes

δΔx :

the error in the measured fringe spacing

∈ x :

bending strain along thex direction

∈ y :

bending strain along they direction

μ∈:

microstrain

ν:

Poisson's ratio

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Authors and Affiliations

  1. Los Alamos Scientific Laboratory, University of California, 87545, Los Alamos, NM

    J. E. Sollid (Staff Scientist)

  2. Instrumentation Laboratory (81), United Technologies Research Center, 06108, East Hartford, CT

    K. A. Stetson (Sentor Research Engineer)

Authors
  1. J. E. Sollid
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  2. K. A. Stetson
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Sollid, J.E., Stetson, K.A. Strains from holographic data. Experimental Mechanics 18, 208–214 (1978). https://doi.org/10.1007/BF02328415

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  • DOI: https://doi.org/10.1007/BF02328415

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