treme ends of the loading bar; the necessary
loading was delivered by a dead weight and
a pulley system. All loads, in every mode,
were monitored by specially designed load
cells, via a digital strain indieator.
The turbine blade was painted flat white to
enhence its reflectivity and its surface was
covered with a rectangular grid pattern to
help in identifying points of interest on
surface of the blade, Fig. 3. The three-di-
mensional coordinates of the points of inter-
section of the grid lines were determined
with respect to the coordinate systemof Fig. 4.
For each loading condition, a double-exposure
hologram was made, recording initial and final
positions of the blade, that is, its deforma-
tion due to a change in the applied load.
Every deformation, for a given recording/re-
construction geometry, resulted in a unique
fringe pattern, overlying the surface of the
blade, that was observed during reconstruc-
tion of a hologram. These fringe patterns
were analyzed, for the displacements incurred
by the blade, using the least-squares tech-
nique of holographic analysis described in
the following section.
THE LEAST-SQUARES TECHNIQUE OF HOLOGRAPHIC
DISPLACEMENT ANALYSIS
The problem of extracting displacements di-
rectly from the fringes of hologram inter-
ferometry has been solved in a number of ways,
for example, see Ref. 3. In the following,
however, we will discuss only the method known
as the least-squares technique which was found
to be most suitable for the present study and
which was, for the first time, used in holo-
graphic analysis by Dhir and Sikora [1] and
was later advanced in Refs 2 and 3.
The Illumination, Observation
and Sensitivity Vectors
The definitions of the illumination and ob-
servation vectors K, and K,, respectively,
can best be understood with respect to Fig. 5.
In this figure, the x, y, and z represent the
axes of a rectangular coordinate system with
an arbitrarily chosen origin. Let us define
à space vector R as
(1)
R=x}+y}+:zkK,
where i, J, and Kk are the unit vectors in the
$£-y-2 coordinate system. If P 1s a point on
the object corresponding to a space vector BA,,
then, for the object illuminated from a point-
source defined by R, the reconstructed virtual
image can be observed from a point described
by Ra. Let us also define the illumination
K, and observation vector K, as the propaga-
tion vectors of light from the point-source
to the object and from the object to the ob-
server, respectively. Then, using the defi-
nition of the space vector, we can write
R,-R, (x,-x Ji + 03d + (2 c ZUR f
Ek (2)
"IR -R) [px P+ (yp-yi)2 + (zp- ze
23
R,-R, (x,- xt +{y, - yp) J ?(2,- z,)k x
zk 2 2 2 Dr Ky .
IR, - RJ [xx] +0" 4) + (22 - 2) ]
Kzk (3)
In the above ‘equations, X, and X, are the unit
illumination and observation vectors, respec-
tively, and k is the magnitude of these vec-
tors defined as
23m
Kl! SZ, (4)
with A being the wavelength of the laser light.
The sensitivity vector X is defined as a dif-
ference between the observation and illumina-
tion vectors:
K-K-K. (5)
where
^ ^ ^
Kk- KA +1 + KK (6)
The components of the illumination and obser-
vation vectors needed to evaluate K can be
either obtained from Eqs 2 and 3, or using
procedures developed in Ref. 5, or by any
other suitable method.
The Displacement as a Function of the
Holographic Parameters
The least-squares technique of holographic
displacement analysis utilizes multiple obser-
vations of the reconstructed image and the
fact that the fringes and the surface of the
object can be vieved in focus by simply using
à sufficiently small observing aperture, i.e.,
one having sufficiently large f/number. If
such an objective of the optical instrument
is focused on the surface of the object for
observation, then, the path difference of the
light arriving at a given point of the foeal
plane of the objective from associated point
of the object and its displaced copy, can be
easily determined from the difference of two
sensitivity vectors x! and X^, corresponding
to the 1-st and the m-th observations, along
directions gi and K,, respectively, as shown
in Fig. 6, Chat is, -
m | m | l,m
(K -K):d = (K,-K,)-d = Q
vn AT)
im = 2,3, ..,7"
where d is the displacement vector, r is a
total number of observations, and © is the
fringe-locus function [6] defined as
ly
Q
m
"orn" (8)
In Eq. 8, nt»? is the fringe shift, that is,
the number of fringes that pass across the
point of interest on the object as the viewer
moves from observation along Kl direction to
K, direction, see Fig. 6, while continuously
observing the point on the object.
Ideally, three equations of the type of Eg. 7
Would be sufficient to determine the compo-
nents of the displacement vector d. However,
the value of the determinant of these equa-
