tions is very small, resulting from a limites 
size of the hologram plate, and substantial 
errors would result. Use of more than one 
hologram is not feasible because a continuous 
observation of the fringe shifts at a point 
of interest is not possible as one moves from 
observation point on one hologram to the next 
hologram. An alternative to this is to gen- 
erate an overdetermined set of linear simul- 
taneous equations relating the fringe shifts 
to three unknown components of displacement; 
one equation for each change in the direction 
of observation. Therefore, we perform multi- 
ple observations (more than three) of the 
holographically reconstructed image. For 
each observation, we compute the sensitivity 
vector X and also, we determine the corre- 
sponding fringe shift n. From multiple views, 
we obtain à set of equations of the form of 
Eq. 7 (one equation for each view), with the 
vector d eommon to all, which ean be combined 
into a Single matrix equation 
[K] (d) = (8) , (9) 
where the elements of the rectangular r x 8 
matrix [K] are differences between components 
of the corresponding observation vectors. 
Finally, we multiply both sides of Eq. 9 by 
the transpose of the rectangular matrix [X]; 
this procedure decreases the rank of the rec- 
tangular matrix [X] to a square, three by 
three, matrix and yields the solution of the 
displacement vector d which has the least- 
Squares error, = 
« » [I] (]] (1 (») 
Expanding Ed. 
(10) 
10, we obtain 
I 
m=. 
r im el, r l,m l,m Lm Lm of r Lm Im 
d,| | XAKo,AKo, YAKo, AKo, DAKo, AKo, | YQ AK, 
m=2 m=2 2 m=2 
r gm. bm. Lo im. bm Qo bm. hm T Lm hm 
dy p AK», 2D, AK2, Sz, Kp, m AK, 
C Vm. . bm L lm. im rh 
AK, AK AK» AK Q 
25 er, = 2,72. 2 
m 
l,m 
r l,m lm m 
d; 28 Kp z AK, AK 2; 
(11) 
where the elements of the square matrix are 
defined as 
l,m m | 
= KT — 12 
AKT eK KS ois (12) 
hm -_ m | 
ly | 
AK, = Ke,” Ka, (14) 
The components of the observation vector a 
entering Eqs 12-2! are as defined in Eq. 2. 
The least-squares method, described in the 
foregoing discussion, can be used as follows. 
Decide as to the number of observations r to 
be made through a single hologram, e. g 
nine as shown in Fig. 6. 
oy 
Determine the di- 
24 
rections of the observation vectors using, e. 
g., Eq. 3. Assume the sign convention for 
fringes moving across the point of interest 
on the surface of the object. Obtain fringe 
shifts while moving from an observation 
through the center of the hologram to obser- 
vation through any other point, one at a time. 
Compute the components of the observation vec- 
tors and substitute them into Eqs 12-14 and 
these results, in turn, into Eq. 11 which must 
be solved for the Cartesian components of the 
displacement vector d. 
A computer program for the solution of Eg. 11 
was developed [3] and used in the present 
Study. 
DISCUSSION OF EXPERIMENTAL RESULTS 
The maximum tensile load applied to the tested 
airfoil produced a displacement in excess of 
1/4 mm. Obviously, the displacement of this 
magnitude is too great to be successfully re- 
corded by one double-exposure hologram; it 
would result in fringe patters of frequencies 
too high to be resolved using standard proce- 
dures and construction of special facilities, 
for this purpose, would be impractical. 
Therefore, in order to circumvent this prolem, 
the maximum tensile load, required in these 
tests, was applied in increments of approxi- 
mately 200 lbs each and for each increase in 
the applied load a double-exposure hologram 
was recorded. This resulted in a total of 
ten holograms, one for each increment in the 
tensile load from zero lbs to 2,061 lbs. Now, 
the fringe patterns observed during recon- 
struction of these holograms were very dis- 
tinct and, therefore, easy to resolve. A 
typical fringe pattern due to a tensile load 
applied at tip of the blade, for airfoil No. 
1, is shown in Eig. T. 
The holographic interference patterns were ob-. 
served from five different directions through 
eaeh hologram; in particular, observation 
directions 7, 2, 4, 6, and 8, see Fig. 6, 
were used to obtain necessary parameters to 
solve Eg. 11. 
Partial displacements from each of the holo- 
grams were added up algebraically and the fi- 
nal tensile test results for airfoil No. 1 
are presented in Figs 8 and 9, where the co- 
ordinates of the airfoil were nondimensiona- 
lized with respect to the cord at the corre- 
sponding grid level. Figure 8 gives the re- 
sults at a grid level 26 (see Fig. 3) whereas 
Fig. 9 shows deflections at level 11. The ex- 
perimental results presented in these figures 
show that for a blade in tension, maximum dis- 
placements occured at the leading edge. More 
specifically, for a tensile load of 2,061 lbs 
applied at tip of the blade, the components 
of the displacement vector at level 26 were: 
d, = -79.52 um, d, = 160.56 yum, and d, s 
204.51 um, while those at level 11 were: d, = 
-25.28 um, d, = 104.00 ym, and d, = 120.23 um. 
For this reason, the deflections presented in 
Figs 8 and 9 were magnified sixty (60) times 
in order to amplify the differences between 
the initial and final positions of the airfoil. 
Furthermore, the experimental results presented 
in Fig. 9 are compared with the theoretical 
results obtained from finite element analysis 
of the airfoil. As is clearly seen from this 
figure, the agreement between the experiments 
and theory is very good at the leading edge,