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A ligtenberg method for plate bending studies using laser speckles
MECHANICS RESEARCH COMMUNICATIONS 0093-6413/80/040241-06502.00/0
Vol. 7(4),241-246,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd.
A LIGTENBERG METHOD FOR PLATE BENDING STUDIES USING LASER SPECKLES
C. J. Lin and F. P. Chiang Department of Mechanical Engineering, State University of New York at Stony Brook, Stony 8rook, Long Island, New York 11794, U.S.A. (Received 21 April 1980; accepted for print 13 May 1980)
Introduction
The classical Ligtenberg method [1] is one of the better experimental methods of solving plate-bending problems. It yields slope rather than deflection contours, thus reducing errors associated with numerical differentiation. Although it requires a mirror-surfaced model, ordinary material such as Plexiglas can be utilized without special treatment. The experimental procedures and apparatus are simple. Originally, the Ligtenberg method was developed using line grating, and later Chiang and Jaisingh [2] modified it by using a cross grating so that it can be applied to transient wave problems. Its basic concept was also employed when Chiang and Juang introduced a laser speckle method [3] using matte surface models. In this paper we increase the versatility of the original Ligtenberg method by using laser speckles as a random grating.
Theory of the Method
I \ LASER
LIGHT
,.
~.Z.Z ,
, GROUND
fSPECKLE Si ~1~
~
, % - . ~ . ~~- f- _- - - - ~ i ~ , s 2~ -
p--------.::.t-~ ~
[/-,--MIRROR ~
~ ---D
/
GLASS
/BEAMSPLT I TER ~
~
_
CAMERA
~ IZM I/ AGN I ARYGRATN I GPLAN
.
~ii,Ji
FIG. 1 Experimental Arrangement for Recording Laser Speckles 241
242
C.J. LIN and F.Po CHIANG
The experimental
arrangement
is as shown in Fig. i.
A ground glass is illumi-
nated by an expanded laser beam generating a volume of speckles multiple interference it.
from each point o11 the ground glass)
(the result of
in the space behind
A partial mirror is used to direct the speckles to the mirror-surfaced
model which in turn reflects them back into the space in front of it.
A camera
is used to focus on the speckles at a short distance D in front of it. can also focus the virtual
speckles behind the model.)
(One
These speckles tilt
when the mirror surface is tilted in much the same way as light rays are tilted. Assuming that under load the plate deflection is small such that the same spatial coordinates
can be used to represent points before and after deformation
(see
insert in Fig. i), it can be seen from Fig. 1 that a speckle displacement d at the focused plane due to a local plate tilt 4 is given by the following equation: d = D tan 24(1 + tan 2 0) 1 tan @ tan 24 ' where @ is the viewing angle.
(i)
When both @ and 4 are small, Eq.
(1) may be
approximated by (with added vector notation) d :
2 D4
(2)
This is the same equation as given by Ligtenberg except that it is in vector form.
The vector form is necessary because unlike the Ligtenberg method where
only the component of ~ and ~ along the principal direction of the grating is detectable,
the present method can give slope components along any direction.
If the speckles at p l a n e D a r e
photographed
twice by double exposure,
one before
and one after the load, the slope information is registered on the specklegram in terms of speckle shifts.
The information can be delineated by inserting the
specklegram in a convergent coherent light field as shown in Fig. 2.
The dif-
fraction spectrum of the specklegram can be shown to have an intensity distribution given by the following equation l(r) = 4cos 2 k
[4]:
Is(r)
(3)
where L is the distance between the specklegram and the transform plane, k the wave number,
Is(r) the diffraction spectrum of a single exposure recording of
the speckle pattern,
and r tile position vector of a filtering aperture through
which the image of the plate is reconstructed.
Thus, the result of the second
exposure is the presence of a series of cos 2 fringes modulating the original diffraction halo.
These fringes are governed by the following equation, using
Eq. (2), 2D(r • ~) = NXL ,
(4)
LIGTENBERG
/I I
(( ~
~---
.~
METHOD
n
USING
LASER
~
SPECKLES
243
FILTERING
~ ( A P E R T U R E
TRANSFORM L .NS
~
'
\
./
I
I
-
TRANSFORMAND ~ I I FILTERINGPLANE RECORDING LENS IMAGE PLANE FIG. 2
Optical Arrangement for Spatial F i l t e r i n g where N = 0, ±i, ±2, ... for light fringes and ±½, ±1½, ±2½, ... for dark fringes These fringes, However,
in general,
are not visible because ~ varies from point to point.
if a filtering aperture is placed at r x and ry, respectively,
and recon-
struction of the plate image is made through them, fringes of constant partial slope ~x = ~w/3x and ~y = ~w/3y are seen to cover the plate.
Thus from Eo.
(4)
with the angle between r x and ex' and ry and ev being zero, one has 3w N~L - ~ = 2Dr x
and
3w N~L ~-y 2Dry
(S)
It can be seen that these equations are identical to the field equation of the Ligtenberg method with an equivalent grating pitch p = ~L/r.
Unlike the Ligten-
berg method wherein the pitch and orientation of the grating are fixed, the laser speckle method has a continuously variable equivalent grating pitch and orientation;
and they can be selected at will at the postrecording
stage.
Thus,
one can choose a suitable sensitivity for the problem at hand, and the easy change of sensitivity can also be used as a means to determine the sign of fringes
[5].
An example of postrecording
sensitivity change is shown in Fig. 3
where the fringe patterns are the ~w/3x (or ~w/~y due to the radial symmetry) contours of a clamped circular plate under central point load. The capability of postrecording demonstrated
in Fig. 4.
change of equivalent grating orientation
The fringe patterns are the partial
along three different directions
slope contours
(0 °, 45 ° , and 90 ° from x axis) of a clamped
triangular plate under central point load. patterns,
is
the fringes have identical
Since r is the same for all the
sensitivity.
The advantage of being able
to obtain slope contours along three or more directions
is the fact that one
244
C.J. LIN and F.P. CHIANG
•
_/aw~ -~,y-;/%,
o---a
,:
_
(0)
a
(b/
b
F.T.P.
FIG. 3 Slope Contours of Different Sensitivity from a Single Specklegram are Obtained at Different Filtering Distance r
(a]
(b)
"\
,;
(c)
FIG. 4 Slope Contours Along Three Directions from a Single Specklegram
can use rosette
[6] or least square approach to increase the accuracy of the
result. While the full-field filtering approach yields slope components along particular directions, maximum slope at each and every point can also be obtained by the so-called pointwise filtering technique
[4].
This is done by pointing a narrow
laser beam at a point (with the beam diameter of about 2 mm, ~ is assumed to be constant) Equation
and receiving the diffraction spectrum at a distance L with a screen. (3) still applies, one has
LIGTENBERG
-
METHOD
USING LASER SPECKLES
245
iqXL 2Dr
(6)
governing a series of straight fringes (the so-called Young's fringes) with r being normal to the fringes.
A series of Young's fringes along a radial direc-
tion of a clamped circular plate under central point load is shown in Fig. 5.
(b)
(a)
FIG. 5 Young's Fringe Patterns at Selected Points
~W
P
00-4)
~:~3jC::~ Wmox =O.028mm
I0"
~,'O'~'~o
5-I.0 k
-0.5 I
"X~
,
I
"~
oX
%\
o~
-
-I-~°
!
FIG. 6 Comparison Between Theoretical and Experimental Results of a Circular Plate
The validity of the method is also demonstrated as shown in Fig. 6 where the experimental result of the clamped circular plate is compared to the theoretical solution.
Agreement is quite good in regions away from the boundary.
boundary, the experimentally obtained slopes have higher values.
Near the
This is to be
expected because the theoretical fixed end condition could not be simulated with precision experimentally.
From Fig. 6 it can be seen that the method has a
sensitivity of the order of 10 -4 .
246
C.J. LIN and F.Po CHIANG
Conclusion and Discussion
We have modified the Ligtenberg method by replacing the original moir~ grating with a laser speckle pattern.
There are several advantages of this approach.
First of all slope contours along any direction can be generated from a single laser specklegram at the postrecording stage. can also be varied at the postrecording stage.
Secondly, the method's sensitivity Unlike the original Ligtenberg
method wherein a higher sensitivity requires the use of a fine grating, the present method has a continuous spectrum of sensztivities from the recorded speckles. Thirdly, the method's experimental apparatus is more versatile in that the focused plane can be easily changed which offers yet another means of changing the method's sensitivity.
The method can also be applied to transient problems
with the help of a pulse laser and to steady state vibration problems.
The
disadvantages are: the necessity of having to have a laser and the fact that the fringe quality is not as good as the conventional one.
The need of a mirror-like
model surface (which is common to both the classical Ligtenberg method and the present one) should not be taken as a disadvantage, for ordinary material such as Plexiglas has a natural finishing which is good enough for this purpose.
Besides,
there may be situations wherein the prototype plate is "naturally" mirror-like and contamination such as painting is not allowed. is uniquely applicable.
For such problems, the method
It is a complementary technique to another laser speckle
method [3] which can only be applied to matte surface models.
Acknowledgment
The authors wish to thank the National Science Foundation for supporting the work reported here.
References
i. 2. 3. 4. 5. 6.
Fo K. Ligtenberg, Proc. SESA, 12, 83 (1954). F. P. Chiang and G. Jaisingh, Ex-~-perimental Mechanics, 14, 459 (1974). F. P. Chiang and R. M. Juang, Applied Optics, 15, 2199 (1976). R. P. Khetan and F. P. Chiang, Applied Optics, 15, 2205 (1976). F. P. Chiang, Progress in Experimental Mechanics, Durelli Anniversary Volume, The Catholic University of America Press, Washington, D.C., 99 (1975). F. P. Chiang, J. of Engrg. Mechanics Div., Proc. ASCE, 96 EM6, 1285 (1970).
Vol. 7(4),241-246,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd.
A LIGTENBERG METHOD FOR PLATE BENDING STUDIES USING LASER SPECKLES
C. J. Lin and F. P. Chiang Department of Mechanical Engineering, State University of New York at Stony Brook, Stony 8rook, Long Island, New York 11794, U.S.A. (Received 21 April 1980; accepted for print 13 May 1980)
Introduction
The classical Ligtenberg method [1] is one of the better experimental methods of solving plate-bending problems. It yields slope rather than deflection contours, thus reducing errors associated with numerical differentiation. Although it requires a mirror-surfaced model, ordinary material such as Plexiglas can be utilized without special treatment. The experimental procedures and apparatus are simple. Originally, the Ligtenberg method was developed using line grating, and later Chiang and Jaisingh [2] modified it by using a cross grating so that it can be applied to transient wave problems. Its basic concept was also employed when Chiang and Juang introduced a laser speckle method [3] using matte surface models. In this paper we increase the versatility of the original Ligtenberg method by using laser speckles as a random grating.
Theory of the Method
I \ LASER
LIGHT
,.
~.Z.Z ,
, GROUND
fSPECKLE Si ~1~
~
, % - . ~ . ~~- f- _- - - - ~ i ~ , s 2~ -
p--------.::.t-~ ~
[/-,--MIRROR ~
~ ---D
/
GLASS
/BEAMSPLT I TER ~
~
_
CAMERA
~ IZM I/ AGN I ARYGRATN I GPLAN
.
~ii,Ji
FIG. 1 Experimental Arrangement for Recording Laser Speckles 241
242
C.J. LIN and F.Po CHIANG
The experimental
arrangement
is as shown in Fig. i.
A ground glass is illumi-
nated by an expanded laser beam generating a volume of speckles multiple interference it.
from each point o11 the ground glass)
(the result of
in the space behind
A partial mirror is used to direct the speckles to the mirror-surfaced
model which in turn reflects them back into the space in front of it.
A camera
is used to focus on the speckles at a short distance D in front of it. can also focus the virtual
speckles behind the model.)
(One
These speckles tilt
when the mirror surface is tilted in much the same way as light rays are tilted. Assuming that under load the plate deflection is small such that the same spatial coordinates
can be used to represent points before and after deformation
(see
insert in Fig. i), it can be seen from Fig. 1 that a speckle displacement d at the focused plane due to a local plate tilt 4 is given by the following equation: d = D tan 24(1 + tan 2 0) 1 tan @ tan 24 ' where @ is the viewing angle.
(i)
When both @ and 4 are small, Eq.
(1) may be
approximated by (with added vector notation) d :
2 D4
(2)
This is the same equation as given by Ligtenberg except that it is in vector form.
The vector form is necessary because unlike the Ligtenberg method where
only the component of ~ and ~ along the principal direction of the grating is detectable,
the present method can give slope components along any direction.
If the speckles at p l a n e D a r e
photographed
twice by double exposure,
one before
and one after the load, the slope information is registered on the specklegram in terms of speckle shifts.
The information can be delineated by inserting the
specklegram in a convergent coherent light field as shown in Fig. 2.
The dif-
fraction spectrum of the specklegram can be shown to have an intensity distribution given by the following equation l(r) = 4cos 2 k
[4]:
Is(r)
(3)
where L is the distance between the specklegram and the transform plane, k the wave number,
Is(r) the diffraction spectrum of a single exposure recording of
the speckle pattern,
and r tile position vector of a filtering aperture through
which the image of the plate is reconstructed.
Thus, the result of the second
exposure is the presence of a series of cos 2 fringes modulating the original diffraction halo.
These fringes are governed by the following equation, using
Eq. (2), 2D(r • ~) = NXL ,
(4)
LIGTENBERG
/I I
(( ~
~---
.~
METHOD
n
USING
LASER
~
SPECKLES
243
FILTERING
~ ( A P E R T U R E
TRANSFORM L .NS
~
'
\
./
I
I
-
TRANSFORMAND ~ I I FILTERINGPLANE RECORDING LENS IMAGE PLANE FIG. 2
Optical Arrangement for Spatial F i l t e r i n g where N = 0, ±i, ±2, ... for light fringes and ±½, ±1½, ±2½, ... for dark fringes These fringes, However,
in general,
are not visible because ~ varies from point to point.
if a filtering aperture is placed at r x and ry, respectively,
and recon-
struction of the plate image is made through them, fringes of constant partial slope ~x = ~w/3x and ~y = ~w/3y are seen to cover the plate.
Thus from Eo.
(4)
with the angle between r x and ex' and ry and ev being zero, one has 3w N~L - ~ = 2Dr x
and
3w N~L ~-y 2Dry
(S)
It can be seen that these equations are identical to the field equation of the Ligtenberg method with an equivalent grating pitch p = ~L/r.
Unlike the Ligten-
berg method wherein the pitch and orientation of the grating are fixed, the laser speckle method has a continuously variable equivalent grating pitch and orientation;
and they can be selected at will at the postrecording
stage.
Thus,
one can choose a suitable sensitivity for the problem at hand, and the easy change of sensitivity can also be used as a means to determine the sign of fringes
[5].
An example of postrecording
sensitivity change is shown in Fig. 3
where the fringe patterns are the ~w/3x (or ~w/~y due to the radial symmetry) contours of a clamped circular plate under central point load. The capability of postrecording demonstrated
in Fig. 4.
change of equivalent grating orientation
The fringe patterns are the partial
along three different directions
slope contours
(0 °, 45 ° , and 90 ° from x axis) of a clamped
triangular plate under central point load. patterns,
is
the fringes have identical
Since r is the same for all the
sensitivity.
The advantage of being able
to obtain slope contours along three or more directions
is the fact that one
244
C.J. LIN and F.P. CHIANG
•
_/aw~ -~,y-;/%,
o---a
,:
_
(0)
a
(b/
b
F.T.P.
FIG. 3 Slope Contours of Different Sensitivity from a Single Specklegram are Obtained at Different Filtering Distance r
(a]
(b)
"\
,;
(c)
FIG. 4 Slope Contours Along Three Directions from a Single Specklegram
can use rosette
[6] or least square approach to increase the accuracy of the
result. While the full-field filtering approach yields slope components along particular directions, maximum slope at each and every point can also be obtained by the so-called pointwise filtering technique
[4].
This is done by pointing a narrow
laser beam at a point (with the beam diameter of about 2 mm, ~ is assumed to be constant) Equation
and receiving the diffraction spectrum at a distance L with a screen. (3) still applies, one has
LIGTENBERG
-
METHOD
USING LASER SPECKLES
245
iqXL 2Dr
(6)
governing a series of straight fringes (the so-called Young's fringes) with r being normal to the fringes.
A series of Young's fringes along a radial direc-
tion of a clamped circular plate under central point load is shown in Fig. 5.
(b)
(a)
FIG. 5 Young's Fringe Patterns at Selected Points
~W
P
00-4)
~:~3jC::~ Wmox =O.028mm
I0"
~,'O'~'~o
5-I.0 k
-0.5 I
"X~
,
I
"~
oX
%\
o~
-
-I-~°
!
FIG. 6 Comparison Between Theoretical and Experimental Results of a Circular Plate
The validity of the method is also demonstrated as shown in Fig. 6 where the experimental result of the clamped circular plate is compared to the theoretical solution.
Agreement is quite good in regions away from the boundary.
boundary, the experimentally obtained slopes have higher values.
Near the
This is to be
expected because the theoretical fixed end condition could not be simulated with precision experimentally.
From Fig. 6 it can be seen that the method has a
sensitivity of the order of 10 -4 .
246
C.J. LIN and F.Po CHIANG
Conclusion and Discussion
We have modified the Ligtenberg method by replacing the original moir~ grating with a laser speckle pattern.
There are several advantages of this approach.
First of all slope contours along any direction can be generated from a single laser specklegram at the postrecording stage. can also be varied at the postrecording stage.
Secondly, the method's sensitivity Unlike the original Ligtenberg
method wherein a higher sensitivity requires the use of a fine grating, the present method has a continuous spectrum of sensztivities from the recorded speckles. Thirdly, the method's experimental apparatus is more versatile in that the focused plane can be easily changed which offers yet another means of changing the method's sensitivity.
The method can also be applied to transient problems
with the help of a pulse laser and to steady state vibration problems.
The
disadvantages are: the necessity of having to have a laser and the fact that the fringe quality is not as good as the conventional one.
The need of a mirror-like
model surface (which is common to both the classical Ligtenberg method and the present one) should not be taken as a disadvantage, for ordinary material such as Plexiglas has a natural finishing which is good enough for this purpose.
Besides,
there may be situations wherein the prototype plate is "naturally" mirror-like and contamination such as painting is not allowed. is uniquely applicable.
For such problems, the method
It is a complementary technique to another laser speckle
method [3] which can only be applied to matte surface models.
Acknowledgment
The authors wish to thank the National Science Foundation for supporting the work reported here.
References
i. 2. 3. 4. 5. 6.
Fo K. Ligtenberg, Proc. SESA, 12, 83 (1954). F. P. Chiang and G. Jaisingh, Ex-~-perimental Mechanics, 14, 459 (1974). F. P. Chiang and R. M. Juang, Applied Optics, 15, 2199 (1976). R. P. Khetan and F. P. Chiang, Applied Optics, 15, 2205 (1976). F. P. Chiang, Progress in Experimental Mechanics, Durelli Anniversary Volume, The Catholic University of America Press, Washington, D.C., 99 (1975). F. P. Chiang, J. of Engrg. Mechanics Div., Proc. ASCE, 96 EM6, 1285 (1970).