In vibration and acoustics, a knowledge of modal fre-quency and shape of vibration is important for gain-ing insight into the vibration behavior of structures.In mathematical terms, these two characteristics arerelated to the eigenvalues and eigenfunctions of thesystem. In a general bending vibration problem,however, they are not readily available, as the closed-form solutions to the fourth-order differential equa-tion may not exist.1 Mode shapes and frequenciesare therefore required to be measured experimen-tally. Traditional methods use accelerometers andsignal analyzers for modal analysis. Although theyare precise and reliable for measuring modal param-eters for structural modeling, the methods are point-wise, time consuming, and contacting. Opticalvibrometers are the modern alternative to accelerom-eters. They are noncontacting. However, they arestill a pointwise instrument and would be time con-suming for modal analysis.Holographic interferometry has been a well-known
The authors are with the Department of Mechanical Engineer-ing, The Hong Kong Polytechnic University, Hung Hom, Kowloon,Hong Kong.Received 10 July 1996; revised manuscript received 28 October
3776 APPLIED OPTICS y Vol. 36, No. 16 y 1 June 1997
noncontacting technique for vibration measurementin the full field. It has been deterred by dark-roomfilm-development practice from wide applications.However, the principle has been employed to developa dry type of interferometry, known as electronicspeckle pattern interferometry ~ESPI!. The furtherdevelopment of the basic technique has led to a moreconvenient setup, shearography, which measures thevibration amplitude gradient directly.2,3 The elec-tronic version of this is often referred to as electronicspeckle shearing interferometry. The capability ofESPI or shearography has now been enhanced by theuse of high-speed computers with modern imageprocessors.3–5 The phase-shift method has permit-ted both techniques for analyzing vibration quantita-tively.6ESPI-basedmethods of measuring vibration have a
few different techniques, for example, the time-averaged technique, the stroboscopic technique, andthe pulsed laser technique. This paper concerns thetime-averaged technique only. In the traditionaltime-averaged ESPI technique, after dc filtering andrectification, the monitored fringe pattern is thesquared zero-order Bessel function of the vibrationamplitude with a very low high-order fringe contrast.Only a few orders can be visualized directly from thescreen, making the quantitative measurement of vi-bration difficult if not impossible. The sinusoidalphase-modulation technique2 permits a completemapping of the amplitude and the phase contours
across any vibrating surface. In this technique, oneof the light paths is modulated with a sinusoidalfrequency equal to the frequency of and in synchro-nization with the vibration of the structure. Thispermits the time-averaged ESPI to measure thephase of vibration.7Comparing ESPI and shearography, one would
sometimes favor shearography. This is becauseESPI is influenced by vibration and the temperatureof the environment, whereas shearography is rela-tively insensitive to rigid body motion and environ-mental factors. Moreover, when higher-orderderivatives of vibration amplitude are measured,shearography is more suitable than ESPI.8Nakadate et al.3 have used a frame subtraction
method in speckle shearing interferometry to mea-sure the vibration amplitude gradient, showing bet-ter fringe contrast but less measurement sensitivitythan that of single-image time-averaged shearogra-phy. In the regions of high vibration gradients, thepresence of nonzerominimal points of the fringe func-tion would limit the fringe visibility. For a givenbackground noise level, the nonzero minima are re-sponsible for the deterioration of the fringe contrast.9Nakadate10 and Saldner et al.6 have employed thephase-step method, trying to measure plate vibrationmode shapes quantitatively. By this method, theyobtained better fringe visibility and quantitative vi-bration values on the plates. The method they usedrequires one to grab several frames of images for onemeasurement and to process the image data point bypoint. Recent advancements in computer andimage-processing technologies may help to improvethe speed. Nevertheless, for certain vibration anal-yses, a quantitative measurement might not be nec-essary. In such a situation, fringe visibility andmeasurement efficiency are relatively more impor-tant.In this paper, a new time-averaged frame subtrac-
tion technique is introduced for vibration analysis bydigital speckle shearing interferometry. The exper-imental setup is similar to the traditional shearo-graphic measurement system, but the techniquepermits the enhancement of fringes by the subtrac-tion of two Bessel fringe patterns at different forcinglevels. As far as the qualitative measurement of vi-bration is concerned, the present technique is moreefficient and easier than the phase-shift method forfringe enhancement. Furthermore, the new tech-nique provides a means for the fast inspection of thevibration behavior of a structure. A fringe interpre-tation method is also described for the convenientidentification of antinodal positions and contours ofzero strains of plate vibration.
2. Theory of Measurement
Transverse vibration is measured by using the time-averaged shearing interferometry method. Theschematic diagram of the optical setup is shown inFig. 1. The object is either a cantilever beam or aclamped circular plate, each with a diffuse surface.Vibration is assumed to be linear in this study. Pho-
tographic films will not be used. Instead a CCD im-aging system is used with a shearing lens of thebirefringent type. A suitable polarizer is employedto bring the two emerged orthogonally polarized wavefronts to the same polarization to ensure the bestinterference. The framegrabber is controlled by amicrocomputer for image processing. A He–Ne la-ser is used as a light source. Its wavelength is l.The shearing lens brings the rays from two points
P and P9 on the object surface to a point O in the CCDcamera. The shearing lens is made and oriented tolet points P and P9 be represented by coordinates ~x,y, z! and ~x 1 dx, y, z!, respectively. That is, theimage of the object is sheared in the x axis; dx shouldbe small enough to obtain a sufficient accuracy of thegradient measurement and a good interference of thelight beams. The intensity of the image recorded bythe camera is written as11
Is 5 2 Io~1 1 cos f!, (1)
where Io is the object image intensity at O and f isthe phase difference between light paths from P9 andP to O. Here Io and f are random in space over theimage surface.Consider vibration at the nth natural mode of fre-
quency vn. The out-of-plane displacement of the ob-ject is expressed as
w~x, y, t! 5 wo~x, y!cos vnt, (2)
where wo~x, y! is the vibration amplitude at ~x, y!,representing a standing wave at frequency vn.The vibrations of P and P9 are, in general, repre-
sented by ~u, v, w! and ~u 1 du, v 1 dv, w 1 dw!,respectively. These produce a further phase shiftbetween the light paths to point O in the image plane.Equation ~1! then becomes
I 5 2Io@1 1 cos~f 1 D!#, (3)
Fig. 1. Schematic setup of time-average speckle pattern shearinginterferometry.
1 June 1997 y Vol. 36, No. 16 y APPLIED OPTICS 3777
Fig. 2. Comparison of differentBessel fringe functions ~normal-ized values!.
where
D 52p
l~Adx 1 Adu 1 Bdv 1 Cdw!, (4)
A 5x 2 xoRo
1x 2 xsRs
,
B 5y 2 yoRo
1y 2 ysRs
,
C 5z 2 zoRo
1z 2 zsRs
, (5)
Ro2 5 xo
2 1 yo2 1 zo
2, and Rs2 5 xs
2 1 ys2 1 zs
2.According to Hung,12 A, B, and C are sensitivity fac-tors relating to the illumination position point S~xs,ys, zs! and the camera image point O~xo, yo, zo!.Because dx is small, Eq. ~4! can be rewritten as
D 52p
l SA 1 A]u]x
1 B]v]x
1 C]w]xDdx. (6)
With normal illumination and normal viewing xs 5 ys5 xo 5 yo 5 0, and for Ro and Rs to be large comparedwith the object dimensions, A > 0, B > 0, and C > 2.Then, by substitution of Eq. ~2! into Eq. ~6!,
D 54p
l~dx!
]w]x
54p
l~dx!
]wo
]xcos~vnt!. (7)
For vn very much higher than the grabbing rate ofthe imaging system, the intensity distribution re-corded by the CCD camera will be the average inten-
3778 APPLIED OPTICS y Vol. 36, No. 16 y 1 June 1997
sity value expressed by
Iavg 51T *
0
T
2IoF~1 1 cos~f 1 D!#dt
51T *
0
T
2IoH1 1 cosFf 14p
l~]x!
]wo
]wcos~vnt!GJdt
5 2IoH1 1 cosf JoF4p
l~dx!
]wo
]xcos~vnt!GJ, (8)
where Jo is the Bessel function of the first kind of thezeroth order and T is the grabbing time of one imagerecord.Subtracting Eq. ~1! from Eq. ~8!, one obtains
DI 5 Is 2 Iavg
5 2Io cos fH1 2 JoF4p
l~dx!
]wo~x, y!]x GJ . (9)
To evaluate the average brightness of the fringe onthe TV monitor, one may assume that the resultantlight amplitude at the CCD obeys zero-mean circularcomplex Gaussian statistics.13 The assumption isjustified as the object in this study has a diffuse sur-face. According to Nakadate et al.,3 if the signalafter subtraction is square-law detected, the averagebrightness of the fringe is proportional to
E~x, y! 5 cH1 2 JoFk ]wo~x, y!]x2 , (10)
where c 5 ~^Is&2 T2y2! and k 5 4p~dxyl!. In time-
averaged ESPI, the vibration analyzed is usually anormal mode. Therefore, wo~x, y! in Eq. ~10! is themode shape written as rFn~x, y!, with r depending onthe magnitude of the force excitation. Here Fn~x, y!is referred to as the normalized mode shape; c is a
function of x and y. For a specified x and y, c is aconstant with a value depending on the optical imagesetup and the grabbing time.Equation ~10! and Fig. 2 show that the method of
Ref. 3 has a disadvantage. It is because the fringevisibility described by the Bessel function is decreas-ing at higher-order loops because of the presence ofnonzero minima of the fringe function. From Eq. ~8!and the same figure, the traditional single-frame Jo
2
fringe pattern also has low visibility at higher-orderloops. This is due to the presence of the self-interference term, 2Io, of the equation.In view of the above, an attempt is made here to
improve the Bessel fringe visibility by a new fringe-generation method. The method generates time-averaged vibration fringes by subtracting two Bessel
fringe patterns at two different forcing levels. Byrewriting Eq. ~2! for these two levels, we see that thelinear vibration displacements are
w1~x, y, t! 5 r1Fn~x, y!cos vnt, (11a)
w1~x, y, t! 5 r1Fn~x, y!cos vnt, (11b)
where r1 and r2 are amplitudes determined by theforcing levels used andFn is the nth normalizedmodeshape. The brightness of the new fringe function isthen proportional to
1 June 1997 y Vol. 36, No. 16 y APPLIED OPTICS 3779
Quite clearly, Eq. ~10! can be treated as a special caseofEq. ~12!when r1 is zero. Takeacantileverbeamasanexample. Figure 2 shows the brightness function com-puted by Eq. ~12!. As we can see in the figure, for thehigher-order loops, the visibility is markedly improvedcompared with the contrast of the fringes calculated byEq. ~10! or by Jo
2. Such enhancement can be obtainedby adjusting the ratio between r1 and r2. In this case, r15 1 and r2 5 0.8. From our analysis, the result showsthat electronic noise and camera noise would exert ap-preciable influences on the fringe contrast value. A sep-arate paper has already been prepared to study this andwill be published elsewhere.According to the mathematical table, J0
2@k~]Fny]x!# is close to @2y~pk]Fny]x!#cos2@k~]Fny]x! 2 ~py4!#when the argument k~]Fny]x! is very large. Thefringe function described by Eq. ~12! has zeros atintervals approximately equal to 2py~r1 1 r2!.Thus, not only has the fringe contrast been improved,but also the loss of sensitivity of the subtractionmethod of Nakadate et al.3 has been recovered.With this enhancement of fringe pattern, the numberof fringes that can be resolved by this method is in-creased dramatically, leading to a better spatial res-olution of the vibration mode measurement.
3. Shearographic Image Pattern: Interpretation
A. Fringe Pattern Caused by Beam Vibration
As an illustration, the third bending vibration modeshape of the cantilever beam of length L and its de-rivative are considered first. They are calculatedand plotted as curves D and C in both Figs. 3~a! and3~b!. The curves are simply for reference in the de-scription of the features of the fringe functions, whichare given below. The vertical axes are the ratios ofthe local vibration amplitudes or their gradients tothe maximum values at the free end of the cantileverbeam. Figure 3~a! is for the brightness functionscalculated by Eq. ~10!, shown as curvesA andB in thediagram for two forcing levels. Figure 3~b! is for thesame functions calculated by Eq. ~12!, also shown ascurves A and B, respectively, for two forcing levels.We can see in Fig. 3~b! that, except for the location
of contraflexure, the minimum brightness value as itoccurs is zero. As mentioned above, the fringe visi-bility in the case of Fig. 3~b! should be much betterthan that of Fig. 3~a!.There are two interesting features about the fringe
patterns. One, Eqs. ~10! and ~12! show that ~]Fny]x! 5 0 at the antinodal positions, implying that thefringes should be completely dark because E 5 0.~For a cantilever beam, of course, the gradient at thebuilt-in end is also zero. A dark fringe also appearsthere.! Two, at the points of contraflexure of thevibrating beam, the brightness values are eithermaxima or minima. We can see this by using thefollowing equations derived from differentiating Eqs.
3780 APPLIED OPTICS y Vol. 36, No. 16 y 1 June 1997
~10! and ~12!, respectively:
]E]x
5 2ckH1 2 JoFk ]Fn~x!]x GJJ1Fk ]Fn~x!
]x GF]2Fn~x!]x2 G ,
(13)
]E]x
5 2ckHJoFr1k]Fn~x!]x G 2 JoFr2k]Fn~x!
]x GJ3 Hr1J1Fr1k]Fn~x!
]x G 2 r2J1Fr2k]Fn~x!]x GJ
3 F]2Fn~x!]x2 G , (14)
where J1 is the Bessel function of the first order. Atthese locations of maxima and minima, the secondderivative ]2Fny]x2 5 0. The surface strains in-duced by bending at these locations should be zero.As vibration of the beam at a natural frequency be-haves as a standing wave, the locations of zero-bending strains and antinodes should be stationary.The fringes formed there should be unmoved.As shown in Figs. 3~a! and 3~b!, the zeroth-order
fringes of high contrast appear at the antinodeswhile thehigh-order fringes of low contrast appear at the con-traflexure locations. As we can see from the figures, inbetween the consecutive zero-strain and antinodal loca-tions, we can increase the number of fringes by increas-ing the amplitude of the standing wave. Because k 54p~dxyl! in Eqs. ~10! and ~12!, we can conclude that wecan also increase the number of fringes by increasing theshearing amount, dx, reducing the light wavelength, orboth. Thus, a continuous increase of any one of thesefactors, e.g., vibration amplitude, will apparently gener-ate higher-order fringes starting from the locations ofcontraflexure to the antinodes. Therefore, the zero-strain positions can be identified as the emanators ofhigher-order fringes and antinodal positions can be iden-tified as attractors of them.
B. Fringe Pattern Caused by Flexural Plate Vibration
The first asymmetrical bending vibrationmode shapeof a circular plate is considered here for illustration.
Fig. 4. First asymmetrical bending mode shape of a circular platewith a clamped boundary condition.
The equation used to calculate themode shape shownin Fig. 4 is from Chen and Zhou.14 For flexural platevibration, antinodal points occur when ]Fny]x 5 0and ]Fny]y 5 0. Substituting the first conditioninto Eq. ~12! gives
Equations ~15! and ~16! indicate that high-contrast dark-fringe curves represent the zero gradient contours in thex and y directions, respectively. If these two fringe pat-terns are made to overlap properly, the intersections ofthese contours will give the locations of the plate vibra-tion antinodes. These contours would be unmovedwhen the plate vibration is of a normal mode.For a plate in flexure, the relationships between
surface strains and deflection are given as
εx 5 h]2w]x2
,
εy 5 h]2w]y2
,
gyx 5 2 h]2w]y]x
, (17)
where εx, εy are the flexural strains of the plate in thex and y directions, respectively; gyx is the surfaceshear strain; and h is half the thickness of the plate.Differentiating Eq. ~15! with respect to x gives
]E1
]x5 2ckHJoFr1k]Fn~x!
]x G2 JoFr2k]Fn~x!
]x GJ Hr1J1Fr1k]Fn~x, y!]x G
2 r2J1Fr2k]Fn~x!]x GJ F]2Fn~x, y!
]x2 G . (18)
Furthermore, ]E1y]x 5 0 gives local maxima orminima for the brightness function E1 at the loca-tions where the second derivatives, ]2Fny]x2 5 0,form unmoved locations of contraflexural ~in the xdirection! points of the vibrating plate at a naturalfrequency. At these contraflexural locations, εx 50. The loci tracing these contraflexural pointswould represent the zero-strain contours on the sur-face of the vibrating plate. These points can belocated by finding the points on the shearographicfringe pattern, ]Fny]x, that have tangent lines par-allel to the x axis. They are points of zero slope in
Fig. 5. Contours of the vibration amplitude gradient and zeroflexural strain of a circular plate with a clamped boundary.
Fig. 6. Contours of the vibration amplitude gradient and zeroshear strain of a circular plate with a clamped boundary.
Fig. 7. Contours of the surface shear strain of a circular platewith a clamped boundary.
1 June 1997 y Vol. 36, No. 16 y APPLIED OPTICS 3781
the fringe pattern, as illustrated in Fig. 5. Simi-larly, the loci of zero shear strain, gyx 5 0, can betraced by finding the points on the similar fringepattern that have tangent lines parallel to the yaxis, as shown in Fig. 6. Figures 7 and 8 give
Fig. 8. Contours of the surface flexural strain of a circular platewith a clamped boundary.
3782 APPLIED OPTICS y Vol. 36, No. 16 y 1 June 1997
fringe patterns, representing the contours of ]2Fny]y]x and ]2Fny]x2, respectively; each is computedby numerical differentiation of Fig. 4 directly.From the two figures, we can see that the contoursfor ]2Fny]y]x 5 0 and ]2Fny]x2 5 0 are exactly thesame curves as in Figs. 6 and 5, traced by themethod introduced above. By the same procedureas above, differentiating E2 of Eq. ~16! with respectto y or x will give a similar conclusion as for thecontours of εy 5 0 and gxy 5 0.
4. Experiments
Experiments are carried out to verify the theory andto prove the observations based on the above analysesof the computed fringe patterns.Figure 1 shows the experimental shearographic
system setup. Shearograms are generated by a real-time frame subtraction method at a video rate of 30framesys. They can be stored in a hard disk or dis-played on a monitor. Hard copies of the fringe pat-terns can be printed by a video printer.Figures 9 and 10 show the shearographic results for
the third bending mode of the cantilever and the firstasymmetrical mode of the clamped plate, respectively.That is, wo 5 rFn, n 5 3 for the beam, or n 5 2 for the
Fig. 9. Time-averaged fringes ofa cantilever beam vibrating atthe third bending mode: ~a!shearographic fringe pattern of]w0/]x, ~b! shearographic fringepattern of ]w0/]x of the object vi-brating at a larger amplitudethan that of ~a!, ~c! enhancedfringe pattern of ~b!.
Fig. 10. Time-averaged fringes of a circular plate with aclamped boundary vibrating at the first asymmetric mode:~a! shearographic fringe pattern of ]woy]x, ~b! shearo-graphic fringe pattern of ]woy]y, ~c! enhanced fringe patternof ~a!.
plate. The fringe patterns are experimentally ob-tained with the time-averaged subtraction method,based on the principle of Eq. ~9!. The fringe density ofthe pattern shown in Fig. 9~b! appears to be higherthan that shown in Fig. 9~a! because the vibrationamplitude for the Fig. 9~b! case is larger. By increas-ing the excitation force level, we can observe that thefringes appear to be unmoved at the locations num-bered by the arrows. The fringes marked with nu-merals 1 and 3 are the dark fringes corresponding tothe zero gradient locations, i.e., antinodal positions.The fringes marked with numerals 2 and 4 havebrightness eithermaxima orminima, corresponding tozero-strain locations, i.e., contraflexural regions. Thefringes at these two locations are somewhat obscure.However, when the method of Eq. ~12! is used, thefringe contrast is markedly improved, as we can seefrom Fig. 9~c!. In the regions between 1 and 2, 2 and3, or 3 and 4, fringes appear to be generated from 2 and4, travel through the in-between regions, and approach1 and 3 with a denser and denser fringe appearance.In Figs. 10~a! and 10~b!, the fringe patterns are also
experimentally generated by the use of the time-averaged subtraction method. Figure 10~a! is opti-cally sheared in the x direction, whereas Fig. 10~b! issheared in the y direction. The fringe pattern in Fig.
10~a! is quite similar in pattern to that computed~Fig. 5!. As we can see in Figs. 10~a! and 10~b!, thedark fringe circles can be understood to be caused bythe clamped condition along the rim of the plate.They represent the lines of zero gradients, i.e., ]woy]x 5 0 for the former figure and ]woy]y 5 0 for thelatter. Away from the clamped rim toward the cen-tral regions, the thick crossed lines in Fig. 10~a! rep-resent contours of ]woy]x 5 0. The thick darkelliptic curve in Fig. 10~b! represents the contour of]woy]y 5 0. Overlapping the two patterns can lo-cate the antinodal position of the vibration mode.The locations are circled as shown in the figures.To apply the method of Eq. ~12!, the fringe quality is
markedly enhanced for the higher-order fringe loops.As shown in Fig. 10~c!, the fringes of the four eyes areclearly visible. These eyes provide a means to tracethe elliptic loci of the zero-strain or shear contours asplotted in Figs. 5 and 6. The improvement of thefringe visibility will also facilitate the tracing of suchloci in the x as well as in the y directions.Quite obviously in Fig. 10, the number of fringes in
Fig. 10~c! is more than that in Fig. 10~a!, although thetwo patterns represent the same plate vibration level.Similarly, the same is true when comparing Figs. 9~c!with 9~b!. This is attributed to the increase in sen-
1 June 1997 y Vol. 36, No. 16 y APPLIED OPTICS 3783
sitivity with the use of the present frame subtractionmethod compared with the use of the traditional sub-traction method.3
5. Conclusion
The principle of a new time-averaged frame subtrac-tion digital shearographic technique for vibrationanalysis has been described, and experiments havebeen done for demonstration. The method of sub-tracting two Bessel fringe patterns at different forc-ing levels has been found to be capable of enhancingthe higher-order Bessel fringes. It facilitates thetracing of the zero-shear-strain and zero-flexural-strain contours. Compared with the phase-shiftmethod, this new frame subtraction method has beenfound to be more efficient and easier to implement forqualitative vibration measurement. This provides ameans for fast inspection of plate vibration behavior.A fringe interpretation method has been introduced
for identifying the antinodal positions and contours ofzero strains in beam and plate vibrations, includingzero-shear-strain contours in the plate vibration case.A good agreement between theoretical and experimen-tal results has been found in this study.
This research was supported by The Hong KongPolytechnic University and acts as part of the doc-toral research work of the first author. We are in-debted to the Head of the Department of MechanicalEngineering, R. M. C. So, for permission to use var-ious resources in the preparation of this paper.
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