The knowledge of resonance frequencies and modeshapes of engineering structures subjected to harmo-nic excitation is of interest in industrial applications.Vibration testing and evaluation provide informationon the material properties at the micrometer level,validation of the design and simulation, preventionof fatigue failure, detection of noise generating parts,and information for optimization of the manufactur-ing process. TV holography or digital or electronicspeckle pattern interferometry (DSPI or ESPI) al-lows both static and dynamic measurements withhigh sensitivity and resolution [1–3]. Two commonlyemployed modes of operation of TV holography arethe speckle correlation subtraction method, for staticdisplacement fringe analysis, and the time-averagemethod, for studying vibrating objects. In either case,fringe overlaid images of the object under study areobtained. “Speckle fringes” denote contours of eitherconstant surface displacement or constant vibration
amplitude. Recent studies have demonstrated pro-mising complementary approaches for vibration ana-lysis using TV holography with stroboscopic [4,5] andcontinuous [6–17] illuminations, which provide time-resolved or time-averaged visualization, respectively,of the vibration modes. The size of the objects thatcan be evaluated by using these methods covers awide range, from few meters as in spacecraft engi-neering structures [1] to few hundred micrometersas in microelectromechanical systems [5,18,19]. Formicrosystems analysis, the TV holography arrange-ment uses a microscopic imaging system [18,19].
In this paper we describe a time-average TV holo-graphic method that can be used for the measure-ment of out-of-plane amplitudes of vibration atresonant frequencies. For quantitative vibrationfringe analysis, we have combined the time-averagerefreshing image mode of operation with the time-average reference phase bias modulation method[20,21]. The time-average refreshing frame methodinvolves subtraction of a π phase-shifted (contrastreversal) time-average frame from the precedingtime-average frame during the object excitation. This
approach removes the high level of the DC term andenhances the contrast of the time-averaged J0fringes [20,21]. For vibration mode shape visualiza-tion and measurement, the experimental setupprovides, in the object and the reference beams, aphase-shifting mirror and a bias phase modulationmirror, respectively. This will allow the phase biasmodulation to be introduced to the reference beamto shift the time-average zero order J0 fringes, simi-lar to the conventional phase shifting of cosinefringes. The proposed method needs only four phase-shifted J0 fringe frames compared with the existingtime-average vibration fringe analysis methods thatuse a larger number of phase-shifted frames forquantitative evaluation [6–8,17]. Further, we use anerror-compensating four-step phase-shifting algo-rithm for generating the phase map [22]. Since weuse the time-averaged J0 fringes in the four-stepphase calculation algorithm, which is based on cosinefringes, the phase map generated has error thatneeds to be corrected by using the lookup tables cor-responding to the algorithm used. Section 2 describesthe basic theory for vibration fringe visualization andquantitative analysis. Section 3 describes the time-average TV holography arrangement. The experi-mental results for a square aluminium plate withone edge clamped at the bottom and excited sinusoid-ally at different resonant frequencies are presentedin Section 4.
2. Theory
A. Visualization of Time-Averaged Speckle CorrelationFringes
In a TV holography system working in the time-average mode of operation, the output intensity dis-tribution with a sinusoidally vibrating object with afrequency much higher than the video frame rate canbe written as [1]
Iavg ¼ Io½1þ V cosðϕÞJ0ðΩÞ�; ð1Þ
where Io is the bias intensity, V is the visibility, ϕ isthe random speckle phase difference, J0ðΩÞ is thezero-order Bessel function of the first kind, and Ωis the fringe locus function, which is related to ampli-tude of vibration A as Ω ¼ ð4π=λÞA.Equation (1) represents poor contrast fringes be-
cause of the high level of the DC term. Several meth-ods have been proposed to improve the contrast asdiscussed below.
i. Four-frame method. In this method, fourphase-shifted time-averaged fringe frames are ob-tained by shifting the phase of the object beam bynπ=2 such that ϕ and ϕþ nπ=2 (n ¼ 1; 2; 3…). Thefour frames can be written as
I1 ¼ Io½1þ V cosðϕÞJ0ðΩÞ�;I2 ¼ Io½1 − V sinðϕÞJ0ðΩÞ�;I3 ¼ Io½1 − V cosðϕÞJ0ðΩÞ�;I4 ¼ Io½1þ V sinðϕÞJ0ðΩÞ�:
ð2Þ
These equations are used to remove the randomspeckle noise term cosðϕÞ and the high level of theDC term by means of the following equation [6–8,13]:
T ¼ ðI1 − I3Þ2 þ ðI2 − I4Þ2 ¼ 4Io2V2J02ðΩÞ: ð3Þ
Effectively, a new intensity frame, we shall call it theT-frame, is generated from the four frames in Eq. (2).The T-frame represents the time-averaged J0
2ðΩÞfringe pattern without the term cosðϕÞ and DC bias.Equation (3) contains three unknowns (Io, V , and Ω),so, at least three phase-shifted equations are re-quired to solve the equation for Ω. This procedureis explained in Subsection 2.B.
ii. Subtraction of the initial frame from the excitedstate of the object. The DC-term in Eq. (1) can also beremoved by subtracting the image corresponding tothe steady state, Ir, from that corresponding to theexited state of the object. The real-time subtractionof the time-averaged image from the reference imageresults in [20,21]
Is ¼ Ir − Iavg
¼ Ioð1þ V cosðϕÞÞ − Ioð1þ V cosðϕÞJoðΩÞÞ¼ IoV cosðϕÞð1 − JoðΩÞÞ: ð4Þ
Equation (4) shows that the real-time subtractionwill produce fringe pattern of a vibrating objectand is modulated by a system of fringes describedby (1 − J0ðΩÞ). As the time interval between the cur-rent time-averaged frame and the reference frame islong, the fringe pattern is very sensitive to externaldisturbances. Also, the fringe pattern is modulatedby (1 − J0ðΩÞ) rather than J0ðΩÞ, and thus the con-trast of the fringe is rather poor.
iii. Automatic refreshing reference frame method.Equation (2) can be used to generate a new intensityframe; we shall call this an R-frame, as follows:
R ¼ jI1 − I3j ¼ j2IoV cosðϕÞJ0ðΩÞj: ð5Þ
Effectively, we build a difference between theframe I1 and its π phase-shifted frame (contrast re-versed) frame, I3. The procedure has been called theautomatic refreshing reference frame method be-cause of the way it is implemented in practice[20,21]. In this procedure the high-level DC termis removed, but the random phase term cosðϕÞ re-mains. As is shown below, this term is eliminatedduring fringe analysis. The fringe pattern repre-sented by Eq. (5) and displayed in real time on themonitor corresponds to contours of out-of-plane vi-bration amplitudes of the object excitation. Figure 1
shows the fringe pattern displayed on the monitor inthe three cases discussed above. Four π=2 phase-shifted frames [Eq. (2)] have been used in Eqs. (3)–(5)to generate the fringes shown in Figs. 1(a)–1(c),respectively. The contrast in Fig. 1(b) is poor as ex-pected. Again, as expected, Figs. 1(a) and 1(c) showgood contrast fringes, with contrast in Fig. 1(a) beingmarginally better. It is important to note that Eq. (5)gives fringes with significantly better contrast thanthose obtained with Eq. (4).
B. Determination of Fringe Locus Function
For quantitative evaluation of the amplitudes of vi-bration, a procedure similar to the phase-shiftingtechnique in conventional interferometry has beensuggested [6–8,17]. To implement this, multiplephase-shifted intensity frames are required. If thephase of the reference wave is modulated sinusoid-ally at a frequency and phase identical to the objectvibration, the zero-order J0 fringe shift is similar tothe case in which the phase modulation shifts cosinu-soidal fringes [6–8]. Thus if B is the amplitude of thesinusoidal modulation of the reference wave, Eq. (3)can be used to write different intensity frames as[6–8]
T1 ¼ 4Io2V2J02ðΩÞ; T2 ¼ 4Io2V2J0
2ðΩþ BÞ;T3 ¼ 4Io2V2J0
2ðΩþ 2BÞ: ð6Þ
In general we may write
Tðnþ1Þ ¼ 4Io2V2J02ðΩþ nBÞ; ð7Þ
where Tnþ1 represents the phase-shifted time-averaged J0
2ðΩÞ fringe patterns, B is the magnitudeof the bias modulation, and n (¼ 0; 1; 2; 3…) is thenumber of phase-shifted frames. Equation (7) repre-sents the phase-shifted frames, similar to the phase-shifted frames of the conventional phase-shiftingtechnique, where the intensity distribution is a co-sine function.Since the Bessel function is not separable in terms
of Ω and B, it is not possible to get an analytical solu-tion for Ω by using Eq. (7). The fringe analysis can,however, be done by assuming that the J0 function isidentical to the cosinusoidal function of conventional
two beam interference and by using one of the sev-eral algorithms available for calculation of phasein phase-shifting interferometry [6–8]. The calcu-lated phase value of Ω will therefore be erroneousand hence is represented by Ω″. The three step algo-rithm will therefore give [6–8]
Ω″ ¼ arctan�T1 − 2T2 þ T3
T1 − T3
�: ð8Þ
The arctan function used in Eq. (8) as well as inEqs (10) and (11) is an extended (four-quadrant) arc-tan 2 function. The error in the calculated value of Ω″
can be corrected to give Ω with the help of lookup ta-ble created for specific algorithms as explained inSubsection. 2.C. Equation (8) represents the simplestalgorithm. Since each T-frame is obtained from fourraw frames, such as in Eq. (2), a total of 12 frames arerequired to implement Eq. (8). Further, a three-stepalgorithm is likely to be more sensitive to error in thebias value B, as is known in the phase-shifting tech-nique. Higher-step algorithms that are known to beerror compensating can be used but will require moreframes of the type in Eq. (2), four for each step. Theobjective of this paper is to use a higher-step algo-rithm, but the same time use fewer raw frames ofthe type in Eq. (2). In order to reduce the numberof raw frames required, here we use the automaticrefreshing reference frame method as described inSubsection 2.A [Eq. (5)]. It requires two frames perstep, instead four frames per step as in the other pro-cedure [13]. Equation (5) can be used to describe thephase-shifted frames in the present case as
Rðnþ1Þ ¼ j2IoV cosðϕÞJ0ðΩþ nBÞj: ð9Þ
We propose use the following four-step (4-step A)phase-shifting algorithm (phase shift ¼ π=2), whichshows improved error compensation [22]:
Ω″ ¼ arctan�R1 − 3R2 þ R3 þ R4
R1 þ R2 − 3R3 þ R4
�: ð10Þ
Fig. 1. Fringes obtained with a vibrating sample at resonance, using Eqs (3)–(5).
The more frequently used four-step algorithm (4-step B) is given by
Ω″ ¼ arctan�R4 − R2
R1 − R3
�: ð11Þ
Although the intensity frames R given by Eq. (5)have a random cosðϕÞ term, this noise term is elimi-nated while evaluating Ω″, as substitution fromEq. (9) into Eq. (10) will show. This approach of usingthe refreshing frame technique with an error-compensating four-frame algorithm is an improve-ment, as it requires fewer frames for visualizationand quantitative evaluation of vibration fringes.The calculated phase distribution Ω″ is wrapped inthe range −π to π. The evaluated phase Ω″ will haveerrors if B ≠ π=2 as assumed. This error is algorithmdependent, and Fig. 2 shows the simulated error inΩ″ as a function of percentage bias error for the three-step algorithm [Eq. (8)], 4-step A [Eq. (10)], and4-step B [Eq. (11)]. A multigrid phase unwrappingprocedure is used to unwrap the phase [23].
C. Generation of Lookup Table
The value ofΩ″ differs from the correct argumentΩ ofJ0 because of the difference between the cosine andJ0 functions and can be expressed as [6,8]
Ω″ ¼ Ωþ ε; ð12Þ
where ε is the error due to the difference between J0and cosine functions. The error ε can be calculated forany value ofB to create a lookup table for any specificalgorithm. The lookup table is used to correct thephase calculated from Eq. (10). Figure 3(a) showsthe lookup table in graphical form for the algorithmin Eq. (10). The actual error due to the assumption ofthe Bessel function as cosine function is shown inFig. 3(b). The corrected phase Ω can be related tothe amplitude of object vibration A as
Ω ¼ 4πλ A: ð13Þ
3. Time-Average TV Holographic Arrangement
Figure 4 represents the schematic a time-average TVholographic arrangement. The narrow beam from acontinuous wave He–Ne 632:8nm (Coherent Inc,20mW) is divided into two beams by using a beamsplitter (BS1). One beam is expanded by using a spa-tial filtering setup (SF1) to act as an object beam toilluminate the object via mirror M1. The second beamserves as a reference beam, and it illuminates thephase modulation reference mirror (PMRM, KarlStetson Associates, PZT7K). The PMRM allows ana-lyzing the vibration amplitudes of frequencies up to7KHz. The reflected reference beam from the PMRMis expanded with the help of a spatial filtering setup(SF2), and it enters the imaging system via mirrorM2, a ground glass plate (GG) giving a diffuse refer-ence beam, and a cube beam splitter (BS2). Similarly,the scattered object beam enters the imaging systemvia the same cube beam splitter (BS2) and a piezo-electric transducer mirror, PZTM (STr 25/150/6PZT from Piezomechanik). The PZTM is connected
Fig. 2. Phase error as a function of percentage bias error for thethree-step [Eq. (8)], 4-step A [Eq. (10)], and 4-step B [Eq. (11)]phase-shifting algorithms.
Fig. 3. (a) Lookup table in graphical form for the error ε between Ω and Ω″ (B ¼ π=2); (b) phase error introduced (ε) when the Besselfunction assumed to be a cosine function.
to an amplifier (Piezo-Mechanik–LE150) and it isinterfaced to a personal computer with a digital-to-analog card (NI6036E). The PZTM in the setupallows for shifting the scattered object beam by a ca-librated phase shift of π. For harmonic excitation ofthe object and the reference mirror (PMRM), we haveused channel 1 (CH1) and channel 2 (CH2) of the ex-ternal dual function generator (DFG) that consists oftwo NI PXI5402 cards. The channel CH1 is connectedto an amplifier A2 (Spranktronics, India) to controlthe amplitude of the object excitation. Channel CH2allows excitation of the PMRM mirror at the samefrequencies as that of the object excitation to intro-duce the bias vibration. The imaging system consistsof a zoom video lens (Thales Optem—Model 34-11-10) and a CCD, (Jai CV-A1 CCD camera). The CCDhas 1384 ðHÞ × 1035 ðVÞ pixels, and the size of eachpixel is 4:65 μm. The scattered object wave and thediffused reference wave are combined coherentlyonto the CCD plane. A neutral-density filter (NDF)is used to adjust the intensity ratio between the ob-ject and the reference wave. The CCD is interfaced toa PC with a NI1409 frame grabber card. The imagingzoom video lens provides 18–108mm manual zoom.The DFG is activated through the computer, and theLabVIEW program is used for varying the frequency,the phase, and the amplitude between the two chan-nels (CH1 and CH2). In addition, the PZTM in thesetup is also used for shifting the scattered objectbeam by a constant phase π for introducing the con-trast reversal to remove the background DC in orderto enhance the contrast of the time-average fringepatterns [Eq. (5)]. Programs based on LabVIEW have
been developed for real-time visualization and stor-ing the phase-shifted frames for vibration fringeanalysis.
4. Experimental Results
Experiments were carried out on a square alumi-nium plate (80mm × 80mm × 1mm) with one edgerigidly clamped at the bottom. The plate is excitedsinusoidally at different frequencies by means of anexternal speaker connected to an amplifier (A2) andCH1 (channel 1) of a DFG. To make the surface areadiffusively reflecting we use a spray (SKD-S2 spot-check developer) that deposits a thin soft layer ofpowder, which can be wiped away easily after the ex-periment. As explained in Subsection 2.A, for real-time visualization of the vibration mode shapes atresonant frequencies, we use the automatic refresh-ing reference frame technique [Eq. (5)] at a video rateof 25 frames=s. For this, software is developed in Lab-VIEW in such a way that it will first store a time-average frame during the object vibration. It willthen shift the mirror PZTM for a π phase shift andsubtract the new frame from the first one to obtainthe time-averaged fringe pattern. The subtractionof the two time-average frames results in a vibrationmode shape at a resonant frequency on the monitor.The above process will be continuously repeatingduring the vibration of the object in short intervalsof time (depending upon the speed of the cameraand the response time of the PZTM shifter) andcan be controlled through the software. The reso-nance frequencies and their mode shapes on the alu-minium plate are shown in Fig. 5. The results shown
Fig. 4. (Color online) Schematic of a time-average TV holography arrangement.
here with the setup are qualitative in nature. Bysweeping the excitation frequency from zero on-wards, one can find all the resonance frequenciesfor the object to be tested within the range. It is tobe noted from the fringe patterns that the bright re-
gions on the subtracted frame are the positions ofzero displacement, i.e. the nodal lines.
To demonstrate the working of the fringe evalua-tion by using the Eq. (9) we consider the vibrationof the aluminium plate at a frequency 1772Hz. Bothchannels CH1 and CH2 of the DFG are now activated.In implementation of the method for vibration fringeanalysis, calibrating the amplitude of the bias vibra-tion to introduce known amounts of the phase shift tothe time-averaged J0ðΩÞ fringe function, as well assetting the phase (in-phase condition) of the biasphase modulation equal to that of the object excita-tion, is an important step.
Figure 6(a) shows the time-averaged fringes at thefrequency of 1772Hz. A curser generated line isdrawn across the image to act as a reference forthe purpose of phase-shift calibration. The calibra-tion of the J0ðΩÞ fringe shift is carried out by initiallykeeping the phase value of the reference mirror(PMRM) through channel CH2 of the DFG at 0°and varying the amplitude of the reference mirrorin a controlled manner. At a particular bias ampli-tude of the reference mirror, the location of the zero-order bright fringe is occupied by the immediate darkfringe and is shown in Fig. 6(b) by the alignment ofthe curser drawn line with the dark fringes. This isthe position where the reference bias modulation mintroduces a π phase shift to the time-average J0ðΩÞfringes. However, the brightness of the zero-orderfringe is degraded owing to the phase mismatch be-tween the object and the reference wave excitation.Now the in-phase condition between the object andthe reference mirror excitation has to be achieved.This is done by varying the reference phase from0°, using channel CH2 of the DFG via the referencemirror (PMRM) to a value that returns the zero-order fringe to maximum brightness. During thisprocess the zero-order J0ðΩÞ fringe remains fixedat its π-shifted position as shown in Fig. 6(c). Wecan also get the in-phase condition by additionof 180° to the current phase value as shown inFig. 6(d). In this case the J0ðΩÞ fringes are π shiftedin the other direction. It is necessary to note that thesame procedure has to be followed whenever we
Fig. 6. (Color online) Phase shift calibration using the phase bias modulation method with the PMRM reference mirror. The aluminiumplate (80mm× 80mm× 1mm) is rigidly clamped at the bottom and is vibrating at a frequency of 1772Hz. (a) J0ðΩÞ fringes without the biasmodulation; (b) π phase-shifted J0ðΩÞ fringes with the phase bias modulation and an arbitrary phase relation between the object and thereference beam excitations; (c) π phase-shifted J0ðΩÞ fringes with the phase bias modulation and an in-phase relation between the objectand the reference beam excitations; (d) π phase-shifted J0ðΩÞ fringes with the phase bias modulation and another in-phase relation byaddition of 180° to the current phase value between the object and the reference beam excitations.
Fig. 5. Mode shapes at different resonant frequencies of analuminum plate (80mm× 80mm× 1mm) rigidly clamped at thebottom.
change the frequency or amplitude of the objectexcitation.After the initial calibration procedures described
above, we use the bias reference modulation valuem that is calibrated for the π phase shift to thetime-averaged J0ðΩÞ fringes using channel CH2 ofthe DFG. By keeping all other parameters in the set-up the same and changing only the variable refer-ence mirror amplitude such that m ¼ 0;m=2;m; 3m=2, one can shift the J0ðΩÞ fringes. The stored phase-shifted J0ðΩÞ fringes for these values are shown inFigs. 7(a)–7(d), respectively. The raw and the filteredwrapped phase maps evaluated from Eq. (10) areshown in Figs. 8(a) and 8(b), respectively. Thewrapped phase map is unwrapped, corrected by
using the lookup table, and converted into amplitudevalues A by Eq. (13). The 3D plot shown in Fig. 8(c)represents the out-of-plane vibration amplitude of analuminium plate at the fundamental frequency1772Hz. A similar analysis has also been carriedout on the same aluminium plate at 2921Hz, andthe results are shown in Fig. 9.
5. Conclusion
We have discussed a modified TV holography proce-dure for the analysis of the time-averaged fringe pat-tern of vibrating objects. The method is based onphase-shifting interferogram analysis technique asapplied to J0 fringes. It is shown that good resultscan be obtained even as we use fewer frames.Further, the use of a higher-order algorithm ensures
Fig. 7. Phase-shifted J0ðΩÞ fringe patterns on the sample at the vibration frequency of 1772Hz. (a) Initial fringes, (b) π=2-phase-shiftedfringes, (c) π-phase-shifted fringes, (d) 3π=2-phase-shifted fringes.
Fig. 8. (Color online) Results of analysis of phase-shiftedJ0ðΩÞ fringe patterns shown in Fig. 7. (a) Raw wrapped phasemap, (b) filtered wrapped phase map using median filtering with3 × 3 window, and (c) 3D plot of out-of-plane vibration amplitude.
Fig. 9. (Color online) Results of analysis when the object isvibrating at a frequency 2921Hz. (a) Raw wrapped phase map,(b) filtered wrapped phase map, and (c) 3D plot of out-of-planevibration amplitude.
that the influence of the bias modulation error issmaller.
This work is supported by the Indian Space Re-search Organization (ISRO), government of India.We thank the reviewers for useful suggestions thathelped us to improve the quality of the paper.
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