Mode-shape measurement of piezoelectric plateusing temporal speckle pattern

interferometry and temporal standard deviationChing-Yuan Chang and Chien-Ching Ma*

Department of Mechanical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan*Corresponding author: ccma@ntu.edu.tw

Received August 31, 2011; accepted September 21, 2011;posted October 3, 2011 (Doc. ID 153594); published November 1, 2011

This study proposes an image processing method to improve the quality of interference fringes in mode-shape mea-surement using temporal speckle pattern interferometry. A vibrating piezoelectric plate at resonance was investi-gated, and the full-field optical information was saved as a sequence of images. According to derived statisticalproperties, an algorithm was developed to remove noise from both the background and disturbance, resultingin high-resolution images of excellent quality. In addition, the resonant frequency and mode shape obtained usingthe proposed algorithm demonstrate excellent agreement with theoretical results obtained by the finite elementmethod. © 2011 Optical Society of AmericaOCIS codes: 100.2000, 100.2650, 100.2980, 100.3175, 120.6160, 120.7280.

In the research on actuators and sensors, speckle in-erferometry is a well-established technique used to inves-tigate vibrations of small amplitude and high frequency[1,2]. Time-averaged electronic speckle pattern interfero-metry (ESPI) produces interference fringes that illustratethe full-field displacement associated with a vibratingobject. The fringes represent mode shapes and indicatethe amplitude of vibrations at resonance, and they areenhanced through image processing [3,4]. Amplitude-fluctuation ESPI (AF-ESPI) is a popular enhancementmethod with the capability of real-time measurement[5]. In the investigation of piezoelectric ceramic (PZT)plates, AF-ESPI has frequently been applied to the mea-surement of dynamic characteristics [6,7]. Unfortunately,background noise and noise from disturbance limit theefficacy of AF-ESPI, particularly at high resonant fre-quencies. Most of the noise is the result of air disturbanceand electronic fluctuations [8,9], and the influence ofthese disturbances is difficult to remove by physicalmeans. Noise detracts from image quality and under-mines the integrity of experimental results. This studycombines temporal speckle pattern interferometry(TSPI) and temporal standard deviation (TSTD) to re-duce this noise and enhance the image quality of interfer-ence fringes. The basic idea behind TSPI is to record theentire history of a deforming specimen, and the phase ateach pixel is measured as a function of time. TSPI fo-cuses mainly on temporal information associated withfluctuations in amplitude, presenting rich informationfor phase measurement and recovery [10].This study first considers a PZT rectangular plate vi-

brating at its resonant frequency ω. The full-field ampli-tude of vibration is Aðx; yÞ, and TSPI records thesequential images as

Iiðx; yÞ ¼ Ioðx; yÞ þ IRðx; yÞ þ 2ffiffiffiffiffiffiffiffiffiIoIR

pj cosðϕðx; yÞ

þ ψ iðx; yÞÞJ0ðΓAðx; yÞÞj; ð1Þ

where subscript i is the specific number of the image inthe sequence and J0 is the zero order Bessel function. Γ

is the sensitivity factor, equal to 2πð1þ cos θÞ=λ for out-of-plane measurement and equal to 4π sin θ=λ for in-planemeasurement. λ and θ are the wavelength and illumina-tion angle of the laser light, respectively. ϕðx; yÞ is therandom phase resulting from surface roughness. At thecorresponding exposure time iτ, ψ iðx; yÞ presents distur-bance between the object beam and reference screen,represented as

ψ iðx; yÞ ¼1τ

Z ðiþ1Þτ

iτψðx; y; tÞf ðψðx; y; tÞÞdt; ð2Þ

where ψðx; y; tÞ is the disturbance in temporal distribu-tion at specific point ðx; yÞ. f ðψðx; y; tÞÞ denotes the prob-ability of ψðx; y; tÞ and is one form of the Gaussiandistribution [11]. The statistical properties of disturbanceare presented as

�ψ s ¼1n

Xni¼1

ψ i; ð3Þ

ψs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n − 1

Xni¼1

ðjψ ij − j�ψsjÞ2s

: ð4Þ

In the statistical expression of disturbance, �ψ sðx; yÞ andψsðx; yÞ are the mean value and TSTD value, respectively,at specific point ðx; yÞ. In the macroscopic aspect of thevibrating object, both of these values are constants thatdescribe the invariant position for the infinite time. A firstorder Taylor expansion is applied to expand Eq. (1),given by

Ii ¼ Io þ IR þ 2ffiffiffiffiffiffiffiffiffiIoIR

pjðcosϕþ ψ i sinϕÞJ0ðΓAÞj ð5Þ

and simplified as

I�i ¼ Ibg þ jðcosϕþ ψ i sinϕÞJ0ðΓAÞj: ð6Þ

To further simplify the expression of the equation, theðx; yÞ dependence in each term was dropped from

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Eq. (5) and subsequent equations. In the sequence ofimages presented by Eq. (6), Ibg is background noise,which is a time-invariant constant for all of the images.ψ i is the disturbance in the discrete images unique ineach frame. Both forms of noise detract from the imagequality of interference fringes J0ðΓAÞ, and mean andsubtractive methods are the conventional solutions to re-move them. Subtractive methods mainly reduce back-ground noise, and mean methods mainly reduce noisefrom disturbance; however, erroneous image processingmay result from an improper combination of the twomethods. This study proposes a novel (to our knowledge)approach to preserving the advantages of both methods.Themain algorithm is used to calculate the TSTD value ofthe sequential images, given by

I� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n − 1

Xni¼1

ðI�i − �I�Þ2s

¼ ψ sj sinϕJ0ðΓAÞj; ð7Þ

where I� is the image result obtained using the TSTD al-gorithm, and �I� is the mean intensity of the sequentialimages, given by

�I� ¼ 1n

Xni¼1

I�i ¼ Ibg þ jðcosϕþ �ψ s sinϕÞJ0ðΓAÞj; ð8Þ

where n is the total number of sequential images. Theproposed method calculates the TSTD value in a se-quence of images and reduces noise according to theirstatistical properties. Based on the Gaussian distributionfor infinite time, the mean value and TSTD value at pointðx; yÞ both converge to constants. In applying this trait tothe discrete condition, a large number of images in eachsequence results in �ψs and ψs converging to constants. Inthis manner, the TSTD method removes noise from dis-turbance while diminishing background noise.Equation (7) indicates a further quantitative relation-

ship between the amplitude of vibration and interferencefringes. The relationship is presented in Fig. 1, and the fol-lowing parameters were applied in this analysis: the illu-mination angle of the laser lightwas 68°; thewavelength ofthe laser light source was 632:8nm; and roughness phaseϕ and disturbance ψs were both constants. This figure pre-sents the normalized intensity of light, in which brightfringes in the resonant vibration indicate local maxima.Each fringe corresponds to quantitative amplitude andprovides functional information with which to analyzefull-field vibration. The first bright fringe indicates nodallines with zero displacement, and the four subsequentbright fringes are at 208, 381, 553, and 724 nm. To providepractical confirmation of the theoretical description of theTSTDmethod, a TSPI system assembled in our laboratorywas applied to empirically measure interference fringes.The system provided sinusoidal voltage to excite a PZTrectangular plate (60mm × 30mm × 1mm) at the eighthin-plane resonance under traction-free boundary condi-tions. Both mode shapes and resonant frequencies wererecorded and compared with results obtained using finiteelement method.Figure 2 presents the TSPI setup for in-plane vibrations

using the PZT specimen. A coherent light source was

provided by a 30mWHe–Ne laser (λ ¼ 632:8nm), dividedinto two parts of equal light intensity, illuminating thespecimen at symmetrical angles. The PZT specimenwas excited through the application of sinusoidal voltagevia a function generator (Hewlett Packard HP-33120A)and a power amplifier (NF Electronic Instruments4005 Type). A digital CCD (Allied Vision TechnologiesPike F-505B, 2452 × 2054 pixels) camera was used to re-cord 50 sequential images, and a personal computer wasused to execute the TSTD algorithm through the sequen-tial images. All the sequential images were normalized inaccordance with Eq. (6) to present clear patterns, for thereason that the resultant intensity associated with vibra-tion amplitude is degraded from constant

ffiffiffiffiffiffiffiffiffiIoIR

p. The

threshold selected in our algorithm was 0.67 of the max-imum intensity, and the intensity of each pixel rangedfrom 0 to 255 following normalization. The finite elementmethod (FEM) was applied to further confirm the reso-nant frequency and mode shape obtained from the TSTDmethod. Reissner–Mindlin formulations for a thin PZTplate were used in the simulation [12,13]. Commercialsoftware, ABAQUS, was used to compute the resonantfrequencies and the corresponding mode shapes.

Figure 3 provides a qualitative comparison to illustratethe eighth mode of in-plane resonance. The numerical re-sults obtained from the FEM are presented in Figs. 3(a)and 3(b), and the experimental results obtained from theTSTD method are presented in Figs. 3(c) and 3(d). In the

Fig. 1. (Color online) Fringe analysis of in-plane vibrationamplitude.

Fig. 2. (Color online) TSPI setup for in-plane measurement.

4282 OPTICS LETTERS / Vol. 36, No. 21 / November 1, 2011

FEM results, the bold lines represent nodal lines withzero displacement. All other solid lines and dotted linespresent vibration amplitude in the positive and negativedirections, respectively. In the TSTD results of modeshape, the brightest lines of the figure indicate nodal lineswith zero displacement. The subsequent four fringes in-dicated constant vibration amplitudes at 208, 381, 553,and 724 nm, based on the analytical results of Fig. 1.The U and V directions defined in this study denotethe horizontal and vertical directions along the resultantimage, respectively. Mode shapes in the U direction arepresented in Figs. 3(a) and 3(c); mode shapes in the Vdirection are presented in Figs. 3(b) and 3(d). In thesefigures, identical results are represented in both direc-tions. This agreement in results indicates that the TSTDmethod is capable of presenting consistent results, andthe traits provide clear fringes of excellent image qualityfor the mode shapes. Moreover, the high quality of thefringes provides reliable information regarding phase re-covery [10] and electrode design [6,7]. In the comparisonof resonant frequencies, the theoretical value providedby the FEM was 88; 193Hz, and experimental results ob-tained using the TSTD method were 88; 300Hz in the Udirection and 87; 700Hz in the V direction. The excitationvoltages in the experimental results were 50V and 60V inthe U and V directions, respectively. The discrepancieswere −0:18% in the U direction and 0.50% in the V direc-tion. This excellent agreement indicates that the TSTDmethod is capable of accurately determining the reso-nant frequency, as well as the mode shape in between.The high quality of the obtained mode shapes can be

partly attributed to the high resolution of the camera(2452 × 2054 pixels), but the main contribution was theremoval of both background and disturbance noise bythe TSTD algorithm. In addition, Eq. (7) provides a pro-gressive confirmation that the greater the number of

images, the greater the reduction in the two forms ofnoise. Subtractive methods reduce the influence of back-ground noise but are unable to remove noise from distur-bance, while mean methods reduce the noise fromdisturbance but leave background noise. This statementcan be further proven by applying the mean or subtrac-tive methods in Eq. (6). Nonetheless, the TSTD methodrequires several images to execute the algorithm, and fivesequential images was the minimum number required inour real-time measurement. Initial results of mode shapeand resonant frequency were provided by real-time mea-surement, and nonsynchronous measurement requiredlarger CPU time while providing high-resolution resultsfor intensive analysis. Nonsynchronous measurementis used to record a large number of sequential images,followed by execution of the TSTD algorithm to obtaina high-resolution image. To obtain a high-quality imageof mode shape, 50 sequential images were included inour experiment, the results of which are presented inFigs. 3(c) and 3(d).

To summarize, we established a TSTD method basedon theoretical, numerical, and experimental analyses,the traits of which provide clear fringes for mode shapesand accurate values for resonant frequencies. The pro-posed method combines the advantages of both the sub-tractive and mean methods, resulting in a considerableimprovement in image quality.

The authors would like to thank the National ScienceCouncil of the Republic of China (NSCT) for financialsupport of this research under the contract number 99-2221-E-002-036-MY3.

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Fig. 3. Comparison of the eighth in-plane mode: mode shape in(a) the U direction and (b) the V direction obtained using FEM;mode shape in (c) the U direction and (d) the V direction ob-tained using the proposed method.

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