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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 20 — Sep. 28, 2009
  • pp: 18014–18025
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Multimode vibration analysis with high-speed TV holography and a spatiotemporal 3D Fourier transform method

Cristina Trillo, Ángel F. Doval, Fernando Mendoza-Santoyo, Carlos Pérez-López, Manuel de la Torre-Ibarra, and J. Luis Deán  »View Author Affiliations


Optics Express, Vol. 17, Issue 20, pp. 18014-18025 (2009)
http://dx.doi.org/10.1364/OE.17.018014


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Abstract

The combination of a high-speed TV holography system and a 3D Fourier-transform data processing is proposed for the analysis of multimode vibrations in plates. The out-of-plane displacement of the object under generic vibrational excitation is resolved in time by the fast acquisition rate of a high-speed camera, and recorded in a sequence of interferograms with spatial carrier. A full-field temporal history of the multimode vibration is thus obtained. The optical phase of the interferograms is extracted and subtracted from the phase of a reference state to yield a sequence of optical phase-change maps. Each map represents the change undergone by the object between any given state and the reference state. The sequence of maps is a 3D array of data (two spatial dimensions plus time) that is processed with a 3D Fourier-transform algorithm. The individual vibration modes are separated in the 3D frequency space due to their different vibration frequencies and, to a lesser extent, to the different spatial frequencies of the mode shapes. The contribution of each individual mode (or indeed the superposition of several modes) to the dynamic behaviour of the object can then be separated by means of a bandpass filter (or filters). The final output is a sequence of complex-valued maps that contain the full-field temporal history of the selected mode (or modes) in terms of its mechanical amplitude and phase. The proof-of-principle of the technique is demonstrated with a rectangular, fully clamped, thin metal plate vibrating simultaneously in several of its natural resonant frequencies under white-noise excitation.

© 2009 OSA

1. Introduction

TV holography, also known as ESPI, is a well-established speckle interferometry technique based on the electronic recording of full-field interferograms, formed from the interference of a reference beam and the light scattered by the object to be measured [1

1. Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000). [CrossRef]

]. Among the wide range of applications, vibration analysis of resonant modes has been carried out with a variety of illumination schemes and phase evaluation methods [2

2. D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995). [CrossRef]

, 3

3. F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000). [CrossRef]

]. Several authors have already addressed the problem where an excited object vibrates in several modes simultaneously. This is a realistic scenario for an object that is subjected to natural excitation under its normal operating conditions. Different approaches have been proposed to separate the vibration into its constituent resonant modes by using time-average [4

4. O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, “Measurement of higher armonics in periodic vibrations using phase-modulated TV holography with digital image processing,” Appl. Opt. 33(22), 4997–5002 (1994). [CrossRef] [PubMed]

, 5

5. A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994). [CrossRef]

] or stroboscopic methods [6

6. J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997). [CrossRef]

]. However, difficulties usually arise if more than two modes are present or if the ratio of the frequencies is an integer number, though a solution for the latter has been proposed for time-averaging [7

7. A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000). [CrossRef]

]. The outputs are fringe patterns or one map of the amplitude and phase of the vibration obtained with a phase-stepping procedure.

TV holography is a full-field technique, so the measurement is performed simultaneously in all the field of view, without the need to scan the surface. The spatial sampling rate is then very high, but the temporal sampling rate achievable with standard cameras is rather low (usually 30 frames per second (fps) or less). With the advent of high-speed cameras, the acquisition of time-resolved measurements was possible. Naturally, they were applied to the study of transient or random vibrations [8

8. A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999). [CrossRef]

]. Buckberry et al. [9

9. C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000). [CrossRef]

] have reported the use of a high speed camera for the time-resolved measurement of multimode vibrations, but no attempt was made to separate the contributions of the constituent resonant modes.

In this work we present the combination of a high-speed TV holography system and a 3D Fourier-transform processing of the data for the analysis of multimode vibrations in plates. In a first stage, the out-of-plane displacement of the multimode vibration is resolved in time by the fast acquisition rate of a high-speed camera, and recorded in a sequence of interferograms. These are processed with a 2D-Fourier transform method to obtain their optical phase and a sequence of optical phase-change maps, that are proportional to the out-of-plane displacement undergone by the object between any given state and a reference state. In a second stage, these data are processed with a 3D Fourier transform that separates the constituent modes of the multimode vibration in the frequency space, and permits their isolation by means of bandpass filters. Contributions to the dynamic behaviour of the object due to single resonant modes or the superposition of several modes can be isolated and studied separately. The structure of the paper is as follows: first, a brief overview of resonant modes and their nomenclature is given in section 2.1. The method used to calculate the optical phase-change maps is sketched in section 2.2. Section 3 is devoted to the 3D Fourier transform processing applied to the data. Section 4 contains a detailed explanation of the experimental setup and procedure. Finally, some results obtained with a rectangular, fully clamped, thin metal plate excited with white noise are presented and discussed in section 5.

2. Background

2.1 Vibration modes

The equation of the deflection of a simply supported rectangular plate during vibration [10

10. W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), Chap. 5.

] can be written as
u3=l=0Lm=0Mu3e,lm(x1,x2,l,m)sin[(l+1)πax1]sin[(m+1)πbx2]cos(ωlmt)
(1)
with
ωlm=π2Dρh[(l+1)2a2+(m+1)2b2],D=Eh312(1ν2)
(2)
where u 3=u 3(x 1,x 2,t,l,m) is the deflection of the plate along direction x 3 (see Fig. 1
Fig. 1 Scheme of the reference system and nomenclature of some vibration modes. The dotted lines represent nodal lines.
), a and b are the horizontal and vertical dimensions of the plate respectively, E and ν are the Young’s modulus and Poisson’s ratio, h is the plate thickness and ρ is the mass per unit volume or volumetric mass density. Each pair of integers (l,m) corresponds to a particular vibration mode, that we name according to the number of nodes (out of the edge of the plate) in the horizontal and vertical directions. For example, for l=0 and m=0 we have the vibration mode (0,0), designated as M00 henceforth. Although our object is a fully clamped plate, its thickness is much smaller than its transversal dimensions, so the simpler model of a simply supported plate can be used to a first approximation to the problem.

2.2 Measurement method. Calculation of optical phase-change maps

With the illumination and observation geometry that we have selected for our experiments, the term φ o ,n is proportional to the instantaneous out-of-plane displacement field u 3 at the plate surface
ϕo,n=4πλu3
(4)
Taking into account Eq. (1), this dependency can be rewritten as
ϕo,n=4πλl=0Lm=0Mu3e,lmsin[(l+1)πax1]sin[(m+1)πbx2]cos(ωlmtn)=l=0Lm=0Mϕ3e,lmcos(ωlmtn)
(5)
where φ3 e ,lm=φ3 e ,lm(x 1,x 2,l,m) is the mode shape (only the factors with spatial dependence) expressed in optical terms.

An established procedure based on the spatial Fourier transform method [11

11. H. O. Saldner, N. E. Molin, and K. A. Stetson, “Fourier transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996). [CrossRef] [PubMed]

] is then used to calculate the optical phase-change ΔΦ between two interferograms, as described by Eq. (3), one of them taken as a reference state. The result is a 2D map of the instantaneous out-of-plane displacement field between the state at the instant tn and the reference state
ΔΦn=ϕo,nϕo,ref=l=0Lm=0M[ϕ3e,lmcos(ωlmtn)ϕ3e,lmcos(ωlmt0)]
(6)
with ΔΦn=ΔΦn(x 1, x 2, tn, l, m). A time sequence of these optical phase-change maps is the input for the second stage of the measurement method that we will explain in the following section.

3. Theory. Data processing with a spatiotemporal 3D Fourier transform

This data processing method was first devised for ultrasonic travelling waves [12

12. C. Trillo, and Á. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).

] but it can be applied to any 3D array of data with spatial and/or temporal carriers. A scheme of the procedure is shown in Fig. 2
Fig. 2 Scheme of the spatiotemporal 3D Fourier transform method. (i) Sequence of optical phase-change maps of a multimode vibration. (ii) Several planes of the spatiotemporal 3D spectrum. The spectral content of one mode (with temporal frequency fk) of the multimode vibration is shown in planes ±fk. Planes with the spectral content of other modes are omitted. Planes at temporal frequencies f0 and fmax do not contain information. The filter used to select one of the side lobes is also shown. (iii) Mechanical amplitude and phase of the selected mode (modulus and argument of the complex data respectively).
. A sequence of N 2D optical phase-change maps of the vibration, acquired in consecutive instants, constitute a 3D set of data that fulfil these requirements. Indeed, Eqs. (5) and (6) show that the data have a spatial carrier and, more importantly in our present case, that there is one or several temporal carriers given by the vibration frequencies ωlm/2π of the modes.

The discrete 3D set of experimental data of P×Q×N sampled points can be expressed as
ΔΦ(x1p,x2q,tn,l,m)=ΔΦ(x10+pΔx1,x20+qΔx2,t0+nΔt,l,m)
(7)
with p=0 … P-1, q=0 … Q-1 and n=0,…, N-1. Δx 1 and Δx 2 are the spatial sampling distances in the horizontal and vertical directions respectively, and Δt is the temporal sampling interval. The discrete spatial and temporal sampling frequencies in x 1, x 2 and t are given by
fp'=p'PΔx1,fq'=q'QΔx2,fn'=n'NΔt
(8)
respectively, with p'=-P/2,…, P/2, q'=-Q/2,…, Q/2 and n'=-N/2,…, N/2.

The 3D discrete Fourier transform of this set of data is then

ΔΦ'=DFT(ΔΦ)=p=0P1q=0Q1n=0N1{l=1Lm=1M[ϕ3e,lmcos(ωlmtn)ϕ3e,lmcos(ωlmt0)]}×exp(j2πpp'/P)exp(j2πqq'/Q)exp(j2πnn'/N)
(9)

This is equivalent to calculating the sum of the 3D discrete Fourier transforms of each mode (l,m) so, in the following, we restrict the calculation to just one mode for simplicity.

Since φ3 e ,lm does not depend on time, the Fourier transform may be separated
ΔΦ'lm=p=0P1q=0Q1ϕ3e,lmexp(j2πpp'/P)exp(j2πqq'/Q)n=0Ncos(ωlmtn)exp(j2πnn'/N)p=0P1q=0Q1ϕ3e,lmexp(j2πpp'/P)exp(j2πqq'/Q)n=0Ncos(ωlmt0)exp(j2πnn'/N)
(10)
and we can write
ΔΦ'lm=DFT(ϕ3e,lm)DFT[cos(2πflmtn)]DFT(ϕ3e,lm)DFT[cos(2πflmt0)]=ϕ'3e,lm2[δ(n'n'lm)+δ(n'+n'lm)]ϕ'3e,lmδ(n')
(11)
where n'lm is the frequency index of the discrete temporal frequency which is nearest to the temporal frequency flm of mode (l,m). The spectral content of the mode is thus symmetrically shifted with respect to the zero temporal frequency, and a bandpass filter can be applied to select one of the side lobes

ΔΦ'lmF=ϕ'3e,lm2δ(n'n'lm).
(12)

The inverse Fourier transform of the filtered data in Eq. (12) yields N complex-valued maps corresponding to the instants tn

Αlm(x1p,x2q,tn,l,m)=DFT1(ΔΦ'lmF)=12ϕ3e,lm(x1p,x2q,l,m)exp(j2πflmtn)
(13)

The modulus of this complex amplitude is proportional to the mechanical amplitude of the mode
mod(Αlm)=Re2(Alm)+Im2(Alm)=ϕ3e,lm2=2πλu3e,lmsin[(l+1)πax1p]sin[(m+1)πbx2p]
(14)
The argument yields the mechanical phase
arg(Αlm)=arctanIm(Alm)Re(Alm)=2πflmtn
(15)
and the real part is proportional to the instantaneous plate deflection

Re(Αlm)=2πλu3e,lmsin[(l+1)πax1p]sin[(m+1)πbx2p]cos(2πflmt)
(16)

If the same process is applied to all the modes in the corresponding temporal frequencies, Eq. (13) becomes

A(x1p,x2q,tn,l,m)=l=0Lm=0M12ϕ3e,lm(x1p,x2q,l,m)exp(j2πflmtn)
(17)

4. Experimental procedure

4.1 Experimental setup

The TV holography system is shown in Fig. 3
Fig. 3 Experimental setup.
. It consisted of a continuous wave laser Coherent Verdi v6 (λ=532 nm), a high-speed camera NAC Memrecam GX-1 and a speckle interferometer with sensitivity to out-of-plane displacements. The test object was a thin aluminium plate with clamped edges and dimensions 190 mm × 140 mm × 1 mm, sprayed with dry developer powder. A rectangular aperture and a lens of focal length 100 mm were used to form an image of the plate on the CMOS sensor. The pixel size of the camera was 21.7 μm × 21.7 μm. A spatial carrier to extract the optical phase was introduced in every interferogram by tilting the reference beam slightly off axis.

Several of the lowest frequency vibration modes of the plate were simultaneously excited by means of a loudspeaker driven by an audio amplifier. The audio signal was white noise, generated with the free software Audacity and exported to a .wav file of duration 40 s. An ordinary mp3/wav player connected to the amplifier was used to play the file. The frequencies of the first four vibration modes were known to be approximately 320 Hz for M00, 560 Hz for M10, 777 Hz for M01 and 890 Hz for M20.

4.2 Data acquisition and processing of optical phase-change maps

A full-field temporal history of the vibration was recorded during a time interval of 272.47 ms in Nrec=970 interferograms with a bit depth of 12 bits per pixel. The frame rate of the camera f frame is programmable in steps of one frame per second and was set to f frame=4f 20=3560 frames per second, four times the frequency of M20. To freeze the vibration, the exposure time during each frame (Texp) was set to 100 μs by the electronic shutter of the camera. The effective integration time was thus kept below 10% of the vibration period Tlm=1/flm for modes up to M20, so the condition of pulsed illumination assumed in section 2.2 is acceptable. The acquisition of the interferograms was manually triggered several seconds after starting the excitation, to ensure that the vibration modes had time enough to build up. Therefore, a stationary motion was assumed and an small subset Nset<Nrec of consecutive interferograms was used for the calculations. The first interferogram of the subset was taken as the reference state.

To compute the Fourier transform (both 2D and 3D), a fast Fourier transform (FFT) algorithm was used [13

13. H. W. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Fourier transform spectral methods” in Numerical Recipes in C, (Cambridge University Press 1988).

]. It required that all the dimensions were integer powers of two, so the interferograms were padded with zeroes from the original size of 856 pixel×848 pixel up to 1024 pixel×1024 pixel, and a value of Nset=65 was selected. The optical phase-change ΔΦn between the reference and the remaining interferograms was calculated as explained in section 2.2, and N=64 optical phase-change maps were obtained. The value of the optical phase-change in the padded region of every map was deliberately set to zero, because it contained spurious values that affected the subsequent computation of the 3D Fourier transform. Figure 2(i) shows five of these optical phase-change maps for consecutive instants. It can be noticed that white noise was capable of exciting simultaneously several low-order modes with enough amplitude to be detected by our system.

4.3 Data processing with the spatiotemporal 3D Fourier transform

A 3D Fourier transform was applied to the N maps, and a set of 64 2D complex-valued spectra were obtained. Each 2D spectrum corresponds to a set of spatial frequencies fp’ and fq’ and one temporal frequency fn’ (see Eq. (8)). Figure 4
Fig. 4 (i) Modulus of several complex-valued spectra obtained after processing N=64 optical phase-change maps with the spatiotemporal 3D Fourier transform. The images were equalized and only a small region of 225 pixel × 225 pixel around the center of the spectra is shown. Black and white represent zero and the maximum modulus respectively. A filter is shown on plane n’=14. (ii) 3D representation of a smaller region of 25 pixel×25 pixel.
shows the modulus of several spectra for different values of n’. The dots replace planes corresponding to intermediate values of n’ that were removed from the figure, and also indicate that the represented spectra are not equally spaced in the temporal frequency axis. In Fig. 4(i) black and white represent zero and the maximum value of the modulus respectively. Peaks near the center of the frequency spectrum appear for n’=0 (DC term), n’=6 (f6=333,75 s−1) and n’=14 (f14=778,75 s−1). The latter are the spectral content of modes M00 and M01, whose natural frequencies are very close to f6 and f14 respectively. Figure 4(ii) is a 3D representation that makes it easier to compare the relative height of the maxima.

The proposed 3D spatiotemporal Fourier transform method permits, in general, mode separation in space and time. For example, Deán et al. [14

14. J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008). [CrossRef] [PubMed]

] separated spatially two propagating Lamb modes with the same temporal frequency taking advantage of their different wavelengths. The case herein presented is the opposite: the spatial frequencies of the excited modes are very low (see Fig. 4); therefore, the separation of different modes is generally possible only if their spectral content is distributed in different temporal frequencies. The number N of optical phase-change maps necessary to achieve the required resolution in temporal frequency depends then on the excited modes and their relative temporal frequencies. The higher the value of N, the better the resolution in the temporal frequency axis, at the cost of more RAM memory requirements and computational effort.

To assess the suitability of the chosen value N=64, the maximum modulus of the spectra was plotted versus n’ in Fig. 5
Fig. 5 Maximum modulus of the spectra versus n’. The values of temporal frequency (computed for Δt=(1/3560) s, N=64) for the main peaks are indicated. They correspond to M00, M01, M20 and M02.
. Peaks for n’=6, 14, 16 (f16=890 s−1) and 26 (f26=1446.25 s−1) are apparent. The first three match with the known frequencies of M00, M01 and M20. The last one turned out to be M02. On the contrary, M10 was not properly excited, since the peak at the expected frequency (n’=10) was not discernable. The separation between peaks indicates that N=64 optical phase-change maps is enough to separate the different excited modes in the frequency space.

4.4 Data processing: separation of vibration modes

The resonant modes M00, M01, M20 and M02 were separated from the same set of spectra by the repeated application of the following two steps: firstly, a filtering stage; secondly, an inverse Fourier transform of the filtered data. To separate each individual mode, a single filter of rectangular shape and profile, located on the corresponding frequency plane, was used (see for example Fig. 4, n =14). Two different combinations of resonant modes were also separated, following the same two-step procedure. However, in this case, multiple filters located on the corresponding frequency planes were simultaneously applied. In all the cases, square filters of size 14 pixel×14 pixel, symmetrical with respect to the center of the spectra, were used. As a result, six sequences of N=64 complex-valued maps, corresponding to the same instants than the original sequence of optical phase-change maps, were obtained. Table 1

Table 1. Summary of results

table-icon
View This Table
summarizes the information regarding this data processing and labels the obtained sequences. SM stands for “single mode” and MM for “multimode”.

The modulus of the complex data is proportional to the mechanical amplitude and the argument is equal to the vibration phase at the recording instant. The real part combines amplitude and phase and is proportional to the instantaneous deflection of the plate. An example of the modulus and argument of the complex data was shown in Fig. 2(iii). In the following figures, only a pseudo 3D representation of the real part of the complex-valued data is shown.

Regarding the data processing time, the calculation of the N=64 optical phase-change maps from the 65 interferograms (section 4.2) takes about 60 s, whereas each 3D Fourier transform takes about 30 s. The whole procedure (which also involves reading from and writing data to disk) takes less than 4.5 minutes. These calculation times were obtained with a personal computer equipped with an AMD Athlon 64 3000+ processor at 1.81 GHz and 1 GB of RAM.

5. Results and discussion

Figures 6(a)
Fig. 6 Instantaneous displacement of (a) mode M00, (b) mode M01, (c) mode M20 and (d) mode M02. (e) Operating deflection shape (ODS) due to modes M00, M01, M20 and M02.
, 6(b), 6(c) and 6(d) show the instantaneous displacement of modes M00, M01, M20 and M02 respectively. The data were taken from sequences SM1, SM2, SM3 and SM4 and correspond to the same instant. The out-of-plane displacement shown in Fig. 6(e) was taken from sequence MM1, and is due to the superposition of all the modes at that same instant. Since, according to Fig. 5, these are the main resonant modes that were excited in the plate, Fig. 6(e) is the operating deflection shape (ODS) of the plate at that instant.

Rows (i), (ii) and (iii) in Fig. 7
Fig. 7 Instantaneous displacement of (i) mode M00 (Media 1), (ii) mode M01 (Media 2) and (iii) mode M20. (Media 3). (iv): Instantaneous displacement due to the superposition of modes M00, M01 and M20 (Media 4).
show respectively the instantaneous displacement of modes M00, M01 and M20 in three consecutive instants. Row (iv), taken from sequence MM2, shows the superposition of those three modes in the same instants. This result is what we call a filtered deflection shape (FDS), since it is an ODS where the contribution of one of the modes present in the plate has been removed. Four multimedia files that show the 64 frames of the temporal history in these four cases are available in the online version of the journal.

All the images have a 160 mm×159 mm field of view, and were trimmed to remove the padded region. In all cases, the displacement (in nm) was obtained by applying a scale factor of λ/2π (see Eq. (16)) to the real part of the complex-valued maps.

These results show that the proposed technique can be used to analyze multimode vibrations. Individual vibration modes can be extracted from the multimode vibration and studied separately. Also, the instantaneous displacement due to any superposition of modes can be obtained by the simultaneous application of multiple filters to the data. This filtering stage also improves significantly the signal to noise ratio of the final output compared to the original optical phase-change maps.

The method does not require the synchronization of acquisition and vibration to separate the modes, neither does it impose conditions on the relative ratios of the mode frequencies. The only requirement is that the frame rate of the camera is at least twice the frequency of the highest mode of interest, to comply with the sampling theorem. In principle, no repetitions of the experiment are required to obtain a true temporal history of the vibration and, therefore, the technique is promising for the study of nonrepetitive phenomena.

6. Conclusions

We have presented a novel technique for the analysis of multimode vibrations in plates. Firstly, a TV holography setup equipped with a high speed camera records a full-field temporal history of the plate deflection —i. e., its operating deflection shape, ODS— as a sequence of interferograms, from which a sequence of optical phase-change maps is calculated. A 3D array of data (two spatial dimensions and time) is thus obtained. Secondly, a 3D Fourier transform of these data separates the modes in the frequency space and allows the extraction of the individual resonant modes (or any combination of resonant modes) by means of bandpass filters. An inverse Fourier transform of the filtered data yields a sequence of complex-valued maps that contain the mechanical amplitude and phase of the selected mode or modes in the recording instants. If all the modes are considered, a smoother version of the originally ODS is obtained. The technique was successfully tested with a fully clamped aluminium plate excited with white noise and vibrating simultaneously in several of its natural resonant modes.

Acknowledgments

This work was co-funded by the Spanish Ministerio de Ciencia e Innovación and by the European Comission (ERDF) in the context of the Plan Nacional de I+D+i (project number DPI2008-02709) and by the Dirección Xeral de Investigación, Desenvolvemento e Innovación da Xunta de Galicia in the context of the Plan Galego de IDIT (project number INCITE08PXIB303252PR). Supplementary co-funding from Universidade de Vigo (project number I608122F64102) is also acknowledged. C. Trillo gratefully acknowledges the mobility grants received from Xunta de Galicia and Universidade de Vigo. Messrs. J. Alfredo Espinoza Pinto, Ricardo Valdivia Hernández and Jorge I. Bermúdez Rodríguez are acknowledged for their help with the excitation system.

References

1.

Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000). [CrossRef]

2.

D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995). [CrossRef]

3.

F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000). [CrossRef]

4.

O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, “Measurement of higher armonics in periodic vibrations using phase-modulated TV holography with digital image processing,” Appl. Opt. 33(22), 4997–5002 (1994). [CrossRef] [PubMed]

5.

A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994). [CrossRef]

6.

J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997). [CrossRef]

7.

A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000). [CrossRef]

8.

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999). [CrossRef]

9.

C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000). [CrossRef]

10.

W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), Chap. 5.

11.

H. O. Saldner, N. E. Molin, and K. A. Stetson, “Fourier transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996). [CrossRef] [PubMed]

12.

C. Trillo, and Á. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).

13.

H. W. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Fourier transform spectral methods” in Numerical Recipes in C, (Cambridge University Press 1988).

14.

J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008). [CrossRef] [PubMed]

15.

R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), Chap. 8.

OCIS Codes
(100.2000) Image processing : Digital image processing
(120.6160) Instrumentation, measurement, and metrology : Speckle interferometry
(120.7280) Instrumentation, measurement, and metrology : Vibration analysis

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: July 1, 2009
Revised Manuscript: September 3, 2009
Manuscript Accepted: September 7, 2009
Published: September 23, 2009

Citation
Cristina Trillo, Ángel F. Doval, Fernando Mendoza-Santoyo, Carlos Pérez-López, Manuel de la Torre-Ibarra, and J. Luis Deán, "Multimode vibration analysis with high-speed TV holography and a spatiotemporal 3D Fourier transform method," Opt. Express 17, 18014-18025 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18014


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References

  1. Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000). [CrossRef]
  2. D. P. Towers, C. H. Buckberry, B. C. Stockley, and M. P. Jones, “Measurement of complex vibrational modes and surface form – a combined system,” Meas. Sci. Technol. 6(9), 1242–1249 (1995). [CrossRef]
  3. F. M. Santoyo, G. Pedrini, Ph. Fröning, H. J. Tiziani, and P. H Kulla, “Comparison of double-pulse digital holography and HPFEM measurements,” Opt. Lasers Eng. 32(6), 529–536 (2000). [CrossRef]
  4. O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, “Measurement of higher armonics in periodic vibrations using phase-modulated TV holography with digital image processing,” Appl. Opt. 33(22), 4997–5002 (1994). [CrossRef] [PubMed]
  5. A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. 107(1-2), 28–34 (1994). [CrossRef]
  6. J. D. R. Valera, J. D. C. Jones, O. J. Løkberg, C. H. Buckberry, and D. P. Towers, “Bi-modal vibration analysis with stroboscopic heterodyned ESPI,” Meas. Sci. Technol. 8(6), 648–655 (1997). [CrossRef]
  7. A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. 174(5-6), 347–353 (2000). [CrossRef]
  8. A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999). [CrossRef]
  9. C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “The application of high-speed TV-holography to time-resolved vibration measurements,” Opt. Lasers Eng. 32(4), 387–394 (2000). [CrossRef]
  10. W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), Chap. 5.
  11. H. O. Saldner, N. E. Molin, and K. A. Stetson, “Fourier transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996). [CrossRef] [PubMed]
  12. C. Trillo, and Á. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).
  13. H. W. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Fourier transform spectral methods” in Numerical Recipes in C, (Cambridge University Press 1988).
  14. J. L. Deán, C. Trillo, Á. F. Doval, and J. L. Fernández, “Determination of thickness and elastic constants of aluminum plates from full-field wavelength measurements of single-mode narrowband Lamb waves,” J. Acoust. Soc. Am. 124(3), 1477–1489 (2008). [CrossRef] [PubMed]
  15. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), Chap. 8.

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