## Multimode vibration analysis with high-speed TV holography and a spatiotemporal 3D Fourier transform method

Optics Express, Vol. 17, Issue 20, pp. 18014-18025 (2009)

http://dx.doi.org/10.1364/OE.17.018014

Acrobat PDF (498 KB)

### 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

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]

*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

*u*

_{3}=

*u*

_{3}(

*x*

_{1},

*x*

_{2},

*t*,

*l*,

*m*) is the deflection of the plate along direction

*x*

_{3}(see Fig. 1 ),

*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 M

_{00}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

*φ*

_{o}

*is proportional to the instantaneous out-of-plane displacement field*

_{,n}*u*

_{3}at the plate surfaceTaking into account Eq. (1), this dependency can be rewritten aswhere

*φ*

_{3}_{e}

*=*

_{,lm}*φ*

_{3}_{e}

*(*

_{,lm}*x*

_{1},

*x*

_{2},

*l*,

*m*) is the mode shape (only the factors with spatial dependence) expressed in optical terms.

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]

*t*and the reference statewith ΔΦ

_{n}*=ΔΦ*

_{n}*(*

_{n}*x*

_{1},

*x*

_{2},

*t*,

_{n}*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

*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

*ω*/2π of the modes.

_{lm}*P*×

*Q*×

*N*sampled points can be expressed aswith

*p*=0 …

*P*-1,

*q*=0 …

*Q*-1 and

*n*=0,…,

*N*-1. Δ

*x*

_{1}and Δ

*x*

_{2}are the spatial sampling distances in the horizontal and vertical directions respectively, and Δ

*t*is the temporal sampling interval. The discrete spatial and temporal sampling frequencies in

*x*

_{1},

*x*

_{2}and

*t*are given byrespectively, with

*p'*=-

*P*/2,…,

*P*/2,

*q'*=-

*Q*/2,…,

*Q*/2 and

*n'*=-

*N*/2,…,

*N*/2.

*l*,

*m*) so, in the following, we restrict the calculation to just one mode for simplicity.

*φ*

_{3}_{e}

*does not depend on time, the Fourier transform may be separated*

_{,lm}*n'*is the frequency index of the discrete temporal frequency which is nearest to the temporal frequency

_{lm}*f*of mode (

_{lm}*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

*N*complex-valued maps corresponding to the instants

*t*

_{n}## 4. Experimental procedure

### 4.1 Experimental setup

*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 M

_{00}, 560 Hz for M

_{10}, 777 Hz for M

_{01}and 890 Hz for M

_{20}.

### 4.2 Data acquisition and processing of optical phase-change maps

*N*=970 interferograms with a bit depth of 12 bits per pixel. The frame rate of the camera

_{rec}*f*

_{frame}is programmable in steps of one frame per second and was set to

*f*

_{frame}=4

*f*

_{20}=3560 frames per second, four times the frequency of M

_{20}. To freeze the vibration, the exposure time during each frame (T

_{exp}) 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 T

*=1/*

_{lm}*f*for modes up to M

_{lm}_{20}, 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

*N*<

_{set}*N*of consecutive interferograms was used for the calculations. The first interferogram of the subset was taken as the reference state.

_{rec}*N*=65 was selected. The optical phase-change ΔΦ

_{set}*between the reference and the remaining interferograms was calculated as explained in section 2.2, and*

_{n}*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

*N*maps, and a set of 64 2D complex-valued spectra were obtained. Each 2D spectrum corresponds to a set of spatial frequencies

*f*and

_{p’}*f*and one temporal frequency

_{q’}*f*(see Eq. (8)). Figure 4 shows the modulus of several spectra for different values of

_{n’}*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 (

*f*=333,75 s

_{6}^{−1}) and

*n’*=14 (

*f*=778,75 s

_{14}^{−1}). The latter are the spectral content of modes M

_{00}and M

_{01}, whose natural frequencies are very close to

*f*and

_{6}*f*respectively. Figure 4(ii) is a 3D representation that makes it easier to compare the relative height of the maxima.

_{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]

*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.

*N*=64, the maximum modulus of the spectra was plotted versus

*n’*in Fig. 5 . Peaks for

*n’*=6, 14, 16 (

*f*=890 s

_{16}^{−1}) and 26 (

*f*=1446.25 s

_{26}^{−1}) are apparent. The first three match with the known frequencies of M

_{00}, M

_{01}and M

_{20}. The last one turned out to be M

_{02}. On the contrary, M

_{10}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

_{00}, M

_{01}, M

_{20}and M

_{02}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 summarizes the information regarding this data processing and labels the obtained sequences. SM stands for “single mode” and MM for “multimode”.

*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

_{00}, M

_{01}, M

_{20}and M

_{02}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.

_{00}, M

_{01}and M

_{20}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.

## 6. Conclusions

## Acknowledgments

## References

1. | Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. |

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. |

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. |

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. |

5. | A. R. Ganesan, P. Meinlschmidt, and K. D. Hinsch, “Vibration mode separation using comparative electronic speckle pattern interferometry (ESPI),” Opt. Commun. |

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. |

7. | A. R. Ganesan, K. D. Hinsch, and P. Meinlschmidt, “Transition between rationally and irrationally related vibration modes in time-average holography,” Opt. Commun. |

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. |

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. |

10. | W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, |

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. |

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 |

13. | H. W. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Fourier transform spectral methods” in |

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. |

15. | R. McCluney, |

**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

- Á. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11(1), 201 (2000). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- W. Weaver, Jr., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), Chap. 5.
- 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]
- 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).
- 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).
- 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]
- R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994), Chap. 8.

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