FULL-FIELD VIBRATION MEASUREMENT BY TIME-AVERAGE SPECKLE
INTERFEROMETRY AND BY DOPPLER VIBROMETRY – A COMPARISON
Aurélien Moreau1, Dan Borza1, Ioana Nistea1, Mariana Arghir2
Institut National des Sciences Appliquées de Rouen, LMR-FIMEN
BP 8, avenue de l'Université, 76800 Saint-Etienne du Rouvray, France
[email protected], [email protected], [email protected]
2
Université Technique de Cluj,
Str. C. Daicoviciu nr 15, 400020 Cluj - Napoca, Roumanie
[email protected]
1
ABSTRACT
Identification of dynamic material properties, non-destructive testing and study of vibroacoustic behaviour of different
structures impose complex, pointwise and full-field measurements. Among other techniques, optical non-contact techniques
are representing today the favourite choice since they do not add any mass to the structure under test. When the range of
vibration amplitudes is small, most of these techniques are based on interferometric principles. Laser sources and detectors
lead to a continuous improvement of vibration measurement techniques. However,a choice has to be made between spatial
resolution and temporal resolution. Another difficult choice is between space bandwidth product and energetic sensitivity of the
detector. While the number of pixels of a camera is continuously increasing, the pixel size seems limited at its lower end. The
paper presents a comparative study of the most representative full-field non-contact techniques for vibration measurement, as
applied at the Mechanics Laboratory of INSA Rouen. The practical case of measurements concerns the free and the forced
vibrations of a thick, non-metallic plate used in a study aiming to identify its complex damping.
Introduction
The various optical, acoustical and numerical techniques used in vibroacoustics produce results which are different in terms of
spatial resolution, outputs (measured quantities), temporal resolution. As long as the spatial distribution map of vibration
amplitudes is concerned, a comparison is necessary so as to allow estimating the errors and their causes. It should also help
specialists in different branches (acoustics, mechanical structures, dynamics) to choose and use the experimental tools.
In the field of vibration measurement, the most widely known full-field coherent optical technique is speckle interferometry [1].
Digital holography is also a very promising technique, but for the moment being its applications are mostly in the study of small
– microscopic or nanometric – objects. Both techniques are measuring full displacement fields, either static or dynamic. At the
recording stage, the main difference is related to the absence, in the digital holography setup, of the camera lens. In speckle
interferometry each acquired image corresponds to an image-plane hologram, and the interferometric fringe pattern is usually
reconstructed with the help of a temporal or spatial phase-stepping procedure. In digital (Fresnel) holography, the
interferometric fringe pattern is obtained with the help of the digitally reconstructed complex object-wave.
As to the pointwise optical measurements, the instrument of choice is the classical interferometer, mostly known today as
"vibrometer" [2]. The Doppler vibrometer may be configured to measure either displacements (vibration amplitudes) or speeds.
As a step towards full-field techniques, scanning may allow successive measurements to be done in a relatively small number
of points, usually at regularly disposed angular directions. Both vibration amplitude and phase maps may be computed in
these points, by using spatial Fast Fourier Transforms (FFT) techniques and a reference phase related to the excitation. The
spatial FFT processing is computationally efficient but may introduce undesired spatial effects.
Apart from these optical techniques, near-field acoustic holography (NAH) is yet another technique to be considered. A
microphone array, usually plane, is sampling spatially the acoustic pressure field radiated from the vibrating object. The
reference phase is recorded by a reference microphone placed near the structure. Numerical procedures are then applied in
order to reconstruct the complex velocity field at different planes between the object and the hologram. If the reconstruction
plane is very close to the object, as in NAH, the result is the velocity map at the object surface. It should ideally be the same as
that recorded by a scanning vibrometer. If the object is vibrating at a single frequency at one of its modes, the NAH result
should also ideally be proportional to the displacement map obtained by speckle interferometry.
Finally, the results offered by any of these experimental, computer-assisted techniques should be comforted by the
comparison with the results offered by a numerical model of the vibrating structure (e. g. a finite-element model).
From the point of view of the vibration amplitudes (displacements) the results with the highest spatial resolution are offered by
speckle interferometry; that is why some aspects related to the measurement conditions and sources of errors will be
mentioned in the next section.
Full-field vibration measurement by speckle interferometry
The typical setup used in out-of-plane sensitive speckle interferometry is shown in Figure 1. The vibrating object is illuminated
by the CW laser beam transmitted by the beamsplitter BS, expanded by a lens and finally redirected by Mirror3. The part of the
laser beam reflected by BS1 and by Mirror1 is expanded by a spatial filter and directed, through Mirror2 and the beamsplitter
BS2 toward the CCD detector. It is the reference beam. The PZT actuator placed behind Mirror 1 is implementing the temporal
phase stepping, producing a 4-step reference phase variation given by:
π
∆ϕ i = (i − 1) ; i = 1,2,3,4,...
(1)
2
Figure 1. Speckle interferometry setup
The basic four-frame bucket acquired by the computer is described by:
( )
rr
I i = A + B cos(ϕ or + ∆ϕ i )J 0 dK
(2)
where A and B are speckled images of the object, ϕ or is the high spatial frequency, random phase difference between the
rr
object wave and the reference wave, J 0 dK is the Bessel function of zeroth order and first kind having as argument the
r r
r r
projection of the displacement vector d = d (M ) on the sensitivity vector K = K (M ) of the holographic layout, for any
( )
point M on the object surface.
The sensitivity vector has a magnitude
r 2π
K =
and is directed, for each point M on the object, along the direction
λ
bissecting the illuminating direction and the observation direction corresponding to that point. The holographic processor
computes and displays an image representing the real-time, time-averaged hologram of the vibrating object given by:
[
I TAV = (I 1 − I 3 ) + (I 4 − I 2 )
2
]
2 1/ 2
( )
rr
= C J 0 dK
(3)
C is a constant. As seen in Figure 2, the result consists in the image of the test object (a plane plate, sized 100 x 150 mm, 5.1
rr
mm thickness) covered by alternate bright and dark fringes, loci of constant values of the product dK . The image is
reproduced at a 1:2 spatial scale, so as to allow comparison with parametric images obtained by other techniques, e. g.
Doppler vibrometry.
Figure 2. Time-averaged hologram obtained by speckle interferometry, of plate vibrating at 1733 Hz.
In order to compare the holographically measured vibration amplitudes with those obtained by other techniques, one may
either try to process the hologram in order to compute
methods able to produce results in the form
( )
rr
dK
d (M ) using as initial data J 0 (M )
modulo2π
[3], or to use other interferometric
or else to present the data obtained from the other techniques as
simulated holograms, as in [4]. Figure 3 presents the vibration amplitude map obtained for the hologram shown in Figure 2 by
using the method described in [3]. At right, at 1:1 scale, Figure 3(b) shows the velocities map obtained by Doppler vibrometry.
a
b
Figure 3. At left (a), full-field of vibration amplitudes d(M) obtained from the holographic fringe pattern in Figure 2. Values are
given in radians of optical phase,
ϕ vib =
2π
λ
d (M ) . At right (b), 1:1 scaled velocities map obtained by Doppler vibrometry
The main factors able to affect the values of the calculated displacement field are:
•
the speckle noise;
•
neglecting the radial distorsions of the objective lens and the correct relations between the 3d metric coordinates of a
point M(x, y, z) and the corresponding pixel position (i, j);
•
neglecting the values of the sensitivity vector
and
r
r
d (M )K (M )
r
K (M )
over the object surface and the differences between
d (M )
Most of the speckle noise may be eliminated during the procedure applied in extracting the argument of the Bessel function by
using high-resolution time-averaged holography [5]. The objective lens distorsions are measured by a camera calibration
procedure, see for example [6]. The sensitivity factor may be minimised by a proper choice of the lateral distance between
camera and illuminating source, then may be estimated for each object point. In the examples presented in this paper the
sensitivity factor varies across the plate between 0.97 and 0.98.
Vibration velocity measurement by scanning pointwise Doppler vibrometry
For the test plate presented, the excitation was done with an instrumented impact hammer. Data were acquired for a number
of 38*16 points then upsampled to 23*38 points. The experimental velocity FRF allows selecting the amplitude and phase in
the frequency interval 100-3000 Hz with a 2 Hz resolution. The selection of frequencies at which the amplitude maps were
compared with the full-field ones was based on the criterion of a maximum standard deviation in the phase map, shown in
Figure 4 (upper row). In these conditions, velocity data from Doppler Vibrometry and Near-Field Acoustic Holography are
proportional to the displacement data measured by Speckle Interferometry.
Figure 4. Vibrometer results: upper row, phase map and profile; lower row, amplitude map and profile
The upsampled amplitude map at the frequency of 1774 Hz is shown in Figure 3 at a 1:1 scale so as to be compared, after
further scaling, with the amplitude map in Figure 3, obtained from speckle interferometry.
For some Doppler vibrometry measurement systems the built-in scanning mechanism produces data which are not exactly
samples of a rectangular grid: such an example is shown in Figure 5 (a) and (b). In such cases, attention has to be paid at the
correct data ordering and scaling, and the location errors have to be taken into account.
a
b
Figure 5. (a) Part of a grid showing data locations for a scanning Doppler vibrometer; (b) enlarged portion of a data line
Comparing data sets from different techniques
The data processing should allow comparing full-field amplitude maps from different sources: Speckle Interferometry (SI),
Doppler Vibrometry (LDV), Near-Field Acoustic Holography (NAH) [7], Finite Element Model (FEM). The different origins of the
data impose a series of preliminary operations, some of them shown in Figure 6. Both data sets being compared are supposed
to be samples of a rectangular grid.
Figure 6. Preliminary data processing
The main purpose of the comparison is to help improving data processing, and further on understanding and quantifying the
confidence in the velocity maps furnished by LDV and NAH, as well as validating the FE results. All these experimental and
numerical techniques have different measurement or modelling capabilities which go far beyond those of speckle
interferometry in many but one particular field: the amplitude maps. LDV is able to get highly precise temporal histories related
to the amplitude and phase of vibration for particular points of the object, but the spatial distribution is obtained sequentially.
NAH provides a fine analysis of acoustic behaviour of a vibrating object, but digital retropropagation of the complex velocity
field from the antenna array to the source is much less direct and may be affected by many experimental and numerical errors.
FE results may predict a detailed mechanical analysis of the object provided the boundary conditions and material properties
are correctly introduced in the model.
The data considered as most reliable for the spatial maps are those offered by SI, since they are obtained simultaneously for
all object points, directly from the linear relation between the displacement amplitude and the optical phase. They have the
highest spatial resolution which also means highest possible frequency for vibration amplitude measured distributions, no
frequency-dependent sensitivity, and have only be corrected for the sensitivity vector variation across the object surface, which
is a simple operation, at least for a plane test object as the plate used in this study.
The comparison of the two spatially scaled and range normalized amplitude maps
B
Amn
and
Bmn
of average values
A
and
may produce global or local results. A good global parameter is the correlation coefficient C defined as:
C=
∑∑ [(A
mn
m
]
− A )(Bmn − B )
n
(4)
⎛
⎞⎛
⎞
⎜ ∑∑ (Amn − A )⎟⎜ ∑∑ (Bmn − B )⎟
⎝ m n
⎠⎝ m n
⎠
This may of course not be enough in the case of the analysis of different particular acquisition or data processing procedures
under study. The correlation may also be calculated ([8], [9]) between all the modes in the two series, leading to the so-called
Mode Shape Correlation Coefficient (or Modal Assurance Criterion, MAC).
At the other end, the pointwise residue (difference) map
R (m, n ) = Amn − Bmn
between the two spatially scaled and range normalized amplitude maps
(5)
Amn
and
Bmn
may help in an initial stage to detect
regional differences due, for example, to a scanning problem in LDV or a numerical model in FE.
Calculated for each pixel in the coherently scaled series of mode pairs to be compared, the local or regional differences may
also be evidenced by calculating the spatial shape correlation using [10] the Coordinate Modal Assurance Criterion (CoMAC).
This criterion is commonly used in modal analysis and identification to detect sensors with poor correlation. In the case of k
modes detected by the two experimental techniques, CoMAC is defined as:
2
⎡
⎤
⎢∑ Ak (i, j )Bk (i, j )⎥
⎦
CoMAC (i, j ) = ⎣ k
⎛
⎞⎛
⎞
⎜ ∑ Ak2 (i, j )⎟⎜ ∑ Bk2 (i, j )⎟
⎝ k
⎠⎝ k
⎠
(6)
For the case of NAH and in particular when the vibroacoustic analysis is concerned, a parameter of interest is the position of
maxima in the vibration amplitude map. The analysis of the disparity between the position of maxima as detected by SI and by
NAH may show, for example, a systematic radial component due to the numerical retropropagation procedure. A related
indicator may be the average displacement of the nodal lines between the two maps.
Some results
The different comparison procedures aiming at estimating these quantities were programmed in a GUI software developped in
Matlab. Figure 7 shows the graphical interface allowing to implement the procedures proposed in order to characterize the
differences between the two amplitude and phase maps. The main global and semi-global results are displayed in the frame
placed into the lower right part of the window (correlation coefficient 0.988, average displacement of antinodes 1.73 pixels,
maximum antinode displacement 4 pixels, average percentage error 10.12 %, average displacement of nodal lines 0.316
pixels).
Figure 7. Comparison of the (0, 2) mode as measured by vibrometry (left) and by speckle interferometry (right)
The "Bi-profile" tool allows choosing interactively a line on one of the images and examine the profiles along that line on both
data fields. In the example presented in Figure 8 it helps detecting a discontinuity between the lines of the LDV data map,
whose origin has to be looked for either in the scanning mechanism or in data processing.
Figure 8. Arbitrary line and profiles across the two data sets: upper row, LDV; lower row, SI
Similar procedures are applied to the data furnished by AH and FE, allowing to use the holographic results in order to improve
data processing in AH and to validate the numerical model simulation.
Figure 9 presents the vibration amplitude maps obtained by LDV (upper row) and the interferograms (lower row) obtained by
SI. The mode orthogonality is shown in Figure 10 (a) by the Modal Assurance Criterion (MAC). The analysis of the correlation
by using the CoMAC is shown in Figure 10 (b). The regions with lower values for CoMAC usually correspond to sources of
discrepancies (or greater effects of discrepancies) between the two series of modes.
Figure 9. Amplitude maps obtained from LDV and corresponding SI interferograms (upper two raws); Enhanced CoMac
(ECoMAC) between the SI and LDV results, obtained from all but the amplitude map on the same column and EcOMAC
obtained only with the amplitude map on the same column (lower rows). All images in a column correspond to the same mode.
a
b
Figure 10. (a) Modal Assurance Criterion (MAC);
(b) Coordinate Modal Assurance Criterion (CoMAC) for the four modes in Figure 9
The ECoMAC [12] is sometimes preferred in estimating the local differences between two series of modes determined by
different techniques. It uses N modeshapes
Ak , Bk
normalized between -1 and 1:
⎡
⎤
⎢∑ Ak (i, j ) − Bk (i, j ) ⎥
⎦
ECoMAC (i, j ) = ⎣ k
2N
(7)
The lower the local matching between the two modes, the higher (brighter on the image) the ECoMAC value. The results
shown in Figure 9 allow acquiring very useful information about the performances of different data processing procedures as
well as about experimental particularities. For example, one may conclude that the third mode (which was not well isolated
from adjacent modes) should better be ommitted from theseries used in the comparison. Next, one may also see that LDV
data raise some local problems because of local discontinuities in the amplitude maps.
Such criteria (CoMAC, ECoMAC, …) were initially created in the field of numerical modal analysis. The interest of the
generalization of their use in comparing the results offered by two different experimental techniques is obvious: one may have
a unitary approach and coherent results in dealing with numerical and experimental techniques.
Conclusions
A comparative study of the vibration amplitude maps estimated from different non-contact measurement techniques may bring
to light specific drawbacks whose origins are either in the hardware, in the setups or in the data processing. Of particular
interest is the speckle interferometry method, a very direct and high spatial resolution technique, which may help in bringing
developments to near-field acoustic holography, pointwise vibrometry and also to finite element modelling. Different aspects
related to data preparing and processing are presented. Correlation coefficients are in the range 0.96 – 0.99 for LDV and FE,
and 0.8-0.9 for NAH.
The use of criteria initially developped for modal analysis and identification by using numerical (FE) models may be of great
help in identifying specific problems related to different non-contact experimental techniques.
As a conclusion for future development of full-field optical techniques, the author believes that high-speed speckle pattern
interferometry [12] would achieve an important increase of its use in vibroacoustics by finding a practical way towards a higher
temporal resolution at each pixel, even with the price of a certain decreasing (at least in an initial phase) of the spatial
resolution.
References
1. Lokberg, O. J, Hogmoen K., "Use of modulated reference wave in electronic speckle pattern interferometry", J. Phys. E: Sci.
Instrum., vol. 9, 847-851 (1976)
2. Castellini, P., Revel, G. M., Tomasini, E. P., "Laser Doppler Vibrometry: A Review of Advances and Applications", The
Shock and Vibration Digest, Vol. 30, No. 6, 443-456 (1998)
3. Borza, D. "Mechanical vibration measurement by high-resolution time-averaged digital holography", Measurement Science
and Technology, 16, 1853-1864, (2005)
4. Borza, D., "New holographic method and numerical model graphic output for parameter estimation and damage detection",
International Conference "Applied Simulation and Modelling", ASM 2005, Benalmadena (2005)
5. Borza, D., "Full-field vibration amplitude recovery from high-resolution time-averaged speckle interferograms and digital
holograms by regional inverting of the Bessel function", Optics and Lasers in Engineering, 44, Issue 8, 747-770 (2006)
6. Bouguet, J-Y., "Visual methods for three-dimensional modeling", PhD Thesis, California Institute of Technology, Pasadena,
California (1999)
7. Martarelli, M. and Revel, G. M., "Laser Doppler vibrometry and near-field acoustic holography: Different approaches for
surface velocity distribution measurements", Mechanical Systems and Signal Processing, 20, 6, 1312-1321 (2006)
8. Bathe, K. J., Wilson, E. L., " Numerical methods in finite element analysis", Prentice-Hall Inc., 1st edition (1976)
9. Ewins, D. J., "Modal testing: Theory, Practice and Application", Research studies press limited, Baldock, Hertfordshire (2000)
10. Lieven, N. A. J. and Ewins, D. J. (1988). "Spatial correlation of mode shapes, the coordinate modal assurance criterion",
Proc., 8th Int. Modal Analysis Conf., Society for Experimental Mechanics, Bethel, Conn., 690-695 (1998)
11. Williams, R., Crowley, J. and Vold, H., “The multivariate mode indicator function in modal analysis,” International Modal
Analysis Conference, pp. 66–70 (1985)
12. Kilpatrick, J. M. et al, "Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle
o
pattern interferometry ", Optics Letters, vol. 25, n 15 , 1068-1070 (2000)