Analyzing the total structural intensity in beams using a ...

299

Analyzing the total structural intensity inbeams using a homodyne laser dopplervibrometer1

Agnaldo A. Freschia, Allan K.A. Pereiraa,Khaled M. Ahmidaa, Jaime Frejlichb andJose Roberto F. ArrudaaaDepartamento de Mecanica Computacional, FEM,Universidade Estadual de Campinas C.P. 6122,Campinas, SP, 13083-970 BrazilbLaboratorio deOptica, IFGW, UniversidadeEstadual de Campinas, Brazil

Received 11 June 1999

Revised 26 June 2000

The total structural intensity in beams can be considered ascomposed of three types ofwaves: bending, longitudinal, andtorsional. In passive and active control applications, it is use-ful to separate each of these components in order to evaluatetheir contribution to the total structural power flowing throughthe beam. In this paper, a twistedz-shaped beam is used inorder to allow the three types ofwaves topropagate. Thecontributions of the structural intensity, due to thesewaves,are computed from measurements taken over the surface ofthe beam with a simple homodyne interferometric laser vi-brometer. The optical sensor incorporates some polarizingoptics, additional to a Michelson type interferometer, to gen-erate two optical signals in quadrature, which are processedto display velocities and/or displacements. This optical pro-cessing scheme is used to remove the directional ambiguityfrom the velocity measurement and allows nearly all back-scattered light collected from the object to be detect. Thispaper investigates the performance of the laser vibrometer inthe estimation of the differentwavecomponents. The resultsare validated by comparing the total structural intensity com-puted from the laser measurements, with the measured inputpower. Results computed from measurements using PVDFsensors are also shown, and compared with the non-intrusivelaser measurements.

Keywords: Structural power flow, vibration intensity, laservibrometers,wavecomponents,wavepropagation

1. Introduction

Predicting and measuring elastic waves, propagatingthrough a structure, can be of foremost importance invibroacoustic problems. The prediction and the mea-surement of the propagating elastic waves within astructure is usually referred to as structural power flowor structural intensity. The energy flow to localizeddampers or to neighboring structures and supports canbe cardinal mechanisms through which structural vi-bration is damped out, which partly explains the prac-tical difficulty in estimating internal damping coeffi-cients in structures using ground vibration tests. It canalso be a key for solving structure-borne noise prob-lems, by channeling vibrations to where they do notradiate noise, instead of trying to suppress them.

Structural intensity refers to the active part of vibra-tion energy. As active energy is usually only a smallfraction of the total vibration energy, estimating it frommeasured vibration is not always a simple task. Mea-suring structural intensity is more elaborate than mea-suring acoustic intensity. Sound propagates through airin compression-type waves only, while vibration prop-agates through solids in two basic types of waves: com-pression and shear waves. Depending on the geome-try of the continuum, these two basic types of wavescombine into different types of waves, such as bending,torsional, and longitudinal waves, which must all bemeasured.

Both acoustic intensity and vibration intensity mea-surements are associated with cross measurements be-tween closely-spaced transducers: microphones in the

1An earlier form of this paper was presented at the 3rd Interna-tional Conference on Vibration Measurements by Laser Techniques,Ancona, Italy, June 1998. The conference was organized by theItalian Association of Laser Velocimetry. The articles were selected,edited, and reviewed by a committee chaired by: Professor EnricoPrimo Tomasini, Depertment of Mechanics, Faculty of EngineeringUniversity of Ancona, Italy.

Shock and Vibration 7 (2000) 299–308ISSN 1070-9622 / $8.00 2000, IOS Press. All rights reserved

300 A.A. Freschi et al. / Analyzing the total structural intensity in beams using a homodyne laser doppler vibrometer

39o

39o

45o

x''

-z'

45o

y' y''

y

x

-z = -z'

y'

x'

-z''Sand box

7 3

8 4

Section 2 Section 1

Back viewMeasurement ZoneMeasurement Zone

1 5

2 6

Section 1 Section 2

Front view

0.0259m

0.0035m

Cross sectionBeam

PVDF

0.03m

0.012m

Fig. 1. Z-shaped beam with a quasi-anechoic termination (sand box).

former case and, usually, accelerometers or strain sen-sors in the latter. Following the original work byNoiseux [10], many authors have investigated differ-ent ways of computing the bending vibration intensityfrom measured accelerations in beams and plates [1,11,13]. Most authors use finite-difference approximationsto compute the partial spatial derivatives that are nec-essary to compute the power flow, but the wave compo-nent approach [6] is more general and does not requireclosely spaced measurements.

In this paper, az-shaped beam (see Fig. 1), wherethe three kinds of waves – bending, longitudinal, andtorsional – propagate, is used to investigate the difficul-ties in estimating the total structural intensity. The con-tributions of the structural intensity, due to these kindsof waves, are computed from measurements taken overthe surface of the beam, with a simple homodyne inter-ferometric laser vibrometer and with piezoelectric film

sensor patches (PVDF). The optical sensor, which wasdeveloped in our laboratories, is briefly described.

2. Structural intensity components

The total structural intensity (SI) depends on thekinds of forces and moments that excite it. Each com-ponent of these forces and moments is associated witha different kind of wave propagating through the struc-ture. In general, the total structural intensity, in astraight beam segment, along thex-axis can be consid-ered to be composed of the following kinds of waves:a bending wave in thexy-plane, a bending wave in thexz-plane, a longitudinal wave in thex direction and atorsional wave (angleθ in theyz-plane). In vibrationcontrol applications, it can be useful to find a way toseparate these components, in order to calculate the

A.A. Freschi et al. / Analyzing the total structural intensity in beams using a homodyne laser doppler vibrometer 301

total SI. Berthelot et al. [3] used a Laser Doppler Vi-brometer (LDV) to separate the contributions from thestructural intensity carried by the longitudinal waves inthe presence of strong flexural waves.

Due to the geometry of the cross section of thez-shaped beam and due to the low frequency range thatwas investigated, bending in thexz-plane was assumednegligible (the beam is much stiffer in this plane). Inthis configuration all three of the kinds of waves men-tioned propagate in the last segment of the beam (themeasurement zone). The purpose here is therefore toseparate the contributions of the SI due to the threekinds of waves using an LDV and compare the resultsobtained with those obtained using PVDF patches.

First, consider the bending waves using Bernoulli-Euler’s theory, which is suitable for thin beams at lowfrequencies. The bending SI component may be ex-pressed as [10]

Pb = EI

⟨∂3w

∂x3

∂w

∂t− ∂2w

∂x2

∂2w

∂x∂t

⟩t

(1)

whereE is Young’s modulus,I is the inertia momentof the cross section of the beam,〈〉t denotes the timeaverage, andw is the displacement in they direction.Considering only the far field components and usingPVDF sensors of length 2d spaced by∆x, bonded tothe surface of the beam, it is possible to show that [6]

Pbff =2EIω

t2kbr2b sin(kb∆x)

�{ε∗1ε2} (2)

whererb = sin(kbd)kbd , k4

b = ω2 SρEI is the flexural wave-

number,ρ is the density,S is the cross-section area,t is half of the beam thickness,εi is the complex am-plitude of the strain measured at positioni, � is theimaginary part of a complex function, and argumentωis suppressed for clarity. The PVDF sensors are usedto measure the surface strain.

The SI component, due to longitudinal waves, canbe written as [4]

PL = −SE

⟨∂u

∂x

∂u

∂t

⟩t

(3)

whereu is the displacement in thex direction.Again, using PVDF sensors, it is possible to show

that

PL =SEω

2kLr2L sin(kL∆x)

�{ε∗1ε2} (4)

whererL = sin(kLd)kLd andk2

L = ω2 ρE is the longitudinal

wavenumber.When using the LDV, both out-of-plane and in-plane

velocities are measured. Therefore, we will now review

the expressions for the structural intensity in beamsusing velocities. The flexural SI component can bewritten as [10]

Pbff =2EIk3

b

ω sin(kb∆x)�{W ∗

1 W2} (5)

whereW is the complex amplitude of the transversevelocity component associated with bending.

The longitudinal SI componentcan be written as [10]

PL =SEkL

2ω sin(kL∆x)�{U∗

1 U2} (6)

whereU is the complex amplitude of the longitudinalvelocity. The correction in the velocity, proposed byBerthelot et al. [3], was used in order allow for themeasured longitudinal velocity component induced bythe flexural waves.

Finally, the torsional SI component may be expressedas [4]

PT = −T

⟨∂θ

∂x

∂θ

∂t

⟩t

(7)

whereT = Gbh3/3 is the torsional stiffness,b is thewidth of the beam, andG is the shear modulus. Usingthe LDV measurements

PT =T ωkT

2b2ω sin(kT ∆x)�{V ∗

1 V2} (8)

wherek2T = ω2 b2ρ

16h2G andV is the complex amplitudeof the rotational velocity.

The input power was computed using the expression:

Pin =12�{F U∗} =

12ω

�{F U∗} (9)

whereF is the excitation force,U is the velocity, andU is the acceleration in the direction of the appliedforce. Notice that in the above expressions, the com-plex real and imaginary components are in phase andin quadrature with the input force, respectively.

3. The homodyne laser Doppler vibrometer

Laser Doppler Vibrometers are instruments used tomeasure velocities and/or displacements of points ona vibrating surface by focusing a laser beam onto theobject. The operation principle of all LDVs relies onthe detection of the Doppler shift in the frequency ofthe light back-scattered by the surface. In most appli-cations, the Doppler shifts are too low to be measuredby a spectrometer, so the light collected from the object


Fr.St. 1mW He-Ne LaserPBS

BS

L1

PD1

D2

Z Shaped Beam

M

λ/4

λ/4

λ/4

P

R

S

Sand BoxExcitation

L2

Fig. 2. Simplified scheme of the homodyne LDV.

is mixed with an optical reference beam and projectedonto a detector. The detector is a nonlinear device withan output proportional to the power of the incident light,i.e., the square of the optical electric field. Thus, thecoherent addition (interference) of the scattered beamcollected from the object with the reference beam re-sults in a voltage output proportional to the difference ofthe two optical frequencies. The frequencyνD (calledDoppler frequency) of this alternate signal can be writ-ten asνD = 2u/λ, whereu is the component of thevelocity in the direction of the incident beam andλ isthe light wavelength. For He-Ne laser light in vacuumλ = 0.6328 µm andνD/u ≈ 3, 16 MHz/m/s.

One of the simplest optical sensors for a LDV is theMichelson interferometer [2]. In its original configu-ration (using a fixed mirror in the reference arm), theMichelson interferometer can be employed for measur-ing vibrations when the displacement of the object issomewhat higher than the laser wavelength and whenthe knowledge of the direction of the velocity is of noimportance. However, most applications require thediscrimination of the direction of the velocity, whichis impossible by this scheme. In fact, although theDoppler frequency shift depends on the direction of themotion, the output of such an interferometer is sym-metric with respect to the sign of velocity. The mostcommonly used approach to discriminate the directionof the motion is to use a frequency shifting device in thereference laser beam to break the symmetry of the op-tical output. These are called heterodyned systems [7,

12]. In this present experiment, an optical setup basedon the scheme used by Jackson et al. [8] was imple-mented. The homodyned system mixes the Dopplershifted laser beam from the object with a referencebeam, which is not frequency shifted, as in the case ofthe classical Michelson interferometer. The directionalambiguity problem is solved by generating two opticalsignals in phase quadrature.

A simplified scheme of the optical setup is shownin Fig. 2. The linearly polarized light from a 1 mWfrequency-stabilized He-Ne laser is split into orthogo-nal linear polarization components by the polarizationbeam splitter (PBS). The reference beam (R) is circu-larly polarized by using a quarter wave plate (λ/4).After passing again through the plate, the polarizationof the reflected light from the mirror (M) is rotated by90◦ with respect to the polarization of the incomingwave and it is transmitted by the PBS cube. A similarprocedure is used in the S direction, where a beam-expanding telescope is used to focus the laser beamonto the point of interest. A system of two slidingorthogonal mirrors allows easy selection of the mea-surement point and the angle of incidence. In thisway, the light collected from the mirror and the ob-ject are transmitted and reflected by the PBS cube, re-spectively. These two orthogonal linearly polarizedbeams are then split into two channels by the non-polarizing beam splitter cube (BS), resulting in twoexactly-similar samples of reference and scattered lighthaving the same phase difference in each channel. A


quarter wave plate is placed in one channel to producean additional 90◦ phase shift between the beams. Toobtain interference from these two beams, 45◦ compo-nents are taken with polarizers (P), and the mixed op-tical signals are projected onto the photodetectors (D).The alternate terms of the resulting voltages fromD1

andD2 can be written asVX = V0 cos(2πνDt + ϕ0)andVY = V0 sin(2πνDt + ϕ0), whereϕ0 is a con-stant phase term (neglecting noise effects introduced bythermal fluctuations and mechanical instabilities in theinterferometer) andV0 is a voltage depending, amongother parameters, on the scattered light collected fromthe object [5].

The LDV used in this work is being developed ina collaboration between the Vibroacoustics Laboratoryand the Optics Laboratory of the University of Camp-inas. TheVX andVY signals are acquired by a digitaloscilloscope and numerically processed for displayingvelocities, using a procedure similar to that describedby Koo et al. [9]. First attempts to make these experi-ments employed a first-generation heterodyne Dopplervibrometer, which uses a rotating disk to modulate thefrequency of the optical reference beam. Although use-ful and inexpensive, this method of modulation addsrandom phase and amplitude noise in the resultingDoppler signals. Compared with this first-generationinstrument, the results obtained using our home-madevibrometer had superior signal-to-noise ratios. Cer-tainly, the experiments here described could also beperformed using any state-of-the-art commercial laservibrometer. However, the development of an instru-ment, besides its educational value, potentially allowsthe implementation of special optical setups which maybe necessary in particular applications.

4. The experiment

4.1. Experimental set-up

An aluminum beam was shaped in such a way thattorsional, longitudinal, and flexural waves would de-velop in the measurement zone indicated in the Fig. 1.The twisted z-shaped beam consists of one straightbeam originally 2 m long which was folded at threedifferent locations along its length so that four straightspans (AB, BC, CD, and DE) were formed. Span ABis parallel to thex axis and is 1.4 m long, with 0.4 mplunged in the sand box. Span BC is in thexy plane;it is 0.3 m long and was produced by a rotation of 39◦

in the positivez direction. Span CD is 0.28 m long

and was generated by folding the beam along a linethat makes 45◦ with respect to the direction normal tothe neutral axis of span BC. The folding consisted ofa rotation of−45◦ with respect to the axis defined bythe folding line (z ′′). Finally, in order to have a flatsurface where to bond a force transducer, span DE wasobtained by folding the beam along the line orthogonalto its neutral axis with a rotation of 90◦. The excitationwas applied normally to span DE, which correspondsto exciting longitudinally the free end of span CD.

Using different combinations of measurements takenusing PVDF patches and LDV, it is possible to sepa-rate the contribution of each one of the displacementsassociated with different types of waves and, therefore,with different SI components. Eight PVDF patcheswere used in the configuration shown in Fig. 1.

By combining the response measured at each of thePVDFs, it is possible to separate the contributions ofthe different types of waves. The measured straindistribution can be decomposed into the sum of ananti-symmetric distribution – the bending component– with respect to the center of the beam (e.g., PVDF1-PVDF3+PVDF2-PVDF4 on cross-section 1 in Fig. 1)and a uniform strain distribution - the longitudinal com-ponent (e.g., PVDF1+PVDF3+PVDF2+PVDF4 oncross-section 1 in Fig. 1). With this PVDF configura-tion it is not possible to measure the torsional contribu-tion, which is expected to be small in this example, aswill be shown with the LDV measurements.

Now, using LDV and measuring at 4 points in 2 dif-ferent sections of the beam (corresponding to the centerof the PVDFs in Fig. 1, at the edges of the beam), aimedat with different angles, it is possible to separate thecontribution due to the three different waves: bending,longitudinal and torsional. At each of the points twomeasurements were taken at+45◦ and at−45◦, on thexy-plane, relative to they direction. The out-of-planevelocity was calculated by adding the measurementstaken at each location at+45◦ and−45◦ and dividingthe result by

√2. The in-plane velocity is simply the

difference between the two measurements divided by√2. The torsional component is obtained by subtract-

ing the out-of-plane components measured at the upperand lower edges of the beam section and dividing theresult by the beam width. Thus, the separated veloc-ity components associated with the different kinds ofwaves are obtained.

4.2. Experimental results

In order to apply the techniques described above, thez-shaped beam structure was excited by an electrody-


Fig. 3. Input power versus active SI measured with PVDF patches. — Input power;· · · active SI.

namic shaker using a periodic chirp signal. The periodof the chirp was 250 ms in the case of the LDV testand 1 s in the case of the PVDF test, corresponding toa frequency resolution of 4 Hz and 1 Hz, respectively.The time length has to be restricted to accommodatethe LDV signal processing requirements. Due to thefact that the LDV signal processing was done off-line,with digitized data, there was a compromise betweenthe high sampling rate necessary to resolve the highfrequency Doppler signal, and a sufficient signal lengthfor the resolution of the low frequency velocity signal.10 blocks of 15000samples each were acquired at a rateof 50 kHz. Also, because of this limitation (which canbe overcome using an analog signal processing boardto extract the velocity signal from the photo-detectorsignals), the magnitude of the excitation force was ad-justed so that the maximum Doppler shift was approx-imately 3 kHz (corresponding to a maximum velocityof about 10−3 m/s). Therefore, the active power inputfed to the structure by the shaker was much smaller inthe LDV test than in the PVDF tests.

In order to improve the signal-to-noise ratio, Fre-quency Response Functions between the force trans-ducer signal and the LDV or PVDF signals were calcu-lated. Structural intensity spectra were computed us-ing the cross spectra between the input force and eachmeasured signal, which is the FRF multiplied by theauto power spectrum of the input force.

Figure 3 shows a comparison of the input power andthe total structural intensity computed from measure-ments made with PVDFs. It can be observed that at

higher frequencies the computed SI is smaller than theinput power. This can be explained partly by the influ-ence of the PVDFs which, together with their cabling,absorb part of the incoming energy. Figure 4 shows therelative contributions of the bending and longitudinalwaves to the total intensity. It can be observed that,at these test force levels, the longitudinal waves carrymore active power than the flexural waves.

Figure 5 shows the results obtained with LDV mea-surements (with the PVDF patches and associated ca-bling removed). There is a better agreement betweeninput power and total structural intensity. The fact thatthe total SI does not decrease with frequency, can beexplained by the fact that the LDV is non-intrusive andthus, the energy flow is preserved. Figure 6 shows therelative contributions of the different waves to the to-tal SI. It can be observed that, in this test, with muchsmaller power levels (compare the magnitudes withFig. 4), the flexural and longitudinal waves have contri-butions of similar magnitudes to the total SI. This canbe explained by the different behavior of the sand boxat different vibration levels. At lower levels most of theenergy is dissipated through flexural motion, while athigher amplitude levels, most of the dissipation is dueto longitudinal motion. To verify this hypothesis, an-other test was carried out at an intermediate force level.Results in Fig. 7 confirm that, for lower force levels, thecontribution of the longitudinal waves decreases withrespect to that of the flexural waves.


Fig. 4. SI components measured using PVDF patches. — Flexural SI;· · · longitudinal SI.

Fig. 5. Input power versus active SI – LDV. — Input power;· · · active SI.

The torsional contribution, as expected (due to thegeometry of thez-shaped beam), is much smaller thanthe other two components.

5. Conclusions

Structural intensity measurements are difficult tocarry out because the active part of the total vibrationenergy is usually very small compared with the reac-

tive part. Furthermore, conventional instrumentationmay alter the structural intensity pattern by absorbingenergy locally. In this paper, the formulation of thestructural intensity propagated via the three types ofwaves in beams (flexural, longitudinal, and torsional)was reviewed. Sensor placement and signal combina-tions that separate the three types of waves were de-scribed. Piezoelectric film sensor (PVDF) and LDVmeasurements were used.

A z-shaped beam was used to investigate the pro-


Fig. 6. SI components measured using the LDV. - - - Flexural SI; — longitudinal SI;· · · torsional SI.

Fig. 7. SI components at a lower vibration level measured using PVDF patches. — Flexural SI;· · · longitudinal SI.

posed techniques experimentally. With the LDV, thethree types of waves were measured, while with thePVDF arrangement, only the longitudinal and flexuralwaves were obtained. It was shown that, in the fre-quency range investigated (DC-300 Hz), the torsionalwave contribution was very small.

The wave component approach was used to estimatethe structural intensity components. It is more accuratethan the finite difference because it does not requireapproximation of the spatial derivatives necessary to

evaluate the structural intensity. Besides, it is straight-forward to obtain one expression from the other. Thefinite difference approximation is obtained simply byreplacing “sin(k∆x)” by “ k∆x” in the denominatorof the expression obtained using the wave componentapproach. When using the wave component, attentionmust be paid to the frequency range under investigation,as the functionsin(k∆x) appears in the denominatorand may cause singularity in the intensity estimation.

Besides the wave component approach, the far-field


0 50 100 150 200 250 300 350 400

10

dB

frequency [Hz]

Fig. 8. Ratio between in-plane component to the 2◦ angle error component.

approximation was assumed in the evaluation of thestructural intensity components. Some authors [14]state that for calculating the flexural component the far-field consideration is valid whend > λf /2, wheredis the distance between the excitation and the measure-ment points andλf is the flexural wavelength at theexcitation frequency. Therefore, in real structures, inorder to make the far-field assumption, attention hasto be paid to the excitation frequency range and to thedistance between the measurement point and any dis-continuities.

The LDV used in this work, developed in our lab-oratories, was briefly described. Due to the fact thatthe signal processing of the photo-detector signals wasdone numerically and off-line, there was a compro-mise between the sampling frequency, necessary to ac-quire the high frequency Doppler carrier frequency andthe low frequency modulation. Thus, in the tests per-formed, using LDV, the vibration levels had to be con-siderably lower than in the tests performed using PVDFsensors. This limitation will be overcome with theimplementation of an analog signal processing circuit.

In order to verify the sensitivity of the techniquepresented here with respect to angle errors in the lasermeasurements, an error of 2◦ was considered and thein-plane component variation associated with this er-ror was evaluated. Figure 8 shows the ratio betweenin-plane component and the 2◦ angle error component.

It was considered that for ratios larger than 3 the mea-surements were meaningful. In thez-shaped beam in-vestigated here this ratio happens above 170 Hz.

It was observed that, at higher vibration levels, withthis particular set up, the longitudinal waves were re-sponsible for the propagation of most of the input en-ergy, while at lower vibration levels, the flexural waveswere as effective as the longitudinal waves in convey-ing the energy. A test at an intermediate vibration levelconfirmed this hypothesis. This tendency is presum-ably linked to the behavior of the imperfect anechoictermination used in the experiments (sand box).

Acknowledgments

The authors wish to acknowledge the Brazilian re-search funding agency CNPq (Proc. No. 620705/94-9) and FAPESP (Proc. No. 97/00511-0 and Proc.No. 98/15720-7) for their financial support.

References

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[3] Y.H. Berthelot, M. Yang and J. Jarzinski, Recent progresson laser Doppler measurements in structural acoustics, in:Proceedings of the 4th International Congress on IntensityTechniques,Senlis, France, 1993, pp. 199–206.

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[5] J.W. Goodman, Some fundamental properties of speckles,Journal of the optical society of America66(11) (1976), 1145–1150.

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with passive quadrature detection,Electronics Letters20(10)(1984), 399–401.

[9] K.P. Koo, A.B. Tveten and A. Dandridge, Passive stabilizationscheme for fiber interferometers using (3×3) fiber directionalcouplers,Appl. Phys. Lett.41(7) (1982), 616–618.

[10] D.U. Noiseux, Measurement of power flow in uniform beamsand plates,Journal of the acoustical society of America47(1970), 238–247.

[11] G. Pavic, Measurement of structure borne wave intensity, partI: formulation of methods,Journal of Sound and Vibration49(2) (1976), 221–230.

[12] T.A. Riener, A.C. Goding and F.E. Talke, Measurement ofhead/disk spacing modulation using a two channel fiber opticlaser Doppler vibrometer,IEEE Transactions on Magnetics24(6) (1988), 2745–2747.

[13] J.W. Verheij, Cross-spectral density methods for measuringstructure-borne power flow on beams and pipes,Journal ofSound and Vibration70 (1980), 133–139.

[14] G.P. Gibbs,Simultaneous active control of flexural and exten-sional power flow in thin beams,Ph.D. thesis, Virginia Poly-technic Institute and State University, USA, 1995.

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