Mass Normalized Mode Shapes Using Impact Excitation and 

Continuous­Scan Laser Doppler Vibrometry

Matthew S. Allen & Michael W. Sracic

University of Wisconsin-Madison8th Intl. Conference on Vibration Measurements by Laser

Techniques, Ancona, Italy, June, 2008

The Challenge: Limitations of Current Scanning Laser Vibrometry Methods:

Automated excitation (shaker) needed, which takes additional time to set up and may modify the structureLong time required to acquire measurements on low frequency structuresDifficulty if system changes with time.But, high spatial resolution measurements are needed:

Structural Health Monitoring:Some schemes are much more effective with high resolution measurements [Ho & Ewins1999]

Finite Element Model Validation:Spatial data may reveal (and perhaps allow one to explain) discrepancies between model and test.

Solutions: New LDV systems are under development with dozens of lasers in parallel (http://www.metrolaserinc.com/ )Speckle-Pattern InterferometryContinuous-Scan Laser Doppler Vibrometry (CSLDV): Velocity is measured as the laser spot sweeps continuously over the structure.

First presented by Sriram & Hanagud (1990)Later extended by Stanbridge, Martarelli & Ewins

Sinusoidal ExcitationTransient (Impact) Excitation

Vanlanduit et al. (2002) explored CSLDV with multi-sine input.Allen & Sracic, 2008, CSLDV for Transient Response

http://www.airforce-technology.com/projects/predator/

LDV with scanning mirrors

drive signal for scanning mirrors

data acquisition

Movie: link 1, link 2

…A Useful Laser Show?

Outline:Overview of Allen & Sracic’s Transient CSLDV Method (IMAC 2008)

Comparison to Other Relevant ApproachesResampling Technique and Consequences

Application: Free-Free BeamNatural Frequencies & Mode Shapes

Proposed Mode Scaling AlgorithmConclusions

Outline:Overview of Allen & Sracic’s Transient CSLDV Method (IMAC 2008)

Comparison to Other Relevant ApproachesResampling Technique and Consequences

Application: Free-Free BeamNatural Frequencies & Mode Shapes

Proposed Mode Scaling AlgorithmConclusions

TheoryFree response of a linear time-invariant structure.

Free response measured by CSLDV with scan pattern (x, y) = (x(t), y(t))

Identical to the free response of a linear time-periodic system (Allen & Ginsberg 2006)

Fourier Series Expansion (FSE) Method

Expand ψ(t) in a Fourier Series:

This is identical to the response of an LTI system with (2*NR+1)*N modes.

fundamental frequency of scan pattern (x(t), y(t))

Ewins, Stanbridge & Martarelli

apparent eigenvalues

( ) ( )1 43 3A Ay T y T⎡ ⎤

⎣ ⎦

Consider sampling 3 times per scan period TABreak response into 3 sets:

Each response describes an LTI system with N modes.Responses can be processed using a global modal parameter identification routine.Note! This aliases the nat. frequencies if ωn < ωscan/2

( ) ( )2 63 3A Ay T y T⎡ ⎤

⎣ ⎦

MDTS Method (Allen & Ginsberg)

( )33(0) Ay y T⎡ ⎤

⎣ ⎦

0 200 400 600 800 100010

0

101

102

103

Frequency (rad/s)

Mag

nitu

de

0 5 10 15 20 25 30 3510

0

101

102

Frequency (rad/s)

Mag

nitu

de

Speckle Noise Considerations

When scanning periodically, speckle noise is approximately periodic (Rothberg, Martarelli).MTDS method places the periodic speckle noise at the 0 Hz line, so it is not confused with modal data.

CSLDV Signal Pseudo FRF - MTDS

…

Speckle Noise

4 Channel DAQ

+ Power Source

Xin

Yin

Dev

ice

Laser Control

Function Generator

Impulse Hammer

Laser Head

i)

ii)

iii) iv)

Experimental Validation

1D Scanning of a Free-Free Aluminum beam.Beam is supported with bungee cords.

0 5 10 15-0.05

0

0.05

Time, (s)

Vel

ocity

, (m

m/s

)

Time History For an Impulsive Hammer Pulse to a Free-Free Beam

0 10 20 30 40 50

101

102

103

Frequency

Mag

nitu

de o

f Sin

gula

r Val

ues

Complex Mode Indicator Function

Processing with MDTSData for 100 Hz scan frequency.Resampled synchronous with laser scan period.CSLDV response decomposed into 205 pseudo-FRFs using the MDTS method.Five inputs were used, so the FRF matrix is 205x5Most modes are aliased.CMIF does not show any close (aliased) natural frequencies.

Effect of Aliasing

1 10

1

10

230.3

1.5 2.217.4

47.7

153.7

322.193.2

Vel

ocity

(mm

/s)

Compos ite FR Fs for 5 Inputs 3 S can Frequencies

21 Hz

1 10

1

10

153.7

1.7 2.7

47.6

93.1 322.1

17.5

428.1

230.4

Vel

ocity

(mm

/s) 51 Hz

1 10

1

10 1.7 2.7

4.2 92.8

17.6

322.1

427.8

230.3

153.9

47.4

Frequency(Hz)

Vel

ocity

(mm

/s) 100 Hz

100 HzAliased

5X

51 HzAliased

10X

21 HzAliased

24X

Mode Shapes – Modes 1‐3

0 20 40 60 80 100-4

-2

0

2

4

6

5

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

6

Mode 1Mode 2Mode 3

6

51 Hz

21 Hz

Mode Shapes – Modes 4‐6

0 10 20 30 40 50 60 70 80 90 100-6

-4

-2

0

2

4

6

Mode 4Mode 5Mode 6

5

0 20 40 60 80 100-5

0

5

6

51 Hz

21 Hz

Mode Shapes at hundreds of

points obtained from five time

records!

Mode Scaling Procedure:Fit the late time, free response portion of the response to identify the structure’s natural frequencies and unscaled mode shapes.Now consider the entire response and the measured input, related in the frequency domain by:

The mode shapes are known up to a scale factor Cr, so the CSLDV signal becomes the following and one can solve a least squares problem for the Cr’s

{ } { }1 1) ) )

o oN NY(ω H(ω U(ω

× ×

= { } { } ,

2 2( )2

Nr r dp

r r r r

Hi

φ φω

ω ω ωζ ω=

− −∑

{ } { }2

, , ,2 2

( ))

2

Nr r ex r ex dp

r r r r

C UY(ω

iφ φ ω

ω ω ωζ ω=

− −∑

( ) ( )2,

/ , 2 2

( )

( )2

N

CSLDV k r r ex r kkr

r r dp exr r r

y t C q t

Uq IDFTi

φ

ωφω ω ωζ ω

=

⎛ ⎞= ⎜ ⎟− −⎝ ⎠

∑

Mode Scaling: Results

Lr = effectiveness of the drive point.MSFs are within 40% of the analytical MSFswhenever the drive point mode shape is reasonably large.Decreasing trend: ?? Due to approximating H(z)as H(ω)??

Freq.(Hz) MSF Lr MSF Lr MSF Lr

1.7 1.16 0.9 1.17 0.9 1.11 0.92.7 1.09 0.2 1.18 0.8 0.95 0.3

17.5 1.10 0.9 1.12 0.5 1.12 0.747.6 0.88 0.4 1.12 0.1 0.92 0.693.1 0.91 0.5 0.88 0.3 3.62 0.0

153.7 0.87 0.6 0.83 0.5 0.88 0.7230.4 1.13 0.3 0.84 0.8 0.72 0.6322.1 1.00 0.8 1.01 0.8 1.77 0.1428.1 0.42 0.1 0.65 0.7 0.62 0.7

DP-1, 55.9 cm DP-2, 85.7 cm DP-3, 62.9 cm MIMOMSF1.161.161.110.940.910.890.811.030.62

{ }( ),

maxr dp

rr

Lφ

φ=

Result of Modal Parameter Identification

100 Hz scan speed.Tails of each mode’s FRF alias into the frequency band.Responses should be fit to an H(z)transfer function, which accounts for this aliasing.

0 10 20 30 40 50

100

101

Frequency (Hz)

Composite of Residual After Mode Isolation & Refinement

DataFitData-Fit

ConclusionsUsing the proposed technique, one can acquire the mode shapes of a structure at hundreds of points from a few transient responses

CSLDV hundreds of times faster than the conventional, point by point scanning method for the system studied.Identified mode shapes agree very well with analytical shapes for an Euler-Bernoulli beam.Impact excitation used => easy setupMDTS processing method for time-periodic systems simplifies modal parameter extraction.

Scaling procedure gives modal scale factors accurate to within 40%. Investigating possible reasons for the remaining discrepancy.

References:Y. K. Ho and D. J. Ewins, "Numerical Evaluation of the Damage Index," Structural Health Monitoring 2000, pp. 995–1011, 1999.V. K. Sharma, S. Hanagud, and M. Ruzzene, "Damage index estimation in beams and plates using laser vibrometry," AIAA Journal, vol. 44, pp. 919-23, 2006.M. Allen and J. H. Ginsberg, "Floquet Modal Analysis to Detect Cracks in a Rotating Shaft on Anisotropic Supports," in 24th International Modal Analysis Conference (IMAC XXIV), St. Louis, MO, 2006.Sriram, P., J. I. Craig, et al. (1990). "Scanning laser Doppler vibrometer for modal testing." International Journal of Analytical and Experimental Modal Analysis 5(3): 155-167.Martarelli, M. (2001). Exploiting the Laser Scanning Facility for Vibration Measurements. Imperial College of Science, Technology & Medicine. London, Imperial College. Ph.D.Stanbridge, A. B., M. Martarelli, et al. (1999). Scanning laser Doppler vibrometer applied to impact modal testing. 17th International Modal Analysis Conference -IMAC XVII, Kissimmee, FL, USA, SEM, Bethel, CT, USA.S. Vanlanduit, P. Guillaume, and J. Schoukens, "Broadband vibration measurements using a continuously scanning laser vibrometer," Measurement Science & Technology, vol. 13, pp. 1574-82, 2002.