Abstract

Ladar-based vibrometry has been shown to be a powerful technique in enabling the plant identification of machines. Rather than sensing the geometric shape of a target laser vibrometers sense motions of the target induced by moving parts within the system. Since the target need not be spatially resolved, vibration can be sensed reliably and provide positive identification at ranges beyond the imaging limits of the aperture. However, as the range of observation increases, the diffraction-limited beam size on the target increases as well, and may encompass multiple vibrational modes on the target's surface. As a result, vibration estimates formed from large laser footprints illuminating multiple modes on a vibrating target will experience a degradation. This degradation is manifest as a spatial low-pass filtering effect: high-order mode shapes, associated with high-frequency vibrations, will be averaged out while low-frequency vibrations will be affected less. A model to predict this phenomenology is proposed for both pulse-pair and cw vibrometry systems. The cw model is compared to results obtained using an off-the-shelf laser vibrometry system.

© 2007 Optical Society of America

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References

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  1. R. R. Ebert and P. Lutzmann, "Vibration imagery of remote objects," Proc. SPIE 4821, 1-10 (2002).
    [CrossRef]
  2. N. Pepela, "Effect of multi-mode vibration on signature estimation using a laser vibration sensor," Master's thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, 2003).
  3. S. W. Henderson and S. M. Hannon, "Advanced coherent lidar system for wind measurements," Proc. SPIE 5887, 58870I (2005).
  4. M. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).
  5. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
  6. T. A. Sturm, R. Richmond, and B. D. Duncan, "A technique for removing platform vibration noise from a pulsed ladar vibration sensor," Opt. Laser Technol. 27, 343-350 (1995).
    [CrossRef]
  7. E. Kreyszig, Advanced Engineering Mathematics, 7th ed. (Wiley, 1993).
  8. Polytec, "Single point vibrometers," http://www.polytec.com/usa/158_421.asp.
  9. F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, 1992).
  10. MathWorks, "Signal processing and communications," http://www.mathworks.com/applications/dsp_comm/.
  11. K. Leentvaar and J. H. Flint, "The capture effect in FM receivers," IEEE Trans. Commun. Comm- 24, 531-539 (1976).
    [CrossRef]
  12. Reflexite, "Products--photoelectric controls," http://www.reflexiteamericas.com/photoelectric.htm.
  13. D. Chambers, "Modeling heterodyne efficiency for coherent laser radar in the presence of aberrations," Opt. Express 1, 60-67 (1997).

2005 (1)

S. W. Henderson and S. M. Hannon, "Advanced coherent lidar system for wind measurements," Proc. SPIE 5887, 58870I (2005).

2002 (1)

R. R. Ebert and P. Lutzmann, "Vibration imagery of remote objects," Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

1997 (1)

1995 (1)

T. A. Sturm, R. Richmond, and B. D. Duncan, "A technique for removing platform vibration noise from a pulsed ladar vibration sensor," Opt. Laser Technol. 27, 343-350 (1995).
[CrossRef]

1976 (1)

K. Leentvaar and J. H. Flint, "The capture effect in FM receivers," IEEE Trans. Commun. Comm- 24, 531-539 (1976).
[CrossRef]

Chambers, D.

Duncan, B. D.

T. A. Sturm, R. Richmond, and B. D. Duncan, "A technique for removing platform vibration noise from a pulsed ladar vibration sensor," Opt. Laser Technol. 27, 343-350 (1995).
[CrossRef]

Ebert, R. R.

R. R. Ebert and P. Lutzmann, "Vibration imagery of remote objects," Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

Flint, J. H.

K. Leentvaar and J. H. Flint, "The capture effect in FM receivers," IEEE Trans. Commun. Comm- 24, 531-539 (1976).
[CrossRef]

Hannon, S. M.

S. W. Henderson and S. M. Hannon, "Advanced coherent lidar system for wind measurements," Proc. SPIE 5887, 58870I (2005).

Henderson, S. W.

S. W. Henderson and S. M. Hannon, "Advanced coherent lidar system for wind measurements," Proc. SPIE 5887, 58870I (2005).

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics, 7th ed. (Wiley, 1993).

Leentvaar, K.

K. Leentvaar and J. H. Flint, "The capture effect in FM receivers," IEEE Trans. Commun. Comm- 24, 531-539 (1976).
[CrossRef]

Lutzmann, P.

R. R. Ebert and P. Lutzmann, "Vibration imagery of remote objects," Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

Pepela, N.

N. Pepela, "Effect of multi-mode vibration on signature estimation using a laser vibration sensor," Master's thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, 2003).

Richards, M.

M. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).

Richmond, R.

T. A. Sturm, R. Richmond, and B. D. Duncan, "A technique for removing platform vibration noise from a pulsed ladar vibration sensor," Opt. Laser Technol. 27, 343-350 (1995).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

Stremler, F. G.

F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, 1992).

Sturm, T. A.

T. A. Sturm, R. Richmond, and B. D. Duncan, "A technique for removing platform vibration noise from a pulsed ladar vibration sensor," Opt. Laser Technol. 27, 343-350 (1995).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

IEEE Trans. Commun. Comm- (1)

K. Leentvaar and J. H. Flint, "The capture effect in FM receivers," IEEE Trans. Commun. Comm- 24, 531-539 (1976).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

T. A. Sturm, R. Richmond, and B. D. Duncan, "A technique for removing platform vibration noise from a pulsed ladar vibration sensor," Opt. Laser Technol. 27, 343-350 (1995).
[CrossRef]

Proc. SPIE (2)

R. R. Ebert and P. Lutzmann, "Vibration imagery of remote objects," Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

S. W. Henderson and S. M. Hannon, "Advanced coherent lidar system for wind measurements," Proc. SPIE 5887, 58870I (2005).

Other (8)

M. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).

N. Pepela, "Effect of multi-mode vibration on signature estimation using a laser vibration sensor," Master's thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, 2003).

E. Kreyszig, Advanced Engineering Mathematics, 7th ed. (Wiley, 1993).

Polytec, "Single point vibrometers," http://www.polytec.com/usa/158_421.asp.

F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, 1992).

MathWorks, "Signal processing and communications," http://www.mathworks.com/applications/dsp_comm/.

Reflexite, "Products--photoelectric controls," http://www.reflexiteamericas.com/photoelectric.htm.

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Figures (11)

Fig. 1
Fig. 1

Block diagram of a pulse-pair laser vibrometer.

Fig. 2
Fig. 2

Pulse pairs with doublet separation 1 / f PRF , pulse separation Δ T s , and width τ.

Fig. 3
Fig. 3

Normalized frequency response of a pulse-pair vibrometer as a function of normalized aperture radius, shown for three circularly symmetric surface vibration modes.

Fig. 4
Fig. 4

Heterodyne cw vibrometer system capable of measuring both positive and negative velocities.

Fig. 5
Fig. 5

Normalized frequency response of a cw vibrometer as a function of normalized aperture radius, shown for three circularly symmetric surface modes.

Fig. 6
Fig. 6

Cw vibrometer target mount assembly.

Fig. 7
Fig. 7

Cw vibrometer experimental setup used for measuring the spatially averaged frequency response due to a truncated Gaussian target illumination beam.

Fig. 8
Fig. 8

This grid demonstrates the actual locations used in sampling the rms velocity across the surface of the target. Positioning errors occured due to the accuracy and precision of the motorized pan-and-tilt platform.

Fig. 9
Fig. 9

Interpolated rms velocity profile of the lowest-order mode present on the target. This mode was present at a vibrational frequency of 375   Hz .

Fig. 10
Fig. 10

Plot of Eq. (16). Peaks occur when surface displacement is zero and all scatterers constructively induce a peak Doppler frequency shift. When surface displacement is maximum (and velocity approaches zero) destructive interference and surface roughness cause a sharp reduction in signal strength.

Fig. 11
Fig. 11

Experimental results for the normalized frequency response as a function of normalized aperture radius. The solid curve represents the noisy cw model after incorporating the experimental 01 mode shape. The asterisks represent the average of ten experimental runs at each aperture radius value, while the error bars indicate one standard deviation in the experimental data.

Equations (22)

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M ( t , p , n ) = Re { exp ( j ω IF t ) M ˜ } ,
M ˜ = C   exp ( t 2 τ 2 ) exp ( j ω IF ( p f PRF + n Δ T s ) ) .
S ( t , p , n , Z 0 ) = C   exp ( t 2 τ 2 ) Re { exp ( j ω IF ( t p f PRF n Δ T s 2 Z 0 c ) ) A exp ( x 2 + y 2 σ 2 ) × exp ( j ϕ target ( x , y ) ) d A } ,
exp ( j ϕ target ( x , y ) ) = exp ( j 2 π 2 Δ T s λ V ( x , y ) × cos ( ω T ( t p f PRF n Δ T s Z 0 c ) ) ) ,
V amb = λ 2 Δ T s .
S ( t , p , n , Z 0 ) = Re { exp ( j ω IF t ) S ˜ } ,
S ˜ = C   exp ( t 2 τ 2 ) exp ( j ω IF ( p f PRF + n Δ T s + 2 Z 0 c ) ) × A exp ( x 2 + y 2 σ 2 ) exp ( j 2 π 2 Δ T s λ V ( x , y ) × cos ( ω T ( t p f PRF n Δ T s Z 0 c ) ) ) d A .
Δ ϕ = angle ( ( M ˜ 0 M ˜ 1 * ) ( S ˜ 0 S ˜ 1 * ) * d t ) ,
M ˜ 1 M ˜ 2 * = | C | 2   exp ( 2 t 2 τ 2 ) exp ( j ω IF Δ T s ) .
S ˜ 1 S ˜ 2 * = | C | 2   exp ( 2 t 2 τ 2 ) exp ( j ω IF Δ T s ) × A exp ( x 2 + y 2 σ 2 ) exp ( j 2 2 π λ Δ T s V ( x , y ) × cos ( ω T ( t p f PRF Z 0 c ) ) ) d A A exp ( x 2 + y 2 σ 2 ) × exp ( j 2 2 π λ Δ T s V ( x , y ) × cos ( ω T ( t p f PRF Δ T s Z 0 c ) ) ) d A .
Δ ϕ = angle [ A exp ( x 2 + y 2 σ 2 ) exp ( j 2 2 π λ Δ T s V ( x , y ) × cos ( ω T ( p f PRF + Z 0 c ) ) ) d A A exp ( x 2 + y 2 σ 2 ) × exp ( j 2 2 π λ Δ T s V ( x , y ) × cos ( ω T ( p f PRF + Δ T s + Z 0 c ) ) ) d A ] ,
Δ ϕ Δ T s = 2 π f Di = 2 π λ 2 Δ z eff Δ T s = 4 π λ V i ,
V ( x , y ) = V 0 J 0 ( Z n r r 0 ) , r r 0 ,
Δ ϕ = 2 4 π λ Δ T s   sin ( ω T Δ T s 2 ) V 0 × cos ( ω T ( p f PRF + Δ T s 2 + Z 0 c ) + π 2 ) .
Δ ϕ Δ T s = 4 π λ V i [ 2   sin ( ω T Δ T s 2 ) ] ,
V i = V 0   cos ( ω T ( p f PRF + Δ T s 2 + Z 0 c ) + π 2 ) .
H e q ( ω ) = 1 sin ( ω Δ T s 2 ) , [ 0 ω π / Δ T s ] .
U R ( x , y , t ) = E 0 ( x , y ) exp ( j ( ω 0 + ω AOM ) t ) ,
U O ( x , y , t ) = E 0 ( x , y ) exp ( j ω 0 t j 4 π λ ω T V ( x , y ) cos ( ω T t ) ) ,
I a c ( t ) = 2 A | E 0 ( x , y ) | 2 × cos ( ω AOM t + 4 π λ ω T V ( x , y ) cos ( ω T t ) ) d A = 2 A exp ( x 2 + y 2 σ 2 ) × cos ( ω AOM t + 4 π λ ω T V ( x , y ) cos ( ω T t ) ) d A ,
V est ( t ) = λ f D ( t ) 2 ,
I a c ( t ) = 2 A exp ( x 2 + y 2 σ 2 ) cos ( ω AOM t + 4 π λ ω T V ( x , y ) × cos ( ω T t ) + N s ( x , y ) + N t ( t ) ) d A ,

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