Abstract

In a previous paper [ R. Tonin and D. A. Bies, J. Sound Vib. 52 (3), 315 (1977)] the theory of time-averaged holographic interferometry was extended to include simple harmonic motion in three orthogonal directions at a single frequency. The amended characteristic function formula was used to calculate the radial and tangential components of a vibrating cylinder by first determining the radial component and from this the tangential component of vibration. In this paper the analysis of the previous paper is improved by making use of a technique originally introduced for the investigation of static deflection using time-averaged holography [ S. K. Dhir and J. P. Sikora, Exp. Mech. 12 (7), 323 (1972)]. The improved procedure allows simultaneous determination of all vibration amplitude components. The procedure is used for the investigation of the low order resonant vibration modes of four cylinders of various sizes and materials with shear-diaphragm end conditions with good results. The procedure is quite general in its application and not restricted to the study of cylinders. It lends itself easily to the study of coupled-mode vibration problems and in fact many complex resonance phenomena.

© 1978 Optical Society of America

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References

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  1. G. M. Brown, M. Grant, G. W. Stroke, J. Acoust. Soc. Am. 45, 1166 (1969).
    [CrossRef]
  2. P. C. Gupta, K. Singh, Indian J. Pure Appl. Phys. 14, 367 (1976).
  3. K. A. Stetson, J. Opt. Soc. Am. 62, 297 (1972).
    [CrossRef]
  4. P. C. Gupta, K. Singh, Appl. Opt. 14, 129 (1975).
    [PubMed]
  5. C. S. Vikram, R. S. Sirohi, Phys. Lett. A 35, 460 (1971).
    [CrossRef]
  6. C. S. Vikram, Opt. Commun. 8, 355 (1973).
    [CrossRef]
  7. C. S. Vikram, Phys. Lett. A 45, 426 (1973).
    [CrossRef]
  8. C. S. Vikram, Opt. Commun. 10, 290 (1974).
    [CrossRef]
  9. C. S. Vikram, Gouranga Bose, Optik 43, 253 (1975).
  10. A. D. Wilson, D. H. Strope, J. Opt. Soc. Am. 60, 1162 (1970).
    [CrossRef]
  11. A. D. Wilson, J. Opt. Soc. Am. 61, 924 (1971).
    [CrossRef]
  12. C. S. Vikram, Optik 43, 65 (1975).
  13. C. S. Vikram, Opt. Commun. 11, 360 (1974).
    [CrossRef]
  14. N. E. Molin, K. A. Stetson, J. Phys. E. 2, 609 (1969).
    [CrossRef]
  15. M. Zambuto, M. Lurie, Appl. Opt. 9, 2066 (1970).
    [CrossRef] [PubMed]
  16. W. J. Dallas, A. W. Lohmann, Opt. Commun. 13, 134 (1975).
    [CrossRef]
  17. K. A. Stetson, J. Opt. Soc. Am. 61, 1359 (1971).
    [CrossRef]
  18. C. S. Vikram, Optik 45, 55 (1976).
  19. R. Tonin, D. A. Bies, J. Sound Vib. 52 (3), 315 (1977).
    [CrossRef]
  20. S. K. Dhir, J. P. Sikora, Exp. Mech. 12(7), 323 (1972).
    [CrossRef]
  21. P. A. Tuschak, R. A. Allaire, Exp. Mech. 15, 81 (March1975).
    [CrossRef]
  22. R. A. Buckingham, Numerical Methods (Pitman Press, New York, 1962), p. 301.
  23. D. P. Bartlett, The Method of Least Squares (Rumford Press, Concord, N.H., 1915), p. 63.
  24. A. W. Leissa, “Vibration of Shells,” NASA-SP288 (U.S. Govt. Printing Office, Washington, D.C., 1973), Chap. 2.
  25. R. K. Erf, Ed., Holographic Nondestructive Testing (Academic, New York, 1974), p. 202.
  26. This method was suggested to one of the authors (D.A.B.) by I. Rudnick at U. California at Los Angeles.
  27. R. Pryputniewicz, K. A. Stetson, Appl. Opt. 15(3), 725 (1976).
    [CrossRef] [PubMed]

1977

R. Tonin, D. A. Bies, J. Sound Vib. 52 (3), 315 (1977).
[CrossRef]

1976

P. C. Gupta, K. Singh, Indian J. Pure Appl. Phys. 14, 367 (1976).

C. S. Vikram, Optik 45, 55 (1976).

R. Pryputniewicz, K. A. Stetson, Appl. Opt. 15(3), 725 (1976).
[CrossRef] [PubMed]

1975

P. C. Gupta, K. Singh, Appl. Opt. 14, 129 (1975).
[PubMed]

P. A. Tuschak, R. A. Allaire, Exp. Mech. 15, 81 (March1975).
[CrossRef]

W. J. Dallas, A. W. Lohmann, Opt. Commun. 13, 134 (1975).
[CrossRef]

C. S. Vikram, Gouranga Bose, Optik 43, 253 (1975).

C. S. Vikram, Optik 43, 65 (1975).

1974

C. S. Vikram, Opt. Commun. 11, 360 (1974).
[CrossRef]

C. S. Vikram, Opt. Commun. 10, 290 (1974).
[CrossRef]

1973

C. S. Vikram, Opt. Commun. 8, 355 (1973).
[CrossRef]

C. S. Vikram, Phys. Lett. A 45, 426 (1973).
[CrossRef]

1972

S. K. Dhir, J. P. Sikora, Exp. Mech. 12(7), 323 (1972).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Am. 62, 297 (1972).
[CrossRef]

1971

1970

1969

N. E. Molin, K. A. Stetson, J. Phys. E. 2, 609 (1969).
[CrossRef]

G. M. Brown, M. Grant, G. W. Stroke, J. Acoust. Soc. Am. 45, 1166 (1969).
[CrossRef]

Allaire, R. A.

P. A. Tuschak, R. A. Allaire, Exp. Mech. 15, 81 (March1975).
[CrossRef]

Bartlett, D. P.

D. P. Bartlett, The Method of Least Squares (Rumford Press, Concord, N.H., 1915), p. 63.

Bies, D. A.

R. Tonin, D. A. Bies, J. Sound Vib. 52 (3), 315 (1977).
[CrossRef]

Bose, Gouranga

C. S. Vikram, Gouranga Bose, Optik 43, 253 (1975).

Brown, G. M.

G. M. Brown, M. Grant, G. W. Stroke, J. Acoust. Soc. Am. 45, 1166 (1969).
[CrossRef]

Buckingham, R. A.

R. A. Buckingham, Numerical Methods (Pitman Press, New York, 1962), p. 301.

Dallas, W. J.

W. J. Dallas, A. W. Lohmann, Opt. Commun. 13, 134 (1975).
[CrossRef]

Dhir, S. K.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12(7), 323 (1972).
[CrossRef]

Grant, M.

G. M. Brown, M. Grant, G. W. Stroke, J. Acoust. Soc. Am. 45, 1166 (1969).
[CrossRef]

Gupta, P. C.

P. C. Gupta, K. Singh, Indian J. Pure Appl. Phys. 14, 367 (1976).

P. C. Gupta, K. Singh, Appl. Opt. 14, 129 (1975).
[PubMed]

Leissa, A. W.

A. W. Leissa, “Vibration of Shells,” NASA-SP288 (U.S. Govt. Printing Office, Washington, D.C., 1973), Chap. 2.

Lohmann, A. W.

W. J. Dallas, A. W. Lohmann, Opt. Commun. 13, 134 (1975).
[CrossRef]

Lurie, M.

Molin, N. E.

N. E. Molin, K. A. Stetson, J. Phys. E. 2, 609 (1969).
[CrossRef]

Pryputniewicz, R.

Sikora, J. P.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12(7), 323 (1972).
[CrossRef]

Singh, K.

P. C. Gupta, K. Singh, Indian J. Pure Appl. Phys. 14, 367 (1976).

P. C. Gupta, K. Singh, Appl. Opt. 14, 129 (1975).
[PubMed]

Sirohi, R. S.

C. S. Vikram, R. S. Sirohi, Phys. Lett. A 35, 460 (1971).
[CrossRef]

Stetson, K. A.

Stroke, G. W.

G. M. Brown, M. Grant, G. W. Stroke, J. Acoust. Soc. Am. 45, 1166 (1969).
[CrossRef]

Strope, D. H.

Tonin, R.

R. Tonin, D. A. Bies, J. Sound Vib. 52 (3), 315 (1977).
[CrossRef]

Tuschak, P. A.

P. A. Tuschak, R. A. Allaire, Exp. Mech. 15, 81 (March1975).
[CrossRef]

Vikram, C. S.

C. S. Vikram, Optik 45, 55 (1976).

C. S. Vikram, Gouranga Bose, Optik 43, 253 (1975).

C. S. Vikram, Optik 43, 65 (1975).

C. S. Vikram, Opt. Commun. 11, 360 (1974).
[CrossRef]

C. S. Vikram, Opt. Commun. 10, 290 (1974).
[CrossRef]

C. S. Vikram, Opt. Commun. 8, 355 (1973).
[CrossRef]

C. S. Vikram, Phys. Lett. A 45, 426 (1973).
[CrossRef]

C. S. Vikram, R. S. Sirohi, Phys. Lett. A 35, 460 (1971).
[CrossRef]

Wilson, A. D.

Zambuto, M.

Appl. Opt.

Exp. Mech.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12(7), 323 (1972).
[CrossRef]

P. A. Tuschak, R. A. Allaire, Exp. Mech. 15, 81 (March1975).
[CrossRef]

Indian J. Pure Appl. Phys.

P. C. Gupta, K. Singh, Indian J. Pure Appl. Phys. 14, 367 (1976).

J. Acoust. Soc. Am.

G. M. Brown, M. Grant, G. W. Stroke, J. Acoust. Soc. Am. 45, 1166 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. E.

N. E. Molin, K. A. Stetson, J. Phys. E. 2, 609 (1969).
[CrossRef]

J. Sound Vib.

R. Tonin, D. A. Bies, J. Sound Vib. 52 (3), 315 (1977).
[CrossRef]

Opt. Commun.

W. J. Dallas, A. W. Lohmann, Opt. Commun. 13, 134 (1975).
[CrossRef]

C. S. Vikram, Opt. Commun. 11, 360 (1974).
[CrossRef]

C. S. Vikram, Opt. Commun. 8, 355 (1973).
[CrossRef]

C. S. Vikram, Opt. Commun. 10, 290 (1974).
[CrossRef]

Optik

C. S. Vikram, Gouranga Bose, Optik 43, 253 (1975).

C. S. Vikram, Optik 43, 65 (1975).

C. S. Vikram, Optik 45, 55 (1976).

Phys. Lett. A

C. S. Vikram, R. S. Sirohi, Phys. Lett. A 35, 460 (1971).
[CrossRef]

C. S. Vikram, Phys. Lett. A 45, 426 (1973).
[CrossRef]

Other

R. A. Buckingham, Numerical Methods (Pitman Press, New York, 1962), p. 301.

D. P. Bartlett, The Method of Least Squares (Rumford Press, Concord, N.H., 1915), p. 63.

A. W. Leissa, “Vibration of Shells,” NASA-SP288 (U.S. Govt. Printing Office, Washington, D.C., 1973), Chap. 2.

R. K. Erf, Ed., Holographic Nondestructive Testing (Academic, New York, 1974), p. 202.

This method was suggested to one of the authors (D.A.B.) by I. Rudnick at U. California at Los Angeles.

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Figures (10)

Fig. 1
Fig. 1

Orthogonal components of vibration at a single frequency. The sensitivity vector which has been omitted for clarity would lie midway between the illumination and observation vectors.

Fig. 2
Fig. 2

Time-averaged hologram reconstructions of a small steel cylinder vibrating in the M = 4, N = 3 mode. The cylinder was rotated 15° between holograms. The line indicates the cross section at which data points were taken (line of varying ξ).

Fig. 3
Fig. 3

Details of end mount and support.

Fig. 4
Fig. 4

Optical system: (L) laser; (M) mirror; (B.S.) beam splitter; (T) turntable; (S.F.) spatial filter; (H) holographic plate; (CYL) cylinder; (E.M.) electromagnet; (C) camera.

Fig. 5
Fig. 5

Modal driving system.

Fig. 6
Fig. 6

Geometry for interpretation of hologram photographs; cylinder vertical.

Fig. 7
Fig. 7

Geometry for interpretation of hologram photographs; cylinder horizontal.

Fig. 8
Fig. 8

Least squares procedure determination of radial and tangential components of a circular cylinder: squares, experimentally determined radial component; circles, experimentally determined tangential component; — least squares curves, (a) Circumferential order N = 2 (small steel cylinder); (b) circumferential order N = 3 (small steel cylinder); (c) circumferential order N = 4 (small steel cylinder).

Fig. 9
Fig. 9

Least squares procedure determination of radial and longitudinal components of a circular cylinder: squares, experimentally determined radial component; circles, experimentally determined longitudinal component; — least squares curves, (a) Longitudinal order M = 1 (aluminum cylinder); (b) longitudinal order M = 2 (aluminum cylinder); (c) longitudinal order M = 3 (copper cylinder).

Fig. 10
Fig. 10

Least squares procedure determination of radial and tangential components of a circular cylinder vibrating in coupled modes (copper cylinder): squares, experimentally determined radial component; circles, experimentally determined tangential component; — least squares curves, (a) Circumferential order N = 4 single-mode analysis; (b) multimodal analysis technique N1 = 2 and N2 = 4.

Tables (5)

Tables Icon

Table I Physical Properties of Test Cylinders

Tables Icon

Table II Theoretical and Experimental Results (Experimental Results in Parenthesis); Small Steel Cylinder

Tables Icon

Table III Theoretical and Experimental Results (Experimental Results in Parenthesis); Large Steel Cylinder

Tables Icon

Table IV Theoretical and Experimental Results (Experimental Results in Parenthesis): Aluminum Cylinder

Tables Icon

Table V Theoretical and Experimental Results (Experimental Results in Parenthesis); Copper Cylinder

Equations (22)

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Ω k = a ( cos θ 1 + cos θ 2 ) + b ( sin θ 1 sin θ 2 ) ,
a ( ξ ) = n = N 1 N 2 [ x n sin ( n ξ ) + y n cos ( n ξ ) ] , b ( ξ ) = n = N 1 N 2 [ w n sin ( n ξ ) + z n cos ( n ξ ) ] .
a 1 C 1 + b 1 S 1 = R 1 , a 2 C 2 + b 2 S 2 = R 2 , · · · · a t C t + b t S t = R t ,
C i = cos θ 1 i + cos θ 2 i , S i = sin θ 1 i sin θ 2 i , R i = Ω i / k .
n = N 1 N 2 x n sin ( n ξ i ) C i + n = N 1 N 2 y n cos ( n ξ i ) C i + n = N 1 N 2 w n sin ( n ξ i ) S i + n = N 1 N 2 z n cos ( n ξ i ) S i = R i ,
[ n = N 1 N 2 x n sin ( n ξ i ) C i + n = N 1 N 2 y n cos ( n ξ i ) C i + n = N 1 N 2 w n sin ( n ξ i ) S i + n = N 1 N 2 z n cos ( n ξ i ) S i R i ] 2 = ( R i R i ) 2 .
F ( x N 1 , , x N 2 , y N 1 , , y N 2 , w N 1 , , w N 2 , z N 1 , , z N 2 ) = i = 1 t ( R i R i ) 2 ,
A [ i , 4 ( n N 1 + 1 ) 3 ] = C i sin ( n ξ i ) , A [ i , 4 ( n N 1 + 1 ) 2 ] = C i cos ( n ξ i ) , A [ i , 4 ( n N 1 + 1 ) 1 ] = S i sin ( n ξ i ) , A [ i , 4 ( n N 1 + 1 ) ] = S i cos ( n ξ i ) ,
x N 1 y N 1 w N 1 z N 1 R 1 R 2 · X = · R = · · · · · x N 2 R t y N 2 w N 2 z N 2
A X = R .
A T A X = A T R ,
σ = | ( R R ) 2 p [ t 4 ( N 2 N 1 + 1 ) ] | 1 / 2 ,
U = A cos ( λ s ) cos ( N θ ) cos ( ω τ ) , V = B sin ( λ s ) sin ( N θ ) cos ( ω τ ) , W = C sin ( λ s ) cos ( N θ ) cos ( ω τ ) ,
γ = tan 1 ( H · R / A A · R υ ) ,
θ = cos 1 ( A A sin γ / R ) + γ ,
θ 2 = π / 2 θ + γ ,
θ 1 = tan 1 { P cos ( α + θ ) / [ R P sin ( α + θ ) ] } ,
γ = tan 1 [ H · L / ( A A R ) · L υ ] ,
ϕ = tan 1 | R sin ( α + γ ) A A sin γ cos ( S α ) A A sin γ sin ( S α ) R cos ( α + γ ) | ,
l = R · tan ( α + ϕ S ) ,
θ 2 = γ + S ,
θ 1 = tan 1 | l P · sin ( α S ) R P · cos ( α S ) | .

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