Abstract

The characteristic fringes of hologram interferometry can be analyzed by the use of density functions, i.e., functions that specify what fraction of the exposure time an object point spends in any one place. In many cases, this method of analysis avoids cumbersome series solutions for fringe functions; in this paper the method is applied to nonlinear vibrations and to combinations of sinusoidal oscillations whose frequencies are related by rational numbers. For nonlinear vibrations, results are obtained that allow direct calculation of nonlinear spring functions, without presumption of a solution to the nonlinear differential equation of motion.

© 1971 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. L. Powell, in The Engineering Uses of Holography, edited by E. R. Robertson and J. M. Harvey (Cambridge U. P., Cambridge, 1970), pp. 333–340.
  2. A. D. Wilson, J. Opt. Soc. Am. 60, 1068 (1970).
    [CrossRef]
  3. A. D. Wilson and D. H. Strope, J. Opt. Soc. Am. 60, 1162 (1970).
    [CrossRef]
  4. K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).
    [CrossRef]
  5. R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [CrossRef]
  6. A. Kozma, private communication.
  7. K. A. Stetson, Optik 29, 386 (1969).
  8. H. Cramer, The Elements of Probability Theory (Wiley, New York, 1958), Ch. 5, pp. 57–86.
  9. E. M. Hofstetter, in Mathematics of Physics and Chemistry Vol. II, edited by H. Margenau and G. M. Murphy (Van Nostrand, Princeton, New Jersey, 1964), Ch. 3.
  10. K. A. Stetson, Optik 31, 576 (1970).
  11. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill, New York, 1958), Ch. 4, Sec. 3, p. 67.
  12. J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley–Interscience, New York and London, 1950), Ch. 2, p. 18.
  13. Reference 12, Ch. 4, pp. 83–90.
  14. Reference 8, Ch. 9, p. 136.
  15. Reference 8, Ch. 3, p. 37, and Ref. 11, p. 43.

1970 (4)

1969 (1)

K. A. Stetson, Optik 29, 386 (1969).

1965 (1)

Cramer, H.

H. Cramer, The Elements of Probability Theory (Wiley, New York, 1958), Ch. 5, pp. 57–86.

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill, New York, 1958), Ch. 4, Sec. 3, p. 67.

Hofstetter, E. M.

E. M. Hofstetter, in Mathematics of Physics and Chemistry Vol. II, edited by H. Margenau and G. M. Murphy (Van Nostrand, Princeton, New Jersey, 1964), Ch. 3.

Kozma, A.

A. Kozma, private communication.

Powell, R. L.

R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
[CrossRef]

R. L. Powell, in The Engineering Uses of Holography, edited by E. R. Robertson and J. M. Harvey (Cambridge U. P., Cambridge, 1970), pp. 333–340.

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill, New York, 1958), Ch. 4, Sec. 3, p. 67.

Stetson, K. A.

Stoker, J. J.

J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley–Interscience, New York and London, 1950), Ch. 2, p. 18.

Strope, D. H.

Wilson, A. D.

J. Opt. Soc. Am. (4)

Optik (2)

K. A. Stetson, Optik 31, 576 (1970).

K. A. Stetson, Optik 29, 386 (1969).

Other (9)

H. Cramer, The Elements of Probability Theory (Wiley, New York, 1958), Ch. 5, pp. 57–86.

E. M. Hofstetter, in Mathematics of Physics and Chemistry Vol. II, edited by H. Margenau and G. M. Murphy (Van Nostrand, Princeton, New Jersey, 1964), Ch. 3.

A. Kozma, private communication.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill, New York, 1958), Ch. 4, Sec. 3, p. 67.

J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley–Interscience, New York and London, 1950), Ch. 2, p. 18.

Reference 12, Ch. 4, pp. 83–90.

Reference 8, Ch. 9, p. 136.

Reference 8, Ch. 3, p. 37, and Ref. 11, p. 43.

R. L. Powell, in The Engineering Uses of Holography, edited by E. R. Robertson and J. M. Harvey (Cambridge U. P., Cambridge, 1970), pp. 333–340.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (1)

Tables Icon

Table I Functions for calculating joint characteristic fringe functions. This table lists the functions sn to be substituted into Eq. (19) to evaluate the joint characteristic fringe functions for four examples of rational vibration modes. The cases tabulated are for the frequency ratios 2:1, 3:1, 4:1, and 3:2.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

M ( Ω ) = ( 1 / T ) 0 T exp [ i Ω f ( t ) ] d t ,
M ( Ω ) = - + p ( f ) exp ( i Ω f ) d f = F Ω [ p ( f ) ] ,
p ( f ) = lim Δ t 0 ( 2 Δ t B τ Δ f ) = 2 B τ d t d f ,
- + p ( f ) d f = 1 ,
B = - 1 + 1 2 τ d t d f d f = - τ / 2 + τ / 2 ( 2 / τ ) d t = 2 ,
p ( f ) = 1 / [ τ ( d f / d t ) ] .
[ d 2 f ( t ) ] / d t 2 + C g ( f ) = 0.
[ d f ( t ) ] / d t = ( 2 C G 0 ) 1 2 [ 1 - G ( f ) / G 0 ] 1 2 ,
p ( f ) = 1 / { τ ( 2 C G 0 ) 1 2 [ 1 - G ( f ) / G 0 ] 1 2 } .
τ = d f / v = 2 / ( 2 C G 0 ) 1 2 - + d f / [ 1 - G ( f ) / G 0 ] 1 2 .
M ( Ω ) = ( 1 / C 0 ) F Ω { 1 / [ 1 - G ( f ) / G 0 ] 1 2 } ,
g ( f ) = G 0 d d f ( 1 - { C 0 2 π F - f [ M ( Ω ) ] } - 2 ) ,
g ( f ) = ( 2 π ) 2 τ 2 C d d f { F - f [ M ( Ω ) ] } - 2 .
M ( Ω 1 , Ω 2 ) = 1 T 0 T exp [ i Ω 1 f 1 ( t ) ] exp [ i Ω 2 f 2 ( t ) ] d t .
M ( Ω 1 , Ω 2 ) = - + - + p ( f 1 , f 2 ) × exp ( i Ω 1 f 1 ) exp ( i Ω 2 f 2 ) d f 1 d f 2 ,
p ( f 1 , f 2 ) = p ( f 1 ) p ( f 2 f 1 ) = p ( f 2 ) p ( f 1 f 2 ) ,
p ( f 1 ) = [ 2 π ( 1 - f 1 2 ) - 1 2 ] .
p ( f 2 f 1 ) = ( 1 / 2 N ) 1 ± N δ [ f 2 - s n ( f 1 ) ] ,
N / M = ω 1 / ω 2 ,             where             N < M ,
p ( f 1 , f 2 ) = 1 4 N π ( 1 - f 1 2 ) 1 2 1 ± N δ [ f 2 - s n ( f 1 ) ] .
M ( Ω 1 , Ω 2 ) = 1 4 π N - + - + 1 ± N δ [ f 2 - s n ( f 1 ) ] ( 1 - f 1 2 ) 1 2 × exp ( i Ω 1 f 1 ) exp ( i Ω 2 f 2 ) d f 1 d f 2 .
M ( Ω 1 Ω 2 ) = 1 4 N π 1 ± N - + exp [ i Ω 2 s n ( f 1 ) ] ( 1 - f 1 2 ) 1 2 × exp ( i Ω 1 f 1 ) d f 1 .
p ( f 1 , f 2 , f 3 ) = p ( f 1 ) p ( f 2 f 1 ) p ( f 3 f 2 , f 1 ) .