Opto-electronics 6 (1974) 305-311

Holographic interferometry of nonsinusoidal vibrations P. C. G U P T A ,

K. S I N G H

Department of Physics, Indian Institute of Technology, Delhi, New Delhi- 110029, India Received 18 January 1974 In this paper, the techniques of extended pulse stroboscopic holography and holographic subtraction have been applied to the study of periodic, non-sinusoidal vibrations represented by a Jacobian elliptic function. Fringe irradiance distribution in reconstructed images has been evaluated for the two cases. For this purpose we make use of an expression for the characteristic fringe function derived from considerations of the effect of motion on the coherence. It is shown that the abovementioned techniques are more advantageous for the measurement of periodic, non-sinusoidal vibrations than in the case of pure sinusoidal vibrations.

1. I n t r o d u c t i o n Time-average hologram interferometry, first introduced by Powell and Stetson [1], provides a very useful and accurate method for studying moving or vibrating objects of different shape and surface finish. However, due to the rapid decrease in intensity of the reconstructed fringes with increasing amplitude of vibration, the direct use of the method is not advisable for measurement of severe vibrations, where a completely dark reconstruction may be observed. Many methods of overcoming this limitation have been suggested in the literature by the holographers engaged in motion and vibration problems. These are the technique of holographic addition or subtraction [2-4], pre- or post-exposure [2, 5], phase modulation of reference and/or object beam [6-8], two and three beam stroboscopic holography [9, 10], extended pulse stroboscopic holography [11, 12] and bleaching [13]. A large number of references can be found in articles by Singh [14, 15]. In the case of pure sinusoidal vibrations most of these methods have been applied to investigate severe vibrations. Each method has its own advantages and disadvantages and has to be judged on its merits for a given application. Pure sinusoidal vibrations can be achieved in practice under laboratory conditions only. So it is important to study the situations in which the structure activity departs from pure sinusoidal form [16]. It is interesting to note the work of Powell [17], Molin and Stetson [18], Wilson [19, 21], Wilson and Strope [20], Stetson [22-24], Stetson and Taylor [25, 26] and Janta and Miller [27] in this connection. A particular case of periodic, non-sinusoidal vibrations (represented by a Jacobian elliptic function) has been briefly outlined by Thornton and Kelly [28] in connection with measurement of vibrations by multiple beam classical interferometric technique. Non-sinusoidal vibrations are quite important in mechanical engineering. They arise for example when a non-linear system is excited by sinusoidal force or when a linear system is excited by a periodic, non-sinusoidal force. Further, it is a well-known fact that even linear systems rarely execute simple harmonic motion. These considerations led us to investigate non-sinusoidal vibrations in detail. In this paper we analyse the techniques of extended pulse stroboscopic holography and holo9 1974 Chapman and Hall Ltd. 305

P. C. Gupta, K. Singh graphic subtraction for the study of a particular case of non-sinusoidal vibration represented by a Jacobian elliptic function. Characteristic fringe functions for the cases have been evaluated and the results for the intensity distribution in the fringes in reconstruction are presented.

2. Theory It has been shown by Zambuto and Lurie [29] that in the case of holographic recording of a vibrating object, by virtue of the Doppler effect the motion of the object slightly changes the frequency of light scattered from the object. This gives rise to the loss of coherence of the object beam compared to the reference beam. It has been shown that if m is a point on the object and m' is the corresponding point in the reconstructed holographic image, the radiance of m' is proportional to the intensity of light from m reaching the photographic plate during recording multiplied by the square of the magnitude of the complex degree of spatial coherence 7,,R(0) between this light and the light received from the reference source. Assuming that vibrations are normal to the plane of object, the complex degree of spatial coherence 7mR(0) can be related to the motion of the object by an expression,

7mR(O) = l l ; exp [i4-~ X(t)]dt ,

(1)

where Tis the exposure time, 2 is the wavelength of light used, and x(t) defines the law of motion of the object. The function given by Equation 1, also known as the characteristic fringe function, can be used to analyse the motion quantitatively. For the case of an object executing pure sinusoidal vibrations the conventional time-average hologram interferometry yields reconstructed images in which the fringe irradiance is governed by the square of the zero-order Bessel function of an argument proportional to the vibration amplitude. Therefore for larger amplitudes of vibrations completely dark reconstruction is obtained. To overcome this limitation of time-average holography many methods [-2-13] have been Suggested. Two of these, namely extended pulse stroboscopic holography and holographic subtraction, have been used by us to analyse periodic, non-sinusoidal vibrations represented by a Jacobian elliptic function.

2.1. Extended pulse stroboscopic holography In the case of conventional stroboscopic holography, the motion of the vibrating object is frozen at a certain displacement. Usually this is done at the maximum displacement position, since at this point the velocity is zero and the largest pulse-width may be used. In this case the reconstruction displays equally spaced fringes of constant intensity, the spacing of which is a measure of the amplitude of vibration. A method for extending the useful amplitude range of normal time-average holography has been discussed by Moffatt and Watrasiewicz [11] and Saito et al. [12]. In their technique, called extended pulse stroboscopic technique, a finite pulse-width is used for exposure, Equation 1 then becomes,

C, = 1 fr/4+~r

[ 4~

]

2~TJr/4-~T exp i--s x(t) dt ,

(2)

where 0 < c~ < 0.25. Thus the intensity in the reconstructed image is given by the expression, r ( x , y) = [o(X, y) c '2 , or

i'

I'(x, y) _ C, 2 -

Io(X, y )

9

(3)

2.2. Holographic subtraction In this method two holograms of the object are recorded in succession on the same plate with the phase of one of the beams shifted by 180~ between the two exposures. The areas of the 306

Holographic interferometry of non-sinusoidal vibrations ,o

08

A ~mo //f y Y

o~ 0.4

\\ \ \ \\ \ \ \\k \

/

\ \ %

,

I

,

I

,o

\

\

\

s c

\

,

0.2

04 o06

\

E o~

I

I

,

2~

0.2

I

6o tt

_.~__~\\\ \

\

.

O.4 0,6 0,8 I.Q

Figure I Jacobian elliptic function sm (u/m) plotted against u

for various values

of m.

object that remain unchanged between the two exposures then disappear and only that portion of the object field which differs in the two exposures is seen in the reconstruction [4]. This technique was applied by Hariharan I-4] for the measurement of small amplitudes of vibration for which the ordinary time-average hologram interferometry is not suitable. Let us assume that the first exposure is made while the object is vibrating and the second exposure is made while the object is stationary in its mean position and the phase of the reference beam shifted by 180 ~. The intensity in the reconstructed image is then given by the expression

I(x, y) = Io(x , y) [1 - C] 2 , or

i -

I(x, y)

Io(x, y) - [1 - C] z .

(4)

where Io(x, y) is the intensity of the reconstructed image when the object is stationary and C, the characteristic fringe function of motion, is given by Equation 1. 2.3. Evaluation of C and C' for periodic, non-sinusoidal vibrations

We consider periodic, non-sinusoidal vibrations represented by an equation of motion

x(t) = Asn(cot l m) ,

(5)

where A is the maximum amplitude of vibration, sn(cot [ m) (Fig. 1) is a one-valued, doubly periodic Jacobian elliptic function [30], co is the angular frequency of vibrations in radians s- 1, and m is a measure of departure from pure sinusoidal form. The numbers K and iK', given by

k(m) = i k'(m,) = i

fao~/z (1 fl/2

m dO sin 2 0) 1/2 I dO

(f - - r n ~

,

(6)

O-~,

are the real and imaginary quarter periods, ml -- 1 - m is known as a complementary parameter. Equation 1 now becomes 307

P. C. Gupta, K. S#~gh c = l f 2 exp [i ~ - sn(ogt l m) ] dt .

(7)

Assuming T to be an integral multiple n of the period of vibration (i.e. T = 4reAl2 = z and mt = u Equation 7 takes the form

C = 4--n-k

4nk/o)) and putting

/2 sin[zsn(ulm)] du .

cos[zsn(ulm)] du + ~-~

(8)

From Fourier-Bessel-Jacobi series,

1 f4nk

-cos[zsn(u] m) du 4nk ao =Jo(z) + 2 2

f 1 [4"k } Jzn'(z) [ 1 - 4n'2" 2! ~ 4 - ~ J o snZ(uim) du

+4n,2(4n,Z_2z)( 1 ~4,t~ } 4I ~nk ao sn4(u/m) du 4n'2(4n'2--22)(en'2--4z){4~fl "k sn6(u I m) du} -6!

+

"

..]

(9)

and 1 f4nk

4nk a o

sin[zsn(u]m)]du = 2

Y2,,_l(z)

(2n' - 1 ) | 4 - ~ j ~

f~ ,

sin3(u ] m) du

sn(ulm) du

n'=l

-

3t

+ [(2n'-1)2-123[(2n'-1)z-32]f 5!

ao

)

1 fl "k } ] ~nk snS(u l m) du . . . . . . . 9

(10)

Evaluation of the integrals [31] occurring in Equations 9 and 10 shows that Equation 10 is equal to zero and thus Equation 8 becomes

+~

~m g

+ T

+ 2J6(z)

1 - ~-

m~

3 2 { ( 8 + 3 m + 4 m Z15m ) F z- ( 2 - 7 m ) E } ] k +

....

,

(11)

where F and E are complete elliptic integrals of the first and second kind, i.e. f =

f? ~/(1 - d0 m sin 2 0) and E = f? ~/(1 -

m sin 2 0) dO ,

(12)

and z = 4hA~2. It should be noted that for m = 0, i.e. when the vibrations are of pure sinusoidal form, Equation 11 reduces to the well-known Jo(4nA/2). An alternative way, suitable for numerical evaluation of Equation 7 is given below. The Jacobian elliptic function sn (u [ m) can be expanded in the form [32] 308

Holographic interferometry of non-sinusoidal vibrations

1

rc

snu

2n

---

-

-

nu + 4 sin ~-~

slq2s, ~, oo 1 -

(I] nu

q Z - ~ - a s i n ( 2 s - 1) ~

(13)

9

q(m) = e x p ( - nk'/k) is called the nome of the function. Substituting this expansion in Equation 7 and putting T = 4nk/o) Equation 7 becomes

where

ff C =

0

{ ,kA [ exp i T

1

sin(2nnt) + 4

~

Cl2s-I

s=l

]-1}

1 --~i

sin(2s - 1)(2nnt)

dt .

(14)

However, it should be noted that our results do not agree with those given by Thornton and Kelly [-28]. Their results are difficult to reproduce in the form given by them, though the integral evaluated by us is of the same form. For the extended pulse stroboscopy of vibrations under consideration, Equation 2 becomes,

C'

1 ~r/4+~r = 2-~-TJr/4 - ~r

[ 4 nA

exp i

" 2

sn(cotlrn)

]

dt

(15)

Using a method similar to that used to evaluate the expression for C, the above integral has been evaluated analytically. However the details are omitted here. In an alternative form, suitable for numerical computation it can be written as C'=

lr88 j 88 ~

{8kA[

exp i

-2-

1 sin (2nnt)

+ 4

~q2S-I _~;-1

s=l 1 -

sin(2s-

1)(2nnt)

]--1}

dt.

(16)

3. Results and discussion Equations 14 and 16 were evaluated using 64-point Gauss quadrature scheme on an ICL 1909 electronic digital computer of our institute. As the integrands are periodic in nature, values of C and C' are independent of n provided n is an integer. However, the results are also valid for the case when the time of exposure is very large compared to the period of vibrations. Five values 1.0

B 0.2 Cs D 0.6 EEQ8

0.8

o6 0.4 E'

0.2

0.O

0.5

I.O

-& X

1.5

2C)

2.5

Figure2 Intensity variation of fringes as a f u n c t i o n

of amplitude, Curves A to E are for extended pulse s t r o b o scopic h o l o g r a p h y for ~ = ~ and curves A ' , C ' and E' are for normal time average holography.

309

P. C. Gupta, K. Singh

LO--

m A C

O.O 0.4

E 0.8

O~TO.6.T

-

0.2l 0.0

I

0.5

,

I

hO

,

I

1.5

,

-

[

2.0

,

I

2.5

X

Figure 3 Intensity variation of fringes as a function of amplitude for the case of holographic subtraction,

of parameter m were chosen and the corresponding values of K and q were utilized in Equations 14 and 16. The fringe irradiance distribution in the normal time-average hologram interferometry of periodic, non-sinusoidal vibrations, is given in Fig. 2. Curve A' represents the case for m = 0, i.e. for pure sinusoidal vibrations. Curves C' and E' represent the cases when in = 0.4 and 0.8 respectively. It is observed that with the departure of vibrations from pure sinusoidaI form, the intensity in the fringes goes up. Further, the non-sinusoidal nature of the oscillations results in a small shift of the position of the corresponding maxima and minima towards the smaller amplitude region. Computed curves for ~ = 1/8 (labelled A to E) for the case of extended pulse stroboscopic holography are also shown in Fig. 2. Curve A shows the case for pure sinusoidal vibrations and agrees with the results given by Moffat and Watrasiewicz [11] and Saito et aL [12]. Curves labelled B to E represent the cases of non-sinusoidal vibrations for various values of m. For pure sinusoidal vibrations the technique extends the measurable amplitude range by a factor of three or four. For other values of m, the measurable range is increased further. The quantitative interpretation is however, somewhat more difficult than in conventional time-average holography. In the case of holographic subtraction, curves computed from Equation 4 and showing i versus A/Z are plotted in Fig. 3. Curve A shows the case for pure sinusoidal vibrations and agrees with the results of Hariharan [4]. The number of intensity maxima is halved and there is only a single intensity zero which occurs at zero vibration amplitude. As a result, stationary parts of the object do not appear in the reconstructed image and the zero intensity fringe permits unambiguous identification of nodal areas in the object. Other curves B to E represent the cases of periodic, non-sinusoidal vibrations for various values ofm. It is noted that small amplitude vibrations, for which normal time average holography is not suitable, can be measured with ease. The measurement is not affected with the departure of vibration from pure sinusoidal form. For higher amplitude of vibrations the departure from pure sinusoidal form gives rise to better fringe contrast. Thus the amplitude measurement range is slightly extended. 310

Holographic interferometry o f non-sinusoidal vibrations

Acknowledgements O n e o f u s ( K . S . ) w i s h e s t o e x p r e s s his a p p r e c i a t i o n t o D r K . A . S t e t s o n , w h o s e a s s o c i a t i o n w i t h h i m in t h e y e a r 1970 a t N . P . L , T e d d i n g t o n ( U K ) p r o v e d v e r y s t i m u l a t i n g .

References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32.

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