concerned with observing fringe pattern anomalies over the surface of an object when it is loaded. During the process of loading the object may undergo rigid body translations and tilt resulting in high-spatial-frequency fringes that mask the region of defect. Fringe control techniques1-7 have been successfully used for the removal of high-spatial-frequency fringe patterns arising as a result of rigid body translations/or gross structural deformation of the object when it is subjected to a stress field. During a course of study on diaphragms7
using holography interferometry (HI), it was observed that large out-of-plane deformation is difficult to compensate fully. The deflection curve of the diaphragm when uniformly pressurized is not uniform; it varies continuously from edge to center.8 The fringe control techniques can therefore not be equally effective over the whole surface of the diaphragm at the same time. The large out-of-plane deformation of the center of the diaphragm results in change of slope; i.e., tilt is introduced. Archbold and Ennos9 showed that in case of defocused speckle photography, the speckle movement is related to the surface tilt. A method10 has been proposed to use the available speckle pattern just in front of the plate surface to produce slope contours of a bent plate.
This Letter demonstrates the suitability of speckle photography for NDT of diaphragms in the case of large displacements (of the order of 30 μm) as it is difficult to use HI in this situation. Figure 1. shows the schematic of the experimental setup for deformation studies of the diaphragm subject to uniform pressure difference. The camera is focused
Fig. 1. Schematic setup for recording spatial speckles for diaphragm deformation studies. The camera is focused on a plane PP' either
in front of or behind the diaphragm surface.
Fig. 2. Arrangement for optical Fourier filtering of specklegram.
Fig. 3. Slope fringes after Fourier filtering; good diaphragm.
Fig. 4. Slope fringes after Fourier filtering; defective diaphragm.
on a parallel plane PP' at a distance A from the diaphragm plane. When the diaphragm is deformed by internal pressures, under the classical assumption of the thin plate theory the prominent speckle movement will be caused by the change of surface slope β(x,y) in comparison with the in-plane strain which is small compared with bending (Fig. 1). The local surface tilt moves the speckles in a fashion analogous to a mirror tilting the light rays.9 From Fig. 1 it is easy to see that the displacement of the speckles d can be related to the surface slope by
where A is the distance from the diaphragm plane to the parallel plane in front of it where the speckles are to be photographed.
For normal viewing and small pressures, i.e., small β, we can approximate tanaα ≃ α and tan2β ≃ 2β. Further, if we assume tan2β tan2α ≪ 1 Eq. (1) can be simplified to
From Eq. (2) it is clear that the speckle movement d is a function of the slope. If the speckle pattern present in plane PP' is photographed before and after deformation of the diaphragm using double-exposure techniques and the resulting specklegram is Fourier filtered as shown in Fig. 2, the light intensity of the diffraction spectrum can be shown to be10
where k = (2Π ) /Λ , d is the displacement vector of the speckle, T is the position vector in the transform plane, L is the distance between the transform plane and speckle interferogram, and I1(T) is the diffraction hollow of the speckle pattern depicting the spatial frequency content. The modulating cos2
fringes will be governed by
where n = 0, ± 1 , ±2 . . . ; d = d; T = T; and θ = the angle between T and d.
Using Eq. (2) for bright fringes:
If spatial filtering is performed as suggested by Khetan and Chiang10 by putting a mask with the aperture at T (Fig. 2), bright fringes will be observed in the image at those places where the slope component of cos0 is such Eq. (5) is satisfied. Contour lines of the partial slope will be obtained when β 2 /A 2
≪ 1. If apertures are situated at Tx and Ty along the x and y directions, we obtain the following relations
where ω is the deflection of the diaphragm. Therefore, we have slope contours along any direction simply by changing the aperture orientation. From Eq. (6) it can be concluded that:
(1) The sensitivity of the method is directly proportional
to the distance between the diaphragm and the focused plane provided the validity of the assumption in arriving at the equation is maintained.
(2) The contrast of the fringes decreases with the increase of the distance and, therefore, beyond a certain defocus plane contrast of the fringes becomes unacceptable.
A large number of experiments were conducted to demonstrate use of this method for the NDT of diaphragms. The experiments were conducted on both good and defective diaphragms. To achieve defective diaphragms the local thickness variations in good diaphragms were created by electrodischarge machining. The defect in the diaphragm used for experimental demonstration is a 10-mm diam and 0.18-mm depth notch. The diaphragms were made of phosphor bronze (60-mm diam, 0.65-mm thick). In our experiment the recording camera of 150-mm focus length was situated ~580 mm away from the diaphragm and focused on a parallel plane 60 mm away from it. A large number of experiments were conducted to fix the optimum value of the distance between diaphragm and focused plane. The diaphragm was subjected to a pressure difference of 1000 mm of water which produced a central deflection ω = 42.72 μm. Double exposures of the speckle pattern were made with an f/8 aperture and Agfa Scientia 10E75 plates. After processing the resulting specklegram was Fourier filtered into the scheme shown in Fig. 2. Figure 3 shows the slope fringes obtained after Fourier filtering. This experiment was repeated with a same-thickness defective diaphragm over a wide range of pressure differences (p = 600-1000 mm of water). Figure 4 shows the slope fringes, obtained by Fourier filtering, of the specklegram of the defective diaphragm (p = 800 mm of water). Filtering is done by keeping the aperture in the direction of the defect.
An important feature to note is that slope contours are symmetrical about the center line for good diaphragms. However, symmetry is lost in defective diaphragms, and the defective side has more fringes. Thus, the asymmetry in the slope fringes for symmetrical problems such as diaphragms can be taken as indicative of a defect.
The work was done when the author was CSIR Senior Research Fellow at IIT, Madras. Discussions with R. S. Sirohi and financial assistance from CSIR, India are gratefully acknowledged.
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(1969). 2. E. B. Champagne "Engineering Uses of Holography," Proc. Soc.
Photo-Opt. Instrum. Eng. page 133 (1972). 3. L. A. Kersch, Mater. Eval. 29, 125 (1971). 4. L. A. Kersch, in Holographic Non-Destructive Testing, R. K. Erf,
Ed. (Academic, New York, 1974), p. 303. 5. N. Abramson, Appl. Opt. 14, 981 (1975). 6. C. Shakher and R. S. Sirohi, Canadian. J. Phys. 57, 2155
(1979). 7. C. Shakher and R. S. Sirohi, J. Phys. E. 13, 284 (1980). 8. E. O. Doebelin, Measurement Systems Application and Design
(McGraw-Hill, New York, 1966), p. 366. 9. E. Archbold and A. E. Ennos, J. Strain Anal. 9, 10 (1974).
10. R. P. Khetan and F. P. Chiang, Appl. Opt. 15, 2205 (1976).
