Vibration analysis of circular cylinders by holographic interferometry

Journal of Sound and Vibration (1973) 29(4), 475-481

VIBRATION

ANALYSIS O F C I R C U L A R

HOLOGRAPHIC

CYLINDERS

BY

INTERFEROMETRY

S. D. LIEMAND C. R. HAZELL

Department of Mechanical Engineering, The University of British Cohtmbia, Vancouver 8, Canada AND J. A. BLASKO

,4tomic Energy of Canada Limited, Sheridan Park, Ontario, Canada (Received 29 December 1972, and in revisedform 24 April 1973) The application of holographic interferometry to the analysis of vibrating fiat surfaces is now a widespread and very powerful experimental technique. This study extends holographic interferometry to the vibration analysis of curved surfaces. To verify that a single hologram was sufficient for the quantitative determination of vibration amplitudes of curved surfaces, with use of the time-averaged method, two holograms were made simultaneously of the same vibrating circular cylinder and compared with one another. The results showed that a single hologram is sufficient for obtaining quantitative data within a limited accuracy since the plane of localization of the fringes was not exactly coincident with the cylinder surface. Experimentally the error was found to be negligible until both the cylinder illumination and viewing angles became simultaneously larger than 50 degrees. Live fringe visualization of the vibrations of the same circular cylinder from a single hologram was also quite feasible. However, due to inherent problems of unwanted initial fringes, quantitative data was difficult to obtain. By intentionally introducing initial fringes in a controlled manner, live nodal line visualization and live relative phase information was possible. 1. INTRODUCTION

Holographic interferometry was first applied to vibration analysis by Powell and Stetson [1] in 1965. Over the years, modification of the original time-averaged technique has resulted in the possibility of live fringe study by strobing the laser beam [2], and phase determination by modulating the reference beam [3, 4] or using an initial static fringe pattern [5, 6]. The vibration analysis of fiat surfaces has been extensively treated and accurate amplitude calculations have been obtained. However, most of the studies of vibrating curved surfaces have been limited to qualitative treatment. Static deformation studies have been done on cylinders [7, 8, 9] and recently a detailed study of the deformation measurement on carbon biaxial specimens [10] was published. Evensen and Aprahamian [11] discussed the effects of curvature on the fringe formation. 2. THEORY Because of the inherent residual fringes present in the live fringe method, only the timeaveraged method was used to obtain quantitative data. 475

476

s.D. LIEM, C. R. HAZELLAND J. A. BLASKO

As shown by Powell and Stetson [I], the intensity distribution from a time-averaged hologram is given by I" IR=Io

do

d(cosOl +cos02

.

(1)

As can be seen from equation (1), Io is modulated by the square of the zero-order Bessel function. A dark fringe will appear whenever the argument of the Bessel function equals zero. By equating the argument of equation (1) to the zeroes of the Bessel function, the amplitude of vibration of a dark fringe is found to be f2u

For a vibrating flat surface, the displacement direction is essentially normal to the surface for small amplitudes. O~ and 02 will remain approximately constant, varying slightly with the divergence of the illumination beam and the location of the viewing position.

R

t

\ ~/~~'~b

~ P

"' l

Figure 1. Geometry of the object, illumination, and viewing positions. I = position of the illumination pinhole, V= position of the camera aperture, H = hologram, O = centre of the object, R = intersection point of the illumination beam IO and the surface of the object, Q = intersection poifit of the viewing beam O Vand the surface of the object, P = position of a dark fringe. However, for a curved surface, when all displacement is assumed to be normal to the surface (a reasonable assumption for small amplitude motion [12]), the displacement direction varies f r o m point to point and thus the corresponding changes in 0t and 02 in equation (2) must be taken into account [11]. 01 and Oz can be expressed in terms of the geometrical parameters Vl, 72, i, r and v, respectively. With the aid of Figure 1, which shows the relation between the object, illumination, viewing positions and the geometrical parameters, cos 0~ can be calculated by the following derivation: cos 01 = - c o s (180 ~ -- 01),

(3)

i 2 = a z + r 2 -- 2arcos (180 ~ -- 0a),

(4)

a 2 = i z + r 2 --

2ir cos 71-

(5)

Substituting equation (5) into equation (4), and simplifying, yields icos 71 -- r cos0z = V,i2 + r 2 _ 2ircos71 J"A list of symbols is given in the Appendix.

(6)

HOLOGRAPHY OF VIBRATING CYLINDERS

477

similarly, cos02 =

13COS72 - - r

(7)

~ v 2 + r 2 _ 2vr cos 72 A graduation was painted at the cross-section of interest; R, Q and P were located with respect to this graduation so that the angles 71 and ?2 could be calculated from the arc lengths and the curvature of the surface. By measuring i and v from the geometry of the holography system and calculating 71 and ?2 from the arc lengths, the amplitude of vibration d at every dark fringe was obtained. A simple computer program was written to calculate these values. Note that this amplitude calculation is only accurate if the fringes are localized on the surface of the specimen. 3. EXPERIMENTAL APPARATUS A schematic of the apparatus employed for the time-averaged and live fringe vibration analysis is shown in Figure 2. Coherent light from a CW Argon laser is divided into the "reference" and "object" beam by being transmitted and reflected, respectively, by the beam splitters VBS. The "reference" beam, after being reflected by mirror M is further split into two beams by another beam splitter VBS. After passing through the beam expanders BE, these two beams form the "reference" beam for the left (LH) and right hologram (RH). The "object" beam illuminates the specimen.

I

M

Figure 2. Schematic o f the holography system. L = argon laser, Spectra Physics Model 165-3, A O M = acousto-optic m o d u l a t o r , VBS = variable b e a m splitter, BE = beam expander, M = mirror, L H = left hologram, R H = right h o l o g r a m , C = camera, O = centre of the object, H = h o r n driver.

The specimen, in this case the circular cylinder, is made diffusely reflecting with a flat white paint to scatter light in all directions so that every point on the specimen reflects light to every point on the hologram. After the photographic emulsion is exposed to these two interfering beams, developed and fixed, a three-dimensional image of the object can be reconstructed in its original position simply by replacing the hologram in the same reference beam and viewing through the hologram as shown in Figure 2. The original object and its image are then exactly superimposed. If any relative motion now occurs between object and image, interference fringes become visible across the object, their density, shape and location in space depending upon the type and amount of motion. The object excitation was provided by a 30 watt horn driver. The driving head was placed within 3 mm of the cylinder surface and driven by a digital audio oscillator with a controlled

478

s.D.

LIEM, C . R . H A Z E L L A N D J. A . B L A S K O

output to within 1 Hz. This selectivity was required to locate the very sharp resonances encountered particularly in the higher modes. The audio oscillator simultaneously drove a pulse generator which in turn drove an acousto-optic light modulator through a +180 deg phase delay. This provided a strobing of the laser beam in synchronization with the object excitation. For a given object resonance, any point of the vibration cycle could be observed by using the phase delay feature. In the time-averaged method, a hologram is made while the object is vibrating at a resonant frequency. The resulting hologram shows a fringe pattern indicative of the mode shape superimposed on the image of the object. 4. EXPERIMENTAL RESULTS Both the live fringe stroboscopic and the time-averaged holography method were used in this study. Because of the residual fringes present in the live fringe method due to inherent errors involved in repositioning the hologram, only the time-averaged method was used to obtain quantitative data. Two holograms (LH and RH) were made simultaneously of the object vibrating at a resonant frequency. Pictures were taken of the reconstructed image and the amplitudes obtained from the left and the right hologram were plotted on the same graph. The following specimens Were investigated: (i) (ii) (iii) (iv)

a cantilevered flat panel (15• 20 cm), a 15 cm radius cl~imped-clamped curved panel (15 • 17.5 era), a 7.5 cm radius cantilevered cylinder (35 cm long), a 4.5 cm radius cantilevered cylinder (9.5 cm long).

Only the 7.5 cm radius cylinder will be discussed in detail, while a summary of the results for the other specimens will be given. Time-averaged holograms were made of the flat panel vibrating at 550 Hz and the 15 cm radius cui:ved panel vibrating at 1791 Hz. Amplitude.plots of the left and the right hologram reconstructions were exactly superimposed, verifying that for these objects the fringe theory and assumptions of normal motion and coincidence of the fringe localization plane with the object surface were valid. i

I

O7 5

0

"~

9 /o

J

I

i

9

25

- 0 25

o,

\.

-I 0

I 2-5

//I

I 5 0

Circumference

T 5

I0

12 5

(cm)

Figure 3. Amplitude plot of the 7"5 cm radius cylinder vibrating at 1155 Hz. F o r illumination beam, i = 152.5 cm. F o r ]eft hologram, cL = 157-5 cm. For right hologram, CR = 118"8 cm. r Left hologram; o, right hologram; A, inductance probe.

e'-

0

"6 e.0 o_ I_

~o

o

E et~

"6 d::

-r

ol

"7o~

o~

.=

e/) f-

i..,

(facing p. 478)

! 9

r.

9

(ol

Plate 2. Live fringes of the 7-5 cm radius cylinder vibrating at 1155 Hz. (a) Live stroboscopic fringe pattern without initial fringes; (b) live fringe nodal line visualization; (c) live stroboscopic fringe pattern with initial fringes; (d) live stroboscopic fringe pattern with initial fringes [180 degrees out of phase with (c)].

HOLOGRAPHY OF VIBRATING CYLINDERS

479

The next specimen investigated was the 7.5 cm radius cylinder. Plates I(a) and (b) show the left and the right hologram reconstruction of the time-averaged fringe pattern of the cylinder vibrating at 1155 Hz. Figure 3 shows a plot of the amplitudes of vibration along the silvered graduation cf the cylinder (located 15 cm from the fixed end) calculated from Plates I(a) and (b). The amplitudes of vibration were measured experimentally by using a non-contacting inductance probe and these data are shown in Figure 3 as well. A comparison of the plots shows that t iie amplitude plot of the left hologram is displaced to the right side of the amplitude plot of the right hologram: i.e., shifting the viewing position from the left to the right results in an amplitude shift from the right to the left. To deter nine the reason for this shift, two sets of experiments were done whereby first the object rxcitation and next the geometry of the holography system was varied. Various holograms were made of the 7.5 cm radius cylinder with different resonant frequencies and amplitudes, but all amplitude plots of these holograms showed the same shift. Next the geometry of the holography system was varied by relocating the illumination beam between the left and the right hologram and the effect of the diverging illumination beam was also investigated by repeating the experiment with a collimated beam being used. The shift was always present and always in the same direction, indicating that it is independent of the system geometry or the object mode of vibration. With a collimated reference beam being used, the real image was observed to determine the fringe localization plane. It was found that the fringes were localized very close to the object surface. Due to the large depth of field of the reconstructed ~mage, it was not possible to separate the object surface from the fringe localization plane. Various holograms were made of the 4.5 cm radius cylinder. The same shift of the amplitude plots was observed with no appreciable change in the amount of shift as compared to the 7.5 cm radius cylinder. Plate II shows the live fringe patterns observed through the left hologram of the 7.5 cm radius cylinder vibrating at 1155 Hz. A hologram was made of the object in its static position and repositioned after developing. When the object was excited and observed under stroboscopic illumination, which was synchronized with the excitation signal, a fringe pattern similar to the time-averaged method was observed as shown in Plate II(a). Note that there is no decrease in fringe contrast with increasing amplitude which is typical of the timeaveraged fringe patterns. When the hologram was slightly rotated a static fringe pattern was observed, due to the small misalignment between the actual object and the reconstructed image. When the object was excited and observed under continuous illumination, the nodal areas became visible as those parts of the static fringes that remained stationary (Plate II(b)). If instead of continuous illumination stroboscopic illumination was used, a complete mapping of the cylinder was obtained as shown in Plates II(c) and (d). The fringe patterns of Plates II(c) and (d) are of opposite curvature and were obtained by delaying the strobe by 180 deg with respect to the excitation. Thus, the relative phase of the object can be determined by observing the direction of curvature of the fringes.

5. SUMMARY AND CONCLUSIONS In this experiment a formula was developed to calculate the amplitude of vibration of circular cylinders due to sinusoidal excitation from time-averaged holographic interferograms. The formula takes into account the divergence of the illumination beam, the change in direction of the normal to the curved surface and a point observation; it is only valid for normal displacement and is based on the assumption that the fringes are localized on the object surface.

480

s . D . LIEM, C. R. HAZELL AND J. A. BLASKO

The amplitude plots of the left and the right hologram for the fiat panel and 15 cm radius curved panel agreed quite well. For the 7.5 cm and 4.5 cm radius cylinder, the amplitude plots of the left and the right hologram were shifted in such a way that when the viewing position is changed from the left to the right, the amplitude plot is shifted from the right to the left. This shift was found to be independent of the system geometry or object mode of vibration and can possibly be attributed to the fact that the fringe localization "plane" for curved surfaces of relatively large curvature is not on the object surface, but slightly in front of it. From the shift in the amplitude plots and on the assumption that the fringe localization "plane" is independent of the location of the hologram it was estimated that the distance between the object surface and the fringe "plane" is only of the order of 4 mm. As was observed, this small gap makes it impractical to investigate the fringe localization "plane" by reconstructing the real image. From this study it is concluded that for fiat and curved surfaces of small curvature, the fringes are localized on the object surface. A simple formula based on the geometry of the holography system will give accurate amplitude plots. Experimental results indicate that for curved surfaces o f large curvature, the fringes are localized slightly in front of the object, which results in a slight shift of the amplitude curve depending on the viewing position. Because of the minute displacement of the fringe localization plane from the object surface, it was not possible to investigate the reason for the larger shift at the right part of the object. From the experimental data, it was found that the larger shift of the amplitude plots coincided with the fact that the illumination and viewing angles simultaneously became larger than 50 deg. ACKNOWLEDGMENT The work described here was undertaken in the course of research supported by the National Research Council of Canada under Grant Number A-3331. REFERENCES 1. R. L. POWELLand K. A. STETSON1965 Journal of the Optical Society of America 55, 1593-1598. !nterferometric vibration analysis by wavefront reconstruction. 2. E. ARCHBOLDand A. E, ENNOS 1968 Nature 217, 942-943. Observation of surface vibration modes by stroboscopic hologram interferometry. 3. C. C. ALEKSOFF1969 Applied Physics Letters 14, 23-24. Time average holography extended. 4. D. B. NEUMANN, C. F. JACOBSONand G. M. BROWN1970 Journal of Applied Optics 9, 1357-1362. Holographic technique for determining the phase of vibrating objects. 5. G. M. MAYER 1969 Journal of Applied Physics 40, 2863-2866. Vibration phase measurement of rotation--strobe holography. 6. C. R. HAZELLand S. D. LIEM 1970 Symposhtm on the Application of Holography, Besancon, France. Vibration analysis by interferometric fringe modulation. 7. I. K. LEADBE'r'rERand T. ALLAN1968 Proceedings of the Symposium on the Engineering Uses of Holography, University of Strathclyde, Glasgow, Scotland. Holographic examination of the pre-buckling behaviour of axially loaded cylinders. 8. A. D. WmsoN 1970 Applied Optics 9, 2093-2097. Holographically observed torsion in a cylindrical shell. 9. O. J. BURCHE'I'rand J. L. IRWIN 1971 Mechanical Engineering 27-33. Using laser holography for nondestructive testing. 10. C. G. MURPHY, O. J. BURCHETrand C. W. MATI'HEWS1972 Proceedings of the Symposium on the Engineerhlg Applications of Holography, Los Angeles, California, 177-186. Holometric deformation measurement on carbon biaxial test specimens. 11. D. A. EVENSENand R. APRAHAMIAN1970 Report No. 70-11, TRIV Systems Group, Redondo Beach, California. Applications of holography to vibrations, transient response, and wave propagation. 12. HARRYKRAUS 1967 Thin Elastic Shells. New York: John Wiley and Sons, Inc. See Chapter 8.

HOLOGRAPHYOF VIBRATINGCYLINDERS APPENDIX LIST OF SYMBOLS d i v r Io la Jo P R Q

QL QR 2 f~s 0t 02 ~'1 72

amplitude of vibration distance between illumination pinhole and centre of object distance between camera aperture and centre of object radius o f object intensity distribution of reconstructed image of static object intensity of Io modulated by fringe function zero-order Bessel function fringe position centre of the illumination beam on the object centre of the viewing beam on the object centre of the viewing beam of the left hologram on the object centre of the viewing beam of the right hologram on the object wavelength of light zeroes of Jo angle between direction of object illumination and direction of vibration angle between direction of object viewing and direction of vibration angle between the centre line of illumination and fringe position P angle between the centre line of viewing and fringe position P

481