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Vibration analysis of singly curved surfaces by holographic interferometry. Blasko, James Alexander 1972

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VIBRATION ANALYSIS OF SINGLY CURVED SURFACES BY HOLOGRAPHIC INTERFEROMETRY BY JAMES ALEXANDER BLASKO B.A.Sc., University of Toronto, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF-APPLIED SCIENCE in the Department of MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1972 In present ing th i s thes is in pa r t i a l f u l f i lmen t o f the requirements fo r an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes i s fo r s cho l a r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on o f th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT Time-average and stroboscopic real-time holography have become established as the prominent tools of holographic vibration analysis. Both are able to provide a quantitative determination of the amplitudes of standing waves of a vibrating f l a t surface, but have to date been limited to qualitative examination of curved surfaces. The purpose of this thesis has been to investigate the problems of extending the holographic technique to the analysis of surfaces of single curvature. The presence of several component motions in curved surface vibration prompted an examination of the fringe behaviour of the individual components of motion. This was performed by studying pure rigi d body step displacements using double exposure holography. The experimental work of previous investigators was duplicated and the results tabulated according to the sensitivities of the fringe period density to a change in displacement magnitude. This was then related to the rotations and translations occurring in the motion of a vibrating surface. A fully clamped curved panel was studied using the time-average method with the assumption that a l l points on the surface were displaced normal to i t . Good agreement was obtained between the amplitudes calculated from the fringes and the measured amplitudes, by accounting for the variation in displacement direction across the surface. A fixed-free cylinder was examined simultaneously from two overlapping views using both the time-average and stroboscopic real-time methods. Comparison of the calculated and measured amplitudes of the time-average indicated that the interference fringes were not localized upon or very near the cylinder's surface. The real-time results had reinforced these findings. However, the inherent presence of residual fringes in this method caused d i f f i c u l t i e s in obtaining an accurate quantitative evaluation of the vibration amplitudes. The real-time fringes were analyzed with attention paid to the effects of the i n i t i a l residual fringes upon the resultant fringes. iv TABLE OF CONTENTS Chapter Page 1 INTRODUCTION ........... .. .... .... . V... ; 1 1.1 Preliminary Remarks ......................... 1 1.2 Statement of the Problem .................... 2 Literature Survey 1. 3 Vib ration Analysis ............. ... 3 • 1.4 Rigid Body Step Displacements .......... 6 1.-5 Curved Surface Analysis 9 2 THEORY .• ........ . .... 13 2.1 Principle of Holography ..................... 13 2.2 Formation of Interference Fringes and Double Exposure Holography 18 2.3 Rigid Body Step Displacements 21 2.4 Vibration Analysis by Time-Average Holography 30 2.5 Vibration Analysis by Stroboscopic Real-Time Holography 32 2.6 Variation of Displacement Direction in Curved Surfaces 34 3 EXPERIMENTAL APPARATUS AND PROCEDURE 40 3.1 Holographic Apparatus • 40 3.2 Vibration Excitation, Beam Strobing and Displacement Measurement .................... 43 3.3 Specimen 46 3.4 Procedure ....>. - 49 V Chapter Page 4 EXPERIMENTAL RESULTS AND DISCUSSION .............. 52 4.1 Rigid Body Step Displacements ............... 52 4.2 Time-Average Study of Curved Panel ..........< 72 4.3 Time-Average and Stroboscopic Real-Time Study of a Fixed-Free Cylinder .............. 80 5 SUMMARY AND CONCLUSIONS ..................... . 99 5.1 Summary ... ........ ........ ...... 99 5.2 Conclusions .... v ...............v ..'. 101 5.3 Suggestions for Future Research.............. 103 BIBLIOGRAPHY ' • ....... . ....... . • . 106 APPENDIX A Fringe Location by Use of Microdensitometer ...... 110 B Differences in Displacement Magnitude for Assumption of Collimation i... ^ . . . . . . H7 C Operation of Fo tonic Sensor 123 v i LIST OF FIGURES Figure Page 2.1 Typical Holographic Arrangement .........-. ...... 14 2.2 Image Recording and Reconstruction 14 2.3 Path Difference of Light 19 2.4 Orientation of Hologram and Specimen for Rigid Body Step Displacement Studies ...... .... 23 2.5 Top View of Rotation About y-axis .................. 23 2.6 Displacement of Elemental Area in y-axis Rotation... 24 2.7 Fringes due to Rotation about y-axis ............... 27 2.8 Fringes due to Rotation about z-axis 27 2.9 Fringes due to Translation along x-axis ............ 28 2.10 Fringes due to Translation along ^z-axis .........—• 28 2.11 Variation of 6^  and % 2 on a Flat Panel ............. 35 2.12 Variation of Displacement Direction on a Curved Panel 37 2.13 Variation of Displacement Direction on a Cylinder... 39 3.1 General Holographic Arrangement for Vibration Studies of Cylinder .. ........ 42 3.2 Sketch of Curved Panel .... ............. j48 3.3 Cross-section of Fixed-Free Cylinder 4wi Holographic Arrangement for Rigid Body Displacement S tudies 4.2 Determination of Fringe Plane Location for x-axis Translation ................................... 4.3 Comparison of Rotation and Translation in a Vibrating Membrane ................ ......... v i i Figure Page 4.4 Holographic Arrangement for Curved Panel Vibration Studies ............... 73 4.5 Plot of Amplitudes for Curved Panel Comparing Results of "Fringe Location by Eye" and by Microdensitometer ............... • ...... 74 4.6 Panel Orientations used in Fig. 4.5 75 4.7 Plot of Amplitudes from Time-Average Fringes of Curved Panel Vibrationj 78 4.8 Holographic Arrangement for Cylinder Vibration Studies 82 4.9 Plot of Amplitudes from Time-Average Fringes of -.//tv. Cylinder Vibrations ................ ......... 84 4.10 Plot of Amplitudes from Stroboscopic Real-Time Fringes of Cylinder. Vibrations .......... 87 4.11 Proposed Fringe Localization. Plane for Cylinder Vibrations .................. ....... 89 4.12 Sketch of Fringes Shown in Photos '4.7 ........ 95 A.l MD Plot .................. ............. 112 A.2 Example of Perspective when Viewing Cylinder from a Point .-...« ............................ H3 A. 3 Geometry for Locating Fringes from MD Plot onto Cylinder ....... 115 B. l Variation of Illumination and Observation Directions on a Flat Panel ........ 117 B.2a Plots of DIFF vs. LOCATION for Specimens ... 120 B.2b Orientations of Specimens for Plots of Fig. B.2a.... 121 v i i i Tables Page 4.1 y-axis rotations 56 4.2 z-axis rotations .••....:. 55 4.3 x-axis translations ............................ • 62 4.4 x-axis translations (corrected) .................... 62 4.5 z-axis translations 67 4.6 Comparison of Fringe Sensitivities ................. 69 Photographs 3.1 General. Holographic Arrangement for Vibration Studies of Cylinder ..................... ...... 41 3.2 Translation and Rotary Tables on Magnetic Support 45 Basec ^ 3.3 Specimen in Place on Translation and Rotary Tables.. 47 3.4 Curved Panel Specimen 47 3.5 Cylinder Specimen with Horn Driver ........... ^ 4.1 Fringes, y-axis Rotation .................. 57 4.2 Fringes, z-axis Rotation ^8 4.3 Fringes, x-axis Translation 63 4.4 Fringes, z-axis Translation ............. ^6 4.5 Time-Average Fringes of Curved Panel ................ 77 4.6 Time-Average Fringes of Cylinder ................... 4.7 Stroboscopic Real-Time Fringes of Cylinder .-... 86 4.8 Residual Fringes, Before and After Being Minimized.. 92 4.9 Residual Fringes Appearing Before Photos 4.7 ........ 93 A.10 Curved Panel with Rulers Used to Determine Location of.P. . ... >.-...•.-...•. 114 ix LIST OF SYMBOLS a d d n P r i ; s x,y,z x ,y o'^o A 1 ' A 2 BE BS20 D DIFF D x FC H H L \ HC HO hologram to object distance displacement displacement calculated assuming collimation displacement calculated with divergence taken into account point on cylinder where illumination beam is centred integer number centre point of f l a t and curved" panel where illumination and observation vectors are centred, point on cylinder where observation vector i s centred general point on a surface radius fringe period in y-axis rotation general co-ordinate system x,y axis components of fringe period in z-axis rotation Amplitude of object and reference beam beam expander non-variable beam sp l i t t e r (20% transmittance) path difference % difference of displacement calculated assuming collimation',: relative to actual displacement which accounts for divergence^1 fringe spacing' measured at Z Q fringe plane to camera distance hologram hologram of l e f t view hologram of right view hologram to camera distance hologram to object distance intensity of reconstructed image intensity" of I q modulated by fringe function zero order Bessel function-lens mirror microdensitometer integer number object object beam point of observation reference beam variable beam sp l i t t e r distance of fringe plane behind object i n x-axis translation linear phase variation of reference beam at plate y-axis rotation angle between reference beam and normal to hologram angle between observation and illumination vectors angle between displacement directions at o and p angle between illumination vector (object beam) and displacement direction angle between observation vector and displacement direction angle change of illumination vector due to divergence angle change of observation vector due to divergence z-axis rotation phase variation at plate, of object and reference beam o wavelength of He-Ne laser, 6328 A = 24,9 uin. x i phase difference -*r angle between illumination vector and displacement direction at reference point o. angle between observation vector and displacement direction at reference point o. angle between illumination and z-axis angle between fringe plane and z-axis in z-axis rotation zeroes of J o x i i ACKNOWLEDGEMENTS The author would like to sincerely thank his supervisor, Dr. C R . Hazell. His guidance given throughout the course of the experimental work and preparation of this thesis has been invaluable. The author also wishes to thank Mr. S.D. Liem for his advice on experimental procedures, and Mr. L.E. Dery for his technical assistance in the photographic work. This study was supported by research grant No. 67-3331f°f the National Research Council of Canada. 1 1. INTRODUCTION 1.1 Preliminary Remarks Before the extension of holography to vibration work in 1965, the available tools for such studies could only provide a point by point determination of the response of an oscillating specimen. There were both contact and non-contact transducers in use, but knowledge of the amplitudes at a given point required the appropriate placement of a probe. Consequently, simultaneous examination of several points required an equal number of probes. This was especially disadvantageous i f the transducers happened to be of the contact variety, where then additional weight altered the vibration characteristics of the specimen at the exact point of measurement. The recent development of holographic interferometry as an ideal laboratory method of vibration analysis, has brought with i t inherent properties that overcame the previous drawbacks associated with such studies. Holography i s a highly sensitive technique particularly suited to the examination of amplitudes on the order of the wavelength of light. But most.significantly i t is a non-contact method which allows the response of a l l regions of the specimen to be viewed simultaneously. 2 1.2 Statement of the Problem The "time-average" and "stroboscopic real-time" methods had by 1969 emerged as the prominent tools of holographic vibration analysis. With these processes holographic studies of vibrational characteristics had been carried out, for the most part, on f l a t surfaced specimens such as beams,arid clamped panels and membranes. Such examinationsswere able to produce accurate quantitative results. The direction of displacement of these vibrating surfaces could be assumed normal to the equilibrium surface, and to be constant in direction at a l l points. This a p r i o r i knowledge of the displacement direction, coupled with the ease of viewing the entire vibrating surface through one hologram, presented the practical feature of being able to obtain a l l the necessary data from a single hologram.' The extension of holographic analysis to curved surfaces (i.e. turbine,blades, cylinders), becomes more involved when the consideration is made that the displacement direction normal to the surface varies due to the curvature of the surface, and furthermore, that in cylinders there are components of motion present other than normal to the surface. The purpose of this thesis is to investigate the procedural requirements of applying the holographic method to the study of surfaces curved in one direction, (i.e. cylinders or segments thereof). Attempts w i l l be made at determining a l l the amplitude and mode shape information from the interference fringe pattern of a single hologram. To the best of the authors knowledge, the analysis of curved vibrating surfaces by the iise', of a single hologram has been limited to qualitative interpretation. This thesis w i l l present a program of study undertaken 3 to examine the problems of analyzing a singly curved vibrating surface by holographic interferometry, and to attempt quantitative measurement of the vibration amplitudes. The program of investigation i s outlined below: (1) The properties and sensitivities of interference fringes due to each of four rig i d body step displacements were studied using double exposure holography. A preliminary study of these motions was prompted by the understanding of several components of motion being involved in curved surface vibrations. (2) A singly curved f u l l y clamped panel of 6" radius was examined using time-average holography. The vibration amplitudes calculated from the experimental results were compared to the values measured\ith a Fotonic sensor, a commercially available non-contacting displacement transducer. (3) A fixed-free cylinder of 3" radius was studied by "time-average" and "stroboscopic real-time" holography. In each case, the investigation was performed with a simultaneous double view to allow observation of as great a portion of the cylinder as possible. The d i f f i c u l t i e s involved with real-time holography were to be analyzed. Literature Survey 1.3 Vibration Analysis The development of holography as a method for vibration analysis began with a study of a film can bottom made by Stetson and Powell in 1965 [4]. A time-average process was used where the hologram was 4 exposed for a length of time while the specimen vibrated at a natural frequency. A discussion of a real-time method followed shortly thereafter [5], with the theory describing the time-average and real-time methods presented in a 1966 paper [6]. The real-time method outlined, required a hologram of the static specimen to be repositioned exactly as before i t s exposure. Then with the specimen vibrating, interference occurred between light from the vibrating object and i t s static image. The resultant fringes were in effect due to a time-averaging taking place in the eye of the observer. The two methods mentioned here w i l l be defined in derail in Chapter 2. Archibold and Ennos continued from this point with a further extension of the real-time process. In two papers [7] and [8], they discuss having stroboscopically illuminated an oscillating specimen at one'extreme of i t s vibration. (This i s i n contrast to the real-time continuous illumination of Powell and Stetson.) With the strobing procedure, interference occurred between light emanating from the oscillating specimen at a particular point in i t s oscillation and the holographic image representing the static or equilibrium position. The experimental setup was described in detail in a paper presented at the 1968 Strathclyde Symposium [12]. Independent works by Mayer [10],Hazell and Liem [18,19], further developed the real-time to include a f a c i l i t y for determining the relative phases of oscillation for different regions of a vibrating specimen, whereas previously, only the amplitudes could be quantitatively determined. The phase determination was accomplished by rotating the repositioned hologram (for example, rotation about the bottom edge of 5 the hologram). As a result, the holographic image was rotated with respect to the real object. Equi-spaced interference fringes resulted which paralleled the rotation axis. The specimen was then vibrated and the illumination strobed as per Archibold and Ennos. The fringes then deformed to a U-shaped curve, i n a direction either towards or away from the rotation axis. If the vibrating specimen was strobed such that i t was illuminated at a position displaced i n the direction of rotation, the curved portion of the U was pointed away from the'vrotation axis, and vice versa. In these previous vibration studies i t was observed that the interference fringes appeared to be,on or very near the surface of the specimen. That i s , no parallax could be detected as the direction of viewing was varied within the borders of a single hologram. Certain types of motion do however result in fringes that appear to be located elsewhere in space. In other words they can be brought into focus by some lens system, such as a camera, at a position in space other than the object's surface*. A study of rigi d body step displacements, as a preliminary to curved surface studies, displays fringes with this property. The analysis of vibrating specimens i s concerned with obtaining amplitude and phase information. Since the ensuing interference fringes appear to be located upon the object's surface, there i s a direct relation between the point at which a given fringe i s located and the amplitude of vibration at that point. However, in curved surface vibration there are several component motions present which, when individually considered, produce fringes that do not localize upon the object. 6 1.4 Rigid Body Step Displacements In the case of interference fringes being localized at a position in space ether than the object surface, .a calculation of displacement magnitude requires knowledge of the position of the localization plane and a consequent examination of the fringes at the plane [25,31,44,28]. If one could focus simultaneously on the object and fringe plane, this requirement may be avoided. Aleksandrov and Bonch-Bruevich [26], describe such an analysis. One of the earliest theoretical studies of fringe behaviour related to rigid-body motions was by Haines and Hildebrand {25]. Two categories of fringes are defined; those in focus on or near the object (i.e. within several thousand wavelengths) and those at a greater distance. The general case is examined where the object motions are described by cartesian co-ordinates and euler angles. The theory is reduced to a determination of the fringe periods in two perpendicular lateral directions in a plane normal to the observation vector, then solving for the rotations by substitution. For these lateral motions the fringes are rot localized on the object. Aleksandrov and Bonch-Bruevich [26], propose; ',' a system with a viewing aperture sufficiently small so that the object and fringes ^ a r e i n focus simultaneously. Then with a large enough hologram the viewing direction is varied to three different orientations while noting the number of fringe orders that pass a given point on the object. Relating the order change to change of observation direction provides three equations to determine fu l l y the three components of motion for each point. No consideration is made of the type of surface contour. This process contrasts that of a p r i o r i knowledge of the displacement 7 vector requiring only a single hologram to calculate the magnitude. Vienot et a l . [28], have viewed the combination of a translation and rotation noting that the translation component may'be neglected for a resultant displacement less than 10 microns.- A theory of homologous rays is described to define conditions and regions of fringe localization. Expressions for several displacements were derived and these were later added to by Froehly et.al. [30], a paper which included the same authors as above. A completely general theory by Stetson [29], covering a l l cases of rotation and translation i s derived particularly for collimated illumination and reference beams. A further modification to the theory accounts for spherical illumination. Analysis and legationof the plane of fringe localization i s emphasized with the magnitude;;, of motion being consequently determined. A table relating types of motion to fringe behaviour is presented in conclusion. Experimental verification of the theory i s performed in a later paper by Mplin and Stetson [35]. These two articles provide one of the more extensive works on fringe behaviour. The general expression derived by Molin and Stetson, defines the phase change at the hologram plate due to object displacement. In the experimentation this expression was specialized for the rig i d body motions considered.- A simultaneous study of a plane surface, and concave and convex surfaces of 10 cm radius revealed no difference in fringe contour or localization. Time-average vibration and double exposure holograms were included, demonstrating that the time function of the displacement had similarly no influence upon the fringe behaviour. 8 Froehly et a l . [30], returned to divergent illumination and a more simplified approach. The homologous ray concept considers that an object undergoing displacement causes a change of direction of the reflected illumination beam from a given point. These reflected rays intersect the hologram plane before and after the displacement. : Observing where the two rays themselves intersect w i l l define the plane of fringe localization. In experimental verification of their theory, Haines and Hildebrand [25] encountered discrepancies of up to + 60% between measured and calculated values of displacement. Boone and Verbiest [31], duplicated these tests with certain modifications. Previously the fringe plane was located by focusing on the fringes with a telescope then measuring the fringe period at this plane. This was performed viewing the v i r t u a l image of the reconstruction, (see Section 2.1). Boone and Verbiest noted that the accuracy was dependent upon locating the fringe plane precisely, which in turn is a function of the depth of f i e l d . To eliminate this and ether inherent d i f f i c u l t i e s they examined the real image by use of a collimated reference beam, thus the hologram was turned 180° for reconstruction and the real image projected on a screen. This advantage allowed accurate location of the fringe plane and measurement, of the inter-fringe spacing. It was observed that the closer the fringe period is measured to the image, the calculated value of translation by Haines and Hildebrand diminishes. Thus a corrected expression had been derived and a comparison of the two theories tabulated for several lateral displacements. Tsujiuchi [33] considers the localization conditions in a respect 9 similar to Stetson [29], except that the theory is not general, but was derived for each motion studied. The derivatives of the path length change of the illumination beam with respect to three cartesian axes are set to zero, whereas Stetson took the derivatives of the phase equations with respect to the direction cosines of the viewing vector. In extending his theory, Tsujiuchi analyzed fringe patterns due to a combination of motions in an attempt to separate the contributions by each to the resulting fringes. This is described for a surface deformation accompanied by a r i g i d body in-plane rotation then an out-pf-plane rotation. 1.5 Curved Surface Analysis A survey of the available literature revealed l i t t l e information on holographic vibration studies of curved surfaces. Works that have been undertaken were qualitative in nature and usually accompanied more rigorous-quantitative analyses of f l a t surfaced specimens. Such an example is an investigation by Sampson [17], who performed a qualitative interpretation of the time-averaged interference fringes produced by a vibrating turbine rotor. This study revealed the blades to be acting as an integral part of the rotor much as the outside edge of a centrally clamped circular plate. Hazell, Liem and Olson [45], have made qualitative studies of individual turbine blades and a cantilevered cylinder. This work was performed using a stroboscopic real-time method with an i n i t i a l fringe pattern [10,18,19]. Concerning the cylinder, i t was observed 10 that in the region of resonance of a shell mode a non-symmetrical oscillation was indicated. This was concluded from a shifting of the modes from the mode shape just previous to resonance, to that, shape immediately after resonance. Some static deformation studies have been performed upon cylinders in the hope of obtaining quantitative results. Leadbetter and Allan [46], examined the pre-buckling in s t a b i l i t y of a thin-walled steel cylinder of 3 5/8" dia. Two holograms were used, both to allow two viewing directions to cover as wide an area of the cylinder as possible, and to have an overlap between the views to relate what i s occurring in one area to that in the other. Each viewing direction had a separate illuminating source. Two methods of study were mentioned, the real-time and doublet .exposure. In the real-time the specimen was deformed and the fringe order determined at a point by counting the fringes as they passed that point. The double exposure method required knowledge of the zero order fringe to determine the fringe order at a point. Otherwise, as described by Aleksandrov and Bonch-Bruevich [26], several views would be required to calculate the displacement without a p r i o r i knowledge of i t s direction. Leadbetter and Allan used the double exposure technique. The cylinder was compressed axially i n equal increments of displacement, and the double exposure fringes recorded the deformation change in the cylinder between compression increments. This resulted in a series of fringe patterns describing the deformation from one compression increment to the next, as the cylinder, was loaded t i l l buckling failure. The study was able to yield only a qualitative analysis. While no mention was 11 made of the region of localization of the fringes i t was this author's opinion that Leadbetter and Allan had observed and assumed that they were on or very near the actual object's surface. Burchett and Irwin [40], discussed a method of nondestructive testing of carbon structures. The specimens were 3" dia. carbon cylinders loaded internally by a pressurized bladder. Of significance is the manner of fringe analysis that was used. A photograph was taken of the reconstructed fringe pattern, and the task was to locate the fringes from the picture to the appropriate angular position on the cylinder. For a f u l l size photo of the cylinder i t would have been a simple matter of direct projection from a plane surface (the photograph) to the cylindrical surface. However, for a photo that i s smaller than f u l l size, perspective angles must be considered. This i s due to the fact that when a cylinder is viewed from a point, say through one eye or a camera, the f u l l 180° of the cylinder is not visible but is actually a few degreesffiless (see Appendix A, Fig. A.2, page ). .Consequently the location of the fringes from the smaller sized photo mustrbe projected to their appropriate location upon the cylinder through consideration of perspective angles. While geometric expressions have been derived for these angles they were not made use of in the data reduction. Of note was the mention of errors occurring due to di f f i c u l t i e s in determining the location (on the photograph) of the extreme edges of the cylinder, that is* the two opposite sides that would be tangential to the viewing vector. These were the points to which the fringes were referenced in locating them properly upon the cylinder i t s e l f . Though quantitative results were presented, i t 12 was s t a t e d t h a t q u a l i t a t i v e d e t e r m i n a t i o n of p o s s i b l e f r a c t u r e a r e a s was w e l l e s t a b l i s h e d , w h i l e problems e x i s t i n the q u a n t i t a t i v e r e d u c t i o n o f d a t a . W i l s o n [ 3 9 ] , made a r i g i d body st u d y of a s o l i d s h a f t i n a s t e p r o t a t i o n . T o r s i o n was a l s o i n c l u d e d u s i n g s i l a s t i c r u b b e r . - The r i g i d s h a f t was s t e e l and b o t h were 1.2 cm i n d i a m e t e r . A t h e o r y was d e r i v e d to r e l a t e t h e a n g u l a r d i s p l a c e m e n t t o the d ouble exposure h o l o g r a p h i c f r i n g e s . I t was o b s e r v e d t h a t f o r r o t a t i o n the f r i n g e s were p a r a l l e l t o the r o t a t i o n a x i s and l o c a l i z e d v e r y n e a r i f n o t c o i n c i d e n t w i t h the image of the c y l i n d e r i t s e l f , t h a t i s , d i r e c t l y on the o b j e c t . I f the f r i n g e s were p r o j e c t e d on to a p l a n e s u r f a c e tangent to the c y l i n d e r , they would be e q u i - s p a c e d . The f r i n g e p e r i o d was independent o f the c y l i n d e r r a d i u s or the a n g u l a r p o s i t i o n around the c y l i n d e r . T h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s showed v e r y good agreement. 13 2. THEORY 2.1 Principle of Holography Holography i s a process of image recording and reproduction such that the reproduced (or reconstructed) image retains a l l three-dimensional properties of the original scene. The essence of holo-graphy lie s in i t s a b i l i t y to record phase as well as intensity variations of a wavefront of light. (Conventional photography records only intensity variations). It i s this property that allows reconstruction of a three dimensional image. A hologram i s a high resolution photographic plate, that when exposed, records a very fine interference fringe pattern resulting 1 from interaction of two coherent light beams at the hologram plate. One is the reference beam R and the other i s the object beam 0,.both derived from a common laser source. A typical holographic set up i s shown in Figure 211. The laser beam i s divided into two by a variable beam sp l i t t e r VBS. The reference beam is guided by a mirror; M, to pass through a beam expander BE, which expands the beam to a spherical wavefront arriving at the hologram undisturbed.. Similarly, the object beam illuminates the specimen, and i s in turn diffusely reflected (thereby being modulated to contain 14 object' diffuse/;''' reflection-hologram .1 • F i g . 2.1 T y p i c a l Holographic Arrangement light from virtual imago undiffractod component 'i r Fig. 2..2-C' 4^jAa.^^c^t^±ng\-^6,..Reconstruction .;J 15 information r e l a t e d to surface shape) to a r r i v e at the hologram. Because the ILlumination i s d i f f u s e l y r e f l e c t e d , l i g h t from any one point on the object a r r i v e s at a l l regions of the hologram. / A Interference of the beams produces, on the hologram, a very'^cqmplex g r i d of dark and l i g h t f r i n g e s , i n v i s i b l e to the naked eye because of t h e i r high density. This grid,contains a l l the necessary information required to reconstruct an image of the o r i g i n a l object. In the reconstruction process the developed hologram i s re-il l u m i n a t e d by the reference beam alone. The interference g r i d acts as a d i f f r a c t i o n grating, d i f f r a c t i n g the reference beam (reconstruction beam) into several component wavefronts (Fig. 212). One of these i s a duplicate of the o r i g i n a l wavefront r e f l e c t e d by the object i n the recording process. The phase recording i s made possible by the use of two beams. One acts as .a c a r r i e r while the other i s the s i g n a l beam. These being the reference and object beams re s p e c t i v e l y . Although photographic emulsions are s e n s i t i v e only to i n t e n s i t y v a r i a t i o n s ( i n t e n s i t y i s proportional to the l i g h t amplitude squared), the phase information i s also retained.' I t i s recorded i n the interference g r i d on the hologram, where the i n t e n s i t y information i s stored i n the i n t e n s i t y v a r i a t i o n s of the interference f r i n g e s , and the phase i s contained i n the spacing of these frin g e s . Holography can be described mathematically r e f e r r i n g to F i g . 2.2. This and s i m i l a r treatments can be found i n most books and a r t i c l e s on basic holography [23,24]. The;j de r i v a t i o n presented here assumes a collimated reference and i l l u m i n a t i o n beam. R and 0 are re s p e c t i v e l y •16 the reference and diffusely reflected object beams. A^, are the ^ i't'-.Cx) 0 = A^e 14»2(x) R = A 2e light beam anplitudes and ij) ^ , cj> 2 define the phase variation of each beam at the hologram. Since they are coherent in time the temporal term iwt . T . , e i s neglected. The reference beam makes an angle & with the hologram normal, and since i t arrives undisturbed the phase variation cj)2(x) i s linear across the plane of the hologram and can be expressed as <j>2(x) = 6x (r^) = ax a = A therefore the reference beam becomes, lax R = A 2e The total amplitude distribution of light at the plate i s the vector sum, a ^ !<•>•, (x) iax 0 + R = A xe + A 2e The photo-plate (hologram) records intensity, which i s proportional to 17 the amplitude squared. Denoting the intensity proportional to I, I = 10 + R | = (0 + R) (0 + R) 2 ? i(cj>1-ax) i ( ax-<{>.,) = A x + A 2 + A-jA^e + e ] It i s assumed that there is a linear relationship between the intensity of light recorded by the photo-plate, and the intensity that i s transmitted afterwards by the developed plate. Consequently when the developed hologram is re-illuminated by the reference beam alone, the light transmitted i s proportional to the transmittance term (1 - KI). K is a constant which is a function of the emulsion properties, relating the recorded intensity to the density (or darkness) of the developed plate. (The greater the intensity, the darker the plate, and the less the transmittance). The amplitude of the transmitted light i s TL = R(l - KI) Substituting for the appropriate terms and carrying out the multiplica-tion yields three separate terms ~ ~ l a x TL = [1 - K(A^ + Ap] A 2e ' (1) 2 i ( f , l •[KApAje 1 • * ' (2) „ i(2ax -cj) ) •[KA^lAje X (3) 1 8 Each term represents a separate wavefront and they can be analyzed as follows. The first term represents an undiffracted component of the reconstructing reference beam which passes through the hologram as an attenuated DC component. The second term i s the original object beam 0, multiplied by a scalar. This component is the reconstructed duplicate of the original wavefront as reflected by the object during exposure of the hologram. It can be viewed with the naked eye as a virtual image.y- and is the image used in any holographic study of vibrations. The third term closely resembles the second in that i t is also the original object beam but i t i s the conjugate thereof due to the negative (f)^(x) -term. Consequently this wavefront converges to form a real image, but is located in space (by the 2ax term) at an angle 2<S from the axis of the virtual image. Any region of the real image may be brought into focus upon a screen without use of a lens. 2.2 Formation of Interference Fringes and Double Exposure Holography Vibration studies make,',1 use of holography as an interferometer. The interferometric process can be described as a comparison of two very nearly similar wavefronts. At a region in space where the two wavefronts interact, interference w i l l occur between a point on one wavefront and the same point on the other wavefront. A dark fringe w i l l occur when the two points-are out of phase by half a wavelength or odd multiples thereof. Similarly a bright fringe appears when the points are i i phase or differ by an even multiple of wavelengths. This 19 point to point correlation i s the underlying principle of fringe formation. Although interference effects occur between a l l points on one wavefront and a single point on the other, this i s of such a random nature that an averaging takes place resulting in no specific contribution to the overall fringe pattern. Double exposure holography results in two images being stored in the hologram. In the reconstruction process the wavefronts from these two images interact to form fringes that represent.the differences between the two wavefronts. The phase expression to be derived i s the basis of the formation of a l l these interference fringes, as i t defines when light reflected from a given point w i l l cause a light or dark fringe to appear, and as such, i t i s related to the displacement the images have undergone, between exposures. Consider a plat panel which w i l l be stati c a l l y deformed. Referring — . JT. • to observer illumination beam 20 to Fig. 2.3, the hologram records an image of the panel at state (a). The panel is-then deformed by some means to position (b), and the hologram is again exposed. Contained in this hologram are then two images, each representing the panel surface at a certain position. From each image there emanates corresponding wavefronts slightly out of phase with one another.' As a consequence, interference fringes form and are related to the phase differences which can further be related to the amount of deformation. In Fig. 2.3, point i on surface (a) is displaced normally to occupy point j on (b). With collimated illumination the path length change in distance travelled to the observer, due to the deformation, is calculated as D. 9^, are the incident and reflected angles respectively of the illumination beam. For a fl a t surface these angles remain constant at a l l points. The phase change between points i and j on the two wavefronts is then expressed as ty. The displacement i s determined from the expression I r = I Q[1 + cosGlO] (2.21) where 'I is the intensity of the reconstructed image modulated by the fringe function [1 + c o s ( i j j ) ] , [9]. This results in I the intensity of ftie actual image viewed including the interference fringes. Whenever 1 = 0 there i s a dark fringe. This occurs when r ° cos{ty) = -1 or whenever 21 ty = (T1) d (cose. + cos6 0) = .(2N-1)TT .. N = 1,2,3 etc. (2.22) where N is the order of a dark fringe. Then at a dark fringe the displacement is calculated as d = (^f1) (coseA cosej ( 2 , 2 3 ) Though the phase change ty has been derived for collimated illumination, i t is equally valid for divergent beams. In the experi-mental work of this thesis, the radius of divergence has been typically 30 to 40 inches, while the displacements encountered are generally of 150 micro-inches. Thus any angle change in the illumination vector in going from point i to j in Fig. 2.3 is extremely small and may be neglected. 2.3 Rigid Body Step Displacements The following theories of r i g i d body step displacements have been derived for both collimated and divergent illumination. As the holographic set ups used in the work of this thesis have employed divergent illuminating beams, emphasis has been put on this aspect of the theory. Four separate motions are presented, in each case the object or specimen referred to i s f l a t surfaced. Referring to Fig. 2.4, the four motions are: in-plane rotation about the z-axis and in-plane lateral translation along the x-axis; out-of-plane rotation 22 about the y-axis and out-of-plane longitudinal translation along the z-axis. There are two degrees of motion not accounted for here, these being translation along the y-axis and rotation about the x-axis. The x-axis rotation has the same fringe properties as rotation about the y-axis, except that the fringes are horizontal lines instead of vertical. Considering translation along the y-axis, i t is seen that the y-axis is perpendicular to the x-z plane wherein' . l i e the illumination and observation vectors. ' Referring bri e f l y to Fig. 2.3, 0^ and are the angles between the illumination and displacement vector, and the observation and displacement vector respectively. Thus in Fig. 2.4 then, 6^ and would be 90°, and in this case ty = 0, independent of the displacement magnitude. That i s , ho fringes w i l l appear for such a displacement examined with the beam geometry shown in Fig. 2.4. If the effects of such a motion were to be studied, the beam geometry or specimen orientation would have to be modified so that the y-axis would assume the relative direction of the x-axis in Fig. 2.4. For certain of.the motions the resultant fringes do not localize on or near the objects surface, [29] . In this case, knowledge of the position of the fringe localization plane is required before the displacement, can be determined [25]. The theories describe the fringes as occur in double exposure holography. However, they may be extended to vibration work as has been shown by Molin and Stetson in the study of the equivalence between sequential (double exposure) and simultaneous (time-averaged) recordings [35], ' The time-averaged method i s described in the following Section 2.4. "S8 r 1 =—r-— V 24 2.31 Rotation about y-axis This i s the simplest motion to analyze i n terms of fringe behaviour In F i g . 2.5 the surface OM^  rotates through and angle g such that moves to point M^. If g i s small and OM^  not large ( l i m i t s to be' defined) the path difference i s found as i n F i g . 2.3, D = d(cos0^ + co s 0 2 ) , where here d = M^M2.: The r e s u l t i n g fringes p a r a l l e l the ro t a t i o n axis, are equally spaced, and l o c a l i z e d on the object (Fig. 2.6) [25,33]. The amount of r o t a t i o n i s found by noting that for a fringe spacing of s, there i s a displacement of d = sg i n going from one dark fringe to an adjacent one [28]. The change of path length between fringes i s i n integer multiples of the l i g h t wavelength X. Fig. 2.6 Displacement of an elemental area in y-axis . rotation, a translation M, plus rotation' 25 Thus D = A = sg(cos9 1 + cos6 2) (2.31) and the rotation is calculated as * - ( * / s ) (cose, + cos6 2) ( 2 - 3 2 ) Considering an elemental area of which is a point, (Fig. 2.6), Vienot et al. 1[28] state that as OM, increases or 3 becomes greater, the displacement of the element is a result of a translation to M2 combined with a rotation 3. The translation effect on'the fringe pattern may be neglected for M,M2 < 10 microns. The fringes have been observed to be localized on the surface for both collimated illumination [33] and divergent illumination [25,28]. 2.32 Rotation about z-axis A study by Molin and Stetson [35], has found that with collimated illumination, the fringes are parallel, equispaced horizontal lines, which l i e on a vertical plane that passes through the object at the origin. The localization plane is so oriented as to be very nearly the mirror image of the illumination vector (Fig. 2;8). Aleksandrov* and Bonch-Bruevich [26], have noted that with divergent illumination the fringes have a slight bending accountable to the sphericity of the beam. That is they appear to be divergent from a point. For collimated illumination Tsujiuchi et a l . [33] predict the rotation about the z-axis 26 (defined as <j>) , to be calculated from the equation 2,1/2 = A/[<f> sine, ] (2.33) where x^, are the x, y axis components of the fringe spacing. Thus when the fringes are horizontal X Q = 0. The fringe spacing (i.e. the normal distance between fringes), is represented by the term 2 2 1/2 (x + y ) , and 0, is the incident angle of illumination, o J o • * 1 & 2.33 Translation along x-axis Haines and Hildebrand [25], have derived an elaborate theory for any combination of rotation and translation with divergent illumination. Considering a pure translation alone, the displacement i s calculated as where d = displacement in x direction Z q = distance from object to fringe plane = fringe spacing at Z Q The localization plane is behind the object and the fringes are equir spaced parallel lines, perpendicular to the translation vector (Fig. 2.9). It was found that the location of the fringe plane is independent of the displacement magnitude [35,28]. Molin and Stetson [35J, compared the fringe behaviour in (i) collimated, and ( i i ) partly (2.34) '•27 Fig. 2.7 -jt^p [Fringes, due to / rotation (3~ about L y-axis . -(a) (b) i i 0 = A s(cose,+ cos'©, Fig. 2.8 Fringes due to rotation (J) about z-axis (a) fringe plane H illumination vector Tor horizontal fringes '(J)=- y sin0. a 1 ong- x-ax i s'' -(a) fringe plane ^ •to); k H / d • = _ x.Z£ 2 . Pig. 2 e10 Fringes due .-to translation .••d" along z-axis .'."" (a) i Cb) - z H fringe plane -F z d = (f 7f 2m X note: Cross-hatched area in {a) represents a poorly definedx ; localization plane in front of the object. An arbitrary fla^ s0%0$ plane i s shown in (b) to show contour of resultant f r i n g e s f , " . ^ ^ ^ The mth ring has a.radius "r", and "a" i s distance between object and hologram,, r i s measured relative to object dimeWsi$n^'J 29 divergent (cylindrical) illumination. The fringes were localized at in f i n i t y for ( i ) , while^(ii) resulted in a,fringe plane located at a fi n i t e distance behind the object. For ( i i ) the location was a function of the divergence of the illumination, such that as the radius of divergence approached i n f i n i t y so did the location of the fringe plane. ft 2.34 Translation along z-axis Gates [37] and Tsujiuchi et a l . [33] found this motion produced concentric rings where the radius r of the mth ring is given by r = a /2mA/d * • • (2.35) a = hologram - object distance d = displacement. • Vienot et a l . [28] found the/'fringes to be visible i f the viewing aperture through the hologram was sufficiently reduced. "But the location plane was poorly defined and varied with the oVder- of any given ring. And the fringes were situated in front of the image when a divergent wave was incident upon the object. With collimated illumination Tsujiuchi [33] observed the fringes to be localized at- i n f i n i t y . Fig. 2.10 shows the localization region of the fringes where the shaded area represents the poorly defined localization plane. The central ring lies along an axis that passes through the centre of the viewing aperture, parallel to the displacement direction (the z-axis in this case). 30 2.35 Summary The fringe properties of each motion are sketched in Figs. 2.7 to 2.10. View (a) is the top view, (b) is a perspective view. They show the localization plane of the fringes in reference to the object surface and hologram, and the motions that have caused them. The equation used to calculate each motion is rewritten from the text. 2.4 Vibration Analysis by Time-Average Holography Time-average holography is a process whereby a specimen, continuously illuminated, is vibrated at a natural frequency while the hologram is simultaneously exposed by the object and reference beams. At a natural frequency an element undergoes sinusoidal oscillation. With this .motion i t can be seen that any specimen element w i l l spend a greater portion of time at or near either extreme of oscillation than at any region between. Consequently light from either, extreme w i l l have'the greatest contribution in the formation of interference fringes. In effect this is similar to a two-step double exposure process in that the two wavefronts that interfer are those reflected by the object at either extreme. However, due to the continuous illumination, the mathematics of the process becomes more complex. The time-average theory was f i r s t derived by Powell and Stetson in 1965, [4,6], The intensity distribution of the reconstructed holographic image is given as V - yy*'»2 C2.41) 31 where I Is the intensity of the static object which is modulated by the squared term of the zero order Bessel function. This latter term, > (the fringe function as mentioned in Section 2 . 2 ) , represents the interference fringes due to the specimen's oscillation. As found previously, the phase difference Whenever IJJ equals a zero of J q , a dark fringe w i l l appear. The local maximum value of the Bessel function squared is a bright fringe, and this value continuously decreases in magnitude as increases. Thus for large displacement values of "d", the contrast between dark and light fringes i s decreased. The amplitude at any dark fringe i s found by determining the order number of that fringe, (the f i r s t dark fringe is order 1, the second is order 2, etc.) which in turn represents that same zero order of J q . Substituting the value of this zero order for the phase difference"ty, with the appropriate values of 0,, 0 2, into equation 2.42 w i l l allow calculation of the displacement "d". \\> = (2Tr/X)d(cos0 + cose ) (2.42) d = (X/2TT) n (2.43) (cos61 + cos6 0) Q = zeros of the zero order Bessel function n n =.order no. of dark fringe. 32 2.5 Vibration Analysis by Stroboscopic Real-Time Holography The stroboscopic real-time method was^discussed by Arichbold and-Ennos in 1968 [7,8], A hologram is made of the static specimen then replaced after development to the exact position i t occupied during the recording. The specimen is then excited to vibrate at a natural frequency while the laser output is strobed at a time when the oscillating surface is at either of i t s extremes of motion. : In this case two wavefronts interfer representing the static object (holographic image) and the actual vibrating specimen. Interpretation of the resultant interference fringes however, is dependant upon the type of hologram used. The two types used are, (1) A negative or absorptive hologram. When developed normally, i t appears dark somewhat like a smoked glass. This is causedvby the very fine dark and light interference fringes forming the diffraction grating which diffracts the reconstruction beam into i t s several wavefronts (see Section 2.1). (2) A positive hologram, which is developed the same as (1), except afterwards i t i s bleached and appears clear with a slight bluish tint.-In the bleaching process [43] the developed silver crystals are removed from the emulsion, (these form the dark fringes of the grating). Furthermore, a hardening action takes place which causes a "pulling together" of the emulsion in regions of less exposure, (i.e. areas of light fringes). Consequently there is a higher concentration of the unexposed original silver halide crystals in these regions thereby increasing the thickness of the emulsion layer relative to other areas. 33 The result i s that a positive hologram modulates the reference beam by phase variations rather than by the density variation present with the dark and light: fringes in an absorptive hologram. The difference of interpretation between the interference fringes viewed through each hologram arises from a ir phase reversal which the image wavefront undergoes when reconstructed with a negative hologram. This occurs because an image wavefront i s compared to an actual wavefront reflected by the specimen. In methods such as the double^exposure or time-average, a l l the information is stored in the holographic image and no real-time comparison is made. The bleaching of the real-time hologram eliminates the IT phase reversal and the fringes may be interpreted as in the doublegexposure or time-average methods; that i s , a bright fringe would appear at a region of zero displacement in double'jexposure, time-average or positive real-time holograms, while i t would be represented by a dark fringe in a negative hologram. For a positive hologram the amplitude of vibration is determined in the same manner as a step displacement in Section 2.2. Rewriting equation 2.21, I R = I q [ 1 + cos(»] (2.21) and similarly from equation 2.23 d ( 2 ). (cos8 1. + -cose2). (2.23) where in this case "d" is the amplitude of vibration where the peak to 34 peak displacement i s twice the amplitude. N = 1,2,3 etc. as. earlier and i s the order number of the dark fringes. Ennos had discussed the ir phase reversal of a negative hologram [42] and Archibold and Ennos [12] have shown that the amplitude can be calculated from the expression T -r • 2,2nd cos0, 1 = 1 sin ( ) (2.51) r O A where "d" is the amplitude and the object illumination i s incident and reflected at the angleSO. Put in a form consistent with the previous notations of this thesis, the result' / is 6, = 0O = 0 for the above conditions. Or for 0,, variable I = 1 s i n 2 (-^ -[ cose, + cosej) (2.52) r o A 1 2 2 then a dark fringe appears whenever the argument of sin equals IT or an integer multiple th e r e°f• Therefore, NA d = -7 T " — ; — - e (2.53) (cose, + cos0 2) where here N = 0,1,2, etc. for a dark fringe, and at zero displacement N = 0. 2.6 Variation of Displacement Direction in Curved Surfaces In a vibrating f l a t panel i t is valid to assume that the displacement direction is normal to the surface at a l l regions of the 35 surface for small amplitudes (Fig. 2.11). Therefore assuming collimation for the incident and reflected rays, that i s , the illum i n a t i o n and observation vectors, 6^  and respectively remain constant at a l l points of the surface i n the equation for ty. ty = (| I L)d(cos6 1 + cos6 2) (2.42) However for a curved surface, the displacement d i r e c t i o n varies from • Fig. 2.11 Variation of 9, and 0z on a f l a t panel.'Disprac^menij;-: j, direction shown at centre point "o", 'and any point "p".•f'3^S35;"; ; Direction remains constant, therefore 0| , &z remain • constanV'T^i. at a l l points with assumption of parallel rays. • x :' ; 36 p o i n t t o p o i n t on the assumption a l l d i s p l a c e m e n t s a r e normal t o the s u r f a c e f o r s m a l l a m p l i t u d e s . Thus the c o r r e s p o n d i n g changes i n 0, and must be taken i n t o account i n e q u a t i o n 2.42. These v a r i a t i o n s w i l l be d e r i v e d i n the f o l l o w i n g s u b s e c t i o n s f o r a clamped p a n e l and c y l i n d e r . 2.61 F u l l y clamped c u r v e d p a n e l The specimen used i n t h i s p o r t i o n o f the e x p e r i m e n t a l work i s shown i n photo 3.4, page47 . The d i s p l a c e m e n t d i r e c t i o n i s assumed to be r a d i a l and thus normal to the s u r f a c e a t a l l p o i n t s . I n F i g . 2.12, a l l measurements are r e f e r r e d t o the p a n e l c e n t r e a t " 6 " . The a n g l e ' between t h e i l l u m i n a t i o n and o b s e r v a t i o n v e c t o r i s " 5 " , "p" i s any p o i n t on the p a n e l , and the change i n d i s p l a c e m e n t d i r e c t i o n from o t o p i s g i v e n by the a n g l e "y". Then.at any p o i n t p, 6, = ? - e 2 - Y (2.62) Q2 = y + e2 However, i f the o b s e r v a t i o n v e c t o r i s p a r a l l e l t o the d i s p l a c e m e n t d i r e c t i o n a t o ; ) £ 2 = 0.0 and 8 1 = C - Y (2.63) 02 = Y T h e r e f o r e when d i s p l a c e m e n t c a l c u l a t i o n s a r e made f o r a p o i n t p, the Fig. 2 ..12 Variation of displacement'direetion^g^ja.^cttrwe'dV;,,.. panel. Angle"V is positive measure^c^l^k^sie:;'^^; from o to p. Illumimation and obser^^i^o^yeQ'tpr'iS''' centred at o, then 0. and 0 2 deteEm>n;edva't;''p^y-;; assumption of collimated vectors. 38 corresponding values of 9^, 62 are substituted into the phase change \p, equation 2.42, which after substitution into the appropriate fringe function (i.e. time-average, real-time), allows calculation of the displacement magnitude. The angles £, E , , are determined from the holographic geometry, and y is found by determing the arc length op from the gradations across the panel (see Section 3.3, examples in Section 4.2). • •'. -2.62 Cylinder The displacements of a cylindrical surface vibrating in a shell mode are also predominantly radial. Though a tangential motion i s present at the node, the amplitude ratio of the tangential motion to the radial motion is very small and may be neglected on order of magnitude [21]. Thus the displacement direction in a vibrating cylinder also varies as the surface normal.' Variations in 6, and 0^ are determined in a manner very similar to the curved.panel,however, the reference points for the beams are slightly different. In a f l a t or curved panel the illumination beam was aimed at the centre of each specimen at point "o". The hologram was oriented such that the vector,^r5presentingthe displacement direction at o, (Fig. 2.12),passed-through the center of the hologram plate, perpendicular to i t (Fig. 2.4, page 2.3). In this case the observation vector l i e s / " ' • along the displacement vector at',or, consequently the observation and V illumination vectors are said to be.centred at o. However with the cylinder, the illumination and observation vectors are centred at different points. 39 The illumination vector in Fig. 2.13 i s centred at point i and the observation vector at point o. The displacement at any point p is to be determined. As before, z, is the angle between the illumination and observation vectors, and 0, and 0^  are calculated as per equation 2.63. The point o in Fig. 2.13 was chosen so that - 0.0°. Obtaining the arc lengths oi and op from gradations on the cylinder allows calculation of the angles t, and y respectively. Displacements at a point p-are determined as outlined for the curved'panel., The reasons for choosing different reference points on the panel and cylinder were for a matter of convenience that are outlined in the experimental results.' 40 3. EXPERIMENTAL APPARATUS AND PROCEDURE 3.1 Holographic Apparatus The holographic components consist of a vibration isolated table upon which are situated the specimen, the laser, and the optical components, such as mirrors^ and beam expanders used to direct the laser beam. Photo 3.1 shows the holographic arrangement as used to study the cylindrical specimen. Two holograms were employed to provide simultaneous viewing of a greater portion of the cylinder. Fig. 3.1 i s a schematic of the same arrangement. The vibration isolated table i s a 6' x 8' x 6" concrete slab supported by four c o i l springs to give a natural, frequency of 3 hz. Eppxied to the top,, and lying parallel to the short ends of the table are 1/4" steel, plates of 6" width. These plates provide a platform for the magnetic based optical components. The coherent light source was provided by a Spectra-Physics helium-neon gas laser, model 125 A, which had been tuned to provide 78 mw output at peak power. The laser emits a 2 mm dia. beam which i s made divergent by a convergent lens and pinhole arrangement (the beam expander) . The lens focuses the beam through the pinhole and'j the beam is in turn diffracted by the pinhole into several bright and dark rings. The central bright region, called the Airy's disc, i s the 41 Photo 3*1 Gteneral Holographic Arrangement for Vibration Studies of Cylinder f i l t e r e d p o r t i o n of the beam, wh i l e the higher order d i f f r a c t i o n r i n g s are produced by the noise components of the beam. The A i r y ' s d i s c i s used i n the i l l u n i n a t i o n and reference beams. The beam s p l i t t e r i s a v a r i a b l e r e f l e c t i v i t y mirror which transmits 0.7% to 90% of the i n c i d e n t beam. Besides d i v i d i n g the l a s e r beam i n two, the VBS i s adju s t a b l e to vary the i n t e n s i t y r a t i o of the reference beam to the d i f f u s e l y r e f l e c t e d object beam, t y p i c a l l y set at 5:1. A f u r t h e r method of i n t e n s i t y adjustment i s obtained by p l a c i n g a divergent lens a f t e r the beam expander to f u r t h e r expand the beam. The hologram h o l d e r , made by Jodon Engineering A s s o c i a t e s , pulse generator J acousto-optic modulator He-No laser cylinder \ BS20 Foton^c| sens'orXl o s c i l l o - C scope H = hologram BE = beam- expander -4 = observer M = mirror . ' L = lens, . , VBS = variable beam splitte r BS20 = nonvariable beam splitter (20% transmittance) Fig. 3.1 General Holographic-Arrangement for Vibration Studies of Cylinder 43 consists of a stainless steel frame which holds the hologram plate in place under spring loading. This frame is in turn held in a larger frame, referred to as the hologram holder, seated upon a magnetic base. The holder contains two micrometer heads intended to produce two perpendicular displacements in a plane parallel to the hologram. Such adjustments are necessary in real-time holography, as the developed hologram has to be repositioned exactly to eliminate or minimize residual fringes (Sections 2.5, 4.3). The residual fringes resulting from any repositioning errors could be corrected by adjustment of the micro-=heads\ However each micro-head was indirectly coupled to the hologram frame, in the holder, by a f l a t cantilevered n spring, which resulted in complex displacements of the frame whenever a displacement was introduced through the micrometer. Consequently this hologram holder was incapable of correcting a l l but the minor repositioning errors without producing further fringes due to the complex movements of the frame. Besides repositioning errors, residual fringes can result from emulsion shrinkage in the hologram drying process. To eliminate this, a liquid gate hologram holder can be used, which is constructed the same as the previous holder, except the hologram plate is surrounded by a water bath. The hologram is soaked in the bath in d i s t i l l e d water for half an hour before exposure and is immediately replaced after development. Thus there i s no emulsion shrinkage due to drying. 3.2 Vibration Excitation, Beam Strobing and Displacement Measurement The curved panel and cylinder described in the next section were excited acoustically by a Universal horn driver, model ID-60, rated at. 44 60 watts output continuous duty. The frequency response of the d r i ve r extended from 70 hz to 12 Khz. The s i g n a l to the horn dr i ve r was provided by a Hewlett Packard o s c i l l a t o r , model 4204A, through a Bogen A m p l i f i e r , model CHB100. The e l e c t ron i c s equipment i s sketched i n F i g . 3.1. Photo 3.5 shows the horn dr i ve r with the cy l i nde r specimen. In r e a l t ime-v ibrat ion a n a l y s i s , a method i s requi red whereby both the i l l um ina t i on and reference beams can be strobed i n synchron iza -t i on with the v i b r a t i ng ob jec t , ( t y p i c a l l y at e i t he r one of i t s extremes of o s c i l l a t i o n , that i s , at a maximum ampl i tude) . This i s done by s t rob ing the beam jus t a f t e r i t leaves the l a s e r . The s t rob ing was performed by a Zenith Acq'usto-Optic Modulator, model M40-R. I t operates i i such a manner as to de f l ec t the l ase r beam away from the pinhole of the beam expander, and thus i n e f f e c t ^ s w i t c h the beam o f f f o r a given dura t ion . A t y p i c a l r a t i o (duty cycle) between the on and o f f time i s 1:20. In opera t ion , the s t rob ing un i t i s t r iggered by a Hewlett Packard pulse generator model 214A, which i s i n turn dr iven by the 4204 A o s c i l l a t o r mentioned p rev ious l y . This pulse generator allows con t ro l of the pulse l o ca t i on (the point i n the specimen's o s c i l l a t i o n at which the beam i s switched on) , and pulse width (durat ion of time fo r which the beam i s on). The o s c i l l a t o r has a frequency s t a b i l i t y of + 0 . 0 1 % . Such s t a b i l i t y i s a requirement i n real-t ime holography f i r s t l y because of the narrow resonant v i b r a t i o n peaks (band width of 2 to 3 h z ) , and secondly because the 10 second exposures used to record the f r inge patterns on f i l m would show washed out f r inges i f the o s c i l l a t i o n was not strobed at the same point each t ime. 45 Vibration amplitudes were measured with a fibre optic Fotonic sensor model KD38-A. The sensor measures reflected light as an indication of displacement between the sensor probe and a surface. The use of the sensor is outlined in Appendix C. A Kinamatic micro translation table, model TT-102, was coupled with a rotary table, model RT-200. The three were stacked one atop the other on a heavy magnetic base support (Photo 3.2). This arrangement provided the out-of-plane rotation and also the lateral and longitudinal Photo 5.2 Translation and Rotary Tables on Magnetic Support Base 46 translations in the rig i d body studies, (Section 2.3). In-plane-rotation was obtained by setting the rotary table i n a vertical orientation, such that the rotation axis was horizontal. In calibrating the Fotonic sensor i t was found that the displacements provided by the translation table could be repeated to + 7 mv as read from the oscilloscope. Using the Fotonic sensor calibration this gave a bi-directional repeatability of the translation tables of approximately + 40 uiti; The rotary table had gradations of 5 sees of arc. 3.3 Specimens A l l specimens were thinly coated with a f l a t white enamel paint to provide a diffusely reflecting surface. The four r i g i d body motions of Section. 2.3 were studied using f l a t surfaced specimens. For the two translations and the out-of-plane rotation a 3" x 4" object was employed. It is shown atop the translation and rotary tables in photo 3.3. For in-plane rotation, the object measured 3" x 3" and was attached to the rotary table, with this table alone connected to a base such as to orient the rotation axis horizontally (rotation <j> in Fig. 2.4, p. 23 ). The particular arrangement of holographic components as employed in this study is shown and discussed in Section 4.1. A curved panel of 6" radius was examined and is shown in photo 3.4 and Fig. 3.2. The panel i t s e l f was of 0.053" thick aluminum while the frame and base were of mild steel. A l l measurements were taken at mid-height of the panel where a strip of metal was l e f t unpainted to provide a reflective surface for use of the Fotonic sensor. 47 Photo 5.5 Specimen i n p l a c e on T r a n s l a t i o n and Rotary Tables Photo 5 . 5 C y l i n d e r Specimen wit h Horn D r i v e r Photo 5 . U Curved Panel Specimen Fig, 3.2 Sketch of'Curved Panel X 13 75 15 t. 2 0 i 'wall t h i e k n e s ^ ^ ^ § < . R = outer r i n i ^ ^ t f c ' ' P = plug ;'. ...... . 5 ; ,£f B = baseC^ith'-Tegs.i^^' screwed to "P . U..^  Fig. 3»'3>f'Crossr?secti6n of"Fixed-Free Cylinder 49 Gradations were marked across this centre: strip in 0.050"£increments. Fringes located upon the panel were referenced to these gradations. The cylindrical specimen was 6" O.D. aluminum tubing of 0.058" wall thickness. A plug was snuggly fit t e d into one end of the cylinder then the ring afterwards was shrink-fitted to act as the clamped end. Photo 3.5 shows the painted specimen while Fig. 3.3 is a schematic cross-section of the unit. The bare strip i s vi s i b l e and was located at that particular height to l i e mid-way between the clamped end and the circumferential node visible in Photo 4.6c, p. 83. 3.4 Procedure A l l holograms pertaining to the rigi d body motions were taken with Kodak 649 F, 4" x 5", holographic photo-plates. Agfa Scientia 10E-75 plates were 55 times faster than the Kodak plates with typical exposures of approximately 21 sees for Kodak and 1/3 sec for Agfa. The exposure times were obtained in a t r i a l and error correlation between the m i l l i v o l t output of an available photo-diode and the light intensity required for proper exposure. The holograms were developed as recommended by each manufacturer. In the case of real-time holography the amount of light available for inage reconstruction and specimen illumination was drastically reduced 'by the fact that the beams were strobed. With a 1:20 duty cycle the available light"was reduced by a factor of 1/20. Additional light could be made available for image reconstruction by increasing the reconstruction efficiency of the hologram by bleaching i t [43], using the cupric halide bleach described in Reference [47]. A l l holograms 50 employed on the -vibrating cylinder were bleached unless noted otherwise in the experimental results. No difference was observed in^.the quality of the reconstructed image obtained with a bleached or unbleached hologram, except that the bleached hologram provided the brighter image. The liquid gate hologram holder described in Section 3.1, increased the speed of the photo-plate emulsions (the holograms) by a factor of 3. Thus the typical exposure times had to be reduced (to .'1/3. When a positive (bleached) hologram was used, the reconstructed image appeared very blurred, though brightness of the image was on a par with previous reconstructions using the "dry" holder. No d i f f i c u l t i e s appeared with a negative hologram. Attempts made at on-site development (where the hologram remained untouched in the liquid gate with the appropriate development chemicals being added and removed as required), showed the density of the residual fringes that appeared, to be very temperature sensitive. A decrease in room temperature of 4°F more than doubled the density. In addition, the residual fringes could not be reduced to a reasonable minimum of 1 or 2. The reconstructed image however was of good quality. Films used to record the interference fringes were Kodak Tri-X 400 ASA (exposed and developed at 1200 ASA), Polaroid 3600 ASA, and Polaroid P/N 55 ASA. This last film provided a negative together with the printed picture, but was very insensitive to the red laser light. Exposure times had to be increased by a factor of 7.0. The other films proved satisfactory at their recommended exposure index. In working with the curved panel and cylinder i t was found that photographing the interference fringes and subsequently determining 51 their location relative to the scale on either specimen was made easier by simultaneously illuminating the specimen with white light. Thus the photograph would show a superposition of the holographic image and the illuminated real object. It was d i f f i c u l t to distinguish the scale gradations at certain regions due to the speckle effect of the laser light. This method increased the contrast between the fringes and the gradations. The light source was a standard light bulb whose intensity was adjusted by a variac.' 52 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.1 Rigid Body Step Displacements The four motions examined are outlined in Section 2.3. The particular holographic layout employed for a l l motions is shown in Fig. 4.1 with the essential dimensions. Distance measurements given in Fig. 4.1 are accurate to +0.5" and the angles to + 0.5°. Because the holographic apparatus had been moved between certain tests then relocated, the geometry varied from test to test. From Fig.s 2.3, 2.4 and 4.1, i t is seen that 0, = 57.6°, and 0 2 = 0.0°. This applies to a l l calculations involved in the following subsections. Though the following motions are step displacements examined with double exposure holography, the findings can be extended to vibratory motion with respect to fringe contour and localization. This equivalence between sequential (double exposure), and simultaneous (time-average) recording has been verified by Molin and Stetson [35]. The similarity between stroboscopic real-time and double exposure holography has been described in Section 2.5. 4.11 Out-of-plane rotation about y-axis The interference fringes due to this motion are shown in Photos 4.1. • They were obtained in real-time with the liquid gate hologram described in Section 3.1. This was not stroboscopic real-time as no F i g . U.1 Holographic arrangement for r i g i d body-step displacement studies. 54 vibratory motion was involved, and therefore no strobing was necessary to "stop" the motion. Otherwise the displacement calculations are carried out in the same manner as in double exposure holography. The effect of residual fringes is discussed below.1 An example of the calculation of the rotation magnitude is given with reference to Photo 4.1b. The fringe period was found by dividing the specimen width by the number of fringes across the specimen at mid-height. The rotation g is calculated in radians from equation 2;32 sCcose, + cos02X, (2.32) A = 24.9 pin. s = 0.0758 in. (fringe period) cose,' = cos 57.6° = 0.535 cos0 2 .= cos 0.0° = 1.000 (24.9 x 10 6) _ -4 , (0.0758) (1.535) " X 1 U r a d -44.4 sec. of rotation The measured rotations were read from the vernier scale of the rotation table. The measured and theoretical results are presented in Table 4.1, where "+"*'denotes that the actual measured value can be estimated at approximately 1 to 2 sees, greater than shown. This was due to a sticking micrometer head which caused the set reading to be'over run. 55 In (real-time holography an exact r epos i t i on ing of the developed hologram i s r equ i red , otherwise r e s i dua l f r inges appear as an i n d i c a t i o n of the degree of mis-alignment between the ac tua l specimen and i t s holographic image (Sect ion 2.5) . However, vttie hologram may be adjusted i n i t s holder' so that the densi ty of the r e s i dua l f r inges can be reduced or e l iminated (Sect ion 3.1). Before the ro ta t ions of th i s sec t ion were app l ied to the specimen, the r e s idua l f r inges were adjusted t i l l i t appeared that a single^ dark f r inge covered approximately ha l f the specimen. The .rotations were then appl ied i n sequence with the resu l tant f r inges shown i n Photos 4 .1 . Pa ra l l ax was observed between the f r inges and specimen only fo r the 30 sec . r o t a t i on and th i s was not i ced inthe upper quarter of the specimen. The parallax c^ountecL to a f r inge s h i f t of approximately h a l f an order i n moving the .viewing po int from one edge of the 5" wide hologram to the other edge. No pa ra l l ax was observed fo r the higher r o t a t i on magnitudes. Tests of y-axis ro ta t i on with double exposure holography, previous to the l i q u i d gate work, showed no signs of pa ra l l ax at any time. This leads to the conc lus ion that the f r inges were l o c a l i z e d on or , avery near to the ob jec t ' s sur face . 4.12 In-plane ro t a t i on about z-axis The observed f r i nge patterns are shown i n Photos 4 .2 . No pa ra l l ax was not i ced between the object and the f r i n g e s . Thus, i t appeared the f r inges were l o c a l i z e d on or very near the ob j ec t ' s sur face . An example of the ro ta t i on c a l cu l a t i on i s made with reference to Photo 4.2a. The f r inge per iod was determined along the v e r t i c a l cen t re l ine Table 1+.1 Rotation about y-axis measured calculated fringe period 30 sees. 31.6 0.1061+ i n . l+0+ 1+1+.1+ 0.0758 50 + 55.1 0.0610 Table 1+.2 Rotation about z-axis p^asured calculated fringe period 30 sees. 2811 0.2175 i n . 60 59.3 0.103 120 111.5 0.051+8 21+0 226 0.027 57 58 Photo U.2a Interference fringes due to rotation about z-axis. Measured rotations are: (a) 30 sec (b^ 60 sec. Photo U.2b 59 by dividing the object height by the number of fringes on the object. Using equation 2.33, ( xo + y o ) 1 / 2 = X / [ < J ) s i n 0 l ] ( 2 , 3 3 ) where x =0.0 for horizontal fringes o ° y o = 0.2175 in. (vertical fringe period) sin = sin 57.6° = 0.844 Rearranging equation 2.33, the rotation § i s , y0Csine1) (24.9 x 10 6) , , -4 (0.2175)(0.844) 1.36 x 10 rad. (4.11) 28.1 sees. The measured and theoretical results are presented in Table 4.2. The bending of the fringes due to the divergence of the illumination beam [26], can be seen in Photos 4.2. Tsujiuchi et a l . [33], derived the location of the fringes to be on the surface of the object for collimated illumination. While Molin and Stetson [35] had derived, and also observed experimentally, the fringes to be on a plane of localization as shown in Fig. 2.8, p.27, (also with collimated illumination). The derivation of the fringe 60 localization theory by Tsujiuchi et a l . was for the particular motion studied (in this case rotation about the z-axis). The expression for the displacement magnitude was given i n - i t s simplest form as equation 2.33. On the other hand, Molin and Stetson had derived a general theory for a l l motions and reduced i t to be applicable to z-axis rotation. However, further reduction of their theory resulted in the same expression for rotation as equation 2.33. 4.13 In-plane translation along x-axis As discussed in Section 2.3, the fringes do not always localize on the surface of the object. In the case of translation along the x-axis, the fringe plane is behind the object and i t s location must be known relative to the object before any displacement calculation can be(mad'e(Fig. 2.9, p.28 ). The location of this plane was determined by use of a Nikkon FTN camera, with split-image focusing and a 50 mm lens set at i t s largest aperture of f 1.8. The camera was positioned at a convenient location behind the hologram (Fig. 4.2), and the lens /) adjusted un t i l the fringes appeared in focus. The position of the camera with respect to the. hologram was noted, (distance HC in Fig. 4.2), then the camera was set upon another table. A white card with black lines was placed before the camera and the card's location varied u n t i l i t also appeared in focus. The distance between the card and camera represented the distance from the fringe plane to the camera (i.e. distance FC). with HO and HC known, Z was determined, o With reference to Section 2.33 and Fig. 2.9, p.28 , i t is noted 61 that the fringe spacing D j>. in equation 2.34 must be determined at the fringe localization plane. Photos 4.3 show the fringes resulting from translation along the x-axis. The camera was focused on the fringe plane in each case, thus the object appeared out of focus. The spacing of the fringes at the appropriate fringe plane was determined by taking a picture of an object of known dimensions set at the distance FC, which was found to be different for each motion.' The spacing of the fringes at the fringe plane was then referenced to the dimensions of the object. This was done by comparing photographs a l l enlarged to the same size. The fringe spacings obtained i n this manner are given i n Table 4.3 with the corresponding distance ZQ.• An example of the displacement- calculation, i s made with reference to Photo 4. 3b;. The displacement "d" is then determined from D~ d = ^ (2.34) x Z = 77 in. o D = 20.4 in. x , _ (24.9)(77) _ d " (2.04) = 9 4 1 y l n ' The theoretical and experimental values are also shown in Table 4.3. The discrepancy between the measured and calculated values i s the largest at higher magnitudes, or for lower values of Z q. And from Section 2.3, the value of Z was predicted to remain constant for o a given divergence of the illumination beam. Fig. J4./2 Fringe plane location. FC = Z0 + HO + HC, is distance from fringe plane to camera. ' fringe plane 4 •f i : object hologram camera HO KHC-H Table U.3 Translation along x-axis. % difference i s of calculated value relative to the measured. measured calculated difference fringe period 1000 >tin. 11+00 2000 91+1 1075 1658 -6 •23 -17 2.0Vlnv,, 77 in, 1.305 57 1.053 70 Table 1+.1+ Translation along x-axis. A l l corrected values calculated wilth Z0 = 77 in. % difference i s of corrected value relative to measured. measured corrected difference 1 0 0 0 i n , 11+00 2000 9M W Q 1820 -6 +5 -9 t Photo U.3a Photo U.3c Interference fringes due to translation along x-exis. Measured translations are: (a) 500 micro-inches, (b) 1000, (c) 1U00, (d) 2000. Photo U.3b Photo U.5d 64 In attempting to explain these differences i t appeared that the manner of locating the fringe plane would contain the greatest room for error. At a typical distance of FC = 10 feet (corresponds to Z q = 73 in . ) , i t was found that the depth of f i e l d under normal room lighting conditions was approximately +1.0 foot. Therefore, an object placed anywhere from 9' to 11' from the camera appeared in focus. It was.further found.by Boone and Verbiest [31], that the displacements calculated by equation 2.34, (from Haines and Hildebrand [25], diminished in value as the fringe spacing was measured at a plane closer to the object (i.e., at a smaller value of Z D ) • From Table 4.3 i t is seen that the measured and calculated values have the best agreement for Z = 77 i n . In the other cases, the • o ' percent deviation of the calculated from the measured values increases as Z . becomes smaller. Then with the assumption that Z = 77 in. o o is the value most closely representative of the "true Z^", the displacements were correspondingly calculated. The results are shown in Table 4.4. < 4.14 Translation along z-axis The fringes due to this motion are shown, in Photos 4.4. In {each case the camera was focused on the object.' As predicted by the theory, portions of rings are visible. However since the centre of the ring system was not v i s i b l e , no calculation of the displacement magnitudes could be made from equation 2.35. Parallax was observed between the object and fringes,and considering that the fringes were indeed rings, the manner of parallax 65 observed had indicated that the plane of localization was behind the object.Cl££cusing upon the fringes with the Nikkon camera showed them to be approximately 6" behind the object. (The fringe-plane to camera distance in this case was 53", compared to FC = 10' in the previous Section 4.13. At this distance of 53", the depth of f i e l d was short enough to be able to distinguish between the object or Vfjringes>appearing in focus.) The findings here at f i r s t seem to contradict those of Vienot et a l . [28], (Section 2.31). However, a paper by Gates [37] shows photos of fringes due to translation along the' z-axis with the centre of the. ring system v i s i b l e . Then superimposed upon this was a lateral displacement along the x-axis. This caused the ring system to shift laterally in the direction of the x-axis to the l e f t . (No mention was made of the direction of x-axis translation, i.e. to the l e f t or right). Furthermore, Vienot et a l . have discussed the effect on the resultant fringe pattern due to the rotational and translational components of a complex motion (for example see Section 2.31). With these previous works i t was then considered that the motions indicated by the fringe patterns of Photos 4.4 are due to a translation along the z-axis, superimposed by a lateral x-axis translation. The effect of the x-axis translation was to laterally shift the centre of the ring system. It also/.appeared that the lateral displacement had the greater influence on the location of.the fringe plane, since i t was located behind rather than in front of the object as would have been the case i f the translation was purely along the z-axis. The lateral, component of translation was possibly due to the translation 66 Photo U.Ua Photo k.kc Interference fringes due to translation along z-axis. Specimen was displaced continuously "between pictures, change of fringe slant in (c) i s attributed to non-uniform displacing action of translation table. Measured translations (with Fotonic sensor) are: (a) 217 micro-inches, (fr) 533, (c) 990, (d) 15U0. Photo U.i+b Photo U,hd 67 .•Table U»5 . TranslationCalong z-axis measured translation .fringe period 533>(in. 990 151+0 2.53 i n . 1.96 •1 ;-ii5,. table not being able to provide a totally pure translation along a single axis. The fringe periods were determined with reference to the dimensions of the object and are given with the measured displacements in Table 4.5. 4.15 Discussion, of. results The deviation of the observed fringe patterns from the predicted results is very l i k e l y due to the i n a b i l i t y of the translation tables to provide a uni-axial displacement (e.g. the shifted ring system in Section 4.i4 and the slanted fringes in Section 4.13,.Photos 4.3). The decreasing severity of the slat of the fringes in Section 4.13, for increasing magnitudes of x-axis displacement, indicates the fringe properties due to this x-axis motion are becoming more influential. 68 Thus the fringes begin to approach parallel vertical lines as predicted for&a pure x-axis displacement.' The experimental work of this section has provided an understanding of the behaviour of fringe patterns due to pure r i g i d body motions, and also to a combination of such motions. In spite of the presence of an unwanted component of motion such as that which caused the slanting of the fringes i n Photos 4.3, the displacement calculations and measured values ."compared reasonably well after the corrections made in Table 4.4. The preceding should lead to an examination of the relative sensitivities of the fringe densities due to each motion. Though such a comparison should also indicate the corresponding effects each motion has on the position of the localization plane (e.g. Section 4.14), the quantitative determination of the magnitude of the motion that has caused a complex displacement with fringes not localized upon the object, is d i f f i c u l t to evaluate from a single hologram [37]. In concluding this section onrigid body studies a comparison is made between the fringe periods on the specimen for each motion (Table 4.6). The product of the fringe period and measured displacement is given as the index of comparison, such that for a given type of motion and beam geometry, this product would remain constant. For each type of motion, the lower the index'value the smaller the fringe period, and the more sensitive are the fringes to any change in displacement in that motion mode. The f i r s t three motions listed have fringe periods inversely proportional "to the displacement.magnitudes (consequently the product of displacement and fringe period is a constant), however, for the z-axis translation i t is the square of the fringe, period that 'j, Table It-.6 Comparison of Fringe Sensitivities motion measured displacement fringe period on object (fringe period)x (displacement) x-axis ^ transsC3 lation 500 x i n . 1000 1.66 i n . O.696 830 ^ i n f .696 11+00 0.528 739c 2000 0.373 71+7 y-axis rotation 30, sec i+o — w 0.1061+ i n . 0.0758 3*19 sec.-in. 3.03: 50 (363 >iin.)* 0.0610 3.05 (-21.5 >(in?) z-axis rotation 30 sec. 60 (i+36xin.)* 0.2175 i n . 0.103 6*53 sec.-in, 6;18 (1+5.0 ^ i n ? ) 120 0.05U8 6.58 21+0 0*027, 6.1+8 z-axis trans-lation 533 >Uin. 990 2.53 i n . 1.96 1350 ju.in? 19U0 151+0 1.1+5 2230, * calculated equivalent linear translation at 1.5 in. radius 70 varies inversely as thedisplacement. Because, of this squared.., -relationship the product of fringe, period and displacement magnitude does rot remain constant, but decreases for a decreasing displacement and conversely, increases for an increasing one. Comparison between the two translations shows that for magnitudes under 500 pinches, the z-axis translation increases in sensitivity and approaches that of the x-axis translation, while at higher magnitudes the z-axis translation becomes less sensitive. A comparison between the rotations shows the y-axis component approximately twice as sensitive as the z-axis component. • The significant difference .occurs between the rotations and translations. Making the comparison on the basis of ^ Linear displacements can be done as follows. The fringe periods are uniform across the centrelines for each rotation component (vertical centreline in Photos 4.2, horizontal in Photos 4.1). The rotation displacement of a point at each centreline can be represented by a linear translation which is the product of a rotation and radius arm. (The radius arm in each case \i s/1.5 inches). Thus for the y-axis this linear translation is 363 uin. at 50 sees, of rotation, and 436 uin. for the z-axis rotation of 60 sees. The corresponding index value is also shown. It i s seen that the rotary motions are 20 ,to 40 times more sensitive than the translatory motions for the holographic geometry used in this study. 1 To provide a comparison to panel vibrations, the out-of-plane rotation and translation have been calculated for a f l a t clamped membrane of 3" square dimension (Fig. 4.3). The maximum rotation at a node vibration in/.--'*-''^  fundamental. . mode (3, increases to ^  at a higher mode'; -••••P, I"'^^ma^i^^ut-b^-plane rotation; at|ran '-••;•{ ' •'. edge^pr no^>;-:>^:^*-;'B^ f>br A =/j1 OO^ in,.. in' a .3 . sq. membrane.;;;,. '•~T£f"'=" 21 *'6vs.ec8.£>2. ~;|+3'.-2.';/secs.:' V ' Fig; U.3- ^ Comparison of out-of-plane rotation and", .linear translation in a, clamped vibrating ' V .-\ vmembrane>;^irosi^sec^ion i s "ahp^^;^;Kf:;^' 72 i s proportional to the maximum vibration amplitude. For vibration i n the fundamental mode and assuming a typical maximum amplitude of 100;.''yin. s..the maximum rotation at the nodes (edges) is 21.6 sees. Asythe mode number increasesj the slopes and thus the rotations at the nodes become proportionately larger for a fixed amplitude of vibration. 4.2 Time-Average Study 'of Curved Panel The experimental work of this section was performed by observing the panel from several views in sequence, and from two views simultaneously. Fig. 4.4 shows the holographic geometry for the two simultaneous views with the essential dimensions shown. Using sequential viewing, the location of the fringes on the panel were made in reference to the scale at mid-height (Photo 3.4, p. 4-7 ) , by viewing the reconstructed image of the panel and pin-pointing the position of each fringe by eye. This method was employed because the gradations on the panel were d i f f i c u l t to distinguish in any photograph taken of the reconstructed image, (In work with the simultaneous views this was overcome). In an attempt to obtain increased accuracy i n the method of locating fringes a microdensitometer was used. This machine measured the light intensity transmitted through a film negative by passing a beam of light through the negative and referencing the intensity of the transmitted beam to the incident beam.- A picture was taken of the reconstructed image and the positioning of the fringes were referenced to the edges of the panel. (The use of the micro-densitometer is described in Appendix A). <The.calculated displacements•", obtained with data from the microdensitometer agreed very well with the perious results by eye. These are plotted in Fig. 4.5. BS20 "5 ~ ~ curved panel 27.8 q -: OHL =.'.36'/;.... 21 ,k° O H R =;3i+.5" / B%HLQ /BE^HftO / M ^ H R - Z 65.U°" ;: BEiM,0.= : U3.1". /M,OHL. =18.7 ° .^ . V. • , ; • v»5, »v The surface.normal at 0 i s along the line-.OH[_; / 1 Prom f i g . :2.12 page 3 7 for" Et\i"^z = 0.0 £, = 18.7 =18.7 ), for. H R (£ 8;,;= 18.7 , ^ = 65.38-)i = 1+6.7 . j Fig. k.»h- Holographic, arrangement' for .curved panel,, 'j .., vibration studies. v-.:->,-.r; •' _ _ _ i _/ "•. - • • •"'' y ' - • . ' -• / . . ' ! * denotes amplitudes calculated from data of "fringe location" f. by microdensitometer. Amplitudes were determined with paiSel at the different orientations shown iri'.fig.l+.6. The " l e f t " and : "right" of LOCATION refer to l e f t ,and right side of panel"as viewed through hologram'.;".LOCATION]=;;0 i s center" of panel, with , edges, at LOCATION = 3il6^ in^onVreither^siddvbf 0. > '« ' :y • . j 20 .0 Lege* ld^^^-^Vr!^ • ;.o--^|vyO^ .^ ; : ; •lA^ ;cp;=:1: '6.;;*:;;: • Peptonic ;sens v^; .< • • ^ •• or; • -:_vn ;: • % • o>». 'd-'. ,2.-:. If": * * of • . \ Yd 3.Left 2 . : .1 . 0 • / v • ; :- 2 - r i g h t 3 «ffflicg;:^ff5s:^ Curved-P^n^lteomp^^ri^/R^sults ^^ v'-^ l-V:^ '^ -^ ^^ .t^ *-~**3F,ic>^ in^ :je^ i/^ in. by eye'"' and by^MlcrpdienBitomete^v 75 observation t vector L B E o H L = '5)#'« ••Fig. U.6' 'Panel-'-'p^fentatloltis' as used in. fig.. i+ • 5.> Views were varied''in f i g . k*5 by changing angle;^. The, hp.ipgraphic set-up i s similar ' to f ig. ; l+.U; exciiepjt&W used and mirror -M, was' left'out of the reference beain>;y ;-' While only single views were used in sequence to obtain the data of Fig. 4 .5 , the holographic set-up employed was very similar to that in Fig. 4 .4 , except that only hologram H was used, and the orientation of the panel varied with respect to the reference beam and observat ion direction.' The data of Fig. 4.5 corresponds to the panel orientations shown in Fig. 4 .6. The calculations have been compared to values measured by the Fotonic sensor at. several points. In Fig. 76 4.5, * denotes that these displacements were calculated from "fringe locating data" obtained with the microdensitometer, while the other displacements were the results of "fringe locating by eye". For a l l the data of this plot, the panel was vibrating at i t s lowest natural frequency of 1791 Hz. Fig. 4.5 has shown that the method of fringe location by eye had proved quite satisfactory in comparison with the microdensitometer results. The analysis of the results from the simultaneous views was performed by obtaining the fringe positions from photographs of the reconstructed image, taken while the actual panel was illuminated with white light. Thus there was a superposition on the photographs, of the holographic image of the panel with the interference fringes, and the real panel. The increments of the scale gradations were made finer than were used with the sequential views to allow more exact fringe positioning. The fringes as obtained for both views of this method are shown in Photos 4.5. The fringe positions were located by reference to the scale across the centre of the panel. Fig. 4.7 is a plot of the vibration displacements as calculated using the fringe ' patterns ofthe previous photos. An example of the calculations involved i s given below. In Photo 4.5a (the l e f t view, from hologram H^), there is a dark fringe of order 4 at location 0.72 of the scale on the right side. From Fig. 4.4 and with reference to Section 2.61, t, = 18.73°. Y is calculated from the arc length op where o represents 0.0 on the panel scale and p is -0.72 (clockwise from o to p is positive). Thus ry = op where r = 6.05" i s the radius to the panel surf ace, .". y = -6.82°. Calculated amplitudes from H L and H R are.,compared,, to amplitudes measured by Eotonic/sensor. 'The ."left" and "right.":.?of.LOCATION refer to le f t , and right side of the panel as viewed through hologram. LOCATION = 0 i s centre of the panei, with panel edges at LOCATION = 3.16 i n . on either side of 0. AO •rl;' 'mi . 1 1 +4 1 / \ A i • si \ \ iliOCXTrpN; (ih..)l Legend ; i A r i glit .viewy •fi^left^view, ;HL;: VQ . Fotonic _s ens or / 1 3 l e f t 0 2 r i g h t rFlg.^lU7- ;'-,Pl^'t;^df.vAmplitude8 from Time-Average ; Fringes of Curved Panel Vibration 79 From equation 2.62 ) 2 = Y =' ~6.82c >1 = ? " Y = 18.73° + 6.82° = 25.55c These values are then substituted into equation 2.43 to calculate the displacement at p = -0.72. •'\ ••••Sln d = (\/2ir) 2 (2.43) •\ . (cose^^ + cose 2) • 'h = 4, fringe order Q, = 11.79, 4th zero of J 4 o cos0 1 = .902 cos9 2 = .993 , ,24.9N (11.79) ' 0. n • d = . ^  T1T895T " 2 4 * 7 V i n ' thus the displacement d at p is 24.7 uin. This operation has been carried out for every dark fringe on the panel for both views, and the results are compared with the measured values in Fig. 4.7. The reader i s reminded at this point that a l l the theory derived and Implemented so far, has been on the assumption of collimation for both the illumination and observation vectors. However, i t can be.seen that observation of .the holographic image by eye, takes place at some point in space, and that the illumination beam is also divergent from a point at the beam expander. Consequently a difference occurs between the actual values of displacement and those calculated 80 assuming collimation. This difference i s due to a relative change in the angles 6^  and as the point of interest i s moved across the panel surface (this is in addition to that resulting from a variation in the displacement direction, Vwhich has already been taken into account). The amount of this difference has been determined in Appendix B for a f l a t surface, the curved panel of this section and the cylinder of the following section. The results of Appendix B show that the differences that occur for the curved panel are on the order of 3% to 5% and may be neglected (Fig. B.2, p.120 ). 4.3 Time-Average and Stroboscopic Realr-Time Study of a Fixed-Free Cylinder The cylinder of this section was studied as a continuation of the curved panel work, by going to a surface of greater curvature (i.e. a smaller radius). The cylinder was examined by both time-average and stroboscopic real-time holography. Because the time-average method stores a l l fringe information in the holographic image, there are no residual fringes due to repositioning errors that occur i n any real-time method (Sections 2.4, 2.5, 3.1). Consequently any errors in displacement calculations due to these fringes are eliminated and thus the time-average i s the more reliable of thetwo methods. The stroboscopic real-time method, however, has the advantage of requiring only one hologram (taken of the specimen at equilibrium) with which the whole frequency range of an oscillating specimen can be examined, while the time-average process requires one hologram for every natural frequency. As a result, the two methods were employed in the analysis of the cylinder both to reveal d i f f i c u l t i e s inherent in determining the vibration 81 amplitudes, and problems involved with the real-time method. The arrangement of the holographic apparatus is shown in Fig. 4.8. Two simultaneous views were used with each method (as with the'^ y panel), but the illumination beam was incident at an angle mid-way between the observation directions. This set-up was employed so that as large a section of the cylinder as possible could be observed, at the same time providing a region of overlap between the two views. The fringe patterns obtained with time-average holography are shown in Photos 4.6. • The cylinder was vibrating at a natural frequency of 1086 Hz with 6 longitudinal nodes and one circumferential node. The scale was placed so as to be mid-way-between the fixed end and the circumferential node. The displacements as calculated from each hologram (left view.H , right view H D), are plotted with the measured values,in Fig. 4.9. • An example of the displacement calculations is given below. In Photo 4.6b at location 8.07" on the scale in the right view, . there i s a dark fringe of order 6. In reference to Fig. 2.13,' p.39 the observation vector is centred at point o, which corresponds to 9.38"'on the cylinder scale, and the illumination vector is centred at i =8.18".' .'. op = 9.37 - 8.07 = 1.30" oi = 9.37 - 8.18 =1.19" since ry = op rt, = oi (where r = 3.0") - time-average • strop."real-time • :.;.:/HL.6M.4- ,..22.5° ' 25.0^;^'•>_'•;•'• . . /HRCM = - 23.0 ,20/0:... ' ": ,' HLc..= 36.o'!,v; 3U-5° : • • • v . ' H Rc = / 31 * 5" t " r;"" ' 31..:0 , , :" r '; ;"v- > ' ' v '. | BEMC = . *+3.p": • " U3.0" i Fig. U.8 Holographic arrangement" for•'c^ll'nWr''TY!itoration,Vl I studies. A portion of f i g . 3.1 i s shown^with:^| • • dimensions used with time-average':' arid-.,':'.^-v/'Sf • : stroboscopic real-time -methods. :.-„. ;v ; '' ;^j 83 Photo U . 6 a l e f t view, H L Photo l u 6b r i g h t view, H R Time-Average F r i n g e s of C y l i n d e r Photo U.6c Amplitudes of f i g . k>9 determined from photos (a) and ( b ) . Photo (c) i s a d i f f e r e n t example of the r i g h t view showing the f u l l specimen. .50 20 !'" 0' Calculated-amplitudes from H L and Hp/ are compared to amplitudes measured by Potoriic sensor. On a b s c i s s a .are marked LOCATION'S where i l l u m i n a t i o n and observation v e c t o r s were centred. Legend \v 3*: ,.;! A r i g h t view, • l e f t view., ;:HLv ;D" Potbnic' sensor ! ML 8 10 IV "i n • -• ti •H <D p.. +> •H r-l ft •3. • V \ D 1 i \\ i • / ' J t _ $ATlfoN.,(lri ti \ > \ J 12' oo 85 find = 24.8° Y ? = 22.7° and from equation 2.63 8 1 = Y = 24.8° ; e2 = c - Y = -2.1° Then as for the panel, the displacement at p = 8.07" is found from equation 2.43, d - (cose,"" costfT) < 2-«> n = 6, fringe order. fi, = 18.07, 6th order of J 6 ' o . , (24.9) (18.07)- _ ' . d ' T ^ O - d.907) . r 3 7 ' 6 . y i n ' This operation was performed for each dark fringe of each view and the results are plotted in Fig. 4.9. Analysis of the cylinder with stroboscopic real-time holography resulted in the fringes shown in Photos 4.7, where a negative hologram was used. Because of the reduced sensitivity of the real-time method, a larger vibration amplitude was required to provide several fringe orders^ such that a sufficient number of points could be plotted to give a reasonable indication of the amplitude distribution. The plotted results aregiven in Fig. 4.10, where the calculations were the same as 86 Stroboscopic Real-Time Fringes of Cylinder Photo U.7c Amplitudes of f i g . U . 1 0 determined from photos (a) and (b). Photo (c) i s a different example of the right view showing the f u l l specimen. L5J. Calculated;^amplitudes T from HL andv.H^lirjs-' "0 compared to amplitudes measured by "Fo tonicf;, sensor.^i>nr abscissa >^  are m a r k e d LOCATION1 js J where iilumination CX5**: and observation vectors i were centred... . 88 for time-average, except that equation 2.53, p.34- > was used in place of equation 2.43 p. 31 . The presence of residual fringes requires a discussion of their interpretation, which is presented in Section 4.32. The plot of Fig. 4.10. displays a pattern very similar to the time-average results of Fig. 4.9. An analysis of these plots i s presented in the following section. 4.31 Discussion of plotted results Referring to Fig. 4.9, the pattern of the plotted data can be described with two significant observations. F i r s t l y , as the viewing direction is shifted from the l e f t to the right.hologram, the fringes shift from the right to the l e f t . Secondly, the calculated plots tend to spread out further than the measured values at either end of the graph (indicating that the theory predicts the anti-node of vibration at the l e f t side, and the one beginning on the right, to be "further around" the cylinder or spread out more than indicated by the measured data). The same shift in the calculated curves is seen in Fig. 4.10, however, the relation between the measured and calculated results in the real-time method are yet subject to interpretation as outlined in the next section. As for the curved panel, a l l displacements calculated in Figs. 4.9 and 4.10 have been based on the assumption of collimation. The differences due to divergence that should be taken into account have been determined in Appendix B. And although the differences are more significant than found for the curved panel, they do not account for the spread of data betweeen the calculated and measured displacements of Figs. 4.9 and 4.10. 89 The author has proposed the following explanation of the behaviour of the calculated displacements. With the understanding that interference fringes do not ne c e s s a r i l y l o c a l i z e upon the object's surface (Section 4.1), i t i s hypothesized that i n the case of the cylinder, the fringes have become l o c a l i z e d upon some c y l i n d r i c a l surface of s l i g h t l y greater radius than the cylinder i t s e l f (Fig. 4.11). i Fig. Proposed fringe localization plane for- ' cylinder vibrations. 90 Assuming collimation, a point b on the fringe plane i s examined. Seen from the l e f t , b appears located at b^ on\the; cylinder, while from the right i t appears at b . Consequently, shifting the view from l e f t to right shifts the apparent location of b from right to l e f t . Consider now a view from a single hologram, with the further assumption that a point b on the fringe plane i s related radially to a corresponding point b Q on the cylinder. Then b w i l l appear located at b R, and the further around the fringe plane, point b moves, (an increasing angle J) the greater i s the distance between b and b . Thus i n the ' R o calculations, the displacement Information contained at b was attributed to the location bR- instead ofthe assumed proper point b Q. As a result the calculated displacements appear "further around" the cylinder than f,_/} the measured ..values. The preceding appears to have accounted for the behaviour of the calculated displacements by the use of an idealized cylindrical shape as a fringe localization plane. From the study.of rig i d body motions i t is understood that because the fringes are located on other than the object's surface, the position of the fringe localization plane in space i s related in someway to the motion causing the fringes. This relation i s relatively simple when pure motions are involved, such as with rig i d body displacements, where unique localization planes are the result. A cylindrical surface for a localization plane can be considered of a relatively simple nature (and thus would possibly lend itself to a straight forward correlation between i t s fringes and the displacements that have caused them). However at this point the 91 shape of the localization plane can only be speculated, although the results indicate i t to be in front of the cylinder. Since no parallax was observed by varying the viewing direction within the limits of the hologram, the distance of the fringe plane from the surface of the cylinder would be small, on the order of a quarter of an inch or less. To the best of the author's knowledge, these previous 1 findings are the f i r s t to indicate the possibility that the time-average fringes due !to"/^]indef>vibrations .are not localized upon the surface of the\cylinde'r. 4.32 Discussion of^Str^oscopjx/Rgj^l-Time Results The d i f f i c u l t i e s in this method of holography arise from inexact repositioning of the developed hologram. As a consequence residual interference fringes appear which are an indication of the repositioning displacement error (Section 2.5). Ideally a l l traces of such fringes should be eliminated, and provisions for such attempts are available with the adjustable hologram holder (Section 3.1). In practice however, these fringes can at best be minimized unti l there are only one or two covering the object (Photos 4.8). An interpretation of the fringe patterns of Photos 4.7 i s described below. TheCresTdual fringes photographed before the cylinder was vibrated, are shown for each view in Photos 4.9. The fringes observed on the cylinder during vibration are shown in Photos. 4.7. Since a negative hologram was-used, a region of zero displacement was. denoted by a Photo U . 8 a Residual fringes, before and after being minimized, (a) shows the i n i t i a l residual fringes appearing immediately after repositioning the developed hologram. The same hologram has been adjusted to minimize the residual fringes to state (b). (b) i s repeated as photo k*9a. Photo U . 8 b 93 Photo U.9a l e f t view, HL ?A°.t9 M>9h right view, HR Photo J4-.9c Residual fringes appearing before the fringes of photos U.7 respectively. 94 zero order dark fringe, with the higher orders having increased integer values. The f i r s t bright fringe was a 1/2 order fringe with the higher orders increasing in odd integer multiples of 1/2. Whenever a system of fringes were examined, they were.done so by grouping them into a closed fringe system where the lowest order fringe that closed on i t s e l f (be i t a dark or bright fringe), defined a system. This may be better seen by reference to Fig. 4.12 which contains sketches of the fringes in Photos 4.7. The sketches' have been simplified by showing a cross-hatched area representing a closed system of fringes where the fringe outlining the system i s a dark one: In Fig. 4.12a, "a" is a bright fringe which is also the lowest order closed fringe of system (1). The same applies to "b" of system (2), except that "b" is not quite closed." The node between systems (1) and (2) appears to be.occupied by a dark fringe, which.if assumed to be a Oth order fringe, would represent zero displacement. As a result, this fringe was so labeled and the adjacent dark fringes became 1st order and so on. The displacement associated with each dark fringe was then calculated and plotted in Fig. 4.10. In Fig. 4.12b, a very similar pattern as the preceding occurs. The bright fringe "c" is very nearly a closed fringe, however, the same can be said for the dark fringe between "b" and "c". Then this dark fringe can be defined as the outlining fringe of system (2). The bright fringe "a" i s not closed, consequently the dark fringe outlining the cross-hatched area defines system (1). There then remains two bright fringes "a" and "b", and a dark fringe between, none of which \ j Sketch of fringes shown in photos | 7a and l+.7b respectively, (page 86). j 96 are closed. In this respect then, the central dark fringe has been defined to be the Oth order with the adjacent fringes of appropriately increased order. The resultant displacement calculations are.?1 ; plotted in Fig. 4.10. The residual fringes in Figs. 4.12 are those that do not form a closed loop system (compare to time-average Photos 4.6, where there are no residual fringes). They are related in some manner to the residual fringes of Photos 4.9. If the type of motion that has caused them is.known (i.e. the repositioning misalignment), then the displacement magnitude of the misalignment can be taken into account in the displacements calculated from the fringes of Photos 4.7. The repositioning misalignment is a r i g i d body displacement that has occurred between the holographic image and the real object. But i t is only for a pure rig i d body motion that a fringe pattern arises with a simple relation between the pattern and the displacement magnitude [37] (Section 4.1). The residual fringes of Photos 4.9. take the form Of oval rings. The motion most'closely associated with such a fringe pattern is z-axis displacement (where the expected fringe contours are concentric rings). It i s not unreasonable then to assume that the repositioning misalignment has been, for the most part, a z-axis displacement,' accompanied by a further component of motion causing the concentric rings to become oval. The result i s a complex displacement where the determination of i t s magnitude from the observed fringe pattern would be very d i f f i c u l t . However, there is a method that may provide an adequate evaluation of the effect of the residual fringes on the f i n a l fringe pattern, and consequently a more accurate calculation of the 97 actual'vibration amplitude. This .can be performed by strobing the laser to coincide with f i r s t one, then the other extreme of oscillation of the surface. Then i f the strobing coincides with the extreme that is farther from' the misaligned holographici'image than the equilibrium surface of the cylinder, there w i l l be an addition of the fringe patterns. F i r s t l y due, to the residual fringes indicating the misalignment, and secondly the fringes arising from actual deformation of the surface from the equilibrium position.. On the other hand, i f the vibrating surface is strobed on the same side of the equilibrium surface as the holographic image, the residual fringes are ^sub^tracted from the deformation fringes. As a result, an overall increase in the fringe density is expected for the former case and a decrease for the latter.. However, this operation was-not performed due to practical limitations. When the strobing was adjusted to occur at the opposite phase used to obtain Photos 4.7, the v i s i b i l i t y of the fringes was greatly reduced, such that no evaluation could be made. The preceding analysis of residual fringes was based onvthe assumption that they were a l l due to repositioning errors. However, such fringes also result from emulsion shrinkage in the drying process of the hologram (Section 3.1). In this case no amount of adjustment of the hologram holder w i l l eliminate them. The following discussion is in reference to the plotted results of Fig. 4.10. Here, the calculated data does not display the same, "spread" with respect to the measured values (at either edge of the 98 graph) as occurred i n Fig. 4.9. This can be explained with reference to the r i g i d body motions, especially Section 4.14. It was shown here that when a displacement is compsed of several components of motion, . each notion has' an effect on the location of the fringe localization plane. The-repositioning error results in the holographi image being rigidly displaced with respect to the reai object. This i n i t i a l displacement then influences the position of the localization plane that was proposed'in Section 4.31. For this reason no definite statement can be made relating.the calculated and measured values of Fig. 4.10 as has led to the proposal in Section 4.31 for Fig. 4.9. As a further point, the fringe patterns of Photos 4.7, being influenced by the residual fringes, were interpreted in a similar manner. Consequently, the two calculated curves have displayed a behaviour relative to one another, that can be considered independent of the effect of the residual fringes. Thus Fig. 4.10 would appear to reinforce the findings of Fig. 4.9, that i s , as the view is shifted from l e f t to right, the curves shift in the opposite direction 99 5. SUMMARY AND CONCLUSIONS 5.1 Summary The purpose o f t h i s t h e s i s has been t o i n v e s t i g a t e the e x t e n s i o n of h o l o g r a p h i c i n t e r f e r o m e t r y t o the v i b r a t i o n a n a l y s i s o f cu r v e d s u r f a c e s . The s t u d y was performed w i t h the aim of o b t a i n i n g a l l n e c e s s a r y i n f o r m a t i o n f o r d i s p l a c e m e n t c a l c u l a t i o n s from the i n t e r f e r e n c e f r i n g e p a t t e r n s of a s i n g l e hologram. The specimens s t u d i e d were a f u l l y clamped p a n e l c u r v e d i n , o n e d i r e c t i o n , and a f i x e d -f r e e c i r c u l a r c y l i n d e r . Q u a n t i t a t i v e d e t e r m i n a t i o n of the v i b r a t i o n a m p l i t udes was attempted, whereas p r e v i o u s s t u d i e s o f c u r v e d s u r f a c e s have been l i m i t e d t o q u a l i t a t i v e e v a l u a t i o n . The s e v e r a l components of motion i n v o l v e d i n the v i b r a t i o n s o f a c u r v e d s u r f a c e ( c y l i n d e r ) prompted an i n i t i a l e x a m i n a t i o n o f the f r i n g e b e h a v i o u r due t o each of the i n d i v i d u a l m o t i o n components. A f l a t s u r f a c e d specimen was examined by double exposure h o l o g r a p h y undergoing s t e p d i s p l a c e m e n t s . F o u r pure motions were s t u d i e d ; r o t a t i o n about and t r a n s l a t i o n a l o n g an a x i s l y i n g on the s u r f a c e o f the specimen, r o t a t i o n about and t r a n s l a t i o n a l o n g an a x i s n ormal to the surface o f the specimen. The stu d y was c o n c l u d e d by a l i s t i n g o f 100 the relative sensitivities of the interference fringes (with respect to their fringe periods) due to each motion. It was found that the fringes due to the two rotations were by far the more sensitive to any change in displacement magnitude. Time-average studies were made of the fu l l y clamped curved panel. For a portion of the work a microdensitometer was used to determine the exact location of the fringes ontthe panel from a film negative. However, the results showed that fringe location by eye (and later from enlarged photographs), could provide equivalent accuracy for the displacement calculations without use of such a device. Thejaniplitudelcalculations' f or the curved panel assumed that the displacement direction at a l l points was normal to the surface. However, due to the curvature the surface normal varied in direction < across the panel. This was accounted for in the displacement calculations as a variation in the incident and reflected angles 8^ and 82' The fixed-free cylinder was examined both by time-average and stroboscopic real-time holography. The variation of displacement direction across the surface was similarly accounted for fas with the curved panel. The time-average results indicated that the interference fringes were not localized upon the cylinder's surface, while they have beeri~fbuifd located on the- surfacet'for the•xufve^pane'lli^d^previojas investigations concerning f l a t panel vibration?[18]. The behavior of'~the data was explained.with the proposal that the interference fringes lay.-.on a cylindrical surface having a radius slightly greater than, that of the cylinder. 101 The stroboscopic real-time results tended to verify the time-average findings. However, this was subject to interpretation of the real-time fringes, which were affected by the presence of residual fringes due to misalignment in repositioning of the hologram. 5.2 Conclusions In r i g i d body step displacements the resultant interference fringes can be localized at a plane in space other thanthe surface of the specimen. Only for out-of-plane rotation do the fringes localize on the object. Determination of the displacement magnitude from the fringe pattern of a'.single hologram requires knowledge of the position of the localization plane i f the fringes are not on the object. This becomes very d i f f i c u l t for other than the simplest motions for which a unique localization plane exists. To the extent that a measurement of fringe periods is an indication of the relative influence each motion has on the overall fringe behaviour due to a complex motion, the rotational components are highly sensitive and thus have the greatest effect. The preceding may be usedtto explain the properties of fringes due to f l a t panel vibrations. In this case the fringes are localized upon or very near the panel's surface. If each elemental area is considered displaced as a linear translation in combination with a rotation, the fringes become localized upon or very near the object due to the large influence of the rotational component. Time-average studies of the curved panel have shown that a good holographic evaluation of the vibration amplitudes i s obtained from a single hologram by taking into account the change in direction of 102 the surface normal as a variation in the incident and reflected angles 6^ and e^ . It was found thaJ-> adequate^precision in amplitude calculations can be obtained without the use of a microdensitometer for fringe location.' The time-average fringes appeared localized upon the air face of the-curved panel.. The vibration, amplitudes determined from the time-average analysis of the cylinder were calculated from the fringe pattern of a single hologram.• However there was a discrepancy between the calculated and measured results, and also between.the two calculated results determined from two different views of the cylinder. This has been related to the fringes not being localized upon the cylinder's surface. A single hologram did not provide sufficient information to indicate that the fringes were localized elsewhere than the surface, but two holograms did. Examination of the plotted amplitudes shows that there is reasonably good agreement between the calculated and measured data for an arc length of about 3", (centred around the point at which the corresponding observation vectors are centred). 3 Inches represents approximately a 60° arc of the 6" ( d i a v cylinder. It can consequently be concluded that an arrangement of several holograms spaced to provide a view covering 60° segements of the surface, would provide a reasonable evaluation of the vibration amplitudes occurring in each section. The above method however, overlooks the finding that the fringes are not on the cylinder's surface, and use i s only made of a portion (the 60° segment) of the cylinder v i s i b l e through each hologram, in which there is a good correlation between measured and-calculated amplitudes. 103 Thus this would be a good approximate method at best. With knowledge of the shape and position of the fringe localization plane,the fringes can more accurately be related to the vibration amplitudes. Residual fringes appear in real-time holography due to repositioning errors and emulsion shrinkage. In practice, these fringes cannot be eliminated, but may be minimized unt i l there are one or two covering the specimen. A l l the residual fringes in the cylinder study were reduced to one or two oval rings centred upon the specimen. Therefore; the effect of the residual fringes on the vibration fringes w i l l be similar in each case. In addition, a similar method of interpretation can be performed, thereby giving a more reliable comparison between the vibration amplitudes as calculated from each view. With analysis of the resultant fringes the stroboscopic real-time data appeared to reinforce-the time-average findings. A reduction in the influence of the residual fringes can be effected by examining the cylinder at a higher amplitude of vibration, thereby increasing the number of fringes due to the vibration of the specimen as compared to the repositioning error. 5.3 Suggestions for Future Research Real-time holography has inherent problems of analysis in the form' of residual fringes. The application of the stroboscopic real-time method to the study of curved surfaces has furthermore been d i f f i c u l t because of the unknown behaviour of the interference fringes that arise in the vibrations of such a surface. It is recommended that a separate detailed study be made of real-time holography to develop the 104 process into a reliable tool to perform quantitative vibration analysis. The time-average method has so developed, and as such, i t is recommended that any investigation requiring quantitative holographic measurements (as in the extension of holography to curved surface analysis) be performed with this method. It i s recommended that an analysis be carried out to determine the shape and position of the localization plane that has occurred in the study of the vibrating cylinder. A possible experimental method li e s in the analysis of the fringes of several holograms, providing different views of the same region of the specimen. The shift of the amplitude plots corresponding to each view can then be related to the position of the fringe localization plane. A consistent pattern in the shifting of the curves can possibly be attributed to a localization plane of simple shape (e.g. a circular cylinder). An inconsistent behaviour of the amplitude curves would be d i f f i c u l t to correlate to a localization plane of regular shape. In such a case i t i s suggested that a solution l i e s in determining ai,region of the cylinder that i s visible through the hologram for a given viewing direction, for which a reasonable agreement can be obtained between calculated and measured amplitudes. As an example, such a region could be the 60° segments mentioned previously. It has been determined that the interference fringes have appeared on the surface of the vibrating panel of 6" radius, yet not on the cylinder of 3" radius. This would seem to indicate that the severity of the curvature influences the position of the localization plane. A study of this behaviour would be in order to 105 ascertain the degree of curvature up to which the fringes can be considered localized upon the surface. It i s suggested that in the stroboscopic real-time analysis of the cylinder, the vibration be strobed at opposite phases of i t s oscillation. A reduction in the effect of the residual fringes is anticipated when the amplitudes as calculated from the overall fringe pattern obtained.at the two opposite extremes of oscillation are averaged. 106 BIBLIOGRAPHY 1. Leith, E.N. and Upatnieks, J., "Reconstructed Wavefronts and Communication Theory," J. Opt. Soc. Am. Vol. 52, No. 10, 1962 pp. 1123-1130. 2. Leith, E.N. and Upatnieks, J., "Wavefront Reconstruction with Continuous-tone Objects,", J. Opt. Soc. Am., Vol. 53, No. 12, 1963, pp. 1377-1381. 3. Leith, E.N. and Upatnieks, J., "Wavefront Reconstruction with Diffused Illumination and Three-Dimensional Objects," J. Opt. Soc. Am., Vol. '54, No. 11, 1964, pp. .1295-1301. 4. Powell, R.L. and Stetson, K.A. ,•. "Interferpmetric Vibration Analysis by Wavefront Reconstruction," J. Opt. Soc. Am., Vol. 55, No. 12, 1965, pp. 1593-1598.' 5. Stetson,.K.A. and Powell, R.L., "Interferometric Hologram Evaluation and Real-Time Vibration Analysis of Diffuse Objects," J. Opt. Soc. Am., Vol. 55, No. 12, 1965, pp. 1694-1695.' 6. Stetson, K.A. and Powell, R.L., "Hologram Interferometry," J. Opt. Soc. Am., Vol.'56, No. 9, 1966. 7. Archibold, E. and Ennos, A.E., "Observation of Surface Vibration Modes by Stroboscopic Hologram Interferometry," Nature Vol. 217, 1968, pp. 942-943. 8. Ennos, A.E. and Archibold, E. "Application Note: Vibrating Surface Viewed in Real-Time by Interference Holography," Laser Focus, 1968. 9. Brown, G.M., Grant, R.M. ond Stroke, G.W., "Theory of Holographic Interferometry," J. Acoust. Soc. Am. Vol. 45, No. 5, 1969, pp. .1166-1179. 10. Mayer, G.M., "Vibration Phase Measurement by Rotation-Strobe Holography," J. Appl. Phys.,Vol. 40, No. 7, 1969, pp. 2863-2866. 11. Fryer, P.A., "A Scanning Technique for Allowing Whole Vibration Cycles to be Stored on One Hologram," Applied Optics Vol. 9, No. 5, 1970. 12. Archibold, E. and Ennos, A.E., "Techniques of Hologram Inter-ferometry for Engineering Inspection and Vibration Analysis," Proceedings of the Symposium on the Engineering Uses of Holography, University of Strathclyde, Glasgow (Scotland), 1968. 13. Powell, R.L., "Measurement of Vibration by Holography, Proceeding of the Symposium on .tit-he Engineering Uses of Holography, University of Strathclyde, Glasgow (Scotland), 1968. 107 14. Stetson, K.A., "Vibration Measurement by Holography," Proceedings of the Symposium on the Engineering Uses of Holography," University of Strathclyde, Glasgow (Scotland), 1968. 15. Monahan, M.A. and Bromeley, K., "Vibration Analysis by Holographic Interferometry," J. Acoust. Soc. Am. Vol. 44, No. 5, 1968, pp. 1225-1231. 16. Wilson, A.D. and Strope, D.H., "Time-Average Holographic Interferometry of a Circular Plate Vibrating Simultaneously in Two Rationally Related Modes," J. Opt. Soc. Am., Vol. 60, No. 9, 1970, pp. 1162-1165. 17. Sampson, R.C., "Holographic-Interfer-ometry Application i n Experimental Mechanics," Experimental Mechanics, Vol. 10, 1970, pp. 313-320. 18. Liem, S.D.,'"Vibration Analysis by Holographic Interferometry," M.A.Sc. Thesis, University of British Columbia, 1970. 19. Hazell, CR. >and Liem, S.D.,' "Vibration Analysis by Interferometric Fringe Modulation," Symposium on the Application of Holography, Besancon, France, 1970. 20. Wilson, A.D., "Computed Time-Average Holographic Interferometric Fringes of a Circular Plate Vibrating Simultaneously in Two Rationally or Irrationally Related MOdes," J. Opt. Soc. Am. Vol. 6, No. 7, 1971, pp. 924-929. 21. Harry Kraus, "Thin Elastic Shells" (John Wiley and Sons, Inc., New York, 1967), Chap. 8. 22. Smith, B.C. and Haft, E.E., "Natural Frequencies of Clamped Cylindrical Shellsj" AIAA Journal Vol. 6, No. 4, 1968, pp. 720-721. 23. H.J. Caulfield and Sun Lu, "The Application of Holography" (John Wiley and Sons, Inc., New York, 1970). 24. TRW Systems Group, No. AM-70-11, "Applications of Holography to Vibrations, Transient Response and Wave Propagation," Appendix A, p. 97, 1970.-25. Haines, K.A. and Hildebrand, B.P., "Surface Deformation Measurement Using the Wavefront Reconstruction Technique," Applied Optics, Vol. '5, No. 4, 1966, pp. 595-602. 26. Aleksandrov, E.B. and Bonch-Bruevich, A.M., "Investigation of Surface Strains by the Hologram Technique," Soviet Phys.-Tech. Phys. Vol. 12, No. 2, 1967, p. 258. 108 27. Tsuruta, T. , Shiotake, N. and Itoh, Y., "Hologram Interferometry Using Two Reference Beams,"'Japanese J. Appl. Phys., Vol. '7, No. 9, 1968, pp. 1092-1100. 28. Vienot, J.Ch., Froehly, Cl. , Monneret, J. ,- and Pasteur, J. , "Hologram Interferometry: Surface Displacement Fringe Analysis as an Approach to- the Study of Mechanical Strains and Other -Applications to the-Determination of Anisopjtropyi i n Transparent Objects," Proceedings of the Symposium on the Engineering Uses of Holography," University of Strathclyde, Glasgow (Scotland), 1968. 29. Stetson, K.A., "Vibration Measurement by Holography," Proceedings of the Symposium on the Engineering Uses of Holography, University of Strathclyde, Glasgow (Scotland), 1968.' 30. Froehly, C., Monneret, J., Pasteur, J., and Vienot, J.Ch., "A Study of Slight Displacements of Opaque Objects and Optical Distortions in Solid Lasers' by Holographic Interferometry," Optica Acta, Vol. 16, No. 3, 1969, pp. 343-362. 31. Boone, P. and Verbiest, R. ,- "Applications of Hologram Interf erometry to Plate Deformation and Translation Measurements," Optica Acta, Nol.^16, No. 5, 1969, pp. 555-567. 32. Tsuruta, T., Shiotake, N. and Itoh, Y., "Formation and Localization of Holographically Produced Interference Fringes," Optica Acta, Vol. 16, No. 6, 1969, pp. 723-733. 33. Tsujiuchi, J., Takeya, N., and Matsuda, K., "Measurement of Strain in a Material by Holographic Interferometry," Optica Acta, Vol. 16, No. 6, 1969, pp. 709-722. 34. Solli d , J.E.,. "Holographic.Interferometry Applied to Measurements of Small Static Displacements of Diffusely Reflecting Surfaces," Applied OPtics, 1969, p. 1587. 35. Molin, N.E.,< and Stetson, K.A.,- "Measurement of Fringe Loci and Localization in Hologram Interferometry for Pivot Motion, In-Plane Rotation and In-Plane Translation," Optik (31), (The Institute of Optical Research, Stockholm Sweden), 1969. 36. Gottenberg, W.G^., "Measurement of 'Rigid Body Motion by Holographic Mechanics," University Southern California, 1971. 37. Gates, J.W.C., "Measurement of Displacements in Three Dimensions from a Single Hologram", Symposium on the Applications of ?~ _J Holography, Besancon, France, 1970. 38. Zambuto, M. and Lurie, M. "Holographic Measurement "of General Forms of Motion," Applied Optics, Vol. 9, No. 9, 1970, pp. 2066-2072. 109 39. Wilson, A.Di#j, "Holographically Observed Torsion in a Cylindrical Shell, "Applied Optics, Vol. 9, No. 9, 1970, pp. 2093-2097. 40. Burchett, O.J. and Irwin, J.L., "Using Laser Holography for Non destructive Testing," Mechanical Engineering, 1971, p. 27-33. 41. Shibayama, K. and Uchiyama, H., "Measurement of Three-Dimensiorial Displacements by Hologram Interferometry," Applied Optics, Vol. 10, No. 9, 1971, pp. 2150-2154. 42. Ennos, A.E., "Measurement of In-Plane Surface Strain by Hologram Interferometry," J. Sci. Instr. (J. of Phys; E.) Vol. 1, Series 2, 1968, pp. '731-734. 43. Data Release, "Reversal Bleach Process for Producing Phase Holograms," Eastman Kodak Co., Rochester, N.Y. 44. Sollid, J.E.,' "A Comparison of Out-of-Plane Deformation and In-Plane Translation Measurements made with Holographic Interferometryj" SPIE Seminar Proceedings, Boston, Mass., Vol. '25, 1971. 45. Hazell, C.R., Liem, S.D., and Olson, M.D. "Real-Time Holographic Vibration Analysis of Engineering Structural Components," SPIE Seminar Proceedings, Boston Mass., Vol.'25, 1971. 46. Leadbetter, I.K. and Allan, T., "Holographic Examination of the Pre-Buckling Behaviour of. Axially Loaded Cylinders," Proceedings of the Symposium on the Engineering Uses of Holography, University of Strathclyde, Glasgow (Scotland), 1968. 47. Pennington, K.S. and Harper, J.S.,- "Techniques for Producing Low-Noise, Improved Efficiency Holograms," Applied Optics, Vol. 9, No. 7, 1970, pp. 1643-1650. 48. Gottenbefg, W.G., "Experimental Study of the Vibrations of a Circular Cylindrical Shell," J. Acoust. Soc. Am. Vol. 32, No. 8, 1960, pp. 1002-1006. 110 APPENDIX A Fringe Location by Use of a Microdensitometer In the early part of the, experimental work on the curved panel (Section 4.2), i t was-seen that the accuracy of determining at exactly what point on- the panel a given fringe was located could be improved. Originally the reconstructed image with the fringes was viewed through the hologram, and the location of the fringes were made in reference to the scale on the panel. ; The gradations were in increments of 1/10". Photographs of the fringes could not be used as the v i s i b i l i t y of the scale gradations was degraded due to the speckle effect of the laser light, though later this problem was overcome (Section 3.4). In the meantime a microdensitometer (MD) was employed, model MK III cs, manufactured by the Joyce Lockel Co. of England. A microdensitometer i s a machine that measures the amount of light that passes through a transparent sheet, such as a film negative. A light beam, the diameter of which may be adjusted to a suitable size, is emitted by a source in the MD. The intensity of the beam is measured with reference to the source intensity. Consequently when a negative (with varying degrees of dark and bright areas) is passed through this beam, the MD gives a reading of the transmitted intensity of the beam relative to .the incident intensity. The negative i s placed on a sliding glass table which passes through the beam. The position of the negative on. the table can. be finely adjusted to have the beam pass through the desired region of the negative. Coupled to this table i s another table, upon which is placed graph paper. The MD guides I l l a pen to trace out variations in the transmitted light. There is provision to adjust the ratio of the movement of the graph table, to that of the specimen table, such that the scale of the graph can be varied from the actual size of the negative up to a maximum of 20:1. The Fig. A.,1 was•made with a ratio of,5:1. A photograph was taken of the time-average image of the vibrating panel using polaroid 4" x 5" 55 P/N film (a negative was provided with the print). The negative was passed through the MD. Fig. A.l shows a typical plot. The high peaks indicate darker areas of the negative^ • On the negative, a dark fringe appeared as a bright area, and a node as a dark area. The use of this plot in locating fringes on the panel can now be described. The d i f f i c u l t y in the use of this plot is related to perspective. This is shown in Fig. A.2. When a curved surface, a cylinder for example, is viewed from a point, a 180° section o f ' i t is not v i s i b l e . It i s somewhat less, say 170°, depending on how far back the point of observation P.'is situated. The problem is one of relating fringes on a f l a t two dimensional plane (the negative), to the proper location on a curved three-dimensional surface. This i s done by assuming that the Ught gathered by a camera lens converges to a point, called the point of 'gfeervation. Then somewhere between this point and the cylinder, the negative is placed, such that a-a on the negative represents b-b on the actual cylinder in Fig. A.2. Consequently any point c on the negative is related to the appropriate point c' on the cylinder by drawing a line from the point of observation, through point c, to intersect the cylinder at c'. Microdensitometer plot made from the negative of a picture of the curved panel similar to photo U.5b, page 77. The plot i s reversed and the anti-node regions are marked " l e f t " , "right" and "centre" as they appear in the photo. Dark fringe orders are numbered and E i s the panel edge. NODE VYWWW 4 3 2 1 1 2 c e n t r e l e f t Fig. A. 2 Example, of .perspective' when, viewing a cylinder; from ..appoint • L - V ^ V ; ^ * 114 Determination of the p o s i t i o n of P with respect to the cylinder w i l l allow the above method to be put into operation. A l l necessary information for t h i s i s obtained from a photograph of the specimen (and with the camera i n the same p o s i t i o n a p icture i s taken of the interference f r i n g e s ) . As i n Photo A . l there are two rulers and R^, placed as shown i n F i g . A.3. In the case of a normal view the following operations were c a r r i e d out. The r u l e r s were placed a distance i j apart, where i was the centre of the panel. Consequently at one edge of the photograph, d on R^ matched up with d' on R^. With t h i s information the l o c a t i o n of point P was determined. The MD plot was now used with the l o c a t i o n of the fringes marked. The p l o t was scaled to represent Photo A.1 Curved p a n e l w i t h r u l e r s u s ed t o determine l o c a t i o n o f P i n f i g . A . 3 . Fig. A,3 Geometry to.^locaiiftig^v^^en- • (.any,,p.pint k) . frpm microdensitometer -plot-'• (length..^'-b') ? >' -pn'to 'cylinder (point c 1 ) ; " • ' -•; 116 the width 'b'-b1 (distance E-E in Fig. A.1), on the baseline (which y ' • - passed through the centre of curvature 0) . The poinok/, representing a fringe on the plot (and point c on the negative) was related to position c' on the panel. This was done by letting OP be the y-axis, and the baseline the x-axis. The equation of the line l^ kp and the circle of radius iO was solved for co-ordinates of intersection at c'. The angle c'Oi was calculated, giving the arc length c ' i . Thereby the location of a dark fringe at c' relative to the centre of the panel at i was determined. This evaluation was performed for each C o i -ncide and dark fringe. Since the centre point 0 does not appear on the MD plot, the /V. point^iwas referenced to either edge of b'-b' (which represents the panel edges b-b). The location of these edges were effected by painting the rim. of the white panel black, thus the sharp contrast was picked up by the MD. The edges are marked E in Fig. A.1. The position of the camera or negative was not required as a l l information could be obtained from a photograph. The greatest room for error li e s in determining the exact locations of the rulers and R^, and their position with respect to some point on the panel such as centre i . " Since there were several arithematic calculations involved that had to be repeated for each fringe, the operations were carried out on a computer. The comparison of the results of this method with those of "fringe location by eye" is given in Fig. 4.4, p. 73 . The variation of the location of the panel edges for the MD data are most likely due to some error in determining the location of the rulers. 117 APPENDIX B Di f f e r e n c e s i n Displacement Magnitude f o r Assumption of C o l l i m a t i o n A l l the previous theory derived f o r c a l c u l a t i n g v i b r a t i o n amplitudes has been w i t h the assumption of c o l l i m a t i o n . However, the experimental work of t h i s t h e s i s has been c a r r i e d out w i t h divergent i l l u m i n a t i o n , and p o i n t observation of the specimen ( P i g . B . l ) . The observation and i l l u m i n a t i o n vectors are centred at the p o i n t o. The displacement d i r e c t i o n i s shown normal to the surface w i t h and the i n c i d e n t and r e f l e c t e d angles r e s p e c t i v e l y . Thus at o, 6 ^ and 6 ^ equal and (See F i g . 2 . 1 1 , p. 3 5 ) . I f c o l l i m a t i o n i s assumed as i n F i g . 2 . 1 1 , 6 ^ and 6 2 remain unchanged i n going to p o i n t p. However, in-,Fig. B . l , the i l l u m i n a t i o n diverges from a p o i n t (beam point Of . v , . - ' i . . observation. ,1 \: point of. beam divergence . ;^B.i,:l. Variation, of-'^pto^a'tio^'-and• 6b>ervatlbiaV| i t^^-^''.<Cr- .^••d^re^tioris^bn' a fla-£'''paneX;' •  *"• • •, '., ^A'.:. j 118 expander) and observat ion i s from point S q (the eye or camera viewing through the hologram). Consequently the angles 0^, ^ change by the amount A0^, A0£ i n going from o to p. ' Then the values of 0^, % 2 a t any point p with reference to the panel centre at o i s found as & 1 + A 6 1 *2 = e 2 + ^ 2 (B. l ) Where the appropriate s igns fo r A0-^ > hQ2 a r e used i n any ca l cu l a t i ons . With determination of the distances SQp and S^p, and angles e^, e^ the angle changes A0^, L02 c a n be ca l cu la ted fo r any po in t p. In a l l equations fo r c a l cu l a t i ng the displacement magnitude i t i s seen that the displacement d. i s p ropor t iona l to the fo l lowing term, 3 a ( c o s e i + cos6 2 ) ( B * 2 ) From equation B.2 the d i f fe rences occurr ing i n assuming co l l ima t ion may be determined as fo l lows . Consider a point, p where the displacement ca lcu la ted with co l l ima t ion i s , d = -r -~- : . (B.3) a (cose + cose„ ) wi th divergence the displacement becomes 119 C (B.4) cos(e + Ae ) + cos(e 0 + A 6 0 ) (It was mentioned previously that 0^ and equal e^, a t point o, and remain unchanged for collimation when going to point p). C is some constant appropriate to the method used, i.e. for stroboscopic real time, equation 2.53, p. 34 c = NX. Thus the percent difference in assuming collimation is DIFF where, The variation of DIFF has been calculated for the curved panel and cylinder used in this thesis. A f l a t panel of 6" width is included for comparison with the curved panel. Though DIFF was derived for a f l a t surface, i t s extension to curved surfaces just involves accounting for Ihe change in displacement direction at each point p. The distances S o and S.o were taken as 40", this being a mean value used in o x » & practice. (For S QO, the camera was generally 4" to 8" behind the hologram. This was added to the hologram to object distance such as OH = 36" in Fig. 4.4. Thus S o for this holographic arrangement would Li o be 40" to 44".) As the point p was moved across the surface, the distances S Qp and S^p varied and were calculated appropriately from the norm of 40" at o. The results of the calculations are presented in Fig. B.2. As an example of i t s use, reference is made to LOCATION = 5" for the l e f t view, DIFF = -6.8%. This means that the displacement DIFF = x 100 (B.5) curved panel 3 Left 2 cylinder -12 Legend , A right view, HR • l e f t view, HL DIPP i s {%) i LOCATION i s (in.) Fig. B.2a •. Plots of DIPP vs. LOCATION for specimens oriented as shown in f i g . B.2b* ^LOCATION ,'refers„ to scale T -:; ' on the surface of each;- specimefa* ' Fig. B.,2b Orierrt^io^^jbf;.: specimens for plots of fig.-fi.2a, flat panel HL = hologram of •'" l e f t r v i e w v . ' : ..','-' - HR = hologram ,of ''••'[,• '• right v.iewt.;;x: ... for:.'all'case's^'' I s0o::f.. sL o .= s-i =;-'kq^'.j^ j for f l a t andcurved panel i a l l ;vectorstcen:tre Q'.. for cylinder; each : : observation., vector is centred at i t s bwri local poiht o, illumination " •', illumination vector i s• |:''centred at 1 . "' '• V." illumination beam' . diverges from" S-L, > ' ••. ;ob;se'rvation by eye or v camera made'at S6 . '-.'V • • „ r-curved -panel cylinder 122 c a l c u l a t e d , assuming c o l l i m a t i o n , at t h i s l o c a t i o n i s 6.8% l e s s than the va lue a c t u a l l y determined by t a k i n g divergence i n t o account . .Thus i n F i g . 4.S, the displacement i s shown as 38 u i n . at LOCATION =<5' The a c t u a l v a l u e should be , 123 APPENDIX C Operation of the Fotonic Sensor Vibration amplitudes were measured with a Fotonic sensor model KD38-A, a non-contacting displacement transducer. The sensor probe consists of a 1/10 in. dia. head containing several hundred glass optic fibres randomly arranged. Half of the fibres direct light from a bulb to the probe end, while the remainder receive the reflected light from the specimen's surface and direct i t to a photo-diode. The specimen's surface musti'be sufficiently reflective, a polished metal surface is appropriate. The closer the probe i s to the surface, the greater w i l l be the'amount of reflected light captured by the probe. The sensor output i n mill i v o l t s i s related to the displacement. For vibration amplitude measurement the sensor i s set to operate in the linear region of the displacement-mv curve. Originally the sensor had been calibrated for use with a f l a t surface, thus recalibra-tion was required for use with curved surfaces. The calibration values were 5.85 yin./mv for a fl a t surface, 5.59 uin./mv for a cylindrical surface of 6" radius, and 5.51 uin./mv for a s i m i l a r surface of 3" radius. The sensor output was displayed at the AC setting on an oscilloscope.' The output contained 2 mv noise, the readings were repeatable to +0.5 mv. The frequency response of the sensor extended from DC^to 10 KHz. A Kinamatic micro-translation table, model TT-102, provided the displacements in the static calibration procedure of the Fotonic sensor. The micrometer gradations were in 100 uin. increments, and sensor 124 readings were taken at 1000 uin. intervals. Ah output vs. displacement curve was plotted and the linear region selected for amplitude measurement. 

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