Holographically Observed Torsion in a Cylindrical Shaft A. D. Wilson Using holographic interferometry, the rotation of a rigid cylinder and the torsion of a flexible shaft (RTV silastic rubber) have been observed. The experimentally generated fringe patterns are compared with computed ones and good agreement is obtained. Introduction Holographic interferometry,1, 2 introduced by Hor- man 3 in 1965, is currently being applied to many mea- surement problems. Since then, numerous papers dealing with holographic interferometry have appeared and with the notable exceptions of the works of Gotten- berg 4 and that of Holds and Fuhs, 5 few papers have attempted to correlate holographically determined and known phase changes caused by body deformation, displacement, or refractive index change. This paper compares experimental and computed fringe patterns for a special inplane body displace- ment-that of the rotation of a rigid cylinder and the torsion in a flexible cylindrical shaft. Qualitative and quantitative agreement is excellent. Analysis To determine the method of fringe formation due to interference of two wavefronts from a rotatable cy- lindrical body, one should consider the optical paths of the incident (source) and scattered (received) waves from a general diffuse surface as depicted in Fig. 1. At points P and P', the source and receiver are in directions of the unit vectors, i and , respectively. The surface of the body is shown in two states: S and S'. With the perturbation of the surface S to S', point P experiences a vector displacement to P'. The physical path difference in the two wavefronts is equal to the sum of projections of a onto directions i and P. The optical path is k (27r . X) times the physical path difference. Thus, 27 d (A + ), The author is with IBM Corporation, Systems Development Division, P.O. Box 6, Endicott, New York 13760. Received 23 January 1970. where X is the wavelength of the light in the medium surrounding S. The formation of visible or macroscopic fringes with a phase relationship of Eq. (1) implies a fundamental assertion: two separate complex wavefronts emerging at different times from a diffuse surface (composed of a collection of point radiators) can be considered spatially coherent (if at some later time these two wavefronts are regenerated simultaneously) as long as the topology of the diffuse surface is unchanged. Upon reconstruc- tion of a double exposure hologram made of a diffuse object, the only wavefronts which tend to interfere over an area sufficiently large so as to produce visible fringes are the two wavefronts originating from point P and the point P displaced, P'. Moreover, inter- ference does not occur between wavefronts emerging from any other two points (i.e., P and P", Fig. 1) with enough regularity to produce visible fringes. Visible fringes are produced when a complex wavefront emanating from a collection of points P interferes with a similar (but not necessarily identical) complex wave- front emanating from a collection of points identical with the original point set but perturbed in some way (e.g., displaced). For a cylindrical surface (Fig. 2) rotated around its axis through an angle a, the phase differential becomes P. (1) \N Fig. 1. Illuminating and receiving geometry of a generalized surface is shown in initial and perturbed states. September 1970 / Vol. 9, No. 9 / APPLIED OPTICS 2093
Transcript
Holographically Observed Torsion in a Cylindrical Shaft
A. D. Wilson
Using holographic interferometry, the rotation of a rigid cylinder and the torsion of a flexible shaft (RTVsilastic rubber) have been observed. The experimentally generated fringe patterns are compared withcomputed ones and good agreement is obtained.
IntroductionHolographic interferometry,1, 2 introduced by Hor-
man3 in 1965, is currently being applied to many mea-surement problems. Since then, numerous papersdealing with holographic interferometry have appearedand with the notable exceptions of the works of Gotten-berg4 and that of Holds and Fuhs,5 few papers haveattempted to correlate holographically determinedand known phase changes caused by body deformation,displacement, or refractive index change.
This paper compares experimental and computedfringe patterns for a special inplane body displace-ment-that of the rotation of a rigid cylinder and thetorsion in a flexible cylindrical shaft. Qualitative andquantitative agreement is excellent.
Analysis
To determine the method of fringe formation due tointerference of two wavefronts from a rotatable cy-lindrical body, one should consider the optical pathsof the incident (source) and scattered (received) wavesfrom a general diffuse surface as depicted in Fig. 1.
At points P and P', the source and receiver are indirections of the unit vectors, i and , respectively.The surface of the body is shown in two states: S andS'. With the perturbation of the surface S to S',point P experiences a vector displacement to P'.The physical path difference in the two wavefronts isequal to the sum of projections of a onto directions iand P. The optical path is k (27r . X) times the physicalpath difference. Thus,
27 d (A + ),
The author is with IBM Corporation, Systems DevelopmentDivision, P.O. Box 6, Endicott, New York 13760.
Received 23 January 1970.
where X is the wavelength of the light in the mediumsurrounding S.
The formation of visible or macroscopic fringes witha phase relationship of Eq. (1) implies a fundamentalassertion: two separate complex wavefronts emergingat different times from a diffuse surface (composed of acollection of point radiators) can be considered spatiallycoherent (if at some later time these two wavefronts areregenerated simultaneously) as long as the topology ofthe diffuse surface is unchanged. Upon reconstruc-tion of a double exposure hologram made of a diffuseobject, the only wavefronts which tend to interfereover an area sufficiently large so as to produce visiblefringes are the two wavefronts originating from pointP and the point P displaced, P'. Moreover, inter-ference does not occur between wavefronts emergingfrom any other two points (i.e., P and P", Fig. 1)with enough regularity to produce visible fringes.Visible fringes are produced when a complex wavefrontemanating from a collection of points P interferes witha similar (but not necessarily identical) complex wave-front emanating from a collection of points identicalwith the original point set but perturbed in some way(e.g., displaced).
For a cylindrical surface (Fig. 2) rotated around itsaxis through an angle a, the phase differential becomes
P.
(1)\N
Fig. 1. Illuminating and receiving geometry of a generalizedsurface is shown in initial and perturbed states.
Fig. 2. Illuminating and scattering geometry of a rotatablecylinder is shown in initial and angularly displaced states.
_ _
l fl+H
-I
f _, _l
Fig. 3. Computed fringe pattern is shown for axially symmetricrigid body rotated 349 grad (72 see of arc). Solid lines are thecenters of the dark fringes, and the dashed lines are the centers of
the bright fringes.
-1.0 -n.8 -c'.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0
Fig. 4. Shown is the fringe intensity vs normalized positionprojected onto a plane perpendicular to viewing and illuminatingdirections for angular rotations of cylinder of (a) 23 rad, (b) 47prad, and (c) 93 urad (4.8 see of arc, 9.6 see of arc, and 19 see of arc).
Fig. 5. Computed fringes are plotted for a shaft 1.2 cm diam andin torsion. Assumed is a linear distribution of twist along theshaft, except a section of the right end is held rigid. The left endis clamped. The maximum rotation of the right end is 223 grad.Dashed lines are the centers of the bright fringes (phase changenfr, n even); solid lines are the centers of dark fringes (phase
When 0 = 0 = 0, i.e., normal illumination and view-ing, Eq. (2) reduces to
4ir *a* /ad = X-
2 p sin sine+ 2/ (3)X 2
If the angular rotation a is small (a << 1), then Aqfurther reduces to
AO (4 r/X)pa sin-y. (4)
Noting that the projection of point P onto a planeparallel to the plane tangent to the shaft at y = 0 isAx = psiny, it is evident that
Jlo/Ax (4-,,/X)a. (5)
Hence, the fringe frequency is independent of both thecylinder radius and the angular position around thecylinder (Fig. 3). For small angular rotations, thefringe frequency is proportional to the amount of rota-tion (Fig. 4), and the fringes run axially on the cy-lindrical body. If torsion exists, fringes tend to con-verge toward the central portion of the cylinder cor-responding to regions of greater total angular displace-ment (Fig. 5).
ApparatusThe experimental apparatus (Fig. 6) consists of a
bearing, rigid shaft, rotating de-vice, flexible shaft, anda clamping collet. A bearing is required which will notintroduce any tilt of the object in the planes of the axis
around which a small angular displacement is to occur.A split bearing 2.5 cm long and 1.2 cm diam was fash-ioned from brass and adjusted by a tie bolt for a verytight fit on a lightly lubricated centerless ground toolsteel shaft. This single bearing limits the total amountof tilt of a double ended shaft to less than 1 see of arc.To rotate the shaft, a force of approximately 500 N isrequired on a 5.45-cm long lever. The shaft is angu-larly displaced by applying a force to the lever armwith a microinch differential screw micrometer. Thisassembly permits the rotation of the shaft in minimumincrements of about 0.5 X 10-6 rad (0.1 see of arc).The resolution is more than adequate for this experi-ment. The shaft is centerless ground with three dif-ferent diameter sections, two of which are joined by aconical segment. The shaft is ground circular to withinabout 2.5 /im, and the diameter of a straight sectionis uniform along the cylinder axis also within about2.5,um. No attempt is made to make the shaft per-fect. Further comment on the effects of geometrical
and topological properties shaft are deferred to thediscussion section of the paper.
A 1.2-cm diam flexible shaft, made of white RTVsilastic rubber rod 3.6 cm long, is fastened to the ro-tatable rigid shaft and a stationary shaft by means ofcollet-type clamps.
The machined components of the apparatus, whichare viewed holographically, are spray painted with aflat white paint. This process insures that the objectsare diffuse reflectors. To reduce flexing of the silasticrod during rotation, it is placed in both slight tensionand torsion when assembled into the fixture. Further-more, it is noted that by placing the RTV silastic undersome stress, its smooth cast surfaces begin to act as adiffuse rather than specularly reflecting surface.
Carrier beam Fresnel amplitude holograms were madeon Eastman Kodak 649F emulsion using a cw He-Ne laser operating at 6328 A and 5 mW. The holo-graphic apparatus is more or less similar to that foundin current holographic laboratories. A sketch of thelayout is shown in Fig. 6(b). Objects are held mag-netically or bolted to a rigid cast-iron surface platewhich is supported by a granite slab mechanically iso-lated by air mounts from the building. Exposure timesare on the order of 20 see (total) for development inD-19 at 230 C for 6 min.
The illuminating light, which is incident on the cyl-inder, is spread out in one dimension by a series ofspherical and cylindrical lenses and then recollimatedinto a beam approximately 8 cm high by 1.5 cm wide.A large (12 cm X 12 cm) beam splitter between theemulsion and cylinder allows illumination and viewingin conjugate directions.
To study the rotation of a shaft by holography thedouble exposure technique is used. First, the totalexposure time for an efficient hologram is divided intotwo sequential exposures-the first holographic ex-posure made with the shaft in an arbitrary position andthe second holographic exposure made with the shaftrotated a selected amount from the initial arbitraryposition. Second, the doubly exposed photographicplate is developed, stopped, fixed, washed, and dried.Third, the plate is reilluminated with a wavefront simi-lar to the reference wave [(Fig. 6(b) ] and the primaryholographic image, which is a virtual image in this case,is visually observed and/or photographed with a con-ventional camera.
Results
Rotation of a two-diameter rigid cylinder throughangles (in grad) of 0, 70, 93, 186, and 349 (or in see of arc0, 14, 19, 38, and 72, respectively) results in the fringes ofFig. 7 which were obtained by double exposure holog-raphy. Figure 7(e), an experimental fringe pattern,
with y = 7r/2. Some small amount of the twist of thesilastic shaft occurs within the collet clamp and thus outof view; hence, it is not shown in Fig. 9.
Discussion
If the cylindrical object is perfect (i.e., completelycircular) and has only minute (compared to the wave-length of light) surface irregularities, then the objectwould not be a diffuse reflector and no fringes would beobserved upon rotation. However, if the cylindricalobject can be considered (1) perfectly circular in amechanical macroscopic sense (i.e., any variation inthe diameter is small with respect to itself), and (2) theobject is still a diffuse reflector, then fringes will beformed as shown here. This is true in spite of the factthat when a nearly perfect cylinder is rotated throughan angle a, point P is replaced, or nearly so, by a pointP" located at an angle negative a. This occurs forall points on the cylinder. Since good, high contrastfringes were obtained for all rotation angles in the
e
Fig. 7. Rigid cylinder is rotated through angles of (in prad) (a) 0,(b) 70, and (c) 93, (d) 186, and (e) 349 (or in see of arc 0, 14, 19,
:38, iid 72).
compares well with the theoretically computed fringepattern, Fig. 3. A cylinder of silastic rubber in torsionproduces the fringes of Fig. S. Here the differential ro-tation between exposures is (in grad) 0, 139, 233, and 279(or in see of arc 0, 29, 48, and 58). The computed fringepattern for a uniform distribution of twist (Fig. ) com-pares favorably with the experimental patterns (Fig. 8).The fringes are localized very near to if not coincidentwith the image of the cylinder itself.
The fringes of Fig. may be used to determine thedistribution of surface twist along the axis of the shaft.Figure 9 depicts this distribution for Fig. 8(c) and isobtained by locating the fringes at the edge of the cyl-inder with respect to axial position. Figure 9 resultsfrom a plot of the fringe number vs position. Theactual angular twist may be computed from Eq. (4)
b
d
Fig. 8. Cylinder of silastic is rotated through angles of (in Arad)(a) 0, (b 139, (c) 233, and (d) 279 (or in see of arc 0, 29, 48, and
Fig. 9. Shown is distribution of twist for Fig. 8(c). Silasticradius = 0.603 cm; maximum angular displacement is 233 rad.
torsion experiments, the topology of the silastic rubberwas apparently not significantly altered by the maxi-mum angle of rotation used.
The author gratefully acknowledges the skillfulassistance of D. Strope, RTV sample preparation byW. Hamm, fixture fabrication by C. Ochs and C.Fetcinko, and helpful discussion with B. D. Martin andT. Young.
This paper was presented at the Annual Fall Meetingof the Optical Society of America in October 1969.
References1. K. A. Haines and B. P. Hilderbrand, Appl. Opt. 5, 595 (1966).2. L. 0. Heflinger et al., J. Appl. Phys. 37, 642 (1956).3. M. L. Horman, Appl. Opt. 4, 333 (1965).4. W. G. Gottenberg, Exp. Mech. 8, 405(1968).5. J. H. Holds and A. E. Fuhs, J. Appl. Phys. 38, 5408 (1967).
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