An Innovative Polariscope for Photoelastic Stress Analysis

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1 An Innovtive Polriscope for Photoelstic Stress Anlysis Jon R. Lesnik, Michel J. Zickel, Christopher S. Welch, Deonn F. Johnson Abstrct An innovtive polriscope involving single rotting opticl element nd digitl cmer for full-field imge cquisition llows utomted dt to be cquired quickly nd efficiently. Softwre nlysis presents the dt in n esy to interpret imge formt depicting the mgnitude of the sher strins nd the directions of the principl strins. Introduction Photoelstic stress nlysis hs long been fithful nd productive technology offering the mesurement community one of the erliest forms of full-field stress nlysis. Unfortuntely, its mturity hs led to the misconception tht the technology is no longer fertile ground for interesting developments, nd consequently, photoelstic stress nlysis is often overlooked by younger scientists. Severl new utomted polriscope systems imed t improving the efficiency nd ccurcy of photoelstic stress nlysis hve been introduced over the pst severl yers [,, 3, 4, 5]; however, ech of these systems involves one or more complicted procedures such s fringe counting or phse stepping [6, 7] often mking the polriscope hrd to use nd the results difficult to interpret nd not very intuitive. It is certinly true tht mny tlented scientists hve invented numerous photoelstic techniques, estblishing mny s fundmentl concepts, but there re still mny fields to plow. The importnce of sher bsed filure criterion persists s does the need to vlidte numericl models with experimentl techniques. Photoelstic stress nlysis is powerful nlysis tool especilly when combined with thermoelstic stress nlysis. The combintion mkes vilble full-field stress tensor dt. For photoelsticity to thrive it needs to keep pce with the turn key er. At the sme time, it is importnt tht dvnces in the technology do not exclude other recent improvements to the technology; therefore, fully utomted polriscope with immedite nd esily interpreted results, nd the verstility to be integrted into current reserch is prmount. Furthermore, simple nd esily understndble description of the mechnics of the polriscope is essentil. One must feel comfortble with the mesurement being mde without requiring extensive study. By combining rotting nlyzer nd video lock-in system n utomted grey-field polriscope hs been devised tht pinlessly collects both mgnitude nd direction dt without user intervention. Although the new polriscope is cpble of mking multiple fringe mesurements, it hs its gretest impct in sub-fringe pplictions where the output prllels the description of the stress stte of n object given by Mohr s circle. Grey-field Polriscope Figure is simplified portryl of the grey-field polriscope. The object is illuminted with circulrly polrized light. Upon propgtion through birefringent medium, or strined photoelstic coting, the light exits ellipticlly polrized. The resulting ellipticl light is nlyzed by n utomtic ellipsometer in the form of rotting nlyzer, nd the resulting light oscilltions re nlyzed by video lock-in system. In order to better describe the science behind the grey-field polriscope ech of the criticl steps in the evolution of the light signl is broken out nd explined through the following figures nd mthemticl discussion. The circulrly polrized light entering the model (Fig. ) is seprted into two components, one component (A ) prllel to the fst xes nd nother component (A ) prllel to the slow xes. The fst xis mkes n ngle β with respect to reference direction, in this cse the horizontl xes. The circulrly polrized light components re described by After propgting through the coting whether in trnsmission or reflection, the slow xes cn be modeled to hve phse lg with respect to the fst xes. The overll retrdtion experienced by both pths is ignored for simplicity. The light exiting the coting now tkes the form where = cos( ηt) A = sin ( η t) A A = cos ηt + A = sin ηt () () (b) Stress Photonics Inc., 300 Progress Rd., Mdison, WI 5376 College of Willim nd Mry, Applied Science, Willimsburg, VA 386

2 where k is the reltive strin optic coefficient h is the thickness, λ is the wvelength of light f is the fringe vlue of the coting [8] (3) The light of Eqs. (,b) is described s ellipticlly polrized, which is pprent when Eqs. (,b) re rewritten s A mj A min 4πhk = λ π π = cos cos ηt π π = cos + sin ηt (4) (4b) The mjor xis of the ellipse is lwys shifted π/4 rdins from the direction of the first principl strin for odd fringes nd π/4 for even fringes (Fig. 3). If n nlyzer is plced in the system t n orienttion α (Fig. 4) from the mjor xis the resulting light mplitude is described by A = cosαamj sinαa α + ( ε ε ) = f ( ε ε ) The intensity of the resulting light is relted to the squre of the mplitude which fter severl trigonometric mnipultions yields The penut shpe of the light intensity results from squring the ellipticl light. If the nlyzer is llowed to rotte t n ngulr frequency ω then It is ssumed tht the nlyzer is prllel with the reference orienttion t t=0. In terms of time, the intensity mp is described by From this reltion it cn be seen tht in the bsence of birefringence the output is neutrl grey. It cn lso be seen tht in the presence of birefringence the mplitude of the signl oscilltes bout this neutrl grey level. The oscillting portion of the signl is zero when the xis of nlyzer coincides with the principl strin xes. It should lso be noted tht only the mplitude mesurement is ffected by the color of the input light. The orienttion of the mjor xis is unffected. min I = Aα = sin [ + cos( α ) ] π α = ωt β 4 I = sin [ + sin ( ωt β) ] (5) (6) (7) (8) Sub-fringe PSA ssumes smll retrdtion ngles so tht the mplitude of the oscilltions cn be directly relted to the mplitude of the sher strins. For very smll retrdtion ngles Eq. 8 cn be rewritten s For lrger ngles some lineriztion my be required. Video Lock-in I = t In order to mke extremely ccurte mesurements of subfringe birefringence video lock-in lgorithm is employed. The nlyzer nd n bsolute position encoder re rotted t constnt ngulr velocity nd the computer monitors the redout of the encoder (Fig 5). When prescribed ngle is crossed n imge is cptured nd stored for subsequent processing. In this study eight imges re cptured for every one hlf revolution of the nlyzer. The ngles t which the computer triggers the cpture of n imge re defined by (9) (0) It is importnt to recll tht full period of signl oscilltion is represented by one hlf of n nlyzer revolution. By controlling the imge smpling in this mnner level of synchroniztion is chieved nd the lock-in lgorithm tht is performed tkes the simple form described by Χ Υ [ + sin ( ω β )] where J n is smpled imge. θ n = π ( x, y) = Jn ( x, y) cos( πn / ) n= () (b) From Eqs. (,b) the mplitude nd direction of sher strins cn be determined by the following reltions γ = K Χ () where K is clibrtion constnt including system gins, light intensity vritions nd residul strins in the coting. The phse is clculted from the X nd Y imges s The orienttion of the first principl strin direction to the reference direction (β) is relted to φ by φ = Tn ( n ) φ = β + Υ n= ( x, y) = Jn ( x, y) sin( πn / ) n= Υ Χ (3) (4) Sub-fringe PSA

3 For odd fringes β points in the direction of the first principl strin, nd for even fringes β points in the direction of the second principl strin. The results of Eqs. 4 re identicl to the equtions describing Mohr s circle (Fig 6) where X is the bse, Y is the height nd the ngle β represents the physicl ngle on the specimen. This mens tht the results of this polriscope directly yield in-plne sher strin components for ny rbitrry direction. Also, this implies tht in the linerized re, the XY imge sets cn be operted on s vectors. The superposition of two strin sttes cn be clculted by dding two XY imge pirs then pplying Eqs. nd 3 to find mgnitude nd direction. To remove the effects of residul strin in the coting n initil XY pir cn be cptured nd subtrcted from the imges collected fter loding. The mgnitude nd direction re then clculted fter the subtrction. Experimentl Setup As seen in Fig. 5, simple yet flexible opticl setup ws used. The light source consists of stndrd light bulb which projects white light through condenser lens, filter (optionl), polrizer nd, finlly, qurter wveplte. Circulrly polrized light is projected onto the re of interest. The polriscope is positioned t shllow ngle with respect to the light source. Within the projector, rely lens projects virtul imge of the smll video detector into free spce. This cretes physicl spce to incorporte the rotting nlyzer nd other clibrtion devices. The rotting nlyzer is positioned ner the imge plne of the virtul detector in order to eliminte jitter effects cused by the refrctive vritions in the polrizer sheet. A field lens condenses the opticl pth so tht stndrd cmer lens with c-mount dpter cn be used. The position of the nlyzer is monitored by the bsolute position encoder. The computer monitors the sttus of this encoder nd triggers the frme-grbber t predetermined ngles. As mentioned bove the use of predetermined ngles llows the use of simple lock-in lgorithm. The smpled imges re processed by the computer nd displyed in both mgnitude nd direction. Results A bem in four-point bending ws used s the test specimen. The bem (3.75 mm thick luminum) ws coted with commercilly vilble photoelstic sheet coting (0.54 mm thick) nd loded t pproximtely k. The stress stte of the bem is depicted in the schemtic of figure 7. The shded res illustrte one of the key dvncements introduced with the grey-field polriscope. With stndrd circulr or plne polriscope it is not possible to distinguish between tension nd compression, but the grey-field polriscope uses wht cn be thought of s n offset to dd sign to tensile nd compressive stress mking the two distinguishble. Results using the polriscope re shown in Figs. 8 nd 9. The horizontl or x-xis ws chosen s the reference xis, so tht, for this exmple, the Y, or cosine, imge is zero. The signl to noise rtio ws improved by collecting dt over four oscilltions of the nlyzer, so tht ech X nd Y imge is sum of eight imges (J - J 8 ). The mgnitude nd phse imges in Fig. 9 nd 9b re clculted ccording to Eqs nd 3. The mgnitude imge hs the chrcteristic neutrl xis down the center of the smple. Direction lines re superimposed on the direction dt ccording to the clculted pixel vlues in the imge, nd they clerly indicte horizontl (0 ) nd verticl (90 ) orienttion on either side of the neutrl xis, just s one might nticipte. Opertionl Modes There re mny procedurl options tht cn improve the collected dt. Severl gols leding to the improvement re: Determine system gin Compenste for illumintion vritions Compenste for coting residul strins Compenste for qulity of projected light There re severl methods tht might ccomplish these gols. The following outlines one experimentl procedure tht cn eliminte ll of the fore mentioned concerns. Polriztion Qulity For typicl opertion, monochromtic light is used nd the qulity of the circulr polriztion is quite good, eliminting ny need for compenstion. In other pplictions, polychromtic light my be desired, nd in these cses, perfect circulr polriztion my not be chievble. A correction for this problem my be mde by plcing diffuse reflective trget in the field-of-view, nd performing the video lock-in procedure. The nlysis will yield the degree of imperfection in the polrized light, thus providing mens of compenstion. Essentilly, the imperfection will be constnt full-field offset tht is esily determined. In the cse of polychromtic light n offset for ech color must be cquired. Light Intensity nd System Gin Light intensity vritions nd pixel by pixel system gins cn be determined seprtely or together. The following is simple pproch to clibrte the bove mentioned prmeters nd to estblish the correct phse of the lock-in, which effectively sets the reference orienttion. To ccomplish ll of the bove in the sub-fringe mode clibrtion plte of birefringence δ cn be introduced into the system to ugment the existing birefringence. This pprent strin cn be resolved by subtrction of the loded imge from the loded-ugmented

4 imge. From this informtion light intensity cn be clculted free of reflection error. An lterntive to this pproch is to use the verge of the imges s the intensity nd to clibrte the system gins seprtely. This pproch will be more importnt s the speed of grey-field polriscope systems increses nd dynmic strins re monitored. Residul Strins nd Lod Rmping Residul strins cn be eliminted simply by lod differencing. Essentilly n imge is run through the pces s described bove then the lod is incresed nd the process is repeted. The difference between these imges will compenste for residul strins nd/or light source imperfections. Lod rmping is lso useful for imging lrge or complex components tht exhibit wide rnge of strin vlues. With this technique low lod is initilly used so tht regions exhibiting high strin fields cn be interrogted with good resolution nd without exceeding sub-fringe limittions. The lod is then incresed so tht regions with lower strin vlues cn be smpled with optiml resolution. By using the lod rmping technique ll regions on the smple cn be inspected with optiml resolution. The extent of the dynmic rnge of the response defines the number of lods tht should be used. This method cn significntly increse the pprent dynmic rnge. Conclusion A fully utomted polriscope system tht determines the mgnitude nd direction of the sher strins hs been developed. This grey-field polriscope lso utomticlly distinguishes between the directions of the principl strins, nd indictes ech direction in the resultnt dt. The dt re presented in n intuitive formt so tht they re esy to interpret. One imge depicts the sher strin mgnitude with grdients represented in greyscle levels, nd the other indictes the directions of the principl strins. Interprettion of the dt prllels the strin stte description given by Mohr s circle. The grey-field plolriscope utilizes video lock-in lgorithm to chieve high degree of sensitivity, which mkes it idel for subfringe opertion. This does not exclude higher order fringe opertion, where techniques such s lod rmping cn be used to gret dvntge. Acknowledgments The uthors would like to cknowledge ASA Lngley Reserch Center nd in prticulr Mr. K. Elliott Crmer for supporting this reserch through the STTR (Smll Business Technology Trnsfer) progrm. In ddition, Mr. Ted Tuttle of Stress Photonics Inc. contributed extensively in the design nd construction of the polriscope nd light source. References [] Ptterson, E. A., Wng, Z. F., Towrds Full-field Automted Photoelstic Anlysis of Complex Components, Strin, My, 49-56, 99. [] Redner, S., ew Automtic Polriscope System, Experimentl Mechnics, December, , 974. [3] Redner, S., Compenstion Method Using Synchronized Polrizer-Anlyzer Rottion, Experimentl Mechnics, June, -4, 976. [4] Robert, A. J., ew Methods in Photoelsticity, Experimentl Mechnics, My, 4-3, 967. [5] Voloshin, A. S., Burger, C. P., Hlf-fringe Photoelsticity: A ew Approch to Whole-field Stress Anlysis, Experimentl Mechnics, September, , 983. [6] Hecker, F. W., Morche, B., Computer Aided Mesurement of Reltive Retrdtion in Plne Photoelsticity, Proceedings of the VIIIth Interntionl Conference on Experimentl Stress Anlysis, My, , 986. [7] Asundi, A., Phse Shifting in Photoelsticity, Experimentl Techniques, Jnury/Februry, 9-3, 993. [8] Mesurements Group Tech ote, T-70, Mesurements Group, Inc., P.O. Box 7777, Rleigh, C, 977.

5 A A β Figure. Grey-field reflection polriscope Figure. Circulrly polrized light Amj Anlyzer Amj A A α Amin π/4 β A Amin π/4 β Θ A Figure 3. Ellipticlly polrized light Figure 4. Light intensity resulting from squring the mplitude of the ellipticlly polrized light Rotting Anlyzer Field Lens Rely Lens Frme Grbbing Εight Imges (J-J8) Frme Buffer Stndrd Zoom Lens /4 wveplte & polrizer Filter Motor Source Video Cmer Detector Encoder AC Cmer Control Trigger In Θn Σ Ν { J n Sin X Jn Cos} = { Y} n = Circulrly Polrized Light Source Figure 5. Experimentl system Computer Processing

6 Circulr or Plne Polriscope Grey-field Polriscope ε β X γ/ Y ε Fst Sher Component Fst Figure 6. Mohr s circle Figure 7. Schemtic of bem in bending b Figure 8. X () nd Y (b) imges before processing b Figure 9. Imges showing the mgnitude () nd the direction (b) of the principl strins

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses