Measuringgeometricimperfectionsofvariable–angleﬁlament–woundcylinderswithasimpledig- ital image correlation setup Saullo G. P. Castro,José Humberto S. Almeida Jr.,Luc St-Pierre,Zhihua Wang Measuring geometric imperfections of variable–angle ﬁlament–wound cylinders with a simple digital image correlation setup Saullo G. P. Castroa,<, José Humberto S. Almeida Jr.b,c,<, Luc St-Pierreband Zhihua Wanga,d aFaculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands bDepartment of Mechanical Engineering, Aalto University, Espoo, Finland cAdvanced Composites Research Group, School of Mechanical and Aerospace Engineering, Queen’s University Belfast, UK dSchool of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, China ARTICLE INFO Keywords : geometric imperfection digital image correlation experimental characterization cylindrical shells variable-angle ﬁlament windingABSTRACT Measuringthegeometricimperfectionsincylindricalshellsisacriticalstepnecessarytocreateaccu- ratenumericalmodelsthatcancapturetheimperfection-sensitivebehaviorofthesestructures. Mod- ern composite structures, such as variable-–angle ﬁlament-wound (VAFW) cylinders, have a unique imperfection signature that is still unknown to the scientiﬁc community. This new class of variable– stiﬀness structures developed by our research group combine wide tailoring capabilities with the ef- ﬁcient manufacturability enabled by ﬁlament winding process. The present study proposes an im- perfection measurementmethod thatis simpleand applicable toboth smalland large structures. The topographic data is measured with only a pair of cameras. Practical aspects of using digital image correlation (DIC) are described and discussed in detail, such as lighting, focus adjustment, and cali- bration. State–of–the–artbest–ﬁtroutines,basedonleast–squaresoptimization,areusedtotransform rawdataintoacommoncoordinatesystem. Finally,thetransformeddataisstitchedtogethertobuild a full 3D imperfection pattern that can be readily used in a nonlinear ﬁnite element analysis. The developed method is used to measure the imperfections of 12 VAFW cylinders. The mass of the cylinders is used to validate the geometric imperfections and evaluate the variability of the proposed methodology. Highlights •A new method is proposed to measure geometric imperfections in composite cylinders •The setup requires only a pair of digital image correlation cameras •Optimization steps are used to convert the DIC raw data into a 3D geometric imperfections •The method developed is tested on 12 variable-angle ﬁlament-wound cylinders 1. Introduction The buckling performance of thin-walled structures can be signiﬁcantly aﬀected by uncertainties coming from load deﬁnitions, material properties, geometric variables and boundary conditions [1]. Fiber-reinforced composite ma- terials are often employed in primary aeronautical and aerospacecomponentsbecauseoftheiroutstandingspeciﬁc strength and stiﬀness, corrosion resistance, and high tai- lorability that allows design solutions not achievable with metals [2]. Aerospace structures usually consist of thin- walled shells that can buckle when subject to compressive loads. Curved shells, especially cylinders, can build a high membrane stress level under compression before buckling, becoming peculiarly sensitive to geometric imperfections [3]. ThisphenomenonwasﬁrstobservedbySouthwell[4]in 1914,whenisotropiccylindricalshellswerefoundtobuckle at considerably lower loads than theoretically predicted (for <Corresponding Author S.G.P.Castro@tudelft.nl (S.G.P. Castro); humberto.almeida@aalto.fi (J.H.S. Almeida Jr.); luc.st-pierre@aalto.fi (L. St-Pierre); zhwanglbb@gmail.com (Z. Wang) ORCID(s): 0000-0001-9711-0991 (S.G.P. Castro); 0000-0002-9408-7674 (J.H.S. Almeida Jr.); 0000-0003-3857-5398 (L. St-Pierre); 0000-0001-8930-4817 (Z. Wang)geometrically perfect cylinders). The ﬁrst buckling design criteria for imperfect structures were proposed by Flügge [5]andDonnell[6],buttheseguidelinesweremainlybased on empiricism. The classical reports from von Kármán and Tsien [7] and Koiter [8] conﬁrmed the assumption that ini- tialgeometricimperfections,asdeviationsfromanidealized geometry, are the primary source of discrepancy between predictions and experiments. These empirical factors, cre- atedtopenalizethegeometryandthenreferredtoasknock- down factors (KDF), were applied with the objective of re- ducingtheconservativepredictionsfortheloadcarryingca- pacity. Although recognizably conservative, this method is still largely applied in preliminary design, such as with the NASA SP-8007 [9]. The ﬁrst work on imperfect ﬁber-reinforced composite shells was reported by Sheinman and Simitses [10]; their analyticalmodelincludedaxisymmetric-typegeometricim- perfections. They concluded that the imperfection sensitiv- ityoforthotropicshellsinaxialcompressiondecreaseswith bothincreasingradius-to-thickness( r_tratio)andlength-to- thickness (l_tratio) values. Their assumptions were later conﬁrmed[11]withaxialcompressiontestsonboron/epoxy composite cylinders [12]. Nevertheless, these assumptions SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 1 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup were still considered as semi-empiric. The ﬁrst direct mea- surementofgeometricimperfectionswasreportedbyChrys- santhopoulos et al. [13] for aramid/epoxy composite cylin- ders. They employed a linear voltage displacement trans- ducer(LVDT)tomeasureinnerandoutersurfaces,inwhich the imperfections were recorded in an interval of 10mm axially and20mm circumferentially. The external surface of the cylinder was adjusted using the so called “best-ﬁt” procedureoriginallyproposedbyArboczandBabcock[14]. Thenceforth, the number of reports and methods increased substantially and it is well established that considering im- perfections is vital to accurately predict the actual load car- rying ability of shells [15]. Numerousapproacheshavebeendevelopedtoincludeim- perfections in numerical simulations. One of the most dis- seminatedmethodisthelinearbucklingmode-shapeimper- fections (LBMI), initially proposed by Khot and Venkayya [16]. This approach uses axisymmetric mode-shapes, ob- tained through a linear buckling analysis, as geometric im- perfections in a nonlinear analysis [17]. An advantage of the LBMI method is that the imperfection pattern is ob- tainedfromacomputationallyinexpensiveeigenvaluebuck- linganalysis,anditisstraightforwardtoincludesuchimper- fections as an initial state to create a load path to reach the post-bucklingstate[18]. Anotherrecent,butperhapsequally well–disseminatedapproach,isthe“singleperturbationload approach(SPLA)”,originallyproposedbyHühneetal. [19]. This technique uses a small lateral load, prior or in addi- tion to the axial compression, to trigger buckling. How- ever, the SPLA does not consider other types of imperfec- tions, e.g. load asymmetries [20]. Although LBMI, SPLA, and axisymmetric imperfections approaches can be realis- tic for shells with well-known or smooth surfaces, they can provide, nevertheless, inaccurate predictions for structures with complex architectures, such as modern cylinders with variable–angletow(VAT)and/orvariable–thicknesscharac- ter [21] produced via advanced manufacturing techniques, such as ﬁlament winding. In these cases, it would be more appropriate to measure the imperfections after manufactur- ing,andthiswouldrequirecapturingmid–surfaceimperfec- tions (MSI) and variations in thickness [22]. Degenhardt et al. [23] carried out an in-depth imper- fections measurement of carbon ﬁber–reinforced polymer (CFRP) unstiﬀened cylinders. Firstly, they used automatic ultrasonictestingwithwatersplitcouplingtodetectanyde- fects in the structures. This method can also be employed to take full ﬁeld thickness measurements. Secondly, they used an optical 3D digitizing system based on photogram- metrytoextract,usingthebest–ﬁtmethod,theactualradius ofthecylindersandtheirinitialgeometricimperfections. To achieve this, they used four high–speed cameras simultane- ously to scan the whole cylinder, each covering 90 °of the shell. Althoughprecise,thismeasurementsystemisveryex- pensive and complex to put in place. A similar system was successfullyemployedbyKhakimovaetal. [24]tomeasure thicknessimperfectionsinunstiﬀenedCFRPcylinders,also using the best–ﬁt method. Otherwise, Eberlein [25] mea-sured the imperfection signature of CFRP cylinders using a light scanner, a system also used by NASA. The measure- ments led to propose a KDF of 0.91, signiﬁcantly less con- servativethanthevalueof0.59recommendedinNASASP- 8007 guideline. Another approach is to use a laser scanner, asproposedbySkukisetal. [26]. Thebest-ﬁt-cylinderalgo- rithm was used to eliminate rigid body motion modes from the measurements. Their method, however, was not able to cover360ýof the cylinder and an error of 15% between ex- periments and predictions was found. Labans and Bisagni [27] measured the imperfections of VAT cylinders using a hybridsystem,inwhichtheoutersurfacewasscannedusing digitalimagecorrelation(DIC),andtheinnersurfacewitha laserdistancesensor. Asakeyachievement,theyfoundthat VAT cylinders are less sensitive to geometric imperfections thanconstant–stiﬀnessshells. Finally,afewconclusionscan be drawn from these studies: •thesetupsnecessarytomeasureimperfectionsarevery expensive and complex; •thereisnotawellestablishedproceduretocapture 360ý of a 3D structure; and •there is no work detailing the complete step-by-step procedure to measure the mid-surface imperfection of a 3D structure via DIC. Inadditiontotheseidentiﬁedissues,itiswell–established that ﬁlament winding (FW) [28] is the most suitable and fastest manufacturing method to produce composite cylin- ders. Moreover, after the very ﬁrst report in the literature on variable–angle ﬁlament–wound (VAFW) cylinders car- ried out by Wang et al. [21], on reliability–based design andoptimization,itbecomesevidentthatrealgeometricim- perfectionsandtheireﬀectsonthemechanicalperformance of ﬁlament–wound shells is yet unknown and unexplored. Almeida Jr. et al. [29] demonstrated the superior tailoring potential and mechanical properties of VAFW cylinders. Therefore, this work aims at overcoming all these is- sues by proposing an original and less costly methodology to measure geometric imperfections of VAFW composite cylinders through DIC using only a pair of stereo cameras, stillbeingabletocoverthewholecircumferenceofthestruc- ture. The methodology on how to stitch each individual measurementtoreconstructthefullshellimperfectionisde- scribed. First, a new best–ﬁt algorithm is developed to ac- curatelydeterminetheoutersurfaceofthecylinderforeach measurement. Thus, each measurement is rotated and the adjacentimperfectiondatacomparedtodeterminetheexact rotationanglethatperfectlystitcheseachmeasurement. Af- ter trimming each rotated measurement, the complete shell imperfection is achieved. 2. Digital Image Correlation (DIC) System Digital Image Correlation (DIC) is an optical method mostly used to measure deformations on a surface of a 3D object. The method tracks the gray value pattern in small SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 2 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup neighborhoods, called subsets, during deformation. This technique has been proven to be accurate when compared to FE predictions. The commercially available VIC-2D and VIC-3D systems from isi-sys Optical Measurement Solu- tionsGmbH[30]areusedinthisstudy. Inthepresentstudy, the images are captured by a pair of 9 mega-pixel cameras with 50mm lenses. 3. Imperfection Measurements 3.1. Cylinders and Speckle Patterns Fourtypesofﬁlament-woundcylinderconﬁgurationsare considered in this study, all having an angle–ply ,layer that can vary along the axial direction. The four designs optimized in a previous study [29] are: i) MArepresent- ing the minimum winding angle, with a constant winding angle of,50ý; ii)CAwith a constant angle of ,58ý; iii) VAFW4wherethecylinderisdividedintofourframesover the length, and each frame has an angle of ,59ý; and iv) VAFW8witheightframesanditthefollowingwindingan- gle sequence: <55|57|61|57|57|61|57|55>. All cylinders are symmetric with respect to their mid–length, have a length L=300mmandahomogeneousinnerradius rcyl=68mm. The material is a carbon/epoxy Toray T 700-12K-50C fromTCRComposites. TheFWsystemconsistsofaKUKA robotmodelKR140L100withMFTechcontrolandperiph- eral devices and it has seven degrees of freedom (six axes plus the mandrel rotation). The cylinders are designed in CADWINDFWsoftware[31]. Themandrelpluscomposite laminate system is cured in an oven with air circulation at 105ýCfor24h. Thesystemiscooleddowntoroomtemper- ature, and the cylinder is extracted from the mandrel. The four types of cylinders are shown in Figure 2, and three cylinders are produced for each family. The cylinders arecastintoametal–ﬁllertoughenedepoxyresintoimprove theloaddistributionduringthecompressiontests(Figure1). Aftercasting,allcylindersarepaintedwithawhitelayerfol- lowed by a speckled pattern. In all cases, the speckle paint- ing is performed by positioning the black painting spray at about0.5mawayfromthecylinders,pointingitslightlytan- gentially to the cylinder and gently pressing the spray noz- zle. Theexhaustionaircarriedthepaintingmisttowardsthe cylinder. Nonetheless,thereisaconsiderablediﬀerencebe- tween the speckled patterns for diﬀerent samples, as shown in Figure 3. For instance, cylinder VAFW4-1 and MA-2 have almost no speckles, whereas MA-1, CA-3, VAFW4- 2, VAFW4-3, VAFW8-1 and VAFW8-2 have rather dense speckled patterns. All other samples have patterns within this qualitative range. The size of the speckles is kept small tominimizethelossofimperfectiondataduetolargeblack- dotted regions. 3.2. DIC Setup TheDICcamerasarepositionedatabout700mmfromthe cylinder, and their height is approximately at the cylinder’s mid-plane. A sturdy table is used to support the cylinders during measurements. The computer is placed in a conve- nient way to look at the monitor while adjusting light and lp = 15 mm ��p = 15 mmlp = 15 mm ��p = 15 mm lc = 270 mmFigure 1: Resin potting details, where lcis the cylinder length, lpis the resin potting length, and  pis the potting diameter variation. MA-1 MA-2 MA-3 CA-1 CA-2 CA-3 VAFW4-1 VAFW4-2 VAFW4-3 VAFW8-1 VAFW8-2 VAFW8-3 Figure 2: Filament-wound cylinders used in the imperfection measurements. focus, as shown in Figure 4. The position of the cylinder is marked on the table with a pen; this allows to rotate the samples without changing their original position for which the cameras are calibrated. 3.3. Lighting and Focus Adjustments Withthecylinderinplace,thefollowingstepsaretakento adjust the ﬁnal camera position, camera exposure, artiﬁcial light and focus. 1) Initial adjustments in artiﬁcial light and camera ex- posuretime . Thecamerasarepositionedaccordingtotheir aperturesizeandavailablespaceinthelab,asdiscussedbe- fore. Withthecapturesystemon,theinitialbrightnessisver- iﬁed by adjusting the LED panel intensity and position. In SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 3 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup MA-1 MA-2 MA-3 CA-1 CA-2 CA-3 VAFW4-1 VAFW4-2 VAFW4-3 VAFW8-1 VAFW8-2 VAFW8-3 Figure 3: Speckled patterns applied to each sample. Sparse and dense speckled patterns are preferred, avoiding large black areas to prevent loss of measured imperfection data. Figure 4: Experimental setup for the DIC measurements. thepresentexperimentalsetup,theambientlightisenhanced withtheLEDpanel. Thecameraexposuretimerangesfrom 20 to 150ms, depending on the varying ambient light con- ditions that is described in the following paragraph. 2) Positioning the cameras . With suﬃcient brightness, the ﬁnal positioning of the cameras takes place. The cameras are moved laterally and rotated along two axes to make the cylinders appear in the middle of the image, and with both sidesalignedwiththeverticaledgesofthephotographs. The positionissecuredbyscrewingtheadjustmentboltstightly. 3) Focus adjustment . The focus adjustment for a cylin- drical surface requires ﬁnding a compromise between sur- face depth-coverage and accuracy. Therefore, the optimal focusisnotadjustedtobeattheouter-mostface,butslightly deeper to increase the accuracy on surface points that are Location of experimental setupSunlight in the morningSunlight in the afternoon Figure 5: DelftAerospaceStructuresand MaterialsLaboratory (DASML) ambient lighting. The varying light aﬀects the mea- surements, requiring on-the-ﬂy adjustments of artiﬁcial light and camera exposure time. more distant from the cameras, see Figure 6. Even with thiscompromise,thesurfacetopographyislessaccuratefor pointsthatarefurtherfromthecamera,wherethepredicted pixel noise can increase up to 5%. Figure 6: Adjusted focus with a compromise between accuracy and surface depth coverage. 4) Final brightness adjustment . Ideally, one should use a camera exposure time as short as possible, with suﬃ- cient diﬀuse, ambient, light to provide suﬃcient brightness. Lighting devices, however, usually generate a focused light that can cause undesirable reﬂections on a cylindrical shell. Therefore, ﬁnding the perfect lighting conditions can be somewhatchallenging,especiallyiftheambientlightcondi- tionsvaryduringtheexperiments. Forinstance,intheDelft Aerospace Structures and Materials Laboratory (DASML), the sunlight switches from one side in the morning to the other side in the afternoon. Moreover, the light conditions change quickly during the experiments because sunlight is SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 4 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup often blocked by passing clouds, as illustrated in Figure 5. Futuresolutionsforabettermeasurementsetupcouldmake use of a tent to minimize the eﬀect of ambient light and ad- ditionalLEDpanelstoguaranteesuﬃcientartiﬁciallightto outshine the ambient one. This would also have the beneﬁt of allowing minimal exposure times, making the measure- mentslesssusceptibletodisturbances. TheLEDpanelillus- trated in Figure 4 is used to compensate these fast changing light conditions, such that the methodology proposed here can be employed in nearly any environment with varying natural light conditions. It is worth mentioning that in the worst case one picture took 15min to be taken until suﬃ- ciently good brightness conditions are achieved. Figure 7 compares the various brightness conditions as presented by the VIC SNAP 3D software [30]. Too dark Dark Good Bright Too bright Figure 7: DICbrightnessconditions. Darkconditionsarefound if the exposure time is too small and there is no suﬃcient envi- ronmental light. Focused light produces undesirable reﬂections on the cylindrical surface. 3.4. Calibrating the DIC System A calibration is performed before each measurement. Three measurements are carried out per cylinder shown in Figure 2, resulting in a total of 36 measurements. The cali- brationishighlyaﬀectedbythefocusadjustmentsexplained earlier, such that before each calibration the focus must be checked and adjusted whenever necessary. The marked po- sition of the cylinder on the table guarantees that the cylin- drical shell remains in the region for which the focus was calibrated, which is crucial since the measurement process requiresrotatingthecylinderinincrementsof 60ý. Asillus- trated in Figure 8, a calibration pattern of 5mm is used and placedatninepositionsthatdeﬁnetheextremitiesofthere- gion of interest being measured. The calibration pattern is always rotated to follow the cylindrical surface to improve theaccuracyofthefurthestmeasuredpoints. Thepixelerror producedbythecalibrationprocedurecanbecalculated,and among all measurements herein reported this error ranges from0.014pixelsto0.049pixels. Alongthe300mmlength of the cylinder there are approximately 3300 pixels, corre- spondingto11pixels/mm;hence,theexpectedmeasurement error ranges from 0.001mm to 0.004mm. This is the ex- pectederrorintheregionscoveredbythecalibrationpattern. Surfacepointslayingbeyondthecalibratedareashowlargererrors and are ignored in the stitching procedure discussed next. 3.5. DIC Raw Data The DIC raw data collected in this study is publically available [32] and consists of a cloud of points xi;yi;zifor i=1;2;§;npointsrepresenting the topography of the mea- sured cylinders at each circumferential position. All points areexpressedintermsoftheDICcoordinatesystem XYZ, depicted in Figure 9. The position of the cylinders is marked on the measure- ment table by drawing the contour of the sample with a markerpen,asmentionedearlier. Anarbitraryreferencefor thecircumferentialpositionthatcorrespondstothe 0ýangle isdeﬁnedonthetable. Meanwhile,allcylindersaremarked attheresinpottingregionatevery 60ý,suchthatatotalof6 circumferential position markers are created. The DIC sys- tem is calibrated once when the cylinder is at the circum- ferential position 0ý. When the cylinder is rotated, bright- ness and focus conditions are always veriﬁed, as explained earlier. One DIC picture is taken from each circumferential position, as shown in Figure 10. The next step consists of stitching the six images into a closed topographic measure- ment of the entire cylinder. Note, as an example, that De- genhardt et al. [23] used a high-speed DIC setup consisting of 4 pairs of synchronized cameras for a complete coverage of the cylinder circumference, collecting up to 1000 images per second. A similar system could have been used to mea- suredthemid-surfaceimperfection,butsuchhardwaresetup is considerably more costly than the one proposed here. 4. Best-ﬁtting the DIC raw data For each stereo picture, a partial topography of the cylin- der is obtained as a cloud of points. A very large number ofpointsispresentsincethecamerasusedhavearesolution of 9 mega–pixels. To reduce the amount of data, 1 out of every 4 points was used in both horizontal and vertical di- rections. Even with this data reduction, the resolution is or- dersofmagnitudesmallerthanatypicalﬁniteelementmesh used to predict the collapse of imperfect cylindrical shells [18, 33]. With the aim to facilitate the subsequent stitching proce- dure,andtocreateastandardformatforallgeometricimper- fections, all DIC raw data points are transformed to a com- mon cylindrical coordinate system, see Figure 11. The data pointsxi;yi;ziaretransformedtonewcoordinates xc;yc;zc according to: h n l njxc yc zci n m nk=[Rz][Ry][Rx]h n l njxi+x0 yi+y0 zi+z0i n m nk+h n l nj0 0 z1i n m nk(1) wherex0;y0;z0representtranslationoﬀsetsbeforerotation, [Rz][Ry][Rx]arerotationmatrices,and z1isasecondtrans- lationoﬀsetappliedafterrotation. Thistwo-steptranslation SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 5 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup Figure 8: DIC calibration images for sample MA1-1. The calibration pattern of 5mm is placed at nine positions deﬁning the extremities of the region of interest. xyzDIC coordinate system DIC raw data xyz Figure 9: DIC coordinate system and example of raw data. The raw data consists of a topological representation of the cylinder outer surface, discretized by points. Position 1Position 2Position 5Position 6 Position 4 Position 3 Figure 10: DIC raw data extracted at each circumferential position. All this data is obtained on the same coordinate system shown in Figure 9. All adjacent positions are 60ýapart. strategyisnotstrictlynecessary,butitenabledasigniﬁcantly faster convergence of the surface ﬁtting algorithm. The ro- tation matrices [Rz][Ry][Rx]are given by: [Rx]=b f fd1 0 0 0 cos *sin 0 sin cos c g ge(2)[Ry]=b f fdcos 0 sin 0 1 0 *sin 0 cos c g ge(3) [Rz]=b f fdcos *sin 0 sin cos 0 0 0 1c g ge(4) where ; ; are transformation angles around the x;y;z axes,respectively,andmeasuredaccordingtotheright-hand rule, see Figure 11. xyz Raw data First TranslationRotation and second translationxyz x yz Figure 11: Best-ﬁt cylinder. Each set of raw data measured at diﬀerentpositionsistransformedtoareferencecylinderdeﬁned using the DIC coordinate system of Figure 9. A total of seven parameters ( ; ; ;x0;y0;z0;z1) are needed in Equation 1 to transform the DIC raw data points xi;yi;zito the desired coordinate system xc;yc;zc. When ﬁtting a perfect cylinder, is not needed and can be set to = 0. The other six parameters are obtained from a ﬁrst SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 6 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup best-ﬁtting optimization governed by: minimize³npoints c=1t x2 c+y2 c*rcyl with respect to rcyl; ; ;x0;y0;z0 subject to rminfrcylfrmax *f ; ; f+ *Øfx0;y0;z0f+Ø(5) wherercylcan vary between rminandrmaxor be set to a constant value, and in the present study it is ﬁxed at rcyl= 68mm, which is the radius of the mandrel used to manu- facture the cylinders. Similarly to Wang et al. [34], the optimization of Eq. 5 is solved using a nonlinear least- squaresmethod,whichinthecaseofthepresentstudyisthe scipy.optimize.least_squares function available in SciPy [35]. Becausethefocusisnotuniformthroughoutthecylindri- cal surface, points outside the focus plane have a relatively largererrorthanthoseinfocus. Thesevariationscanbecor- rected using the best-ﬁt elliptic cylinder equation, detailed inAppendixA.Forthebest-ﬁtellipticcylinder,thetransfor- mation angle must be included as a design variable in the optimizationprocedure. Therefore,afterthebest-ﬁtcylinder optimization (Eq. 5 with a constant value of rcyl), a second optimization is conducted to determine ;a;baccording to: minimize³npoints c=1t x2 c+y2 c*r./ with respect to ;a;b subject to *f f+ 0:9rcylfa;bf1:1rcyl(6) where= arctanyc_xc; the elliptical radius r./is calcu- lated with Eq. A.2; a;bare respectively the major and the minor radii of a best-ﬁt elliptic cylinder, to be determined by the nonlinear least-squares optimizer. The next best-ﬁt step consists of calculating the axial translationz1that is used in Eq. 1. This is done based on: minimize³npoints c=1zc with respect to z1 subject to *Øfz1f+Ø(7) withzccalculated using Eq. 8 that represents the sum of distances of all points laying outside the inner cylindrical shell domain delimited using the cylinder height H, and onthe length of the resin potting lpillustrated in Fig. 1. zc=h n l njzc=zc*.H*lp/zc>.H*lp/ zc=0 lp<zc<.H*lp/ zc=lp*zc zc<lp(8) Note that the DIC raw data may contain points located in the resin potting areas or even on the table, and these can have a negative eﬀect on the best-ﬁt optimization. To avoid this,theDICrawdatapointsareclippedbeforetheoptimiza- tion with the following clipping box: xmin;xmax=*Ø;+Ø ymin;ymax=*130mm;+120mm zmin;zmax(9) wherezmax=zimax+1[mm]andzmin=zmax*R.1*cos45ý/*1[mm]. The value zimaxis determined for each DIC raw data corresponding to one circumferential position, as shown in Figure 9. The limits ymin;ymaxare chosen to remove spurious points from the resin potting areas. Despite the points outside the clipping box are ignored during the best-ﬁt optimization steps, they alsoundergothetransformationgivenbyEq. 1andarekept for the stitching operation, which is described next. 5. Stitching methodology Afterbest-ﬁttingtheDICrawdatapointstothedesiredco- ordinate system illustrated in Figure 11, each measurement is rotated to its nominal circumferential position, as shown in Figure 12. Simply trimming the data of each angular po- sitionandthenassemblingeachsectionwouldcreateatopo- graphicpatternwithsigniﬁcantdiscontinuities,asillustrated in Figure 13a. To correct this issue, an automated stitching scheme is proposed to create smooth three-dimensional im- perfection patterns, as shown in Figure 13b. Each pair of adjacent DIC measurements, named posi- tionsiandi+1,arestitchedseparately. Aprobinglineisde- ﬁned at the circumferential position that corresponds to the frontierbetweenpositions iandi+1. Forinstance,between positions1and2theprobinglineislocatedat *30ý;between 2and3at*90ýand so forth. The probing line is located at a coordinate that corresponds to the mid-surface, here as- sumedtobeatthemeanvalueofall rc=t x2 c+y2 c,withrc representing the radial coordinates of all points among po- sitionsiandi+1. During the stitching process, the points belongingtoposition i+1areadjustedcircumferentiallyby avaluei+1andlongitudinallybyavalue zi+1,according SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 7 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup Figure 12: Initial rotation of each DIC transformed data to their nominal position. to: minimize³np c=1rci*rci+12 with respect to i+1;zi+1 subject to *10ýfi+1f+10ý *10mmfzi+1f+10mm(10) wherenpis the number of points along the probing line, here ﬁxed at np= 1000;rci;rci+1are the out-of-plane distances between the probing line and the positions iand i+1, respectively. Figure 14 illustrates the stitching vari- ablesi+1;zi+1, and includes a plot of rciandrci+1 as a function of position along the probing line. For each cylinder, the optimization in Eq. 10 is run six times. First, position 1 is taken as a reference (Figure 15a) and the ﬁrst stitching optimization ﬁnds 2;z2(Figure 15b). This is repeated for each probing line until the ﬁnal stitching step ﬁnds the adjustment required for position 1, 1andz1. In the end, all iandzifori=2;3;4;5;6, can be oﬀset such that1=0andz1=0. The initial guess for i;ziin the optimization of Eq. 10 can have a signiﬁcant inﬂuence on the result. Therefore, theoptimizationforeach i;ziisbasedonatotalof121 (a) Wrong stitching (b) Correct stitching Figure 13: Comparison between a wrong and a correct stitch- ing. The wrong stitching is simply achieved by rotating and trimming. The correct stitching requires rotating, adjusting according to the optimization given in Eq. 10, and ﬁnally trimming. xyzi+1 probing line Length along the probing line, [mm]Position i Position i+1Out-of-plane distance from probing line, [mm] Figure 14: Variablesi+1andzi+1used to adjust position i+1inordertominimizethediﬀerenceofout-of-planedistance between positions iandi+1, measured from the probing line. equallyspacedinitialpoints,coveringcircumferentialoﬀset values ranging from *10ýto+10ý, and axial oﬀset values ranging from *20mm to+20mm. The best initial guess is determined by ﬁnding the initial point i;zithat re- sults in the minimum value of³np i=1ri*ri+12, which isthesamefunctionbeingminimizedintheoptimizationde- scribed in Eq. 10. 6. Stitching Results The steel-based mandrel used to manufacture the ﬁlament–woundcylindershasasmoothsurface,resultingin a very low imperfection signature at the inner surface. Fig- ure 16 shows the inner surface of cylinder CA-2, where the diﬀerent tow orientations can be seen. Assuming a smooth inner surface, the three-dimensional stitched outer-surface SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 8 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup Figure 15: Stitching steps of each DIC measurement towards the ﬁnal three-dimensional geometric imperfection. topology can be used to calculate the thickness imperfec- tion patterns generated by the FW manufacturing process. Figures18–21showthestitchedimperfectionsforthefour familiesofcylindersconsideredinthisstudy(seeFigure2). Ineachﬁgure,theresultsareshownforthreediﬀerentcylin- ders(rows)andeachmeasurementwasrepeatedthreetimes (columns) to evaluate the variability of the proposed tech- nique. The diﬀerences between each column are negligible, whichdemonstratesthattheproposedtechniqueoﬀersahigh repeatability. For each family of cylinder, there are notice- ablediﬀerencesbetweenthethreesamples(rows),indicating that the technique has the accuracy needed to detect small diﬀerences in the imperfection patterns. Finally, note that these results are available in a public dataset [36]. The thickness averaged over the circumference havg.z/is plottedinFigure17asafunctionoftheaxialcoordinate zfor one cylinder in each family. Results are given for 25mmf zf275mmonly to exclude the regions close to the resin potting. Oneapproachtoverifythesethicknessproﬁlesisto use them to calculate the mass of the cylinder and compare this result to their actual mass. The mass of each cylinder Figure 16: Inner surface of ﬁlament-wound cylinder VAFW4-3. The smooth mandrel produces a signiﬁcantly smoother inner surface on the cylinder compared to the outer surface. Table 1 Massesin[g]ofeachcylindercalculatedfromthereconstructed imperfection data. The values mrealrepresent the real masses of each cylinder measured after manufacturing. Cylinder mrealm1m2m3mavg,mavg MA-1 148148.05 148.06 148.08 148:06,0:01 MA-2 148.13 148.18 148.14 148:15,0:02 MA-3 148.03 148.07 148.05 148:05,0:02 CA-1 146146.04 146.07 146.06 146:06,0:01 CA-2 145.97 145.98 145.97 145:97,0:01 CA-3 145.97 145.97 145.97 145:97,0:00 VAFW4-1 149149.01 149.02 149.03 149:02,0:01 VAFW4-2 149.12 149.16 149.10 149:13,0:03 VAFW4-3 149.07 149.08 149.02 149:06,0:03 VAFW8-1 165164.77 164.77 164.77 164:77,0:00 VAFW8-2 164.67 164.66 164.65 164:66,0:01 VAFW8-3 164.71 164.71 164.69 164:70,0:01 can be calculated by: mcyl=Ê300mm z=0mmÊ2 =0Êrcyl+h.r;/ r=rcylrdrddz (11) whereh.r;/is the local thickness measurement; = 1600kg/m3isthedensityofthematerial;and rcyl=68mm is the mandrel radius. Assuming an average thickness SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 9 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup havg.z/, this equation can be approximated as: mcylù2rcyl300 250Êz=275mm z=25mmhavg.z/dz (12) where the ratio 300_250 must be used to compensate the missing region close to the edges. Table 1 presents the cal- culated mass for all reconstructed geometric imperfections. The columns m1;m2;m3represent the three DIC measure- mentsforeachsample. Thestandarddeviationofthecalcu- lated mass is extremely small, with a maximum of ,0:03g. The real mass of each sample, measured after manufactur- ing, is also included in Table 1. In all cases, there is an excellent agreement between the calculated and real mass, whichisevidencethatthemethodologyproposedhereoﬀers a high level of accuracy to measure geometric imperfection proﬁles. /uni00000018/uni00000013 /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000018/uni00000013 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 /uni00000026/uni00000024/uni00000010/uni00000016/uni00000010/uni00000016 /uni00000039/uni00000024/uni00000029/uni0000003a/uni00000017/uni00000010/uni00000016/uni00000010/uni00000016 /uni00000039/uni00000024/uni00000029/uni0000003a/uni0000001b/uni00000010/uni00000016/uni00000010/uni00000014 Figure 17: Average thickness distributions havg.z/for diﬀerent VAFW designs. 7. Conclusions A novel methodology was presented to measure the ge- ometric imperfection pattern of composite cylinders using digital image correlation (DIC). The experimental setup is simple and requires only a pair of cameras. DIC is largely applied as a technique to measure displacement and strain ﬁelds, and the present study is the ﬁrst one to extend its applicability towards measuring geometric imperfections. The method proposed here covers all steps of the process from image acquisition to the post-processing and stitching needed to reconstruct three-dimensional imperfection pat- terns from DIC raw data. The proposed methodology is applicable to any shell of revolution and assumes that the inner mold surface is smooth, such as in the case of ﬁlament-wound structures, and a large class of shells manufactured by means of a smooth inner mold. Future studies should focus on uncertainty quantiﬁca- tion and reliability-based analyses to assess the sensitivity of the stability behavior of variable-angle ﬁlament-wound(VAFW) cylinders to the error of the herein proposed im- perfection measurement technique, and preferably cross- checking with other imperfection measurement techniques, for small and large structures. Acknowledgements TheauthorswanttothankDaveRuijtenbeek,JohanBoen- der, Victor Horbowiec, Fred Bosch and Berthil Grashof for theirgreatsupport,andforalltheteaminvolvedwiththead- ministrationandoperationoftheamazingAerospaceStruc- tures and Materials Laboratory (DASML). Also, thanks to JavierGutierrezAlvarezforgivingimportanthintsaboutthe DASML DIC system. CRediT authorship contribution statement Saullo G. P. Castro: Conceptualization, methodology, software, formal analysis, investigation, validation, data cu- ration,supervision,resources,writing-originaldraftprepa- ration.José Humberto S. Almeida Jr.: Formal analysis, investigation,fundingacquisition,validation,writing-orig- inal draft preparation, writing - reviewing and editing. Luc St-Pierre: Writing-reviewingandediting,resources. Zhi- hua Wang: Investigation, methodology. A. Elliptic cylinder The stitching procedure can be improved by ﬁtting the data points to an elliptic cylinder instead of a circular one. UsingthecoordinatesysteminFigure11,theperfectelliptic perimeter can be represented in Cartesian coordinates as: x2 a2+y2 b2=1 (A.1) wherex;yarepointslayingperfectlyontheellipse’ssurface, aisthemajorand btheminorradiioftheellipse. Substitut- ingxandywith the polar relations: x=r./cos y =r./sin into Eq. A.1 gives the perfect ellipse’s radius r./: r./=abt .asin/2+.bcos/2(A.2) First,allmeasuredDICpointsaretransformedto xc;yc;zc coordinates according to Eq. 1. Second, the measured ra- diusofeachpoint, rc,anditscorrespondingcircumferential positionc, can be calculated as: c=tan*1yc_xc rc=t x2 c+y2 c(A.3) Themeasuredradius rcdiﬀersfromtheperfectellipticcylin- der radius (for a speciﬁc circumferential position c). This SGP Castro, JHS Almeida Jr, L St-Pierre, Z Wang Preprint Page 10 of 14 Measuring geometric imperfections of VAFW cylinders with a simple DIC setup MA-1-1 MA-1-2 MA-1-3 MA-2-1 MA-2-2 MA-2-3 MA-3-1 MA-3-2 MA-3-3 Figure 18: Stitched thickness imperfection for MA- i-j:ithsample,jthmeasurement using the proposed DIC scheme. diﬀerence represents a geometric imperfection for this data point, and it is here represented by rc: rc=rc*r.c/ (A.4) This imperfection value can be remapped to a best-ﬁt cylinder, as the one illustrated in Figure 11, by adding the geometricimperfectiontothebest-ﬁtradius,suchthatanew radial position rcnewis calculated: rcnew=R+rc (A.5) Finally,thenewCartesiancoordinatesforeachdatapoint can be retrieved using: xcnew=rcnewcosc ycnew=rcnewsinc zcnew=zc(A.6)Data Availability Statement The raw data [32] and ﬁnal three-dimensional imperfec- tions [36] obtained in the present work are already publicly available. Thescriptsusedtobest-ﬁtandstitchtheDICdata can be made available under request per email to Saullo G. P. Castro.