A CAE Approach for the Stress Analysis of Gear Models

Int J Interact Des ManufDOI 10.1007/s12008-013-0201-4

ORIGINAL PAPER

A CAE approach for the stress analysis of gear modelsby 3D digital photoelasticity

Paola Forte Alessandro Paoli Armando Viviano Razionale

Received: 9 September 2013 / Accepted: 14 December 2013 Springer-Verlag France 2013

Abstract The use of numerical and experimental methodsto determine the stress field of mechanical components is wellknown. In particular, 3D photoelasticity can be consideredthe only experimental technique for the complete stress stateevaluation of 3D components. The advent of rapid prototyp-ing techniques has allowed the manufacturing of complexmodels in a matter of hours by using birifrangent materials.The present paper is focused on the description of a ComputerAided Engineering (CAE) approach which combines FiniteElement (FE) simulations and automatic photoelastic inves-tigations for the stress analysis of face gear drives, made bystereolithography. Computer Aided Design (CAD) geome-tries, used to manufacture the stereolithographic models, aredirectly used to perform FE analyses, thus allowing the stressanalysis process to become simpler and easier. The substan-tial agreement observed between experimental and numeri-cal results proved the potentialities of the adopted approachand the usefulness of FE simulations to optimize photoelasticanalyses through cost- and time-effective experiments evenfor complex 3D shapes.

Keywords 3D photoelasticity Stereolithography Face gear models FEA

P. Forte A. Paoli (B) A.V. RazionaleDepartment of Civil and Industrial Engineering, University of Pisa,Largo Lucio Lazzarino, n.1, 56126 Pisa, Italye-mail: [email protected]

P. Fortee-mail: [email protected]

A.V. Razionalee-mail: [email protected]

1 Introduction

Photoelasticity is a full-field technique which directly pro-vides the information of principal stress difference and theorientation of principal stress direction by fringe analysisof components made of birifrangent materials [1]. Nowa-days, numerical techniques, such as the finite element method(FEM), have progressively replaced experimental stressanalysis during prototype design. However, photoelasticitymay still play an important role in supporting numericalmethods with the aim of increasing confidence when complexanalyses must be carried out. In particular, three-dimensional(3D) photoelasticity can be considered the only techniquefor the experimental assessment of the complete stress statethroughout complex mechanical components.

3D photoelasticity typically presents two sequentialphases: the preparation of the models to be used for theexperimental tests and the fringe analysis step. The advent ofdigital image processing techniques (digital photoelasticity)[2] has allowed photoelastic analyses to become faster andmore accurate. On the other hand, the manufacturing andslicing of complex 3D photoelastic models still representsa troublesome task. In particular, the traditional methods formodel fabrication involve casting to shape or machining fromsolid blocks using thermo-setting resins. These processes areexpensive and time-consuming, with models typically requir-ing days to be completed.

In the last decades, rapid prototyping (RP) techniques haveprovided additive manufacturing processes, such as stere-olithography, which allow 3D components to be built in amatter of hours [3]. Recently, some photoelasticians haveanalyzed the optical and mechanical properties of stereoli-tographic resins (usually acrylate- and epoxy-based resins)[4,5]. These studies have been mainly focused on the inves-tigation of the influence of processing parameters, such as

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position and orientation of specimens in the resin bath, postcuring treatments, build style, and on the material shrinkageas well as mechanical and birifrangent properties. The resultspublished in the literature have evidenced that an interestingscenario can be opened within the full-field stress analysis ofcomplex models. In this context, few researches have aimedat analyzing the stress distribution in three-dimensional stere-olitographic complex specimens [6,7]. In particular, whenexperimental analyses concerning 3D photoelasticity are car-ried out, one of the most challenging issues relies on thedetermination of the most suitable load to be used for thestress freezing process. Large values of the applied load mayimproperly distort the model while too low values may resultin a poor optical response from slices cut from the 3D model.In addition, the mechanical slicing process requires the orig-inal model to be destroyed. A careful experimental planningprocess should then be guided by FE simulations in orderto determine the proper loading conditions and speed up thewhole measurement process.

The purpose of the present paper is to describe a CAEapproach, which combines FE simulations and photoelas-tic investigations for the stress analysis of complex 3D pro-totypes made by stereolithography. CAD geometries, usedto manufacture RP models, are directly used to performFE computations, thus allowing numerical simulations tobecome simpler and easier. In this paper, the photoelasticstress analysis of gear drives (spur gears and face gears) wascarried out by exploiting models made of an epoxy-basedresin through the stress-freezing technique. The face geartransmission, typically used in high performance aerospacedrive systems, consists of a driving spur involute pinion andone or two conjugated face gears. The lack of publisheddesign experience and design standards, the complex geome-

tries and unconventional operating conditions have led to theapplication of numerical simulations, mainly using finite ele-ment codes, for the analysis of their performances [8]. Suchan analysis involves multiple contacts, edge contacts, nonlinearity and high stress gradient areas and therefore it is agood example of how experimental data can help to integratenumerical results obtained during the design process. In par-ticular, the present work is aimed at highlighting potentiali-ties and criticalities of the photoelastic analysis of complex3D models guided by a FEM approach.

2 Materials and methods

Figure 1 schematizes the entire process of manufacturingand experimental testing of 3D complex stereolithographicgear models adopted in the present work. Stereolithographywas used to manufacture the models, geometrically definedby CAD. A simple but effective test apparatus was set-upwith the aim of loading gear samples and FE analyses wereused to support and plan the whole experimental activity.The experimental stress analysis was carried out by exploit-ing the principles of 3D photoelasticity which require stressfreezing and subsequent slicing of the resin models. Resultsaccomplished by numerical simulations have been also usedas comparison to assess those experimentally obtained.

2.1 Digital photoelasticity

Photoelasticity enables experimental stress analysis of two-and three-dimensional components to be performed byextracting information from two groups of interference fringepatterns emerging from a polariscope: isochromatics, which

Fig. 1 Scheme of the adopted process for the stress field experimental analysis of complex 3D gear models manufactured by stereolithography

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represent the contours of principal stress differences, and iso-clinics, which are related to principal stress directions. Iso-clinic and isochromatic fringe analysis yields two parametersfor each point of the test piece:

1. The relative retardation (known also as fringe order)introduced by the photoelastic material between twopolarized light beams transmitted through it, expressedin multiples of its wavelength. The relative retardation isproportional to the difference of principal stresses and isgiven, for a plane-stress state, by the relation:

= d (1 2)f (1)

where 1 and 2 (1 2) are the principal stresses, d isthe model thickness and f is the stress-optic coefficientfor the wavelength of the light source used.

2. The angle of orientation, , between the maximum prin-cipal stress 1 and the horizontal reference axis.

When 3D photoelastic analyses are carried out, the mod-els can be stress frozen and then mechanically sliced in orderto evaluate the stress field. In particular, the model is loadedand allowed to go through a thermal cycling process knownas stress freezing. At the end of the process, when the loadsare removed, the model retains the stress field induced by theapplied loads due to the bi-phase chemical structure of theresin [1]. Proper thin slices are then cut from the model andanalyzed by using the principles of 2D photoelasticity. Theslices retain the same stress distribution of the 3D model,provided the slicing operation is carried out carefully so thatno machining stresses are introduced. This restriction is usu-ally accomplished by using single-point cutting tools andproviding appropriate coolant while cutting.

A large class of problems, such as determination of stressconcentration factors, stress intensity factors and contactstress parameters, require only the information of isochro-matic fringe order [1]. In this work, among the various pro-cedures proposed in literature in order to recover the isochro-matic parameter [2], the photoelastic analysis of the fringepatterns was done by using the RGB method as proposedby [9]. RGB photoelasticity can give the total fringe orderfrom a single color isochromatic full-field image obtainedby using a white light source. This characteristic turns usefulnot only for the analysis of time varying phenomena but alsofor recovering small fringe orders from stress frozen slices.

2.1.1 RGB method

In RGB photoelasticity, the model is observed by a circularpolariscope (Fig. 2), generally configured in dark field. Theisochromatics are acquired in white light by a color camera

Fig. 2 Arrangement of the circular polariscope used for the RGB tech-nique

and decomposed into three primary colors, red, green andblue, by means of three wide band filters. The three levelsof intensities, R, G and B, are digitized and used to per-form full-field automatic analyses of isochromatics throughcomparison with calibrated values stored in a Look-Up-Table(LUT). The methodology is based on a calibration procedurewhich exploits a known stress field. The procedure consistsof acquiring RGB triplets at each pixel along a transversesymmetry section of a calibration bending beam and storingthe intensity values into an array (LUT). The RGB tripletscorrespond to retardations increasing linearly from a pointwhere the retardation is zero to a point subjected to a maxi-mum known retardation. In general, the material of the cali-bration specimen must match the material used in the test inorder to take into account the dispersion of birefringence.If the first pixel of the calibration specimen correspondsto a zero fringe order (retardation 0 = 0), the retardationi corresponding to pixel i , increasing with retardation, isgiven by:

i = Nm iim (2)

where Nm is the maximum relative retardation and im is thenumber of the acquired RGB triplets calibrating the system.The R, G and B values in the LUT are directly linked withthe retardation in the bending beam using monochromaticlight. During the photoelastic analysis, the levels Rm, Gm andBm are measured at those pixels in the specimen where theretardation has to be determined. Using the calibration array,the retardation corresponding to each set of Rm, Gm and Bmlevels can be evaluated. The general procedure consists of [9]:

1. comparing the Rm, Gm and Bm values digitized at eachpoint of the specimen with the Ri , Gi and Bi values storedin the LUT by means of the error function defined as:

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Ei = (Ri Rm)2 + (Gi Gm)2 + (Bi Bm)2 (3)

for each index i of the LUT;2. searching the value of the index i that minimizes the error

function (3);3. evaluating i by means of Eq. (2).

Due to the attenuation of the RGB levels for retardationshigher than 4 orders, the maximum measurable retardation islimited to three orders if incandescent lamps are used [9]. Inthis case, the retardation search procedure provides exact val-ues with the exception of a few pixels where a measurementerror can occur due to the similarity of colors of differentorders. In cases where the light source is a fluorescent lamphaving a discrete emission spectrum, it would be possible todetermine higher fringe orders (up to 12) as reported in [10].

2.2 Photoelastic stress analysis for stereolithographic gearmodels

The analysis of two different meshing conditions was carriedout in order to test the developed framework. Firstly, themeshing of a pair of involute spur gears was considered inorder to tune the procedure on a simple, known case. Then,the meshing of a face gear drive, consisting of a driving spurinvolute pinion and one conjugated face gear, was analyzed.

Fig. 3 CAD model of the face gear drive used for the stress fieldanalysis

2.2.1 Geometric models

Three-dimensional photoelastic gear models were directlymanufactured through a stereolithographic technique byusing the StL models obtained from CAD models (Fig. 3),thus reducing time and costs. While involute spur gears andpinions have a simple profile, face gear modeling involves thedescription of highly complex surface geometry. In this work,solid modeling of gear flank surfaces was done by interpolat-ing data obtained from an enveloping generation code [11].The face gear and the mating spur pinion were both modeledwith the commercially available solid modeler Pro/Engineer.Points from the generation code were imported in an ASCIIfile as a set of ordered points belonging to contact lines ofthe shaper and the face gear. These points were interpolatedwith NURBS surfaces approximating the flank and the filletof the face gear [8].

2.2.2 Stereolithographic models

All the models used in this work (Fig. 4) were manufacturedby stereolithography using the 3DSystems stereolithographicapparatus SLA 250/50 and an epoxy resin, CIBATOOLSL5170, which is photoelastically sensitive. In the currentwork the minimum possible layer thickness of 0.1 mm wasused with the Accurate Clean Epoxy Solid (ACES) build stylewhich produces extremely accurate, optically clear parts. Inorder to obtain models free of residual stress and birefrin-gence it is essential to achieve homogeneous material prop-erties. The standard build parameters employed in the SLAlead to substantially different material properties in the edgeand core of models. For this reason, various parameters wereadjusted to customize the process for our specific task accord-ing to [4]. However, a further high-temperature thermal cyclewas used to relieve the residual stresses after manufacture byphoto-curing.

Among the different specimen configurations and load-ing arrangements known for the calibration of the stress-optical coefficient, the simply supported prismatic beam inpure bending was chosen. For this purpose some prismatic

Fig. 4 Stereolithographic models used for the experimental analyses. a Meshing of involute spur gears, b, c spur involute pinion and mating facegear sector, respectively

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Fig. 5 a CAD model of theexperimental apparatus, baluminum profile bars used tofix the face gear sector to thereference plate

Fig. 6 Pictures of the experimental apparatus housed inside the oven:a involute spur gear meshing, b meshing of a face gear drive

specimens, having dimensions of 11 30 150 mm, weretested in addition to the gear prototypes. When RGB pho-toelasticity is applied, the same specimen material, lightingconditions as well as heat treatment (stress-freezing, anneal-ing) should be used for both the calibration and applicationsamples in order to avoid a color adaptation procedure whichsuitable modifies the LUT [1214]. For this reason, the speci-mens used to determine the calibration parameters (i.e., LUTfor the RGB method, and the stress-optic coefficient) wereobtained from the same container of the test gear models.

2.2.3 Experimental set-up

The experimental apparatus (Fig. 5a), designed to apply theload to the meshing gears, is very simple and small since ithas to be housed in an oven. Figure 6 shows two photos ofthe apparatus in the oven with two meshing spur gear sectors(Fig. 6a) and the pinion and face gear sectors (Fig. 6b). Itconsists of a reference plate (420 300 20 mm) used to

fix the face gear sector and of a horizontal drum (250 mmdiameter) to refer the spur involute pinion. The drum is free torotate around its axis and to move along its vertical positionthus allowing the relative distance of the gear pairs to beadjusted.

While the pinion sector is fixed to the flanges of the drumby means of threaded bolts, the reference features and fixingconditions to the plate are different in the cases of spur gearand face gear. The spur gear is fixed and referenced to theplate by means of an interface made of a U-shaped profileprovided with 4 slotted holes, for fixing the sector by meansof bolts in different meshing positions (Fig. 6a).The face gearsector is instead fixed and referenced directly to the plate bymeans of aluminum profile bars (as shown in Fig. 5b) andby a pin defining its radial placement. The torque is simplyapplied by calibrated weights hanged directly to the flangesof the drum or to cantilever beams fixed to the flanges inorder to increase to load arm (Fig. 6b).

The drum is hollow and some holes were drilled in theflanges with the aim of avoiding excessive heat accumula-tion in the model areas in contact with the apparatus, thuspreventing thermal distortions. Figure 6 also shows the place-ment of the calibration specimen, along with its four-point-bending setup, inside the oven. In order to avoid erroneousevaluations of the fringe order through the RGB method, theprismatic calibration specimen undergoes the same stressfreezing treatment of the test gear drives. This conditionalso allows a reliable and customized computation of thestress-optic material coefficient to be used for photoelasticanalyses. The images acquisition process was performed byusing the digital photocamera Nikon D1 having a reso-lution of 1,300 2,000 pixels for 16 millions of colors.The apparatus used for the thermal cycle is a forced air-heating oven equipped with a probe PT100 for temperaturecontrol and governed by the automatic controller ASCONXS3001/CAA/99999999.

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2.2.4 Stress freezing and model slicing process

Three-dimensional photoelasticity using the stress-freezingtechnique is effective on resin models that do not possessany residual stress due to the manufacturing process. Forthis reason, in order to remove the residual stresses in theprototypes caused by the post-curing resin shrinkage, all thesamples were annealed at resin glass transition temperature(Tg) before experimental tests. The stress-freezing cycle wasdivided into four steps (Fig. 7):

1. A linear temperature rise from room temperature to a tem-perature slightly below Tg . A rate of 4 C/h was used.

2. A constant temperature interval (from 5 to 10 h) to assureheat uniformity throughout the models. The interval widthwas determined accordingly to the size of the prototypesused.

3. A first cooling rate (2 C/h) down to 65 C.4. A second higher cooling rate (3 C/h) down to room tem-

perature. Once the critical stress freezing temperature (Tc)is reached, secondary bonds are re-formed between theelastically deformed primary bonds. For this reason, aslightly faster cooling rate can be used in order to speedup the thermal cycle.

Fig. 7 Stress-freezing thermal cycle adopted for the experimental tests

Even if loads are usually not applied to the test modelsuntil the stress-freezing temperature is reached, in this workthe loads were applied before the beginning of the stress-freezing and kept throughout the complete thermal cycle.This choice was adopted in order to avoid possible thermalshocks.

Model slicing was then performed in order to carry out adirect transmission photoelastic analysis. If the slice does notcorrespond to a principal plane, the stresses in the slice planeare secondary principal stresses. Since it is not always pos-sible to identify the principal planes, the slicing was carriedout through planes parallel to the free surfaces of the three-dimensional models. The slice thickness must be determinedto comply with two opposite requirements: the slice must be(1) sufficiently thin with respect to the model size in orderto ensure that the stresses do not change in either magnitudeor direction through the slice thickness, (2) sufficiently thickto assure a minimal number of fringe orders for an accuratephotoelastic analysis.

Slices having constant thickness (Fig. 8) were thenobtained by using a milling machine with a 2 mm thick cut-ter, rotating at high speed (1,500 rpm). The cutter must becooled with a large amount of coolant to prevent the modeloverheating, which would result in distortion or annealing ofthe fringe pattern.

After the slicing process, each slice was manually finishedby using 500-grit sandpaper. Moreover, to further augmenttheir transparency and the optical clarity of the fringe pat-terns, the slices were coated by a film of mineral oil.

2.2.5 Calibration of the model material

A prismatic specimen, in a four-point-bending set-up(Fig. 9a), was used to calibrate the photoelastic material prop-erties (stress-optic coefficient) at Tg [15]. If l2 is the loadingspan and (l2 + 2 l1) the support span, then in the central por-tion, at a distance of one beam thickness from the loadingpoints, the stress state is uniaxial and can be written as:

Fig. 8 Slicing process regarding involute spur gear a, calibration specimen (b) and face gear sector (c) respectively

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Fig. 9 a Prismatic specimen in a four-point-bending set-up used for the calibration of the material properties, b isocromatics emerging from adark field polariscope (in monochromatic and white light) after a stress-freezing cycle

x (y) = 6 P l1d h3 y (4)

where P is the total applied load. From Eq. (1), considering2 = 0, it derives that:

f = 6 P l1h3 |y|(y)

(5)

The fringe order varies linearly from 0, on the neutral axis,to its maximum values on lower and upper extremities ofthe specimen. A plot relating integer values of the fringeorder, extracted from isochromatics as observed with a cir-cular dark-field polariscope (Fig. 9b), with the distance fromthe inferior edge of the specimen can be obtained (Fig. 10).A linear regression is then performed by using the integerfringe orders in the central region of the specimen and usedto extrapolate the fringe orders Ni and Ns on the inferiorand superior specimen edges, respectively. The stress-opticcoefficient can then be computed by using relation (5) as:

f = 6 P l1h3 h

Ni + Ns (6)

Given h = 30 mm, l1 = 25 mm, d = 11 mm, l = 150 mmand P = 45 N, a mean value for the stress-optic coefficientf = 0.82 N/mm/fringe was determined.

Fig. 10 Fringe order plot with respect to the distance from the inferiorspecimen edge

The resin Elastic Modulus at Tg must also be determinedin order to carry out a finite element analysis. The elasticdeformation induced in the prismatic specimen at the stress-freezing temperature becomes permanent when it returns toroom temperature (Fig. 9b) and can be used to determine theresin elastic modulus at the stress-freezing temperature [15].A mean value for the elastic modulus E = 18 MPa was thusdetermined through experimental tests.

2.2.6 Finite element analysis

The finite element investigation was carried out in orderto define the loading set-up for the 3D photoelastic analy-sis. Loading conditions must be accurately estimated sincemechanical slicing destroys the model and only a single load-ing case can be considered for each model.

Tooth modeling and complete gear solid modeling, as wellas the relative placement of the gear pair in the meshingcycle, was performed with Pro/Engineer. The CAD mod-els were imported into the FEA environment by the IGESstandard data exchange to obtain a better control on meshgeneration. Mesh generation, pre-processing, and FE analy-sis were accomplished with Ansys. Meshing simulationimproves considerably if a regular mesh is used on contactsurfaces. This was achieved by using isoparametric brick ele-ments with 8 nodes and generating a volumetric solid meshby sweeping the elements created on the tooth front surface.

Fig. 11 Meshed involute spur gears (a) and face gear sector (b)

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Fig. 12 Constraints applied tothe involute spur sector (a) andface gear sector (b). c Masterand slave nodes used toconstrain the pinion sector torotate around its axis

Fig. 13 FEM principal stress difference maps for spur (a) and facegear (b) sectors

Contact and target elements, necessary to build the contactingpair FE model, were then defined to simulate gear meshingcondition. Whole toothed sectors were finally modeled using

coarser meshes for the unloaded teeth (Fig. 11). This allowsfor the numerical model to be as similar as possible to theexperimental one.

The spur gear sector model was constrained at its holes(Fig. 12a) while the face gear sector was constrained on thereference plate and the aluminum profile bars (Fig. 12b). Therotation of the pinion sector was obtained by defining a masternode on the pinion axis, free only to rotate, and linking it toslave nodes on the bolt holes through constraint equations(Fig. 12c). The loading condition is simulated by applying atorque around the rotation axis.

Particular care was taken in dealing with the initial contactcondition because it could cause convergence problems dueto rigid body motion or to highly non-linear contact load.In simulations where rigid body motions are constrainedonly by the presence of contact condition, contacting bod-ies should be modeled in just touching position to gen-

Fig. 14 Dark-field isochromatic fringes in white light observed for a 1.5 mm (a), 2 mm (b) and 3 mm (c) thick slices. d calibration array used tocreate the LUT for RGB photoelasticity and derived from a slice of the same specimen used for the stress-optic calibration

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erate a numerically stable problem and the torque should beapplied gradually, increasing it by steps. The material proper-ties were defined according to those of the resin at the stress-freezing temperature, obtained by the calibration procedures(E = 18 Mpa, = 0.48). Moreover, since the experimentaltests are carried out in the oven, temperature and its effectswere considered in simulations through a thermal expansioncoefficient.

Figure 13 shows an example of principal stresses differ-ence maps obtained by the FEM simulations. For the spurgear pair the applied torque to the pinion was 9,960 N mm,for the face gear pinion 5 104 N mm. The FEM modelswere also sectioned correspondingly to the slices obtainedfrom the resin models and the numerical results in terms ofprincipal stress difference distribution were compared to theexperimental ones, as will be shown in the following section.

Fig. 15 Full-field maps of the retardation obtained by processing the images shown in Fig. 14b (a) and Fig. 14c (b), by using the RGB method

Fig. 16 Isochromatics in white light relative to the contact area (a) and to the root of the tooth (b) in the 3 mm thick slice along with the respectiveretardation maps obtained with the RGB method (c, d)

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Fig. 17 Numerical andexperimental results of thedifference of the principalstresses along the contact line(a) and the direction normal tothe free edge of the tensile sideof the gear tooth fillet (b). Theexperimental results have beencalculated by using the RGBmethod for the 2 mm thick slice

3 Results

Figure 14 shows the isochromatic patterns in white light,as observed from slices cut within the spur gear pair, byusing a circular polariscope in the dark field configura-tion. The results are relative to a 1.5 mm (Fig. 14a), 2 mm(Fig. 14b) and 3 mm (Fig. 14c) thick slices. The higherfringe order observed in the thicker slice is due to the lin-ear dependence of the retardation with respect to the modelthickness.

The calibration array used for RGB photoelasticity wasderived from the isochromatics observed in the same spec-imen used for the stress-optical calibration and the integerfringe orders are obtained by using monochromatic light.The load to be applied to the calibration specimen mustbe evaluated with particular attention: the fringes of themodel can be compared to those of the calibration speci-men only if the light intensities are the same. It is there-fore necessary to obtain slices from the calibration spec-imen with thickness comparable to those to be analyzedalso containing the same fringe order to make the studyof the whole slice possible. Figure 15 reports, as example,

the full-field maps of the retardation obtained by process-ing, with the RGB method, the images shown in Fig. 14b, c,respectively.

Due to the high difference of the fringe gradient in themodel from that in the calibration specimen in the areas ofstress concentration, the assignment of the right value of theretardation can be difficult. That can bring the rise of spots onthe maps of the retardation (as shown in Fig. 15b) in boundaryareas that are themselves more subjected to disturbances. Tosolve this problem the image of these areas can be magnified(Fig. 16) to make the image fringe gradient slower than 0.1order/pixel.

Figure 17 presents the comparison between numerical andexperimental results along the contact line (Fig. 17) and alongthe direction normal to the free edge of the tensile side ofthe gear tooth fillet (Fig. 17). The curve of the experimentalresults is referred to data obtained by the RGB method byprocessing the 2 mm thick slice.

Except for the boundary areas where a certain data dis-persion can be observed, the agreement of the two curvesis quite satisfactory. The experimental curve stays below thenumerical one. This circumstance was almost expected since

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Fig. 18 a Dark-field isochromatic fringes in white light observed for 1.8 mm thick slices obtained along the whole longitudinal dimension of theteeth, b contact area visualized on the face gear flank surfaces

the difference of the principal stresses observed experimen-tally is, actually, averaged through the thickness of the slice,and an interpolation is made.

Finally, the experimental test involving face gear driveswas carried out by simulating the meshing of pinion andgear sectors with two teeth. Because of the resin large strainvalue, the contact area results quite large on both teeth (ascan be seen in Fig. 18), causing an excessive load distributionthat hinders a 3-D photoelastic analysis. Therefore the mini-mum torque to be applied in order to visualize isochromaticfringes in slices not thicker than 2 mm was determined by apreliminary finite element analysis. 1.8 mm thick slices were

then extracted along the whole longitudinal dimension of theface gear to verify the meshing of the two teeth (Fig. 18).

In Fig. 19 some examples of retardation values determinedby using the RGB photoelasticity along with a comparisonwith the results obtained with FEM are reported. Differencesare due to the experimental gear position error, unavoidableand not easily measurable with the used fixture.

As demonstrated by Fig. 20, the fringe order for a 0.8 mmslice seems to be too low for a reliable photoelastic analy-sis and a higher load combined with a resin having differentmechanical properties, in particular higher stiffness to avoidexcessive strain, should be used. In such cases, the limit of

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Fig. 19 a Dark-field isochromatic fringes and b relative retardation maps obtained with the RGB method for a 1.8 mm thick slice. c Difference ofthe principal stresses obtained by the finite element analysis

Fig. 20 a Example of dark-field isochromatic fringes for a 0.8 mm thick slice and b relative retardation map obtained with the RGB method

the used stereolithographic resin emerges as well as the use-fulness of an accurate experimental planning carried out withproper numerical simulations.

4 Discussion and conclusions

In this paper, a CAE approach for the stress analysis of com-plex mechanical components was proposed by integratingFEM simulations with 3D photoelasticity. In particular, thestress field distribution of face gear models made by stere-olithography was studied. Even if many are the parametersinvolved in the problem (misalignments in the gears meshingdue to the loading apparatus, discrepancies between CADgeometries and real manufactured shapes, finite thicknessof the slices used for photoelastic analyses), the obtainedresults show a substantial agreement between numerical sim-ulations and experimental data. The main noticeable differ-ences between numerical and photoelastic analyses can be

ascribed to the experimental stress field average which isintrinsically caused by the finite slice thickness. Moreover,residual stresses introduced by manufacturing, and not com-pletely annealed by the stress relieving process, affect thefinal stress field within the contact areas. Surface finishingproperties, not accounted for in the ideal geometry, may fur-ther contribute to the differences between results especiallyin the proximity of the contact areas.

Despite the above described criticalities, the present studyattests the potentialities of the adopted approach as well asthe usefulness of such stereolithographic models to perform3D photoelastic analyses guided by numerical simulations.Moreover, FE analyses demonstrated their usefulness to min-imize the number of experiments and optimize the slicingprocess thus providing cost- and time-effective experimentalphotoelastic analyses.

It is worth mentioning that potential differences betweenCAD geometries and stereolithographic models have notbeen taken into account in the present work. However,

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real manufactured geometries could be easily considered byacquiring the resin prototypes with non-contact reverse engi-neering techniques, as, for example, a full-field optical scan-ner [16]. The direct use of the optically acquired shapes for FEsimulations would produce numerical results which are basedon the same geometry that originates those experimentallyobtained. Moreover, this would allow for the extension of thepresented framework applicability for the analysis of hand-crafted components when CAD models are not available.

Future works will investigate the framework applicabil-ity in new research fields, particularly where the use of RPtechnique is increasing. RP models have been traditionallyused as complementary design tools. In recent times, how-ever, there has been a growing interest in the use of RP tech-niques to produce real models which are adopted within manyfields. For example, in biomedical applications (prosthesesdesign, orthodontic treatments, surgical dental guides) thereis an extensive use of customized prototypes, and the possi-bility to easily integrate numerical and experimental analy-sis could offer high potentialities for the optimization andvalidation of components design. Also in the field of indus-trial applications, there is a growing use of direct 3D printedmodels as actual components which could benefit from thepresented framework (i.e. small plastic gearbox, generic cus-tomized components). The introduction of customized pho-toelastic analyses for real components would greatly enhancethe detection of possible criticalities arising from challengingapplications. The described framework, even if presented fora specific test case, can be easily extended to different appli-cations without losing its effectiveness.

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A CAE approach for the stress analysis of gear models by 3D digital photoelasticityAbstract 1 Introduction2 Materials and methods2.1 Digital photoelasticity2.1.1 RGB method

2.2 Photoelastic stress analysis for stereolithographic gear models2.2.1 Geometric models2.2.2 Stereolithographic models2.2.3 Experimental set-up2.2.4 Stress freezing and model slicing process2.2.5 Calibration of the model material2.2.6 Finite element analysis

3 Results4 Discussion and conclusionsReferences

A CAE Approach for the Stress Analysis of Gear Models

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