Ignacio Gonzalez-Perez1

Department of Mechanical Engineering,Materials and Manufacturing,

Universidad Politecnica de Cartagena,Cartagena 30202, Spain

e-mail: ignacio.gonzalez@upct.es

Pedro L. Guirao-SauraDepartment of Mechanical Engineering,

Materials and Manufacturing,Universidad Politecnica de Cartagena,

Cartagena 30202, Spaine-mail: pedro.guirao@upct.es

Alfonso Fuentes-AznarDepartment of Mechanical Engineering,

Rochester Institute of Technology,Rochester, NY 14623e-mail: afeme@rit.edu

Q1¶

Application of the Bilateral Filterfor the Reconstruction of SpiralBevel Gear Tooth Surfaces FromPoint CloudsReconstruction of gear tooth surfaces from point clouds obtained by noncontact metrologymachines constitutes a promising step forward not only for a fast gear inspection but alsofor reverse engineering and virtual testing and analysis of gear drives. In this article, a newmethodology to reconstruct spiral bevel gear tooth surfaces from point clouds obtained bynoncontact metrology machines is proposed. The need of application of a filtering processto the point clouds before the process of reconstruction of the gear tooth surfaces has beenrevealed. Hence, the bilateral filter commonly used for 3D object recognition has beenapplied and integrated in the proposed methodology. The shape of the contact patternsand the level of the unloaded functions of transmission errors are considered as the criteriato select the appropriate settings of the bilateral filter. The results of the tooth contact anal-ysis of the reconstructed gear tooth surfaces show a good agreement with the design ones.However, stress analyses performed with reconstructed gear tooth surfaces reveal that themaximum level of contact pressures is overestimated. A numerical example based on aspiral bevel gear drive is presented. [DOI: 10.1115/1.4048219]

Keyword: gear geometryQ2¶

1 IntroductionReconstruction of gear tooth surfaces from coordinate measure-

ment through a contact metrology was presented in Ref. [1] forgear inspection and computerized simulation of meshing. A non-contact methodology for gear inspection was proposed in Ref. [2]through a laser holographic measurement. Nowadays, reconstruc-tion of gear tooth surfaces from point clouds obtained throughoutthe use of noncontact metrology machines constitutes a promisingstep forward not only to perform the traditional gear inspectionprocess but also to apply virtual testing and simulation of theto-be inspected gears by means of the tooth contact analysis(TCA) and the stress analysis. Noncontact metrology machinesallow a fast measurement of a whole gear in few minutes.However, one of the main drawbacks of laser noncontact metrologymachines at this moment is the precision and dispersion of the mea-sured points that form the point cloud of the gear tooth surfaces.Such a dispersion is a general behavior in 3D scanners as it isrevealed in Ref. [3].Some previous publications consider the raw data provided by

noncontact metrology machines [4–6], mainly point clouds forgear inspection, tooth contact analysis, and stress analysis of thereconstructed gear drives. However, the deficiency in the resultsobtained from tooth contact and stress analysis is revealed inRefs. [5,6], demanding the application of filters to the point cloud.Filtering of point clouds is an usual technique that is being

applied in 3D object recognition to reduce the noise and preservethe shape of a being-inspected object. First applications of surfacereconstruction from unorganized points can be found in Ref. [7]where the underlying theory to get the tangent plane at a givenpoint of the sough-for surface was introduced. Such a plane

constitutes the regression plane of the neighboring points to thegiven point of the point cloud. In Ref. [8], the bilateral filter pro-posed in Ref. [9] for image processing is adapted for mesh recon-struction, although the normal to the tangent plane is not obtainedas it is proposed in Ref. [7], but from triangulation betweenpoints of the first ring in the neighborhood. Here, in Ref. [8], theEuclidean distance of the neighboring points to the given pointand the normal distance to the tangent plane are considered in theequation to denoise each given point, considering Gaussian func-tions to weigh the contribution of such distances. Later, inRef. [10], the bilateral filter is adapted considering the determina-tion of the normal to the tangent plane as it was presented inRef. [7], proposing the use of a function of the sphere radius,which determines the size of the neighborhood, for the variancesin the Gaussian functions.This article establishes a methodology to reconstruct spiral bevel

gear tooth surfaces from filtered point clouds by using the bilateralfilter. The active tooth surfaces are successfully reconstructed fromthe obtained point clouds after the application of the proposed meth-odology. However, due to the poor quality of the point cloud in thefillet areas, these areas are reconstructed throughout the applicationof Hermite curves that approach these curves to the available cloudpoints in such areas. This does not diminish the proposed method-ology of reconstruction that can be easily extended to the fillet areasas soon as the point data would be available. Conversely, the aim ofthis article is to prove the effectiveness of the bilateral filter toreconstruct gear tooth surfaces and to perform tooth contact andstress analysis of gear drives with the reconstructed gear tooth sur-faces. Furthermore, some criteria are established for the applicationof the bilateral filter based on the results of the tooth contact anal-ysis. Basically, the shape of the contact pattern, the unloaded func-tion of transmission errors, and the level of the unloadedtransmission errors are taken into account to judge whether theparameters of the bilateral filter are appropriate for the reconstructedgear. However, stress analysis by application of the finite elementmodels shows that the technology of noncontact metrologymachines require improvements to approach contact stresses in

1Corresponding author.Contributed by the Power Transmission and Gearing Committee of ASME for

publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 12,2020; final manuscript received July 27, 2020; published online xx xx, xxxx. Assoc.Editor: Mohsen Kolivand.

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reconstructed geometries to those obtained from computational geargeneration of the design gears.A numerical example of a spiral bevel gear drive is considered

later in this study. The design and manufacturing data of thespiral bevel gears are known. A spiral bevel pinion, once it hasbeen manufactured and inspected, is reconstructed following theprocedure described in this article. Then, it will be considered invirtual meshing with a master spiral bevel gear. The results provethe effectiveness of the bilateral filter to reproduce, with the recon-structed pinion tooth surfaces, similar contact patterns, andunloaded transmission errors to those obtained with computation-ally generated gear tooth surfaces. The reconstruction is performedwith the point clouds for both tooth sides of just a tooth, leaving thereconstruction of all the gear teeth for future research.

2 Reconstruction of Gear Tooth SurfacesReconstruction of a gear tooth from point clouds obtained by a

noncontact metrology machine requires the application of theprocess described later. A summary diagram is visualized in Fig. 1.

Step 1: The axial location of the pitch cone apex Q of the to-bereconstructed gear is determined. Figure 2 shows the to-beinspected gear, the point clouds corresponding to eachtooth side, and the point cloud corresponding to the backside of the gear that need to be inspected. These points onthe back side of the gear approach a plane that can be desig-nated here as back plane. Figure 2 shows a well coordinatesystem Sm where point coordinates of all the point cloudsare obtained. Point Om represents the origin for the measure-ments of the noncontact metrology machine. Location of thepitch cone apex in the axial direction means to determine thecoordinate z(Q)m in system Sm. Such a coordinate can beobtained as follows:

z(Q)m = z plane + md (1)

where zplane is the mean value of coordinates z of the pointcloud that approaches the back plane and md is the so-calledmounting distance. The value of the mounting distance isrequired in this procedure. Otherwise, the location of thegear in its virtual assembly with a master gear wouldrequire a trial and error process since the origin Om do nothave to coincide with the pitch cone apex.Figure 2 also shows the coordinate system Sg where the

to-be reconstructed gear will be defined. The origin ofsystem Sg is located at the pitch cone apex Q. A coordinate

transformation from system Sm to system Sg is required. Itis suggested to perform that at step 10 once the reconstructedgear tooth surfaces have been obtained.

Step 2: The radial location of the pitch cone apex Q of the to-bereconstructed gear is determined. Figure 3 shows the pointclouds corresponding to each tooth side and the pointcloud corresponding to an internal cylinder of the gear tobe inspected. The axis of such a cylinder is the axis wherepoint Q is located. Such an axis may not pass through theorigin of the coordinate system Sm since some positionerror Δrm may exist. To determine it, a least-square minimi-zation problem is established following the Levenberg–Marquardt algorithm [11,12] where the objective function

Fig. 1 Summary diagram of the proposed methodology to reconstruct spiral bevel geartooth surfaces

Fig. 2 Illustration of coordinate systems Sm where point cloudsare obtained and Sg where the to-be reconstructed gear isdefined

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is defined as follows:

f (x) =∑m

i=1

[(x(Pi)m − Δrm,x)2 + (y(Pi)

m − Δrm,y)2 − ρ2] (2)

where x = [Δrm,x, Δrm,y, ρ] is the vector of optimization var-iables, where Δrm,x and Δrm,y allow the determination of theradial location of point Q and ρ is the radius of a circle thatadjusts to the cylinder point cloud in the plane (xm, ym). Con-versely, (x(Pi)

m , y(Pi)m ) represent the coordinates of any point Pi

of the cylinder point cloud in the coordinate system Sm. Theobjective function is constructed on the m points of the cyl-inder point cloud. Just one of the variables is restricted to bepositive

0 ≤ ρ (3)

An additional coordinate transformation may be requiredfrom system Sm to system Sg in case Δrm,x≠ 0 or Δrm,y≠ 0.It is suggested to perform that at step 10 once the recon-structed gear tooth surfaces have been obtained.

Step 3: Cleaning the point clouds corresponding to both toothsides of a given tooth. The point clouds obtained by noncon-tact metrology machines contain large amounts of pointsrelated to different parts of the gear tooth (top land, frontand back surfaces of the tooth, and root land) that are notof interest for the reconstruction of the active gear tooth sur-faces. Figure 4(a) shows the raw point cloud correspondingto the concave side of a spiral bevel gear tooth. Figure 4(b)shows such point cloud after removing those points at thementioned areas of not interest. This task saves time forthe next step. The same procedure has to be repeated withthe point cloud for the convex tooth side.

Step 4: Filtering of the point clouds is done. The bilateral filterapproach is applied by using the following algorithm (seeRef. [10] for more details):

(a) A spatial radius r, a normal offset rn, and a number of itera-tions niter are chosen as input data for the algorithm. Variabler represents the radius of a sphere to obtain the neighbors foreach given point p of the point cloud. Variable rn representsan offset value in both directions of a to-be calculated normalnp at the point p. Variable niter represents the number of iter-ations the algorithm is applied. At the end of each iteration,the points of the point cloud are located in new positions.

(b) Neighbors pi of point p are determined using a kd-structureof the points [13] to find those points, where ∥pi − p∥ < r.Then, the unit normal np is calculated from the locations ofthe neighbors as follows:(i) A mean value of the neighbors or centroid cp is deter-

mined as follows:

cp =1Np

∑

i=1,Np

pi (4)

where Np is the number of neighbors of point p insidethe sphere of radius r.

(ii) A matrix A is determined for each point p consideringthe locations of the neighbors pi

A =1Np

∑

i=1,Np

(pi − cp) · (pi − cp)T (5)

(iii) The normal np is obtained as the eigenvector corre-sponding to the minimum eigenvalue λmin of such amatrix A since

(x − cp) · np = 0 (6)

represents a plane that minimize the sum of squared dis-tances between the plane and the neighbors pi.

(c) Each point p is filtered using the following equation:

p′ = p + δp · np (7)

where

δp =

∑i=1,Np

wd(∥pi − p∥ )wn((pi − p) · np)(pi − p) · np∑i=1,Np

wd(∥pi − p∥ )wn((pi − p) · np)(8)

Here, wd and wn are decreasing Gaussian functions, whichare defined as follows:

wd(x) = e−x2/2σ2d (9)

wn(x) = e−x2/2σ2n (10)

Here, σ2d is the Gaussian variance for the distance from theneighbor pi to point p, whereas σ2n is the Gaussian variancefor the distance from the neighbor pi to the tangent planedefined by normal np. The Gaussian variances σ2d and σ2n

Fig. 3 For determination of a circle that adjusts to the internalcylinder point cloud in the plane (xm, ym) and position error Δrmfor the location of pitch cone apex Q

(a)

(b)

Fig. 4 Cleaning the point clouds: (a) point cloud before cleaningand (b) point clouds after cleaning

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are fixed considering σd = r/3 and σn = rn/3 since for thoseneighbors pi whose distance is beyond 3σd and normal dis-tance is beyond 3σn, the weights wd and wn are negligible(see Ref. [10]).

(d) Steps b and c are repeated several times depending on thechosen value of number of iterations, niter.

Step 5: This is the representation of the radial projection of thepoint cloud considering a plane formed by an axis X directedalong the gear axis and an axis Y that represents the radialdistance of each point of the point cloud. Figure 5(a)shows a radial projection of the filtered point cloud wherepoints p′(x, y, z) are represented through the following rela-tions:

X(p′) = z (11)

Y(p′) =��������x2 + y2

√(12)

Step 6: Definition of the flank edges on the radial projection of thepoint cloud is presented. These flank edges are based on a topedge, a bottom edge, a root edge, a front edge, and a back edge.In addition, a back-to-top edge and a front-to-top edge can beconsidered as being illustrated in Fig. 5(a) for the case of aspiral bevel gear tooth surfaces in which the back-top andfront-top corners have been cut off. The flank edges can bedefined considering that the to-be derived active toothsurface has to be delimited by all these edges with the excep-tion of the root edge. Figure 5(a) also shows that some pointsaremissing in the point cloud, whichmay force the location ofthe bottom edge for the to-be derived active tooth surface tohave points where a radial grid can be created.

Step 7: Creation of a grid of m × n points on the radial projectionis carried out as shown in Fig. 5(b). Points m(X, Y ) areobtained from a regular grid defined by the aforementionededges delimiting the to-be derived active tooth surface.

Step 8: The closest point of the point cloud for each pointm(X, Y )of the radial grid is found. This closest point of the point

(a) (b)

Fig. 5 Radial projection of the filtered point cloud with illustration of (a) flank edges and(b) computed grid for a spiral bevel gear tooth

Fig. 6 Reconstructed spiral bevel gear from filtered point clouds

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cloud can be found using a kd-tree structure of the points[13].

Step 9: A NURBS (nonuniform rational B-splines) surface [14] isbuilt considering the obtained points in the previous step ascontrol points. Most of the graphic libraries include thisfeature to interpolate a regular grid of points through asecond-order derivative surface. This surface constitutesthe active tooth surface of the gear tooth.

Step 10: The fillet surface is built to entrench the bottom edge ofthe active tooth surface with the root edge. Hermite curvesare applied here for the determination of the control pointsthat will allow to build a NURBS surface for the fillet area.Hermite curves require basically the initial and final pointpositions, the initial and final unit tangents, and theweights (or modules) of such tangent vectors (see Ref. [15]for a complete review of application of Hermite curves infillet construction). The initial point and initial unit tangentcan be derived from the active tooth surface. The finalpoint and final unit tangent can be derived from the locationof the root edge on the radial projection and the angular pitch2π/Ng of the gear. Here, Ng is the tooth number of the to-beinspected gear that is considered as known. The weights arederived through an optimization process that follows theLevenberg–Marquardt algorithm [11,12] and minimizes thesum of distances of a sample of cloud points in the filletarea to the to-be computed NURBS surface.

Figure 6 shows a reconstructed spiral bevel gear with the illustra-tion of the NURBS surfaces for the concave and convex tooth sidesand the NURBS surfaces of the fillet areas. Here, a detail view of anHermite curve is illustrated as well on the front section, where P0 isthe initial point, t0 is the unit initial tangent, P1 is the final point, andt1 is the unit final tangent.

3 Tooth Contact and Stress AnalysisIn this work, TCA is applied for a reconstructed spiral bevel

pinion in mesh with a master spiral bevel gear. The TCA algorithmis based on the work [16] and has been applied later in Refs. [17,18].It is based on the minimization of the distance between the mastergear driven tooth surface and the reconstructed pinion driving toothsurface. It considers the surfaces as rigid, so that it is necessary to seta value for the so-called virtual marking compound thickness tv,which is usually fixed to 0.0065mm, to obtain the contact patternand the unloaded function of transmission errors. Figure 7 shows

both surfaces at a given step of the algorithm. A coordinatesystem Sb is predefined in the master gear tooth surface with itsaxis zb directed along the normal at a given point Ob on themiddle of the surface. For each given position of the reconstructedtooth surface, distance Δzcosμ is minimized by approaching themaster gear tooth surface toward the reconstructed tooth surface.Such a distance is obtained following a kinematic procedure, as itis explained in Ref. [16], using the direction of velocity vectorv (G) at point G. Point G is obtained from the projection of pointP of the reconstructed tooth surface on the master tooth surface.A set of points like P are taken from a grid of points for the appli-cation of the algorithm. Angle μ is formed by the directions ofvector −v (G) and GP

⇀. Once the contact is established, those

points like P and G whose distance Δz is equal to the given valuetv form the contours of the instantaneous contact patterns.In previous works where TCA was applied in reconstructed gear

tooth surfaces (see Refs. [5,6]), the value of tv had to be increased tobe able to obtain acceptable contact patterns. An acceptable contactpattern means that the contact pattern present a regular shape withno holes or isolated areas. The need to increase tv was due to theirregularities of the NURBS surfaces since filtering approacheswere not applied. In this study, the magnitude of tv will be fixedto 0.0065mm, as it has been set in the previous study (seeRef. [19]), where TCA was applied with theoretically generatedgear tooth surfaces, and the results of the filtering approaches willbe judged in terms of the shape of the contact pattern and thelevel of the function of transmission errors.Conversely, the stress analysis is based on the application of the

finite element method and constitutes another tool to judge thegoodness of the filtering approach. A finite element model is illus-trated in Fig. 8, which is built with elements C3D8I (see Ref. [20]).The foundation for the construction of the finite element model isfound in Ref. [19]. The nodes on the bottom part and both sidesof the rim of each gear constitute rigid surfaces that are rigidly con-nected to the respective reference nodes. Here, at the referencenodes, the degrees-of-freedom are established. Although the gearis held at rest at each given contact position, a torque T is appliedto the pinion reference node. It is observed that the model of thereconstructed pinion is provided with additional flank edges onthe toe and the heel top corners. To evaluate the quality of the recon-structed gear tooth surfaces for the stress analysis, the maximumcontact pressure at the middle tooth of the reconstructed pinionwill be obtained along two cycles of meshing and compared withthe maximum contact pressures obtained with the designed andmanufactured pinion, all in mesh with the same master spiralbevel gear.

Fig. 7 Schematic view for the explanation of the minimization ofdistances between the master gear tooth surface and the recon-structed pinion tooth surface

Fig. 8 Finite element model

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Journal of Mechanical Design XX 2020, Vol. XX / 003400-5

4 Numerical ExampleA numerical example is being considered based on a known

design of a spiral bevel pinion and gear, both manufactured in aPhoenix II 275G machine. Furthermore, the manufactured pinion,for which the machine tool settings are known, has been inspectedin a HNC3030 noncontact metrology machine whose precision is±5 μm. The to-be inspected pinion will be in virtual mesh, afterreconstruction, with a master gear. Here, the term master refers toa gear that has been computationally generated based on itsknown design.Once the to-be inspected pinion has been scanned, the recon-

struction of the pinion will be performed following the ideas pre-sented in Sec. 2. The proposed approach shown in Sec. 2 will bejudged in terms of the shape of the contact pattern, the level ofunloaded transmission errors, and the level of contact stressesobtained from the analysis of the reconstructed pinion in meshwith the master gear. Additional information of micro-geometrydeviations after manufacturing of the pinion has been obtained byusing a coordinate measurement machine GMS350 whose precisionis ±1 μm. These micro-deviations can be added to the computation-ally generated geometry corresponding to the known design of thespiral bevel pinion, obtaining the virtual manufactured geometry ofthe pinion. The results of TCA and stress analysis of the recon-structed spiral bevel pinion in mesh with the master gear can becompared with those obtained from the tooth contact and stressanalysis of the virtual manufactured geometry of the pinion inmesh with the same master gear.

4.1 Design Data. The design of a spiral bevel gear drive is pre-sented. The basic transmission and blank data are summarized inTable 1. The cutter data and basic machine tool settings are pre-sented in Table 2. A detailed description of the basic machinetool settings is available in Ref. [19].

4.2 Manufactured Data. The averaged micro-geometry devi-ations for the pinion concave tooth side are given in Table 3 for agrid of 5 × 9 points. Such a grid covers 80% of the tooth surface.Once those micro-deviations are incorporated into the virtual com-putationally generated pinion, the tooth contact and stress analysisresults obtained considering such a pinion in mesh with themaster gear will establish the reference baseline results for compar-ison with those obtained with the reconstructed pinion in mesh withthe same master gear.

4.3 Preliminary Tooth Contact Analysis Results. Theresults of the TCA between the designed concave tooth side ofthe pinion in mesh and the convex tooth side of the master spiralbevel gear is shown in Fig. 9. The results of TCA between the man-ufactured concave tooth side of the pinion in mesh with the samemater gear convex tooth surface is also illustrated in Fig. 9. Thelevel of unloaded transmission errors is about 9 arc-seconds forthe designed gear tooth surfaces, whereas it is about 26 arc-secondswhen the manufactured pinion tooth surface is used instead of thedesigned one.

4.4 Tooth Contact Analysis With Reconstructed Gears.The manufactured pinion has been reconstructed considering theprocedure described in Sec. 2. At step 1, a mounting distancemd = 58.1mm and a zplane=−56.4mm provide the valuez(Q)m = 1.7mm to locate the pitch cone apex and the origin ofsystem Sg, where the to-be reconstructed pinion is defined (seeFig. 2). At step 2, the Levenberg–Marquardt algorithm is appliedupon m= 243,062 points of the internal cylinder point cloud toobtain Δrm,x=−1.8182×10−5mm, Δrm,y=−4.1794×10−6mm andρ = 17.175mm, which reflects negligible errors in the radial loca-tion of the pitch cone apex. A grid of 10 × 15 points has been con-sidered in the radial projection in step 6 for all the cases considered

here. The previous study (see Ref. [5]) has shown that a largeramount of grid points may lead to obtain reconstructed surfaceswith more irregularities. Figure 6 shows an example of a recon-structed pinion.Figure 10 represents a comparison of the unloaded functions of

transmission errors for the manufactured pinion and a reconstructedpinion without using any filter. The contact patterns are also repre-sented in Fig. 10 as a set of instantaneous contact areas obtainedfrom the procedure exposed in Sec. 3. When no filter is applied,

Table 2 Cutter data and basic machine tool settings of the spiralbevel gear drive

DataPinion (concave

tooth side)Gear (convex tooth

side)

Cutting method Face millingCutter type Fixed setting Spread bladePoint radius (mm) 46.672 43.701Blade angle (deg) 17.760 22.0Blade profile Circular StraightBlade profile radius (mm) 762.0 Not applicableBottom relief height (mm) 1.5 1.5Bottom relief parabolacoefficient (mm−1)

0.015 0.010

Machine center to back (mm) −0.493 0.0Sliding base (mm) −0.540 −0.934Blank offset (mm) 0.0100 0.0Radial distance (mm) 36.220 36.771Cradle angle (deg) 60.403 58.735Machine root angle (deg) 41.767 43.916Velocity ratio 1.4526 1.3826Modified roll coefficient C 0.006269 Not applicableModified roll coefficient D −0.01633 Not applicable

Table 1 Basic transmission and blank data of the spiral bevelgear drive

Data Pinion Gear

Shaft angle (deg) 90.0Tooth number 22 23Pitch angle (deg) 43.727 46.273Spiral angle (deg) 45.0Hand of spiral Right hand Left handOuter transverse module (mm) 3.7398Face width (mm) 18.0Tooth taper DuplexOuter addendum (mm) 2.823 2.344Outer dedendum (mm) 2.906 3.384Face cone angle (deg) 46.084 48.233Root cone angle (deg) 41.767 43.916

Table 3 Deviations in μm on the concave tooth side of themanufactured spiral bevel pinion along 5×9 grid points

Points along facewidth/profile

1(near topedge) 2 3 4

5 (near bottomedge)

1 (near toe) −2.1 −1.9 0.3 0.2 −1.72 −2.0 0.3 0.5 1.1 −1.23 −2.1 0.8 0.0 1.5 −1.34 0.5 −0.3 0.6 1.2 −2.35 −0.6 0.1 0.0 0.0 −3.36 −1.2 0.4 0.0 −1.3 −4.37 −3.5 −0.6 −1.3 −2.7 −5.78 −4.1 −3.6 −3.1 −3.9 −7.39 (near heel) −2.7 −3.9 −3.4 −5.1 −8.3

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the contact pattern of the reconstructed pinion shows isolated areas.Also, the shape of the function of transmission errors and the levelof transmission errors are far from the function of transmissionerrors and the level of transmission errors of the manufacturedpinion. These results (Fig. 10) show the need to apply a filter tothe point clouds before the reconstruction of the gear tooth surfaces.The bilateral filter is applied to the point clouds of the manufac-

tured spiral bevel pinion providing different reconstructed pinionsfor different values of the filter parameters. The first question thatarises here is how large the sphere radius r and the offset value rnshould be chosen for a given point cloud. This depends on howthe points are distributed all over the point cloud. If the radius istoo small, the fitting regression plane could not be enough accurate.A set of values r= rn= {25, 50, 75, 100} μm combined with niter={1, 2, 3, 4, 5} has been tested to reconstruct 20 spiral bevel pinions.For each reconstructed pinion in mesh with the master gear, TCA isapplied.Figure 11 shows the unloaded functions of transmission errors

when the bilateral filter with r= rn= 25 μm is applied. The levelof transmission errors for any number of iterations from 1 to 5 isfar from the level of transmission errors of the manufactured

Fig. 9 Contact patterns and unloaded functions of transmission errors for the designed and the virtual manufac-tured spiral bevel pinion in mesh with the master spiral bevel gear

Fig. 10 Contact patterns and unloaded transmission errors forthemanufactured pinion and the reconstructed pinionwithout fil-tering in mesh with the master gear

Fig. 11 Contact patterns and unloaded transmission errors forthe manufactured pinion and the reconstructed pinions usingthe bilateral filter with r= rn=25 μm in mesh with the master gear

Fig. 12 Contact patterns and unloaded transmission errors forthe manufactured pinion and the reconstructed pinions usingthe bilateral filter with r= rn=100 μm in mesh with the mastergear

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pinion in mesh with the master gear. The shapes of the contact pat-terns are also illustrated for niter= 1 and niter= 5 in Fig. 11. It can beobserved that the shape of the contact patterns is also far from thereference shape of the contact pattern corresponding to the manu-factured pinion.Figure 12 shows the unloaded functions of transmission errors

when the bilateral filter with r= rn= 100 μm is applied. Theshapes of the contact patterns are also illustrated for niter= 1,niter = 3, and niter= 5 in Fig. 12. It is observed that the contactpattern with niter= 1 presents holes and isolated areas in the for-mation of the bearing contact. Such features start to disappearfrom niter= 3 on. The minimum number of iterations from whichthe contact pattern does not present holes or isolated areas is niter= 3. A comparison of the functions of transmission errors fromniter= 3 to niter = 5 shows that the function for niter= 3 presents alower peak-to-peak level than the function for niter= 4 and the func-tion for niter= 5.Figure 13 shows a comparison of the peak-to-peak level of the

unloaded functions of transmission errors for the 20 reconstructedpinions where the bilateral filter is applied and for the recon-structed pinion when no filter is applied. The peak-to-peak levelof transmission errors of the manufactured pinion is also illustratedby means of a black horizontal line. The shapes of the contact pat-terns for some reconstructed pinions are also illustrated. It isobserved that for the size r= rn= 25 μm, the level of transmission

errors fluctuates along the different number of iterations and it iseven larger than the level of transmission errors when no filteris applied. For the size r= rn= 50 μm, the level of transmissionerrors decreases from niter= 2 although the shape of the contactpattern is not appropriate for any number of iterations from 1 to5 (the contact pattern for r= rn= 50 μm and niter= 3 is illustratedin Fig. 13 and shows holes and isolated areas over the contactpattern). For the sizes r= rn= 75 μm and r= rn= 100 μm, thelevel of transmission errors reach similar values to the manufac-tured pinion when niter= 3. However, it is necessary to reachniter= 4 for r= rn= 75 μm to eliminate any isolated areas in thecontact pattern (the contact pattern for r= rn= 75 μm and niter=3 is illustrated in Fig. 13 and an isolated area is still observed inthe contact pattern).Finally, Fig. 14 shows the best results in terms of the unloaded

function of transmission errors and shape of the contact patternsfor the reconstructed pinions using the bilateral filter with r= rn=75 μm and niter= 4 and with r= rn= 100 μm and niter= 3. Bothcontact patterns are similar to each other, without any holes or iso-lated areas on them. The level of transmission errors is also similar.However, the function of transmission errors for the case of r= rn=75 μm has more fluctuations than for the function for r= rn=100 μm. Both functions of transmission errors appear shiftedrespect to the function of transmission errors corresponding to themanufactured pinion. This shift may be due to an inaccurate loca-tion of the 5 × 9 micro-deviations, presented in Table 3, over thedesigned pinion tooth surface.

Fig. 15 Contact pressure field on the contact elements of themiddle tooth of the pinion at a given step for (a) themanufacturedgeometry, (b) the no-filter reconstructed geometry, and (c) thereconstructed geometry with the bilateral filter using r= rn=100 μm and niter=3

Fig. 13 Contact patterns and level of unloaded transmissionerrors for the manufactured pinion and the reconstructedpinions using no filter and using the bilateral filter

Fig. 14 Contact patterns and unloaded transmission errors forthe manufactured pinion and the reconstructed pinions usingthe bilateral filter with r= rn=100 μm and r= rn=75 μm in meshwith the master gear

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4.5 Stress Analysis. For each tooth contact analysis, a stressanalysis is performed using the finite element model shown inFig. 8. Such a model is provided with 204,500 elements and248,304 nodes. A torque T= 197.0Nm is applied at the referencenode of the pinion, while the gear is held at rest at each step. Ageneral purpose computer program (see Ref. [20]) is used for thefinite element analysis. Figure 15 shows the contact pressure distri-bution at the same intermediate step of the stress analysis for threegeometries of the pinion: the virtual manufactured one, a recon-structed geometry without application of the bilateral filter, and areconstructed geometry with application of the bilateral filter withr= rn= 100 μm and niter= 3. A shape very close to an ellipse isobserved in Fig. 15(a). However, Figs. 15(b) and 15(c) show acontact area that contains isolated areas, especially when the geom-etry is not filtered (Fig. 15(b)). Although filtering improves theshape of the contact pattern (see Fig. 15(c)), it is not enough andthe level of stresses is still higher than the one shown for the man-ufactured geometry (see Fig. 15(a)). It is also observed that somecontact areas appears in the fillet area for the reconstructedpinions (see Figs. 15(b) and 15(c)) due to the impossibility to dis-place down the bottom edge (see Fig. 5) since no available datawere provided in some areas of the point cloud below this line.Figure 16 shows the variation of the contact pressure at the

middle tooth of the pinion along 41 contact positions spreadalong two angular pitches of rotation of the reconstructed pinionwith several types of geometry: (i) the designed and the manufac-tured ones provide similar level of contact pressure along twoangular pitches of rotation, (ii) the reconstructed one without appli-cation of the filter shows the highest values of contact pressures, and(iii) several reconstructed geometries using the bilateral filter withniter= 3 and different values of radius r= rn show that the level ofstresses decreases as the radius r is increased, reaching a similarlevel for the reconstructed geometries with r= rn= 75 μm and r=rn= 100 μm. Figure 16 shows in gray color the overestimation ofcontact pressures for the reconstructed geometry with r= rn=100 μm and niter= 3 respect to the manufactured geometry.Figure 16 shows as well, in green color,Q3

¶those contact positions

where contact on fillet does not exist at the central tooth.

5 ConclusionsA methodology to reconstruct spiral bevel gears from point

clouds has been proposed and applied for the virtual simulationand the analysis of reconstructed gears from filtered point cloudsin terms of contact patterns and unloaded functions of transmission

errors when in mesh with a master gear. Based on the performedresearch and results obtained, the following conclusions can bedrawn:

(1) Point clouds obtained from noncontact metrology machinesshould be filtered before gear tooth surfaces are reconstructedfor virtual simulation and analysis. The bilateral filter hasbeen applied and their parameters r, rn, and niter have beenchosen considering the shape of the contact pattern, andthe shape and peak-to-peak levels of the unloaded functionof transmission errors.

(2) Contact patterns and unloaded functions of transmissionerrors obtained as results of tooth contact analysis with thereconstructed pinion show a satisfactory agreement withthose obtained when using the reference manufacturedpinion, both in mesh with the same master gear, if the appro-priate parameters of the bilateral filter are chosen.

(3) Stress analysis by the finite element method reveals that thereconstructed pinion yields higher values of contact pressurethan the reference manufactured pinion, mainly due to thequality of the point clouds obtained from noncontact metrol-ogy machines. Although the application of the bilateral filterhelps reducing the dispersion of points in the point cloud, it isnot enough to avoid high values of contact pressure in thereconstructed pinion. The missing points in the point cloudfor areas close to the to-be reconstructed active toothsurface is another important factor that limits the applicationof the proposed methodology for the stress analysis of recon-structed gears.

AcknowledgmentThe authors express their deep gratitude to the Spanish Ministry

of Economy, Industry and Competitiveness (MINECO), theSpanish State Research Agency (AEI), and the European Fundfor Regional Development (FEDER) for the financial support ofresearch project DPI2017-84677-P.

Conflict of InterestThere are no conflicts of interest.

Data Availability StatementThe authors attest that all data for this study are included in the

paper. No data, models, or code were generated or used for thispaper.

References[1] Kin, V., 1994, “Computerized Analysis of Gear Meshing Based on Coordinate

Measurement Data,” ASME J. Mech. Des., 116(3), pp. 738–744.[2] Fujio, H., Kubo, A., Saitoh, S., Hanaki, H., and Honda, T., 1994, “Laser

Holographic Measurement of Tooth Flank Form of Cylindrical Involute Gear,”ASME J. Mech. Des., 116(3), pp. 721–729.

[3] Mahan, T., 2019, “Pulling at the Digital Thread: Exploring the Tolerance StackUp Between Automatic Procedures and Expert Strategies in Scan to PrintProcesses,” ASME J. Mech. Des., 141(2). Q4

¶[4] Peng, Y., Ni, K., and Goch, G., 2017, “A Real Evaluation of Involute Gear FlanksWith Three-Dimensional Surface Data,” American Gear ManufacturersAssociation Fall Technical Meeting 2017, AGMA Paper No. 17FTM08.

[5] Fuentes-Aznar, A., and Gonzalez-Perez, I., 2018, “Integrating Non-ContactMetrology in the Process of Analysis and Simulation of Gear Drives,”American Gear Manufacturers Association Fall Technical Meeting 2018,AGMA Paper No. 18FTM21.

[6] Gonzalez-Perez, I., and Fuentes-Aznar, A., 2020, “Tooth Contact Analysis ofCylindrical Gears Reconstructed From Point Clouds,” New Approaches to GearDesign and Production, Mechanisms and Machine Science, Vol. 81, Goldfarb,V., Trubachev, E., and Barmina, N., eds., Springer, New York, pp. 219–237.

[7] Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., and Stuetzle, W., 1992,“Surface Reconstruction From Unorganized Points,” Comput. Graph. (ACM),26(2), pp. 71–78.

[8] Fleishman, S., Drori, I., and Cohen-Or, D., 2003, “Bilateral Mesh Denoising,”ACM SIGGRAPH 2003 Papers, SIGGRAPH ’03, pp. 950–953. Q5

¶

Fig. 16 Variation of contact pressure at the middle tooth of thepinion for the designed, manufactured, no-filtered and filteredreconstructed pinions using the bilateral filter with niter=3

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[9] Tomasi, C., and Manduchi, R., 1998, “Bilateral Filtering for Gray and ColorImages,” Proceedings of the IEEE International Conference on ComputerVision, pp. 839–846.

[10] Digne, J., and Franchis, C. D., 2017, “The Bilateral Filter for Point Clouds,”Image Process. On Line, 7, pp. 278–287.Q6

¶ [11] Gill, P. E., Murray, W., and Wright, M. H., 1997, Practical Optimization,Academic Press, London.

[12] Nocedall, J., and Wright, S. J., 1999, Numerical Optimization, Springer, Berlin.[13] Samet, H., 2006, Foundations of Multidimensional and Metric Data Structures,

Morgan Kaufmann, San Francisco, CA.[14] Piegl, L., and Tiller, W., 1995, The NURBS Book, Springer-Verlag, New York.[15] Fuentes-Aznar, A., Iglesias-Victoria, P., Eisele, S., and Gonzalez-Perez, I.,

2016, “Fillet Geometry Modeling for Nongenerated Gear Tooth Surfaces,International Conference on Power Transmissions, Chonqing, China,pp. 431–436.Q7

¶

[16] Sheveleva, G. I., Volkov, A. E., and Medvedev, V. I., 2007, “Algorithms forAnalysis of Meshing and Contact of Spiral Bevel Gears,” Mech. Mach. Theory,42(2), pp. 198–215.

[17] Fuentes, A., Ruiz-Orzaez, R., and Gonzalez-Perez, I., 2014, “ComputerizedDesign, Simulation of Meshing, and Finite Element Analysis of Two Types ofGeometry of Curvilinear Cylindrical Gears,” Comput. Methods Appl. Mech.Eng., 272, pp. 321–339.

[18] Gonzalez-Perez, I., and Fuentes-Aznar, A., 2017, “Analytical Determination ofBasic Machine-Tool Settings for Generation of Spiral Bevel Gears andCompensation of Errors of Alignment in the Cyclo-Palloid System,”Int. J. Mech. Sci., 120, pp. 91–104.

[19] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory,Cambridge University Press, New York.

[20] Dassault Systemes, 2019, Abaqus/Standard Analysis User’s Guide, DassaultSystemes Inc., Waltham, MA.

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