Mechanism and Machine Theory 71 (2014) 64–78

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Mechanism and Machine Theory

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An approximate method to predict surface wear of hypoidgears using surface interpolation

D. Park 1, M. Kolivand⁎, A. KahramanGear and Power Transmission Research Laboratory, Department of Mechanical and Aerospace Engineering, The Ohio State University, 201W. 19th Avenue,Columbus, OH 43210, USA

a r t i c l e i n f o

⁎ Corresponding author at: American Axle and MaE-mail address: mohsen.kolivand@aam.com (M.

1 Currently with Samsung Heavy Industries Co., Lt

0094-114X/$ – see front matter © 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mechmachtheory.2013.09.

a b s t r a c t

Article history:Received 3 January 2013Received in revised form 10 August 2013Accepted 2 September 2013Available online 29 September 2013

In this paper, an approximate and fast wear computation methodology is proposed for predictingtooth surfacewear of hypoid gears. Thismethodology combines loaded tooth contact results froma semi-analytical hypoid gear contact model with Archard's wear model. In order to reduce thecomputational time required compared to previous hypoidwearmodels, (i) a patching techniquethat reduces the rotational span of the analysis to a single mesh cycle from a complete loadingcycle is implemented, (ii) a surface interpolation technique is employed to surface-fit wear depthsat a coarser rotational increment to an approximate wear distribution, and (iii) a sliding distancecomputation based on the contact velocities is devised rather than a coordinates-based methods.Differential geometry formulations based on surface normal curvatures and their directions areused to determine surface velocities and contact line directions. The results of this approximatemodel are compared to those from a previous model using the same semi-definite tooth contactmodel to demonstrate its accuracy, especially at lower torque values, as well as its computationalefficiency.

© 2013 Elsevier Ltd. All rights reserved.

Keywords:Hypoid gearsGear wearContact mechanics

1. Introduction

Hypoid gears, as any other type of gearing, exhibit several failure modes including scoring, pitting, tooth breakage and surfacewear [1–4]. Among them, gear surface wear is perhaps the least studied one, partly because it takes place in a relatively longperiod of time to impact the functionality of a gear pair under mixed or boundary lubrication conditions. The worn tooth surfacesoften accelerate the occurrence of other more immediate failure modes such as scoring and pitting. Surface wear also changes thedynamic behavior of a gear pair to result in an increase of structure-borne noise and dynamicmesh forces that are often attributedto tooth bending and contact fatigue failures as well [5].

Screwing motions associated with distinct geometry of hypoid gears with shaft offset cause higher sliding action in both facewidth and profile directions compared to other gear types [6,7]. This makes hypoid gears more susceptible to wear problems thanother types of gears. In fact, this wear tendency has long been recognized and used as a tool to conform the hypoid gear toothsurfaces through a process known as lapping. During lapping, a hypoid gear pair is operated under lightly loaded conditions withan abrasive lapping paste applied to the gear mesh. At the same time, the mounting positions (misalignments) of the pinion orgear are varied according to a predefined schedule to move the instantaneous contact zone to different areas of the tooth flankallowing the entire tooth surface to be worn to a more desirable geometry.

Wear, in general, is a very complex phenomenon, caused by “plastic deformations, fatigue cracking, ductile extrusion andfracture on a scale associated with asperity contact” [8] and “involves chemical and physical interactions with mechanicalcomponents” [9]. From their review of prominent wear studies from 1957 to 1992, Meng and Ludema [9] identified more than

nufacturing, Detroit, MI, USA. Tel.: +1 248 299 6541; fax: +1 248 299 6648.Kolivand).d., S. Korea.

ll rights reserved.002

65D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

180 different wear equations based on (i) empirical means, (ii) contact mechanics or (iii) material fracture mechanisms. Inaddition, they listed more than 150 variables used in these wear models. While there is no universally accepted wear model thatis suitable for any given wear problem, Archard's contact mechanics based wear equation [10] proposed for abrasive wearemerged over the years as one of the most widely used wear model used in an array of applications ranging from piston rings [11],fuel injectors and gears. Spikes et al. [8] state that the Archard's wear equation is “still probably the most widely accepted theoryof wear at least for the severe phase of metallic sliding wear”. Experimental observations of transferred films on wear surfaces anddirect observations of asperity adhesion both add credence to the theory.

Krantz and Kahraman [12] performed a large number of gear wear tests with lubricants having different viscosities result indifferent wear depths. Elastohydrodynamic lubrication (EHL) models like the ones proposed in recent years for hypoid gears(e.g. Xu and Kahraman [13], Kolivand et al. [14] and Mohammadpour et al. [15]) provide little help to the prediction of wear. Asthese models consider either rough surface [13,14] or smooth surface [15] EHL analyses of hypoid gear contacts and employperfectly elastic formulations with the profiles remaining unchanged, they offer no mechanism for material removal caused bywear. Only tangible use of EHL models in wear simulations has been the attempts made to devise a very simple form of wearcoefficient that is a function of the smooth surface film thickness (lambda ratio) [5,11].

For the above reasons, combined with added complexities associated with gear contact mechanics, all of the published gearmodels have used the Archard's wear equation. Among them, spur and helical gear wear models of Ref. [5,16–19] used Archard'swear equation in conjunction with a gear load distribution model. Some of these studies [5,18] provided comparisons tomeasured gear wear profiles to show that this approach is reasonably accurate as long as the wear coefficient can be obtainedempirically. As such, most of the published models on long-cycle hypoid wear also employed the same methodology. Amongthem, Singh [20] and Park and Kahraman [21] both combined a commercially available FE-based hypoid gear contact model [22](based on a hybrid formulation defined in Refs. [23,24]) with Archard's wear equation to predict the initial wear distributionsof hypoid gears with or without gear misalignments. Singh's model was limited to face-milled hypoid gears while the model ofPark and Kahraman [21] included both face-milled and face-hopped hypoid gears. Their predictions of the initial maximumpressure and wear distributions showed a direct influence of gear position errors on the contact patterns and the resultant wearprofiles. These models focused mostly on the initial stage of wear due to computational effort associated with the use of adeformable-body contact model for the prediction of contact stresses. Such wear simulations require contact analyses that mustbe repeated at a large number of rotational gear positions (say more than 100 positions covering one loading cycle of a tooth) at avery small rotational increment. If the progression of wear is to be modeled, then these contact stress predictions must berepeated iteratively at a certain increment of wear accumulation. As such, several thousands of contact analyses might be requiredfor a complete wear analysis, making FE-based methods impractical.

In order to reduce computational demand associated with hypoid gear wear simulations, these authors proposed a hypoidwear model of face-milled and face-hobbed spiral bevel and hypoid gears [25] that employed the semi-analytical loaded toothcontact analysis (LTCA) model of Kolivand and Kahraman [26] in place of a FE model to reduce the computation time about anorder of magnitude. While computationally more efficient, the hypoid wear model of Park et al. [25] still required considerablecomputation time. This might be deemed acceptable for analysis purposes, but not for a practical design tool that must be used forparameter studies or for lapping simulations where the required number of geometry upgrades can be at least an order ofmagnitude more than a long-cycle wear analysis [27].

The main focus of this paper is development of an approximate hypoid gear wear computation methodology with significantcomputational time improvements while the accuracy of the predictions is not compromised especially under lightly loadedconditions. For this, various schemes will be devised to reduce the number of LTCA analyses required for the wear simulationsignificantly in addition to defining sliding distances directly from the contact line velocities. This method will be referred to as an“approximate method” here whose predictions will be compared to the model of Ref. [25] that will be called the “exact method”.Section 2 will provide a brief overview of the application of Archard's wear equation to gear wear through application of anoverall iterative wear simulation scheme. The proposed approximate method will be introduced in Section 3 including therequired formulations to define contact velocities and directions of the contact lines. Finally, an example face-hobbed hypoid gearpair will be introduced in Section 4 andwear predictions of the approximate method will be compared to those of the exact modelin terms of their accuracy and computational efficiency.

2. Brief description of gear wear prediction methodology

The same hypoid wear prediction methodology proposed by Park and Kahraman [21,25] will be employed here with theexception of the computational procedures of contact pressures and sliding distance. Here, wear depth h of a point on a slidingtooth contact interface is given by a first-order differential equation

dhds

¼ kP tð Þ ð1Þ

s is the sliding distance between a contact point and the mating point on the other contact surface, and P is the normal

wherecontact pressure. All other lubricant, material and surface related parameters [20] are lumped into a single constant parameter k

66 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

known as the wear coefficient. As this simplifies the predictions significantly, it also makes the model semi-empirical since kmustbe determined experimentally for the contact conditions in hand.

While the sliding distance s of a given contact point at a given rotational position changes negligibly with surface wear, thechanges in contact pressure P with wear are significant [21,25]. Contact pressure computations must be updated frequently asaccumulation of wear alters the load distributions and the contact pressures. This necessitates an iterative computationalmethodology outlined in the flowchart of Fig. 1 for the prediction of the progression of surface wear of hypoid gear teeth [21].First, the initial (no wear) tooth surfaces are defined. These initial tooth surfaces can either be the nominal theoretical surfacesdefined by the simulation of the cutting process or actual measured ones deviated from their theoretical counterparts due toheat treatment distortions and manufacturing errors. The initial surface geometries of both (driving) pinion p and (driven) gearg are defined at a given tooth surface grid point ij shown schematically in Fig. 2(a) as (Gij

p)ζ = 0 and (Gijg)ζ = 0 where superscript ζ

is the index for geometry updates with ζ = 0 indicating that these are initial surfaces with no wear. With these initial surfacegeometries, a loaded gear contact analysis is performed at each incremental rotational gear position n (n ∈ [1,N]) to determinethe initial contact pressures at each surface grid point ij on both gear surfaces as (Pij

p)nζ = 0 and (Pijg)nζ = 0 as shown schematically

in Fig. 2(b). As illustrated in Fig. 2(a), point ij enters the contact zone at the rotational increment n = ns and remains in contactuntil the position n = ne. For a reasonably refined discretization to capture Pij

p and Pijg profiles in Fig. 2(b) accurately, the set

n ∈ [ns,ne] must contain a reasonable number of positions, requiring a very small computation increment with a large N (sayN ≥ 100). This is one of the main reasons for the computational burden in wear analysis of hypoid gears that will be remedied inthe next section.

According to Fig. 1, the worn gear tooth surfaces (Gijp)ζ and (Gij

g)ζ are then fed into the gear contact model for prediction of theupdated contact pressures (Pij

p)nζ and (Pijg)nζ at all rotational positions n ∈ [1,N] and the same process is repeated until another

Fig. 1. Flowchart of the iterative gear wear prediction methodology [25].

j = 1i = 1 i = I

ij

1 ns ne Nn

Pij

Root

Tip

Toe Heel

Actual distribution

Discretized distribution

nj = J

n = ne

n = ns

(a)

(b)

Fig. 2. (a) Moving contact zones at n = ns and n = ne on the tooth surface grid Gij, and (b) the resultant discrete pressure distribution a grid point ij.

67D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

geometry update is needed. The wear depth accumulated after each geometry (and pressure) update are added to determine thefinal wear depth of point ij as [25]

hp;gij ¼Xζ max

ζ¼0

Cζ Δhp;gij

� �ζ ð2aÞ

(a)

(b)

Fig. 3. (a) Illustration of contact grid cells on a contact line l and (b) definition of tangent planes at contact segments.

Fig. 4. (comple

68 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

(Δhijp,g)ζ represents the wear accumulated at this grid point through one wear cycle during the wear iteration ζ defined

where

as

Δhp;gij

� �ζ ¼ 12kXNn¼1

Pp;gij

� �ζ

nþ Pp;g

ij

� �ζ

nþ1

� �sp;gij

� �n→nþ1

: ð2bÞ

Here ζmax is the user-defined maximum number of geometry iterations, Cζ is the number of wear cycles during the weariteration before the ζ-th geometry update, and (sij

p,g)n → n + 1 is the sliding distance experienced by the grid points ij on the pinionand gear surfaces from rotational position n to n + 1. (sij

p,g)n → n + 1 was computed in earlier studies [20,21,25] by quantifying therelative distances occur between two mating surface points after each rotational increment, requiring time-consuming rotation

(a)

(b)

(c)

a) Simultaneous contact lines on two consecutive pinion teeth covering a mesh cycle, (b) contact lines superimposed on a single tooth covering ate loading cycle, and (c) the contact grid cells constructed on the contact lines.

69D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

and translation transformations amongst various local and global coordinate systems. An alternate method will be devised in thenext section to compute sliding distances directly from sliding velocities.

Actual number of tooth pairs in contact (or number of contact lines Ncl) is defined by several parameters such as gear ratio, rollangle of the pinion and the amount of torque applied. Number of potential contact lines (ℓ∈ 1;Ncl½ �) as well as the load intensitydistributions along each contact lineℓ is predicted by the LTCA model [26], with each line discretized by Ncp number of segments(m ∈ [1,Ncp]). Load on each segment, width and combined radius of curvature of the contact at the location of the line segmentare then used to compute Hertzian pressure distribution (Pij

p,g)nζ of Eq. (2b). Following the methodology of Ref. [25], Fig. 3(a)shows a schematic of the contact line ℓ having Ncp contact segments. The center point of segment m along the contact line isdenoted by Mm. Each contact grid cell defined to correspond to each contact segment is approximated to a rectangular shape onthe planeΠm that is tangent to the tooth surfaces at pointMm as shown in Fig. 3(b). The maximum pressure Pm at the center pointMm of the contact segmentm is predicted by the LTCAmodel. The contact pressure distribution within the same contact cell at thesame time instant is obtained by assuming a Hertzian pressure distribution of a line contact.

The second reason is the need to compute sliding distance experienced by each surface grid point at each rotational increment,which requires various time-consuming rotation and translation transformations. The approximate method provided in thissection focuses on minimizing number of rotational increments N required and on computing sliding distances from slidingvelocities. Specifically:

(i) A patching technique that reduces the rotational span of the analysis to a single mesh cycle from a complete loading cycle(defined as the period from the instant a single tooth enters the contact to the instant it leaves the contact) will beimplemented.

(ii) A surface interpolation technique will be employed to surface-fit wear depths at a coarser rotational increment to anapproximate wear distribution.

(a)

(b)

Fig. 5. (a) Moving velocity vector of a contact grid cell m at a contact point Mmn at rotational increment n, and (b) the pressure time history at the fixed surface

point Mnm having a semi-elliptical Herztian contact profile.

70 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

(iii) A sliding distance computation based on the velocities of the contact lines will be devised rather than a coordinates-basedone.

3. The approximate method

The first step in the approximate method is to perform a load distribution analysis covering a complete mesh cycle through asmall number of time steps Nmc. Fig. 4(a) shows the contact lines on two consecutive teeth of an example hypoid gear pair forNmc = 5. Note that the contact load distribution on tooth #1 at the next time step n = Nmc + 1will be identical to the contact ontooth #2 at n = 1. In general, as long as Nmc number of rotational increments cover exactly one mesh cycle, contact line and loaddistribution on tooth #1 at position n + Nmc (n ∈ [1,Nmc]) are identical to the contact line and load distribution on tooth #2 at n.Likewise, if there are more than two tooth pairs in contact, then contact line and load distribution on tooth #1 at positionn + 2Nmc (n ∈ [1,Nmc]) are identical to the contact line and load distribution on tooth #3 at n. Recognizing this mesh periodicity,load distributions from adjacent teeth within a single mesh cycle can be patched to obtain the loading information covering theentire load cycle instead of running extended analysis to move the contact line through a tooth surface during a complete loadingcycle. In Fig. 4(b), such a patching process is applied to obtain a loading cycle consisting of N = 10 from only Nmc = 5 analysissteps covering a mesh cycle. Here it is seen that the simultaneous contact lines displayed on two teeth in Fig. 4(a) aresuperimposed on a single tooth.

With the contact line discretization scheme described in Section 2 applied to the contacts of Fig. 4(b), the discretized contactsegments are obtained as shown in Fig. 4(c). Focusing on one of the grid cells m (m ∈ [1,Ncp]) along one of these lines atincrement n, the center point Mm

n of the cell m at position n shown in Fig. 5(a) corresponds to a point Mnm defined on the fixed

surface grid shown in Fig. 2(a), whose coordinates can be defined using the global coordinate frame. There are a total of N × Ncp

such points defined in Fig. 4(c). The approximate method aims at finding the wear depth only at these points Mnm instead of

seeking wear along every point of a highly refined fixed tooth surface grid. For this, the wear depth at pointMnm must be computed

as the corresponding moving grid cell m passes through it. At the instant when Mnm and the moving grid point Mm

n coincides inspace,M

nm is on the contact line and experiences its maximum pressure value (Pmn )max.M

nm moves across the moving grid cell m in

the direction of the surface velocity of the cell (vr)mn (velocity of the contact line at pointMmn ) as shown in Fig. 5(a) in the process,

spending (τd)mn period of time as a loaded tooth surface point. The time this surface point Mnm spends within the contact is

given as

where

where

τdð Þnm ¼ Dnm

vrð Þnmð3aÞ

(vr)mn = ‖(vr)mn ‖ and Dmn is the distance traveled within the contact cell, as shown schematically in Fig. 5(a), that is defined

where

as

Dnm ¼ 2bnm

sinχnm: ð3bÞ

χmn is the angle between moving velocity (vr)mn and contact line direction vector (t1)mn on the tangent plane Πm

n . With this,wear accumulated at point Mn

m after one wear cycle is determined in a manner analogous to Eq. (2b) as

Δhnm ¼ kZτdð Þnm

0

Pnm tð Þ vsð Þnmdt ð4aÞ

(vs)mn is the sliding velocity of the contact cell such that the sliding distance (ds)mn = (vs)mn dt and the normal pressure timehistory Pm

n (t) of the same fixed point can be defined by using the Hertzian theory as shown in Fig. 5(b) as

Pnm tð Þ ¼ Pmaxð Þnm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 2t− τdð Þnm

τdð Þnm

� �2s

: ð4bÞ

Finally, the wear depths at all tooth surface points ij (i ∈ [1,I], j ∈ [1,J]) are obtained by an interpolation scheme from thesewear depths computed at discrete pointsM

nm (m ∈ [1,Ncp], n ∈ [1,N]). Akima's [28] surface interpolation method was used for this

purpose. Algorithms and subroutines for this method are available in open literature. Firstly, the tooth surface grid represented bypoints ij needs to be mapped to a plane Q rz(r,z) where r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pand z is the coordinate along the rotational axis in local

coordinate systems Xp(x,y,z) and Xg(x,y,z), which are attached to each pinion and gear, respectively. Delaunay triangulation is

71D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

performed with the given data points at Mnm (m ∈ [1,Ncp], n ∈ [1,N]). Then, on each triangle T defined here, the interpolant h

(wear depth) has the form of a bivariate quintic polynomial on Q rz(r,z) plane such that

Fig. 6. (onto th

h r; zð Þ ¼X

mþnð Þ≤5

Amnrmzn ∀r; z∈T ð5Þ

Amn are the coefficients of the local polynomials.

whereThe tooth surface can be locally approximated to a second polynomial surface at each contact pointM

nm since Hertzian contact

conditions are assumed for the computation of the pressure time history. With this approximation, the contact shape becomeselliptical so that the contact line direction along the major axis of the contact ellipse can be defined. This locally approximatedtooth surface is given as f(xm,ym,zm) = 0where xm, ym and zm are the coordinates ofM

nm on the pinion tooth surface (or on the gear

tooth surface) with respect to coordinate system Xp(x,y,z) or Xg(x,y,z). For any tooth surface nearMnm, f(x,y,z) = 0 can be written

by using Taylor series as [29]

∂ f∂x xnm−x

þ ∂ f∂y ynm−y

þ ∂ f∂z znm−z

þ 12∂2 f∂x2

xnm−x 2 þ 1

2∂2 f∂y2

ynm−y 2 þ 1

2∂2 f∂z2

znm−z 2

þ ∂2 f∂x∂y xnm−x

ynm−y þ ∂2 f

∂y∂z ynm−y

znm−z

þ ∂2 f∂z∂x znm−z

xnm−x þ ::: ¼ 0:

ð6aÞ

(b)

(a)

a) Movement of a contact point M on the surface between two consecutive pinion rotational position θ = θp and θ = θp + Δθp and (b) projection of vrp

e principal axes on the pinion surface.

72 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

This surface function can be simplified by retaining only the first- and second-order terms and by applying a coordinatetransformation from Xp(x,y,z) or Xg(x,y,z) into a tooth local coordinate system (Xt)m = (t1,t2,n)m. Here Xt is attached to thetangent plane Πm on the tooth surface at contact points M

nm and comprises two perpendicular unit vectors t1 (contact line

direction) and t2 (perpendicular to contact line direction), and surface unit normal vector n, as shown in Fig. 5(a) as

whereparamdirecti

zt ¼ a1x2t þ 2a2xtyt þ a3y

2t ð6bÞ

xt, yt, and zt are components of position vector on the surface with respect to the local coordinate system (Xt)m = (t1,t2,n)m. In

wherepractice, this polynomial surface can be simplified further by matching vectors t1 and t2 with the two principal direction vectors e1and e2 at the contact pointM

nm [29–31]. With this, the surface function defined by Eq. (6b) is expressed by principal curvatures κ1 and

κ2 with respect to the local coordinate system (Xe)m = (e1,e2,n)m with coordinates (xe,ye,ze) such that

ze ¼ 12κ1xe

2 þ 12κ2ye

2 ð7Þ

the normal principal curvatures κ1 and κ2 along the principal directions e1 and e2 are known as second order geometriceters at Mn

m [31]. In this local coordinate system (Xe)m on the second order surfaces, moving velocity and contact ellipseon will be expressed in the following sections.

3.1. Computation of the velocity of the contact line

Implementation of the above formulation requires computation of velocity (vr)mn of the contact line segment m at position nand the associated parameters Dm

n and χmn that are required to determine (τd)mn . Well-known differential geometry formulations

based on surface normal curvatures and their directions will be used here to determine (vr)mn . Fig. 6(a) shows the movement ofcontact point on the pinion surface from a rotational position θ = θp to the next consecutive position θ = θp + Δθp. When theposition of the pointM at θ = θp moves toM ' at the next rotational step θ = θp + Δθp, the velocity vector vp of the same point onthe pinion surface with respect to the global coordinate frame Xfg is expressed as

vp ¼ vpr þ vp

ω ð8aÞ

vp = dXp/dt (the first derivative of position vector Xp) vrp is the moving velocity of this point M on the pinion surface, and

wherevωp = ωp × Xp with ωp being the angular velocity vector of the point M on a pinion surface. With point M being common to bothpinion and gear surfaces at every instant,

vg ¼ vgr þ vg

ω: ð8bÞ

In order to obtain vrp and vrg for a contact point M on the pinion and gear surfaces, respectively, two geometric conditions areconsidered [31]. The first one is the condition of contact that is written from Fig. 6(a) as:

Xp ¼ Eþ Xg ð9Þ

E is the position vector representing shaft offset. Condition of contact can be differentiated in time to obtainXp ¼Xg, stating

wherethat vp = vg at point M. With this, Eq. (9) becomes

vpr þ vpω ¼ vgr þ vg

ω: ð10Þ

Here, moving velocities vrp and vrg are expressed on their respective local coordinate systems Xe = (e1,e2,n) on the pinion andgear surfaces as

vpr ¼ vpr1ep1 þ vpr2e

p2 ; ð11aÞ

vgr ¼ vgr1e

g1 þ vgr2e

g2 ð11bÞ

whereLikewthese,

Fig. 7. Isurface

73D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

vr1p and vr2

p are the projections of the vrp onto two principal directions e1p and e2

p on the pinion surface, as shown in Fig. 6(b).ise, vr1g and vr2

g are the projections of the vrg onto two principal directions e1g and e2g on the gear surface, respectively. With

Eq. (10) is written as

vpr1ep1 þ vpr2e

p2 þωp � Xp ¼ vgr1e

g1 þ vgr2e

g2 þωg � Xg

: ð12Þ

The second geometric condition is the collinearity condition of the surface normal vectors: np = ng. It is differentiated in timeto obtainnp ¼ng wherenp ¼np

r þnpω andng ¼ng

r þngω . Heren

pr andng

r are the time derivative of the surface normal vectors due tomovement of the point M on the pinion and gear surface respectively, andnp

ω andngω are time derivatives of the surface normal

vector due to rotation of the point M along the rotational axis (z axis) of the global coordinate frame. Rodrigues' formula [32]relates vr andnr at point M as

−κ1;2vr ¼nr ð13Þ

κ1 and κ2 are the two principal curvatures along the principal direction vectors e1 and e2, respectively. Using Rodrigues'

whereformula, conditionnp ¼ng is expressed by

κp1v

pr1e

p1 þ κp

2vpr2e

p2

−ωp � np ¼ κ g

1vgr1e

g1 þ κ g

2vgr2e

g2

−ωg � ng

: ð14Þ

By taking scalar product of both sides of Eqs. (12) and (14) with e1g and e2

g, respectively, a system of four coupled algebraicequations is obtained, which are solved numerically for the unknown components vr1p , vr2p , vr1g , and vr2

g of moving velocity along thepinion and gear principal directions e1

p, e2p and e1

g, e2g, respectively. With these components, Eqs. (11a) and (11b) are used to

compute the moving velocity vector on the pinion and gear surfaces.

3.2. Computation of the contact line direction

In order to determine (τd)mn to be used in Eq. (4a), the angle χmn between moving velocity (vr)mn and contact line direction

vector t1 (Fig. 5(a)) on the tangent planeΠmn between pinion and gear surfaces should be computed. The major axis of the contact

ellipse is considered to be in the same direction as the contact line direction on each contact point Mmn . In order to search for the

direction of the major axis of the contact ellipse, a difference surface Σpg for both pinion and gear surfaces is defined [31]. Thedifference surface Σpg contacts both pinion and gear tooth surfaces at Mm

n , and its normal curvature along an arbitrary surfacevector t on the tangent plane Πm

n is defined by

κpgt ¼ κp

t−κ gt : ð15aÞ

The major and minor axis directions of the contact ellipse can be defined on Σpg by finding the principal directions, t1 and t2.These principal directions on Σpg represent the extreme (limiting) values of the curvatures as the direction of vector t is varied, asshown in Fig. 7. By using Euler's equation [24], each normal curvature along this vector t on each surface is expressed in terms ofits principal curvatures κ1 and κ2 as

κpt ¼ κp

1 cos2λþ κp

2 sin2λ; ð15bÞ

llustration of the surface local coordinate systems at an instant contact point M and the instantaneous contact ellipse directions t1 and t2 on the tangentΠ.

wherecounteon theIn Fig.(vr)mn .

Table 1Basic gear geometry parameters of the example face-hobbed hypoid gear pair.

Parameter Unit Pinion Gear

Number of teeth 12 41Hand of spiral Left RightShaft angle [Degree] 90.00Shaft offset [mm] 30.00Outer cone distance [mm] 104.21 120.86Face width [mm] 36.80 32.33Root angle [Degree] 24.33 64.66Spiral angle [Degree] 46.99 19.65Pitch diameter [mm] 85.86 218.46

Fig

74 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

κgt ¼ κ g

1 cos2 λ−ψð Þ þ κp

2 sin2 λ−ψð Þ ð15cÞ

λ is the angle between e1p and arbitrary vector t, and ψ is the angle between e1p and e1g, both defined positive in the

r-clockwise direction. In order to search the principal directions that represent contact line direction at contact point Mmn

difference surface Σpg, the angles corresponding to the limiting curvature values κtpg can be found by setting dκtpg/dλ = 0.7, the angle λ corresponding to t1 is denoted by λp. Once this contact line vector t1 is found, the angle χm

n is defined, given

. 8. Principal directions at the contact points of a contact line at position n (dotted arrows are for the pinion, and the solid arrows are for the gear).

Tip

Root

Toe

Heel

Toe

Root

Tip

Pinion Surface

Gear Surface

Heel

(b)

(a)

Fig. 9. Moving velocity vectors of the contact grid cells of a contact line along (a) the pinion surface and (b) the gear surface at Ωp = 2000 rpm.

75D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

4. Comparison between the exact and approximate methods

An example face-hobbed hypoid gear pair whose basic parameters are shown in Table 1 will be used here to demonstrate theproposed approximate wear prediction method and to compare it to the exact method proposed in Ref. [25]. This example hypoidset represents a typical automotive rear axle gear pair with a shaft off-set of E = 30 mm. Number of rotational incrementscovering one mesh cycle was chosen to be Nmc = 5 which resulted in N = 10 increments after applying the patching schemedemonstrated in Fig. 4 to cover one complete loading cycle.

Fig. 8 shows the principal directions e1 and e2 on both the pinion and gear surfaces on each of the contact cells of a contact lineat a certain rotational increment n. It is noted that the principal directions on the pinion and gear surfaces at each contact pointMare different from each other. These principal directions and the corresponding curvatures are used to compute the movingvelocity vectors (vr)mn of each contact grid cell on the pinion and gear surfaces. Fig. 9 shows moving velocities (vr)mn of each linegrid cellm at each position n on the theoretical pinion and gear surfaces for a pinion speed of atΩp = 2000 rpm. The distributionsof the magnitude of the velocity vector ‖(vr)ij‖ along the pinion and gear surfaces for this same condition are shown in Fig. 10 onthe projection plane Qrz, obtained after the interpolation of ‖(vr)mn ‖ values within a contact zone. Both magnitudes and directionsof the moving velocity vector vary for each cell along each contact line at each instant n. Magnitude of moving velocities shown inFig. 10(a) for the pinion surface are almost two times larger than those of the gear surface shown in Fig. 10(b) due to thegeometry of the tooth surfaces of this example gear set with E = 30 mm.

Fig. 11 shows the wear distribution profile that is constructed by using the approximate method under a pinion torque of T =600 N m. Here, solid dots represent the actualwear calculations done byusing Eq. (2a) at each of the discrete contact pointsMm

n while

(a)

(b)

Fig. 10. Distribution of the contact cell speeds on the projection plane Qrz at Ωp = 2000 rpm; (a) the pinion surface and (b) the gear surface.

76 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

the continuous surfaces represent the wear distribution on the pinion and gear surface obtained by using the surface fitting scheme,according to Eq. (5). These fitted wear surfaces follow discrete wear values closely, with some deviations along the edges of thecontact at the tip of the pinion on and the root of the gear. Such deviations are a direct result of using a significantly smaller N in theapproximate analysis. Such deviations are much smaller at lower T values, making this approximate method especially suitable forsimulation of the lapping process.

(a) (b)

Fig. 11. (a) Pinion and (b) gear surface wear distributions. Dots are the wear depths at discrete points Mmn and the continuous surfaces are the wear distributions

constructed by using Akima's [28] interpolation scheme.

(a) Exact (b) Approximate50 Nm

300 Nm

600 Nm

Fig. 12. Comparison of the pinion wear distributions predicted by using (a) the exact method [25] and (b) the approximate method at T = 50, 300 and 600 N m.

77D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

Fig. 12 compares the initial wear distributions (normalized by k, i.e. (hijp)ζ = 0/k) on a pinion tooth surface predicted by using

the exact model of Ref. [25] with N = 100 and the proposed approximate method with Nmc = 5 (corresponding to N = 10) atthree different torque values of T = 50, 300 and 600 N m. The same comparison is presented in Fig. 13 for the initial wear on thegear tooth surfaces. It is seen from these figures that both methods produce almost identical initial wear distributions at T =50 N m while some slight differences are evident especially at the highest torque value of T = 600 N m. The exact method [25]with the semi-analytical contact model of Ref. [26] took 60 s of CPU time on a 3 GHz PC (10 s for the compliance computationsrequired by the LTCA model and 50 s for wear computation with N = 100 at about 0.5 s per position n). In comparison, the

50 Nm

300 Nm

600 Nm

(a) Exact (b) Approximate

Fig. 13. Comparison of the gear wear distributions predicted by using (a) the exact method [25] and (b) the approximate method at T = 50, 300 and 600 N m.

78 D. Park et al. / Mechanism and Machine Theory 71 (2014) 64–78

approximate method with N = 10 took 12.5 s of CPU time (10 s for the compliance calculation and 2.5 s for wear computationsfor Nmc = 5). When a complete wear simulation with ζmax = 20 geometry updates is performed, the exact method requiresabout 17 min of CPU time while the same simulation with the proposed method took 1.8 min. This indicates that the proposedapproximate method, while it might deviate slightly from the exact method under heavier loads, is about an order of magnitudefaster than the exact method. This makes the proposed method suitable for practical long-cycle wear calculations as well assimulation of the lapping process.

5. Conclusion

In this paper, an approximate wear computation methodology was proposed for predicting tooth surface wear of hypoid gears,with the aim of reducing the computational time required for such simulations significantly. This wear methodology combinesloaded tooth contact results from the semi-analytical hypoid gear contact model of Kolivand and Kahraman [26] with Archard'swear model. This methodology followed the formulations of Ref. [25] with three distinct differences implemented to increase thecomputational efficiency. First of all, patching technique that reduces the rotational span of the analysis to a single mesh cyclefrom a complete loading cycle was implemented exploiting the periodicity of the contact conditions at consequent loaded teethat the gear mesh frequency. Secondly, a surface fitting scheme was employed to generate a continuous, approximate surface weardistributions from wear depths computed at a much coarser rotational increment. Finally, sliding distance computationprocedures of earlier models from the changes in the positions of mating contact surfaces was replaced by a new one based onthe contact velocities, in the process eliminating the need for extensive coordinate transformations. Differential geometryformulations based on surface normal curvatures and their directions were used to determine surface velocities required for thissliding distance computation.

Wear of an example face-hobbed hypoid gear pair from an automotive rear axle application was simulated by using theproposed approximate model as well as the exact model of Ref. [25]. As both approaches employed the same loaded tooth contactmodel [26], the comparisons of these predictions were used to assess the accuracy of the proposed approximate method as well asthe computational time improvements. It was shown that the results of the approximate method agree well with those from Ref.[25], especially at lower torque values. It was also shown that the time required for wear computations can be reduced about 10times compared to the model of Ref. [25]. With this, the proposed model is suitable for design optimization and parametersensitivity studies.

Our current work on this topic focuses on application of this approximate hypoid wear model to simulate the lapping processapplied routinely to face-hobbed hypoid gears as part of the manufacturing processes. For this, the proposed wear model and theloaded tooth contact model of Ref. [26] are exercised with time-varying schedules of E, P and G error settings to obtain the finalsurface geometries after lapping. Results of this study will be presented in a separate paper.

References

[1] W.J. Bartz, V. Kruger, Influence of lubrications on the pitting fatigue of gears, Wear 35 (1975) 315–329.[2] W. Coleman, Bevel and Hypoid Gear Surface Durability: Pitting and Scuffing, Gleason Works Publication, 1969. 243–258.[3] T.I. Fowle, Gear lubrication: relating theory to practice, Lubr. Eng. 32 (1976) 205–219.[4] P.M. Ku, Gear failure modes — importance of lubrication and mechanics, ASLE Trans. 19 (1975) 239–249.[5] H. Ding, A. Kahraman, Interactions between nonlinear spur gear dynamics and surface wear, J. Sound Vib. 307 (2007) 662–679.[6] H.J. Stadtfeld, Advanced Bevel Gear Technology, The Gleason Works, 2000.[7] C. Naruse, S. Haizuka, R. Nemoto, T. Nmezu, Limiting load for scoring and frictional loss of hypoid gear, Bull. JSME 29 (1986) 2271–2280.[8] H.A. Spikes, A.V. Olver, P. Macpherson, Wear in rolling contacts, Wear 112 (1986) 121–144.[9] H.C. Meng, K.C. Ludema, Wear models and predictive equations: their form and content, Wear 181–183 (1995) 433–457.

[10] J.F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys. 24 (1953) 981–988.[11] M. Priest, C.M. Taylor, Automotive engine tribology — approaching the surface, Wear 241 (2000) 193–203.[12] T. Krantz, A. Kahraman, An experimental investigation of the influence of the lubricant viscosity and additives on gear wear, Tribol. Trans. 47 (2004) 138–148.[13] H. Xu, A. Kahraman, Prediction of friction-related power losses of hypoid gear pairs, Proc. IMechE. Part K J. Multi-body Dyn. 221 (2007) 387–400.[14] M. Kolivand, S. Li, A. Kahraman, Prediction of mechanical gear mesh efficiency of hypoid gear pairs, Mech. Mach. Theory 45 (2010) 1568–1582.[15] M.Mohammadpour, S. Theodossiades, H. Rahnejat, Elastohydrodynamic lubrication of hypoid gear pairs at high loads, Proc. IMechE Part J. Eng. Tribol. 226 (3) (2012)

183–198.[16] A. Flodin, S. Andersson, Simulation of mild wear in spur gears, Wear 207 (1997) 123–128.[17] A. Flodin, S. Andersson, Simulation of mild wear in helical gears, Wear 241 (2000) 123–128.[18] P. Bajpai, A. Kahraman, N.E. Anderson, A surface wear prediction methodology for parallel axis gear pairs, J. Tribol. 126 (2004) 597–604.[19] A. Kahraman, P. Bajpai, N. Anderson, Influence of tooth profile deviations on helical gear wear, J. Mech. Des. 127 (2005) 656–663.[20] G. Singh, A Wear Model for Face-Milled Hypoid Gears, (M.S. Thesis) University of Toledo, U.S.A., 2003.[21] D. Park, A. Kahraman, A surface wear model for hypoid gear pairs, Wear 267 (9–10) (2009) 1595–1604.[22] Hypoid Face-Milled, Hypoid Face-Hobbed, Advanced Numerical Solutions, Inc., 2011.[23] S. Vijayakar, H.R. Busby, D.R. Houser, Finite element analysis of quasi-prismatic bodies using Chebyshev polynomials, Int. J. Numer.Methods Eng. 24 (1987) 1461–1477.[24] S. Vijayakar, H.R. Busby, D.R. Houser, A combined surface integral and finite element solution for a three-dimensional contact problem, Int. J. Numer.

Methods Eng. 29 (1991) 525–545.[25] D. Park, M. Kolivand, A. Kahraman, Prediction of surface wear of hypoid gears using a semi-analytical contact model, Mech. Mach. Theory 52 (2012) 180–194.[26] M. Kolivand, A. Kahraman, A load distribution model for hypoid gears using ease-off topography and shell theory, Mech. Mach. Theory 44 (10) (2009) 1848–1865.[27] C. Gosselin, Q. Jiang, K.K. Jenski, J. Masseth, Hypoid gear lapping wear coefficient and simulation, AGMA Fall Technical Meeting, 2005, (Paper number: 05FTM09).[28] H.A. Akima, Method of bivariate interpolation and smooth surface fitting for irregularly distributed data points, ACM Trans. Math. Softw. 4 (2) (1978) 148–159.[29] A. Dyson, A General Theory of the Kinematics and Geometry of Gears in Three Dimensions, Clarendon Press, Oxford, 1969.[30] M.L. Baxter, Second-order surface generation, J. Ind. Math. Soc. 23 (2) (1973) 85–106.[31] X.C. Wang, S.K. Ghosh, Advanced Theories of Hypoid Gears, Elsevier Science B. V., 1994.[32] F.L. Litvin, Gear Geometry and Applied Theory, Prentice-Hall, 1994.