Modeling and Analysis of the Meshing Losses of Involute Spur ...

L. Chang1

Fellow ASME

Department of Mechanical and

Nuclear Engineering,

The Pennsylvania State University,

University Park, PA 16802;

Advanced Institute of Manufacturing with

High-Tech Innovations,

National Chung Cheng University,

Ming-Hsiung, Chia-Yi 621, Taiwan

e-mail: [email protected]

Yeau-Ren JengFellow ASME

Advanced Institute of Manufacturing with

High-Tech Innovations,

National Chung Cheng University,



Pay-Yau HuangDepartment of Mechanical Engineering,

Wufeng University,



Modeling and Analysisof the Meshing Lossesof Involute Spur Gearsin High-Speed and High-LoadConditionsA first-principle based mathematical model is developed in this paper to analyze themeshing losses in involute spur gears operating in high-load and high-speed conditions.The model is fundamentally simple with a few clearly defined physical parameters. It iscomputationally robust and produces meaningful trends and relative magnitudes of themeshing losses with respect to the variations of key gear and lubricant parameters. Themodel is evaluated with precision experimental data. It is then used to study the effects ofvarious gear and lubricant parameters on the meshing losses including gear module,pressure angle, tooth addendum height, thermal conductivity, and lubricant pressure-viscosity and temperature-viscosity coefficients. The results and analysis suggest thatgear module, pressure angle, and lubricant pressure-viscosity and temperature-viscositycoefficients can significantly affect the meshing losses. They should be the design parame-ters of interest to further improve the energy efficiency in high-performance, multistagetransmission systems. Although the model is developed and results obtained for spurgears, the authors believe that the trends and relative magnitudes of the meshing losseswith respect to the variations of the gear and lubricant parameters are still meaningfulfor helical gears. [DOI: 10.1115/1.4007809]

Keywords: gear meshing, transmission efficiency, gear lubrication

1 Introduction

Modeling and experimentation of transmission efficiency ofgear systems have been active topics of research in the past deca-des. Some of the recent developments which the authors founduseful and/or informative are Britton et al. [1], Lehtovaara [2],Handschuh and Kilmain [3], Martins et al. [4], Diab et al. [5], Xuet al. [6], Petry-Johnson et al. [7], Magalhaes et al. [8], Li andKahraman [9], and Kuria and Kihiu [10]. Among them, Xu et al.[6] presented an extensive literature summary of the modelingwork and Petry-Johnson et al. [7] presented an extensive literaturesummary of the experimental work.

The transmission efficiency of a precision gear system can bevery high, particularly the meshing efficiency of a pair of spurgears, which can reach above 99.5% [7]. Nevertheless, anydesigns that can further improve the energy efficiency can still besignificant for high-power, multistage transmissions. Because ofthe high efficiency, it can be very difficult to determine accuratevalues of the percentage power losses either in laboratory experi-ment or by theoretical modeling. The task is further complicatedif the loss in the gear system is to be separated into various com-ponents such as meshing, spinning and bearing losses. A goodexample on the experimental side is the work reported by Petry-Johnson et al. [7]. They carried out a very systematic measure-ment program with a well maintained and well instrumented FZGtest rig. The repeatability tests in their experiments yielded a mea-surement scatter of about 0.04% of the total power input. Theysubsequently carried out many power loss measurements with

different gear sets and lubricants under various operating condi-tions. The average of the measured total power losses in all thecases is about 0.45% and that due to gear meshing (with somemeaningful separation calculation) is about 0.25%. Therefore, themeasurement scatter amounts to about 10% of the total power lossand 20% of the meshing loss. Such a measurement uncertaintycan significantly contaminate the results and, more importantly,the trends associated with parameter variations particularly in themeshing losses. The prospect of mathematical modeling of thepower losses does not fare any better. Take as example the mesh-ing loss of a pair of spur gears. The contact geometry, rolling andsliding velocities, load and thus the elastohydrodynamic lubrica-tion (EHL) film thickness and film temperature all vary through-out the meshing cycle. These variations make it very difficult toaccurately calculate the friction forces in the gear contact. Theproblem is further complicated by many other factors as stated inXu et al. [6], such as gear surface roughness, tooth bending, toothtip modification and gear shaft misalignment, to name just a few.Consequently, the meshing loss obtained through mathematicallyintensive modeling such as Li and Kahraman [9] is not necessarilymore accurate than that obtained using empirical friction formulasuch as Kuria and Kihiu [10].

In the authors’ opinion, it is the trends and relative magnitudesof the percentage losses with respect to parameter variations thatplay the key role in the gear design and lubricant selection to fur-ther improve transmission efficiency. These trend and relative-magnitude results may be obtainable without the need to obtainexceedingly accurate power-loss results which are limited by ex-perimental resolution and/or modeling imperfection. A first-principle based modeling appears to be more suitable for this taskas the results are not contaminated by measurement scatters inher-ited in the experiments or hard-to-trace uncertainties associated

1Corresponding author.Contributed by the Tribology Division of ASME for publication in the JOURNAL OF

TRIBOLOGY. Manuscript received July 11, 2012; final manuscript received September22, 2012; published online December 21, 2012. Assoc. Editor: Robert L. Jackson.

Journal of Tribology JANUARY 2013, Vol. 135 / 011504-1Copyright VC 2013 by ASME

Downloaded 19 Jan 2013 to 130.203.244.26. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

with the parameter-laden empirical formulas. This paper aims todevelop such a first-principle based model focusing on meshingefficiency of involute spur gears. The model is fundamentally sim-ple with a few clearly defined physical parameters so that thecause-and-effect results produced by the model can be easilytraced and evaluated. The accuracy of the model is evaluatedagainst well documented experimental data of Petry-Johnson et al.[7]. Subsequently, it is used to study the trends and relative mag-nitudes of the meshing losses with respect to the variations of afew key parameters that describe gear design and lubricantproperty.

2 Modeling

Figure 1, taken from Johnson [11], shows a schematic of themeshing of a pair of involute spur gears. It suggests that the mesh-ing is instantaneously equivalent to the contact of two cylinders.The two cylinders rotate at constant angular velocities equal to theangular velocities of the underline gears but their radii vary as themeshing proceeds along the line of action. Furthermore, the con-tact force between the two cylinders may vary, primarily due tothe change of the number of meshing pairs in the process. Thisequivalent configuration allows the tooth contact and lubricationof the gears to be modeled and friction force calculated usingEHL theory. The EHL problem is basically governed by fourintegral-differential equations [12]:

Reynolds equation

@

@x

h3

12g@p

@x

� �¼ u

@h

@xþ @h

@t(1)

Elasticity equation

hðx; tÞ ¼ hoðtÞ þx2

2RðtÞ þ rðx; tÞ � 2

pE0

ðxout

xin

pðf; tÞ ln f� x

f

� �2

df

(2)

Load equation

wðtÞ ¼ðxout

xin

pðx; tÞdx (3)

Energy equation

kf@2T

@z2� qf cf

@T

@tþ uf

@T

@x

� �þ s _c ¼ 0 (4)

These equations describe a transient problem as the contact force,rolling and sliding velocities and geometry vary through the mesh-ing cycle. Also, the surface roughness of the gear teeth is includedthrough the term rðx; tÞ in Eq. (2). The equations may be solvedby numerical methods with appropriate pressure and temperatureboundary conditions. Various versions of solution have beenobtained for the transient problem as presented in Larsson [13],Wang et al. [14], Li and Kahraman [9,15]. After the equations aresolved at a given point along the line of action, the friction forcein the tooth contact can be calculated by integrating the shearstresses over the contact region:

f ðtÞ ¼ b

ðxout

xin

sðx; z; tÞjz¼0dx (5)

Because of the complexity of the problem and inherited model-ing uncertainties, it is not possible to very accurately determinevarious contact and lubrication variables and thus a very accuratevalue for the small percentage meshing loss. Therefore, it may bewise to reduce the model to a simpler form approximating somevariables that do not substantially affect the magnitude and char-acteristics of the contact friction. First, the surface roughness maybe neglected without losing the result trend, largely reducing thenumerical difficulty to obtain converged solutions. Experimentaldata in Petry-Johnson et al. [7] showed that roughness of signifi-cant magnitude only causes about 10% increase in the meshingloss for a wide range of operating conditions (although Brittonet al. [1] showed significantly higher loss). Calculations by Chang[16] showed that surface roughness causes slight decrease in theEHL traction in the absence of asperity contacts, and calculationsby Johnson et al. [17] showed that asperities carry less than 10%of total load in an EHL junction with a unity k ratio. Thus, the the-ories suggest that surface roughness would only cause a modestincrease in traction under significant mix-lubrication conditionswhen the asperity contacts are well covered by low-frictionboundary films. Second, the problem may be solved in quasisteady state, avoiding the computationally intensive transient cal-culations. Li and Kahraman [9] showed that the transient effectsdo not substantially change the level of the EHL film thickness inthe gear contact. Dama and Chang [18] also showed that the filmthickness does not sensitively affect EHL traction. Third, the EHLpressure may be approximated by the Hertz distribution so thatthe pressure may be obtained in close form instead of throughcomplicated numerical iterations. Dama and Chang [18] showedthat the use of Hertz pressure and Hertz zone as the calculationdomain does not significantly affect the calculated traction undera wide range of operating conditions. Lastly, the film thicknessmay be taken as a parallel gap calculated by the central film-thickness formula of Dowson and Toyoda [19]. With these simpli-fications, Eqs. (1) to (3) are replaced by two straightforward alge-braic equations to calculate pressure distribution and filmthickness at each meshing position along the line of action:

pðxÞ ¼ ph

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðx=aÞ2

q(6)

and

ho ¼ 3:06CpCTRU0:69G0:56W�0:1 (7)Fig. 1 Schematic of the meshing of a pair of involute spurgears (taken from Johnson [11] with permission)

011504-2 / Vol. 135, JANUARY 2013 Transactions of the ASME


where CP is a compressibility reduction factor [19] and CT an inletshear-heating reduction factor [20], which are given by

Cp ¼ 1þ 0:58� 10�9ph

1þ 1:68� 10�9ph

� ��1

(8)

and

CT ¼ exp �0:75bgou2

4kf

� �(9)

Next, under quasi steady state, the energy equation reduces to

kf@2T

@z2� qf cf uf

@T

@xþ s _c ¼ 0 (10)

In fact, the transient term in Eq. (4) can be easily shown to be afew orders of magnitude smaller than the transport term for ahigh-speed, small-size contact. It is well known that the EHL trac-tion is sensitively related to the film temperature. Because thelubricant viscosity is exponentially related to temperature, it canbe very difficult to devise a close-form formula to calculate thermalEHL traction with consistent quality under various speed-and-load conditions, as shown in Dama and Chang [18]. Furthermore,recent research has revealed the development of thermallyinduced shear localization across the EHL film under certain con-ditions [21,22]. The shear localization can dramatically reduce theEHL traction and may also reverse the trend of traction behaviorwith temperature as shown in Willermet et al. [23], Webster et al.[24] and Chang et al. [21]. Therefore, Eq. (10) should be rigor-ously solved by a full numerical method to determine the tempera-ture field and to capture the possible shear localization in the EHLfilm. Equation (10) is a parabolic equation and requires tempera-ture boundary conditions at the Hertz inlet and along the twosurfaces to solve. The temperature at the Hertz inlet, �a, may betaken as the ambient temperature, To, of the supplied lubricant.For a high-speed gear system, the surface velocities, u1 and u2, ofthe tooth contact in the direction perpendicular to the line ofaction are very high throughout the meshing cycle. Therefore, thetwo surface temperatures may be described by the well-knownJaeger equation [25]:

TiðxÞ ¼ To þkfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pqicikiui

pðx

�a

@T

@z

� �i

d1ffiffiffiffiffiffiffiffiffiffiffix� 1p i ¼ 1; 2 (11)

Any well formulated non-Newtonian model may be used todescribe the lubricant rheological behavior such as the Eyringmodel proposed by Johnson and Tevaarwerk [26]:

_c ¼ so

gsinh

sso

� �(12)

The well-known Barus law may be used to describe the viscosity-pressure-temperature relation of the lubricant:

g ¼ goeape�bðT�ToÞ (13)

Define an effective viscosity, ge, such that

s ¼ ge _c (14)

Then, for a lubricant obeying Eq. (12), the effective viscosity isgiven by

ge ¼so

_csinh�1 g _c

so

� �(15)

The shear strain rate in the EHL film is related to the lubricant ve-locity by

_c ¼ @uf

@z(16)

Differentiating Eq. (14) with respect to z and rearranging withEq. (16) yields

ge

@2uf

@z2þ g0e

@uf

@z¼ s0 (17)

where g0e ¼ @ge=@z, s0 ¼ @s=@z and the velocity boundary condi-tions are uf ð0Þ ¼ u2 and uf ðhoÞ ¼ u1. The term s0 describes thevariation of the shear stress across the film. It is equal to dp=dx oris caused by the Poiseuille flow. Under high-load and high-speedconditions, the Poiseuille flow in the Hertz region is very smallcompared to the Couette flow and thus s0 may be neglected.Therefore, Eq. (17) may be reduced to

ge

@2uf

@z2þ g0e

@uf

@z¼ 0 (18)

Equations (10) and (18) govern the temperature and velocityfields of the lubricant in the EHL film. The two equations arecoupled to each other. The coupling may become very pro-nounced under high-load and high-sliding conditions such as ingear meshing, leading to the development of thermal shear local-ization across the EHL film [22,27]. A good solution of themshould ensure a quality calculation of the shear stresses and thusthe meshing friction that is reasonably accurate. More impor-tantly, it should yield the correct trends and relative magnitudes ofthe meshing losses with respect to the variations of key gear andlubricant parameters. The derivatives in Eqs. (10) and (18) areapproximated by various standard finite-difference formulas.Specifically,

@T

@xji;j �

Ti;j � Ti�1;j

Dx(19)

@2T

@z2ji;j �

Ti;jþ1 � 2Ti;j þ Ti;j�1

Dz2(20)

@uf

@zji;j �

ðuf Þi;jþ1 � ðuf Þi;j�1

2Dz(21)

@2uf

@z2ji;j �

ðuf Þi;jþ1 � 2ðuf Þi;j þ ðuf Þi;j�1

Dz2(22)

where i defines the finite-difference grids along x direction, and jdefines the grids along z direction. Equations (10) and (18) arethen converted to two systems of algebraic equations. Asuccessive-iteration scheme is used to solve them with proper ini-tial estimates of the velocity and temperature variables at the gridpoints. A simple choice of the initial estimates is the linear flowfield of u2 � ðu2 � u1Þz=ho and uniform temperature field of To.The solution is obtained by marching from Hertz inlet to outlet,one column of the finite-difference grids at a time starting withi ¼ 2. The velocity variables are calculated first. In the process, ge

and g0e in Eq. (18) are first calculated using the initial estimates ofuf and T (or their values at the previous iteration); ðg0eÞi;j� ½ðgeÞi;jþ1 � ðgeÞi;j�1�=ð2DzÞ and _ci;j � ½ðuf Þi;jþ1 � ðuf Þi;j�1�=ð2DzÞ are used in the calculations. A system of linear equations isthen obtained for the velocities at the ith grid column. It is a tri-diagonal matrix equation and can be easily solved simultaneously.The solution is used to update the velocity variables from theinitial estimates (or the values at the previous iteration); under-relaxation is used in the update to smooth the convergence. Subse-quently, the temperature variables at the grid column are calculated.The velocity variables just calculated are first used to calculate s _c

Journal of Tribology JANUARY 2013, Vol. 135 / 011504-3


in Eq. (10), leading to a system of linear equations for the temper-atures. The equations are also tri-diagonal and can be easilysolved. In the solution, the surface temperatures at i� 1 columnare used. This use does not result in appreciable error with smallDx grid spacing. It gets rid of unnecessary surface-temperatureiterations and enhances numerical stability. The solution of thesystem of equations is used to update the temperature variableswith under-relaxation. This two-step successive-iteration proce-dure for velocity and temperature variables continues until a con-verged solution is obtained. The surface temperatures are thencalculated at the current grid column using Eq. (11) in which½@T=@z�1;2 at the two surfaces are approximated by one-sided fi-nite differences and the integration is carried out using a simplesummation routine from the inlet to i� 1 column. Thereafter, thesolution process advances to the next grid column until it reachesHertz outlet. Upon completion, the friction in the gear contact iscalculated:

f ¼ b

ða

�a

sðxÞdx (23)

In addition, an average temperature rise in the lubricant film isalso calculated:

DT ¼ 1

2aho

ða

�a

ðho

0

Tðx; zÞdzdx� To (24)

It provides a single-variable measure on the level of temperaturein the contact similar to the Hertz pressure on the level ofpressure.

The pressure, film thickness, temperature and friction forcemay be calculated at any meshing position. Referring to Fig. 1, ameshing-position variable, s, may be defined along the line ofaction from left to right. It is negative before the meshing positionreaches the pitch point and positive thereafter. The meshing ini-tiates and terminates, respectively, at

si ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

a2 � r2b2

q� r2 sin w

� �(25)

and

st ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

a1 � r2b1

q� r1 sin w (26)

For a pair of spur gears, the contact ratio, mc, is usually betweenone and two. There are two meshing pairs when the meshing posi-tion is in the range of si � s � s1 or s2 � s � st. There is a singlemeshing pair when s1 < s < s2. It can be shown with gear funda-mentals that

s1 ¼si þ ðmc � 1Þst

mc(27)

and

s2 ¼ðmc � 1Þsi þ st

mc(28)

The radii of the two equivalent cylinders at a given meshing posi-tion, referring to Fig. 1, are given by

R1 ¼ r1 sin wþ s (29)

and

R2 ¼ r2 sin w� s (30)

Thus, the equivalent radius of the EHL contact is

RðsÞ ¼ R1R2

R1 þ R2

¼ ðr1 sin wþ sÞðr2 sin w� sÞc sin w

(31)

The surface velocities of the two cylinders are given by

u1ðsÞ ¼ x1R1 (32)

and

u2ðsÞ ¼ x2R2 (33)

Thus, the entraining velocity of the EHL contact is

uðsÞ ¼ u1 þ u2

2¼ x1r1 sin wþ 0:5ðx1 � x2Þs (34)

and the sliding velocity is

usðsÞ ¼ u1 � u2 ¼ ðx1 þ x2Þs (35)

In the equations, the angular velocities of the gear pair, x1 andx2, are taken to be positive valued with their directions defined inFig. 1. The tooth contact force generally varies as the meshingproceeds along the line of action. The variation is primarily due tothe switch between one meshing pair and two meshing pairs.Other secondary effects may be neglected, which should not resultin meshing-loss calculation error greater than the errors associatedwith other model uncertainties and imperfections. Therefore, thecontact force per unit gear face width is given by

wðsÞ ¼ kTp

br1 cos w(36)

where

k ¼0:5 si � s � s1 or s2 � s � st

1 s1 < s < s2

�(37)

With RðsÞ, wðsÞ, uðsÞ and usðsÞ determined, the pressure, filmthickness, temperature, and friction in the gear contact can now becalculated as the meshing progresses along the line of action. Theenergy loss in the contact, DEm, in a time period, tm, as one toothpair go through their meshing can also be calculated. Referring toFig. 1, the line of action may be wrapped on the base circle of thepinion and thus the meshing point progresses from s ¼ si to s ¼ st

with a constant velocity, which is given by

vm ¼ds

dt¼ x1rb1 (38)

Therefore,

DEm ¼ðtm

0

fusdt ¼ 1

x1rb1

ðst

si

fusds (39)

Because N1x1=2p numbers of tooth pairs go through the meshingin unit time, the meshing power loss is equal to

DP ¼ N1x1

2pDEm ¼

1

pm cos w

ðst

si

fusds (40)

The integration in Eq. (40) can be evaluated using a simple nu-merical summation with a sufficiently small Ds to progress themeshing position and calculate the corresponding friction forceand sliding velocity in the tooth contact.



Figure 2 shows the calculated results of key tooth-contact varia-bles through one meshing cycle of a sample problem with a pairof identical spur gears of standard geometry and steel materials.The problem is defined in Table 1, which lists the geometry andmaterial parameters and operating conditions of the gear systemthat fits in a standard FZG test rig. The calculated meshing loss is0.49% of the input power. The calculation took about five secondsto complete on a PC. One hundred Ds increments were used toprogress through the meshing. A 60� 80 finite-difference gridwas used to discretize the Hertz region along x and z directions,respectively, to numerically solve Eqs. (10) and (18). The conver-

gence was set with 0.001 relative-error tolerance. Calculationaccuracy was evaluated with grid refinements and tighter error tol-erances for a number of problems. In the next section, a series ofevaluations of the model is conducted with the measured meshinglosses reported in Petry-Johnson et al. [7] under a wide range ofconditions. Subsequently, analyses are carried out to study thetrends and relative magnitudes of percentage meshing losses withrespect to variations of a few key gear and lubricant parameters.

3 Results and Analysis

The model developed in this paper is first evaluated with a setof experimental data reported in Petry-Johnson et al. [7]. Petry-Johnson et al. conducted a systematic experimental investigationof spur-gear efficiency using a well maintained and well instru-mented FZG test rig. The experiments were conducted with a pairof identical involute gears. Two sets of gears of significantly dif-ferent modules were used; one set has 23 teeth and the other, 40teeth. A group of gears is ground finished with RMS roughnessaround 0.25 lm, and another group is chemically polished toreduce the roughness to 0.05 lm. Three different lubricants wereused in the study. A temperature-controlled oil circulation systemsupplied the lubricant to the gear meshes through directed jets.The experiments were carried out under high-load, high-speedconditions with a test matrix of three different loads and three dif-ferent speeds. They measured the power loss of the system in eachexperiment and extracted the meshing loss from it by subtractingthe loss measured under zero-load spin condition and the loss dueto bearing friction using a published formula. Table 2 lists thegeometrical parameters of the two gear sets. Table 3 shows the ninetest conditions defined by various combinations of the pinion rpmand torque. Figure 3, taken from Fig. 6 in their paper, shows thepercentage meshing-efficiency results determined from the meas-urements of the nine test conditions. Ground-finished gears wereused. The lubricant is a brand of synthetic 75W90 gear oil and wassupplied at 110 �C. The results show a clear effect of gear moduleon the meshing efficiency. The effects of load and speed are rela-tively small and the trend is not very clear and consistent.

Fig. 2 Key variables through a meshing cycle of the problem defined in Table 1

Table 1 Parameters for the nominal problem

Independent parameters

Gear center distance, c 91.5 mmPressure angle, w 20 degNumber of teeth, N1,2 29Tooth addendum height, A1,2 1.0 Pd

Face width, b 20 mmPinion torque, Tp 500 N-mPinion speed, x1 6000 rpmEquivalent Young’s modulus, E0 2.25� 1011 PaThermal conductivity, k1,2 47 W/(m �C)Density, q1,2 7800 kg/m3

Specific heat, c1,2 460 J/(kg �C)Lubricant ambient temperature, To 100 �CLubricant ambient viscosity, go 0.01 Pa-sPressure-viscosity coefficient, a 1.2� 10�8 Pa�1

Temperature-viscosity coefficient, b 0.04 �C�1

Lubricant thermal conductivity, kf 0.14 W/(m �C)Lubricant density, qf 800 kg/m3

Lubricant specific heat, cf 2290 J/(kg �C)

Calculated parameters

Gear module, m 3.155 mmContact ratio, mc 1.646Input power, P 314 KW



The above experimental gear system and operating conditionswere implemented into the model developed in this paper. Thegear geometry parameters such as the pitch radius and pressureangle can be calculated from the parameters listed in Table 2 byr ¼ c=2 and w ¼ cos�1 rb=r. While not mentioned in Petry-Johnson et al. [7], the gears are assumed to be made of steels.Then the material parameters listed in Table 1 are used. Based onTable 2 in Li and Kahraman [9], which describes the properties ofthe lubricant used in the above experiments, the lubricant viscosityat the supply temperature of 110 �C is taken to be go ¼ 0:009 Pa-s.The pressure-viscosity coefficient is set at a ¼ 9:5� 10�9Pa�1 andthe lubricant density at qf ¼ 790kg=m3. Parameters describingthe lubricant thermal properties are taken from Table 1, which arevery representative of EHL lubricants and are similarly used inWang et al. [14]. Figure 4 shows the meshing-efficiency resultspredicted by the model for the nine test conditions of Petry-Johnson et al. [7]. The results show a clear effect of gear moduleon the meshing efficiency. For the nine cases, the meshing effi-ciency averages at 99.786% for the 40-teeth gears and 99.644%for the 23-teeth gears, which are very close to their experimentalcounterparts of Fig. 3, being 99.784% and 99.643%, respectively.In addition, the differences in the efficiency among the nine casesare visibly smaller for the 40T gears than those for the 23T gears.They range within 0.055% for the former and 0.113% for the

latter, which are similar to and consistent with what observed inthe experiments, being 0.054% and 0.084%, respectively. Themodel predicts a modest decrease of the meshing efficiency asload increases for both sets of gears. It also predicts a modestincrease of the efficiency as speed increases. These trends are notclear in the experiments: the results exhibit an increase of the effi-ciency with the load for the 23T gears but a conflict trend for the40T ones, and no trend with the velocity in both sets of gears.This difference between the model predictions and the experi-ments may be due to the relatively small effects of load and speedon meshing efficiency. These small effects may be overshadowedand trends contaminated in the experiments by other small effectssuch as surface roughness effects, extraction errors of the meshinglosses from the overall losses, and measurement repeatability scat-ters. Despite all the uncertainties and imperfections in the modeland in the experiments, the differences between the predictionsand the corresponding measurements are still very comparable tothe measurement repeatability reported in Petry-Johnson [7]. Thebiggest such difference is 0.03% in Test 6 for the 40T gears and0.08% in Test 3 for the 23T ones; only two of the eighteen casessee the difference exceeds the reported 0.04% measurementrepeatability.

Next, the model is used to study, under a high-load and high-speed condition, the trends and relative magnitudes of the meshinglosses with respect to variations of a few key gear and lubricantparameters. The parameters studied are those which affect mesh-ing losses but have very little effects on other losses such aschurning, windage, bearing and seal losses. The parametersinclude gear module, pressure angle, tooth addendum height, gearthermal conductivity, and lubricant pressure-viscosity andtemperature-viscosity coefficients. In the study, these parametersare varied one at a time in a range of practical interest. The nomi-nal problem used for the study is the one with parameters definedin Table 1 and key variables along the meshing shown in Fig. 2.

The effect of gear module is studied first. Figure 5 shows thepercentage meshing losses with the numbers of gear teeth rangingfrom 19 to 39. For a given gear pitch diameter, the module isinversely proportional to the numbers of teeth. It is 4.816 mm inthis problem with 19T and decreases to 2.346 mm with 39T. Thecalculated meshing loss is about 0.64% of the input power withthe coarsest gears and down to 0.4% with the finest gears. Theresults of Fig. 5 are re-plotted in Fig. 6 to show the loss as a func-tion of the gear module. The loss is fairly linear with the module,varying with a slightly higher rate at the lower module range. Thevariation of the loss with the module is very significant with amagnitude relative to the mean loss of about 50%. Examinationsof the detail meshing results such as those illustrated in Fig. 2reveal that the increased level of sliding velocity is the main causeof the increased loss with module, which was also pointed out inXu et al. [6] and Petry-Johnson et al. [7]. As an example, the

Table 3 Operating conditions for the results of Figs. 3 and 4

Test number Pinion speed (rpm) Pinion toque (N-m)

1 6000 4132 6000 5463 6000 6844 8000 4135 8000 5466 8000 6847 10000 4138 10000 5469 10000 684

Fig. 3 Experimental results of percentage meshing efficiencyby Petry-Johnson [7]

Fig. 4 Calculated results corresponding to the experiments ofPetry-Johnson [7]

Table 2 Geometry parameters of gears used in Petry-Johnson[7]

Geometry parameter (mm) 23T-gears 40T-gears

Gear center distance 91.5 91.5Base-circle diameter 82.34 81.89Addendum-circle diameter 100.34 95.95Face width 19.5 26.7



sliding velocity at the initial point of meshing is 13.8 m/s in the19T gears in contrast to 7.4 m/s in the 39T ones. The friction coef-ficient is lower for the 19T than for the 39T gears with l ¼ 0:025for the former at s ¼ si and l ¼ 0:029 for the latter. However, thismodestly lower friction is not nearly sufficient to compensate forthe much higher sliding velocity to result in significantly highermeshing loss. The results reveal another benefit of smaller modulefor gear contact and lubrication. The Hertz pressure, film thick-ness and temperature rise are ph ¼ 1:25 GPa, ho ¼ 0:29lm andDT ¼ 79:2 oC in the fine gears at s ¼ si while these values areph ¼ 1:62 GPa, ho ¼ 0:23lm and DT ¼ 148:8 oC in the coarsegears. The 39T gears also enjoy a higher contact ratio ofmc ¼ 1:709 in comparison to mc ¼ 1:544 for the 19T gears andthus are expected to run slightly more smoothly. Therefore, itwould be highly desirable to use gears of smaller modules pro-vided that the bending stress in the gear tooth is below the maxi-mum allowable value.

The effect of gear pressure angle is studied next. Figure 7shows the meshing losses of the gears with the pressure anglesranging from 14 deg to 26 deg. The rate of increase of the loss isfairly steep as the pressure angle is reduced below the nominalvalue of 20 deg. The rate of reduction of the loss is somewhatmodest for the angle above 20 deg. The overall variation of themeshing losses in this range of the pressure angle is very pro-nounced with a magnitude above 0.35%, which is about 80% of

the loss with w ¼ 20 deg. Therefore, the pressure angle is a verysignificant design parameter on the meshing efficiency, even moresignificant than the module previously discussed. The significanceof the pressure angle on the meshing efficiency has also beennoted in earlier research by Hohn et al. [28] and Magalhaes et al.[8]. Low loss gears with w ¼ 40 deg were proposed [28]. Exami-nations of the meshing variables suggest that the increased slidingvelocity with a small pressure angle is once again the main con-tributing factor to the increased meshing loss. With w ¼ 26 deg,the sliding velocity is equal to 8.07 m/s at s ¼ si. It is increased to11.88 m/s with w ¼ 14 deg, an increase of about 47%. The frictionforce is also significantly higher with the lower pressure angle. Itincreases by about 21% when the pressure angle is reduced from26 deg to 14 deg. In addition to the meshing efficiency, the pres-sure angle may significantly affect tooth contact and lubrication.The calculation shows that the Hertz pressure, film thickness andtemperature rise are ph ¼ 1:10 GPa, ho ¼ 0:38lm andDT ¼ 75:1 oC at s ¼ si with w ¼ 26 deg. In sharp contrast, thesevariables are very severe with ph ¼ 2:60 GPa, ho ¼ 0:12lm andDT ¼ 205:5 oC when the pressure angle is 14 deg. The muchhigher Hertz pressure puts the tooth surface in significantly higherfatigue risks while the much reduced film thickness and muchincreased contact temperature, in conjunction with high sliding,could lead to lubrication failure and surface scuffing. Therefore, itwould be very undesirable, especially for high-performance trans-mission systems, to use gears with pressure angles below thestandard 20 deg. The main drawback of a large pressure angle isthe increased tooth-contact force, resulting in increased bearingload and bearing friction which would eventually negate the bene-fit of increasing w. A higher pressure angle also leads to a smallercontact ratio. For the problem studied, mc ¼ 1:966 withw ¼ 14 deg. It is reduced to mc ¼ 1:441 with w ¼ 26 deg.

In gear design, tooth tip relief or tip modification is often usedto prevent premature engagement and stress concentration on thetip of the tooth. The tip modification may also reduce the meshingloss as suggested by Diab et al. [5], Xu et al. [6] and Velex andVille [29]. A main argument of the reduction is that the tip modifi-cation shortens the length of action, st � si, removing the two endportions that involve high sliding in the tooth contact. The effectof the tip modification on the meshing loss is also studied in thispaper. For this purpose, the modification is simply modeled byreducing tooth addendum height or using gears with shortenedteeth. Figure 8 shows the calculated meshing losses with the toothaddendum varying from the full stub tooth of A1;2 ¼ 0:8Pd to thestandard tooth of A1;2 ¼ 1:0Pd. The meshing loss is fairly linearwith respect to the tooth addendum height. The reduction of theloss is very modest as the tooth is shortened with a relative magni-tude of about 14% as the addendum is reduced from A1;2 ¼ 1:0Pd

Fig. 7 Effects of gear pressure angle on the meshing lossFig. 5 Effects of gear module: meshing loss versus numbersof gear teeth

Fig. 6 Effects of gear module: meshing loss versus module



to A1;2 ¼ 0:8Pd . Figure 9 shows the tooth contact and lubricationvariables through the meshing with the full stub-tooth gears. Com-parison of the results with their full-addendum counterparts ofFig. 2 reveals two main factors that affect the change of the mesh-ing loss as the addendum is varied. The first factor is that themeshing of the stub-tooth gears initiates at a significantly laterposition along the line of action (and terminates significantlysooner) than does the meshing of the full-addendum gears. As aresult, the sliding velocity at the extreme position is significantlylower for the former than for the latter, being 7.93 m/s and 9.63m/s, respectively. The removal of the meshing segment with highsliding significantly contributes to the reduction of the meshingloss. The second factor is that the central meshing segmentwith one meshing pair (i.e., s ¼ s1 � s2) is significantly widerwith the stub-tooth gears than with the full-addendum gears.For the former, s1 ¼ �0:29 mm at which the sliding velocityis jusj ¼ 3:65m=s. For the latter, s1 ¼ �0:15 mm and

jusj ¼ 1:93m=s. The significantly higher sliding velocity ats ¼ s1;2 multiplied by nearly twice as large a friction force signifi-cantly contributes to the increase of meshing loss for the stub-tooth gears. The two factors act to negate the effect of each otherto a significant degree resulting in only a modest reduction of themeshing loss when the addendum is shortened from the full lengthto the stub length. Therefore, tooth tip modification does not seemto significantly increase gear meshing efficiency, far below whatcould be brought about by module and pressure angle considera-tions. In addition, the reduced length of action may significantlyreduce the contact ratio to potentially increase meshing noise andcontact fatigue.

Another gear property which may significantly affect the mesh-ing efficiency is its thermal conductivity. A number of studiesshowed or suggested that, under high-load and high-sliding condi-tions, large thermally induced shear localization may be devel-oped across the EHL film to significantly reduce the shear stressesthus EHL traction. Furthermore, Chang [27] showed that if one ofthe contact surfaces is insulated, the localization will be developedat the insulated surface due to high local temperature. As a result,the localization can be significantly more pronounced than theproblem with two conductive surfaces and, in conjunction withthe higher overall temperature, the shear stress in the lubricant canbe much lower. The theory suggests a possibility to significantlyreduce the meshing losses in gear transmission by reducing thethermal conductivity of the tooth surface of one of the two gears.Based on some results from the theory, Chang [27] speculatedthat a reduction of as much as 50% may be achieved. This specu-lation is analyzed using the model developed in this paper. Figure10 shows the calculated meshing losses as the thermal conductiv-ity of one of the gears, k1, is varied from its nominal value of 47W/(m �C) to zero. The reduction in the meshing loss is very mildeven after k1 is reduced to less than 10 W/(m �C). The rate ofreduction accelerates as k1 is further reduced, but the total reduc-tion is still not a very significant amount of 12.5% atk1 ¼ 3:0W= m oCð Þ. Thereafter, the loss takes a steep dive to givea total reduction of 45.5% when k1 is reduced to zero. Such a lossresult is likely associated with the behavior of the cross-film ther-mal shear localization. Figure 11 shows the shear strain rate across

Fig. 8 Effects of tooth addendum height on the meshing loss

Fig. 9 Key variables through a meshing cycle with the stub gears of A12 ¼ 0:8Pd



the EHL film at the Hertz center x ¼ 0ð Þ in the tooth contact at theinitial moment of meshing. At this time and location, the shearlocalization, if developed, would be most severe. Results obtainedwith four values of k1 are presented. The strain rates are localizedin a relatively narrow film layer. The localization is around thecenter of the film with two equally conductive surfaces ofk1 ¼ k2 ¼ 47W= m oCð Þ, which is the nominal value representativeof steel materials. The average temperature rise (Eq. (24)) in thefilm at this condition is DT ¼ 99:9 oC. As k1 is reduced to 3.0 W/(m �C), the localization is significantly shifted toward the muchless thermally conductive surface due to higher temperature inthat region. The degree of localization is slightly lessened with apeak strain rate 12% lower than the peak of the previous case. Thetemperature rise in this case is DT ¼ 122:0 oC. The higher DT isprimarily responsible for the 12.5% reduction in the meshing loss.As k1 is further reduced to 1.0 W/(m �C), the localization is fur-ther shifted toward the hotter surface and is more localized with apeak strain rate slightly higher than that in the nominal case. Thetemperature rise is DT ¼ 132:0 oC, which is likely the main causefor the additional 8.5% reduction in the meshing loss. With a per-fectly insulated surface of k1 ¼ 0, the temperature rise is highestright at the surface reaching over 300 �C as shown in Fig. 12; thetemperature rise across the film for the nominal problem is also

shown for comparison. This at-surface highest temperaturereduces the local viscosity dramatically, making the shear of thelubricant much easier right at the surface thereby creating muchmore pronounced shear localization as shown in Fig. 11. As aresult, the shear stresses and thus the friction and meshing loss aremuch reduced. Nevertheless, the use of an insulated gear to muchreduce the meshing loss speculated in Chang [27] is seen to bemore or less a theoretical fantasy as there are hardly any materialsor surface coatings that can offer such perfect insulation. A smallamount of heat conduction through the surface would generatesufficient thermal leakage to diminish almost all the benefit. Inci-dentally, the problem is also studied with simultaneous reductionof both k1 and k2. No significant reduction in the meshing loss isobtained until the thermal conductivity is reduced to a very smallvalue.

The last part of the analysis is on the effects of lubricant proper-ties on the meshing losses. The study focuses on the lubricantpressure-viscosity and temperature-viscosity coefficients as thesetwo parameters, for a given ambient temperature, have littleeffects on the losses outside of tooth meshing. Figure 13 showsthe meshing losses for a reasonable range of lubricant pressure-viscosity coefficient around its nominal value of a ¼ 1:2�10�8Pa�1. The results suggest that the viscosity sensitivity topressure at the EHL pressure is an important property on gear

Fig. 10 Effects of gear thermal conductivity on the meshingloss

Fig. 11 Effects of k1 on the cross-film thermal shear localiza-tion at x 5 0 and s 5 si

Fig. 12 Cross-film temperature distribution at x 5 0 and s 5 si

Fig. 13 Effects of lubricant pressure-viscosity sensitivity onthe meshing loss



meshing efficiency. The meshing loss increases nearly linearlywith a. The magnitude of variation is about 65% relative to theloss at the mean level, which is as significant as the relative mag-nitudes associated gear module and pressure angle studied earlier.Therefore, it is desirable to select a gear lubricant with a relativelysmall pressure-viscosity coefficient to increase meshing effi-ciency. The main drawback of a reduced a is the decrease in theEHL film thickness. For the problem studied, the film thickness isho ¼ 0:325 lm with a ¼ 1:6� 10�8Pa�1 and is ho ¼ 0:220 lmwith a ¼ 0:8� 10�8Pa�1. The decrease is somewhat modest con-sidering the large percentage reduction in the meshing loss andmay be compensated with a better surface finish or surface run-in.Figure 14 shows the meshing loss as a function of lubricantviscosity-temperature coefficient. The study includes a range of bup to 0.09 �C�1 because research such as Paluch et al. [30] andKumar et al. [31] has shown that b can increase significantly atelevated pressures from its ambient value. The results obtainedsuggest that the effect of the lubricant viscosity-temperature sensi-tivity at EHL pressures is fairly significant on the meshing effi-ciency. The reduction in the meshing loss with b ¼ 0:09 oC�1 isclose to 30% of the loss at b ¼ 0:04 oC�1, which is often a repre-sentative value of the viscosity-temperature coefficient at ambientpressure. Therefore, it would be desirable to select a gear lubricantwhich exhibits increased viscosity-temperature sensitivity at highpressures. It would increase gear meshing efficiency with littleside effects in contrast to other gear and lubricant parametersstudied.

4 Conclusion

A first-principle based mathematical model is developed in thispaper to analyze the meshing losses in spur-gear systems operat-ing in high-load and high-speed conditions. The model is funda-mentally simple and computationally robust. It is shown to yieldsufficiently accurate results to produce meaningful trends and rel-ative magnitudes of the meshing losses with respect to the varia-tions of key gear and lubricant parameters. The model is used tostudy the effects of various gear and lubricant parameters on themeshing losses. The results and analysis lead to the followingconclusions:

(1) Gears with reduced module can significantly reduce themeshing loss primarily by reducing the sliding velocity inthe tooth contact. A smaller module also yields a bettertooth contact and lubrication condition and a larger gearcontact ratio. Therefore, small modules should be consid-ered in system design and gear selection provided that the

gears have sufficient tooth bending strength under maxi-mum loading conditions.

(2) Gears with increased pressure angle can significantlyreduce the meshing loss by means of both reduced slidingvelocity and friction force in the tooth contact. A largerpressure angle also yields better tooth contact and lubrica-tion. A pressure angle significantly below the standard20 deg can produce very unfavorable contact and lubrica-tion conditions with high pressure and temperature and lowfilm thickness, putting the gears in high risk of contact fa-tigue and scuffing especially in high load and high speedtransmission. Therefore, a high pressure angle is desirableprovided it does not cause excessive bearing loading andyield too low a contact ratio.

(3) Gears with shortened tooth or gears with tooth tip modifica-tion do no result in significant reduction in the meshingloss. While it helps remove a high-sliding portion of thelength of action to reduce the meshing loss, a shortenedtooth increases the single-meshing portion of the length ofaction with high sliding and high friction to increase theloss. The two effects neutralize each other to a significantdegree.

(4) The effect of gear thermal conductivity on the meshing effi-ciency is insignificant. While the mesh of one conductivetooth with one insulated tooth is shown to reduce the mesh-ing loss by about 50% from its both-conductive counter-part, the reduction is largely diminished with a very smallamount of heat conduction through the surface.

(5) Lubricant pressure-viscosity coefficient can have a pro-found effect on the meshing efficiency. More than 50%reduction may be achieved in the meshing loss from a con-ventional level by using a lubricant with a much reduced a.The decrease in the EHL film thickness with the use of asmall a is somewhat modest and may be compensated withimproved surface finish and/or surface run-in of gear teeth.

(6) Lubricant temperature-viscosity coefficient can also have asignificant effect on the meshing efficiency. It would bebeneficial to select a gear lubricant which exhibits a signifi-cant increase of viscosity-temperature sensitivity underhigh pressures. Such a lubricant property may help reducethe meshing loss by about 30% from a nominal level.

The results presented in this paper are obtained with variationsof gear and lubricant parameters from those of a nominal problemdefined in Table 1. The authors believe that the trends and relativemagnitudes of the meshing losses with respect to the parametervariations would be a meaningful representation for other prob-lems in high-load and high-speed operations. This belief is basedon the authors’ fundamental understanding of the problem and theresults obtained with other systems under other conditions asexemplified in Fig. 4. The authors also believe that the trends andrelative magnitudes of the meshing losses would be meaningfulfor helical gears. This belief is based on the fact that the meshingprocess and the contact and lubrication conditions in helical gearsare fundamental similar to those of their spur-gear counterpartsaside from geometric and kinematic complications.

Notations

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8wR=ðpE0Þ

p, Hertz half width

A1;2 ¼ pinion, gear tooth addendum heightb ¼ gear face widthc ¼ gear center distance

cf ¼ lubricant specific heatc1;2 ¼ pinion, gear specific heatE1;2 ¼ pinion, gear Young’s modulus

E0 ¼ 2½ð1� �21Þ=E1 þ ð1� �2

2Þ=E2��1, equivalent Young’s

modulusf ¼ friction in the tooth contact

G ¼ aE0, EHL material parameter

Fig. 14 Effects of lubricant temperature-viscosity sensitivityon the meshing loss



h ¼ lubricant film thicknessho ¼ central film thicknesskf ¼ lubricant thermal conductivity

k1,2 ¼ pinion, gear thermal conductivitym ¼ 1/Pd, gear module

mc ¼ ½ðr2a1 � r2

b1Þ1=2 þ ðr2

a2 � r2b2Þ

1=2 � c sin w�=ð2pm cos wÞ,gear contact ratio

N1,2 ¼ pinion, gear numbers of teethp ¼ lubricant pressure

ph ¼ 2w/(pa), Hertz peak pressureP ¼ Tpx1, pinion input power

DP ¼ meshing power lossDP% ¼ 100(DP/P), percentage meshing loss

Pd ¼ N1/(2r1), gear diametral pitchr1,2 ¼ pinion, gear pitch-circle radius

ra1,2 ¼ pinion, gear addendum-circle radiusrb1,2 ¼ r1,2, cos w, pinion, gear base-circle radiusR1,2 ¼ pinion, gear tooth radius of curvature at contact

R ¼ R1R2/(R1þR2), contact equivalent radiuss ¼ meshing position along the line of action

s1,2 ¼ meshing positions across which the numbers of meshingpairs change

si ¼ initial meshing positionst ¼ ending meshing positiont ¼ time

T ¼ lubricant temperatureT1,2 ¼ pinion, gear surface temperature in the tooth contact

To ¼ lubricant ambient temperatureDT ¼ average temperature rise in the tooth contact [Eq. (24)]Tp ¼ pinion input torqueu ¼ (u1þ u2)/2, entraining velocity in the tooth contact

u1,2 ¼ pinion, gear surface velocity in the tooth contactuf ¼ lubricant velocityus ¼ u1 – u2, sliding velocity in the tooth contactU ¼ gou/(E0R), EHL speed parameterw ¼ contact force per unit width of gear flankW ¼ w/(E0R), EHL load parameterx ¼ coordinate along tooth contact (Fig. 1)

Dx ¼ grid spacing of finite difference along x directionz ¼ coordinate along film thickness

Dz ¼ grid spacing of finite difference along z directiona ¼ lubricant pressure-viscosity coefficientb ¼ lubricant temperature-viscosity coefficient_c ¼ shear strain rateg ¼ lubricant viscosity

ge ¼ effective viscositygo ¼ ambient viscosity�1,2 ¼ pinion, gear Poisson’s ratioq1,2 ¼ pinion, gear density

qf ¼ lubricant densitys ¼ shear stress

so ¼ Erying stress, so � 5.0 MPax1,2 ¼ pinion, gear angular velocity

w ¼ gear pressure angle

References[1] Britton, R. D., Elcoate, C. D., Alanou, M. P., Evans, H. P., and Snidle, R. W.,

2000, “Effect of Surface Finish on Gear Tooth Friction,” ASME J. Tribol., 122,pp. 354–360.

[2] Lehtovaara, A., 2002, “Calculation of Sliding Power Loss in Spur Gear Con-tacts,” Tribotest, 9, pp. 23–34.

[3] Handschuh, R., and Kilmain, C. J., 2003, “Preliminary Comparison of Experi-mental and Analytical Efficiency Results of High-Speed Helical Gear Trains,”Paper No. NASA/TM212371.

[4] Martins, R., Seabra, J., Brito, A., Seyfert, C., Luther, R., and Igartua, A., 2006,“Friction Coefficient in FZG Gears Lubricated With Industrial Gear Oils: Bio-degradable Ester vs. Mineral Oil,” Tribol. Int., 39, pp. 512–521.

[5] Diab, Y., Ville, F., and Velex, P., 2006, “Prediction of Power Losses due toTooth Friction in Gears,” Tribol. Trans., 49, pp. 260–270.

[6] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, “Predictionof Mechanical Efficiency of Parallel-Axis Gear Pairs,” ASME J. Tribol., 129,pp. 58–68.

[7] Petry-Johnson, T. T., Kahraman, A., Anderson, N. E., and Chase, D. R., 2008,“An Experimental Investigation of Spur Gear Efficiency,” ASME J. Tribol.,130, p. 062601.

[8] Magalhaes, L., Martins, R., Locateli, C., and Seabra, J., 2010, “Influence ofTooth Profile and Oil Formulation on Gear Power Loss,” Tribol. Int., 43, pp.1861–1871.

[9] Li., S., and Kahraman, A., 2010, “Prediction of Spur Gear Mechanical PowerLosses Using a Transient Elastohydrodynamic Lubrication Model,” Tribol.Trans., 53, pp. 554–563.

[10] Kuria, J., and Kihiu, J., 2011, “Prediction of Overall Efficiency in MultistageGear Trains,” Int. J. Aerospace and Mech. Eng., 5:3, pp. 171–177.

[11] Johnson, K. L., 1985, Contact Mechanics, Cambridge University, Cambridge,England.

[12] Hamrock, B. J., Schmid, S. R., and Jacobson, B. O., 2004, Fundamentals ofFluid Film Lubrication, Marcel Dekker, New York.

[13] Larsson, R., 1997, “Transient Non-Newtonian Elastohydrodynamic LubricationAnalysis of an Involute Gear,” Wear, 207 pp. 67–73.

[14] Wang, Y., Li, H., Tong, J., and Yang, P., 2004, “Transient Thermoelastohydro-dynamic Lubrication Analysis of an Involute Spur Gear,” Tribol. Int., 37, pp.773–782.

[15] Li, S., and Kahraman, A., 2010, “A Transient Mixed Elastohydrodynamic Lubrica-tion Model for Spur Gear Pairs,” ASME J. Tribol., 132, p. 011501-1–011501-9.

[16] Chang, L., 1992, “Traction in Thermal Elastohydrodynamic Lubrication ofRough Surfaces,” ASME J. Tribol., 114, pp. 186–191.

[17] Johnson, K. L., Greenwood, J. A., and Poon, S. Y., 1972, “A Simple Theory ofAsperity Contact in Elastohydrodynamic Lubrication,” Wear, 19, pp. 91–108.

[18] Dama, R., and Chang, L., 1997, “An Efficient and Accurate Calculation ofTraction in Elastohydrodynamic Contacts,” Wear, 206, pp. 113–121.

[19] Dowson, D., and Toyoda, S., 1977, “A Central Film Thickness Formula forElastohydrodynamic Line Contacts,” Proc. 5th Leads/Lyon Symp., pp. 60–67.

[20] Greenwood, J. A., and Kauzlarich, J. J., 1973 “Inlet Shear Heating in Elastohy-drodynamic Lubrication (discussion by K. L. Johnson),” ASME J. Lubr. Tech-nol., 95, pp. 417–425.

[21] Chang, L., Qu, S., Webster, M. N., and Jackson, A., 2006, “On the Mechanismsof the Reduction in EHL Traction in Low Temperature,” Tribol. Trans., 49, pp.182–191.

[22] Chang, L., 2006, “A Parametric Analysis of Thermal Shear Localization inElastohydrodynamic Lubrication Films,” ASME J. Tribol., 128, pp. 79–84.

[23] Willermet, P. A., McWatt, D. G., and Wedeven, L. D., 1999, “Traction Behav-ior Under Extreme Conditions,” SAE International Fall Fuels and LubricantsMeeting and Exposition, Toronto, Ontario, Canada, Oct. 25–28, SAE Paper No.1999-01-3612.

[24] Webster, M. N., Lee, G. H., and Chang, L., 2006, “Effect of EHL Conditionson the Behavior of Traction Fluids,” Tribol. Trans., 49, pp. 439–448.

[25] Jaeger, J. C., 1942, “Moving Surfaces of Heat and the Temperature at SlidingContacts,” Proc. R. Soc. N.S.W., 56, pp. 203–224.

[26] Johnson, K. L., and Tevaarwerk, J. L., 1977, “Shear Behavior of EHD OilFilms,” Proc. R. Soc. London, Ser. A, 356, pp. 215–236.

[27] Chang, L., 2009, “Effects of Thermally Induced Inhomogeneous Shear and Sur-face Thermal Boundary Conditions on the Shear Stress in Sliding Elastohydro-dynamic Contacts,” Lubr. Sci., 21, pp. 227–240.

[28] Hohn, B., Michaelis, K., and Wimmer, A., 2007, “Low Loss Gears,” GearTechnol., 24, pp. 28–35.

[29] Velex, P., and Ville, F., 2009, “An Analytical Approach to Tooth FrictionLosses in Spur and Helical Gears – Influence of Profile Modification,” ASME J.Mech. Des., 131, p. 101008-1–101008-10.

[30] Paluch, M., Denzik, Z., and Rzoska, S. J., 1999, “Scaling of High-Pressure Vis-cosity Data in Low-Molecular-Weight Glass-Forming Liquids,” Phys. Rev. B,60, pp. 2929–2982.

[31] Kumar, P., Khonsari, M. M., and Bair, S., 2009, “Anharmonic Variations inElastohydrodynamic Film Thickness Resulting from Harmonically VaryingEntrainment Velocity,” IMechE J. Eng. Tribol., 224, pp. 239–247.



http://dx.doi.org/10.1115/1.555367

http://dx.doi.org/10.1002/tt.3020090104

http://dx.doi.org/10.1016/j.triboint.2005.03.021

http://dx.doi.org/10.1080/05698190600614874


http://dx.doi.org/10.1080/10402000903502279

http://dx.doi.org/10.1080/10402000903502279

http://dx.doi.org/10.1016/S0043-1648(96)07484-4


http://dx.doi.org/10.1115/1.4000270

http://dx.doi.org/10.1115/1.2920859

http://dx.doi.org/10.1016/0043-1648(72)90445-0

http://dx.doi.org/10.1016/S0043-1648(96)07309-7

http://dx.doi.org/10.1115/1.3451844

http://dx.doi.org/10.1115/1.3451844

http://dx.doi.org/10.1080/05698190500544619

http://dx.doi.org/10.1115/1.2125968

http://dx.doi.org/10.1080/10402000600815016

http://dx.doi.org/10.1098/rspa.1977.0129

http://dx.doi.org/10.1002/ls.84

http://dx.doi.org/10.1115/1.3179156

http://dx.doi.org/10.1115/1.3179156

http://dx.doi.org/10.1103/PhysRevB.60.2979

http://dx.doi.org/10.1243/13506501JET669

Modeling and Analysis of the Meshing Losses of Involute Spur ...

Documents

Transcript of Modeling and Analysis of the Meshing Losses of Involute Spur ...