Implementation of Hertz theory and validation of anite element
model forstress analysis of gear drives with localized bearing
contactIgnacio Gonzalez-Pereza,, Jose L. Iserteb, Alfonso
FuentesaaDepartment of Mechanical Engineering, Polytechnic
University of Cartagena (UPCT), SpainbDepartment of Mechanical
Engineering and Construction, University Jaume I, Castellon,
Spainarti cle i nfo abstractArticle history:Received 18 November
2009Received in revised form 20 October 2010Accepted 21 January
2011Available online 22 February 2011An analytical approach for
stress analysis of gear drives with localized bearing contact based
ontheHertztheoryis proposed. Theproposedapproachprovides
acompleteandeffectivesolution of the contact problem but
satisfaction of the hypotheses for application of the Hertztheory
is its main drawback. On the other hand, a finite element model has
been developed andvalidatedinterms of thecontact area,
maximumcontact pressure, pressuredistribution,maximumTrescastress,
andTrescastressdistributionunderneaththecontactingsurfaces.Validation
of the finite element model is provided for those cases wherein the
Hertz theory canbe applied. The obtained results confirm the
applicability of the proposed approach for geardrives with
localized bearing contact wherein edge contact is avoided by
surface modificationsand whole crowning of tooth surfaces is
provided. 2011 Elsevier Ltd. All rights reserved.Keywords:Gear
stress analysisTCAHertz theoryFinite element method1.
IntroductionTooth contact analysis (TCA) and stress analysis are
important tools in the design of gear drives. Application of both
techniquesallows us to determine, among other information, the
following:(i) The contact pattern, that involves determination of
the size, shape, and location of the successive instantaneous
contactareas on the pinion and gear tooth surfaces along a cycle of
meshing.(ii)The contact stresses, that involve determination of the
contact pressure distribution on the gear tooth surfaces, the
stressdistribution underneath the surfaces, and the maximum contact
pressure or the maximum Mises or Tresca effective stress.(iii)The
surface deformation, useful for calculation of transmission errors
under load.Different approaches may be applied for stress analysis
of gear drives. One of these approaches is based on the application
ofthenite element method. Many works based on thenite element
method have investigated the contact area formation fordifferent
surface topologies [13] in order to analyze their behavior in terms
of maximum effective stress or maximum contactpressure. Other works
based on thenite element method are directed to the prediction of
transmission errors [4]. Application
oftheniteelementmethodfortheanalysisof awholecycleof
meshingiscomputationallyexpensiveandtime-consuming,although
automatic generation of thenite element models [5] may reduce the
preprocessing time.Some other approaches are basedonthe
applicationof inuence coefcient methods. These methods
establishthedetermination of the total deformation at each point of
the contact area as the sumof deformations caused by unit loads
located ateach of those points. Some works based on these methods
are directed to the prediction of transmission errors [6],
determinationof pressure distribution [7], or determination of the
contact pattern [8]. Although these methods are computationally
effective, theMechanism and Machine Theory 46 (2011) 765783
Corresponding author.E-mail address: [email protected] (I.
Gonzalez-Perez).0094-114X/$ see front matter 2011 Elsevier Ltd. All
rights reserved.doi:10.1016/j.mechmachtheory.2011.01.014Contents
lists available at ScienceDirectMechanism and Machine Theoryj our
nal homepage: www. el sevi er. com/ l ocat e/ mechmtcontact problem
is not solved completely since stresses underneath the surface are
not calculated. Validation of these methods iscarried out by
comparison of their results with those provided by thenite element
method in terms of transmission errors,pressure distribution, or
contact pattern, respectively.Finally, there are approaches based
on the application of analytical methods, wherein the theory of
elasticity is directly
appliedundercertainhypothesesandboundaryconditions.
ThisisthecaseofHertz'smethod[9],
thatconsidersasemi-ellipsoidaldistribution of pressure on a
half-space with plain strain state. Such hypotheses work properly
with non-conforming surfaces andsmall contact areas compared with
the relative radii of curvature of contacting surfaces [10]. The
advantages of analytical methodsare that the contact problem is
solved completely and their algorithms are computationally
effective. Their disadvantage is thelimitation imposed by the
considered hypotheses and boundary conditions. Sheveleva at al.
[11] have applied Hertz's method fordetermination of maximum
contact pressure at spiral bevel gears, validating such a method
thorough an inuence coefcientmethod.The main goals of this paper
are as follows:(i) Implementation of an approach based on the Hertz
theory for analytical determination of the area of contact,
pressuredistribution, stress distribution underneath the tooth
surfaces, and their elastic deformation, for gear drives with
localizedbearing contact wherein edge contact is avoided by whole
crowning of tooth surfaces.(a) (b)Fig. 1. A spur pinion tooth
surface in case of: (a) partial crowning (areas 2 and 8 are
provided with prole crowning; areas 4 and 6 are provided with
longitudinalcrowning; areas 1, 3, 7, and 9, are provided with
double crowning; area 5 is not modied), and (b) whole crowning
(areas 1, 2, 3, and 4, are provided with proleand longitudinal
crowning).(a) (b)Fig. 2. Results of simulation of meshing and
contact of a gear drive wherein the pinion tooth surface are
provided with whole crowning: (a) path of contact, and(b) function
of transmission errors.766 I. Gonzalez-Perez et al. / Mechanism and
Machine Theory 46 (2011) 765783(ii)Validationof a niteelementmodel
forgeardriveswithlocalizedbearingcontactintermsof
thecontactpressuredistribution, maximum contact pressure, maximum
Tresca stress, stress distribution underneath the surface, size of
thecontact area, and elastic deformation.(iii)Validation of
application of the Hertz theory for stress analysis of gear drives
with localized bearing contact. The validatednite element model
will allowthe approach based on the Hertz theory to be assessed,
especially when the areas of contactare larger.(iv)Application of
the validatednite element model for gear drives with localized
bearing contact in cases wherein the Hertztheory does not work
properly as in modied involute gear drives provided with partial
crowning.2. MethodologyThe following methodology will be considered
in this research
work:1)Ageardriveiscomputationallydesignedbyapplicationofthetheoryofgearing[5].
Forthepurposeofsimplicityoftheproposedapproach,
aspurgeardrivewithlocalizedbearingcontactisconsidered. Fig.
1showstwoversionsofsurfaceFig. 3. Coordinate system Sp dened at
contact point P.(a) (b) (c)Fig. 4. For determination of surface r:
(a) tangent plane , (b) surfaces 1 and 2, and (c) surface r.767 I.
Gonzalez-Perez et al. / Mechanism and Machine Theory 46 (2011)
765783topologies. Fig. 1(a) and (b) shows a spur pinion with
partial crowning and whole crowning, respectively, that allows
thebearing contact to be localized when errors of alignment occur.
In the case shown in Fig. 1(a), line contact exists in zones 258
when there are no errors of alignment but when they occur, the
contact is localized. In the case shown in Fig. 1(b),
theoreticalpoint contact exists when misalignments are presented or
not. The case represented in Fig. 1(b) is similar to those gears
wherewhole crowning is provided, as in spiral or hypoid bevel
gears.Fig. 1(a) shows some boundary lines between nine areas. The
boundary line between areas 123 and 456 is controlled byparameter
uot and the amount of prole crowning is established by a parabola
coefcient apt. On the other hand, the boundaryline between areas
456 and 789 is controlled by parameter uob and the amount of prole
crowning is controlled by aparabola coefcient apb. For longitudinal
crowning, parameters lof and lob control the location of boundary
lines between areas369 and areas 258, and between areas 147 and
258, respectively. The amount of longitudinal crowning is
establishedbyparabolacoefcientsalfandalb, respectively.
Moredetailsabouttheseparameterscanbefoundin[1,12].
Thesurfacetopology shown in Fig. 1(b) is a particular case of the
one shown in Fig. 1(a) wherein parameters uot=uob and
lof=lob.2)Simulationof meshingandcontact of pinionandgear
toothsurfacesisperformedassumingthat point contact exists.The
algorithm of tooth contact analysis [5] is applied for
determination of contact points and the function of
transmissionerrors. Fig. 2 shows the results of application of TCA
to a spur gear drive with a pinion with whole crowning as the one
shown inFig. 1(b).3)An algorithm for application of the Hertz
theory in gear drives with localized bearing contact has been
developed and appliedfor stress analysis of a spur gear drive with
localized bearing contact wherein the pinion tooth surface is
provided with wholecrowning (see Fig. 1(b)). Two cases are
considered: (i) contact at a single point when just one pair of
teeth is in contact, and(ii) contact at two points corresponding to
the contact points of two consecutive pairs of contacting
teeth.4)Finite element models of contacting teeth are automatically
developed for each case of design. A sensitive-contact region
isdened around the contact point in order to control the renement
of the mesh in this zone.5)Different cases of design are considered
for computation of the Hertz theory andnite element analysis.
Results from bothmethods are compared for the case of a single
point of contact and the case of two points of contact. Different
amounts of wholecrowning are considered.Fig. 5. Illustration of
principal directions corresponding to principal curvatures on plane
.(a) (b)Fig. 6. Dimensions of contact ellipse and area of
interference in: (a) pinion and gear tooth surfaces, and (b)
reference surface r.768 I. Gonzalez-Perez et al. / Mechanism and
Machine Theory 46 (2011) 7657836)Finite element analysis is carried
out for a modied spur gear drive wherein pinion tooth surface is
provided with partialcrowning and the Hertz theory is not
applicable.3. Proposed algorithm for application of the Hertz
theoryThe Hertz theory is based on the following hypothesis
[10]:(1)Surfaces are continuous and non-conforming.(2)Contact area
is elliptical.(3)Each body is an elastic half-space loaded over a
small elliptical region of its plane surface.(4)Surfaces are
assumed to be frictionless.For these hypothesis to be satised,
contact area has to be small compared with the dimensions of each
body and the relativeradii of curvature of the surfaces. Such
hypotheses can be satised for gear drives when the bearing contact
is localized inside toothsurfaces and whole crowning is considered.
The hypothesis of frictionless surfaces implies that only normal
pressure is transmittedbetween contacting surfaces.Two approaches
for application of the Hertz theory are considered for a whole
crowned spur gear drive with localized bearingcontact: (i) contact
at a single point, when the load is shared by just one pair of
contacting teeth, and (ii) contact at two points,when the load is
shared by two consecutive pairs of contacting teeth.(a) (b)Fig. 7.
For determination of main and secondary contact points: (a) contact
at the main contact point, and (b) function of transmission
errors.0 2 4 6 8 100246810Main ContactSecondary Contact
1(105rad)Compressions 1 and 2 (m)Fig. 8. Interpolated compression
functions 1(1) and 2(1) at the main and secondary contact
points.769 I. Gonzalez-Perez et al. / Mechanism and Machine Theory
46 (2011) 7657833.1. Contact at a single pointThe application of
the Hertz theory at a single contact point is based on the
following algorithm:(i) Pinion tooth surface 1 and gear tooth
surface 2 contact at a single point P. Position vector r1(P)and
unit normal n1(P)aredetermined in coordinate system S1, rigidly
connected to the pinion tooth surface (Fig. 3).(ii)A newreference
systemSpis dened as follows (Fig. 3). Origin Opis located at point
P. Axis zpresults collinear to unit normaln1(P). Axes xpand ypare
collinear to vectors r1(P)/l and r1(P)/u, respectively, wherein u
and l are the surface parameters forprole and longitudinal
directions. Axes xp and yp dene a common tangent plane to the
pinion and gear tooth surfaces(Fig. 3).(iii)A new surface r is
formed from pinion and gear tooth surfaces as follows (Fig. 4)(a)A
point A on the plane is given in system Sp by vector position
rp(A)(x, y) (Fig. 4(a)).(b)Two projections of point A, A1 and A2,
on tooth surfaces 1 and 2, respectively, are obtained as (Fig.
4(b))rA1 px; y = rA px; y + 1x; y nP p1rA2 px; y = rA px; y + 2x; y
nP p2wherein 1 and 2 are scalar coefcients and np(P)=Lp1n1(P)is the
unit normal at the contact point P in system Sp. Here, Lp1is a
matrix 33 for coordinate transformation from system S1 to system
Sp. Scalar coefcients 1 and 2 are determined(a) (b) (c) (d)Fig. 9.
Illustrationsof: (a)thevolumeof designedbody,
(b)auxiliaryintermediatesurfaces, (c)determinationof
nodesforthewholevolume, and(d)discretization of the volume bynite
elements.(a) (b)Fig. 10. Schematic illustration of the arrangement
for thenite element mesh about contact point P along: (a) prole,
inner, and (b) longitudinal directions.770 I. Gonzalez-Perez et al.
/ Mechanism and Machine Theory 46 (2011) 765783from the
intersection of the projections of point A on the pinion and gear
tooth surfaces, respectively.(c)Surface r is then obtained as (Fig.
4(c))rAr px; y = rA px; y +j 1x; y j+ j 2x; y j nP p3wherein Ar is
a new point that belongs to the new surface r. Function (|1(x, y)|
+|2(x, y)|) is designated as h(x, y) for thepurpose of simplicity
and represents the gap between pinion and gear tooth surfaces when
these surfaces are in contact atpoint P.(iv)Principal directions
and curvatures of surface r are obtained by determination of
maximum and minimum values of itsnormal curvature at point P [5].
Principal directions are given by angles I and II respect to axis
xP (see Fig. 5) and principalcurvatures are designated as I and II,
respectively. The principal curvature radii are given by RI=1/I and
RII=1/II. In aspur gear drive, since I=0 rad and II=/2 rad, axes xp
and yp coincide with the principal directions.(v)
Ratioa/bofthelengthsofmajorandminorsemi-axisofthecontactellipsegivenbytheHertztheoryisrelatedwithprincipal
curvature radii RI and RII as [10]:RIRII=ab_ _2Ee Ke Ke Ee :
4Here,e =1b2a2_is the eccentricity of the ellipse;Ke = = 201e sin2_
_1=2d is the complete elliptic integral of therst kind [13];Ee = =
201e sin2_ _1=2d is the complete elliptic integral of the second
kind [13].For the solution of Eq. (4),the ratioa/b is considered as
unknown.During the iterative process for solution of
Eq.(4),elliptic integrals K(e) and E(e) are determined numerically
as a function of values a/b. The dimensions of the semi-majorTable
1Set of variables for mesh renement control.Number of elements on
the tooth surface in prole direction NpNumber of elements on the
tooth surface in longitudinal direction NlNumber of elements under
the tooth surface in inner direction NsNumber of elements in the
sensitive-contact region in prole direction 2Nb+6Number of elements
in the sensitive-contact region in longitudinal direction 2NaNumber
of elements in the sensitive-contact region in inner direction
Nd(a) (b)Fig. 11. Schematic illustrations of: (a) boundary
conditions for the pinion and the gear, and (b) reference node of
the rigid surface of the pinion.771 I. Gonzalez-Perez et al. /
Mechanism and Machine Theory 46 (2011) 765783and semi-minor axes of
the contact ellipse, a and b, are quite different from the
dimensions of the semi-major and semi-minor axes of the contact
area of interference, A and B, as shown schematically in Fig. 6 for
a given compression .(vi)The dimensions of the contact ellipse, a
and b, may be obtained by consideration of an additional relation
between a and b[10],ab =3FRe4E_ _2=3F1e 2: 5Here,F is the
transmitted load between the pair of teeth and can be obtained from
T1=r1(P)Fn1(P)wherein T1 is the appliedtorque to the pinion.Re =
RIRIIpis the equivalent radius of curvature at the contact
point.ET=1121E1+122E2is the equivalent elastic modulus, that is a
function of elastic modules E1 and E2 and Poisson'sratios 1 and 2
of pinion and gear materials.F1e =4e2_ _1=3ba_ _1=2ab_ _2Ee Ke _ _
Ke Ee _ _1=6is a function that depends on the relation
a/b.(vii)Contact area Ac, maximum contact pressure po, and
compression , are obtained as [10]Ac = ab 6po =32Fab7 =32Fab1E bKe
8(a) (b) (c)Detail ADetail AFig. 12. Finite element model: (a)
pinion and gear tooth models, (b) slice of the pinion tooth model,
and (c) detail A of the pinion tooth model.Table 2Common gear data
for three designs of a spur gear drive.Number of teeth of the
pinion, N121Number of teeth of the gear, N250Module, m [mm]
4.0Pressure angle, [degrees] 25.0Helix angle, [degrees] 0.0Face
width, F [mm] 40.0Parabola coefcient for prole top crowning, apt
[mm1] 0.00025Parabola coefcient for prole bottom crowning, apb
[mm1] 0.00025Parabola vertex location for prole top crowning, uot
[mm] 0.0Parabola vertex location for prole bottom crowning, uob
[mm] 0.0772 I. Gonzalez-Perez et al. / Mechanism and Machine Theory
46 (2011) 765783(viii)Pressure distributions along axis xp and yp,
p(x) and p(y), are determined as [10]px = po1xa_ _1=29py = po1yb_
_1=210(ix)Stress distributions along common normal, x(z), y(z), and
z(z), are obtained in the pinion as [10]xz = po2be2axz + 1xz _
_11yz = po2be2ayz + 1yz _ _12zz = pobe2a1Tz 2Tz 13wherexz = 121Tz
+zaFz ; e Ez ; e xz = 1a2Tz b2+zaa2b2 Ez ; e Fz ; e _ _yz =12+12Tz
Ta2b2+zaa2b2 Ez ; e Fz ; e _ _yz = 1 + Tz +zaFz ; e Ez ; e Tz =b2+
z2a2+ z2_ _1= 2z =arctanazHere, F((z), e) and E((z), e) are the
incomplete elliptic integrals of therst kind and the second kind,
respectively [13].(x)Tresca stress distribution along common
normal, T(z), is obtained asTz = max xz zz ; yz zz _ _14(xi)Maximum
Tresca stress, T, o, is obtained fromTz z= 0zT;oT;o = TzT;o_ _:
15Table 3Different gear data for three designs of a spur gear
drive.Design A Design B Design CParabola coefcient for longitudinal
front crowning, alf [mm1] 0.0001 0.0002 0.0004Parabola coefcient
for longitudinal back crowning, alb [mm1] 0.0001 0.0002
0.0004Parabola vertex location for longitudinal front crowning, lof
[mm] 0.0 0.0 0.0Parabola vertex location for longitudinal back
crowning, lof [mm] 0.0 0.0 0.0Table 4Requested variables.Contact
area, [mm2] AcMaximum contact pressure, [MPa] poCompression, [m]
Pressure distribution along dimension a, [MPa] p(x)Pressure
distribution along dimension b, [MPa] p(y)Principal stress
distributions along the normal direction, [MPa] x(z), y(z),
z(z)Tresca stress distribution along the normal direction, [MPa]
T(z)Maximum Tresca stress, [MPa] T, o773 I. Gonzalez-Perez et al. /
Mechanism and Machine Theory 46 (2011) 7657833.2. Contact at two
pointsThe application of the Hertz theory at two contact points is
based on the following algorithm:(i) Two types of contact point are
dened: (1) the main contact point and (2) the secondary contact
point. The assignation ofthe main and secondary contact points
depends on the angle of rotation of the pinion, 1, and the
corresponding function oftransmissionerrors. Fig.7shows thetwo
typesofcontact point, P1andP2, wherein themain one, P1,
isthatoflowertransmission error.(ii)The main contact point, P1, is
determined by application of the algorithm explained in detail in
[5]. Angle of rotation of thepinion, 1, and angle of rotation of
the gear, 2, are then determined, for pinion and gear tooth
surfaces to be in contact atpoint P1.(iii)The secondary contact
point, P2, is determined by application of a similar algorithm, but
considering as xed the previouslydetermined angle 2. An additional
angle of rotation of the pinion, 1, is required to provide contact
at point P2, causingcompression 1 at the main contact point.
Position vector rP2 1and unit normal nP2 1are also
determined.(iv)Compression functions 1(1) and 2(1) at the main and
secondary contact points, respectively, are determined. A set
ofvalues of 1 is considered for determination of interpolated
functions 1(1) and 2(1) (Fig. 8).(v) The algorithm explained in
Section 3.1 is applied here at each contact point, P1 and P2, from
step (i) to step (v), in order toobtainrelationsa1/b1anda2/b2,
respectively.
Thesubscripts1and2refertothemainandsecondarycontactpoints,respectively.
Relative curvature radii are also obtained at each contact point,
Re1 = RI1RII1p, Re2 = RI2RII2p.vi The dimensions of the contact
ellipses at each contact point depend on the transmitted forces F1
and F2 (see Eq. (5)). Suchforces depend on the compressions 1 and
2, since the Hertz theory provides [10]Fi =16E__2Rei9_ _1=2i2aibi_
_1=2F1ei Kei 1_ _3=2i = 1; 2: 16Since compressions 1 and
2arefunctionsof angle 1(see Fig. 8), an iterative process based
onthe solution of thefollowing non-linear equation is implementedT1
= rP1 1 F111 nP1 1+ rP2 1 F221 nP2 117wherein the unknown is
1.(vii)Once the equilibrium is satised and 1 is known, compressions
1 and 2 are obtained from interpolated functions (seeFig. 8),
forcesF1andF2areobtainedfromEq. (16),
anddimensionsofthecontactellipsesareobtainedfromEq. (5).Relations
fromEqs. (6) to (15) are then considered at each contact point to
obtain Aci, poi, i, pi(x), pi(y), xi(z), yi(z), zi(z),Ti(z), and
Toi, i =1, 2.Table 5Numerical calculations based on the Hertz
theory.Case Ac [mm2] a [mm] b [mm] [m] po [MPa] T, o [MPa] RI [mm]
RII [mm]A1 3.321 8.195 0.129 3.989 593.4 360.0 10,140.2 12.3A2
5.265 10.320 0.162 6.330 747.7 453.6 10,140.2 12.3A3 8.366 13.009
0.205 10.052 942.0 571.5 10,140.2 12.3B1 2.901 6.269 0.147 4.859
678.8 413.7 4945.2 12.3B2 4.606 7.899 0.186 8.187 855.2 521.2
4945.2 12.3B3 7.297 9.952 0.233 12.240 1077.5 656.7 4945.2 12.3C1
2.510 4.734 0.169 5.941 781.9 479.7 2347.6 12.3C2 3.998 5.964 0.213
9.432 985.1 604.4 2347.6 12.3C3 6.322 7.514 0.268 14.970 1241.2
761.5 2347.6 12.3Table 6Ratios RI/a and RII/b for cases A1, B1, and
C1.Case RI/a RII/bA1 1237.4 95.3B1 788.8 83.7C1 495.9 72.8774 I.
Gonzalez-Perez et al. / Mechanism and Machine Theory 46 (2011)
7657834. Development ofnite element modelsThe approach for the
development ofnite element models is already known and has been
proposed in [14]. Some featureshave been incorporated here to the
nite element models for a better control of the mesh renement
around the contact point. Theproposed approach is accomplished as
follows:Step 1. Pinion and gear tooth surface equations and
portions of the corresponding rims are considered for determination
of thevolumes of the designed bodies. Fig. 9(a) shows the volume
for one tooth model of the pinion of a spur gear drive.Step 2. The
volume of each tooth of the model is divided into six subvolumes
using auxiliary intermediate surfaces 16 as shownin Fig. 9(b).Step
3. Node coordinates (Fig. 9(c)) are determined analytically
considering tooth surface equations, portions of correspondingrims,
and the set of variables for mesh renement control that are
described below.Step 4. Discretization of the model bynite elements
using the nodes determined in the previous step is accomplished as
shownin Fig. 9(d).Step5. Thesetofvariables for meshrenement
controlaffectstothe numberof nite elements around thecontact
pointP(Fig. 10). A sensitive-contact region is dened around the
contact point P as follows:(i)A group of elements of regular size
with dimension l l l is created in the sensitive-contact
region.(ii)The size l of the elements is controlled by dimension b
and number Nb. Here, b is the length of the semi-minor axis of
thecontact ellipse that Hertz theory predicts; Nbis the number of
elements that covers dimension b. Length l may be obtainedas l
=b/Nb.(iii)The number of elements of regular size that covers a
fraction of dimension a is controlled by number Na. Here, a is
thelength of the semi-major axis of the contact ellipse that Hertz
theory predicts. Since ab, NaNb for the whole contactarea to be
covered by elements of regular size and this is, in practice,
computationally expensive. Beside this, it is expectedthat pressure
distribution has a larger scope along dimension b than along
dimension a. These are the reasons why Na isthe number of elements
of regular size that covers just a fraction of dimension a.(iv)The
number of elements of regular size underneath the tooth surface is
controlled by number Nd.(v)Three more rows of elements of regular
size are added along dimension b, up and down (Fig. 10(a) shows
just one row upand one rowdown for the purpose of simplicity). This
establishes the number of elements in the sensitive-contact region
as(2Nb+6)NaNd.Table 7Meshes M1, M2, and M3, and corresponding
results for case A1.{Nb, Nd, Ns} Mesh M1 {1, 1, 4} Mesh M2 {2, 2,
5} Mesh M3 {3, 3, 6} Hertz result Relative error [%]Ac [mm2] 3.841
3.606 3.450 3.321 3.884 [m] 4.186 4.201 4.186 3.989 4.939po
[MPa]a665.2 592.9 593.1 593.4 0.051po [MPa]b572.9 582.9 588.3 593.4
0.859T, o [MPa] 482.0 458.7 417.7 360.0 16.028T, o [MPa]c482.0
390.5 372.4 360.0 3.444aThrough variable CPRESS [15].bThrough
viewer [15].cAverage threshold of 35%.CPRESS
(MPa)+0.000e+00+5.883e+01+1.177e+02+1.765e+02+2.353e+02+2.942e+02+3.530e+02+4.118e+02+4.707e+02+5.295e+02+5.883e+02slavesurfaceCOPEN
(mm) + 5.058e-02- 4.186e-04- 8.372e-04- 1.256e-03- 1.674e-03-
2.093e-03- 2.511e-03- 2.930e-03- 3.349e-03- 3.767e-03- 4.186e-03(a)
(b)Fig. 13. Illustration of (a) contact pressure distribution on
pinion tooth surface, and (b) overclosures distribution on pinion
tooth surface.775 I. Gonzalez-Perez et al. / Mechanism and Machine
Theory 46 (2011) 765783(vi)Size of the elements located out of the
sensitive-contact region increase exponentially from such a region
towards theborders or the intermediate surfaces of the tooth model.
Number of elements in prole, longitudinal, and inner directionsare
controlled by numbers Np, Nl, and Ns, respectively.A summary of the
set of variables for mesh renement control is shown in Table 1.Step
6Setting of boundary conditions are accomplished as follows (Fig.
11):(i) Nodes on the two sides and bottom part of the gear rim are
encastred (Fig. 11(a)).(ii) Nodes on the two sides and bottom part
of the pinion rim form a rigid surface (Fig. 11(a)).(iii)A
reference node N located on the axis of the pinion is used as the
reference point of the previously dened rigid surface(Fig. 11(b)).
Reference point N and rigid surface constitute a rigid
body.(iv)Only one degree of freedom is dened as free at the
reference point Nrotation about the pinion axiswhile all
otherdegrees of freedomare xed. Application of a torque T1in
rotational motion at the reference point N allows to apply such
atorque to the pinion model while the gear model is held at
rest.Step 7Elements of rst order have been considered in the nite
element mesh. This type of elements is recommended for
contactsimulations [15].Step 8Contact formulation is based on a
surface-to-surface discretization, anite-sliding tracking approach,
and a slavemastercontact algorithm [15]. Elements of the model
required for the formation of pinion-slave tooth surface and
gear-mastertooth surface are automatically identied.Fig. 12 shows
an example of anite element model of a spur gear drive with one
pair of teeth.1 0.5 0 0.5 10200400600x/ap(x) (MPa)FEMHertz1 0.5 0
0.5 10200400600y/bp(y) (MPa)FEMHertzFig. 14. Contact pressure
distributions p(x) (left) and p(y) (right) in case A1 obtained
fromthe Hertz theory and the fromnite element method through mesh
M3.(Avg: 0%)S, Tresca (MPa)+2.125e
01+4.197e+01+8.372e+01+1.255e+02+1.672e+02+2.090e+02+2.507e+02+2.925e+02+3.342e+02+3.760e+02+4.177e+02detail
AEndStartnormal direction(a) (b)Fig. 15. Tresca stress distribution
in the pinion tooth model for case A1 and mesh M3 without any
averaging threshold: (a) slice of the pinion tooth model, and
(b)detail A showing points along the normal direction for
collecting stress results.776 I. Gonzalez-Perez et al. / Mechanism
and Machine Theory 46 (2011) 765783(Avg: 35%)S, Tresca (MPa)+2.242e
01+3.744e+01+7.466e+01+1.119e+02+1.491e+02+1.863e+02+2.235e+02+2.607e+02+2.979e+02+3.352e+02+3.724e+02detail
A(a) (b)Fig. 16. Tresca stress distribution in the pinion tooth
model for case A1 and mesh M3 with an averaging threshold of 35%:
(a) slice of the pinion tooth model, and(b) detail A.3002001000z/bx
(MPa)x-HertzxFEM-avex-FEM0 1 2 3 4 5 0 1 2 3 4 56004002000z/by
(MPa)y-Hertzy-FEM-avey-FEM6004002000z/bz (MPa)z-HertzzFEM-avez-FEM0
1 2 3 4 50 1 2 3 4 50100200300400z/bT
(MPa)T-HertzT-FEM-aveT-FEMFig. 17. Variation of principal stresses
and Tresca stress along the normal direction z, x(z) (up-left),
y(z) (up-right), z(z) (down-left), and T(z) (down-right), incase
A1, obtained from the Hertz theory and from thenite element
method.777 I. Gonzalez-Perez et al. / Mechanism and Machine Theory
46 (2011) 7657835. Numerical examplesThis section covers stress
analyses based on the Hertz theory and thenite element method for
several cases of design. Ageneral-purpose computer program [15] has
been applied for the stress analyses based on thenite element
method.The approach for the development of the numerical examples
is as follows:(1)Gear data for three designs of a spur gear drive
are considered. Each design of the gear drive is based on a
different amountof crowning in longitudinal direction of their
pinion tooth surfaces. The amount of crowning in prole direction of
theirpinion tooth surfaces is the same for the three designs and
this means that the same function of unloaded transmissionerrors is
provided to each design.(2)Three values of torque T1 are being
applied to the pinion for the same angular position of the pinion
in each of the threedesigns, resulting in nine congurations ofnite
element models.(3)Pinion and gearnite element models are
automatically developed for each analysis or conguration by the
developedcomputer programs as explained in Section 4.(4)Requested
data from numerical calculations and stress analyses are presented.
The variables that are used for the output ofresults in the
general-purpose computer program [15] are presented.(5)Validation
of thenite element model is accomplished for the numerical example
in which the Hertz theory should workthe best. Setting of variables
for the mesh renement control is achieved.(6)Numerical calculations
based on the Hertz theory andnite element analyses are compared for
each design and appliedtorque in order to validate the analytical
approach based on the Hertz theory.0246810 (m)Ac (mm2)Hertz(A)
FEM(A)Hertz(B) FEM(B)Hertz(C) FEM(C)0 5 10 1505001,000 (m)po
(MPa)Hertz-A FEM-AHertz-B FEM-BHertz-C FEM-C0 5 10 15Fig. 18.
Contact area Ac versus compression (left) and maximum contact
pressure po versus compression (right) obtained from the Hertz
theory and from thenite element method for the nine cases of
design.0 5 10 150200400600800T,o (MPa)Hertz(A) FEM(A)Hertz(B)
FEM(B)Hertz(C) FEM(C)(m)Fig. 19. Maximum Tresca stress T, o versus
compression obtained from the Hertz theory and from thenite element
method for the nine cases of design.778 I. Gonzalez-Perez et al. /
Mechanism and Machine Theory 46 (2011) 7657835.1. Input and
requested dataThree designs A, B, and C of a spur gear drive are
considered. The common gear data for all three designs is shown in
Table 2.The different gear data for designs A, B, and C is shown in
Table 3. The applied torques to the pinion in each design are
T1={50, 100, 200}N m. The total number of analysis is nine and is
identied as Ai, Bi, and Ci (i ={1, 2, 3}), where i =1 for T1=50 N
m,i =2 for T1=100 N m, and i =3 for T1=200 N m.The requested data
fromnumerical calculations and stress analyses is shown in Table 4.
The calculation of the variables shownin Table 4 is accomplished
through the Hertz theory and thenite element method for the nine
analyses.1 0.5 0 0.5 10200400600800x/ap(x) (MPa)FEMHertz1 0.5 0 0.5
10200400600800y/bp(y) (MPa)FEMHertzFig. 20. Contact pressure
distributions p(x) (left) and p(y) (right) in case A2 obtained from
the Hertz theory and from thenite element method.0 1 2 3 4
58006004002000z / bx (MPa)x-HertzxFEM-avex-FEM0 1 2 3 4 51,0005000z
/ by (MPa)y-Hertzy-FEM-avey-FEM0 1 2 3 4 51,0005000z / bz
(MPa)z-HertzzFEM-avez-FEM0 1 2 3 4 50200400600800z / bT
(MPa)T-HertzT-FEM-aveT-FEMFig. 21. Variation of principal stresses
and Tresca stress along the normal direction z, x(z) (up-left),
y(z) (up-right), z(z) (down-left), and T(z) (down-right), incase
C3, obtained from the Hertz theory and from thenite element
method.779 I. Gonzalez-Perez et al. / Mechanism and Machine Theory
46 (2011) 7657835.2. Validation of thenite element modelValidation
of thenite element model is accomplished as follows:(1)The Hertz
theory is applied to the nine cases of analysis described
above.(2)Severalnite element analyses are accomplished for the
chosen case of design (in which the hypotheses of application ofthe
Hertz theory apply) by increasing the renement of the mesh through
variables Nb, Nd, and Ns.(3)Convergency in terms of Ac, , po, and
T, o is observed to conrm validation of thenite element model.Table
5 shows the numerical results obtained fromapplication of the Hertz
theory to the nine cases of analysis described above.The material
is steel with the properties of Young's Modulus E=2.068105MPa and
Poisson's ratio 0.29 in both members of thegear drive. The case of
design is chosen by considering that: (i) the areas of contact
should be small (cases A1, B1, and C1) incomparison with the
dimensions of the designed bodies, and (ii) RIa and RIIb in order
for the tooth surfaces to be consideredas elastic half-spaces.
Comparison of the ratios RI/a and RII/b is shown in Table 6 for
cases A1, B1, and C1. Since the largest ratios areobtained for case
A1, this is the chosen case for validation of thenite element
model.Three sets of variables for the mesh renement control are
considered and identied in Table 7 as mesh M1, mesh M2, andmesh M3.
Numbers of elements in prole and longitudinal directions, Np=34 and
Nl=98, are common to the three meshes.Number Na=15 is large enough
and common as well to the three meshes since its variation scarcely
affects the results.Table 7 shows the results Ac, , po, and T,
oobtained fromstress analysis for each mesh. Relative errors
fromthe results obtainedthrough mesh M3 in comparison with Hertz
results are shown in the last column of Table 7.Fig. 13(a) shows
the contact pressure distribution on the pinion-slave tooth surface
for case A1 obtained through mesh M3.Fig. 13(b) shows the
overclosures distribution on the pinion tooth surface. A visual
comparison of the area of contact in Fig. 13(a)and the area of
interference between pinion and gear tooth surfaces in Fig. 13(b)
shows a signicant difference.(Avg: 37%)S, Tresca (MPa)+2.505e
01+5.003e+01+9.982e+01+1.496e+02+1.994e+02+2.492e+02+2.990e+02+3.487e+02+3.985e+02+4.483e+02+4.981e+02CPRESS
(MPa)+0.000e+00+7.571e+01+1.514e+02+2.271e+02+3.028e+02+3.785e+02+4.542e+02+5.299e+02+6.056e+02+6.813e+02+7.571e+02slave
surface 1slave surface 2(a) (b)Fig. 22. Pressure and Tresca stress
distributions in the pinion tooth model for case A3 and mesh M2
when two pairs of teeth are in contact: (a) pressure
stressdistribution on two teeth, and (b) Tresca stress distribution
on a slice of the pinion tooth model.Fig. 23. Maximum contact
pressures po (left) and contact areas Ac (right) for case A3 when
two pairs of teeth are in contact, obtained from the Hertz theory
andfrom thenite element method, along a portion of the cycle of
meshing.780 I. Gonzalez-Perez et al. / Mechanism and Machine Theory
46 (2011) 765783Fig. 14 shows the pressure distribution p(x) and
p(y) along dimensions a and b, respectively, in case A1, obtained
from theHertz theory and fromthe nite element method through mesh
M3. Fig. 14 shows that both methods provide similar distributionsof
pressure.Fig. 15(a) shows the Tresca stress distribution on the
middle slice of the pinion tooth model. Since no averaging
threshold isconsidered in the representation, discontinuities are
observed. Fig. 15(b) shows a detail of the middle slice of the
pinion toothmodel. Fig. 15(b) shows as well the set of points along
the normal direction to the pinion tooth surface for collecting
stress results.The start point coincides with the contact point.
The end point is 5b far from the start point in the normal
direction. The middlepoints are obtained as the intersections of
the normal direction to the pinion tooth surface and the element
faces.Fig. 16 shows the Tresca stress distribution obtained for
case A1 and mesh M3 when an averaging threshold of 35% is
applied.Such an averaging threshold is large enough to smooth the
Tresca stress distribution. The set of points considered in Fig.
15(b) isbeing considered here for representation of averaged
values.Fig. 17 shows the variation of principal stresses and Tresca
stress along the normal direction z, x(z), y(z), z(z), and T(z),in
case A1, obtained from the Hertz theory and from thenite element
method with mesh M3. The set of points represented inFig. 15(b) was
considered here for collection of the averaged and not-averaged
stress results. Fig. 17 shows a good agreement ofthe stress
variations obtained by the Hertz theory and the averaged stress
results obtained by thenite element method.The obtained results for
case A1, where the Hertz theory should work better than in other
cases, conrmthat the nite elementmodel provided with mesh M3 is
validated. Mesh M3 provides 52,788 elements with 58,697 nodes when
one pair of teeth isconsidered. In other cases where hypotheses of
application of the Hertz theory do not apply, the validatednite
element modelwill be applied to solve the contact problem.5.3.
Results5.3.1. Contact at a single pointFig. 18 shows the contact
areas Acand the maximumcontact pressures pofor the nine cases of
analysis, obtained fromthe Hertztheoryandfromtheniteelementmethod,
versusthecompressionobtainedfromtheHertztheory.
Agoodagreementisobtained for the contact areas wherein the maximum
relative error is reached in case A3 with 5.426%. A better
agreement isobtained for the maximum contact pressures wherein the
maximum relative error is reached in case A3 with 1.352%.Fig. 19
shows the maximumTresca stress T, o for the nine cases of analysis,
obtained from the Hertz theory and from the
niteelementmethod(withathresholdof35%),
versusthecompressionobtainedfromtheHertztheory.
Agoodagreementisobtained as well for the maximum Tresca stress,
wherein the maximum relative error is reached in case B1 with
6.4%.Pressure distributions p(x) and p(y) are shown in Fig. 20 for
case A2. Principal stress variations x(z), y(z), and z(z),
andTresca stress variation T(z) are shown in Fig. 21 for case C3.
The good agreement of the results validates the approach based on
theTable 8Specic gear data for design D.Parabola coefcient for
prole top crowning, apt [mm1] 0.00025Parabola coefcient for prole
bottom crowning, apb [mm1] 0.00025Parabola vertex location for
prole top crowning, uot [mm] 1.6Parabola vertex location for prole
bottom crowning, uob [mm] 1.6Parabola coefcient for longitudinal
front crowning, alf [mm1] 0.0006Parabola coefcient for longitudinal
back crowning, alb [mm1] 0.0006Parabola vertex location for
longitudinal front crowning, lof [mm] 10.0Parabola vertex location
for longitudinal back crowning, lof [mm] 10.0CPRESS
(MPa)+0.000e+00+9.496e+01+1.899e+02+2.849e+02+3.798e+02+4.748e+02+5.698e+02+6.647e+02+7.597e+02+8.547e+02+9.496e+02CPRESS
(MPa)+0.000e+00+1.086e+02+2.172e+02+3.259e+02+4.345e+02+5.431e+02+6.517e+02+7.603e+02+8.689e+02+9.776e+02+1.086e+03(a)
(b)Fig. 24. Contact pressure distributions in the pinion tooth
model for case of design D and mesh M3 when shaft angle error is:
(a) 1.0 arcmin, and (b) 2.0 arcmin.781 I. Gonzalez-Perez et al. /
Mechanism and Machine Theory 46 (2011) 765783Hertz theory to solve
the contact problem on those gear drives with localized bearing
contact wherein edge contact is avoided bywhole crowning of the
tooth surfaces.5.3.2. Contact at two pointsCase A3 is considered
for investigation of application of the Hertz theory when the load
is shared by two pairs of teeth. Fig. 22shows the contact pressure
and the Tresca stress distribution for case A3 obtained by
application of thenite element method.Two slave surfaces are
considered for determination of the maximum contact pressure and
the contact area at two locations, themain tooth and the secondary
tooth. Slave surfaces 1 and 2 correspond to the main and secondary
teeth, respectively. Fig. 23 showsthe maximumcontact pressure po
and the contact area Ac at the main tooth and at the secondary
tooth, obtained by application ofthe Hertz theory and the nite
element method, during a portion of the cycle of meshing at which
the transfer of meshing fromthemain tooth to the secondary tooth
occurs. It is observed a difference between FEM and Hertz results
due to the inuence of thebending deection, that is not considered
in the model that applies the Hertz theory.The maximum relative
error of the maximum contact pressure at the main tooth is 3.83%
whereas at the secondary tooth is5.79%. The maximumrelative error
of the contact area at the main tooth is 12.49% whereas at the
secondary tooth is 11.73%. At oneof the contact points, the
maximumcontact pressure at the secondary tooth obtained by the nite
element method (103.8 MPa) isnot reliable due to the presence of
edge contact at the top of the gear tooth. The results show that
the model based on the Hertztheory overestimates the maximum
contact pressure and underestimates the contact area.5.3.3. Contact
at a single point on pinion tooth surface provided with partial
crowningA new case of design (design D) based on partial crowning
of pinion tooth surfaces is considered here. Table 8 shows
thespecic gear data.Fig. 24(a) and (b) shows the contact pressure
distribution when a torque of 200 N m is applied and a shaft angle
error of1.0 arcmin and 2.0 arcmin, respectively, is considered for
design D. It is observed that the shaft angle error affects to the
size of thecontact area and to the maximum contact pressure.Table 9
shows a comparison of the area of contact and the maximumcontact
pressure obtained fromthe Hertz theory and fromthenite element
method. In this case, the Hertz theory does not work properly due
to the existence of an involute area close tothe contact point that
affects to the hypothesis of the Hertz theory, since pinion and
gear tooth surfaces cannot be modeled asquadratic surfaces. Table 9
shows that the Hertz theory provides the same solution when the
shaft angle error is 1 arcmin or if it is2 arcmin. Compliance of
results may be achieved if the contact point is far enough from the
involute area.6. ConclusionsThe obtained results allow the
following conclusions to be drawn:(i) An approach based on the
Hertz theory has been implemented for determination of contact
variables, such as the contactarea, the pressure distribution, the
maximum contact pressure, the stress distribution underneath the
contacting surfaces,the maximum Tresca stress, and the compression,
for gear drives with localized bearing contact wherein edge contact
isavoided by whole crowning of the tooth surfaces. Such an approach
has been applied to the case of one pair of teeth sharingthe load
and the case of two pairs of teeth sharing the load.(ii)Anite
element model has been developed and validated in terms of the
contact area, the maximum contact pressure, thepressure
distribution along major and minor semi-axes of the contact area,
the principal stress distribution underneath
thecontactingsurfaces,theTresca
stressdistributionunderneaththecontactingsurfaces,andthe
maximumTresca stress.Validationhasbeencarriedout byobservationof
theconvergenceof theabovementionedmagnitudesthroughanincreased
renement of the mesh and by comparison of the results provided by
the Hertz theory in such a case whereinHertz hypotheses are
satised.(iii)The validatednite element model has been applied in
gear drives with localized bearing contact wherein tooth
surfacesare provided with partial crowning and the Hertz theory
does not apply.AcknowledgmentsThe authors express their deep
gratitude to the Spanish Ministry of Science and Innovation for
thenancial support of projectRef.DPI2007-63950 andto theFundacin
CajaCastelln-Bancaja for the support received through the Program
of Mobility ofResearchers, Ref. E-2008-11.Table 9Comparison of
contact areas and maximum contact pressures obtained by the Hertz
theory and by thenite element method in case D. [arcmin] Ac-Hertz
[mm2] Ac-FEM [mm2] po-Hertz [MPa] po-FEM [MPa]1 5.828 9.504 1352.3
967.72 5.828 8.610 1352.3 1103782 I. Gonzalez-Perez et al. /
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