Approximate Analytical Model for Hertzian Elliptical Contact Problems

TRIB-05-1037 : FINAL COPY

APPROXIMATE ANALYTICAL MODEL FORHERTZIAN ELLIPTICAL CONTACT PROBLEMS

J-F. Antoine ∗LGIPM-CEMA

IUT de Thionville-Yutz1 impasse Kastler, 57970 YUTZ, France

e-mail: [email protected]

C. Visa†

LGIPM-CEMAIUT de Thionville-Yutz

1 impasse Kastler, 57970 YUTZ, Francee-mail: [email protected]

C. SauveyLGIPM-CEMA

IUT de Thionville-Yutz1 impasse Kastler, 57970 YUTZ, France


G. AbbaLGIPM-CEMA

Ecole Nationale d’Ingenieurs de MetzIle du Saulcy, 57045 METZ Cedex, France


ABSTRACT

In rolling bearing analysis Hertzian contact theory is used to compute local contact stiffness. This theory does not have a

closed form analytical solution and requires numerical calculations to obtain results. Using approximations of elliptical func-

tions and with a mathematical study of Hertzian results, an empirical explicit formulation is proposed in this paper and allows

to obtain the dimensions, the displacement and the contact stress with at least0.003%precision and it can be applied to a large

range of ellipticity of the contact surface.

Keywords: elliptical contact, elastic bodies, hertzian theory, approximate numerical solution, rolling bearings.

1 Introduction

To analyze the behavior of angular-contact ball bearings, the theory of Hertz on contact between two elastic solids is usually used

[1, 2]. It is useful to define the stiffness of each contact or line point [3], the deformation of the contacting bodies (rolling elements and

rings), the maximal stress and also the dimensions of the elliptic contact area. All these values have to be related to the applied force and

the geometry of the two bodies in the contact area.

This generally accepted theory was presented by Hertz [4] and summed up in many textbooks and handbooks [5, 6]. The calculation

of contact properties is usually processed thanks to tables or charts [7]. Numerical solutions to contact problems already exist. They use

arithmetic-geometric mean algorithms as described in [8] to calculate elliptical functions. Implicit equation of contact is usually solved

∗Address all correspondence to this author, phone: +33(0)382820637, Fax:+33(0)382820606.†We thank the French Minister of Research and theConseil Regional de Lorrainefor their financial support.

Journal of Tribology Copyright c© by ASME / 1

by a Newton-Raphson algorithm [1, 9] or numerical algorithm using elliptic functions’ properties [10]. An analytic solution to slightly

elliptical Hertzian contacts has been studied in [5, 11, 12]. An engineering approach to Hertzian contact elasticity dedicated to bearings

has been suggested in [3].

An analytical solution to Hertzian contact problems is useful to know the influence of each parameter on the stiffness or maximal

stress at contact and also allows the optimization of the solids’ geometry to improve the rolling performances.

The scheme proposed in this paper to find an analytical solution consists first in having an explicit expression for elliptic integrals,

then to replace tables with an analytical expression yielding the value of ellipticity parameterκ in connection to the relative principal

curvatures of the contactsA andB.

Hertzian theory’s notations and results are first summed up. Elliptic functions are then calculated by expressions given in [8]. The

exact solution of Hertzian theory is next studied to choose an appropriate function forκ in terms ofB/A. This expression is then fitted

on exact results to finally suggest a highly accurate solution. The results are first confronted to the ones that can be found in literature.

2 Summary of Hertzian Contact theory

It is convenient to sum up the Hertzian contact theory and the notations that are used in this paper in order to correctly set the

problem. Assumptions, geometry of solids, materials’ properties are described in the following part, to finally introduce the main results

of Hertzian theory.

2.1 Main assumptions made by Hertz

In performing his analysis, Hertz made the following assumptions [1, 10, 13]:

• At the point of contact the shape of each of the contacting surfaces can be described by a homogeneous quadratic polynomial in two

variables;

• Both contacting surfaces are ideally smooth;

• Contact stresses and deformations satisfy the differential equations for stress and strain of homogeneous, isotropic and elastic bodies

in equilibrium. All deformations occur in the elastic range;

• The stress disappears at great distance from the contact zone;

• Loading is perpendicular to the surface: the effect of surface shear stresses is neglected;

• Tangential stress components are zero at both surfaces within and outside the contact zone;

• The stress integrated over the contact zone equals the force pushing the two bodies together;

• The distance between the two bodies is zero within the contact zone but finite outside : the contact area dimensions are small

compared to the local radii of curvature of the bodies under load;

• In the absence of an external force, the contact zone degenerates in a point.

The notations and main results commonly used in the Hertzian contact theory are going to be exposed now.

2.2 Geometrical description of contact

Figure 1 presents the contacting surfaces of two bodies.ω is the angle between planes (O,−→xI ,−→z ) and (O,−→xII ,−→z ).

The contacting surface of the two bodies (ellipsoids), subscriptI andII (rolling elements and raceways in case of rolling bearings),

is defined by their principal curvatureρ = 1/r along two perpendicular directionsx andy denoted with subscriptI or II according to the

considered body. The resulting notations of curvature areρIx andρIy for body I andρIIx andρIIy of body II.

Curvature is negative if the center of curvature is outside the body (i.e. the surface is concave in the considered direction).


rIIxrIIy

Body II

Body I

rIx

xII

yII

yIxI

w

Q

Q

rIy

z

Ow

w

Figure 1. Contact problem definition: bodies I and II in contact, planes of principal curvatures, geometrical parameters and applied Force Q [7].

Contact geometry is often defined with the curvature sum∑ρ and curvature differenceFρ [1, 7, 10]:

∑ρ = ρIx +ρIy +ρIIx +ρIIy (1)

Fρ =−(√

R2I +R2

II +2RI RII cos2ω)

/∑ρ (2)

with RI = ρIx−ρIy andRII = ρIIx −ρIIy . The relative curvatures of the contact are obtained from:

A = ∑ρ4

(1+Fρ

)B = ∑ρ

4

(1−Fρ

)(3)

The projected contact surface is supposed to be an elliptical area of contact anda andb denote its semi-axes. In the general case, it

is not possible to know which one of both axes is the semimajor axis. The ellipticity parameter of the ellipse is denotedκ = a/b. Harris’s

formulation [1] of the problem yields a parameterκ greater or equal to1.

2.3 Description of contacting materials

The contacting bodies are made of materials defined by their modulus of elasticityE and Poisson’s ratioν. The materials’ properties

are noted [EI , νI ] and [EII , νII ] for bodies I and II respectively.

Equivalent modulus of elasticityE∗ is defined by [1] in terms of properties of both contacting materials with the following relation-

ship:

1E∗

=1−ν2

I

EI+

1−ν2II

EII


2.4 Main Results of the Theory

The Hertzian contact theory established that the ellipticity parameter was related to the geometrical parameterFρ, by the means of

an implicit equation, which has been found to be the following [1]:

Fρ =B/A−1B/A+1

=(κ2 +1)E(κ)−2F (κ)

(κ2−1)E(κ)(4)

TermsF andE in (4) respectively denote complete elliptic integrals of first and second kinds, which expressions are as follows:

F (κ) =Z π/2

0

[1−

(1− 1

κ2

)sin2 φ

]−1/2

dφ (5)

E(κ) =Z π/2

0

[1−

(1− 1

κ2

)sin2 φ

]1/2

dφ (6)

Solving a contact problem requires to calculate the parameterκ for the studied geometry, defined by the parameterFρ.

Equation (4) is usually solved by numerical means. The elliptic integrals are commonly calculated thanks to the arithmetic-geometric

mean algorithm, as described in [8].

Once the valueκ related toFρ is obtained, the dimensions of the contact area, the displacement (mutual approach of the bodies’

centers) and maximal stress can be calculated. The dimensions of the contact ellipse and the mutual approach of the centers of both

bodies are given by the following formulae [1]:

a =a∗(

3Q2 (A+B) E∗

)1/3

(7)

b =b∗(

3Q2 (A+B) E∗

)1/3

(8)

δ =δ∗(

3Q2E∗

)2/3 (A+B)1/3

2(9)

a∗, b∗ andδ∗ are functions of the only parameterκ and are given by:

a∗ =(

2κ2 E(κ)π

)1/3

(10)

b∗ =(

2E(κ)πκ

)1/3

(11)

δ∗ =2F (κ)

π

(π

2κ2 E(κ)

)1/3

(12)

The maximal stressσ located at geometric center of contact ellipse0 is finally given by:

σ =32

Qπab

(13)


At present, calculation of results of the Hertzian stress theory (i.e values ofa∗, b∗, δ∗) are usually given in tables available in [1, 6].

Equation (9) allows to obtain the force-displacement relationship:

Q =

[25/2

3E∗

(δ∗)3/2 (A+B)1/2

]δ3/2 (14)

The results are at hand for rapid calculations in a design procedure, but in case of iterative calculation such as found in an opti-

mization process, the numerical solution of equation (4) is necessary. For the special case of bearings, Houpert [3] develops an explicit

approach to Hertzian contact elasticity that avoids numerical calculation and quickly provides approximate results.

3 Numerical Approximate Solution for Hertzian Contact Theory

Obtaining an analytical solution consists, on the one hand, in having explicit expressions of elliptic functions and, on the other hand,

in finding an analytical solution to implicit Eq. (4). These two points will be discussed further on.

In the following parts we use theppm(part per million) to quantify the error.

3.1 Elliptic functions

Elliptic integrals of first and second kinds can be approached by the following expressions given in [8]:

F (κ) =(α0 +α1m1 +α2m2

1

)− (α3 +α4m1 +α5m2

1

). lnm1 (15)

E(κ) =(β0 +β1m1 +β2m2

1

)− (β3m1 +β4m2

1

). lnm1 (16)

with the complementary parameterm1 = 1/κ2. Approximate functions are only valid for0≤m1 ≤ 1 (i.e. κ≥ 1). Coefficientsα0 to α5

andβ0 to β4 are given in Tab. 1.

Table 1. Coefficients of the polynomial approximations (15) and (16) for complete elliptic integrals of first and second kinds [8, 14].

α0 1.38629 44 α4 0.12134 78 β2 0.10778 12

α1 0.11197 23 α5 0.02887 29 β3 0.24527 27

α2 0.07252 96 β0 1 β4 0.04124 96

α3 0.5 β1 0.46301 51

Absolute value of the error does not exceed20 ppmand40 ppmfor F andE respectively, according to the authors [8].

3.2 Ellipticity parameter approximation

3.2.1 Exact values of κ To approximate the exact functionκ of B/A, we first solve (4) with the functionfminbndof the

Matlab software to finally obtain the reference results. Theκ parameter has been evaluated for 2,000 different values ofB/A with

1≤ B/A < 1010. A logarithmic distribution is chosen to improve density of points whenB/A is close to1. The tolerance ofκ has been

set to10−16. Fig. 2 first shows the results forκ. As shown in literature [11, 15, 12, 3],κ will be further written under the form(B/A)γ.

The values of exponentγ are also represented on Fig. 2.


100

105

100

101

102

103

104

B/A ratio

Elli

ptic

ity P

aram

eter

κ =

(B

/A) γ

100

105

0.54

0.56

0.58

0.6

0.62

0.64

0.66

B/A ratio

γ E

xpon

ent F

unct

ion

Figure 2. Variations of the ellipticity parameter κ = (B/A)γ and of the exponent γ in terms of B/A ratio.

3.2.2 Approximate expression of function κ(B/A) The functionκ(B/A) has been found to have the following structure

[11, 15]:

κ = (B/A)γ (17)

In the range of smallκ value, Brewe and Hamrock have found forγ a value close to0.636. To perform curve-fitting of Hertzian

results, Houpert [3] has given average values of exponentγ for different ranges ofB/A. Exponent values are 0.6552, 0.6065 and 0.5603

respectively for1 < B/A≤ 8.74, 8.74< B/A≤ 122.44 and122.44< B/A < 13,576. Consequentlyγ can be expressed as a function of

B/A. Figure 2 presents the variation of the exponentγ in terms ofB/A.

It would be convenient to find a model valid even ifA andB are interchanged. In mathematical terms, we should havea/b =(B/A)γ(B/A) andb/a = (A/B)γ(A/B). A variable changeX = log10(B/A) and the use of an even function inX for γ allow us to reach this

goal. Theγ function, we have chosen, is the following:

γ = 2/3.

(1+µ1X2 +µ2X4 +µ3X6 +µ4X8

1+µ5X2 +µ6X4 +µ7X6 +µ8X8

)(18)

whereµ1 to µ6 are constant. This polynomials’ ratio varies between2/3 and2µ4/(3µ8) for high values ofB/A. The polynomial’s order

has been limited to8 to simplify expression.

The function is fitted to exact values (for1≤ B/A < 1010) using the functionlsqnonlinof Matlaband takingµ1 to µ8 as optimization

parameters. The optimized coefficients are given in Tab. 2.

Figure 3 presents the relative error on ellipticity parameterκ made thanks to the model (18) with the above mentioned coefficients.

Maximal relative error is5.86 ppmfor B/A = 278. Mean square relative error is2.66 ppm. As presented on Fig. 4, the accuracy

has been highly improved according to other approximate models in literature. Houpert’s model had only been dedicated toB/A values

up to 13,576.


Table 2. Optimized coefficients for approximate expression of κ (18).

µ1 0.40227436 µ5 0.42678878

µ2 3.7491752×10−2 µ6 4.2605401×10−2

µ3 7.4855761×10−4 µ7 9.0786922×10−4

µ4 2.1667028×10−6 µ8 2.7868927×10−6

10−5

100

105

−6

−4

−2

0

2

4

6

B/A ratio

Err

or o

n E

llipt

icity

Par

amet

er κ

in p

pm

Figure 3. Error on ellipticity parameter κ in terms of B/A made by using (18).

100

102

104

106

108

−5

−4

−3

−2

−1

0

1

2

3

4

5

B/A ratio

Err

or o

n E

llipt

icity

Par

amet

er κ

in %

Our ModelHoupert 2001Brewe & Hamrock 1977Hamrock & Brewe 1983

Figure 4. Error on ellipticity parameter κ in terms of B/A for different models.

4 Application to Hertzian Results

The model is now applied to calculate the dimensionsa, b of the elliptical area of contact given by (7) and (8), the approachδobtained by (9) and the Hertzian contact pressureσ given by (13).


Elliptical integrals are analytically calculated with Eq. (15) and (16) proposed in [8]. Forκ values inferior to1 (i.e. m1 > 1), E and

F are obtained fromE(κ) = 1κ E(1/κ) andF (κ) = κ F (1/κ).

10−5

100

105

−30

−20

−10

0

10

20

30

B/A ratio

Err

or in

ppm

error on aerror on berror on δerror on σ

Figure 5. Errors on Hertzian theory results in terms of B/A.

Figure 5 presents the error of the model in calculating the dimensions ofa andb the deformationδ and the Hertzian contact stress

σ. Relative errors do not exceed30 ppmup to108. Maximal relative errors occur forB/A = 31.6. Thus we can note that all values of

Hertzian theory are very well evaluated.

5 Conclusion

Hertzian theory on contact of elastic solids has been summed up to detail notations and equations commonly used to solve contact

problems. To obtain an approximate analytical model of Hertzian theory, complete elliptic integrals are replaced by already existing

polynomial approximations. The numerical solution of the implicit equation commonly used to obtain the ellipticity parameterκ in

terms of the geometrical parameterFρ has been replaced by an explicit function ofB/A. Coefficients have then been used as optimization

parameters to fit 2,000 values ofκ that have been numerically calculated on the range1≤ κ≤ 108. The analytical model proposed in this

paper gives the exactκ value with a mean square relative error of2.66 ppm(i.e. 2.6610−4%) for 10−8≤ κ≤ 108. Error does not exceed

5.86 ppm in this range. The application of our model to Hertzian theory calculation shows that numerically calculated exact results

can be approximated with less than30 ppmerror for10−8 ≤ κ ≤ 108. This model can be used instead of numerical tables provided in

handbooks. When Hertzian theory remains precise enough for technical applications to calculate maximal contact stress, the proposed

solution will be helpful to reduce calculation time in a design optimization process. Mathematical analyses such as differentiation can

be easily performed with the proposed function. This function can also be helpful in optimization processes especially in bearing design

or in static and dynamic problems with bearings.


ACKNOWLEDGMENT

The authors would like to thank M. Droineau-Eyl for his help concerning the English language, and the reviewer for their construc-

tive remarks.

NOMENCLATURE

Capital LettersA, B = Relative principal curvature of the contact.E∗ = Equivalent modulus of elasticity.E = Modulus of elasticity of a material.Fρ = Geometrical parameter (so called curvature

difference [1]).F , E = Complete elliptic integral of first and sec-

ond kinds.Q = Normal force between contacting bodies.R = Radius of curvature.

Lowercase Lettersa, b = Semi-axes of the projected contact ellipse.a∗, b∗ = Dimensionless semimajor and semiminor

axes of contact ellipse.k = Local contact stiffness.ppm = Error unit (part per million):

1 ppm =10−4%.

Greek Lettersα, β = Coefficients of polynomial approximations

of F andE .δ = Deformation.δ∗ = Dimensionless contact deformation.κ = Ellipticity parameter (κ = a/b).ρ = Curvature (= 1/R).µ = Optimized coefficients of the proposed an-

alytical solution.ν = Poisson’s ratio.σ = Maximal contact stress (at geometric cen-

ter of contact ellipse).ω = Angle between planes of relative principal

curvatures of both bodies.

SubscriptsI , II refers to first and second contact bodies or

materials1, 2 refers to first and second principal planes

REFERENCES

[1] Harris, T., 2001.Rolling Bearing Analysis, fourth ed. Wiley, New York. Chap. 5.

[2] Liao, N., and Lin, J., 2001. “A New Method for the Analysis of Deformation and Load in a Ball Bearing with Variable Contact

Angle”. J. Mechanical.Design,123[June], pp. 1–9.


[3] Houpert, L., 2001. “An Engineering Approach to Hertzian Contact Elasticity - PartI and II ”. ASME J. Tribology.,123 [July],

pp. 582–588, 589–594.

[4] Hertz, H., 1881. “The Contact of Elastic Solids”.J. Reine und Angewandte Mathematik,92 , pp. 156–171.

[5] Timoshenko, S., and Goodier, J., 1970.Theory of Elasticity, third ed. McGraw-Hill, New York.

[6] Young, W., 1989.Roark’s Formulas of Stress and Strain, sixth ed. McGraw-Hill, New York.

[7] Johnson, K., 1985.Contact Mechanics. Cambridge University Press, Cambridge.

[8] Abramovitz, M., and Stegun, I.A., 1972.Handbook of Mathematical Functions, 9th ed. Dover Publication. Chap. 17.

[9] Tanaka, N., 2001. “A New Calculation Method of Hertz Elliptical Contact Pressure”.ASME J. Tribology,123, pp. 887–889.

[10] Deeg, E., 1992. “New Algorithms for calculating Hertzian Stresses, Deformations and Contact Zone Parameters”.AMP

J.Technology,2 [November], pp. 14–24.

[11] Brewe, D., and Hamrock, B., 1977. “Simplified Solution for Elliptical Contact Deformation between two Elastic Solids”.ASME

J. Lub. Technol.,99 , pp. 485–487.

[12] Greenwood, J., 1997. “Analysis of Elliptical Hertzian Contacts”. Tribology International,30 (3) , pp. 235–237.

[13] Adams, G., and Nosonovsky, M., 2000. “Contact Modelling – Forces”.Tribology International,33 , pp. 431–442.

[14] Hastings, C., 1955.Approximations for Digital Computers, fourth ed. Princeton University Press, Princeton, NJ.

[15] Hamrock, B., and Brewe, D., 1983. “Simplified Equation for Stresses and Deformations”.ASME J. Lub. Technol.,105 , pp. 171–

177.

List of Figures

1 Contact problem definition: bodies I and II in contact, planes of principal curvatures, geometrical parameters and applied

Force Q [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Variations of the ellipticity parameterκ = (B/A)γ and of the exponentγ in terms ofB/A ratio. . . . . . . . . . . . . . . 6

3 Error on ellipticity parameterκ in terms ofB/A made by using (18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Error on ellipticity parameterκ in terms ofB/A for different models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Errors on Hertzian theory results in terms ofB/A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8


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Approximate Analytical Model for Hertzian Elliptical Contact Problems