Presion de Hertz

COMPUTATIONAL MECHANICSNew Trends and Applications

S. Idelsohn, E. Oñate and E. Dvorkin (Eds.)CIMNE, Barcelona,Spain, 1998

1

AN ORTHOGONAL UNIFORM/PARABOLIC SURFACE PRESSUREHALF-SPACE ELEMENT FOR THE ANALYSIS OF NON-HERTZIAN

CONTACT

Jan M. de Mul*, Joost J. Kalker** and Joop M. Vree*

*SKF Engineering & Research Centre B.V.Bearing Theory & Testing Function

Postbus 23503430DT Nieuwegein

The Netherlandse-mail: [email protected], web page: http://www.skf.come-mail: [email protected], web page: http://www.skf.com

** Delft University of TechnologyFaculty of Information Technology & Systems

Department of Technical Mathematics & Computer Science Mekelweg 42628CD Delft

The Netherlandse-mail: [email protected]

Key words: parabolic surface pressure, half-space analysis, non-Hertzian contact, finitebodies, experimental comparison

Abstract. An orthogonal uniform/parabolic surface pressure half-space element is derivedfor the analysis of non-Hertzian contact. Restriction is made to parabolic variation of thesurface gap in the contact in one orthogonal direction. The surface pressure is integratedanalytically over the surface element to obtain influence co-efficeints, e.g. for the normalsurface displacement. The co-efficients are assembled into matrices leading to a linear systemof equations. After solving the parabolic pressure variation is transformed to an equivalentsemi-elliptic one co-inciding with Hertzian line contact. Adjustments are introduced toapproximate the finite size of the bodies in contact. The theoretical predictions are comparedto experiments involving the non-Hertzian contact between cylinders of finite lengths.

Jan M. de Mul, Joost J. Kalker and Joop M. Vree

2

1 INTRODUCTION

In the first meaningful analysis of the contact problem for parabolic surface separation byHertz1, the contacting bodies are essentially approximated by elastic half spaces. As aconsequence, the contact compression becomes infinite in the case of line contact. It can bemade finite by involving the dimensions of the bodies, but this is outside the scope of Hertz'analysis.

Taking the contact surface pressure as uniform over a finite length in one direction andsemi-elliptical in the other, perpendicular direction, Lundberg2 achieves a finite surfacedisplacement for a half space in line contact. He also derives the initial surface separationrequired to obtain the assumed pressure distribution under load.

Plane strain analysis of a cylinder with circular cross-section under diametrically opposedloads also yields a finite contact surface displacement, as is shown by Lundberg3, Schwartzand Harper4, and Sternberg and Turteltaub5.

For general non-Hertzian contact analysis Boussinesq's results for a half space loaded by aconcentrated force on its surface are most popular. By taking some surface element with asuitable pressure variation, the contact area and pressure mountain can be modelled. Manyresearchers like Fridman and Chernina6, Conry and Seireg7, Kalker and Van Randen8, Singhand Paul9, Oh and Trachman10, Paul and Hashemi11, Nayak and Johnson12, Hartnett13 andAhmadi et al.14 proceed in this way. A special class in this category are constituted by themulti-grid methods e.g. by Venner15. These are particularly computation efficient for largeproblems and even allow the inclusion of a lubricant fluid in the contact.

Basically using the half space approach, Reusner16 introduced adjustments to approximatethe influence of the finite dimensions of the contacting bodies. A complicated but exactquarter-space method has been developed by Keer et al.17 Chiu and Hartnett18 have publisheda technique for (almost) straight circular cylindrical bodies that is based on known solutionsfor an infinitely long straight circular cylinder. Reusner's relatively simple adjustments for thefinite dimensions of the contacting bodies have been adopted by De Mul et al.19 in essentiallya half space method based on a rectangular surface element with uniform pressure. In thepresent paper the pressure on the rectangular surface element is uniform in only one directionwhile in the other perpendicular direction it is assumed parabolic. The analysis is intended forbodies with parabolic surface separation in at least one direction and is compared toexperiments. For the nomenclature used throughout this paper see section 12.

2 THE HALF SPACE CONTACT MODEL

Two elastic bodies, a and b, are pressed together in the absence of friction. In the region ofthe contact of the bodies a plane (z = 0) can be defined that is so close to the surface of thebodies that the surface displacement of both bodies may be referred to it in the linear theory ofelasticity. A Cartesian co-ordinate system is defined with axes x and y in the mentioned planeand z-axis pointing inward to body a, see Fig. 1. In the special case that the bodies are bodiesof revolution about parallel axes, the y-axis lies in the plane defined by the axes.


3

b

z

b

a

w(

ay

aw

y)

bx

Fig. 1 Contact co-ordinate system, geometry and displacements

In planes of constant y each body has a constant radius of curvature, convex is countedpositive, variation with y is allowed. The centres of the radii of curvature lie in the plane(x = 0). In this plane the bodies have a surface separation h(0,y) = h(y) in the undisplaced statewhich needs not be symmetric about y = 0. With a normal contact displacement � along z anda contact tilt angle � about x (both with respect to body a), the resulting interference at(x = 0, y, z = 0) is given by :

δ(y) = δ + γ y − h(y) (1)

A combined radius of curvature is defined by:

1θ(y) = 1

θa(y) + 1θb(y)

(2)

The undeformed surface gap between the bodies g(x,y) is approximated by parabolicvariation with x as follows :

g(x, y) = x2

2θ(y) − δ(y)

(3)

Since interference is physically not admissible a contact pressure p(x, y) is generated in thecontact area A that causes displacements wa and wb along z, into each body. The sumdisplacement w is given by:


4

w(x,y) = wa(x,y) + wb(x,y) (4)

The deformed surface distance then becomes

D(x,y) = g(x,y) + w(x,y) (5)

The contact/no contact conditions are given by,

contact if (x,y) inside A :p(x,y) ≥ 0 and D(x,y) = 0 (6)

no contact if (x,y) outside A :p(x,y) = 0 and D(x,y) > 0 (7)

The contact load and tilt moment (on body a) follow from :

Q = ∫∫ p(x,y)dxdy (8)

T = °∫∫ p(x,y) ydxdy (9)

Reusner16 uses a surface element with uniform pressure along y and semi-ellipticalvariation over x, with semi-axes p(0,y) and b(0,y) :

p(x,y) = p(0,y) 1 − x2

b(0,y)2

(10)

To obtain the two unknowns p(0,y) and b(0,y) he introduces two conditions. One isinvolving the deformed surface distance D(0,y) at the contact centre line, the other the sum ofcurvatures 1/�(y). In this paper a similar approach is taken. Since the semi-elliptical pressuredistribution cannot be integrated analytically in the expression for the influence co-efficient ofthe element, the semi-ellipse is approximated by a parabola, see Fig. 2. With reduced summitpressure s(y) and contact semi-width c(y) it reads :

p(x,y) = s(y) c(y)2 − x2 (11)


5

i2

i

ci

s c

ciiy

Fig. 2 Uniform/parabolic surface pressure mountain

The deformed surface distance at the contact centre line becomes :

D(0,y) = g(0,y) + w(0,y) (12)

D(x, y) is developed into a Taylor series around x = 0 :

D(x,y) = D(0,y) + x∂D∂x (0,y) + x2

2∂2D∂x2 (0,y) + x3

6∂3D∂x3 (0,y) + .... (13)

Terms of order higher than 3 are neglected. Inside the contact area A it is requiredD(x,y) = 0. In Eq. (13) the uneven terms ∂D/∂x and ∂3D/∂x3 vanish due to the symmetry in theloading, yielding the following requirements from Eqs (3), (5), (6), (12) and (13) :

D(0,y) = g(0,y) + w(0,y) = 0 (14)

∂2D∂x2 (0,y) = 1ρ(y) + ∂2w

∂x2 (0,y) = 0 (15)

With Eqs. (11), (14) and (15) the contact condition Eq. (6) is approximately satisfied.The total contact load is found by integration over x and summing over the elements i, see

Fig. 2 :

Q = ∑

i

43 sici

3∆yi

(16)

Analogously for the total contact moment :

T = ∑

i

43 sici

3yi ∆yi

(17)


6

Boussinesq's results for a half space loaded by a concentrated force are used to find thesurface displacement w :

∫∫−+−

−=22 )'()'(

'')''(

2

1),(

yyxx

dydxyxp

G

vyxw

π

(18)

With Eq. (11) it is found for the displacement in (x,y) due to the uniform/parabolicpressure on rectangle j :

ba

babaj G

vyxw

,

,,

2

1),(

π−

= ∫ ∫−

∆+

∆−

−+−

−j

j

jj

jj

c

c

yy

yy

jj

yyxx

dyxcdxs

2

2

22

22

)'()'(

')'(' (19)

Performing variable transformation and partial integration the following transpires :

ba

babaj G

vyxw

,

,,

2

1),(

π−

=

−−−

++−+

+

−−−

−

−

22

12

22

12

22

''')6

'(

'

')

6

'(

'

''

3

'')(

yxyx

x

y

xsinh

yxc

x

ysinhx

xxxxc

j

j

(20)

with :

xcw j −±=±' (21)

2' j

i

yyyy

∆±−=±

(22)

Defining the compliance co-efficient :

wja,b(0,yi) = Cij

a,bsj (23)It follows from Eq. (20) that :

+

−

−−

+−

+

++

−=

'

'

22

12

213

,

,,

''3

''

3

'2

'

3

4

2

1

y

yj

j

jj

jj

ba

babaij

ycyc

y

csinhy

yc

c

ysinhc

G

vc

π (24)


7

The derivative with respect to cj is required also :

+

−=

∂∂

= −−

''4

'4

2

1 112

,

,,

,

y

csinhyc

c

ysinhc

G

v

c

CC j

jj

jba

ba

j

baijba

dij π (25)

Combining bodies a and b :Cij = Cij

a + Cijb

(26)

Cdij =∂Cij

∂cj= Cdij

a + Cdijb

(27)

From Eqs. (4), (23) and (26) :wj(0,yi) = Cij sj (28)

Adding the influence of all rectangles j :

w(0,yi) = ∑

jCij sj

(29)

Substituting this and Eq. (3) in Eq. (14) and re-arranging, one contact equation is found :

∑j

Cij sj = δ(yi)

(30)

For the other contact equation the second devirative of the surface displacement is needed.Defining :

−∂ 2wj

a,b

∂x2 (0,yi) = Pija,bsj

(31)

it is found from Eq. (19) :

ba

babaij G

vP

,

,, )1(2

π−

=+

−

−

'

'

1

''

y

y

j

y

csinhy

(32)

The derivative with respect to cj will again be needed :

ba

babaij G

vP

,

,, )1(2

π−

=

+

−

+

'

'

22 '

'y

yj yc

y (33)


8

Combining bodies a and b :Pij = Pij

a + Pijb

(34)

Pdij =∂Pij

∂cj= Pdij

a + Pdijb

(35)

From Eqs. (4), (31) and (34) :

−∂ 2wj

∂x2 (0,yi) = Pij sj (36)

Adding the influence of all rectangles j :

−∂ 2wj

∂x2 (0,yi) = ∑j

Pij sj (37)

Substituting this in Eq. (15) the other contact equation is found :

∑j

Pij sj = 1ρ(yi) (38)

Using matrix [M]=[Mij] and vector {v}={vj} notation Eqs. (30) and (38) read :

[C] s = δ (39)

[P] s = { 1ρ}

(40)

Eqs.(39) and (40) may be solved iteratively by linearising with respect to variations in thecontact semi-width cj. Thus :

[ ] [ ][ ]{ } { }δ=∆+ scCC jdij (41)

[ ] [ ][ ]{ }

=∆+ρ1

scPP jdij (42)

From step k to k + 1 in the iteration :


9

{ } [ ] [ ][ ]

∆+= −+ ρ

111 jdijk cPPs

[ ] [ ][ ] [ ]

∆+=−−−

ρ1

1111 PcPP jdij

[ ] [ ][ ][ ]

∆−≈ −−

ρ1

.1 11 PcPP jdij (43)

From Eq. (40) :

[P] −1{ 1ρ} = s k (44)

Into Eq. (43) :

s k+1 λ [1 − [P] −1[Pdij ∆cj]] s k (45)

Substitution into Eq. (39) :

[[C + [Cdij ∆cj]] [1 − [P] −1[Pdij∆cj]] s k≈

≈ [[C] + [Cdij∆cj] − [C][P] −1[Pdij∆cj]] s k≈ [[C] + [Cdij∆cj] − [CP−1Pdij∆cj]] s k = δ (46)

[[Cdij∆cj] − [CP−1Pdij∆cj]] s k = δ − [C] s k (47)alternative :

[[Cdijsj] − [CP−1Pdijsj]] ∆c = δ (48)

With this the basic system of linear equations that needs to be solved for the contact hasbeen established. Once the contact pressures and semi-widths have been found for a particularvalue of approach δ and tilt angle γ between the bodies, the total contact load Q and tiltmoment T are calculated from Eqs. (16) and (17). If the difference from the desired load andmoment values is too large δ and γ are changed and the iteration is continued until thedifference drops below the limit set.

This process is carried out using the Newton-Raphson method, making use of thederivatives of the load and moment with respect to the approach and tilt angle, which can beestablished analytically. This is not further explained since it is not essential for solving thecontact problem.

3 MODELLING THE LENGTH OF THE BODIES

Reusner16 developed adjustments of the influence co-efficient to approximate the effect ofthe finite length of the bodies along the contact axis (y). It is based on mirroring the pressuremountain about the body side faces to cancel the subsurface shear stresses on these side faces.This is explained in detail by De Mul et al.19 and therefore not repeated here. All "influence"matrices [C], [Cd] (defined by Eq.(27)), [P] and [Pd] (defined by Eq.(33)) are affected.


10

4 MODELLING THE DEPTH OF THE BODIES

Also for this Reusner16 developed adjustments of the influence co-efficient to approximatethe effect of the finite depth of the bodies. It is based on the displacement difference betweenthe surface and the location at a certain depth below the surface. This has been adapted andexplained in detail by De Mul et al.19 and is therefore repeated here only for those parts thatare changed. Defining :

−+

+= −

)(

')1(2

)('

''

,

1

2,

2, yd

ysinhv

ydy

yY

baba

ba (49)

Eq. (24) of De Mul et al.19 reads while dropping here the redundant x argument anddefining qj = fxj :

wa,b(y,da,b(y)) ≈ 14π $Ga,b

∑j

qjYa,b ′

(50)

With : qj = 4

3 sjcj3

(51)

Eq. (50) becomes :wa,b(y,da,b(y)) ≈ ∑

j

cj3

3π Ga,bYa,b

′ sj

(52)

The adjusted displacement difference then becomes :

wa,bd (0,y) = wa,b(0,y, 0) − wa,b(y,da,b(y)) (53)

The effect is taken into account by adapting the "influence" co-efficients in Eqs. (24) and(25) :

∆ Cija,b = −

cj3

3π Ga,bYa,b

′ (54)

∆Cdija,b = −

cj2

π Ga,bYa,b

′ (55)

If | yj- y | << da,b(y) the equations above can be approximated to :


11

∆Cija,b = − 3 − 2νa,b

3µGa,b da,b(yi) cj3∆yj (56)

∆Cdija,b = − 3 − 2 a,b

π Ga,b da,b(yi) cj2∆yj (57)

So far mirroring has not been considered. When this is done the additional terms comingfrom the mirroring each cause their own adjustment, see De Mul et al.19 Eq. (32).

Since the elastic displacement at depth d(y) does not depend on x [P] and [Pd] are notaffected.

5 COMPARISON WITH HERTZIAN ANALYSIS

The uniform/parabolic pressure element is first compared to Hertzian line contact. Forequal elastic constants Eq. (32) becomes :

−= −

''

)1(4 1

y

csinhy

G

vP j

ij π (58)

i = j is set and the limit of the element length ∆yj is taken. Using l'Hospital's rule it is found:

iiiy

cG

vPlim

iπ

)1(8 −=∞→∆

(59)

Using Eqs. (10), (16) and (38), De Mul and Van Engelenburg20 Eqs. (1) - (5), and :

G = E

2(1 + ν)(60)

it is found that the Hertzian semi-width and maximum pressure are given by :

b = 2

3 c = 0.81650c

(61)

p0 = 4 63π s c2 = 1.03960s c2

(62)

This compares with p(0,0) = sc2 from Eq.(11).


12

So the parabola underestimates the maximum contact pressure and overestimates thecontact width. The factors shown in Eqs. (61) and (62) can be used to modify the parabolicvalues to achieve the Hertzian values. In the case of rolling bearing fatigue life analysis this isimportant since the fatigue life depends on the 9.3rd power of the subsurface shear stressamplitude, which depends linearly on the maximum elliptic pressure, see De Mul at al.21. Inthe results shown in the present paper the modification of the parabolic values to the semi-elliptic values is made everywhere.

Modification factors having been established comparison may now be made to an ellipticalHertzian point contact example from De Mul et al.21, see Fig. 3. The Hertzian ellipticalpressure and half-width distribution (symmetric quarter shown only) is compared with thecomputed distributions. The correspondence between the present model and the Hertziananalysis is excellent.

0

0.2

0.4

0.6

0.8

1

1.2

y/a

p(y)

/p0,

b(y

)/b

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Hertz

model p-ratio

model b-ratio

Fig. 3 Elliptical contact pressure and semi-width variation - model versus Hertz

6 COMPARISON WITH PLANE STRAIN CIRCULAR CYLINDERS

The approach taken by De Mul and Van Engelenburg20 to fit the depth adjustment isfollowed here. Their Eq. (6) gives the contact centre displacement of a cylinder in plane strain.Analysing two long cylinders with the present model plane strain can be simulated.Comparing the results with the equation mentioned above gave a depth adjustment equal tothe diameter of the cylinder considered to yield excellent agreement for different curvaturesums 1/�. Apart from this simple proportionality to the diameter the depth was found to beunaffected by the load level.


13

Using the same procedure De Mul and Van Engelenburg20 had found a different depthadjustment, namely 3/4 diameter of the cylinder considered. This is attributed to the differentpressure element, i.e. uniform instead of uniform/parabolic

In the next section the uniform/parabolic model with the depth adjustment of the diameteris applied to the contact between rollers or needles and bearing rings.

7 COMPARISON WITH EXPERIMENTS

The B-TAN-2 roller and CON needle experimental cases are taken from De Mul and VanEngelenburg20. The (measured) roller and needle profiles are shown in Fig. 4 and 5. The ringshad straight profiles.

Fig. 4 B-TAN-2 roller profile

Fig. 5 CON needle roller profile trace


14

In Fig. 6 the measured and calculated footprints of the contact are shown in overlay, whilethe latter is also shown separately. Excellent agreement is observed.

Fig. 6 Footprint of B-TAN-2 profiled roller on (plugged) inner ring at 20 kN load - comp. vs. exp.

In Fig. 7 the contact pressure and semi-width distributions over the contact centre line areshown. Fig. 6 and 7 can be seen to agree well with the symmetric quarter of the pressuremountain shown in Fig. 9 of De Mul and Van Engelenburg20, calculated with a uniformpressure rectangular surface element.

-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-2

0

2

4

y [mm]

-b(

y) [m

m/1

0]

p(0

,y)

[GP

a]

pressure

semi width

Fig. 7 Variation of contact presssure and semi-width along contact centre line of B-TAN-2 profiled roller on(plugged) inner ring at 20 kN load - computation


15

In Fig. 8 and 9 the measured and calculated load-compression relationships for B-TAN-2and CON are shown. With the depth adjustment of the diameter good agreement betweenexperiments and calculations is obtained for B-TAN-2 on the stiff side while for CONreasonable correspondence is achieved on the weak side.

Fig. 8 Computed and experimental load vs. compression for B-TAN-2 profiled roller on (plugged) inner ring

Fig. 9 Computed and experimental load vs. compression for CON profiled needle roller on solid cylinder


16

8 MODEL PERFORMANCE CHARACTERISTICS

Compared to models using a uniform/uniform pressure element e.g. De Mul et al.19, theuniform/parabolic element has the advantage that the contact semi-width b(y) does not need tobe known a priori as it is a result of the calculation. In practice this reduces the modellingeffort significantly. Also the lower number of elements increases the computation efficiencywith calculation time according to O(n2), n being the number of elements. Since the elementitself is more complex the theoretical maximum benefit is not achieved. Compared to De Mulet al.19 an improvement by a factor of about 2 in CPU time is obtained. While the contactingbodies in this model may have arbitrary profile in the y-direction (axial), in the x-direction(circumferential) they must be parabolic, consistent with their use as an appriximation tocircular bodies. The model is thus particularly applicable to roller/raceway profilinginvestigations, where the number of elements required is moderate.

A class of methods using a uniform/uniform pressure element that are particularlycomputation efficient for large problems is the multi-grid technique e.g. by Venner15. Thebasis for this advantage is that the calculation effort increases with O(nlogn) rather than O(n2).The multi-grid models are typically used for analysing 3-D surface features/topography, EHLor both and require very powerful PCs or work stations. Compared with multi-grid methods asso far implemented, the present uniform/parabolic element approach enjoys severaladvantages. One is its ability to correct for the finite size of the contacting bodies, shown hereto be of some considerable importance. Another is the ease with which local grid refinementcan be introduced, since no restriction is placed on the element aspect ratio. The latter effectsboost calculation efficiency. In addition to this, if the problem is not large the method benefitsfrom being simpler and therefore having less overhead than multi-grid and shows highercomputation efficiency. Both the present uniform/parabolic element method and theuinform/uniform element method programs are running on a VAX 4100A computer andrequire less than 1 minute CPU time for a typical profile calculation.

9 CONCLUSIONS

An orthogonal uniform/parabolic surface pressure half-space element for the analysis ofnon-Hertzian contact has been developed. The major advantage of the element is that theparabola allows analytic integration of the relevant influence co-efficients and lowers thenumber of elements compared to the frequently used orthogonal uniform surface pressure halfspace element while the contact semi-width is even an explicit part of the solution. Forsubsequent analysis involving the contact pressure, such as the particularly sensitivesubsurface-stress-based rolling bearing fatigue life, the parabolic pressure variation is scaledto the more realistic semi-elliptic distribution. Comparison to Hertzian contact then showsexcellent agreement.

In addition relatively simple adjustments are introduced in the model to approximate thefinite dimensions of the contacting bodies, making it particularly applicable to the finitecylinders occurring in roller bearings. Comparison of the model with experiments shows, forthe most part, good agreement.


17

10 ACKNOWLEDGEMENT

The authors wish to thank Dr. H.H. Wittmeyer, Managing Director of SKF Engineering &Research Centre B.V., for his kind permission to publish this paper, Robert-Jan Buunk forassisting with the computer program runs and preparation of the pictures, and HugoWilleumier for contributing to the conversion to MS Word.

11 REFERENCES

[1] H. Hertz, Über die Berührung fester elastischer Körper, 1881, Über die Berührungfester elastischer Körper und über die Härte, 1882, Gesammelte Werke, pp. 155-196.

[2] G. Lundberg, Elastische Berührung zweier Halbräume, Forschung auf dem Gebietedes Ingenieurwesens, Sep./Oct. 1939, pp. 201-211.

[3] G. Lundberg, Cylinder Compressed between Two Plane Bodies, SKF, Göteborg,Sweden, 1949.

[4] J. Schwartz and E.Y. Harper, On the Relative Approach of Two-Dimensional Bodiesin Contact, International Journal of Solids and Structures, Vol. 7, 1971,pp. 1613-1626.

[5] E. Sternberg and M.J. Turteltaub, Compression of an Elastic Roller Between Two Rigid Plates, Continuum Mechanics and Related Problems of Analysis,Muskhelishvili Anniversary Volume, pp. 495-515, Nauka Publishing House, Moscow,1972, L.I. Sedov et al., Eds.

[6] V.M. Fridman and V.W. Chernina, Iteration Methods Applied to the Solution ofContact Problems in Bodies, (in Russian) Mekh. Tverdogo Tela AN SSSR, Vol. 1(1967), pp. 116-120.

[7] T.F. Conry and A. Seireg, A Mathematical Programming Method for Design ofElastic Bodies in Contact, ASME Journal of Applied Mechanics, Vol. 38 (1971),pp. 387-392.

[8] J.J. Kalker and Y. van Randen, A Minimum Principle for Frictionless Contact withApplication to Non-Hertzian Half Space Contact Problems, Journal of EngineeringMathematics, Vol. 6, No. 2 (Apr. 1972), pp. 193-206.

[9] K.P. Singh and B. Paul, Numerical Solution of Non-Hertzian Elastic Contact Problems, ASME Journal of Applied Mechanics, June 1974, pp.484-490.

[10] K.P. Oh and E.G. Trachman, A Numerical Procedure for Designing Profiled RollingElements, ASME Journal of Lubrication Technology, Oct. 1976, pp. 547-552, 574.

[11] B. Paul and J. Hashemi, An Improved Numerical Method for Counterformal Contact Stress Problems, Report No. FRA-ORD-78/26, US Department of Transportation,Federal Railroad Administration.

[12] L. Nayak and K.L Johnson, Pressure Between Elastic bodies Having a Slender Areaof Contact and Arbitrary Profiles, International Journal of Mechanical Science, Vol.21 (1979), pp. 237-247.

[13] M.J. Hartnett, A General Numerical Solution for Elastic Body Contact Problems,Solid Contact and Lubrication, AMD-Vol. 39, ASME 1980, Eds. H.S. Cheng andL.M. Keer.


18

[14] N. Ahmadi, L.M. Keer and T. Mura, Non-Hertzian Contact Stress Analysis for anElastic Half Space - Normal and Sliding Contact, International Journal of Solids andStructures, Vol. 19, No. 4, pp. 357-373.

[15] C.H. Venner, Multilevel Solution of the EHL Line and Point Contact Problems, Ph.D.Thesis, Feb. 1991, Twente University of Technology, Enschede, the Netherlands.

[16] H. Reusner, Druckflächenbelastung und Oberflächen-verschiebung im Wälzkontakt von Rotationskörpern, Dissertation, May 1977, University of Karlsruhe, Germany.

[17] L.M. Keer, J.C. Lee and T. Mura, A Contact Problem for the Elastic Quarter Space,International Journal of Solids and Structures, Vol. 20 (1984), No. 5, pp. 513-524.

[18] Y.P. Chiu and M.J. Hartnett, A Numerical Solution for the Contact Problem InvolvingBodies with Cylindrical Surface Considering Cylinder Effect, ASME Journal ofTribology, Vol. 109 (July 1987), pp. 479-486.

[19] J.M. de Mul, J.J. Kalker and B. Fredriksson, The Contact Between Arbitrarily CurvedBodies of Finite Dimensions, ASME Journal of Tribology, Vol. 108 (Jan. 1986), pp.140-148.

[20] J.M de Mul and H. van Engelenburg, The Line Contact Between Two Cylinders –Some Experiments and Theoretical Predictions, ASME Journal of Tribology,Vol. 110 (Apr. 1988), pp. 285-291.

[21] J.M. de Mul, J. M. Vree and J.C. Kuijpers, The Influence of Certain Raceway DentGeometries (3-D) on Contact Stresses and Rating Fatigue Life of Rolling Bearings,ASME Journal of Tribology, Vol. 109 (July 1987), pp. 452-461

12 NOMENCLATURE

12.1 Parameters

a Hertzian contact semi-length [L] F = force unit, L = length unitb Hertzian contact semi-width [L]

contact element semi-width (semi-elliptic pressure) [L]c contact element semi-width (semi-parabolic pressure) [L]C compliance co-efficient [L5/F]d depth below surface [L]D deformed surface distance [L]E elastic modulus [F/L2]g undeformed surface gap [L]G shear modulus [F/L2]


19

h undisplaced surface separation [L]p contact pressure [F/L2]p0 maximum Hertzian contact pressure [F/L2]P partial derivative (influence) co-efficient [L3/F]q load intensity [F/L]Q contact load [F]s parabolic summit pressure [F/L4]T contact moment (about x) [FL]w surface displacement [L]x,y,z right hand Cartesian co-ordinate axes [L]

',' yx integration variables (along x and y) [L]

±'x xcx j −±=±' [ ]L

±'y2

' jj

yyyy

∆±−=±

[L]

γ contact tilt angle (about x) [radian]δ contact normal approach [L]∆c increase of c [L]∆y element length [L]ν Poisson's ratioρ radius of curvature [L]

12.2 Subscripts

a,b bodyd derivative with respect to cj

i point consideredj location of pressure elementk iteration step

12.3 Superscripts

a,b bodyd depth adjusted

Presion de Hertz

Documents

Transcript of Presion de Hertz