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Analysis of elliptical Hertzian contacts
Tribologp
SCIENCE:
PII:
Analysis
Vol. 30, No. 3, pp. 235-237, 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. Ail rights reserved 0301~679X/96/$17.00 +O.OO
Internntiond
SO301-679X(96)00051-5
of elliptical
Hertzian
J. A. Greenwood
The paper compares three approximate methods of finding the area of contact, the contact pressure and the deformation in elliptical Hertzian contacts. Copyright 0 1996 Elsevier Science Ltd Keywords: deformation
Hertzian
contact,
elliptical
contact,
contact
pressure,
contact
Some years ago the author published a method’ of calculating the maximum contact pressure and the approas;h for elliptical Hertzian contacts by using the simple circular contact relations together with an effective radius R,. Experience over the years, by the author and ot!lers, has shown the method to be very satisfactory, but has also suggested the following as the most conver ient procedure.
This method works well for mildly elliptical contacts when answers of engineering accuracy are acceptable. For example, for B/A = 5, the area is 0.2% too large, and the Hertz pressure 0.2% too small.
For circular contacts, the contact area nc2 and maximum contact pressure p0 are given by:
(3)
=-
3W 2m2
(1)
1 :I - v: 1 - v$ - := E” E, + E, To determine the effective radius R, the relative principal curvatures of the contact (A,B) are first found as described by Timoshenko and Goodier’ or Johnson3: when the principal axes of the separate bodies are parallel, the values are simply A = l/R, + l/R,; B = l/R’, + l/R’Z. (Note that the authors quoted use twice l:he relative curvature: in the present notation a 10 mm radius ball on a 10 mm radius cylinder will have 1:~= 0.1 mm-‘, B = 0.2 mm-‘). Then the effective radius is defined by: )]-“3
The semi-axes are readily found from their product and ratio. If the ‘approach’ (Y is needed, as well as the Hertz pressure, the following method is more convenient (though less accurate) than the one previously suggested’. Approximate values for the semi-axes are found as above: then the exact equation:
where
R, = [A.B.r$
If the individual semi-axes a,b of the ellipse are needed rather than just their product (ab = c2), their ratio may be found using the simple asymptotic equation:
ca
CR,= 8.736 mm for the case given). It is convenient to remember that the effective curvature, the reciprocal of R,, is a compromise between the arithmetic mean of the relative curvatures, (A + B)/2, and their geometric -mean dAB.
cx= ; (Au’ + Bb’) is used to find (Y.*
An alternative method is to use the exact equations for the semi-axes:
(-3 where l/R =A + B and F is an elliptic integral; approximate equations are used to obtain the values for the axis-ratio k = a/b and the elliptic integral. Two versions of this method exist: Brewe and Hamrock use
*Equations
(3)
cy = ; c”(AB)‘!3[B’/3
Tribology
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International
and
(4)
may
be
combined
to
give
+ A U/7]
Volume
30 Number
3 1997
235
Analysis
of elliptical
Hertzian
contacts:
J. A. Greenwood
This has the disadvantage that for almost circular contracts (B = A), the axis-ratio k is 3.4% too high and F is 1.7% too high. (Brewe and Hamrock specifically exclude the circular contact (B = A), but values near BIA = 1 are almost as bad). Subsequently Hamrock and Brewe5 recommended:
giving the correct values for almost-circular contacts. (The two papers also give approximate equations for the elliptic integral of the first kind, K, to enable the exact equation for the approach to be used). The present method upyears to be only a rough approximation, with its introduction of an ‘effective radius’ and no mention of elliptic integrals; the occasion for the present note was encountering an industrial user who had compared the values found using Equations (1) and (2) with the ‘exact’ values found using the Brewe and Hamrock equations, and deduced that the ‘Greenwood’ approximation was, for his case, only 1.5% in error and so acceptable. This is fine: except that for the case considered, (B/A = 2), Equations (1) and (2) give the correct answer: it was the ‘exact’ answer which was in error. Brewe and Hamrock’s method was originally offered as a method of finding the approach 01 only, so their paper gives only the errors in cy: Hamrock and Brewe record only the errors in the methods recommended in that paper. It seems that a direct comparison of the three methods would be valuable, particularly since the publication by Dyson et n16 of a simple computer routine enables the exact answers to be found readily. Figures 1 and 2 show the results. Figure 1 demonstrates that over the range B/A % 25 the error in the maximum contact pressure is less than 2% whichever method is used (except when using Brewe and Hamrock with B/A 5 1.05). Brewe and Hamrock is impressively good for 2.5 5 B/A 5 22; but Greenwood is equally impressive for BIA i 5, and it is not clear which is the more important range. Hamrock and Brewe pays dearly for getting the correct answer at BIA = 1.
Ratio
of principal
curvatures
B/A
Fig 2 Errors in the approach CL The signs against the curves indicate whether the estimate is too large (+) or too small (-)
Figure 2 shows that the errors in the approach (Y behave similarly, but are larger. The present method gives an answer within 2% over the range BIA 5 10, but should not be used much above this. Errors greater than 2% are obtained if Brewe and Hamrock is used for B/A 5 1.7, or if Hamrock and Brewe is used for 1.7 5 B/A 5 7.5. It is interesting, particularly since Brewe and Hamrock was proposed as a method of finding 01, that, over this range, better answers can be found by using their approximation to find the semiaxes n and h, and substituting these into Equation (4): in other words, their approximations for k and 6 are better than their approximation for K: see Fig 2. (Modifying the Hamrock and Brewe method in the same way reduces the error for B/A 5 5, but increases it for larger values).
Recommendations There will be few occasions when a 3% error is not acceptable: so for moderately elliptical contacts there is nothing to choose between the methods in terms of practical accuracy. If the ‘effective radius’ method is chosen on grounds of simplicity, or taste, no penalty is thereby incurred: in particular, for 1 I B/A 5 5 the effective radius method gives the best value for the Hertz pressure (except for B/A = 3 where the Brewe and Hamrock error changes sign and so passesthrough zero). The author’s opinion is that for mildly elliptical contacts, or when only a hand calculator is available, the present method should be used: otherwise, the exact calculation should be performed using the method recommended by Dyson et n16. Example
A 10 mm radius ball runs on an inner race of radius 25 mm in a groove of radius of curvature 16 mm: the load on the ball is 100 N. Both ball and race are of steel with E = 202 kN/mm’, p = 0.3. Ratio
of principal
curvatures
B/A
Fig I Errors ill the maximum Her-t,: pressure pP The signs against the curves indicate whether the estimate is too high (+) or too low (-) 236
Tribology
International
Volume
30 Number
The contact modulus E* is 1 - 1’2 1 - $ -1 = 111.O kN/mm’ E + E I 3 1997
Analysis
In the axial direction: A = ;k - i
of elliptical
Hertzian
contacts:
J. A. Greenwood
b -- CI
= 0.0375 mm-’
(The exact answer 0.4183) (minus because the groove is concave!). In the plane of the race: B=-!X+?Ij=O.l4mm-’
a = 5 [0.0375*0.3193” + 0.14*0.1327’] = 3.144 pm
(in the present method A and B are interchangeable. but conventionally A is the smaller of the two). Then R, = [Cl.14*0.0375*0.08875]p”3 = 12.90 mm “3
= 1.1273 kN/mm’ 1 (The exact answer is p,, = 1.1284 kN/mm’, so the approximate answer is 0.10% low). If the approach or the semi-axes are required, it is more convenient to find p0 from the mean contact radius ‘Crather than directly: we have c =
,&&
=
0.75”0.1”12.90 111.0
1. Greenwood J.A.. Formulas for moderately contacts. ASME J. Tribal. 1985 107 501-503 2. Timoshenko Hilt, New
S and Goodier York, 1951
3. Johnson K.L. Contact Cambridge, 1985
J.N. Theory
Mechanics,
4. Brewe D.E. and Hamrock B.J. Simplified contact deformation between two elastic Technol. 1977, 99 485387
= 0.2058 mm.
elliptical
of Elasticity,
Cambridge
5. Hamrock B.J. and Brewe D.E. Simplified and deformations. ASME .I. Lub. Technol.
1 “3
(exact answer 3.136 pm). References
and 6”O.l”lll.O’
SO that b = 0.2058*~0.4155 = 0.1327 and a = 0.2058/ do.4155 = 0.3 193. Hence:
Universio
Hertzian McGrcwPress.
solution for ellipticalsolids. ASME J. Lub. solution for stresses 1983 105 171-J77
6. Dyson A, Evans H.P. and Snidle R.F. A simple, accurate method for calculation of stresses and deformations in elliptical Hertzian contacts. J. Mech. Eng. Sci. Proc. I. Mech. E pt C 1991 206 139-111
But
Tribology
International
Volume
30 Number
3 1997
237
SCIENCE:
PII:
Analysis
Vol. 30, No. 3, pp. 235-237, 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. Ail rights reserved 0301~679X/96/$17.00 +O.OO
Internntiond
SO301-679X(96)00051-5
of elliptical
Hertzian
J. A. Greenwood
The paper compares three approximate methods of finding the area of contact, the contact pressure and the deformation in elliptical Hertzian contacts. Copyright 0 1996 Elsevier Science Ltd Keywords: deformation
Hertzian
contact,
elliptical
contact,
contact
pressure,
contact
Some years ago the author published a method’ of calculating the maximum contact pressure and the approas;h for elliptical Hertzian contacts by using the simple circular contact relations together with an effective radius R,. Experience over the years, by the author and ot!lers, has shown the method to be very satisfactory, but has also suggested the following as the most conver ient procedure.
This method works well for mildly elliptical contacts when answers of engineering accuracy are acceptable. For example, for B/A = 5, the area is 0.2% too large, and the Hertz pressure 0.2% too small.
For circular contacts, the contact area nc2 and maximum contact pressure p0 are given by:
(3)
=-
3W 2m2
(1)
1 :I - v: 1 - v$ - := E” E, + E, To determine the effective radius R, the relative principal curvatures of the contact (A,B) are first found as described by Timoshenko and Goodier’ or Johnson3: when the principal axes of the separate bodies are parallel, the values are simply A = l/R, + l/R,; B = l/R’, + l/R’Z. (Note that the authors quoted use twice l:he relative curvature: in the present notation a 10 mm radius ball on a 10 mm radius cylinder will have 1:~= 0.1 mm-‘, B = 0.2 mm-‘). Then the effective radius is defined by: )]-“3
The semi-axes are readily found from their product and ratio. If the ‘approach’ (Y is needed, as well as the Hertz pressure, the following method is more convenient (though less accurate) than the one previously suggested’. Approximate values for the semi-axes are found as above: then the exact equation:
where
R, = [A.B.r$
If the individual semi-axes a,b of the ellipse are needed rather than just their product (ab = c2), their ratio may be found using the simple asymptotic equation:
ca
CR,= 8.736 mm for the case given). It is convenient to remember that the effective curvature, the reciprocal of R,, is a compromise between the arithmetic mean of the relative curvatures, (A + B)/2, and their geometric -mean dAB.
cx= ; (Au’ + Bb’) is used to find (Y.*
An alternative method is to use the exact equations for the semi-axes:
(-3 where l/R =A + B and F is an elliptic integral; approximate equations are used to obtain the values for the axis-ratio k = a/b and the elliptic integral. Two versions of this method exist: Brewe and Hamrock use
*Equations
(3)
cy = ; c”(AB)‘!3[B’/3
Tribology
(4)
International
and
(4)
may
be
combined
to
give
+ A U/7]
Volume
30 Number
3 1997
235
Analysis
of elliptical
Hertzian
contacts:
J. A. Greenwood
This has the disadvantage that for almost circular contracts (B = A), the axis-ratio k is 3.4% too high and F is 1.7% too high. (Brewe and Hamrock specifically exclude the circular contact (B = A), but values near BIA = 1 are almost as bad). Subsequently Hamrock and Brewe5 recommended:
giving the correct values for almost-circular contacts. (The two papers also give approximate equations for the elliptic integral of the first kind, K, to enable the exact equation for the approach to be used). The present method upyears to be only a rough approximation, with its introduction of an ‘effective radius’ and no mention of elliptic integrals; the occasion for the present note was encountering an industrial user who had compared the values found using Equations (1) and (2) with the ‘exact’ values found using the Brewe and Hamrock equations, and deduced that the ‘Greenwood’ approximation was, for his case, only 1.5% in error and so acceptable. This is fine: except that for the case considered, (B/A = 2), Equations (1) and (2) give the correct answer: it was the ‘exact’ answer which was in error. Brewe and Hamrock’s method was originally offered as a method of finding the approach 01 only, so their paper gives only the errors in cy: Hamrock and Brewe record only the errors in the methods recommended in that paper. It seems that a direct comparison of the three methods would be valuable, particularly since the publication by Dyson et n16 of a simple computer routine enables the exact answers to be found readily. Figures 1 and 2 show the results. Figure 1 demonstrates that over the range B/A % 25 the error in the maximum contact pressure is less than 2% whichever method is used (except when using Brewe and Hamrock with B/A 5 1.05). Brewe and Hamrock is impressively good for 2.5 5 B/A 5 22; but Greenwood is equally impressive for BIA i 5, and it is not clear which is the more important range. Hamrock and Brewe pays dearly for getting the correct answer at BIA = 1.
Ratio
of principal
curvatures
B/A
Fig 2 Errors in the approach CL The signs against the curves indicate whether the estimate is too large (+) or too small (-)
Figure 2 shows that the errors in the approach (Y behave similarly, but are larger. The present method gives an answer within 2% over the range BIA 5 10, but should not be used much above this. Errors greater than 2% are obtained if Brewe and Hamrock is used for B/A 5 1.7, or if Hamrock and Brewe is used for 1.7 5 B/A 5 7.5. It is interesting, particularly since Brewe and Hamrock was proposed as a method of finding 01, that, over this range, better answers can be found by using their approximation to find the semiaxes n and h, and substituting these into Equation (4): in other words, their approximations for k and 6 are better than their approximation for K: see Fig 2. (Modifying the Hamrock and Brewe method in the same way reduces the error for B/A 5 5, but increases it for larger values).
Recommendations There will be few occasions when a 3% error is not acceptable: so for moderately elliptical contacts there is nothing to choose between the methods in terms of practical accuracy. If the ‘effective radius’ method is chosen on grounds of simplicity, or taste, no penalty is thereby incurred: in particular, for 1 I B/A 5 5 the effective radius method gives the best value for the Hertz pressure (except for B/A = 3 where the Brewe and Hamrock error changes sign and so passesthrough zero). The author’s opinion is that for mildly elliptical contacts, or when only a hand calculator is available, the present method should be used: otherwise, the exact calculation should be performed using the method recommended by Dyson et n16. Example
A 10 mm radius ball runs on an inner race of radius 25 mm in a groove of radius of curvature 16 mm: the load on the ball is 100 N. Both ball and race are of steel with E = 202 kN/mm’, p = 0.3. Ratio
of principal
curvatures
B/A
Fig I Errors ill the maximum Her-t,: pressure pP The signs against the curves indicate whether the estimate is too high (+) or too low (-) 236
Tribology
International
Volume
30 Number
The contact modulus E* is 1 - 1’2 1 - $ -1 = 111.O kN/mm’ E + E I 3 1997
Analysis
In the axial direction: A = ;k - i
of elliptical
Hertzian
contacts:
J. A. Greenwood
b -- CI
= 0.0375 mm-’
(The exact answer 0.4183) (minus because the groove is concave!). In the plane of the race: B=-!X+?Ij=O.l4mm-’
a = 5 [0.0375*0.3193” + 0.14*0.1327’] = 3.144 pm
(in the present method A and B are interchangeable. but conventionally A is the smaller of the two). Then R, = [Cl.14*0.0375*0.08875]p”3 = 12.90 mm “3
= 1.1273 kN/mm’ 1 (The exact answer is p,, = 1.1284 kN/mm’, so the approximate answer is 0.10% low). If the approach or the semi-axes are required, it is more convenient to find p0 from the mean contact radius ‘Crather than directly: we have c =
,&&
=
0.75”0.1”12.90 111.0
1. Greenwood J.A.. Formulas for moderately contacts. ASME J. Tribal. 1985 107 501-503 2. Timoshenko Hilt, New
S and Goodier York, 1951
3. Johnson K.L. Contact Cambridge, 1985
J.N. Theory
Mechanics,
4. Brewe D.E. and Hamrock B.J. Simplified contact deformation between two elastic Technol. 1977, 99 485387
= 0.2058 mm.
elliptical
of Elasticity,
Cambridge
5. Hamrock B.J. and Brewe D.E. Simplified and deformations. ASME .I. Lub. Technol.
1 “3
(exact answer 3.136 pm). References
and 6”O.l”lll.O’
SO that b = 0.2058*~0.4155 = 0.1327 and a = 0.2058/ do.4155 = 0.3 193. Hence:
Universio
Hertzian McGrcwPress.
solution for ellipticalsolids. ASME J. Lub. solution for stresses 1983 105 171-J77
6. Dyson A, Evans H.P. and Snidle R.F. A simple, accurate method for calculation of stresses and deformations in elliptical Hertzian contacts. J. Mech. Eng. Sci. Proc. I. Mech. E pt C 1991 206 139-111
But
Tribology
International
Volume
30 Number
3 1997
237