The lubricant elastohydrodynamic (EHD) film thickness formula is reduced to a simplified form whereby only the rolling-element bearing inside and outside diameters and speed (in revolutions per minute) and the lubricant type and viscosity (in centipoise) at temperature are required for its use. Additionally, a graph is provided for the first time that is based upon experimental data giving an EHD film reduction factor as a function of contact lubricant flow number. This reduction factor accounts for lubricant starvation within the Hertzian contact. A graph relating the ratio of minimum film thickness to composite surface roughness and a lubrication-life correction factor is also provided. The life correction factor is used to determine resultant bearing life

Zaretsky, Erwin V.

English

NASA Technical Reports Server

NASA Technical Memorandum 10257_5
Bearing Elastohydrodynamic ::i i
Lubrication: A CompleX : :
Calculation Ma_de Simple
Erwin V. Zaretsky
Lewis Research Center
Cleveland, Ohio ................... ......
April 1990
(NASA-IM-102575) BEARING EtASIOHYDRODYNAMIC
LUBRICATION: A COMPLEX CALCULATIJN MADE
SIMPLF (NASA) LI p CSCL 1IN
N90-22041
Uncl as
G3137 0377355
https://ntrs.nasa.gov/search.jsp?R=19900012725 2020-03-19T23:06:20+00:00Z

BEARING ELASTOHYDRODYNAMIC LUBRICATION" A COMPLEX CALCULATION MADE SIMPLE
Erwln V. Zaretsky
Natlona] Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
SUMMARY
The lubricant elastohydrodynamic (EHD) film thickness formula Is reduced
to a simplified form whereby only the rolllng-element bearing Inside and out-
side diameters and speed (In revolutions per minute) and the lubricant type and
viscosity (In centipolse) at temperature are required for its use. Addition-
ally, a graph is provided for the first time that is based upon experimental
data giving an EHD film reduction factor as a function of contact lubricant
flow number. This reduction factor accounts For lubricant starvation withln
the Hertzlan contact. A graph relating the ratio of minimum film thickness to
composite surface roughness and a lubrIcatlon-life correction factor is also
provided. The llfe correction factor is used to determine resultant bearing
llfe.
INTRODUCTION
Until approximately 1960, the role of the lubricant between surfaces in
rolling contact was not fully appreciated. Metal-to-metal contact accompanied
by boundary lubrication was presumed to occur in all applications. However,
the development of elastohydrodynamlc (EHD) theory showed that most rolling-
element bearings have a thin film separating the contacting bodies during
motion. Because of the high pressures in the contact zone, the lubrication
process is accompanied by some elastic deformatlon of the contact surfaces
shown In figure l, and thus, the process Is referred to as elastohydrodynamlc
(EHD) lubrication. Grubin (1949) was among the first to Identify thls phenome-
non, which also occurs for other oil-lubrlcated, rolllng-contact machine ele-
ments such as gears.
HistorlcaIly, elastohydrodynamIc (EHD) ]ubrlcatlon may be viewed as one of
the major trlbologIcal developments of the twentleth century. It not only
revealed the existence of a previously unsuspected regime of lubrlcatlon in
highly stressed, nonconfoFming machine elements, but It also brought order to
the complete spectrum of lubrication reglmes, ranging from boundary to
hydrodynamic.
The most important practical aspect of elastohydrodynamic lubrication
theory is the determlnatlon of the fllm thickness separating the contacting
surfaces. The first satisfactory equation representing the minimum film thick-
ness was reported by Grubln (1949). Other theories are those of Dowson and
Hlgglnson (1966) and Archard and Cowking (1965-66) and Hamrock and Dowson
(1977). Hamrock and Dowson (1981) provide a complete discussion of EHD theory
and give an excellent historical overview of its development. Thelr respective
formulas are
Grubln (1949)
Dowson-Hlgglnson (1966)
H 1.95 GO.73uO.73W -0.091
H _ I 6 GO.6uO.Tw -0.13
(1)
(2)
Archard-Cowklng (1965-66)
-0.71
GO.74uO.74H-O.074
(3)
Hamrock-Dowson (1977) H = 3.63(I - e-0.68 k)GO-49uO.68w-O.073 (4)
The variations in EHD theory and the resulting formulas are beyond the
scope of thls writing. However, suffice it to say, the variations between the
EHD film thicknesses calculated with the dlfferent formulas are less than the
variations between the varlous sets of experimental data used to verify the
theories (Coy and Zaretsky, 1981), Thls is i11ustrated in flgure 2, which
shows the film thickness accordlng to the theories of Hamrock and Dowson
(1977), Grubln (1949), Cheng (1972), and Chlu (1974). A theoretical model for
oli starvation in a ro11Ing-element bearing was formulated by Chlu (1974).
Figure 2 shows the results from Chiu's (1974) analysis and compares his experi-
mental data with the experlmental data of Coy and Zaretsky (1981). From the
correlatlon between the experimental data of Coy and Zaretsky (1981) and the
analytical predictions of Chlu's starvatlon theory, o11 starvation in the Hert-
zlan contacts appears to be the primary reason why the experlmental data devi-
ated from classical EHD theory (Coy and Zaretsky, 1981).
SYMBOLS
El'E2
F
G
G
H
hc
I
h C
ID
k
kG
major seml-axls of Hertzlan contact, In.
minor seml-axis of Hertzian contact, in.
l t - Vl 1 - v
E' reduced modulus of elasticity, E' - 2 El + E2 , psl
Young's modulus for body l and body 2, psi
normal load, Ib
materials parameter, :E'
G0.49
dlmenslonless film thickness, hclR x
central or average film thlckness, in.
hC multlplled by EHD fllm reductlon factor (see flg. 4), In.
Inner-rlng (bearing bore) dlameter, In.
elllptlclty ratlo, alb
lubricant flow factor, mln/cP
kh
N
OD
P
Rx
Ry
U
U
W
X
Y
Zo
o[
n 0
A
_1 ,'°2
0
o I ,0 2
l
2
EHD fl]m Factor, In.l.36/lb 0.68
speed, rpm
bearing outer dlameter, |n.
bearlng pltch dlameter, In.
I _ + _ in.
equivalent radlus In ro111ng direction, _xx = Rx1 Rx2 '
equivalent radlus perpendicular to rolllng dlrection, 1 I l
--+-- , In.Ryl Ry2
speed parameter, uno/E'Rx
average surface speed, In./sec
load parameter, F/E'R X2
coordlnate In rolllng dlrectlon, In.
coordlnate In transverse dlrect|on, In.
absolute vlscoslty, cP
pressure-vlscoslty exponent, psl -l
absolute vlscoslty at temperature, Ib-sec/In. 2
I
f11m parameter, hc/o
Polsson's ratlo For body l and body 2
composite surface roughness, e + o2 , In.
root mean square (rms) surface roughnesses of contactlng bodles, In.
Inner-rlng speed, rad/sec
body I
body 2
EHDSIMPLIFIED
Fllm thickness calculatlons can be greatly slmpllfied for flrst-order
approxlmatlons. Referring to the Hamrock-Dowson(1977) formula, the elllptlclty
ratio k for most bearings can be assumedto be approxlmately 7. Addltionally,
for most ball and roller bearing applications, the resultant maximumHertzlan
stresses on the bearlng raceways wlll be between 200 000 and 350 000 psl. As
the maximumHertzlan stress is increased from 200 000 to 350 000 psi, the EHD
f11m thickness Is reduced only 9 percent. As a result, a maximum Hertzlan
stress level of 275 000 psl can be assumed as a representative stress. Based
upon the assumption that the load parameter N can be considered a constant,
W-0"073 - 2.27 (5)
Values of the materlals parameter G are a function of lubrlcant chemistry.
From equation (4),
: G0"49 (6)
Representative values of G and G are given In table I. The average surface
velocity u at the contact of the rolling element and the race can be wr|tten
as follows:
p _|(OD + ID)
(7)
Using equation (7) In the speed parameter U gives
mi(OD + ID)
U : 8E,Rx no (8)
For most bearings,
OD - ID
Rx = 8 (9)
Comblnlng equations (5), (6), (8), and (9) into the Hamrock-Dowson (]977)
formula (eq. (4)), where k = 7 and E' : 33x106 psl, gives
hc = kh(OD - ID)O'32[mI(OD + ID)]O'68n_'68_ (lO)
For OD, ID, and hC In inches, kh : 7.85x10 -6 In.l.36/lb 0.68
To further simplify equatlon (lO), the equatlon can be rewrltten to
reflect the rotational speed In revolutions per mlnute and the viscosity in
centipolse,
where
hC : kh(OD- ID)O'32[N(O0 + ID)]O'68Z_'68G (11)
For OD, ID, and hc in Inches, kh = 3.BxlO-ll; for ODand ID In mllll-
meters and hc in inches, kh = 1.49xi0 -12
Representative temperature-vlscoslty curves for typical lubricants are
given In figure 3. From these graphs and the bulk bearing operating tempera-
ture (known or assumed), a value of Z0 for a typlcal ]ubrlcant type can be
obtained for use in equation (ll).
STARVATION EFFECT
As stated in reference to figure 2, it was concluded that the devlatlon
between calculated and measured EHD film thicknesses can be attributed to
lubricant starvation in the Hertlzan contact zone. Starvation is caused by the
inability of the lubricant to flow Into the contact zone at a rate necessary
to supply the volume required to establish the theoretical film thickness.
Because the starvatlon effect occurs mostly at higher speeds, where film thick-
ness Is generally large, the reductlon in film thickness by Itself may not
affect bearing performance. However, thlnner f11ms can shorten bearing life.
From figure 2, starvation becomes a factor to be considered when the con-
tact lubricant flow number (G x U) becomes greater than 2xlO -?. Below this
value, the measured film thickness reasonably approximates that whlch is
calculated.
The EHD film thicknesses of figure 2 calculated by using the Grubln (1949)
and Hamrock and Dowson (1977) formulas are based upon the contact being fully
flooded with lubricant. If the bearing Itself is starved for oll because the
oll supply is interrupted or because the oil cannot reach the bearing contact
zone during extremely hlgh-speed bearing operation, then addltlonal reduction
in film thickness can occur at all values of the lubricant flow number. This
reduction Is usually manifested by distress of the bearlng raceway surfaces.
For the purpose of determlnlng an EHD film reduction factor, the experi-
mental data plotted In figure 2 were compared with the Hamrock-Dowson (1977)
prediction. The resultant EHD film reduction factor was plotted agalnst the
contact lubricant flow number in figure 4.
The contact lubricant flow number is as fo]lows.
G x U = kG_ZoN IOD+OD- II_1
(12)
For _ _n square inches per pound, kG = 1.52xi0 -lO mln/cP.
By using the value of G x U from equatlon (12), the EHD fllm reduction
factor can be obtained from figure 4. An estlmate of the reduced EHD film
thickness hc' In the central reglon of the Hertzlan contact zone is then
found by multlplylng the hc (from eq. (ll)) by the EHD film reduction factor.
BEARINGLIFE EFFECTS
The measureof the effectiveness of the lubricant f11m Is the lubricant
film parameter or A ratio (i.e., h_/a, the central film thickness multlplled
by EHD film reduction factor dlvlded by the composite surface roughness of the
rolllng-element surfaces). Usually the root mean square (rms) surface flnlshes
of the contactlng bodies oI and o2 are used to determlned the composite
surface roughness as follows:
(13)
For most commercial off-the-shelf bearings, a can be taken as lO pin. For
most aerospace bearings, _ can be taken as 5 Nin. For very large industrial
bearings, a can be taken as 25 Nln.
The lubricant film parameter or A ratio can be used as an indicator of
rolling-element performance and llfe. For A < I, surface smearlng or deforma-
tion, accompanied by wear, will occur on the rolling surfaces. For l < A < 1.5,
surface distress may be accompanied by superficlal surface pitting. For
1.5 < A < 3, some surface glazing can occur with eventual failure caused by
classical subsurface-orlglnated rolling-element fatigue. At A > 3, minimal
wear can be expected with extremely long llfe; fallure w111 eventually be by
classical subsurface-orlglnated rolling-element fatigue. The most expedient
way of attaining a higher A ratio is to reduce the bearing operating tempera-
ture and, thus, increase the lubricants viscosity. Another way is to select a
lubricant wlth a higher viscosity at operating temperature and/or a larger
pressure-viscosity coefficient. The most expensive way of attaining a higher
A ratio Is to select a hlgh-quallty surface finish on the raceways and ro111ng
elements.
The effect of film thickness on bearing llfe is shown in figure 5. The
life factor obtained from this flgure can be used to modify or adjust the bear-
ing calculated life (Zaretsky, 1986).
REFERENCES
Archard, J.F.; and Cowklng, E.W.: Elastohydrodynamlc Lubrlcation at Point
Contacts. Proceedings of the Institution of Mechanical Englneers; (London),
Vol. 180, Pt. 3B, 1965-66, pp. 47-56.
Cheng, H.S.: Isothermal Elastohydrodynamlc Theory for the Full Range of
Pressure-Vlscoslty Coefflcient. J. Lubr. Technol., vol. 94, no. l,
Jan. 1972, pp. 35-43.
Chiu, Y.P.: An Analysis and Prediction of Lubrlcant Film Starvation in Rolling
Contact Systems. ASLE Trans., vol. 17, no. l, 1974, pp. 22-25.
Coy, J.J.; and Zaretsky, E.V.: Some Llmltatlons in Applying Classical EHD Film
Thlckness Formulae to a High-Speed Bearing. J. Lubr. Technol., vol. IO3,
no. 2, Feb. ]981, pp. 295-304.
Dowson,D.; and H1ggInson, G.R." Elastohydrodynam|c Lubrication" The Funda-
mentals of Roller and Gear Lubrlcatlon. PergamonPress, 1966.
Grubln, A. N.: Fundamentalsof the HydrodynamlcTheory of Lubrlcat|on of
Heavlly LoadedCyllndrIcal Surfaces. Investlgation of the Contact Machlne
Components,Kh. F. Ketova, ed., translatlon of Russlan Book No. 30, Central
SclentIflc Instltute of Technology and Mechanlcal Englneerlng, Moscow,1949.
(Available from Dept. of Sc|entiflc and Industrial Research, Great Br|ta|n,
Trans. CTS-235, and from Speclal L1brarles Associatlon, Chlcago, Transl.
R-3554).
Hamrock, B.J., and Dowson,D." Isothermal ElastohydrodynamIc Lubrlcatlon of
Polnt Contacts. Part III- Fully Flooded Results. J. Lubr. Technol.,
vol. 99, no. 2, Feb. 1977, pp. 264-276.
Hamrock, B.J.; and Dowson,D." Ball Bearing Lubrlcatlon - The Elastohydro-
dynamic of Elllpt|cal Contacts. W11ey,1981.
Roelands, C.J.A." Correlatlon Aspects of the Vlscoslty-Temperature-Pressure
Relatlonshlp of Lubricating Oils. PhDThesis, Unlv. of Delft, Holland, 1966.
Zaretsky, E.V.: Lubrlcant Effects on Bearing Life. NASATM-88875, 1986.
TABLE I. - PRESSURE-VISCOSITY COEFFICIENT AND MATERIALS PARAMETER FOR REPRESENTATIVE LUBRICANTS
Type (speclflcat_on) Pressure-vlscoslty coefflclent, Materials parameter, _ - G0"49
_, G - _E'
m2/N (psi-l)
40 "C I00 "C 150 "C 40 °C I 100 "C 150 "C
(104 "F) (212 °F) (302 "F) (104 °F) I (212 "F) (302 "F)
Automatlc transmission 1.54x10-8 1117x10-8 1.02xlO -8 3498 2673 2310
fluld (GMGI37-M) (1.05x10 -4) (0.81xi0 -4) (0.70xi0 -4)
1.54xi0-8 1.27x10 -8 5709 3498 2871
(0.87x10 -4)(1.06x10 -4)Superreflned mineral o11 (2.5xlO -8
.73xi0 -4)
Type II ester (MIL-L-23699) .24xi0-8 0.99x10-8 0.87xI0 -8 2805 2244 1980
(0.85×]0-4) (O.68xlO -4) (0.60xi0 -4)
01ester (polyo]ester) 1.15xI0-8 0.93xi0-8 0.80x10 -8 2607 2079 i 1815
(MIL-L-7808) (0.79x10 -4) (0.83xi0 -4) (0-55 xlO-4) ;
Synthetic hydrocarbon 1.38xi0-8 1.13x10-8 1.04xi0 -8 3135 2607 i 2376
plus 20 percent polyolester (0.95xi0 -4) (0.79xi0 -4) (0.72xi0 -4)
(MIL-L-2104C, MIL-L-46152)
Synthetic hydrocarbon 1.77x10-8 1.51XI0-B 1.09xlO -8 3861 3432 i 2475
(I.17xI0 -4) (1.04x10 -4) (0.75xi0 -4)
Mineral o11 (MIL-L-6081) 2.19xi0-8 L54xIO-8 1.07xi0 -8 4983 3498 i 2442
(1.51x10-4) (l.06xlO -4) (0.74xi0 -4) _ J
aThe values glven are representative values of _ and may vary from batch to batch or from one lubrlcant brand to another,
40 "C " 100 "C I 150 "C
(104 "F) (212 "F) I (302 "F)
54.5 ' 47.8 I 44.5
69.3 54.5 I 49.5
48.9 43.9 I 41.2
47.2 42.2 I 39.5
51.7 47.2 I 45.1
57.2 54 46 I
64.8 54.5 45.7
I . I
,_"i_ ,,-*-Hlghcontacl
XlI III_ _
_,IIIY '-Pressorlz_
_' lubdcant
Figure 1. - Elastohydrodynamtc film between two bodies In rolling contact.
1 000 000
100 000
10 000
1 000
100
5O
10
5
(3.
O
o
N
:>
10"-3
x
tl
-1-
10 -4
E
_ 10-8
g
_E
Q
10-8
Hamrock and Dowson (1977)
- ,,, _,
_ Cheng (1972)--, // /_...-----.,,,_
_ ",_.,_..,_
Grubin(1949)_. "Z _..,.,O Oo o°OD
- " o"o
% o oo o •'_..--ch_o (1974)
v • Chiu (1974)
" O Coyand Zaretsky (1981)
I =l_lJhl I.hhld I =lllllil 1 Jllltl,l
10-8 10-7 10-6 10-5 10.-4
Dimensionless contact lubricant flow number. G x U
Figure 2. - Theoretical effect of kinematic starvation and inlet sheax
heating on film thickness and comparison with experiment (Coy
and Zaretsky. 1981).
SAE 10 --" "_
1 , I , I
(a) Mineral oils.
I I
100 000
10 000
1 000
100
50
20
1
.05
.04 J
-50 0
Type of oil
0 Polyphenyl ether
[] Chotoronuorocarbon
A Fluorocarbon
O Potyglycal
C_ _ilicone oil
(_ Diester
V Phosphate ester
Silicate ester
t 1 j 1_11
50 100 200 300 400
Temperature,6C
I I I [ I I I t It
-40 0 32 100 200 400500 700
Temperature, °F
(b) Synthetic lubricants.
Figure 3. - Viscosity-temperalure chart for represent-
ative lubricants. (Data for mineral oils courtesy of
BP Amedca, Inc., Cleveland, Ohio; synthetic lubri-
cants from Roetands (1968).)
8
' iiiiililiiiiiii!i!iiiii!iii!iiiii!iiiiiiiiiiiiiiiiiiiiiiili
= ======================= ==================stan,,atloniiiiiiiil..........
o i_'i_:i_:ili_:_ii!i!_!ii!i!':_i_:i_i:.|:ii_,ii_i_:t,l, I,l,l I, i, i, J,I l • l, i, l,l
10--8 10-7 10.6 10-5 10.4
Dimensionless contact lubricant flow number, G x U
Figure 4. - Elastohydrodynamic (EHD) film reduction factor as a
function of contact lubricant flow number.
Calculated from AFBMA
o I _ I = I ,I,I
.6 .8 1 2 4 6 8 10
Lubricant film parameter, A
Figure 5. - Lubrication-life correction factor as function of
lubricant film parameter (ratio of mlntmum film thickness
to composite surface roughness).
9
Report Documentation PageNational Aeronautics and
Space Administration
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
NASA TM-102575
4. Title and Subtitle
Bearing Elastohydrodynamic Lubrication: A Complex Calculation
Made Simple
7. Author(s)
Erwin V. Zaretsky
5. Report Date
April 1990
6. Performing Organization Code
8. Performing Organization Report No.
E-5360
10. Work Unit No.
505-63-1B
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
9. Performing Organization Name and Address
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135-3191
12. Sponsoring Agency Name and Address
Nationai Aeronautics and Space Administration
Washington, D.C. 20546-000t
15. Supplementary Notes
16. Abstract
The lubricant elastohydrodynamic (EHD) film thickness formula is reduced to a simplified form whereby only the
rolling-element bearing inside and outside diameters and speed (in revolutions per minute) and the lubricant type
and viscosity (in centipoise) at temperature are required for its use. Additionally, a graph is provided for the first
time that is based upon experimental data giving an EHD film reduction factor as a function of contact lubricant
flow number. This reduction factor accounts for lubricant starvation within the Hertzian contact. A graph relating
the ratio of minimum film thickness to composite surface roughness and a lubrication-life correction factor is also
provided. The life correction factor is used to determine resultant bearing life.
17. Key Words (Suggested by Author(s))
Rolling element bearing
Bearing lubrication
Elastohydrodynamic lubrication
Bearing life
18. Distribution Statement
Unclassified- Unlimited
Subject Category 37
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Unclassified Unclassified 10
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