NASA/TM—2015-218745

June 2015

An Erratum was added to this report March 2016.

Pradeep K. Gupta

PKG Inc., Clifton Park, New York

Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Comparison of Models for Ball Bearing

Dynamic Capacity and Life
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NASA/TM—2015-218745

June 2015

An Erratum was added to this report March 2016.

Pradeep K. Gupta

PKG Inc., Clifton Park, New York

Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Comparison of Models for Ball Bearing

Dynamic Capacity and Life

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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NASA/TM—2015-218745 iii 

Erratum

Issued March 2016 for 

NASA/TM—2015-218745 

Comparison of Models for Ball Bearing Dynamic Capacity and Life

Pradeep K. Gupta, Fred B. Oswald, and Erwin V. Zaretsky 

June 2015 

Page 16, Figure 4(b) has been corrected. 

NASA/TM—2015-218745 1 

Comparison of Models for Ball Bearing 

Dynamic Capacity and Life 

Pradeep K. Gupta*

PKG Inc. 

Clifton Park, New York 12065 

Fred B. Oswald† and Erwin V. Zaretsky‡

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Generalized formulations for dynamic capacity and life of ball bearings, based on the models 

introduced by Lundberg and Palmgren and Zaretsky, have been developed and implemented in the bearing 

dynamics computer code, ADORE. Unlike the original Lundberg-Palmgren dynamic capacity equation, 

where the elastic properties are part of the life constant, the generalized formulations permit variation of 

elastic properties of the interacting materials. The newly updated Lundberg-Palmgren model allows 

prediction of life as a function of elastic properties. For elastic properties similar to those of AISI 52100 

bearing steel, both the original and updated Lundberg-Palmgren models provide identical results. A 

comparison between the Lundberg-Palmgren and the Zaretsky models shows that at relatively light loads 

the Zaretsky model predicts a much higher life than the Lundberg-Palmgren model. As the load increases, 

the Zaretsky model provides a much faster drop off in life. This is because the Zaretsky model is much 

more sensitive to load than the Lundberg-Palmgren model. The generalized implementation where all 

model parameters can be varied provides an effective tool for future model validation and enhancement in 

bearing life prediction capabilities. 

Introduction 

For over a century, rolling-element fatigue has been the criterion for determining the life of 

rolling-element bearings. Classical rolling-element fatigue is stress- or load-cyclic dependent, beginning 

as a crack at a depth below the race or rolling-element surface. The crack propagates into a crack network 

that reaches the contacting surface to form a spall limited in area and depth of penetration (Ref. 1). This 

failure mode can be classified as high-cycle fatigue where most of the bearing life is related to crack 

initiation with a relatively short time related to crack propagation. Rolling-element fatigue is extremely 

variable but statistically predictable, depending on the steel type, steel processing, heat treatment, bearing 

manufacturing and type, lubricant used, and operating conditions (Ref. 1). Sadeghi, et al. (Ref. 2) provide 

an excellent review of this failure mode. 

In 1924, Palmgren (Ref. 3) began and later together with Lundberg in 1947 and 1952 (Refs. 4 and 5), 

developed what is now referred to as the “Lundberg-Palmgren Model for Rolling Bearing Life Prediction.” 

The Lundberg-Palmgren analysis is based on the Weibull fracture strength and life models (Refs. 6 to 10)

and classical rolling-element fatigue (Ref. 11). In 1953, the Lundberg-Palmgren life model was adopted as 

the rolling bearing life prediction standard by American National Standards Institute (ANSI)/American 

Bearing Manufacturers Association (ABMA) (Refs. 12 and 13) and later by International Organization for 

Standardization (ISO) (Refs. 14 and 15). It is currently the basis for all bearing life prediction worldwide. 

 

*

 Consultant. 

†

 Retired. 

‡

 Distinguished Research Associate. 
NASA/TM—2015-218745 2 

Jones in 1952 (Ref. 16) was the first person to consider the effect of centrifugal loading on the life of 

angular-contact ball bearings. His analysis modified the bearing life analysis of Lundberg and Palmgren. 

Subsequently, Jones (Ref. 17) in 1959 published his method for determining the kinematics of a ball and 

sliding friction in a high-speed, angular-contact ball bearing. Following this analysis in 1960, Jones 

(Ref. 18) published a completely general solution for both ball and roller bearing kinematics, loading, and 

life. For the first time, a bearing analysis incorporated elastic deflection of the bearing shaft and 

supporting structure as well as centrifugal and gyroscopic loading of the rolling elements under combined 

loading and high-speed operation (Ref. 19). The solution, which was accomplished numerically by 

iterative techniques, was programmed by Jones for an IBM 704 digital computer. This was the first 

rolling-element bearing computer code and a major technical achievement.

The Lundberg-Palmgren (Refs. 4 and 5) bearing life analysis as well as that of Jones (Ref. 19) was 

benchmarked to a pre-1940 rolling bearing life database. With advancements in materials, steel 

processing, manufacturing techniques, and lubrication beginning in the early 1950s, rolling bearing 

fatigue life has significantly increased. Fatigue life predictions from the Lundberg-Palmgren model began 

to significantly underestimate bearing life (Ref. 20). 

In 1987, Anderson (Ref. 21) discussed the limitations of the Lundberg-Palmgren life model. He stated 

that “in the decades since its development a number of shortcomings have become apparent. These 

shortcomings, which manifest themselves as discrepancies between predicted and actual bearing 

behavior, are partly due to limitations of the original model in accounting for all relevant phenomena and 

partly due to continuously advancing bearing technology.” These limitations in the Lundberg-Palmgren 

life model have stimulated research on more comprehensive life models such as those of Ioannides and 

Harris (Ref. 22) and Zaretsky (Refs. 23 and 24). The limitations have also resulted in the introduction of 

life factors that attempt to correct for material and lubrication effects in the Lundberg-Palmgren life 

model (Refs. 25 to 27). 

Ioannides and Harris (Ref. 22) modified the Lundberg-Palmgren model by introducing a shearing 

stress fatigue limit, below which the bearing life is assumed to be infinite. Thus, a limiting shear stress was 

subtracted from the base shear stress before computing the life (Ref. 11). 

Zaretsky (Refs. 11, 23, and 24) made some fundamental changes to the Lundberg-Palmgren model. 

First, the Weibull equation was rewritten to make the shear-stress exponent independent of the Weibull 

slope. Second, the maximum shearing stress, rather than the orthogonal shearing stress, was chosen as the 

critical shearing stress. Third, the life dependence on the depth of the critical shear stress was eliminated. 

Awaiting further experimental validation of these fundamental modifications, the Lundberg-Palmgren 

model has continued to be the dominant life prediction model while life correction factors have been 

developed to reflect the behavior of improved materials in varied operating environments (Refs. 26 and 27). 

As the modern bearing steels and manufacturing techniques have advanced, bearing life has increased 

several fold in comparison to that achievable in the 1940s. However, demands on operating environments 

in terms of speed, load, and temperatures have become significantly more adverse. This has resulted in 

renewed interest in fatigue life modeling. Currently, the three most referenced fatigue models are 

Lundberg-Palmgren (Refs. 4 and 5), Ioannides-Harris (Ref. 22) and Zaretsky (Refs. 11, 23, and 24). In the 

Lundberg-Palmgren model, as originally developed, certain material properties and survival statistics are 

embedded in the model constants. In addition, simplifying assumptions were made to implement the 

elastic contact solutions. Thus, the life factor approach has been the only viable approach to update the life 

prediction from this model, as it is currently used. 

The shear-stress fatigue limit concept proposed by Ioannides and Harris (Ref. 22) has been a very 

controversial subject. Several researchers (Refs. 28 to 30) have demonstrated that a realistic fatigue limit 

on rolling bearing materials does not really exist. Thus, an input shear-stress fatigue limit in the Ioannides 

and Harris model (Ref. 22) may lack physical significance. 

The Zaretsky model attempts to physically modify the fatigue process as a whole. Since the actual 

data used to develop the Lundberg-Palmgren model is not available in the open literature to revisit the 

model at a fundamental level, it is essential to reformulate the Lundberg-Palmgren life equation in a more 
NASA/TM—2015-218745 3 

generalized fashion in order to establish the physical significance of each element of the model. Likewise, 

the Zaretsky model required development so that it can be applied to practical bearings. 

Based upon the above discussion, the objectives of the investigation reported were to (a) develop 

generalized expressions for both the Lundberg-Palmgren and the Zaretsky models; (b) implement the 

models in a bearing performance simulation computer code; (c) carry out a parametric model evaluation to 

derive the model constants from available experimental data; and (d) compare the model predictions over 

the operating conditions as a function of key model elements. 

In order to achieve these objectives, the bearing dynamics computer code ADORE (Ref. 31), which is 

based on the bearing dynamic analysis of Jones (Ref. 19), was used as a baseline code. Published 

experimental data for a jet engine main-shaft bearings (Refs. 32 to 35) were used to derive the constants 

in the life equation. Model predictions were then compared with another set of high-speed turbine engine 

bearing life data (Ref. 36). The groundwork developed in this investigation is intended to provide a 

starting point for future model development and validation as experimental data on newer materials 

become available. 

Nomenclature 

A original Lundberg-Palmgren constant in load capacity equation, N/m1.80 (lbf/in.1.80) 

ALP updated Lundberg-Palmgren constant in load capacity equation, N/m1.80 (lbf/in.1.80) 

AZ Zaretsky constant in load capacity equation, N/m1.332 (lbf/in.1.332) 

a semimajor width of Hertzian contact area perpendicular to direction of rolling, m (in.) 

b semiminor width of Hertzian contact area in direction of rolling, m (in.) 

c shear-stress-life exponent 

D diameter of rolling element, m (in.) 

d diameter of rolling-element running track, m (in.) 

dm pitch diameter, m (in.) 

f raceway curvature factor: ratio of groove radius to ball diameter 

h exponent for depth to critical shear stress 

K empirical proportionality constant in fundamental Equation (6) 

L individual contact life, millions of rotating-race revolutions 

LS life of stationary race, millions of rotating-race revolutions 

LR life of rotating race life, millions of rotating-race revolutions 

L10 10-percent bearing life: life at which 90 percent of a population survives, millions of 

 rotating-race revolutions 

m Weibull slope or modulus 

N life, number of stress cycles 

n number of rolling elements 

pH maximum Hertz stress, Pa (lbf/in.2

) 

Q normal force (load) between rolling element and raceway, N (lbf) 

Qc dynamic radial load capacity, N (lb) 

S probability of survival 

u contact frequency of rolling element per revolution of bearing race 

Vo stressed volume, m3

 (in.3) 

z distance below surface to critical shear stress, m (in.) 

α contact angle 
NASA/TM—2015-218745 4 

γ D cos()/dm 

 ratio of critical shear stress to maximum Hertz stress, τ/pH 



 rolling-element orbital velocity 

λE elastic property ratio 

 nondimensional depth to critical shearing stress, z/b 

τ critical shearing stress, Pa (lbf/in.2

) 

Ω angular velocity of bearing race 

Subscripts: 

i designates race (1 = inner race, 2 = outer race) 

j designates rolling-element number (from 1 to n) 

m designates maximum sheer stress or depth of this stress 

o designates orthogonal sheer stress or depth of this stress 

Lundberg-Palmgren Model 

Based on very simple observations, where mechanical failures were caused by some functions of 

applied stress and the stressed volume, Weibull (Refs. 6 and 7) introduced the Weibull distribution 

function for strength and life analysis, which is presently very commonly used in a wide range of 

applications. In very general terms the distribution has three parameters, a location parameter, a scaling 

factor, and a shape parameter. The location parameter defines a base time with respect to which life may 

be measured, the scaling factor is a reference life at a defined survival probability, relative to which 

reliability at any arbitrary survival probability may be measured, and the shape parameter essentially 

defines the shape of the distribution, more commonly known as Weibull slope. The statistical distribution 

of course does not define any functions of stress or the volume stressed. Following this fundamental 

development, Weibull (Refs. 8 and 9) applied this distribution to fatigue and other types of failures. Based 

on this work, Lundberg and Palmgren (Refs. 4 and 5) applied the Weibull distribution to rolling-bearing 

failures. 

After a statistical analysis of a large amount of experimental bearing life data, it was postulated that in 

addition to subsurface shear stress and the stressed volume, as suggested by Weibull, rolling-element 

fatigue was also dependent on the depth at which the fracture is initiated. For this analysis the location 

parameter was set to zero and the characteristic life was determined with 90-percent survival probability, 

commonly known as L10 life. Thus, based on a two-parameter Weibull distribution bearing life, N, in 

stress cycles, with a probability S, was written as a product of an empirical function of the maximum

subsurface orthogonal shear stress, o, depth below the surface, zo, at which the maximum orthogonal 

shear stress occurs, and the volume, Vo, of the material stressed: 

0.90

1 ln 1 ln o h

o c m oN z V

S

  (1) 

Here c, m and h, are empirical exponents determined by fitting experimental data to the model. The factor 

0.90

1 ln results from the fact that life at arbitrary survival probability, S, is normalized to 90 percent 

probability of survival. 
NASA/TM—2015-218745 5 

Consider a concentrated (Hertzian) contact between two elastic bodies, with prescribed geometries 

and material properties, subjected to an applied load Q. For a generalized three-dimensional contact, 

Harris and Kotzalas (Ref. 37) have documented the elastic solutions in terms of contact half widths a and 

b, normal to and along the rolling directions, respectively, the Hertzian contact stress, pH, and the 

subsurface shear stress distribution as a function of depth below the surface. The relation between the 

maximum orthogonal shear stress, o and the depth zo, at which it occurs, is illustrated in Figure 1 as a 

function of the ratio of the contact half widths. 

For most ball bearings, the ratio of the semiminor axis b of the Hertzian contact ellipse to the 

semimajor axis a is approximately 0.15 (b/a ≈ 0.15) (Fig. 1). Thus, 

0.497 0.5 2   

H

o

p

 (2a) 

 0.492  0.5

b

zo (2b) 

Although Lundberg and Palmgren (Refs. 1 and 4) used the orthogonal shear stress, o, it may be convenient 

to generalize the model in terms of any shear stress and the depth at which it occurs. Thus, in Equation (1), 

o and zo may be simply replaced by  and z. Equation (2) may be expressed in general terms as 

  pH andz b (3) 

where the ratio of critical shear stress to maximum Hertz stress,  = 0.25 and the nondimensional 

depth to the critical shearing stress,  = 0.50 for the Lundberg-Palmgren model. 

Now, if the raceway life, L, is expressed in terms of millions of revolutions, while each contact exerts 

u stress cycles per revolution, then 

6 N  uL10 (4) 

Also, if d is the diameter of the rolling-element track on the raceway, then for point contact, the volume of 

material stressed is expressed as (Lundberg-Palmgren (Ref. 4)) 
NASA/TM—2015-218745 6 

V  azd (5) 

where a is the contact half width normal to the rolling direction, and z is the depth at which the failure 

originates, as defined earlier. 

Thus, after combining Equations (3) to (5) and introducing a proportionality constant, K, Equation (1) 

may be rearranged to express raceway life resulting from a single contact as 

   u

S

K p abd b

L

m

c h

H

1/

0.90

1 ln 1 ln

1





















   



 (6a) 

where the constant 106m

 is absorbed in the proportionality constant K. Additionally, depending on 

computational preference, the constant term, ln(1/0.90), may either be absorbed in the constant, or 

consistent with ANSI/ABMA (Refs. 12 and 13) and ISO (Ref 14) standards, a reliability factor may be 

expressed as

  m a S 1/

1  ln ln 0.90 (6b) 

and Equation (6a) may be reduced to 

    

1

1 1

a

u K p abd b

L

c h m H      (6c) 

Equations (6a) or (6c) represents the generalized form of the Lundberg-Palmgren life equation, 

applicable to both point and line contacts. In the event the survival probability, S = 0.90, then of course 

Equations (6a) and (6c) will be identical and the reliability factor in Equation (6b) will be one. Note that 

the Hertzian contact stress, pH and the contact dimensions a and b are related to the applied contact load, 

geometry of the interacting surfaces, and the applicable material properties. Thus, specialized versions of 

Equations (6a) or (6c) for point and line contact configurations may be developed. 

Based on the Hertzian elastic point contact solution (Harris and Kotzalas) (Ref. 37) as applicable to 

ball bearings, the relationships between the applied contact load, Q, and the contact parameters contained 

in Equation (6) may be written as 

ab

Q pH

  2

3 (7a) 

1/ 3

1/ 3

*

2

3 Q E

a a 















  

 and 1/ 3

1/ 3

*

2

3 Q E

b b 















  

 (7b) 

2

2

2

1

2 1 1 1 1

E E E

 

   

 (7c) 
NASA/TM—2015-218745 7 

where  is summation of the principal curvatures of the two interacting surfaces, E1, 1, E2, 2 are, 

respectively, the elastic modulus and Poisson’s ratio of the two surfaces, a* and b* are functions of 

elliptical integrals corresponding to the ratio of the contact half widths, a and b. 

Equations (7) may be substituted into Equation (6) to get a life-load relationship, which can be 

inverted to compute a load, which will sustain 1 million rotating ring revolutions of life. This load is 

defined as the dynamic load capacity of a bearing component. In the original Lundberg and Palmgren 

formulation Equations (4) and (5), E is considered as a constant, corresponding to elastic properties of 

AISI 52100 bearing steel, and it is included in the proportionality constant, K, in Equation (6). In order to 

permit elastic properties variation in the updated Lundberg-Palmgren relation, developed herein, it is 

convenient to define an elastic property ratio, 

o

E E

E



   1/

1/ (8) 

where Eo

 corresponds to the elastic properties of AISI 52100 steel. 

Substitution of Equations (7a) to (7c) and (8) into Equation (6) provides a life equation in terms of 

bearing geometry, the applied loads, speeds, and applicable material properties. The proportionality 

constant, K, is determined by correlating the model predictions with experimental data. While Equation 

(6a) or (6c) can be readily used to compute bearing life, it is often customary to compute a load under 

which the bearing could sustain a life of one million revolutions with a given probability. This load is 

defined as the dynamic capacity of the bearing. Thus, after substituting Equations (7) and (8), 

Equation (6a) can be inverted to provide the following generalized expression for load capacity for ball 

bearings. 

  2

3

2

3

1 1  



 



    c h

m

h c c h b c QcLP ALP kcka k d u (9) 

where the various constants are 

1/ 3

*

2

3



















 



E ak a (10a) 

1/ 3

*

2

3



















 



E

bk b (10b) 

























 



 







 



S

k

c h

c 1 ln

2

2

3 1

 (10c) 

c h

o

LP h c E

K A

  

























  

2

3

3

2 2 (10d) 
NASA/TM—2015-218745 8 

Again, kc in above Equation (10c) is a constant, since the survival probability, S for the L10 life is 0.90, 

and  and  corresponding to the magnitude and depth of maximum orthogonal subsurface shear stress 

are, respectively, 0.25 and 0.50 for the Lundberg-Palmgren model. Thus, kc can be included in the 

proportionality constant. However, kc is left here as a separate term in the generalized expression 

(Eq. 10c) in the event it is desired to change any of these parameters. Also, note that the constant 0.90

1 ln

has been absorbed in the constant K in Equation (10d). The constant ALP is the life constant, which has to 

be determined by fitting experimental data to the model. 

With prescribed race angular velocities for outer and inner races as 1 and 2, respectively, if the 

rolling-element orbital velocity is 

 , then the contact frequency per revolution of faster of the two races 

is 

2 1

2

 1,2 and   



  

u  i

i

i



 (11a) 

1 2

1

 1,2 and   



  

u  i

i

i



 (11b) 

Note that in the entire generalized formulation above, life is modeled at each individual contact. For 

life of the entire bearing these lives are appropriately summed over the total number of rolling elements as 

discussed later in the paper. Thus, while the above contact frequency is for an individual contact, the total 

number of rolling elements does enter in the calculation while estimating life of the entire bearing. In 

commonly used simplified expressions where all rolling elements are assumed to operate with some 

effective load, the expression for contact frequency includes the number of rolling elements and the 

summation, as discussed later in the paper, is eliminated. 

The contact parameters in Equation (9) are readily available in rolling-element bearing computer 

codes. As a result, implementation of Equation (9) is straight forward. The load capacity QcLP is 

dimensional. The resultant units of the constant ALP will depend on the values of exponents c and h. From 

Lundberg and Palmgren, with the values of 31/3 and 7/3 for c and h, respectively, the units reduce to 

(Force/Length)1.80. 

While developing the original model in the 1940s, Lundberg and Palmgren (Refs. 1 and 4) made 

certain simplifications in relating the bearing geometry, material properties to compute the elastic contact 

solutions. Also, they assumed the materials to be air-melt AISI 52100 bearing steel and the values of the 

exponents, c, h, and m were set to 31/3, 7/3, and 10/9, respectively. The result is the following dynamic 

capacity equation for a point contact in ball bearings with a survival probability of 90 percent (Ref. 38). 

  1.8 1/3

0.30

1.39

0.41

1

2 1

2 

















 





 







  D ud

D

f

f Q A

m

c  (12a) 

dm

D 

  cos (12b) 

where the upper sign refers to the inner race while the lower sign denotes outer race contact. 
NASA/TM—2015-218745 9 

The variables, D, dm, , f, and u are, respectively, the ball diameter, pitch diameter, contact angle, 

race curvature factor, and the number of stress cycles exerted by a single rolling element on the raceway 

per revolution of the bearing. After correlating Equation (12a) with experimental data available at the 

time, Lundberg and Palmgren proposed a value of 2.464107

 N/m1.8 (7450 lbf/in.1.8) for the empirical 

constant A. However, the data used to perform the correlation is not available in the open literature. 

Equation (12a) does not have any input for the elastic properties of the materials since they are 

included in the constant A. However, if the value of contact angle is set to that existing under the applied 

loads, the effect of changes in overall bearing deformation under the applied loads, speeds, and 

temperatures, will be accounted for. 

Equation (12a) for its simplicity has been used extensively over many decades. With the advancing 

materials technology, the validity of this simple equation has increasingly been in question. Since 

computerized analysis of rolling contacts is readily available in modern models for rolling bearings, 

Equation (9) may be implemented as a more generalized and updated Lundberg-Palmgren model. The 

model constant, ALP, has to be, of course, determined by correlating the model to experimental failure

data. For the purpose of discussion in this paper, Equation (12a) is referred to as the “original” LundbergPalmgren model while the generalized Equation (9) is called the “updated” Lundberg-Palmgren model. 

Once the dynamic capacity is calculated from either Equation (9) or (12a), the raceway life due to a single 

contact subjected to a contact load, Q, which is an output from a detailed load distributed analysis under 

the applied operating conditions on the bearing, may be written as: 

m

c h

cj

j

j Q

Q

L

3

2

1

 

















 (13) 

With the values of c, h and m as 31/3, 7/3, and 10/9, respectively, the value of the load exponent

3

3

2   

m

c h , for the Lundberg-Palmgren model. Based upon a sensitivity study in Appendix A, the 

values of the Weibull modulus m and c chosen by Lundberg and Palmgren best reflects their database. 

Now the life of the raceway will be calculated from an inverse summation of individual contact lives. 

For the stationary raceway each contact may be subject to a different load condition, thus, the summation 

over n rolling elements will involve the Weibull dispersion slope, m, 

m

n

j

c h

cj

j

m n

j

m

S j Q

Q

L L

1/

1

3

1/ 2

1

1 1







































  

































   



 



 (14) 

For the rotating raceway each rolling element applies an identical loading condition. Thus, the 

raceway life is computed by a simple summation over the n rolling elements 

 



 

 













  

n

j

m

c h

cj

j n

j R j Q

Q

L L 1

3

2

1

1 1 (15) 

Finally, the life of the entire bearing is computed by summation over the races 
NASA/TM—2015-218745 10 

  m m

R m L LS L

1/

10

     (16) 

It should be noted that by computing life individually for each contact in the bearing and then 

carrying out appropriate summation over the raceway eliminates the need for computing an “effective” or 

“equivalent” load as commonly done in simplified implementation of life models. Such a generalized 

implementation permits more precise modeling of load variation on the rolling elements. 

Zaretsky Model 

Zaretsky (Ref. 23) proposed several modifications to the fundamental Weibull and 

Lundberg-Palmgren life equation, Equation (1): 

1. Zaretsky explained that in the Lundberg-Palmgren model the dependence of life on the depth, zo, 

below the surface where the orthogonal shear stress is a maximum, implies that the life is dependent 

on the time it takes for the crack to propagate from the place of origination to the surface. However, 

since rolling bearing fatigue life can be categorized as “high-cycle fatigue,” the crack propagation 

time is extremely small in comparison to the total running time of the bearing for a bearing made 

from modern vacuum-processed material. Thus, Zaretsky (Ref. 23) dispensed the term zo

–h

 in 

Equation (1). Subject to future evaluation and validation, the exponent, c, to the subsurface shear 

stress is kept as 31/3, as in the Lundberg-Palmgren model. 

2. The critical shear stress used in the Lundberg-Palmgren model is the maximum orthogonal shear 

stress, τo. Zaretsky proposed to use the maximum shear stress instead, τm, which in a ball bearing is 

30 percent greater than the maximum orthogonal shear stress, τo. 

3. The volume of material being fatigued in the Lundberg-Palmgren model is based on the depth, zo, at 

which the orthogonal shear stress is a maximum. Zaretsky proposed that the volume should be based 

on the depth, zm, at which the maximum shear stress occurs, which is 57 percent greater than the 

depth of maximum orthogonal shear stress. 

4. It is seen that when Equation (1) is inverted to express life as a function of shear stress, the shear 

stress exponent is dependent on scatter in the life data (the Weibull slope, m). Zaretsky demonstrated 

for most materials, the shear-stress-life exponent, c, is independent of scatter in the data. As a result, 

Zaretsky (Ref. 11) modified the shear-stress exponent to be equal to cm. This makes the shear-stress 

exponent independent of scatter in life data once Equation (1) is inverted to express life. 

Once the Lundberg-Palmgren model is implemented in generalized form (Eq. (9)), the implementation of 

the Zaretsky model is quite straightforward. Modification (1) is implemented by simply setting the depth 

exponent, h, in Equation (1) to zero; modifications (2) and (3) are simply accomplished by setting the 

values of  and  to 0.30 and 0.786, respectively; finally, changing the shear-stress exponent in 

Equation (1) from c to cm satisfies modification (4). Thus, for the Zaretsky model the load capacity 

Equation (9) becomes 

    2

3

2

3

1 







  cm

m

c cm QcZ AZ kcZ kakb d u (17) 

where ka and kb are the same as defined in Equations (10a) and (10b), while the constants kcZ and AZ are 

modified as 

m

cZ

S

k 



 







   2

3

1 ln

2 and  

c

c

o

Z

E

K A

 





























2

3

3

2 1 (18) 
NASA/TM—2015-218745 11 

Single contact, stationary and rotating raceway, and bearing lives are written similar to Equations (13) to 

(14) as 

m

cm

cj

j

j Q

Q

L

3

2

1



















 (19) 

m

n

j

cm

cj

j

m n

j

m

S j Q

Q

L L

1/

1

3

1/ 2

1

1 1







































  

































   







 (20) 

 



 

 













  

n

j

m

c h

cj

j n

j R j Q

Q

L L 1

3

2

1

1 1 (21) 

and the equation of life of the bearing is identical to Equation (16). Note that we used the same 

Lundberg-Palmgren values for the shear-stress-life exponent c and the Weibull modules m of 31/3 and 

10/9, respectively. The load-life exponent p in Equation (19) comes out to be approximately 4.0 for the 

Zaretsky model as compared to p = 3.0 for the Lundberg-Palmgren model. Thus, as the load increases, 

the Zaretsky model shows a much faster dropoff in life. A schematic comparison of the two models is 

shown in Figure 2. 

Appendix A contains a sensitivity study varying the values for c and m on p and n. For the Zaretsky 

model, the values for p and n are less sensitive to variations in the Weibull modulus m than the 

Lundberg-Palmgren model. 
NASA/TM—2015-218745 12 

Model Implementation 

The fatigue life module in the bearing dynamics computer code ADORE (Ref. 31) was rewritten to 

program both the Lundberg-Palmgren and the Zaretsky models in the generalized forms, as presented 

above. In addition, the existing original Lundberg-Palmgren model was preserved for comparison 

purposes. All model coefficients such as shear-stress exponent, c, depth exponent, h, Weibull slope, m, 

and survival probability, S, are kept as variable inputs. Such an implementation eases the development of 

new values for the model coefficients when correlating model predictions with experimental data. The 

coefficients in the original Lundberg-Palmgren model, Equation (12), of course, remain constant. Along 

with applied loads, ADORE models centrifugal and thermal expansion of all bearing elements. Thus, the 

change in internal clearance and contact stresses are taken into consideration when solving for the elastic 

contact solutions. 

Experimental Data 

As stated earlier, model coefficients in the original Lundberg-Palmgren model are based on a large 

amount of unpublished pre-1940 experimental data. For this paper four sets of experimental data reported 

in References (32) and (33) were selected for computing the model coefficients. All the reported data was 

obtained with sets of ABEC-5 grade, split inner race, 120-mm bore angular-contact ball bearings, 

operating with a thrust load of 25,800 N (5800 lbf), at an operating speed of 12 000 rpm (1.44 DN). The 

surface finishes were approximately 0.05 to 0.75 m (2 to 3 in.) rms on the races and 0.025 to 0.05 m 

(1 to 2 in.) rms on the balls. Bearing geometry as input into the bearing dynamics code (Ref. 31), is 

documented in Table 1. 

All the four sets of test bearings were lubricated with polyalphaolefin (PAO) lubricant. Properties of 

the lubricant are tabulated in Table 2. Note that dataset number 2 used less viscous (thinner) oil than used 

in sets 1, 3, and 4. The operating temperatures of the four datasets were respectively, 478 K (400 F), 

492 K (425 F), 533 K (500 F), and 589 K (600 F). The lubricant film thickness to composite roughness 

ratios for the four sets were 5 or above. Standard Weibull analysis (Refs. 20 and 21) of the experimental 

data was carried out to compute the L10 bearing life along with a Weibull slope defined by a least-squared 

regression line through the data points. 

Due to the limited amount of experimental data the statistical variation in both the expected life and 

Weibull slope is significant. Thus, the data is inadequate for computation of expected Weibull slope and 

the expected L10 life. Therefore, the Weibayes method (Ref. 38) is used to compute the expected 

experimental L10 life with the Weibull slope held fixed at a value of 1.11, as recommended by 

Lundberg-Palmgren. This results in four data points, which were fitted to the life models to derive the 

model constant by using a least-squared deviation fit analysis. 

TABLE 1.—ANGULAR-CONTACT BALL BEARING DETAILS 

Bearing bore ........................................................................................... 120 mm

Bearing outer diameter (OD) ...................................................................... 190 mm

Number of balls ............................................................................................... 15

Ball diameter ................................................................................... 20.6375 mm

Pitch diameter ......................................................................................... 155 mm

Contact angle ................................................................................................. 20°

Outer race curvature factor ........................................................................... 0.52

Inner race curvature factor ............................................................................ 0.54

Bearing width ........................................................................................... 35 mm

Outer race shoulder diameter .................................................................. 165 mm

Inner race shoulder diameter ................................................................... 145 mm

Bearing material (balls and races)................................ CEVM AISI M−50 Steel

Lubricant ..................................................................................... Polyalphaolefin
NASA/TM—2015-218745 13 

TABLE 2.—TEST LUBRICANT PROPERTIES 

PAO oil lubricant Test sets 1,3,4 (Ref. 32) Test set 2 (Ref. 33) 

Additives Antiwear 

Antifoam 

Antiwear 

Oxidation inhibitor 

Kinematic viscosity, cSt at 

 311 K (100 F) 

 372 K (210 F) 

 478 K (400 F) 

443.3 

39.7 

5.8 

60 

8.9 

1.9 

Pour point K (F) 236 (–35) 225 (–55) 

Specific heat 2910 J/(kg K) at 533 K 

(0.695 BTU/(lb F) at 500 F) 

2810 J/(kg K) at 477 K 

(0.671 BTU/(lb F) at 400 F) 

Thermal conductivity 

J/(kg K) (BTU/(hr ft F)) 

0.12 at 533 K 

(0.070 at 500 F) 

0.12 at 477 K 

(0.073 at 400 F) 

Specific gravity 0.71 at 533 K (500 F) 0.85 at 477 K (400 F) 

TABLE 3.—EXPECTED L10 BEARING LIVES FOR THE EXPERIMENTAL DATASETS 

Dataset Failure index Weibull least-squared fit analysis Expected L10 life, hr with a 

Weibull slope L10 life, hr Weibull slope of 1.11 

1 10/27 1.724 311.3 195.9 

2 14/27 1.906 195.7 101.5 

3 11/26 3.560 432.2 161.7 

4 6/26 1.920 260.9 205.0 

5 6/30 1.873 1640 1535 

Once the model constants were derived, model predictions were tested against another experimental 

dataset; this one obtained by Bamberger, et al. (Ref. 34) for a set of 30 each 120-mm bore angular-contact 

ball bearings made from VIM−VAR AISI M−50 steel. The geometry of these bearings was identical to 

those outlined in Table 1, except that the free contact angle of these bearings was increased to 24. The 

bearings operated with a thrust load of 22 240 N (5000 lbf) at an operating speed of 25 000 rpm, which 

results in a DN value of three million. The operating temperature was 492 K (425 F) and the bearing was 

lubricated with the MIL−L−23699 lubricant. Lubricant properties are documented in Reference 37. 

Just for the purpose of comparison, the computed Weibull slope, associated L10 life as computed from 

the Weibull least-squared regression along with the estimated lives at a Weibull slope of 1.11 using the 

Weibayes method, are summarized in Table 3. The failure index represents the number of bearings failed 

out of the total number in the set. Datasets 1 to 4 represent the four datasets used to compute the model 

constants, while set 5 represents the high-speed bearing case used to perform the final model validation. 

Typical Weibull plots of the experimental data are shown in Figures 3(a) and (b). Clearly, the life 

estimated by Weibayes method with the Lundberg-Palmgren value of Weibull slope of 1.11 is different 

from the least-squared fit to the experimental data. However, as discussed above, due to the limited sample 

size in the present investigation and large expected variability in Weibull slope, the L10 life estimated by the 

Weibayes method is used to correlate the model predictions and derive the model constants. For a valid 

correlation of experimental life with analytical model predictions in all of the five datasets, it is essential 

to apply life modification factors to determine the corresponding basic subsurface life since the 

experimental data represents life under actual operating conditions. The commonly used Society of 

Tribologists and Lubrication Engineers (STLE) publication (Ref. 26) provides simple life modification 

factors for improved materials and lubrication conditions; the factors are applied as simple multipliers on 

the basic subsurface (Lundberg-Palmgren) life. 

Tallian (Ref. 27) presented a more comprehensive formulation of experimentally validated life 

modification factors; these factors are applied to each individual contact in the bearing and thus, the 

implementation is significantly more complex. In the present investigation since life is computed at each 

contact before summation over the races, no additional work was involved in implementing the Tallian 

life factors. Simply for the purpose of comparison, life factors based on both methods were computed for 

the five experimental datasets. The results are summarized Table 4 along with a summary of bearing 

materials and operating conditions. It is interesting to note that while the Tallian life factors are somewhat 

more conservative, the overall magnitudes of the two sets of life factors are not greatly different. 
NASA/TM—2015-218745 14 

TABLE 4.—SUMMARY OF EXPERIMENTAL DATASETS AND APPLICABLE LIFE MODIFICATION FACTORS 

Dataset Bearing 

material 

Lubricant Operating 

temperature,

K 

Thrust 

load,

kN 

Inner 

race 

speed, 

rpm 

STLE life 

factor 

(Ref. 26) 

Tallian life 

factor 

(Ref. 27) 

1 CVM AISI M−50 PAO 478 (400 F) 25.80 12 000 8.88 6.51 

2 CVM AISI M−50 PAO 492 (425 F) 25.80 12 000 8.25 6.48 

3 CVM AISI M−50 PAO 533 (500 F) 25.80 12 000 8.53 6.39 

4 CVM AISI M−50 PAO 589 (600 F) 25.80 12 000 8.10 6.15 

5 VIM−VAR AISI M−50 MIL−L−23699 492 (425 F) 22.24 25 000 32.19 30.46 
NASA/TM—2015-218745 15 

Therefore, simply due to their more comprehensive and conservative nature, the Tallian life factors were 

used in the present investigation to derive the basic subsurface fatigue life, corresponding to the 

experimental life, for validating the life models. Note that life modification factors are only used to make 

a valid comparison of analytical predictions with actual experimental data. In all other parametric studies 

only the basic subsurface lives free of any life modification factors are presented since the principal 

objective of the present investigation is to compare the various subsurface fatigue life models. 

Based on the experimental data discussed above, the model constants, ALP and AZ in Equations (9) and 

(16), were computed by least-squared fit analysis of the predicted lives against the experimental lives for 

datasets 1 to 4. Once the model constants were established, the experimental life with dataset 5 was then 

compared with model prediction to further establish reliability of model predictions. 

Since the generalized formulations presented herein show sensitivity of life with elastic modulus, the 

physical properties used for the AISI M−50 steel in comparison to those of AISI 52100 steel are 

summarized below in Table 5. 

For VIM−VAR AISI M−50 steel a significant drop in elastic modulus with increased temperature has 

been reported in the literature (Ref. 39). At the operating temperature of 492 K for dataset 5, the elastic 

modulus drops to 166 GPa. No such data is presently available for the CVM AISI M−50 steel. Thus, a 

constant value of elastic modulus for datasets 1 to 4 was assumed. 

It should be pointed out that in the present investigation no thermal analysis was undertaken. The 

available experimental data simply reports one operating temperature. This temperature is used for all 

both races and all the rolling elements. In reality the bearing will have a temperature field, which will 

vary with applied load and operating speeds. So a more advanced modeling of life will require a close 

integration of life equations and thermal interactions. Unlike the original Lundberg-Palmgren model, 

where the empirical life constant includes the elastic properties of AISI 52100 bearing steel, the newly 

developed updated Lundberg-Palmgren and Zaretsky models introduce a new elastic property parameter 

and the empirical life constants are free of any elastic properties. However, the change in contact load and 

geometry resulting from the change in elastic properties and thermal expansion of the bearing elements is 

accounted for in all models. While the operating temperature results in thermal distortion of bearing 

elements, the change in elastic properties affect the centrifugal expansion of the rotating race. Thus, both 

the operating temperature and speed affect the operating internal clearance and hence, the load 

distribution in the bearing. The room temperature and operating internal clearances for the five 

experimental datasets are summarized in Table 6. 

TABLE 5.—PHYSICAL PROPERTIES OF BEARING MATERIALS AT ROOM TEMPERATURE 

Property CVM AISI M−50 VIM−VAR AISI M−50 CVM AISI 52100 

Density (kg/m3) 7830 8027 7827 

Elastic Modulus (GPa) 203 203 201 

Poisson’s Ratio 0.28 0.28 0.277 

Thermal Coefficient of Expansion (m/m/K) 1.29810–5 1.00610–5 1.1510–5

TABLE 6.—ROOM TEMPERATURE AND OPERATING INTERNAL CLEARANCES 

UNDER EXPERIMENTAL CONDITIONS 

Dataset Room temperature 

clearance, 

mm 

Operating 

temperature, 

K 

Operating speed, 

rpm 

Operating clearance, 

mm 

1 0.1493 478 (400 F) 12 000 0.1051 

2 0.1493 492 (425 F) 12 000 0.1022 

3 0.1493 533 (500 F) 12 000 0.09374 

4 0.1493 589 (600 F) 12 000 0.08218 

5 0.2141 492 (425 F) 25 000 0.04734 
NASA/TM—2015-218745 16 

Model Correlation and Validation 

A least-squared deviation analysis between the predicted and experimental lives, for experimental 

datasets 1 to 4, yields the values of model constants, ALP and AZ, as 3.8540106

 N/m1.80 and 

3.6635105

 N/m1.332, respectively, for the updated Lundberg-Palmgren and Zaretsky models. The constant 

A, in the original Lundberg-Palmgren model is, of course, 2.4640107

 N/m1.80 as stated earlier. 

Although, as expected, units of the model constant in the updated and original Lundberg-Palmgren 

models are identical, the difference in the values corresponds to the different variable constituents of 

Equations (9) and (12a). The difference in units of the model constants between the Lundberg-Palmgren 

and Zaretsky models corresponds to elimination of the shear stress depth, h, term and replacement of 

exponent c with cm in the fundamental life Equation (1) for the Zaretsky model. 

With the above model constants, the computed dynamic load capacities for the test bearing are 

compared in Figure 4(a). As expected, the load capacities between the original and updated 

Lundberg-Palmgren models are almost identical. The observed very small difference is attributed to a 

small difference in elastic properties between the AISI M−50 and AISI 52100 steels. The Zaretsky model 

provides a significantly lower load capacity. This is primarily due to a higher load-life exponent in the 

Zaretsky model because the life at 1 million revolutions involves a very high load. 
NASA/TM—2015-218745 17 

Comparison of predicted L10 lives against the observed experimental life for the four test cases is 

shown in Figure 4(b). Again the small difference in life between the original and updated LundbergPalmgren lives is attributed to a small difference in elastic properties. Both the load capacities and 

predicted lives with the original and updated Lundberg-Palmgren models are identical if the elastic 

properties are set to those of AISI 52100 steel. This established analytically sound implementation of the 

models in the computer code. Predictions of the Zaretsky model under the test operating conditions are 

almost identical to those of the updated Lundberg-Palmgren model. Except for the scatter in the 

experimental data, the lives for the four test conditions are almost the same. Statistical analysis of the 

variance of experimental data shows that the differences in lives of the four datasets are statistically 

indistinguishable. 

Once the model constants are established by regression analysis of the above four experimental 

datasets, the model predictions are compared with another experimental dataset obtained with another 

120-mm turbine engine angular-contact ball bearing operating at 3 million DN. The material for this 

bearing is VIM−VAR AISI M−50 steel. At an operating temperature of 492 K (425 F), it has been 

reported that the elastic modulus reduces to 1.661011 N/m2

 from a room temperature value of 

2.031011 N/m2 (Ref. 39). With such a reduction in elastic modulus (about 18 percent) the L10 lives 

predicted by both the updated Lundberg-Palmgren and Zaretsky models are more than three times 

higher than those calculated by the original Lundberg-Palmgren formulation. 

Figure 5(a) shows a comparison of load capacities, while Figure 5(b) compares the predicted lives 

against the experimental data. Note that the lives predicted by both the updated Lundberg-Palmgren and 

Zaretsky models are very close to the life observed experimentally. Life predicted with the original 

Lundberg-Palmgren model, where the elastic properties are fixed to those of AISI 52100 steel at room 

temperature, is significantly lower. Also shown in Figure 5(b) are life predictions at room temperature, 

which are very similar and comparable between the various models. 

It is interesting to note while the predicted lives at room temperature are nearly identical between the 

original and updated Lundberg-Palmgren models, the original model shows a slight drop in life at the 

higher operating temperature where the elastic modulus is significantly lower. This is primarily attributed 

to increased centrifugal expansion of the rotating inner race with the lower elastic modulus at the high 

operating temperature. While the original Lundberg-Palmgren life equation does not provide for variation 

of elastic properties, the increase in contact load resulting from reduced internal clearance due to 

increased centrifugal expansion of the inner race is accounted for when the model is implemented in the 

computer code. As a result, there is a small drop in life at the higher temperature as shown in Figure 5(b). 

Although the Lundberg-Palmgren and Zaretsky models show similar results in the comparisons 

discussed above, the Zaretsky model, due to a higher load-life exponent, predicted a significantly higher 

life at light loads and a much faster drop off in life at the load increases. This is seen in Figure 6(a), where 

the predicted lives are plotted as a function of speed with the VIM−VAR AISI M−50 bearing operating at 

492 K (425 F). As the speed increases, the centrifugal loading increases, which results in a reduction of 

life. Due to the faster drop off in life with increasing load with the Zaretsky model, the predicted life is 

actually somewhat lower than that predicted by the updated Lundberg-Palmgren model. The significantly 

lower life prediction by the original Lundberg-Palmgren model is directly related to the fixed and 

significantly higher value of elastic modulus. 

Just for comparison, the results of Figure 6(a) are re-plotted in Figure 6(b) with the elastic modulus 

set to the room temperature value. Now both the original and updated Lundberg-Palmgren models show 

nearly identical lives while the Zaretsky model predicts a higher life at low speeds. 

Load-life dependence with the various models is somewhat better seen in Figure 7, where the 

predicted lives are plotted as a function of applied load at the lower speed (1.44 million DN) with AISI 

M-50 elastic modulus at room temperature. Now the predicted lives with the original and updated 

Lundberg-Palmgren models are closely identical and the Zaretsky model predicts a much higher life at 

lighter loads. As the load increases the difference in predicted lives reduces. At the highest loads the life 

predicted by the Zaretsky model is actually lower than that estimated with Lundberg-Palmgren model. 
NASA/TM—2015-218745 18 
NASA/TM—2015-218745 19 
NASA/TM—2015-218745 20 

The key contribution of the generalized expressions developed in the present investigation is 

improved modeling of variation in elastic properties of the bearing materials. Although the changes in 

contact geometry and the resulting effect on volume of the material stressed is accounted for in the 

original Lundberg-Palmgren life equation, the empirical life constant is based on constant elastic 

properties corresponding to the common AISI 52100 bearing steel and there is no provision for variation 

of elastic properties in the life equation. The empirical constants in both the updated Lundberg-Palmgren 

and Zaretsky models are free of elastic properties of the bearing materials and the life equations include a 

new elastic property ratio term, which defines the elastic properties variation in terms of a ratio of 

properties of the current material to those of room temperature AISI 52100 bearing steel. 

To more precisely demonstrate the significance of elastic property variation, Figure 8 plots life 

variation as a function of elastic modulus of the bearing material. The test ball bearing of experimental 

datasets 1 to 4 is used for these parametric runs but the elastic modulus of bearing material is varied 

arbitrarily. All runs are made at room temperature with an applied thrust load of 25 000 N and at an 

operating speed of 12 000 rpm. Although the elastic property term in the newly developed equations 

permits independent variation of elastic properties of the rolling elements and both races, just for simplicity 

the same modulus values for the rolling elements and the races are used in these parametric runs. 

As the modulus drops, lives computed with the updated Lundeberg-Palmgren models sharply 

increase; lives with the Zaretsky model are slightly greater in comparison to the updated LundbergPalmgren model but the differences are quite small under the operating conditions for these parametric 

runs. Life variation as a function of the modulus is insignificant with the original Lundberg-Palmgren 

model. A very small drop in life predicted by the original Lundberg-Palmgren model with increasing 

modulus, not clearly seen in Figure 8, is a result of variation of contact geometry and centrifugal 

expansion of the rotating race with elastic modulus. This proves that variations in life due to these effects 

are insignificant in comparison to those contributed by elimination of elastic properties from the empirical 

life constant. It may also be noted that at a modulus value of about 200 GPa, which is the value for 

AISI 52100 bearing steel at room temperature, life computed by all the models is about the same. In fact, 

the lives computed by the original and updated Lundberg-Palmgren models are identical as expected. 

As the modulus increases, the internal clearance in the bearing increases due to reducing centrifugal 

expansion of the rotating inner race, and the contact stress increases primarily due to reduced contact size 

for given load on the bearing. These effects are taken into account in all models. Figure 9 shows the 

variation in internal clearance and contact stress with modulus for the parametric runs. 
NASA/TM—2015-218745 21 

Summary of Results 

The bearing dynamics computer code ADORE (Ref. 31), which is based on the bearing dynamic 

analysis of Jones (Ref. 19), was used to develop generalized expressions for both the Lundberg-Palmgren 

and the Zaretsky rolling-element fatigue life models. Published experimental data for a jet engine bearing 

(Refs. 32 to 34) were used to derive the constants in the life equation. Model predictions were then 

compared with another set of high-speed turbine engine bearing life data (Ref. 36). Subsurface fatigue life 

models for ball bearings were generalized to permit parametric elastic property variation on subsurface 

shearing stress. Model constants were derived by least-squared regression analysis of available 

experimental life data. 

 
NASA/TM—2015-218745 22 

Generalized life equations for both the Lundberg-Palmgren and the Zaretsky models in which, unlike 

the Lundberg-Palmgren model, the shearing stress-life exponent in the fundamental life equation is 

independent of scatter in life data and life dependence on depth of critical failure stress is eliminated, 

were developed and incorporated into the commercially available bearing dynamics code, ADORE. The 

following results were obtained: 

1. For AISI 52100 steel the updated Lundberg-Palmgren model shows identical results when 

compared with the original Lundberg-Palmgren formulation. As the elastic properties deviate 

from those of AISI 52100 the predicted lives between the original and updated LundbergPalmgren models begin to deviate. 

2. For the VIM−VAR AISI M−50 bearing steel at a temperature of 492 K (425 F), where a 

reduction in elastic modulus of as much as 18 percent has been reported, the predicted L10 life of a 

typical 3 million DN gas turbine engine bearing with both the updated Lundberg-Palmgren and 

the Zaretsky models is in complete agreement with experimental data. The life calculated by the 

original Lundberg-Palmgren model is about three times lower. 

3. Parametric evaluation of the Lundberg-Palmgren and Zaretsky models demonstrates that at light 

loads the Lundberg-Palmgren models significantly underestimates the bearing life. As the applied 

load increases, the Zaretsky model shows a much faster drop off in life in comparison to the 

Lundberg-Palmgren model. Such a behavior is a direct result of the ball bearing load-life 

exponent of 4 in the Zaretsky model in comparison to 3 in the Lundberg-Palmgren model. 
NASA/TM—2015-218745 23 

Appendix A.—Effect of Varying the Weibull Modulus m and Shear-Stress 

Exponent c on the Load-Life Exponent p and the Hertz Stress 

Life Exponent n—A Sensitivity Study 

Lundberg-Palmgren Model 

In 1947, Lundberg and Palmgren (Ref. 4) applied the Weibull analysis to the prediction of 

rolling-element bearing fatigue life. In order to better match the values of the Hertz stress-life exponent n

and the load-life exponent p with experimentally determined values from pre-1940 tests on air-melt steel 

bearings, they introduced another variable, the depth to the critical shearing stress z to the h power where 

f(N) in Equation (1) of the text can be expressed using the product law of reliabilities as 

f   N dV

S V  1 ln (A1) 

and where 

  0.90

1 ln h

c m

z

N f N 

 (A2) 

The rationale for introducing z

h

 was that it took a finite time period for a crack to propagate at a distance 

from the depth of the critical shearing to the rolling surface. Lundberg and Palmgren assumed that the time 

for crack propagation was a function of z

h

. 

Applying Equation (A2) to (A1), 

  m h m c m m

S

z

V

N

1 1

0.90

1 ln 1 ln 1 1 



 



 



 



 



 







 (A3) 

where N is the life in stress cycles for an arbitrary reliability S. 

For their critical shearing stress, Lundberg and Palmgren chose the orthogonal shearing stress τo. 

From Hertz theory (Jones) (Ref. 40), 

z  pH (A4a) 

 pH (A4b) 

2 V  d  pH (A4c) 

For point contact, substituting Equations (A4a), (A4b) and (A4c) in Equation (A3), and denoting life as L

(millions of revolutions) instead on N (number of stress cycles), equation (A3) may be written as 

  n

H

m H

m

H

c m

H p

p

p p

L 1 1 h 1

1

2  































 (A5) 

 
NASA/TM—2015-218745 24 

From Zaretsky, et al. (Ref. 41), solving for the value of the exponent n for point contact (ball on a 

raceway) from Equation (A5) gives 

m

c h

n

   2 (A6) 

Lundberg and Palmgren (Ref. 4), using values of 1.11 for m, c = 10.33, and h = 2.33, from Equation (A6) 

for point contact 

9

1.11

10.33 2 2.33    n  (A7) 

From Hertz theory (Ref. 40) for point contact 

3

1

pH  Q (A8) 

Equation (A5) becomes 

n p p H Q

L 1 1   (A9) 

From Equations (A8) and (A9) for point contact, where n = 9, 

3

3

9

3    n

p (A10) 

These values of n and p for point contact correlated to the then-existing rolling-element bearing 

database, which can be assumed were generated in Sweden with air-melt-processed AISI 52100 steel 

prior to the Second World War. In their 1947 paper (Ref. 4), Lundberg and Palmgren stated that their 

database reflected a variation in the Weibull modulus m between 1.1 and 2.1 and a load-life exponent p

equal 3. It is not intuitively obvious how they selected values for the shear-stress-life exponent c and the 

exponent h related to the depth to critical shearing stress z. It was assumed by us that these values were 

made to fit their existing values by trial and error. 

To determine the sensitivity of the Weibull modulus m on the load-life exponent p and the Hertz 

stress-life exponent n, the Weibull modulus m in Equation (A6) was assumed to be 1, 1.5, and 2 with

c = 10.33, and h = 2.33. The resultant values for the load-life exponent p were 3.33, 2.22, and 1.66, 

respectively. For values the Hertz stress-life exponent n was 10, 6.67, and 5, respectively. Clearly, a 

Weibull modulus of approximately 1 best reflects the Lundberg-Palmgren database. 

The shear-stress-life exponent c in Equation (A7) was varied and assumed to be 9, 10, and 11 for a 

Weibull modulus m equal to 1. The resultant values for the load-life exponent p were 2.89, 3.22, and 3.56. 

For these values, the Hertz stress-life exponent n was 8.67, 9.67, and 10.67. Clearly, for the LundbergPalmgren model, life can be more sensitive to variations in the Weibull modulus m than to variations in 

the shear-stress-life exponent c.
NASA/TM—2015-218745 25 

Zaretsky Model 

Applying the Weibull distribution function with reliability S1, the function f (N) expressed by an 

arbitrary life N can be developed by Equation (A11). 

0.9

1 ln 1 ( ) ln

1 63 10

m m

N

N

N

N

S f N 













  













   (A11) 

Where N63 is the characteristic life at which 63.2 percent of the bearings are expected to fail. The 

conventional f(N) functions are almost always lacking in the (ln 1/0.9) term. 

Further, introducing the stress-life relation N10  –c into Equation (A11) and then substituting this 

into Equation (A1), the life N in stress cycles is given by Equation (A12). Here a1 denotes a reliability 

factor. 

c m m S

a

V

N a

1

1

1

1 ln0.9

ln , 1 1 



 



   

 



 



 







 (A12) 

For critical shearing stress , Zaretsky (Ref. 23) chose the maximum shearing stress m. 

In the case where S = 0.9, a1 = 1 and L = N/u denotes life in units of 106

 rev unit for one rotating ring, 

then Equation (A12) can be written as 

n

H

c m

V p

L 1 1 1 1/

 



 



 



 









 (A13) 

From Zaretsky, et al. (Ref. 41), solving for the value of the Hertz stress-life exponent n, for point contact 

from Equation (A13) gives 

m

n c

2   (A14a) 

and from Equation (A10) 

3

n

p  (A14b) 

If Lundberg-Palmgren values are assumed, where c = 10.33 and m = 1.11, p = 4.04 and n = 12.13 for point 

contact. 

To determine the sensitivity of the Weibull modulus m on the load-life exponent p and the Hertz 

stress-life exponent n, the Weibull modulus m in Equation (A14) was assumed to be 1, 1.5, and 2 with 

c = 10.33. The resultant values for the load-life exponent p were 4.11, 3.89, and 3.78, respectively. The 

values for the Hertz stress-life exponent n were 12.33, 11.66, and 11.33, respectively. The values for 

p and n are less sensitive to variations in the Weibull modulus m for the Zaretsky model than for the 

Lundberg-Palmgren model. 

The shear-stress-life exponent c in Equation (A14a) was varied and assumed to be 9, 10, and 11 for a 

Weibull modulus m equal to 1. The resultant values for the load-life exponent p were 3.67, 4, and 4.33. 

For values for the Hertz stress-life exponent n, the values were 11, 12, and 13. As with the Weibull 

modulus m, the variation in the values for p and n are within those of the existing database (Ref. 42). 
NASA/TM—2015-218745 26 

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