## Abstract

This article begins by describing standard bearing life models in continuous rotation before going on to explain how the bearing life can be calculated for roller and ball bearings in oscillatory applications. An oscillation factor a_{osc} is introduced, which accounts for the oscillating and stationary ring. This can be calculated numerically as a function of the oscillation angle θ and load zone parameter *ɛ* as well the parameters γ = D · cos α / d_{m} and the ball-race osculation factors. Critical angles as used by Rumbarger are also employed at low θ values. Appropriate curve-fitted relationships for both roller and ball bearings are then given for a simple calculation of a_{osc} with an accuracy of approximately 10%. Finally, several methods are suggested for estimating the ɛ parameter using a real case with a finite element analysis load distribution accounting for structural ring deformation and ball-race contact angle variations. The results derived in this article allow the lifetime of any arbitrary oscillating ball or roller bearing to be calculated.

## 1 Introduction

In 1999, Houpert [1] provided a model for calculating the roller bearing life of an oscillatory inner ring (IR) as a function of the oscillation angle *θ* and load zone parameter *ɛ*. In 2020, Breslau and Schlecht [2] provided a numerical model for calculating the life of an oscillating roller bearing accounting for both the IR and outer ring (OR) and identified some inconsistent results observed in two figures in Ref. [1]. By coincidence, the same inconsistencies were also noted by Menck in the same year in the scope of a project on the same topic on which Houpert and Menck were collaborating. The 1999 published oscillation factor of the oscillating inner ring was found to increase as the oscillation angle *θ* decreases and to “usually” decrease as the load zone parameter *ɛ* decreases, except in some case (large *θ*, low *ɛ*) where the factor was increasing as *ɛ* decreases, the latter being inconsistent. It will be demonstrated that the survival probability *S* of an oscillating ring is defined using $ln(1/S)\u223c(Qeqp\u22c5N)e$ and hence the product $Qeqp\u22c5N$, where *p* is the life—load exponent, *Q _{eq}* is the equivalent load, and

*N*is the number of stress cycles. Both parameters (

*Q*and

_{eq}*N*) can be defined using either a constant oscillating arc (2

*θ*) or a variable loaded arc

*H(ψ,θ)*, see the upper images in Fig. 2. Since these arcs are used at the denominator when defining $Qeqp$ and numerator when defining

*N*, so that the same results are obtained when using either 2

*θ*or

*H(ψ,θ)*. The mindless error conducted in 1999 was to use 2

*θ*for defining $Qeqp$ and

*H(ψ,θ)*for defining

*N.*This single error in Ref. [1] has been rectified, and as such the first objective of this article is to explain the 1999 error and 2020 correction as well as to present two updated figures. In addition, the original corrected model is expanded to include the outer stationary ring’s lifetime to obtain the lifetime of the entire bearing. Results are given for both ball and roller bearings of arbitrary dimensions by means of an oscillation factor

*a*

_{osc}.

Before describing bearing life in continuous rotation and oscillatory conditions, the authors found useful to share, in a specific chapter, some general comments about life models and concepts and then continue with standard exponents and life model. Then, analytical relationships for calculating the bearing life in continuous rotations are developed as these are required for explaining the correction factor to apply in oscillatory applications. This will be done with appropriate line contact (LC) sets of exponents (LC Dominik [3] or LC ISO) for roller bearings and using the appropriate set of point contact (PC) ISO exponents for ball bearings. The aforementioned analytical relationships can also be used for explaining and deriving the dynamic capacity (or rating) and the dynamic equivalent load using standard set of exponents (*c*, *h*, and *e*) or any new set.

The oscillation factor will then be described, accounting for both rings, as a function of the oscillation angle *θ*, the load zone parameter *ɛ*, geometrical parameter *γ,* and the osculation factors *f _{i}* and

*f*used in ball bearings.

_{o}Finally, some useful curve-fitted relationships will be given for easy calculation of the final oscillation factor as a function of these input parameters. Moreover, approaches to calculate the required parameter *ɛ* from the results of the finite element analysis (FEA) are given.

Therefore, the following sections about life in continuous rotation are not new but have the merit of explaining in an analytical manner how to derive miscellaneous bearing life relationships. Readers who are solely interested in the oscillation factor may proceed directly to Sec. 3.3.

## 2 General Comments About Life Models and Concepts

*S*can be calculated using:

*N*is the number of stress cycles,

*τ*

_{0}is the maximum orthogonal shear stress amplitude found at depth

*z,*and

*V*is the macro stressed volume.

Cste is a constant, and *c*, *h*, and *e* are exponents to be defined experimentally. The exponent *e* is also called the Weibull slope.

The model has since been fine-tuned using elementary small volumes *dV* to integrate through the volume using a triple integration (accounting for possible pressure spikes at the edge of the roller-race contact and pressure increases on the skin of rough surfaces) and/or using an endurance limit when calculating the final life, leading to possible infinite life at low load when the maximum stress is below the endurance limit, see, for example, the work of Ioannides and Harris [6], Houpert and et al. [7], or Gnagy et al. [8]. Ioannides’ approach served as the basis in current ISO standards, see Refs. [9,10], for example, where the factor *a _{ISO}* has been outlined.

In all the previously cited references, the material was considered homogeneous with an implicit uniform distribution of inclusions in the volume, justifying the exponent 1 on the volume *V*.

In this article, the authors offer some general comments about life models and steel cleanliness effects, even if such comments are not the primary objectives of this article.

Ai developed an advanced life model in Ref. [11] accounting for steel cleanliness when simulating the number of inclusions (in a given stressed volume), inclusions acting as stress raisers. A steel cleanliness life reduction factor was then found to be inversely proportional to the cumulative length of inclusions raised to the power 0.865.

Furthermore, it can be demonstrated that for a given steel cleanliness, the number of inclusions, hence also the cumulative length, is proportional to the volume, meaning that one can question the exponent 1 on the volume *V*.

More importantly, it can also be demonstrated that, when simulating a given high quality steel cleanliness (or inclusion density), the probability of finding one inclusion in a batch of 24 small ball bearings (ball bearings of size 6204 tested in Ref. [6]) is almost nil, thus explaining the perceived infinite life and introduction of an endurance limit. However, when simulating bearings 100 times larger (as found in wind turbine applications, for example), the stressed volume is approximately 1 million times greater and many inclusions will be numerically simulated, causing a true risk of a finite life, so that the concept of endurance limit and infinite life can be questioned too. Bearing endurance tests are conducted on small- to medium-sized bearings for testing current ratings, but very large size bearings are not tested so that current dynamic ratings of large size bearing can be questioned. Note that a higher steel quality is used in large bearings for compensating the risk of inclusions effects on life or rating.

Even when dealing with small-size to medium-size bearing, ratings should be redefined using recently defined exponents (*c*, *h*, and *e*) and life-load exponent *p*. Large variations of the ratings and calculated bearing life (with the new set of exponents) can be expected when extrapolating these calculations to large size bearings.

Ai did not use any endurance limit and suggested a set of exponents (*c* = 11.385, *h* = 0.319, and *e* = 1.278), leading to a life-load exponent *p* equal to 4.72 when using Eq. (5) developed later (instead of 10/3 in Dominik’s model or 4 in ISO standards) for matching endurance test results. This exponent is in line with the proposal of Zaretsky et al. of using *p* = 5 in Ref. [12]. For ball bearings, Londhe suggested *p* = 4.1 (instead of 3) in Ref. [19].

## 3 Bearing Life Calculations in Continuous Rotation

In the scope of this study, roller and ball bearing rating *C* and dynamic equivalent load *P*_{eq} relationships will be derived using standard exponents, keeping analytically track of the (*c*, *h*, and *e*) exponents and without using an endurance limit (as initially done by Lundberg and Palmgren). Harris and Kotzalas conducted a similar exercise in Ref. [13], splitting this exercise into steps to define first a contact rating *Q _{c}* and then an equivalent load

*Q*obtained using appropriate summations as also done in ISO 16281.

_{e}However, we can employ more concise analytical calculations here (thereby avoiding additional steps) to derive the roller and ball bearing rating and dynamic equivalent load, drawing inspiration from Dominik’s work [14] (for roller bearings only) and Houpert’s approach (also for roller bearings) described in Ref. [3].

Since the primary focus of this article is to define an oscillation factor, this article describes the main concepts used for calculating the bearing life of the inner ring and outer ring and, in particular, identifies the differences to be taken into consideration when studying oscillatory applications. The details presented in this section could be used in the future should the need arise to revisit rating and dynamic equivalent load relationships with any new set of exponents (*c*, *h*, *e*, and *p*).

*Q*

_{max}that can be calculated as a function of the radial load

*F*or axial load

_{r}*F*, the number of rolling elements

_{a}*Z*, the nominal contact angle

*α*, and the Sjövall integral

*J*or

_{r}*J*, which are only a function of the load zone parameter

_{a}*ɛ*(defined later in Sec. 5 using miscellaneous approaches).

The Hertzian exponent *n* is equal to 10/9 for roller bearings and 1.5 for ball bearings and 2 · *ψ _{l}* is the loaded arc (to be compared later with the loaded arc

*H*in oscillatory application from Sec. 3.1.1).

*J*and

_{r}*J*can also be calculated via the integrals

_{a}*L*and

_{r}*L*used by Houpert in Ref. [15].

_{a}Hertz theory gives for roller-race line contact: *τ*_{0} = 0.25 · *P*_{max} (where *P*_{max} is the maximum contact pressure) and *z* = 0.5 · *b*, where *b* is the half contact width. Slightly smaller ratios are used when calculating ball-race point contact. Note that *τ*_{0} is described next as *τ* for allowing the possibility of introducing the index *i* and *o*, differentiating the inner from the outer ring, respectively. *P*_{max} (therefore also *τ*) and *b* (therefore also *z*) can be calculated as a function of the contact load using Hertz analytical relationships for LC or Hertz numerical results for PC (requiring complex numerical calculations of elliptical integrals), curve fitted by Houpert in Refs. [16,17].

In the following, outer ring results will be expressed as a function of the inner ring ones. For the sake of simplicity, it will first be assumed that the inner ring (index *i*) is rotating and the outer ring (index *o*) is stationary, although more general cases (for example, both rings rotating) could also be considered.

### 3.1 Roller Bearing Life.

*P*

_{max}(for defining

*τ*) and

*b*(for defining

*z*), which are not further discussed here, and also assuming centrifugal and/or gravity force to be negligible, the Hertz LC on the inner (

*i*) and outer (

*o*) race is described as follows:

Here, *E _{eq}* =

*E/(*1 –

*v*

^{2}

*)*= 2.26 × 10

^{6}N/mm

^{2}is the equivalent Young's modulus derived from the Young's modulus

*E*and Poisson’s ratio

*v*of steel,

*L*is the equivalent roller-race length, and

_{i,o}*R*is the equivalent contact radius in the rolling direction

_{x_eq}*x*. Variables

*D*and

*d*are the rolling element diameter and pitch diameter, respectively, and

_{m}*α*is the nominal contact angle.

*S*of an elementary small volume

_{dψ_i,o}*dV*defined by its arc angle

_{i,o}*dψ*, mean race radius

*R*, effective roller-race length

_{i,o}*L*, and depth

_{i,o}*z*.

*S*(corresponding to a given number of stress cycles

_{dψ_i,o}*N*) as a function of the load

_{i,o}*Q*and an elementary volume represented by

*dψ*. For LC, for example, one obtains:

Equation (5) shows that the number of stress cycles *N* for a given survival probability *S _{dψ}* is described by

*N*∝

*Q*So far

^{−}^{p}.*Q*may have been considered constant, but the use of an elementary volume represented by

*dψ*will make it possible to account for a variable load on the rotating ring (via the introduction of an equivalent load

*Q*, in Eq. (6)) or stationary ring (via the product of survival probabilities of all elementary volumes as will be shown in Eq. (11)).

_{eq}#### 3.1.1 Rotating Race.

Let us now consider a varying load *Q* that each elementary volume (defined by *dψ*) of the rotating ring will face during its excursion through the load zone spanning from –*ψ _{l}* to +

*ψ*, the loaded arc being 2 ·

_{l}*ψ*. The percentage of occurrence

_{l}*%*for this elementary volume of facing any load level

_{j}*Q*is constant. Each load

_{j}*Q*corresponds to a potential number of stress cycles

_{j}*N*. The final or weighted number of cycles

_{j}*N*that this elementary volume can endure is defined using Miner’s rule and results in the derivation of an equivalent load

*Q*for the rotating ring.

_{eq_rot}*Z*· (2 ·

*ψ*

_{l}

*)*/(2 ·

*π*)), assumed to be a real number, defines the occurrence $(%=1Z\u22c52\u22c5\pi 2\u22c5\psi l)$ of having one elementary small volume

*dV*experiencing a load

*Q*corresponding to a potential cycle number

_{j}*N*

_{j}=

*N*

_{ref}· (

*Q*

_{ref}/

*Q*

_{j})

^{p}. Miner’s rule then defines the final number of stress cycles (

*N*) the elementary volume can endure:

The load *Q _{eq_rot}* can now be considered as constant for all elementary volumes (represented by

*dψ)*of the rotating ring (as opposed to what will be seen in oscillatory applications) and can be used for calculating

*S*of the rotating ring according to Eq. (5). $Qeq_rotp$ can also be considered as the mean value of

_{dψ_i}*Q*

^{p}over the loaded arc of an elementary volume’s movement with 2 ·

*ψ*appearing at the denominator.

_{l}*freq*, the continuous rotating frequency (in revolutions per second) of the inner and outer rings, the cage or rolling element set frequency reads:

_{i,o}*N*that the volume

_{i}*dV*will endure during a time

_{i}*t*(corresponding to a number of revolutions

_{i}*N*=

_{rev}*freq*) because of its excursion through the load zone is calculated using the relative frequency (

_{i}· t_{i}*freq*) = 0.5 · (1

_{i}− freq_{c}*+ γ*) ·

*freq*. Moreover, only loaded elements in the load zone are considered, as defined earlier:

_{i}Note how the loaded arc 2 · *ψ _{l}* seen by all elementary volumes (appearing at the numerator of

*N*and denominator of $Qeq_rotp$) disappears in Eq. (5) when using the product $(Qeq_rotp\u22c5Ni)e$. This explains why Refs. [4,13] obtain the same result without consideration of the load zone size, replacing 2 ·

_{i}*ψ*with 2 ·

_{l}*π*, for example.

The final survival probability *S _{i}* of the entire rotating ring is obtained using the product of survival probabilities

*S*or the integral of ln(1/

_{dψ_i}*S*) as defined in Eq. (5), explaining the factor 2 ·

_{dψ_i}*π*in front of the mean race radius

*R*when defining the volume

_{i}*V*, see Eq. (9).

_{i}*Q*and rotating ring, the load

*Q*must therefore be replaced with $Qeq_rot=Qmax\u22c5(\pi \psi l\u22c5Kbi2)1/p$, defined by Eq. (6), when calculating

*τ*,

_{i}*z*, and

_{i}*V*

_{i}:

#### 3.1.2 Stationary Race.

*dV*defined by

_{o}*dψ*and located at the angle

*ψ*is always facing the same load

*Q(ψ)*. The final survival probability of the stationary ring is then obtained using the product of surviving probabilities of each each elementary volume

*dV*, hence by integrating ln(1/

*S*) through the load zone, leading to another relationship for defining the equivalent load

_{dψ_ο}*Q*to use instead of

_{eq_stat}*Q*and also the use of 2 ·

*ψ*(instead of 2 ·

_{l}*π*) when calculating the stationary race volume.

*dψ*) in the load zone will endure is calculated using the frequency

*freq*or simply

_{c}–freq_{o}*freq*, since the outer race is assumed to be stationary.

_{c}*S*of the stationary outer ring can be expressed as a function of the rotating inner ring’s survival probability, introducing the important parameter

_{o}*B*, which describes the effect of the stationary ring on the survival probability.

#### 3.1.3 Combining Both Raceways.

*S*of surviving a given number of revolutions

*N*is equal to the product

_{rev}*S*, resulting in:

_{i}· S_{o}*B _{LC}* can also be used for defining the life of the outer race as a function of the life of the inner race, see Eq. (31).

*S*(commonly

*S*= 0.9) as well as any set of exponents (

*c*,

*h*,

*e*, and

*p*), see Ref. [3]:

The second equation in Eq. (15) is important since it can be used to derive analytically the bearing life *N _{rev}* as a function of the survival probability, applied radial load, and load zone, keeping track of any exponents

*c*,

*h*,

*e*, and

*p*selected. It can also be used to derive the dynamic rating and dynamic equivalent load using the selected exponents

*c*,

*h*,

*e*, and

*p*.

When fixing the survival probability to a reference value (*S _{ref}* = 0.9), the number of revolutions obtained is often called

*L*

_{10}(in revolutions) and the corresponding radial load

*F*is the dynamic rating called

_{r}*C*in this article or

_{ref}*C*(or

_{r}*C*

_{1}in this article) at ISO and

*C*

_{90}by Dominik.

*C*

_{1}an

*d C*

_{90}are hence defined using a life of reference (

*L*

_{10}=

*L*= 1 million or 90 million revolutions, respectively), but also with a load zone of reference (2 ·

_{ref}*ψ*= 180 deg for ISO and 150 deg for Dominik) used for defining all integrals (

_{l_ref}*J*,

_{r_ref}*K*, and

_{bi_ref}*K*and the ratio $Kbe/Kbie$ used in

_{be_ref})*B*.

_{LC_ref}*C*

_{90}rating rearranged in Eq. (17) using

*L*= 1.141 ·

_{o}*L*(not explained in Ref. [14]) and Dominik exponents from Eq. (5).

_{i}*M*is called the material factor, while

*H*is called the geometrical factor:

Cste is equal to 1.39 · 10^{−49} for matching *M* = 57.6 (used for defining Dominik *C*_{90} corresponding to 90 million revs) or *M* = 222.18 (used for defining Dominik *C*_{1} corresponding to 1 million revs) with a load zone of reference equal to 150 deg.

*C*

_{1}rating and exponents on (

*L*· cos

*α*),

*Z*, and

*D*described by Eq. (11.156) in Ref. [13] using

*L*=

_{o}*L*and LC ISO exponents from Eq. (5) with a load zone of 180 deg. Cste must then be fixed to 3.8 × 10

_{i}^{−44}to obtain

*M*= 189.43 and

*M · H*=

*b*in the entire range of

_{m}· f_{c}*γ*.

*b*

_{m}is equal to 1 for drawn cup needle roller bearings, 1.1 for cylindrical roller bearings, tapered roller bearings, and needle roller bearings with machined rings, and 1.15 for spherical roller bearings. The geometrical factor

*f*is given in tables in Ref. [9] as a function of

_{c}*γ*.

#### 3.1.4 Differences to ISO.

*P*used for calculating the life (relative to the reference life) as soon as one deviates from reference conditions, either because the load zone deviates from its reference or because the radial load is not equal to the dynamic rating.

_{eq}The latter ratio can be plotted versus the load zone parameter *ɛ* or the ratio *F _{a}*/

*F*tan

_{r}/*α*or

*F*· tan

_{r}*α/F*, see Fig. 1 or 3 in Ref. [3], or can be used for defining the load zone factor using both rings (thus including

_{a}*B*) or only the rotating inner ring (thus fixing

_{LC}*B*= 0), see Houpert’s Figs. 6 and 7 in Ref. [3].

_{LC}Minor differences are observed when using one or two rings for calculating *P _{eq}*, so only

*P*is shown in Fig. 1.

_{eq_IR+OR}A degree 6 polynomial relationship is also suggested in the same figure for easily calculating the curve-fitted ratio *P _{eq_IR+OR_cf}* as a function of the ratio

*F*· tan

_{r}*α/F*.

_{a}Note how *P _{eq}* increases (and therefore the life decreases) when

*F*is either very large or small (the latter causing a narrow load zone) with both cases causing an increase of

_{a}*Q*

_{max}.

The ratio *P _{eq_Catalog}/F_{r}* is fixed to 1 in any “Catalog” approach as soon as the estimated ratio

*F*tan

_{a}/F_{r}/*α*is smaller than 1.5 and is equal to 0.4 · (1

*+ F*tan

_{a}/F_{r}/*α*) otherwise. Such a simplification can result in very erroneous overestimation of the bearing life at low axial force (or narrow load zone). Also, in a Catalog approach, the final axial force

*F*

_{a_}_{1,2}in each row is not calculated precisely, but rather estimated using the axial force equilibrium accounting for both induced (by the radial force) axial forces (calculated using approximately

*±*1.26 ·

*F*

_{r_}_{1,2}· tan

*α*if

*ɛ*= 0.5) and the external axial force, meaning the final axial force is not very accurate in the unseated row (having a narrow load zone).

_{ref}As a reminder, the parameter *B* has been introduced for including the stationary outer ring. A similar exercise could be conducted for introducing another parameter (for example, *C*) accounting for the roller set effects on bearing life and rating using (1 *+ B* + *C*) instead of (1 *+ B*), but such an exercise is out of the scope of this study only describing standards.

In addition, similar relationships could be derived for describing the axial dynamic rating and dynamic equivalent load. Slightly different relationships can be derived when accounting for a rotating outer race with a zero or nonzero inner race rotation.

Finally, an inconsistency can be noted when using the ISO roller bearing life practices, where the final *L*_{10} life is calculated using (*C*_{1}*/P _{eq})*

^{10/3}, hence

*p*= 10/3, while the rating and equivalent load are defined with

*p*= 4.

### 3.2 Ball Bearing Life.

*P*

_{max}and

*b*correspond now to the PC ones. The contact length 2 ·

*a*as suggested in Ref. [16] is also used in lieu of the effective roller-race length:

*CP*,

*CB*, and

*CA*have been curve-fitted in Ref. [16] as a function of the ratio

*k*(ratio of equivalent radius in the

*y*and

*x*directions), for example:

*f*and

_{i}*f*are the osculation ratio of the IR and OR, respectively, typically of the order 0.53, ranging from 0.51 to 0.56.

_{o}When analytically developing a relationship for ln(1/*S _{i}*) (in a similar manner to Eq. (15), but not explicitly shown in this article), the product $CAi\u22c5CBi1\u2212h\u22c5CPic$ will be used when tracing the final exponents applied on (1 ±

*γ*) and osculation ratios.

*B*:

_{PC_ISO}*ɛ*= 0.5 (reference case used for deriving the rating), $KbeKbie=1.0699$ and

*f*, used herein for deriving

_{c}*B*:

_{Harris_ref}Only minor differences (due to the curve-fitted relationships used not only by Houpert but also by Palmgren) are observed.

*C*

_{1}, found in this article to be proportional to:

### 3.3 Final Results to Retrieve for Moving to the Calculation of the Oscillation Factor.

The relationships described earlier are not only important for analytically deriving bearing life models in rotating applications but also for understanding the development of life models in oscillating applications.

*B*can also be used for deriving the life

*t*(in time: seconds or hours, for example) of the stationary outer ring as a function of the rotating inner ring life

_{o}*t*using Eq. (15) with

_{i}*B*or

_{LC}*B*, see the following demonstration:

_{PC}The reader is also reminded that the life of the rotating inner ring is finally calculated using the product $(Qeq_rotp\u22c5Ni)e$ in which the loaded arc 2 · *ψ _{l}* cancels out. Moreover, for a given

*ɛ*, the equivalent load $Qeq_rot$ as well as the loaded arc 2 ·

*ψ*of an elementary volume on the rotating race are constant for any elementary volume

_{l}*dV*on the rotating inner ring.

_{i}In Sec. 4, when investigating oscillatory applications, it will become apparent that the equivalent load of the oscillating ring and the loaded arc *H* of an elementary volume on the oscillating race will be a function of *ɛ*, *ψ*, and the oscillation angle *θ*.

## 4 Bearing Life Calculations in Oscillatory Applications

Based on Sec. 3, we will now derive a factor *a _{osc}* to correct the lifetime for oscillatory behavior.

### 4.1 Oscillating Inner Ring.

An oscillatory inner ring oscillating between ±*θ* at an oscillating frequency *freq _{i}* will be studied first. The loaded zone is still defined by the angle

*ψ*as shown in the upper left corner of Fig. 2. Each elementary volume

_{l}*dV*is defined by its position

_{i}*ψ*, nil in the 12 o’clock position and equal to 90 deg or

*π/*2 rad in the 9 o’clock position in Fig. 2.

#### 4.1.1 Loaded Arc H(ψ).

As explained in Secs. 4.1.2 and 4.1.3, the calculation of the loaded arc *H(ψ)* that an elementary volume will face during one oscillation is conceptually needed for correctly calculating the number of stress cycles that each elementary volume *dV _{i}* (located at the orbital angle

*ψ*on the oscillating inner ring) will endure as well as its equivalent load

*Q*but is not essential because

_{eq}(ψ)*H(ψ)*will cancel out in the final derivation of the oscillation factor. However, it is useful to explain how

*H(ψ)*can be derived for obtaining accurate calculation of the lower and upper bounds of the required integrals used in Eq. (35), Sec. 4.1.3.

It is shown in Fig. 2 that the loaded arc *H(ψ)* is a function of *θ*, but it is now also a function of the initial orbital angle *ψ* and load zone angle *ψ _{l}*.

*H(ψ)*can be calculated numerically as shown and used in Ref. [1]:

An analytical relationship for *H(ψ)*, only applicable in the case *θ > ψ*_{l}, is given in Table 1 of Ref. [1] and can now be extended to all cases (see updated Table 1), including the case not considered analytically in 1999 where two partial incursions of *dV _{i}* in the load zone (by its two ends) occur as shown in the lower left corner of Fig. 2 (with an example corresponding to

*ψ*= 150,

_{l}*θ*= 120, and

*ψ*= 160 deg):

|ψ| ≤ |θ − ψ_{l}| | (θ − ψ_{l}) ≤ |ψ| ≤ (ψ_{l} + θ) | |ψ| ≥ ψ_{l} + θ |

H = 2 · Min(ψ_{l}, θ) | H = (θ + ψ_{l} − |ψ|) + Max(0, |ψ| + θ + ψ_{l} − 2 · π) | H = 0 |

|ψ| ≤ |θ − ψ_{l}| | (θ − ψ_{l}) ≤ |ψ| ≤ (ψ_{l} + θ) | |ψ| ≥ ψ_{l} + θ |

H = 2 · Min(ψ_{l}, θ) | H = (θ + ψ_{l} − |ψ|) + Max(0, |ψ| + θ + ψ_{l} − 2 · π) | H = 0 |

The lower right corner of Fig. 2 shows how *H(ψ)* varies as a function of *ψ* (with *ψ _{l}* = 150 and

*θ*= 120 deg), with the possibility of observing a second plateau corresponding to

*H*= 180 deg in Fig. 2 when a second incursion in the load zone occurs (by the other end of the load zone).

*H*

_{1}and

*H*

_{2}; hence, the final value

*H*=

*H*

_{1}

*+ H*

_{2}:

With the lower and upper bounds accurately defined, the required integrals can be calculated using only a few points (101 points for the results in this article, using an Excel table) and an accurate integration method.

#### 4.1.2 Number of Stress Cycles.

*dV*on the oscillating inner ring will endure is now defined using:

_{i}*N*is the number of oscillations of the inner ring. Note the factor 2 introduced before

_{osc}*Z*because the oscillation angle goes from 0 to

*–θ*, then +

*θ,*and back to zero, causing two crossings through the loaded arc.

#### 4.1.3 Equivalent Load Q_{eq}(ψ).

*Q*that a volume

_{eq}(ψ)*dV*located at orbital angle

_{i}*ψ*on the oscillating inner ring will endure is defined using the average value of

*Q*over the arc

^{p}*H(ψ)*.

Therefore, the equivalent load *Q _{eq}* and loaded arc

*H*are not constant and vary as a function of

*ψ*, as shown in Fig. 3 using a given example.

#### 4.1.4 Final Survival Probability of the Oscillating Inner Ring.

The survival probability *S _{dψ_i}* of the elementary small volume is obtained as in Eq. (5) using the product $[Qeqp(\psi )\u22c5Ni(\psi )]e$, in which the loaded arc

*H*cancels out since it is used at the denominator when defining

*Q*and numerator when defining

_{eq}^{p}*N*. The final results are therefore identical irrespective of whether

_{i}*H*is defined correctly (conceptually at least with the number of stress cycles being proportional to a variable loaded arc

*H(ψ)*), defined as constant and proportional to 2 ·

*θ*(as is the case in Refs. [2,18]) or defined as constant and equal to 1, for example (which makes no sense).

The only error in Ref. [1] is that 2 · *θ* was used for defining *Q _{eq}^{p}*, while

*H*was correctly used for defining

*N*, meaning that cancellation of the denominator and numerator was not possible.

_{i}The following steps are described in Ref. [1], where the final survival probability *S _{i}* of the oscillating inner ring is obtained by integrating ln(1/

*S*) over the entire inner ring volume

_{dψ_i}*V*, leading to the use of a double integral.

_{i}#### 4.1.5 Oscillation Factor a_{osc_IR} of the Oscillating Inner Ring.

*a*describing the life ratio oscillating/continuous of the oscillating inner ring can be defined as follows:

_{osc_IR}These relationships are identical to Eq. (32) in Ref. [1] (where the factor 2 · *π* was missing—a typographical error—but used in the calculations).

*a*can be simply defined using:

_{osc_IR}*n · p*is nil outside of the loaded range (–

*ψ*,

_{l}*ψ*), and

_{l}*n*,

*p*, and

*e*are defined as in Eqs. (5) and (25) for roller and ball bearings, respectively.

*θ*is small.

When fixing the acceptable accuracy to 10%, the aforementioned approximation can be used for *θ* < 36 deg when *ɛ* > 0.05. The accuracy improves to 2.34% when limiting *θ* to 10 deg and improves for any *θ* values as *ɛ* increases.

### 4.2 Oscillation Factor *a*_{osc_OR} of the Stationary Outer Ring.

_{osc_OR}

On the stationary outer ring, the number of stress cycles that any volume *dV _{o}* endures over a time period

*t*can be defined using the cage frequency

_{o}*freq*and is simply equal to 2 ·

_{c}*Z*· (2 ·

*θ)*/(2 ·

*π)*·

*freq*

_{c}·

*t*

_{o}, whereas it was

*Z*·

*freq*

_{c}·

*t*

_{o}in continuous rotation.

*a*applicable to the stationary outer race therefore reads:

_{osc_OR}*a*will also be called

_{osc_OR}*a*in the following, since a similar relationship was described in Ref. [13] without explaining that it is specific to the stationary race, implicitly suggesting its use when accounting for both the inner and outer rings.

_{Harris}### 4.3 Results and Figures.

We will now apply the aforementioned calculations to a variety of cases. Starting with the inner ring factor *a _{osc_IR}*, we will then turn to the factor

*a*

_{osc_IR+OR}for the entire bearing and consider the effect of

*γ*and osculation.

#### 4.3.1 a_{osc_IR} Results Obtained.

It is now possible to correct Figs. 5 and 6 published in Ref. [1] and replace them with the following Figs. 4 and 5. In Fig. 4, the oscillation factor of the rotating IR can be seen as increasing when the oscillation angle decreases. Furthermore, by using any given oscillation angle, the oscillation factor decreases as *ɛ* decreases or increases as *ɛ* increases for reaching asymptotically a plateau when *ɛ >* 4.

Figure 5 shows the same results as Fig. 4, but divided by *a _{Harris}*. The results for

*θ*= 180 are not visible in Fig. 5, because they are constantly equal to 1, therefore being identical to the results for

*θ*= 360.

For the sake of completeness, Figs. 6 and 7 are presented using the ISO line contact (for roller bearings) and point contact (for ball bearings) sets of exponents, respectively, observing a smaller effect of *ɛ* on the final factor, the lowest ratio being of the order of 0.8 for ISO instead of 0.5 for Dominik.

Appropriate curve-fitted relationships for calculating *a _{osc_IR}* are presented in Sec. 4.5.1.

#### 4.3.2 Oscillation Factor a_{osc_IR+OR} Combining Both the Inner and Outer Rings.

*a*

_{osc_IR+OR}to use for the entire bearing, combining the IR and OR. Using Eq. (31) and the previous results, it is relatively easy to define the final oscillation factor using both rings. By calling

*t*the survival time and by using the index

*cont*for “continuous rotation” and

*osc*for “oscillating condition” as well as

*i*for “inner ring” and

*o*for “stationary outer ring”, one can write:

Equation (42) can be used for LC or PC using the appropriate relationships and values for *B*.

Equation (42) is already used in Ref. [13] to produce Figs. 11.28 and 11.29, unfortunately with an incorrect value of *a _{osc_IR}* in some cases.

Figure 11.28 should, for example, be replaced Fig. 8 (corresponding to *L _{o}* =

*L*and

_{i}*γ*= 0.1),

*γ*= 0.1 being a typical geometrical ratio (

*γ*=

*D·cos*) used when deriving Fig. 11.28. The final oscillation factor continuously decreases as

_{a}/d_{m}*ɛ*decreases, while this was not always the case in Fig. 11.28, especially when

*θ*was large. Note that Fig. 8 is also shown in the appendix using the same format as in Ref. [13] (

*A*here called

_{osc},*a*

_{osc_IR+OR_LC}, instead of a ratio relative to Harris), to compare with Figs. 11.28 and 11.29.

Again, for the sake of completeness, the ISO LC and PC results are shown in Figs. 9 and 10 using *γ* = 0.1 (and *f _{i}* =

*f*when defining

_{o}*B*).

_{PC}Appropriate curve-fitted relationships are presented in Sec. 4.5.2.

#### 4.3.3 Effect of γ.

The previously described models can also be used for scanning on *γ* and/or the osculation ratio *f _{o}*, for example, see Fig. 11 (using

*θ*= 1 deg and

*f*=

_{i}*f*).

_{o}#### 4.3.4 Effect of the Osculation Ratio.

When scanning on the outer race osculation ratio, Fig. 12 is obtained:

Moderate, but not negligeable, variations of the oscillation factors can be noticed when scanning on *γ* or the osculation factor.

### 4.4 Use of the Critical Angles.

*dV*during half an oscillation period, therefore with no overlap with the adjacent rolling element producing a second impact. It is defined using the relative race-cage speed as follows:

The oscillation factor to use when *θ* < *θ _{crit}* is described in Refs. [2,18]. Some approximations are used, such as the replacement of

*Q*with the load of a stationary ring

_{eq}(ψ)*Q(ψ)*.

*f*can be derived from Ref. [2] to apply to the previously described oscillation factors:

_{θ_crit_ι,o}Breslau and Schlecht calculated numerically in Ref. [2] the oscillation factor (combining both rings) of a needle roller bearing with *Z* = 23 rollers and *γ* = 0.1429, corresponding to an inner and outer ring critical angles of 13.7 and 18.26 deg, respectively; see Fig. 8 in Ref. [2] obtained with *ɛ* = 0.5.

### 4.5 Curve-Fitted Relationships.

It is important to provide curve-fitted relationships of the oscillation factor to allow application engineers to easily calculate it. Two types of curve-fitted relationships will be suggested:

An accurate set of relationships using a distinct curve-fitted relationship for

*a*and_{osc_IR}*B*, subsequently used for defining*a*_{osc_IR+OR}via Eq. (42). The effects of*γ*and/or the osculation factor (included in*B*) are then considered._{PC}A simplified and single relationship in which

*γ*has been fixed to 0.1 and the osculation factors*f*and_{i}*f*are assumed to be equal._{o}

#### 4.5.1 Curve Fitting of a_{osc_IR}

The error obtained, measured via the ratio $(|aosc_cf\u2212aosc|/aosc)$, is lower than 4.6%, except when curve fitting the results obtained using the *LC_Dominik* exponents, where a maximum error of 18% can be found.

Note the small effect of *θ* on *a _{osc}/a_{Harris}* at low values of

*θ*, the product 0.014818

*· θ*being small relative to 1.4689 when studying ball bearings (index

*PC*), for example. The load zone parameter

*ɛ*can be seen to have more influence on the ratio.

*θ*values only) using Eq. (39) and appropriate curve fitting of $Kbi/Kbe1/e$, leading to:

#### 4.5.2 Curve Fitting of B.

*B* has been defined previously as a function of the ratio $Kbe/Kbie$, which needs to be curve fitted as a function of *ɛ* (for a miscellaneous set of exponents *n* and *p*). This ratio tends toward the asymptote 0.5^{(e−}^{1)} when *ɛ* tends towards infinity, with the result that it is appropriate to curve fit $(Kbe/Kbie\u2212(1/2)e\u22121)$ as a function of *ɛ* (Fig. 14).

*a*is, compared with Eq. (42):

_{osc}*e*= 1.5 or 9/8 or 10/9 when using the

*LC_Dominik*,

*LC_ISO,*or

*PC*exponents, respectively, according to Eqs. (5) and (24) for the respective choice of exponents.

#### 4.5.3 Simplified Relationships.

The simplified curve-fitted relationships have been defined using *γ* = 0.1 and *f _{i}* =

*f*.

_{o}*θ*

_{crit}= 2 ·

*π*/

*Z*) in Eq. (44) for both rings when defining

*f*, leading to:

_{θ_crit}The final oscillation factors obtained using the curve-fitted relationships or the noncurve-fitted approach (with *γ* = 0.1) differ by about 10% maximum for both ISO LC and PC cases.

## 5 Estimation of *ɛ* and Life Example in an Oscillatory Application With an Finite Element Analysis Load Distribution

The oscillation factor described above requires knowledge of the load zone parameter *ɛ* usually described using five relative race center displacements (axial displacement *dx*, radial displacements *dy* and *dz,* and misalignments *θ _{y}* and

*θ*) and circular rigid race assumptions.

_{z}### 5.1 Rigid Races.

*dx*and an equivalent radial displacement

*D*for defining a parameter

_{r}*A*and corresponding

*ɛ*. These displacements also define the orbital angle

*ψ*, where the maximum rolling element load

_{r}*Q*

_{max}is found:

*A*) is due to the specific sign convention used. When

*A*= −1, only one rolling element is loaded and

*ɛ*= 0. When

*D*is nil,

_{r}*A*and

*ɛ*are infinite, and all rolling elements are equally loaded, corresponding to pure thrust force applied on the race.

The parameter *A* or *ɛ* also defines the integral *L _{a}* and

*L*used for calculating the final axial and radial bearing forces

_{r}*F*and

_{a}*F*respectively. Therefore, the ratio

_{r,}*F*tan

_{a}/F_{r}/*α*can be curve fitted versus

*ɛ*; see Fig. 15 using LC or PC exponents.

*F*and

_{a}*F*are known and that the circular rigid race assumption is appropriate, one can use:

_{r}*K*

_{Hertz_i+o}defines the rolling element-race contact stiffness combining both races, see Ref. [16], and

*n*is the Hertzian exponent equal to 10/9 for line contact (or roller bearings) and 1.5 for point contact (or ball bearings).

### 5.2 High Structural Ring Deformation.

When using FEA to obtain the load distribution, the five race center displacements may not be known and structural ring deformations may modify the total deformation to consider (Hertz + structural deformation) as well as the ball-race contact angles (if ball bearings are used), significantly affecting the shape of the load distribution. Bearings with significant structural ring deformation tend to be large slewing bearings, which typically have many rolling elements *Z*.

However, Eq. (53) can be kept for trying to match or curve fit the FEA load distribution using three unknowns: *Q*_{max}, *ψ _{r}*, and, of course,

*ɛ*, the parameter used as the input for calculating the oscillation factor. Due to the structural ring deformations and contact angle variations, it is also possible to select the exponent

*n*as an additional unknown to be determined.

Several approaches can then be suggested for estimating *ɛ*, for example, an “advanced cf” or “exact” (meaning advanced curve fitting with an iterative approach) approach outlined in Eq. (54) and two simplified ones (in Eqs. (55) and (56)). Note that even when using the “advanced cf,” the corresponding *ɛ* is still estimated since the FEA load distribution differs from the one obtained with rigid race. However, the value of *ɛ* obtained is supposed to be more accurate than the ones obtained with the noniterative approaches.

Let’s call next *Q _{FEA_i}* the FEA load and

*Q*the curve-fitted load at each

_{cf_i}*ψ*value.

_{i}*X*variable using three unknowns: $A=Qmax1/n$

*, ψ*, and

_{r}*ɛ*to define in an iterative manner for minimizing

*S*2:

It can be demonstrated that A and *ɛ* can be defined directly (and therefore without any iteration), while *ψ _{r}* must be defined by solving a nonlinear relationship.

Schleich and Menck published different load distributions (corresponding to rows A1, A2, B1, and B2) of a 5000 mm double-row four-point contact ball bearing used as a wind turbine blade bearing in Ref. [21]. Such a large bearing requires many balls: *Z* = 147 (per row) in this case. Figure 16 shows that the estimated *ɛ* value is then 0.532 when using this “exact” approach for the load case of pitch angle *p* = 0 deg, bending Moment *M* = 20 MNm, and load angle *l* = 90 deg given in Ref. [21].

Since Z is large, one can also assume *Q*_{max} and *ψ _{r}* to be close to the most loaded calculated load and corresponding orbital angle. If

*Z*is small, a polynomial interpolation using three points (the most loaded one and its two adjacent loads) should also prove to be a good estimate of

*Q*

_{max}and

*ψ*.

_{r}*Q*

_{max}and

*ψ*, the first simplified approach consists of conducting a standard linear regression between

_{r}*Y*and

*X*now defined as follows:

The intercept will be forced to 1 in the following, leading to an *ɛ* value of 0.492.

Other approaches (also requiring linear regression and no iteration) are possible, but not described here.

*Z*is large, a simple third approach can be suggested for directly deriving

*ɛ*using loaded ball information. The total number of loaded balls (

*N*) is used when

_{LB}*N*< Z, while

_{LB}*Q*

_{max}and

*Q*

_{min}(

*Q*remaining > 0) are used when

_{min}*N*= Z. It can then be shown that:

_{LB}*ɛ* is then equal to 0.569 in the example shown in Fig. 16.

### 5.3 Application Example.

Figure 16 finally shows the load distribution A1 used by Menck et al. in Ref. [22] and given in Ref. [21] for a load case with *p* = 0 deg*, M* = 20 MNm, *l* = 90 deg and the three estimated values of *ɛ*, with values ranging from 0.492 to 0.569 when *n* = 1.5.

The corresponding oscillation factor then varies from 76.92 to 77.23, respectively, if *θ* is taken, for example, to be equal to 1 deg, *γ* = 0.0121, and *f _{i}* =

*f*= 0.53.

_{o}In view of all uncertainties related to bearing life calculations and small *ɛ* variations observed when selecting miscellaneous approaches, it can be recommended to select the simplest approach, i.e., using Eq. (56) for estimating *ɛ*.

The final life in this example can then be calculated using ISO 16281 standards, giving a bearing life of 1.683 × 10^{5} revolutions if the rotating frequency is equal to 1 Hz or 1 revolution/s.

If the oscillating frequency is 1 Hz (hence 1 oscillation/s), and the amplitude is 1 deg, the final life is then equal to 1.3 × 10^{7} oscillations or 3610 h, thanks to the oscillation factor equal to 77.23 in this case.

Note that this life result applies only to the (IR + OR) raceway A1, without consideration of lubrication effects, and for only one operating condition. It cannot be compared to Menck et al.’s results [22]. In Menck et al.’s article, the full load cycle and Harris oscillation factor were considered but not the specific effects of *ɛ* on the oscillation factor, which is the novelty considered in this article.

## 6 Use of a Lubrication Factor

Note that in Ref. [1], the factor 60 was missing (a typographical error) in the last line before the acknowledgments (but was correctly introduced earlier).

## 7 Conclusions

This article provides a comprehensive description of Lundberg and Palmgren’s life models for bearings in continuous rotation, starting with reasonable questions about these old models still used in current standards.

Suggesting new life models is, however, not the objective of this article (which is to define appropriate oscillation factors for roller and ball bearings), which is why standard life models and concepts have been retained and explained.

### 7.1 Bearing Life Calculations in Continuous Rotation.

The equivalent load *Q _{eq_i}* to use on the rotating ring has been explained using Miner’s rule and can be derived using the mean value (or an integral) of

*Q*; hence,

^{p}*Q*appears to be constant for any rotating volume

_{eq_i}*dV*defined by its initial orbital angle

_{i}*ψ*. The integral of

*Q*is divided by the loaded arc 2 ·

^{p}*ψ*

_{l}when defining its mean value.

The number of stress cycles *N _{i}* that any volume will endure during one excursion through the load zone is proportional to the loaded arc 2 ·

*ψ*, with the result that, when calculating the final survival probability

_{l}*S*of the rotating ring (via the product $(Qeq_ip\u22c5Ni)e$), the loaded arc 2 ·

_{i}*ψ*

_{l}used at the denominator and numerator will cancel out.

The life of the stationary ring is calculated as a function of the life of the rotating ring and a parameter *B*, resulting in an analytical derivation of the final life (combining both rings) as well as the dynamic rating and equivalent load.

Analytical relationships for *B* and dynamic ratings for roller and ball bearings have finally been obtained and successfully compared to current standards. These results are therefore not new but are needed for explaining the next steps (life calculations in oscillatory applications). Furthermore, the explanations provided here are hopefully simpler to follow (and relatively concise, since only 30 equations are used) than the ones found in the standard literature. The previously derived analytical relationships could also be used in a future exercise for improving miscellaneous relationships (for the rating and dynamic equivalent load) accounting for any appropriate new set of exponents (*c*, *h*, *e*, and *p*) derived from endurance or field test results.

### 7.2 Bearing Life Calculations in Oscillatory Applications.

When calculating the life of the oscillating ring, the first step (conceptually at least) is to calculate the loaded arc *H(ψ)* that a given volume *dV(ψ)* located at angle *ψ* will endure during one oscillation. *H(ψ)* can be calculated analytically and is not identical for all volumes *dV(ψ)* but varies as a function of *ψ* and *θ*.

The same applies for the equivalent load $Qeq_i(\psi )$, which varies as a function of *ψ* (not only because of the integral bounds *ψ* ± *θ* to be calculated but also because *H(ψ)* appears at the denominator).

The number of stress cycles *N _{i}(ψ)* endured by each elementary volume

*dV(ψ)*remains proportional to

*H(ψ),*with the result that

*H(ψ)*will also cancel out when using the product $(Qeq_ip(\psi )\u22c5Ni(\psi ))e$ defining the survival probability of an elementary small volume

*dV(ψ)*.

The survival probability of the entire oscillating inner ring is the product of the survival probability of each elementary volume *dV(ψ)*, resulting in the calculation of a double integral. The oscillation factor *a _{osc_IR}* of the oscillating ring can be finally calculated and defined as the ratio of the life under oscillation divided by the life in continuous rotation. Some simplified models of

*a*have also been developed, which apply to small oscillation angles only. Appropriate plots and curve-fitted relationships of

_{osc_IR}*a*have then been provided for line contact (or roller bearings) and point contact (ball bearings).

_{osc_IR}The oscillation factor *a _{osc_OR}* of the stationary ring is simpler to calculate and equal to 90/

*θ*deg.

The correction factor for oscillations below the critical angle is also offered before providing a final analytical relationship for calculating the oscillation factor *a*_{osc_IR+OR} (accounting for both rings) as a function of *a _{osc_IR}*,

*a*,

_{osc_OR}*B,*and

*e*, the Weibull exponent. Again, appropriate plots and curve-fitted relationships of the final oscillation factor have been provided and successfully compared to some published results (obtained numerically). Even simpler curve-fitted relationships can be suggested and used by application engineers when

*γ*is fixed to 0.1.

### 7.3 Estimation of *ɛ* and Life Example in an Oscillatory Application With an Finite Element Analysis Load Distribution.

All these calculations require the use of the oscillation angle as well as the load zone parameter *ɛ*. When using rigid race assumptions, *ɛ* can also be derived as a function of five relative race displacements or as a function of the ratio *F _{a}/F_{r}/*tan

*α*using useful curve-fitted relationships.

When using FEA, these displacements may not be known and large structural ring deformations as well as large contact angle variations (when studying ball bearing) may occur. Several simplified methods and relationships are then given for estimating *ɛ*.

Finally, when calculating the bearing life in an oscillatory application (the ring oscillating at a frequency *f _{osc}*), one first calculates the life in continuous rotation using

*L*or any more sophisticated approach such as the one described in ISO 16281 and then simply corrects the former life by

_{ref}· (C_{ref}/P_{eq})^{p}*a*

_{osc_IR+OR}to define the life in number of oscillations or hours.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

### Appendix

Figure 17