## 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 aosc 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 / dm 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 aosc 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 $ln(1/S)∼(Qeqp⋅N)e$ and hence the product $Qeqp⋅N$, where p is the life—load exponent, Qeq is the equivalent load, and N is the number of stress cycles. Both parameters (Qeq and N) can be defined using either a constant oscillating arc (2θ) 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 2θ or H(ψ,θ). The mindless error conducted in 1999 was to use 2θ 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 aosc.

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 fi and fo used in ball bearings.

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.

Lundberg and Palmgren’s models, published in 1947 and 1952, compared with [4,5], remain the basis for current life calculations. They postulated that the survival probability S can be calculated using:
$ln(1S)=Cste⋅τ0c⋅Nezh⋅V$
(1)
where N is the number of stress cycles, τ0 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 aISO 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.

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 Peq 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 Qc and then an equivalent load Qe obtained using appropriate summations as also done in ISO 16281.

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).

Life calculations start with the calculation of the most loaded rolling element contact load Qmax that can be calculated as a function of the radial load Fr or axial load Fa, the number of rolling elements Z, the nominal contact angle α, and the Sjövall integral Jr or Ja, which are only a function of the load zone parameter ɛ (defined later in Sec. 5 using miscellaneous approaches).
$Qmax=FrZ⋅cosα⋅Jr=FaZ⋅sinα⋅JawithJr=12⋅π⋅∫−ψlψl(1−1−cosψ2⋅ε)n⋅cosψ⋅dψandJa=12⋅π⋅∫−ψlψl(1−1−cosψ2⋅ε)n⋅dψψl=arccos(1−2⋅ε)(orπwhenε≥1)$
(2)

The Hertzian exponent n is equal to 10/9 for roller bearings and 1.5 for ball bearings and 2 · ψl is the loaded arc (to be compared later with the loaded arc H in oscillatory application from Sec. 3.1.1). Jr and Ja can also be calculated via the integrals Lr and La used by Houpert in Ref. [15].

Hertz theory gives for roller-race line contact: τ0 = 0.25 · Pmax (where Pmax 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 τ0 is described next as τ for allowing the possibility of introducing the index i and o, differentiating the inner from the outer ring, respectively. Pmax (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.

Starting with the aforementioned Pmax (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:
$Rx_eq_i=D2⋅(1−γ)Rx_eq_o=Rx_eq_i⋅1+γ1−γwithγ=D⋅cosαdmτi=Q⋅Eeq32⋅π⋅Rx_eq_i⋅Liτo=τi⋅1−γ1+γ⋅LiLozi=2⋅Q⋅Rx_eq_iπ⋅Eeq⋅Lizo=zi⋅1+γ1−γ⋅LiLo$
(3)

Here, Eeq = E/(1 – v2) = 2.26 × 106 N/mm2 is the equivalent Young's modulus derived from the Young's modulus E and Poisson’s ratio v of steel, Li,o is the equivalent roller-race length, and Rx_eq is the equivalent contact radius in the rolling direction x. Variables D and dm are the rolling element diameter and pitch diameter, respectively, and α is the nominal contact angle.

Equation (1) is used to study the survival probability Sdψ_i,o of an elementary small volume dVi,o defined by its arc angle , mean race radius Ri,o, effective roller-race length Li,o, and depth z.
$dVi=Ri⋅zi⋅Li⋅dψdVi=Ri⋅2⋅Q⋅Rx_eq_i⋅Liπ⋅Eeq⋅dψdVo=dVi⋅1+γ1−γ⋅1+γ1−γ⋅LoLiwithRi,o=dm2⋅(1∓γ)$
(4)
As shown in Ref. [3], the aforementioned relationships and Eq. (1) can now be used to calculate Sdψ_i,o (corresponding to a given number of stress cycles Ni,o) as a function of the load Q and an elementary volume represented by . For LC, for example, one obtains:
$ln(Sdψ_i,o)∝−Cste⋅Qp⋅e⋅Ni,oe⋅dψwithp=0.5⋅(c−h+1)e=103whenc=343h=73e=1.5(LCDominik)=4whenc=313h=73e=98(LCISO)$
(5)

Equation (5) shows that the number of stress cycles N for a given survival probability S is described by NQp. So far Q may have been considered constant, but the use of an elementary volume represented by will make it possible to account for a variable load on the rotating ring (via the introduction of an equivalent load Qeq, in Eq. (6)) or stationary ring (via the product of survival probabilities of all elementary volumes as will be shown in Eq. (11)).

#### 3.1.1 Rotating Race.

Let us now consider a varying load Q that each elementary volume (defined by ) of the rotating ring will face during its excursion through the load zone spanning from –ψl to +ψl, the loaded arc being 2 · ψl. The percentage of occurrence %j for this elementary volume of facing any load level Qj is constant. Each load Qj corresponds to a potential number of stress cycles Nj. The final or weighted number of cycles N that this elementary volume can endure is defined using Miner’s rule and results in the derivation of an equivalent load Qeq_rot for the rotating ring.

The number of rolling elements in the loaded zone (Z · (2 · ψl)/(2 · π)), assumed to be a real number, defines the occurrence $(%=1Z⋅2⋅π2⋅ψl)$ of having one elementary small volume dV experiencing a load Qj corresponding to a potential cycle number Nj = Nref · (Qref/Qj)p. Miner’s rule then defines the final number of stress cycles (N) the elementary volume can endure:
$1N=∑j%Nj=%⋅∑j1Nj=2⋅πZ⋅12⋅ψl⋅1Nref⋅Qrefp⋅∑jQjp=1Nref⋅1Qrefp⋅2⋅πZ⋅12⋅ψl⋅∑jQjp⏟Qeq_rotpwithQeq_rotp=2⋅πZ⋅12⋅ψl⋅∑jQjp.Replacingnow∑jQjpbyZ2⋅π⋅∫−ψlψlQp,oneobtains:Qeq_rotp=12⋅ψl⋅∫−ψlψlQp⋅dψ=Qmaxp⋅πψl⋅Kbi2andKbi=1π⋅∫−arccos(1−2⋅ε)arccos(1−2⋅ε)[1−1−cosψ2⋅ε]n⋅p⋅dψ$
(6)

The load Qeq_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 Sdψ_i of the rotating ring according to Eq. (5). $Qeq_rotp$ can also be considered as the mean value of Qp over the loaded arc of an elementary volume’s movement with 2 · ψl appearing at the denominator.

By calling freqi,o, the continuous rotating frequency (in revolutions per second) of the inner and outer rings, the cage or rolling element set frequency reads:
$freqcage=0.5⋅freqi⋅(1−γ)+0.5⋅freqo⋅(1+γ)$
(7)
The number of stress cycles Ni that the volume dVi will endure during a time ti (corresponding to a number of revolutions Nrev = freqi · ti) because of its excursion through the load zone is calculated using the relative frequency (freqi − freqc) = 0.5 · (1 + γ) · freqi. Moreover, only loaded elements in the load zone are considered, as defined earlier:
$Ni=Z⋅2⋅ψl2⋅π⋅0.5⋅(1+γ)⋅freqi⋅ti=Z⋅2⋅ψl2⋅π⋅0.5⋅(1+γ)⋅Nrev$
(8)

Note how the loaded arc 2 · ψl seen by all elementary volumes (appearing at the numerator of Ni and denominator of $Qeq_rotp$) disappears in Eq. (5) when using the product $(Qeq_rotp⋅Ni)e$. This explains why Refs. [4,13] obtain the same result without consideration of the load zone size, replacing 2 · ψl with 2 · π, for example.

The final survival probability Si of the entire rotating ring is obtained using the product of survival probabilities Sdψ_i or the integral of ln(1/ Sdψ_i) as defined in Eq. (5), explaining the factor 2 · π in front of the mean race radius Ri when defining the volume Vi, see Eq. (9).

When considering a variable load Q and rotating ring, the load Q must therefore be replaced with $Qeq_rot=Qmax⋅(πψl⋅Kbi2)1/p$, defined by Eq. (6), when calculating τi, zi, and Vi:
$ln(1Si)=Cste⋅τic⋅Niezih⋅ViwithVi=2⋅π⋅Ri⋅2⋅Qeq_rot⋅Rx_eq_i⋅Liπ⋅Eeq∝Cste⋅Qeq_rotp⋅e⋅Nie$
(9)

#### 3.1.2 Stationary Race.

The approach is different when studying the survival probability of the stationary race since an elementary small volume dVo defined by 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/Sdψ_ο) through the load zone, leading to another relationship for defining the equivalent load Qeq_stat to use instead of Q and also the use of 2 · ψl (instead of 2 · π) when calculating the stationary race volume.
$Qeq_statp⋅e=12⋅ψl⋅∫−ψlψlQp⋅e⋅dψ=Qmaxp⋅e⋅πψl⋅Kbe2=Qeq_rotp⋅e⋅KbeKbieandKbe=1π⋅∫−arccos(1−2⋅ε)arccos(1−2⋅ε)[1−1−cosψ2⋅ε]n⋅p⋅e⋅dψ$
(10)
$ln(1So)=Cste⋅τoc⋅Noezoh⋅VowithVo=Vi(1+γ1−γ)1.5LoLi2⋅ψl2⋅π$
(11)
On the outer ring, the number of stress cycles that each elementary volume (represented by ) in the load zone will endure is calculated using the frequency freqc–freqo or simply freqc, since the outer race is assumed to be stationary.
$No=Z⋅0.5⋅(1−γ)⋅NrevorNo=Ni⋅(1−γ1+γ)⋅2⋅π2⋅ψl$
(12)
Finally, the survival probability So of the stationary outer ring can be expressed as a function of the rotating inner ring’s survival probability, introducing the important parameter B, which describes the effect of the stationary ring on the survival probability.
$ln(1So)=Cste⋅τoc⋅Noezoh⋅Vo=ln(1Si)⋅BLCwithBLC=2e−1⋅KbeKbie⋅(1−γ1+γ)c+h−3+2⋅e2⋅(LoLi)1−c+h2$
(13)

#### 3.1.3 Combining Both Raceways.

The final probability S of surviving a given number of revolutions Nrev is equal to the product Si · So, resulting in:
$ln(1S)=ln(1Si)⋅[1+BLC]$
(14)

BLC 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).

It is now possible to develop analytically all previously defined relationships for obtaining the final life (in time or revolutions of the rotating ring) of the roller bearing corresponding to any survival probability S (commonly S = 0.9) as well as any set of exponents (c, h, e, and p), see Ref. [3]:
$ln(1S)=ALC⋅(1+BLC)⋅Kbie⋅Jr−e⋅p⋅Fre⋅p⋅NrevewithALC=Cste⋅22−2⋅c−2⋅e⋅π1−e⋅p⋅Eeqc−e⋅p⋅Ze−e⋅p⋅(1+γ)e⋅(1−γ)1+e⋅p−c⋅Li1−e⋅p⋅D1+e⋅p−c⋅(cosα)1−e⋅p⋅γ−1$
(15)

The second equation in Eq. (15) is important since it can be used to derive analytically the bearing life Nrev 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 (Sref = 0.9), the number of revolutions obtained is often called L10 (in revolutions) and the corresponding radial load Fr is the dynamic rating called Cref in this article or Cr (or C1 in this article) at ISO and C90 by Dominik. C1 and C90 are hence defined using a life of reference (L10 = Lref = 1 million or 90 million revolutions, respectively), but also with a load zone of reference (2 · ψl_ref = 180 deg for ISO and 150 deg for Dominik) used for defining all integrals (Jr_ref, Kbi_ref, and Kbe_ref) and the ratio $Kbe/Kbie$ used in BLC_ref.

An experimental value of the constant Cste is of course necessary for matching either endurance test results or field test results.
$Cref=ALC−1/ep⋅Jr_ref⋅Kbi_ref−1/p⋅(1+BLC_ref)−1/ep⋅ln(1Sref)1/ep⋅Lref−1/p$
(16)
It has been demonstrated in Ref. [3] that the rating obtained with Eq. (16) exactly matches the Dominik C90 rating rearranged in Eq. (17) using Lo = 1.141 · Li (not explained in Ref. [14]) and Dominik exponents from Eq. (5). M is called the material factor, while H is called the geometrical factor:
$Cref=M⋅H⋅Dc−1e⋅p−1⋅Z1−1p⋅(Li⋅cosα)1−1e⋅p$
(17)
with
$M=(Cste)−1e⋅p⋅2−2+2⋅c+2⋅ee⋅p⋅π−1+e⋅pe⋅p⋅Eeq1−ce⋅p⋅Jr_ref⋅Kbi_ref−1p⋅ln(1Sref)1e⋅p⋅Nrev_ref−1p=57.6(usingmmandN)$
(18)
$H=γ1e⋅p⋅(1+γ)−1p⋅(1−γ)c−1e⋅p−1⋅(1+BLC_ref)−1e⋅p$
(19)
$BLC_ref=(1−γ1+γ)c−e⋅p+e−1⋅(Kbe_refKbi_refe)⋅(LoLi)1−e⋅p⋅2e−1$
(20)

Cste is equal to 1.39 · 10−49 for matching M = 57.6 (used for defining Dominik C90 corresponding to 90 million revs) or M = 222.18 (used for defining Dominik C1 corresponding to 1 million revs) with a load zone of reference equal to 150 deg.

It can also be demonstrated that the rating obtained with Eq. (16) matches the ISO C1 rating and exponents on (L · cos α), Z, and D described by Eq. (11.156) in Ref. [13] using Lo = Li and LC ISO exponents from Eq. (5) with a load zone of 180 deg. Cste must then be fixed to 3.8 × 10−44 to obtain M = 189.43 and M · H = bm · fc in the entire range of γ.
$C1=bm⋅fc⋅(i⋅L⋅cosα)7/9⋅Z3/4⋅D29/27$
(21)
where bm 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 fc is given in tables in Ref. [9] as a function of γ.

#### 3.1.4 Differences to ISO.

Equation (15) can also be used to define the equivalent load Peq 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.
$L10=Lref⋅(1+BLC1+BLC_ref)−1e⋅(JrJr_ref)p⋅Kbi_refKbi⋅(CrefFr)p=Lref⋅(CrefPeq)p(revolutions)(FrPeq)IR+OR=JrJr_ref⋅(Kbi_refKbi)1p⋅(1+BLC_ref1+BLC)1e⋅p≈(FrPeq)IR=JrJr_ref⋅(Kbi_refKbi)1p$
(22)

The latter ratio can be plotted versus the load zone parameter ɛ or the ratio Fa/Fr/tan α or Fr · tan α/Fa, see Fig. 1 or 3 in Ref. [3], or can be used for defining the load zone factor using both rings (thus including BLC) or only the rotating inner ring (thus fixing BLC = 0), see Houpert’s Figs. 6 and 7 in Ref. [3].

Minor differences are observed when using one or two rings for calculating Peq, so only Peq_IR+OR is shown in Fig. 1.

Fig. 1
Fig. 1
Close modal

A degree 6 polynomial relationship is also suggested in the same figure for easily calculating the curve-fitted ratio Peq_IR+OR_cf as a function of the ratio Fr · tan α/Fa.

Note how Peq increases (and therefore the life decreases) when Fa is either very large or small (the latter causing a narrow load zone) with both cases causing an increase of Qmax.

The ratio Peq_Catalog/Fr is fixed to 1 in any “Catalog” approach as soon as the estimated ratio Fa/Fr/tan α is smaller than 1.5 and is equal to 0.4 · (1 + Fa/Fr/tan 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 Fa_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 · Fr_1,2 · tan α if ɛref = 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).

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 L10 life is calculated using (C1/Peq)10/3, hence p = 10/3, while the rating and equivalent load are defined with p = 4.

### 3.2 Ball Bearing Life.

The approach for calculating ball bearing life is identical to that used for roller bearings with the exception that the starting relationships for Pmax 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:
$ForPC:Pmax_i,o≈CPi,o⋅Eeq⋅(QmaxEeq⋅Rx_eq_i,o2)13bi,o=CBi,o⋅Rx_eq_i,o⋅(QmaxEeq⋅Rx_eq_i,o2)13ai,o=CAi,o⋅Rx_eq_i,o⋅(QmaxEeq⋅Rx_eq_i,o2)13$
(23)
where the factors 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:
$Forballbearingsor8.74≤k≤122.44:CPi,o≈0.3122⋅ki,o−0.2117,CBi,o≈1.1687⋅ki,o−0.1974,CAi,o=1.3085⋅ki,o0.4091ki,o=Ry_eq_i,oRx_eq_i,o=2⋅fi,o(2⋅fi,o−1)⋅1(1∓γ)(sign−forIR)$
(24)
where fi and fo are the osculation ratio of the IR and OR, respectively, typically of the order 0.53, ranging from 0.51 to 0.56.

When analytically developing a relationship for ln(1/Si) (in a similar manner to Eq. (15), but not explicitly shown in this article), the product $CAi⋅CBi1−h⋅CPic$ will be used when tracing the final exponents applied on (1 ± γ) and osculation ratios.

For PC, the following life-load exponent is ultimately obtained:
$p=c−h+23⋅e=3whenc=313,h=73,e=109(PCISO)$
(25)
As shown in Eq. (14) for line contact, the effect of the stationary outer ring is described using BPC_ISO:
$BPC_ISO=2e−1⋅KbeKbie⋅(1−γ1+γ)1.3649⋅c+1.5922⋅h−4.3649+3⋅e3⋅(fo⋅(2⋅fi−1)fi⋅(2⋅fo−1))0.2117+0.1974⋅h−0.2117⋅c=1.0801⋅KbeKbie⋅(1−γ1+γ)5.5958⋅(fo⋅(2⋅fi−1)fi⋅(2⋅fo−1))−1.5153$
(26)
When ɛ = 0.5 (reference case used for deriving the rating), $KbeKbie=1.0699$ and
$BPC_ref=1.1556⋅(1−γ1+γ)5.5958⋅(fo⋅(2⋅fi−1)fi⋅(2⋅fo−1))−1.5153$
(27)
This can be compared to Eq. (11.106) in Ref. [13] giving fc, used herein for deriving BHarris_ref :
$BHarris_ref=1.1397⋅(1−γ1+γ)5.7333⋅(fo⋅(2⋅fi−1)fi⋅(2⋅fo−1))−1.3667$
(28)

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

The same is true of the final rating C1, found in this article to be proportional to:
$C1∝Z23⋅D1.8⋅(cosα)0.7⋅γ0.3⋅(1−γ)1.3454(1+γ)13⋅[2⋅fi(2⋅fi−1)]0.4546⋅(1+BPC_ref)−0.3$
(29)
Equation (29) is only slightly different than the relationship published in Ref. [13], see Eqs. (11.103) and (11.106), where the ISO rating is finally proportional to:
$C1_Harris∝Z23⋅D1.8⋅(cosα)0.7⋅γ13⋅(1−γ)1.39(1+γ)13⋅[2⋅fi(2⋅fi−1)]0.41⋅(1+BHarris)−0.3$
(30)

### 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.

It is important to mention that B can also be used for deriving the life to (in time: seconds or hours, for example) of the stationary outer ring as a function of the rotating inner ring life ti using Eq. (15) with BLC or BPC, see the following demonstration:
$1tconte=1ti_conte+1to_conte=cte′⋅(1+B)WhenfixingBtozero,hencenotconsideringtheouterring:1ti_conte=cte′⇒1to_conte=B⋅1ti_conteto_cont=ti_cont⋅1B1e$
(31)

The reader is also reminded that the life of the rotating inner ring is finally calculated using the product $(Qeq_rotp⋅Ni)e$ in which the loaded arc 2 · ψl cancels out. Moreover, for a given ɛ, the equivalent load $Qeq_rot$ as well as the loaded arc 2 · ψl of an elementary volume on the rotating race are constant for any elementary volume dVi on the rotating inner ring.

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 aosc to correct the lifetime for oscillatory behavior.

### 4.1 Oscillating Inner Ring.

An oscillatory inner ring oscillating between ±θ at an oscillating frequency freqi will be studied first. The loaded zone is still defined by the angle ψl as shown in the upper left corner of Fig. 2. Each elementary volume dVi is defined by its position 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.

Fig. 2
Fig. 2
Close modal

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 dVi (located at the orbital angle ψ on the oscillating inner ring) will endure as well as its equivalent load Qeq(ψ) but is not essential because 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 ψl.

H(ψ) can be calculated numerically as shown and used in Ref. [1]:
$H(ψ,θ)=∫ψ−θψ+θh(ψ)⋅dψwithh(ψ)=1ifQ(ψ)≠0orh(x)=0ifQ(ψ)=0$
(32)

An analytical relationship for H(ψ), only applicable in the case θ > ψl, 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 dVi in the load zone (by its two ends) occur as shown in the lower left corner of Fig. 2 (with an example corresponding to ψl = 150, θ = 120, and ψ = 160 deg):

Table 1

Analytical relationships valid for −πψπ and θπ

 |ψ| ≤ |θ − ψl| (θ − ψl) ≤ |ψ| ≤ (ψl + θ) |ψ| ≥ ψl + θ H = 2 · Min(ψl, θ) H = (θ + ψl − |ψ|) + Max(0, |ψ| + θ + ψl − 2 · π) H = 0
 |ψ| ≤ |θ − ψl| (θ − ψl) ≤ |ψ| ≤ (ψl + θ) |ψ| ≥ ψl + θ H = 2 · Min(ψl, θ) H = (θ + ψl − |ψ|) + Max(0, |ψ| + θ + ψl − 2 · π) H = 0

The lower right corner of Fig. 2 shows how H(ψ) varies as a function of ψ (with ψl = 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).

It is possible to calculate analytically the lower and upper bounds defining the arc H1 and H2; hence, the final value H = H1+ H2:
$Forψ≥0:ψ11=Max(−ψl,|ψ|−θ)ψ12=Min(ψl,|ψ|+θ)H1=ψ12−ψ11ψ21=|ψ|+θ−Max(0,|ψ|+θ+ψl−360)ψ22=|ψ|+θH2=ψ22−ψ21H=H1+H2$
(33)

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.

Almost like the continuously rotating ring in Eq. (8), the number of stress cycles that a volume dVi on the oscillating inner ring will endure is now defined using:
$Ni(ψ)=2⋅Z⋅H(ψ)2⋅π⋅0.5⋅(1+γ)⋅Nosc$
(34)
where Nosc is the number of oscillations of the inner ring. Note the factor 2 introduced before Z because the oscillation angle goes from 0 to –θ, then +θ, and back to zero, causing two crossings through the loaded arc.

The equivalent load Qeq(ψ) that a volume dVi located at orbital angle ψ on the oscillating inner ring will endure is defined using the average value of Qp over the arc H(ψ).
$Qeq(ψ)=0whenH=0otherwise:Qeq(ψ)=(1H⋅∫ψ−θψ+θQp⋅dψ)1por(1H⋅(∫ψ11ψ12Qp⋅dψ+∫ψ21ψ12Qp⋅dψ))1p$
(35)

Therefore, the equivalent load Qeq and loaded arc H are not constant and vary as a function of ψ, as shown in Fig. 3 using a given example.

Fig. 3
Fig. 3
Close modal

The equivalent load distribution in Fig. 3 can be compared to Breslau’s definition in Ref. [2] defined with 2 · θ (instead of H) at the denominator.

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

The survival probability Sdψ_i of the elementary small volume is obtained as in Eq. (5) using the product $[Qeqp(ψ)⋅Ni(ψ)]e$, in which the loaded arc H cancels out since it is used at the denominator when defining Qeqp and numerator when defining Ni. The final results are therefore identical irrespective of whether 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 Qeqp, while H was correctly used for defining Ni, meaning that cancellation of the denominator and numerator was not possible.

The following steps are described in Ref. [1], where the final survival probability Si of the oscillating inner ring is obtained by integrating ln(1/Sdψ_i) over the entire inner ring volume Vi, leading to the use of a double integral.

#### 4.1.5 Oscillation Factor aosc_IR of the Oscillating Inner Ring.

The ratio aosc_IR describing the life ratio oscillating/continuous of the oscillating inner ring can be defined as follows:
$ifψl+θ<π(case1):aosc_IR=(2⋅π)1e2⋅∫−ψlψlQp⋅dψ[∫−ψl−θψl+θ{∫|ψ|−θ|ψ|+θ[Q(ψ′)]p⋅dψ′}e⋅dψ]1eifψl+θ>π(case2):aosc_IR=(2⋅π)1e2⋅∫−ψlψlQpdψ[∫−ππ{∫|ψ|−θ|ψ|+θ[Q(ψ′)]p⋅dψ′}e⋅dψ]1e$
(36)

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).

Accurate lower and upper bounds can be used when performing the calculation of the double integral with a reduced number of points (101 points, thus 10,201 when conducting the double integral, for the results shown in this article). If more points are used, aosc_IR can be simply defined using:
$aosc_IR=(2⋅π)1e2⋅∫−ψlψl[1−1−cosψ2⋅ε]n⋅p⋅dψ[∫−ππ{∫ψ−θψ+θ[1−1−cosψ′2⋅ε]n⋅p⋅dψ′}e⋅dψ]1e$
(37)
where the function in the square bracket with exponent n · p is nil outside of the loaded range (–ψl, ψl), and n, p, and e are defined as in Eqs. (5) and (25) for roller and ball bearings, respectively.
An acceptable assumption can be used and a double integration can be avoided when the oscillation angle θ is small.
$Whenθissmall:Qeq(ψ)≈Q(ψ)and∫|ψ|−θ|ψ|+θ[Q(ψ′)]p⋅dψ′≈2⋅θ⋅Qp⇒aosc_IR_approx≈(2⋅π)1e2⋅∫−ψlψlQp⋅dψ[∫−ψl−θψl+θ{Qp⋅2⋅θ}e⋅dψ]1e=(2⋅π)1e2⋅12⋅θ.∫−ψlψlQp⋅dψ[∫−ψl−θψl+θQp⋅e⋅dψ]1e$
(38)
It can be demonstrated that the aforementioned approximated factor finally reads:
$aosc_IR_approx≈(πψl)−e−1e⋅π2⋅θ⋅(Qeq_rotQeq_stat)p=(12)e−1e⋅π2⋅θ⋅KbiKbe1/e$
(39)

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 aosc_OR of the Stationary Outer Ring.

On the stationary outer ring, the number of stress cycles that any volume dVo endures over a time period to can be defined using the cage frequency freqc and is simply equal to 2 · Z · (2 · θ)/(2 · π) · freqc · to, whereas it was Z · freqc · to in continuous rotation.

The oscillation factor aosc_OR applicable to the stationary outer race therefore reads:
$aosc_OR=π2⋅θor90θdegree=aHarris$
(40)
aosc_OR will also be called aHarris 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.

### 4.3 Results and Figures.

We will now apply the aforementioned calculations to a variety of cases. Starting with the inner ring factor aosc_IR, we will then turn to the factor aosc_IR+OR for the entire bearing and consider the effect of γ and osculation.

#### 4.3.1 aosc_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.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

Figure 5 shows the same results as Fig. 4, but divided by aHarris. 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.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

Appropriate curve-fitted relationships for calculating aosc_IR are presented in Sec. 4.5.1.

#### 4.3.2 Oscillation Factor aosc_IR+OR Combining Both the Inner and Outer Rings.

It is now possible to calculate the oscillation factor aosc_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:
$1tosce=1ti_osce+1to_osce=1(ti_cont⋅aosc_IR)e+1(ti_cont⋅1B1/e⋅aosc_OR)etosc=aosc_IR+OR⋅tcont=aosc⋅tcont1tosce=1aosce⋅1tconte=1aosce⋅1ti_conte⋅(1+B)$
(41)
$aosc_IR+OR=aosc=aosc_IR(1+(aosc_IRaosc_OR)e⋅B)−1e⋅(1+B)1e$
(42)
It can be demonstrated that:
$aosc>aosc_IRWhenε→∞,aosc_IR→aosc_ORandaosc→aosc_IR=aosc_OR$
(43)

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 aosc_IR in some cases.

Figure 11.28 should, for example, be replaced Fig. 8 (corresponding to Lo = Li and γ = 0.1), γ = 0.1 being a typical geometrical ratio (γ = D·cosa/dm) used when deriving Fig. 11.28. The final oscillation factor continuously decreases as 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] (Aosc, here called aosc_IR+OR_LC, instead of a ratio relative to Harris), to compare with Figs. 11.28 and 11.29.

Fig. 8
Fig. 8
Close modal

Again, for the sake of completeness, the ISO LC and PC results are shown in Figs. 9 and 10 using γ = 0.1 (and fi = fo when defining BPC).

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

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 fo, for example, see Fig. 11 (using θ = 1 deg and fi = fo).

Fig. 11
Fig. 11
Close modal

#### 4.3.4 Effect of the Osculation Ratio.

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

Fig. 12
Fig. 12
Close modal

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.

The critical angle is defined by Rumbarger and Jones in Ref. [18] as the oscillation angle corresponding to one rolling element-race stress cycle for an elementary volume 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:
$θcrit_i,o=2⋅πZ⋅(1±γ)$
(44)
with the upper sign being used for the inner ring.

The oscillation factor to use when θ < θcrit is described in Refs. [2,18]. Some approximations are used, such as the replacement of Qeq(ψ) with the load of a stationary ring Q(ψ).

Although not explicitly mentioned, a correction factor fθ_crit_ι,o can be derived from Ref. [2] to apply to the previously described oscillation factors:
$aosc_IR_or_OR→fθ_crit_i,o⋅aosc_IR_or_ORWhenθ≥θcrit_i,o:fθ_crit_i,o=1Whenθ<θcrit_i,o:fθ_crit_i,o=(θθcrit_i,o)1−1e$
(45)

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.

Breslau and Schlecht’s results and those of the authors are compared in Fig. 13 (also Fig. 22 in Ref. [2]), showing a perfect match between both authors and a moderate effect of the critical angle on the results.

Fig. 13
Fig. 13
Close modal

### 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 aosc_IR and B, subsequently used for defining aosc_IR+OR via Eq. (42). The effects of γ and/or the osculation factor (included in BPC) are then considered.

• A simplified and single relationship in which γ has been fixed to 0.1 and the osculation factors fi and fo are assumed to be equal.

#### 4.5.1 Curve Fitting of aosc_IR

$aosc_IR_LC_Dominik_cf≈fθ_crit_i⋅90θdeg⋅{1−exp(−(0.012289⋅θdeg+0.52891)⋅(ε0.05)0.048993⋅ε+0.24067)}aosc_IR_LC_ISO_cf≈fθ_crit_i⋅90θdeg⋅{1−exp(−(0.014686⋅θdeg+1.3771)⋅(ε0.05)0.036493⋅ε+0.13258)}aosc_IR_PC_cf≈fθ_crit_i⋅90θdeg⋅{1−exp(−(0.014818⋅θdeg+1.4689)⋅(ε0.05)0.035479⋅ε+0.12632)}$
(46)

The error obtained, measured via the ratio $(|aosc_cf−aosc|/aosc)$, 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 aosc/aHarris 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.

The aforementioned relationships can also be simplified (at low θ values only) using Eq. (39) and appropriate curve fitting of $Kbi/Kbe1/e$, leading to:
$aosc_IR_LC_Dominik_approx_cf≈fθ_crit_i⋅90θdeg⋅[1−0.27581⋅ε−0.2586]aosc_IR_LC_approx_ISO_cf≈fθ_crit_i⋅90θdeg⋅[1−0.10342⋅ε−0.3047]aosc_IR_PC_approx_cf≈fθ_crit_i⋅90θdeg⋅[1−0.09381⋅ε−0.30679]$
(47)

#### 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 $(Kbe/Kbie−(1/2)e−1)$ as a function of ɛ (Fig. 14).

Fig. 14
Fig. 14
Close modal

Integrating these curve-fitted relationships into Eqs. (13) and (26) leads to:

$BLC__Dominik__cf≈(1−γ1+γ)6.8333⋅(LoLi)−4⋅(1+0.60371⋅ε−0.48862)BLC__ISO__cf≈(1−γ1+γ)5.9583⋅(1+0.12973⋅ε−0.36716)BPC__cf≈(1−γ1+γ)5.5958⋅(fo⋅(2⋅fi−1)fi⋅(2⋅fo−1))−1.5153⋅(1+0.11491⋅ε−0.36257)$
(48)
The final accurately curve-fitted relationship for aosc is, compared with Eq. (42):
$aosc__IR+OR__cf=aosc__cf=aosc__IR__cf(1+(aosc__IR__cfaosc__OR)e⋅Bcf)−1e⋅(1+Bcf)1ewithaosc__OR=fθ__crit__o⋅90θdegree$
(49)
with 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 fi = fo.

A simplified correction factor is also suggested using the mean critical angle (θcrit = 2 · π/Z) in Eq. (44) for both rings when defining fθ_crit, leading to:
$Forγ=0.1:aosc_IR+OR_LC_Dominik_cf≈fθ_crit⋅aHarris⋅{1−exp(−(0.115196⋅θdeg+0.79902)⋅(ε0.05)0.049966⋅ε+0.16252)}aosc_IR+OR_LC_ISO_cf≈fθ_crit⋅aHarris⋅{1−exp(−(0.015374⋅θdeg+1.6458)⋅(ε0.05)0.034617⋅ε+0.11229)}Forγ=0.1andfi=fo:aosc_IR+OR_BB_cf≈fθ_crit⋅aHarris⋅{1−exp(−(0.015462⋅θdeg+1.7544)⋅(ε0.05)0.033515⋅ε+0.10699)}withaHarris=90θdeg$
(50)

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 θy and θz) and circular rigid race assumptions.

### 5.1 Rigid Races.

Houpert shows in Refs. [15,20] how these five relative race displacements can be reduced into an axial displacement dx and an equivalent radial displacement Dr for defining a parameter A and corresponding ɛ. These displacements also define the orbital angle ψr, where the maximum rolling element load Qmax is found:
$Dy={dy+dθz⋅Ri⋅tanα}Dz={dz−dθy⋅Ri⋅tanα}Dr=Dy2+DZ2A=−dx⋅tanαDrandε=0.5⋅(1+A)ifDr=0,theloadzoneis360deg,thereisnoneedtodefineψrifDy=0,ψr=π2⋅sign(Dz)ifDy≠0ψr=tan−1(DzDy)ifDy>0ψr=tan−1(DzDy)+πifDy<0$
(51)
where the negative sign (used in the definition of A) is due to the specific sign convention used. When A = −1, only one rolling element is loaded and ɛ = 0. When Dr is nil, 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 La and Lr used for calculating the final axial and radial bearing forces Fa and Fr, respectively. Therefore, the ratio Fa/Fr/tan α can be curve fitted versus ɛ; see Fig. 15 using LC or PC exponents.

Fig. 15
Fig. 15
Close modal
For point contact, for example, assuming Fa and Fr are known and that the circular rigid race assumption is appropriate, one can use:
$Usingx=(FaFr⋅tanα−1):Whenx<0.66664εcf≈1.4389⋅x3−3.0838⋅x2+2.9206⋅xWhenx≥0.66664εcf≈0.38641⋅x+0.78125$
(52)
$When−ψl≤ψ−ψr≤ψl:Q(ψ)=KHertz_i+o⋅(Dr⋅cosα)n⋅[1+cos(ψ−ψr)−12⋅ε]n=Qmax⋅[1+cos(ψ−ψr)−12⋅ε]nwithψl−ψr=arccos(1−2⋅ε)ifε<1ψl−ψr=πor180degreesifε≥1$
(53)
where KHertz_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: Qmax, ψr, 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 QFEA_i the FEA load and Qcf_i the curve-fitted load at each ψi value.

When selecting the “exact” or simplified approaches, $QFEA1/n$ is curve fitted versus another X variable using three unknowns: $A=Qmax1/n$, ψr, and ɛ to define in an iterative manner for minimizing S2:
$X=cos(ψ−ψr)−1YFEA=QFEA1nYcf=A⋅[1+X2⋅ε]S2=∑(Ycf_i−YFEA_i)2$
(54)

It can be demonstrated that A and ɛ can be defined directly (and therefore without any iteration), while ψr 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].

Fig. 16
Fig. 16
Close modal

Since Z is large, one can also assume Qmax and ψr 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 Qmax and ψr.

Having estimated Qmax and ψr, the first simplified approach consists of conducting a standard linear regression between Y and X now defined as follows:
$X=cos(ψ−ψr_cf)−1andY=(QFEA(ψ)Qmax_cf)1nY=[1+X2⋅εcf_single′]Ycf≈const+slope1⋅XorYcf≈1+slope2⋅Xifforcingtheinterceptto1εcf_single′=12⋅slope1or2$
(55)

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.

When Z is large, a simple third approach can be suggested for directly deriving ɛ using loaded ball information. The total number of loaded balls (NLB) is used when NLB < Z, while Qmax and Qmin (Qmin remaining > 0) are used when NLB = Z. It can then be shown that:
$εloaded≈1−cos(NLBZ⋅π)2forNLB
(56)

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 fi = fo = 0.53.

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 × 105 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 × 107 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

A lubrication factor can also be used as suggested in Ref. [1] using the mean speed calculated during one oscillation:
$Nmean=60⋅fosc⋅2⋅θπ(rpm)$
(57)

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 Qeq_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 Qp; hence, Qeq_i appears to be constant for any rotating volume dVi defined by its initial orbital angle ψ. The integral of Qp is divided by the loaded arc 2 · ψl when defining its mean value.

The number of stress cycles Ni that any volume will endure during one excursion through the load zone is proportional to the loaded arc 2 · ψl, with the result that, when calculating the final survival probability Si of the rotating ring (via the product $(Qeq_ip⋅Ni)e$), the loaded arc 2 · ψl 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 $Qeq_i(ψ)$, 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 Ni(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 $(Qeq_ip(ψ)⋅Ni(ψ))e$ 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 aosc_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 aosc_IR have also been developed, which apply to small oscillation angles only. Appropriate plots and curve-fitted relationships of aosc_IR have then been provided for line contact (or roller bearings) and point contact (ball bearings).

The oscillation factor aosc_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 aosc_IR+OR (accounting for both rings) as a function of aosc_IR, aosc_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 Fa/Fr/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 fosc), one first calculates the life in continuous rotation using Lref · (Cref/Peq)p or any more sophisticated approach such as the one described in ISO 16281 and then simply corrects the former life by aosc_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.

Figure 17

Fig. 17
Fig. 17
Close modal

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