A New Method of Calculating the Attainable Life and Reliability in Aerospace Bearings

DOI: http://dx.doi.org/10.24018/ejers.2020.5.6.1977 Vol 5 | Issue 6 | June 2020 745  Abstract—The aviation industry made significant progress improving reliability, efficiency and performance throughout the last decades. Especially aircraft engines and helicopter transmission systems contributed significantly to these improvements. The kerosene consumption decreased by 70 % and the CO2 emissions due to air transport decreased by 30 % per passenger kilometer within the last 20 years. Simultaneously, the flight safety was increased with aircraft engine in-flight-shut-downs as low as 1 ppm and „unscheduled engine removals” as low as 4 ppm. Flight safety is equal to the reliability of the systems in service. Failure of these systems directly leads to exposure of human life. Among the most critical aviation systems are aircraft engines including the rolling element bearings which support the rotors. A serious damage to the aircraft engine main shaft bearings during flight requires shout-down of the engine to avoid a further damage escalation subsequently leading to engine fire. Today, it is a requirement for aircraft to operate with one engine shut down. However, each in-flight-engineshut-down typically is connected with flight diversion or abort and immediate landing. Inflight-shut-downs translate into increased risk for passengers and crew and substantial on cost. Therefore, rolling element bearings for aircraft engines are developed – similar to other aircraft engine components – targeting a reliability of nearly 100 % over an operation time of more than 10 000 hours prior to overhaul. To achieve this requirement despite the extreme operating conditions such as high speed and temperatures occurring in gas turbines, special high-performance materials are used for the rolling bearing components which are partially integrated in surrounding engine parts like shafts and housings. These special conditions deviating from conventional industrial rolling element bearing applications are currently not sufficiently considered in the standardized method of calculating the bearing life per ISO 281. A new method of calculating the attainable life of rolling elements bearing in aerospace applications is presented. This method considers the special aerospace conditions and materials and thus enables a higher reliability of the theoretical analysis and life prediction.


I. INTRODUCTION
The calculation of the rolling bearing life is a crucial method to determine the operational reliability and the service life for aerospace bearing applications. Today's aircraft systems require highest reliabilities and service lives of their components. Bearing life calculation represents an important method to substantiate EASA and FAA "type certificate" qualification. Currently ISO 281 does not consider all aerospace bearing operating conditions such as high temperature, lubrication, cleanliness, tangential stresses and special materials. Therefore, a state-of-the-art calculation method is required to address these operating conditions, materials and reliability requirements. The modern calculation method presented herein considers all these requirements, operating conditions and materials and therefore enables a more precise and reliable theoretical determination of aerospace bearing life.

A. The "Classical" Rolling Bearing Life Calculation
Today's calculation method of rolling element bearing life is based on Lundgren and Palmgren [1] theory, published in 1947. Preconditions at that time were constant material properties and the exclusive influence of external loads on the following basic approaches: i. The failure distribution follows a linear curve in the double-logarithmic Weibull-diagram with a slope of 10/9 for point contact and 9/8 for line contact. ii. The number of load cycles at the same failure probability is indirect proportional to the contact load by exponent ε: Z ~ P -ε Based on these approaches, the classic life equation for rolling element bearings was developed:

B. The Modified Rolling Bearing Life Calculation
The experience made during the decades after introduction of equation (1) showed that rolling element bearing operating conditions, especially the lubrication conditions, have significant influence on the bearing service life, leading to the modified bearing lifetime calculation [2], [3].
Further research in the 1980's demonstrated that rolling element bearingssimilar to other machine elementscan achieve endurance life under certain operating conditions [3], [4]. Based on that, the standardized calculation method per ISO 281 was developed [5,6], which is currently used for industrial rolling bearing applications.
This method assumes that i. classical rolling bearing steels are used for all overrolled bearing components, ii. no reduction of the material strength due to increased temperatures, iii. lifetime is limited by a spalling of one of the raceways, i.e. a raceway spalling occurs prior to a potential spalling of the rolling element. The high requirements for reliability and service life for aerospace bearing applications translatedstarting already in 1950'sinto developments of steels with increased hot hardness and high temperature strength and fracture toughness, such as AISI M50 and AISI M50NIL. The properties of these bearing materials were considered in the aerospace calculation methods developed in the 1990's [7,8], which are based on the equations derived in [2,3].
A further performance and reliability increase can be achieved by using special aerospace bearing materials such as 32CDV13, Cronidur30 [9], balls and rollers made of ceramic material [10,11] and special heat treatment procedures such as Duplex Hardening [12]. Further highstrength aerospace bearing materials are currently under development, expected to be introduced within the next 5 to 10 years.
The trend towards ultra-lightweight aircraft requires integration of bearing rings with other aircraft engine parts such as housings, elastic fixations, shafts, sealings, etc. This often translates into different materials used for the components within one bearing, i.e. different treatments for the inner raceway, outer raceway and rollers or balls are required.
For a realistic calculation of the bearing life or its reliability, the various materials and their combination for inner raceway, outer raceway and rollers or balls must be considered.
The "new method of calculating the attainable life and reliability in aerospace bearings as described in the following sections is based on the fundamental Lundberg-Palmgren theory. The new method applies adjustment coefficients to consider the operating conditions and reliability based on ISO 281 [5], but refined to consider the special requirements of aerospace bearing applications. This includes the individual consideration of the components inner ring raceway, outer ring raceway and particularly the ball/roller set, the high rotational speed, the elevated operating temperatures, the manufacturing quality, the lubricant properties, the system reliability, and much else.

III. CALCULATION OF THE MODIFIED ATTAINABLE LIFE IN AEROSPACE BEARING APPLICATIONS
The modified life of the entire bearing is calculated by the modified life of the components inner ring raceway, outer ring raceway and set of rolling elements per equation (2) 10 = ( 10 − + 10 − + 10 − ) − 1 (2) The modified life of the components inner ring raceway, outer ring raceway and set of rolling elements is calculated from the basic rating life multiplied with the life modification factor aAC of the corresponding component.
The basic rating life of the components inner ring raceway, outer ring raceway and rolling element is determined by using systematically the Lundgren and Palmgren-theory [1] in correlation with ISO 281:2007, ISO T/R 1281-1:2008 and ISO/TS 16281:2008 [5,6,13].
For that, the basic dynamic load rating of the entire bearing C is dissolved into the prime element basic dynamic load rating on the inner and outer ring raceway contacts, respectively, Qci/en1. This is the contact load acting on a contact element with a diameter of 1 mm attaining 10 6 load cycles or rolling contacts with 90 % reliability.
Point Contact: Line Contact: The upper signs refer to the rolling element / inner raceway contact and the lower signs refer to the rolling element / outer raceway contact.
Based on the prime element basic dynamic load rating Qci/en1 the elementary basic dynamic load rating for a specific raceway contact diameter per 10 6 load cycles is determined per equations (6) and (7).
Line Contact: Again, the upper signs refer to the rolling element / inner raceway contact and the lower signs refer to the rolling element / outer raceway contact.
The basic life of the inner raceway and outer raceway contacts per 10 6 load cycles is determined from the loading of the individual contacts of the inner ring raceway Fin and outer ring raceway Fen, respectively (equation (8)).
The exponent e' equals 1, if each volume element of the raceway experiences all loads Fi/en, i.e. the load rotates relative to the raceway.
If the ring stands still relative to the load, the entire volume is not loaded uniformly. Therefore, the average loading is compiled from the different loaded volume elements. For this case, e' = e = 10/9 for point contact and e' = e = 9/8 for line contact, applies.
Also based on prime element basic dynamic load rating Qci/en1 (equations (4) and (5)), the elementary basic dynamic load rating per 10 6 load cycles for the rolling elements is determined per equations (9) and (10).  (9) Line Contact: Again, the upper signs refer to the rolling element / inner raceway contact and the lower signs refer to the rolling element / outer raceway contact.
One rolling element (ball or roller) is loaded during its 360° orbit by all contact loads acting on the inner and outer ring raceway contacts. Analogous to equation (8), the basic reference life for one rolling element set L10r is yielded by: It is known from comprehensive rig testing and practical field experience, that the calculated basic life of rolling element bearings per equation (1) is exceeded multiple times in real applications and endurance strength can be achieved depending on the true material stressing due to contact load, the lubrication and contamination conditions ( [14] to [17]) This discrepancy is compensated by introduction of the life factor aAC. This factor is determined for the elements inner ring raceway, outer ring raceway and rolling elements depending on contact stress, temperature, lubricating conditions, system cleanliness and additional tangential stresses as from race deflection or friction. In its form aAC is based on the calculation of the standardized life factor aISO as per ISO 281 [5].
The factor aAC is limited to an upper threshold of 5000 as opposed to an upper limit of 50 for aISO. The lower limit is identical at 0,1. The increase of the upper limit by a factor of 100 is necessary to enable the required calculated reliability for a realistic stressing.
The factors fλ, , and f1 are intensifiers of the nominal contact stress for consideration of specific operating conditions. The method for determination of these factors will be identical for both the point contact and the line contact because the basic life of discrete stressed material in rolling contacts results in minor difference only when calculated either for a point contact or a line contact (Fig. 2).

A. The Endurance Strength at Operating Temperature P0tu
The endurance strength of bearing steels is reduced at elevated operating temperature depending on their hot hardness. This reduction is considered by constant Kt and equation (13): P0u is valid for temperatures up to 100 °C, i.e. Kt is applicable only for operating temperatures above 100 °C.  i.e. the ratio of the minimum film thickness [18,19,20] and the composite roughness of ring raceway and rolling element per [7].
Referring to , the factor fλ having inverse effect on the stress in equation (12) can be expressed by the following function: The slope shown in Fig. 3 is determined by kλ which considers the material behavior of mixed-friction conditions for typical aerospace bearing materials. The values for k currently in use for aerospace bearing materials are shown in Table II.

C. The Influence of the Lubricating System Cleanliness
Per ISO 281, the cleanliness factor for classical rolling bearing steel 100Cr6 (SAE 52100) is exponentiated by 1/3 for point contact (ball bearings) and by 0,4 for line contact (roller bearings).
Likewise, for the new calculation method presented herein, the cleanliness level SAC is exponentiated by ec, the applicable individual aerospace bearing material exponent. Note that ec is multiplied with a factor of 1,2 for line contact and 1 for point contact. The exponent ec considers therefore the sensitivities of the various aerospace bearing materials on hard particle contamination.  The cleanliness level is determined per Fig. 4, depending on filtration rate, filter quality, bearing configuration and rolling element size.

D. The Influence of Tangential Stresses
The influence of tangential stresses (friction) is considered by factor f1. The factor is evaluated depending on friction intensity of each bearing configuration, e.g. for axial bearings.
The material stressing might be further affected by severe operational effects resulting in particular additive tangential stress due to: i. interference fit and / or high rotational speed ii. ring bending of elastically supported races iii. slip effects (spin-to-roll, P*V) iv. ball excursion of combined loaded ball bearings These effects together with considerations of the according additional stresses on life have been described and evaluated in [20], [21], [22], [23] for instance. The factor f1 can be estimated accordingly or on the basis of advanced calculation programs.
Without consideration of these specific operation effects the factor f1=1 is applied.
IV. THE RELIABILITY FACTOR a1 (FAILURE PROBABILITY) As discussed before, a reliability of almost 100 % is expected from modern aircraft engines. Therefore, a rolling bearing life calculation based on the typical failure probability of 10 % is not acceptable.
Currently aircraft operators require one "Unscheduled Engine Removal" (UER) per one million flight hours. As multiple rolling element bearings and other critical components act together with individual reliabilities, the single component reliability must be significant higher. Therefore, the typical reliability requirement is 99,8 % per 10000 operating hours for rolling element bearings in flight critical application.
According to the reliability definition discussed, for a required reliability of 99,8 % (L0,2AC) applies: For a typical operating time of 10000 hours with a reliability of 99,8 %, a calculated bearing life L10AC of more than 80000 hours would be required.
Derived from the required system reliability with systems having multiple bearings and other components, a single bearing reliability of almost 100 % shall be targeted in order to avoid extreme over-dimensioning with detrimental consequences for bearing operability and performance, e.g. for high-speed bearings.
Experimental investigation results from Tallian [4] show that for reliabilities of more than 99,95 % no bearing failures occur.
Based on ISO 281, 100% reliability is achieved if The presented method of calculating the attainable bearing life for aerospace applications considers the properties of the individual materials used for inner ring raceway, outer ring raceway and rolling element set under the applicable operating conditions, i.e. lubricating, friction and cleanliness conditions.
The theoretical bearing life is based on the theoretical calculated life of the components inner ring raceway, outer ring raceway and rolling element set.
The reference or basic life of the individual bearing components is calculated with reference to [5, 6 and 13]. An adjustment factor aAC is introduced which allows for consideration of the properties of each individual bearing component material and the lubricating, friction and cleanliness conditions. Based on the reference or basic bearing life and the factor aAC for the individual bearing components, the modified component life and subsequently the modified bearing life is calculated. The factor aAC is defined similar to aISO in [5].
The method presented in this article allows for incorporation of properties of future bearing materials. L10ACr = modified life of the rolling element set with a failure probability of 10% e = exponent for failure distribution according to Weibull, 10/9 for point contact and 9/8 for line contact L10i, L10e, L10r = basic rating life for inner ring raceway, outer ring raceway and rolling element set Kc = constant for point contact: 121; approximated from 98,1, bm = 1,3 and reduction factor λ = 0,95. Constant for line contact: 564; approximated from 551,1, bm = 1,1 and reduction factor λ = 0,93 (the reduction factor of 0,93 instead of 0,83 is applied under the precondition that the determination of the pressure distribution across the roller length per [13] is calculated with highest precision).  = auxiliary parameter: Dw•cosα/Dpw (Dw = rolling element diameter, α = axial contact angle, Dpw = pitch diameter Ri/e = curvature radius for inner/outer raceway fDW: from Dw 1,8 for Dw≤25,4 or from 3,647•DW 1,4 for DW >25,4, see [5] lW = effective roller length aAC = life factor P0tu = Endurance strength as Hertzian stress at operating temperature P0 = maximum Hertzian stress at the contact element for the inner ring raceway or outer ring raceway at operating condition fλ = Stress factor for consideration of insufficient lubricant separation between the contact elements fsf = Friction factor SAC = Cleanliness level ec = Contamination exponent eL = Stress-Life-Exponent, 9 for point contact, 8 for line contact  = film thickness parameter, ratio of minimum calculated film thickness and the composite roughness of ring raceway and rolling element Qci/en1 = Contact load acting on a contact element with a diameter of 1 mm attaining 10 6 load cycles respectively over-rolling contacts with 90 % reliability Qci/e = Load capacity on the inner or outer contact for any contact diameter attaining 10 6 load cycles with 90 % reliability Qcwi/e = Rolling element load capacity on the inner or outer contact attaining 10 6 load cycles with 90 % reliability.