applied sciences

Article Influence of Hybridization on Lifetime

Christian Habermehl * , Georg Jacobs , Stephan Neumann and Kevin Weißenfels

Institute for Elements and Systems Engineering, RWTH Aachen University, Schinkelstraße 10, 52062 Aachen, Germany; [email protected] (G.J.); [email protected] (S.N.); [email protected] (K.W.) * Correspondence: [email protected]; Tel.: +49-241-80-95606

 Received: 21 September 2020; Accepted: 9 October 2020; Published: 12 October 2020 

Abstract: Parallel hybrid for passenger have additional electric drives compared to conventional drivetrains. In the event of deceleration, the electric drives are operated as generators, thus recovering kinetic energy through regeneration. If these drives are positioned upstream of the transmission input, regeneration power must be transferred by the transmission. This creates additional loads on the individual machine elements, which has a negative effect on the transmission lifetime. This paper investigates the influence of hybridization in terms of regeneration on the lifetime of bearings as highly critical elements in a dual clutch transmission. The vehicle simulation model employed in this study consists of an internal combustion , an , a mechanical drivetrain and the vehicle body, as well as a driver and a simple operating strategy. In this model, a detailed transmission model including its controls is embedded to determine its component loads. The resulting load spectra are used in a methodical approach to calculate the bearing lifetime of the transmission. The results show that the additional regenerative power flow reduces the bearing lifetime so that additional loads must be taken into account in the development and operation of transmission systems.

Keywords: dual-clutch transmission; bearing lifetime; vehicle simulation; hybrid ; fatigue damage; regeneration

1. Introduction Hybrid and battery electric are paving the way to the reduction of the greenhouse gas and pollutant emissions in global individual transport. Purely battery electric vehicles (BEVs) still have the disadvantages of high battery costs and low range [1]. As a transitional , hybrid electric vehicles (HEVs) offer a compromise between low pollutant emissions from BEVs and the high range of conventional vehicles [2]. In principle, they have the potential to be operated more efficiently than purely combustion engine vehicles. In fact, HEVs dominate the European vehicle market, with a market share of 3.7% compared to BEVs with 1% [3]. In HEVs, one or more electrical motors (EMs) can be arranged in the drivetrain and thus define different drivetrain concepts. Parallel hybrid drivetrains form the group of concepts in which a mechanical of the internal combustion engine (ICE) with the is possible and all drives can simultaneously provide for propulsion [4]. Five basic parallel hybrid topologies (P0-P4) are distinguished depending on the arrangement of the EM in the power path from the ICE (internal combustion engine) to the wheels (Figure1). By combining the EM arrangements of the basic topologies, new drivetrain variants can be created, such as the exemplary P0P2 hybrid in Figure1. Dual-clutch transmissions are particularly well suited for HEVs, as they combine high efficiency with a

Appl. Sci. 2020, 10, 7086; doi:10.3390/app10207086 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7086 2 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 19 highdesign level modifications of driving comfort[6]. In the [5 ]P2 and arrangement, meet the additional they find functionala particularly requirements broad application with few in design HEVs modifications[7]. [6]. In the P2 arrangement, they find a particularly broad application in HEVs [7].

Figure 1.1. Arrangements of the EM (electric machine) in basic parallelparallel hybridhybrid topologiestopologies (P0-P4) and exemplary arrangement throughthrough combinationcombination (P0P2).(P0P2).

Among the development goals of the automotive industry,industry, reliabilityreliability is ofof centralcentral importanceimportance alongside eefficiency,fficiency, emissionsemissions andand costscosts [[8].8]. Automotive products are multidisciplinarymultidisciplinary and subject to increasingincreasing complexitycomplexity with aa highhigh numbernumber ofof variants.variants. The later correction of errors that occur during early development phases is particularlyparticularly cost-intensivecost-intensive [[9].9]. For this reason, the reliabilityreliability ofof vehicle transmissionstransmissions is is continuously continuously taken taken into account,into account, from thefrom early the design early phases design to phases the realization to the ofrealization the physical of the product physical [10 product]. In early [10]. product In early creation product phases,creation lifetime phases, calculationlifetime calculation is often usedis often to assessused to reliability assess reliability [11]. In the [11]. process, In the occurringprocess, occurring loads for theloads investigated for the investigated component component are compared are withcompared their bearablewith their loads bearable and convertedloads and intoconverted component into component damage or damage lifetime. or Occurring lifetime. Occurring loads are, forloads instance, are, for extractedinstance, extracted from real from driving real tests, driving or are tests, generated or are generated synthetically synthetically and condensed and condensed to load spectrato load spectra by means by means of a suitable of a suitable classification classification method. method. Xue et Xue al. [et12 al.] present [12] present a method a method to extract to extract real loadreal load data data for the for durability the durability design design of transmissions of transmissions considering considering numerous numerous influences influences on the on loads, the andloads, apply and it apply to a conventional it to a conventional vehicle, a BEVvehicle, and a an BEV HEV. and Approaches an HEV. toApproaches generating to of syntheticgenerating load of datasynthetic canbe load found data in can [13 ,be14 ].found Belingardi in [13] et and al. [[14].15] investigate Belingardi the et al. influence [15] investigate of dynamic the overloads influence onof thedynamic lifetime overloads of gears on in the lifetime transmission of gears of anin the electric transmission vehicle. Forof an this electric purpose, vehicle. the dynamicFor this purpose, factor is determinedthe dynamic analytically, factor is determined using a rigid analytically, and a fully usin elasticg a simulationrigid and a model fully elastic of the gearsimulation pairs. Amodel lifetime of calculationthe gear pairs. of a A six-speed lifetime manualcalculation transmission of a six-speed in a conventionalmanual transmission vehicle drivetrain in a conventional is performed vehicle by Kamperdrivetrain et al.is performed in [16]. The by development Kamper et of al. a simulationin [16]. The model development and a method of a ofsimulation calculating model the damage and a inmethod gears inof acalculating conventional the drivetraindamage in is gears described in a conv by Foulardentional et drivetrain al. in [17]. is An described approach by to Foulard determining et al. thein [17]. remaining An approach lifetime to of determining transmission the gears remaining in real vehicle lifetime operation of transmission is presented gears by in Foulard real vehicle et al. inoperation [18] and is applied presented to aby manual Foulard transmission et al. in [18] inand a conventionalapplied to a manual vehicle. transmission Major influences in a conventional on damage includevehicle. theMajor clutch influences actuation on indamage gearshifts, include the the inclusion clutch ofactuation the lower in gearshifts, drivetrain naturalthe inclusion frequencies of the inlower the drivetrain model and natural special frequencies events such in as the the model loss ofand special adhesion. events Negative such as the loss of from tire ICEadhesion. coast operationNegative torques are indicated from asICE negligible coast operation [19]. A load-relatedare indicated operating as negligible strategy [19]. for A theload-related vehicle is presentedoperating andstrategy discussed for the in vehicle detail inis presented [20], regarding and discussed damage to in gears. detail Haqin [20], et al. regarding [21] use adamage simulation to gears. model Haq to predictet al. [21] the use load a spectrasimulation and model damage to inpredict the gears the ofload a vehicle spectra transmission. and damage in the gears of a vehicle transmission.The computational approaches mentioned so far are suitable for evaluating reliability in product development.The computational In addition, approaches experimental mentioned methods so are far used are suitable to ensure for the evaluating reliability reliability of the implemented in product physicaldevelopment. product. In Foraddition, this purpose, experimental load spectra methods in the are form used of test to cyclesensure are the used reliability in running of tests.the Inimplemented [22], Friedmann physical et al. product. present For a method this purpose, of carrying load out spectra customer-representative in the form of test cycles gear testing are used with in regardrunning to tests. service In [22], life in Friedmann a time- and et cost-optimizedal. present a method manner. of carrying out customer-representative gear testing with regard to service life in a time- and cost-optimized manner. Transmissions in hybrid drivetrains can be subject to additional loads compared to their use in conventional drivetrains, which must be taken into account in order to dimension the components Appl. Sci. 2020, 10, 7086 3 of 19

Transmissions in hybrid drivetrains can be subject to additional loads compared to their use in conventional drivetrains, which must be taken into account in order to dimension the components and ensure reliable operation. In [23], Fugel et al. investigate the influence of a parallel and a power split hybrid drivetrain on the transmission input torque and compare the results to a conventional drivetrain. For this purpose, they use a simulation model and evaluate it for a given vehicle with varying drivers and driving routes. In [24], Lavall summarizes the influences of hybridization on the component loads in the transmission as a “hybrid effect.” To this end, Lavall mentions additional torques in drive and coast operation during boosting and regeneration, additional functions such as electric launch, changed vehicle parameters such as weight and changed component properties such as housing geometry. It is pointed out that it is necessary to take coast operation into account in the lifetime calculation, since regeneration can increase the load in negative power flow direction. An application-oriented reduction of the transmission input torque to achieve a target gear lifetime is presented in [25]. In the presented studies, the critical role of gears on transmission lifetime is well established. Like gears, bearings are among the high-risk transmission components [26] and cause 15% of transmission failures, making them the second most frequent cause of transmission failures after gears [8]. A literature search revealed few studies on bearing lifetime in automotive transmissions and especially hybrid applications. The aim of this paper is to investigate the influence of hybridization in terms of regeneration on the lifetime of bearings as highly critical elements in a dual clutch transmission. For this purpose, we use a detailed drivetrain model of a parallel HEV. In the drivetrain topologies P0, P1 and P2, the EM is placed upstream of the transmission and has to transmit additional power in the coast direction to achieve regeneration. The power transfer leads to additional loads acting on the machine elements and to a distinct lifetime behavior. The remaining part of the paper is structured as follows. Section2 first introduces the use case considered for this study. Afterwards, the underlying method used for calculating the lifetime and the system simulation model are introduced and explained. Section3 presents the results of the simulation and lifetime calculations for a conventional and a hybrid drivetrain, as well as a sensitivity analysis. Finally, Section4 gives a brief summary, a conclusion and an outlook regarding necessary and possible future work.

2. Materials and Methods The investigations of this contribution are carried out by means of a simulation of an exemplary system. For this purpose, the investigated vehicle and transmission are presented in Section 2.1. The approach to determine component lifetimes that fail due to fatigue and the application to bearing lifetimes is explained in Section 2.2. The calculation of the necessary load and motion quantities for the determination of bearing lifetimes is carried out with the help of a 1D drivetrain model. This model is presented in Section 2.3.

2.1. Use Case In this paper, a front- drive C-segment [27] in P2 configuration is considered as a use case. It is driven by a diesel ICE and a PMSM (permanent magnet synchronous motor) EM. The vehicle data are given in Table1. The drivetrain structure is presented in more detail in Section 2.3.

Table 1. Vehicle data.

Parameter Value Vehicle class C-segment Vehicle mass mveh 1600 kg ICE Power 125 kW EM Power 83 kW Appl. Sci. 2020, 10, 7086 4 of 19

Power is transmitted via a six-speed dual-clutch transmission with an integrated differential. TheAppl. layout Sci. 2020 of, the10, x transmission FOR PEER REVIEW the structure of the simulation model presented in Section 2.3 is4 shownof 19 in FigureAppl. Sci.2. 2020 The, 10 EM, x FOR is mountedPEER REVIEW on the input side of the clutch drum. The dual clutch connects4 of 19 the in Figure 2. The EM is mounted on the input side of the clutch drum. The dual clutch connects the clutch drum with one input shaft each. This ensures, with respective actuation from the transmission clutchin Figure drum 2. with The oneEM isinput mounted shaft each.on the This input ensures, side of withthe clutch respective drum. actuation The dual fromclutch the connects transmission the controlcontrolclutch unit unit drum (TCU), (TCU), with that onethat theinput the power power shaft floweach. flow isThisis notnot ensures, interruptedinterrupted with respective during during gear gear actuation shifts. shifts. Thefrom The two the two inputtransmission input shafts shafts are are arrangedarrangedcontrol concentrically; unitconcentrically; (TCU), that input theinput power shaft shaft 2flow is2 designedis is designednot interrupted as aas hollow a hollow during shaft gearshaft for shifts. for this this purpose.The purpose. two input The The shafts idlers idlers are of gearsof 1–6gears andarranged 1–6 the reverseand concentrically; the gearreverse are inputgear mounted are shaft mounted on2 is needledesigned on needle bearings as a bearingshollow on the shaft on intermediate the for intermediatethis purpose. shafts. shafts.The They idlers They are of axially are fixedaxiallygears by thrust fixed1–6 and by washers. thethrust reverse washers. Input gear shaftare Input mounted 1 shaft is mounted on1 is needle mounted in bearings locating in locating on bearing the intermediatebearing 6 and 6 isand supportedshafts. is supported They by are radial by bearingsradialaxially bearings 4 andfixed 6by 4 and andthrust thrust 6 and washers. bearingthrust Input bearing 3 onshaft input 3 on1 is input shaftmounted shaft 2. Input in 2. locating Input shaft shaft bearing 2 is 2 mounted is 6mounted and is insupported in the the housing housing by via nonlocatingviaradial nonlocating bearings bearing bearing 4 and 7. The 6 and7. intermediateThe thrust intermediate bearing shafts 3 onshafts input are are supported shaft supported 2. Input in in theshaft the housing 2 housing is mounted by by locating locatingin the housing bearings bearings 15 and15 17viaand andnonlocating 17 and nonlocating nonlocating bearing bearings 7. bearings The intermediate 16 16 and and 18, 18, shafts respectively. respec aretively. supportedThe The didifferential inff theerential housing is is mounted mountedby locating in in anbearings an adjusted adjusted 15 and 17 and nonlocating bearings 16 and 18, respectively. The differential is mounted in an adjusted bearingbearing arrangement arrangement in in an an X X layout layout (bearings (bearings 11 and 2). The The reversal reversal shaft shaft is is supported supported by by two two floating floating bearing arrangement in an X layout (bearings 1 and 2). The reversal shaft is supported by two floating bearingsbearings and and thrust thrust washers. washers. The The transmissiontransmission has a total of of 20 20 bearings. bearings. bearings and thrust washers. The transmission has a total of 20 bearings.

FigureFigure 2. Transmission2. Transmission layout layout and structure of of its its torsional torsional rigid-body rigid-body model. model. Figure 2. Transmission layout and structure of its torsional rigid-body model. 2.2. Lifetime Calculation 2.2. Lifetime Calculation 2.2. Lifetime Calculation In thisIn this paper, paper, the the bearing bearing lifetime of of the the transmission transmission is calculated is calculated on the onbasis the of basisdamage of due damage dueto materialIn this paper, fatigue. fatigue. the For bearing For this, this, the lifetime theapproach approach of the according transmission according to Figure is to calculated Figure 3 is applied.3 is on applied. theAs shownbasis As of in showndamage [28], the in due [ 28 ], theto approach approachmaterial canfatigue. can also also Forbe be adapted this, adapted the to approach toother other machine machineaccording elem elementsents to Figure that fail that 3 dueis failapplied. to due fatigue, to As fatigue, suchshown as suchingears [28], asor the gears orapproach shafts.shafts. ByBy can makingmaking also be suitable suitable adapted assumptions, assumptions, to other machine wear-related wear-related elem entsfailures failuresthat such fail as suchdue those to as fatigue, of those friction ofsuch frictionclutches as gears clutchesor or orshafts. sealsseals can Bycan alsomaking also bebe calculatedcalculatedsuitable assumptions, [29], [29], but but these these wear-related are are om omitteditted failuresin inthis this paper such paper foras forthosebrevity. brevity. of Thefriction Theapproach clutches approach for or for seals can also be calculated [29], but these are omitted in this paper for brevity. The approach for consideringconsidering component component lifetime lifetime is is basedbased on existi existingng standards standards and and established established procedures procedures in the in the considering component lifetime is based on existing standards and established procedures in the automotiveautomotive industry. industry. The The core core aspect aspect ofof thethe approach is is the the comparison comparison of occurring of occurring and andbearable bearable automotiveloads, which industry. is necessary The forcore the aspect quantitative of the approachassessment is of the component comparison lifetime of occurring [26]. and bearable loads, which is necessary for the quantitative assessment of component lifetime [26]. loads, which is necessary for the quantitative assessment of component lifetime [26].

Figure 3. Approach for determining the system lifetime [28]. Figure 3. Approach for determining the system lifetime [28]. Figure 3. Approach for determining the system lifetime [28]. Appl. Sci. 2020, 10, 7086 5 of 19

Here, a comparison is made computationally using a damage accumulation hypothesis. An alternative to simulation is testing, but this is usually associated with a high expenditure of time and money. Initially, driving cycles are defined to determine the bearing loads for the computational analysis. The cycles represent the assumed use of the transmission consisting of various boundary conditions such as driver, driving distance, special events and others. The influence of the vehicle is represented by a simulation model. The simulation model considers the interaction of the subsystems in a selected level of detail and outputs time-based records of the bearing loads (axial and radial forces) and their frequency of occurrence in the form of the rotational speed. The speed and load of the bearings depend on which gear is engaged, whether the vehicle accelerates or decelerates and whether gearshifts occur. In the case of measurement of the occurring loads, the bearing loads are typically calculated back from an easily measurable variable, such as the input torque of the transmission [26]. Bearings can be subjected to a combined load from radial FR(t) and axial forces FA(t). However, for the comparison of the occurring and bearable loads, a reduced quantity is required, i.e., the dynamically equivalent force P(t). The calculation is carried out according to Equation (1). X and Y are the dynamic radial and axial load factor, respectively. They can either be obtained directly from the bearing manufacturer or calculated using analytical equations of ISO 281 [30].

P(t) = X F (t) + Y F (t) (1) · R · A The resulting time records P(t) per bearing are then simplified to load spectra with the aid of the two-parametric time-at-level counting [31]. The occurring loads and speeds are divided into 64 classes according to the recommendation of [32]. The bearable loads of the bearings required for the accumulation of damage are determined using ISO 281 [30]. The possible number of rotations Li of a load class i with a failure probability of 10% is obtained in 106 bearing revolutions, according to Equation (2).

!p C Li = aISO (2) · Pi

The life modification factor aISO takes into account the type, quality and geometry of the bearing, as well as the fatigue limit of the raceway material [30]. C is the dynamic load rating of the bearing, 10 and can be obtained from the bearing manufacturer. The exponent p is 3 for ball bearings and 3 for roller bearings. Pi is the dynamically equivalent bearing load of a load class. Information on the bearing data, taken from bearing manufacturer catalogues, can be found in Table A1. In the damage accumulation, occurring and bearable loads are compared and summed to yield damage D over all k load classes (Equation (3)). ni denotes the number of bearing revolutions occurring for an individual load class in the cycle. k X n D = i (3) L i=1 i

Based on the cumulative damage D and the cycle duration Tcycle, the component lifetime B10 of a bearing is calculated (Equation (4)). D B10 = (4) Tcycle

The failure distribution of a single bearing Fj(t) over time is described using the Weibull distribution (Equation (5)). t is the failure-free time and can take values between 0.1 B and 0.3 B 0 · 10 · 10 for bearings [32]. In this paper t = 0.2 B is used. b is a shape parameter and equals 1.1 for ball 0 · 10 bearings and 1.35 for roller bearings [32]. The characteristic lifetime Tj of a single j is calculated using Equation (6). t t b ( − 0 ) T t F (t) = 1 e− j− 0 (5) j − Appl.Appl. Sci. 20202020,, 1010,, 7086x FOR PEER REVIEW 66 ofof 1919

B −t T = B10 t0 +t (6) Tj = p−log− (0.9)+ t0 (6) b log(0.9) − The system failure distribution or the failure distribution of all bearings as a group F(t) over The system failure distribution or the failure distribution of all bearings as a group FB(t) over time is calculated by a superposition of the failure probabilities of the individual bearings. Here, it is time is calculated by a superposition of the failure probabilities of the individual bearings. Here, assumed that the group of bearings fails as soon as one bearing fails. The underlying Boolean it is assumed that the group of bearings fails as soon as one bearing fails. The underlying Boolean relationship is given in Equation (7). relationship is given in Equation (7).

F(t) =1−1−FY (t) F (t) = 1 1 F (t) (7)(7) B − − j j The system lifetime or the lifetime of all bearings as a group with a failure probability of 10% is

the timeThe systemt at which lifetime F reaches or the lifetime the value of 0.1. all bearings as a group with a failure probability of 10% is the timeFor tfurther at which information FB reaches on the the value approach 0.1. described, we refer the reader to references [28] and [29]. For further information on the approach described, we refer the reader to references [28,29].

2.3.2.3. System Model ToTo determinedetermine the operating conditions of of the the bear bearingsings in in terms terms of of force force and and rotational rotational speed, speed, a adetailed detailed forward forward oriented oriented overall overall vehicle vehicle model model is is built built in MatlabMatlab/Simulink/Simulink using thethe SimscapeSimscape library.library. TheThe high-levelhigh-level structurestructure ofof thethe modelmodel isis shownshown inin FigureFigure4 4.. The The submodels submodels are are divided divided into into ICE,ICE, EM, mechanical drivetrain, drivetrain, vehicle, vehicle, driver driver and and hybrid hybrid control control unit unit (HCU). (HCU). On On the theinput input side, side, the the model receives a driving cycle-dependent time-based desired velocity v . In the driver model, model receives a driving cycle-dependent time-based desired velocity v. desIn the driver model, the the desired velocity is compared to the actual velocity v and the deviation is converted into an desired velocity is compared to the actual velocity vact and the deviation is converted into an accelerator x or pedal position x by a PI-controller. accelerator xthr or brake pedal position brkx by a PI-controller.

FigureFigure 4.4. Modular structure of the system simulationsimulation modelmodel andand exchangedexchanged quantities.quantities.

The pedal positions are converted into desired torques of the ICE Tdes,ICE, the EM Tdes,EM and the The pedal positions are converted into desired torques of the ICE T,, the EM T, and mechanical brake (MB) Tdes,MB by the HCU. In this paper, a simplified operating strategy is considered, the mechanical brake (MB) T, by the HCU. In this paper, a simplified operating strategy is sinceconsidered, the influence since the of regenerativeinfluence of regenerative braking on the brakin bearingg on lifetime the bearing of the lifetime transmission of the transmission is investigated. is Theinvestigated. vehicle is The driven vehicle purely is driven by the purely ICE. The by decelerationthe ICE. The ofdeceleration the vehicle of is the implemented vehicle is implemented by splitting theby splitting braking the torque braking between torque ICE, between EM and ICE, the EM MB. and For the this, MB. a For required this, a decelerationrequired deceleration torque T dem,brktorque is calculated from the brake pedal position. This deceleration torque is applied with priority by the T, is calculated from the brake pedal position. This deceleration torque is applied with priority dragby the torque drag oftorque the ICE. of the The ICE. desired The desired ICE torque ICE istorq calculatedue is calculated according according to Equation to Equation (8) with (8) the with current the transmission gear ratio iG (see Table A2). current transmission gear ratio i (see Table A2).

Tdem,brkT, Tdes,ICET,== (8)(8) iGi

If the maximum drag torque T, of the ICE is not sufficient for the deceleration of the vehicle, the EM is used as a generator (Equation (9)).

T , T, = −T, (9) i Appl. Sci. 2020, 10, 7086 7 of 19

If the maximum drag torque T of the ICE is not sufficient for the deceleration of the vehicle, Appl. Sci. 2020, 10, x FOR PEER REVIEWmax, ICE 7 of 19 the EM is used as a generator (Equation (9)). For stronger braking which cannot be achieved by the drag torque of the ICE and the generator T dem,brk operation of the EM T,,T, des,EMthe mechanical= brakeT max,is applied ICE (Equation (10)). (9) iG − T, =T, −T,, (10) For stronger braking which cannot be achieved by the drag torque of the ICE and the generator In the ICE model, the desired torque T is converted into an acting torque T by a first- operation of the EM Tmax,EM,ICE, the mechanical, brake is applied (Equation (10)). , order delay with a time constant of t = 0.203 s [33]. The maximum drive torque is limited in the model by a full load characteristicT =curve.T For Tcoast operation, an analytically derived(10) thrust des,MB dem,brk − max,EM,ICE characteristic curve is used [26], which decreases in proportion to engine speed to −10% of the nominalIn the ICE torque model, (Figure the desired5a). torque Tdes,ICE is converted into an acting torque Tact,ICE by a first-order delay with a time constant of tICE = 0.203 s [33]. The maximum drive torque is limited in the modelIn the by EM a full model, load the characteristic desired torque curve. T, For coast is converted operation, into an ananalytically acting torque derived T, thrust by a first- characteristicorder delay curve with is a used time [ 26constant], which of decreases t = 0.04 s in proportion [33]. The tomaximum engine speed deliverable to 10% motor of the and nominal generator − torquetorques (Figure are5a). limited by power hyperbolas and constant torque curves (Figure 5b).

(a) (b)

FigureFigure 5. (a) 5. Full-load (a) Full-load characteristic characteristic curve (blue) curve and (blue) thrust and characteristic thrust characteristic curve (red) curve of the (red) ICE model;of the ICE (b) tractionmodel; (blue)(b) traction and generator (blue) and (red) generator torque limits(red) torque of the EMlimits model. of the EM model.

In the EM model, the desired torque Tdes,EM is converted into an acting torque Tact,EM by a In the vehicle model, the effective driving resistance force F is calculated as a function of the first-order delay with a time constant of tEM = 0.04 s [33]. The maximum deliverable motor and vehicle velocity x according to Equation (11). f and f are driving resistance coefficients which generator torques are limited by power hyperbolas and constant torque curves (Figure5b). cause constant and velocity proportional driving resistances. f causes a resistance force which is In the vehicle model, the effective driving resistance force Fdrag is calculated as a function of the proportional. to the square of the vehicle velocity. The values of the coefficients are listed in Table A3. vehicle velocity xveh according to Equation (11). fd0 and fd1 are driving resistance coefficients which F =f +f ⋅x +f ⋅x cause constant and velocity proportional driving resistances. fd2causes a resistance force which is (11) proportionalThe toforces the square and torques of the vehicledescribed velocity. above The act valueson a torsional of the coe vibrationfficients aremodel listed of inthe Table mechanical A3.

drivetrain (Figure 6). Individual inertias are re.presented as. 2 cylinders connected by spring-damper F = f + f x + f x (11) combinations or clutches. Externaldrag torquesd0 actd1 · thvehroughd2 the· vehICE, EM, MB and driving resistance. For the recommended [19] consideration of the lower natural frequencies of the drivetrain, the dual mass The forces and torques described above act on a torsional vibration model of the mechanical (DMF), the side shafts and the were modeled elastically [34]. The wheel-to-ground drivetrain (Figure6). Individual inertias are represented as cylinders connected by spring-damper contact is modelled ideally (i.e., ideal rolling), since special events such as loss of tire grip will not be combinations or clutches. External torques act through the ICE, EM, MB and driving resistance. For the investigated in this paper. recommended [19] consideration of the lower natural frequencies of the drivetrain, the dual mass flywheel (DMF), the side shafts and the tires were modeled elastically [34]. The wheel-to-ground contact is modelled ideally (i.e., ideal rolling), since special events such as loss of tire grip will not be investigated in this paper.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 19

For stronger braking which cannot be achieved by the drag torque of the ICE and the generator operation of the EM T,,, the mechanical brake is applied (Equation (10)).

T, =T, −T,, (10)

In the ICE model, the desired torque T, is converted into an acting torque T, by a first- order delay with a time constant of t = 0.203 s [33]. The maximum drive torque is limited in the model by a full load characteristic curve. For coast operation, an analytically derived thrust characteristic curve is used [26], which decreases in proportion to engine speed to −10% of the nominal torque (Figure 5a).

In the EM model, the desired torque T, is converted into an acting torque T, by a first- order delay with a time constant of t = 0.04 s [33]. The maximum deliverable motor and generator torques are limited by power hyperbolas and constant torque curves (Figure 5b).

(a) (b)

Figure 5. (a) Full-load characteristic curve (blue) and thrust characteristic curve (red) of the ICE model; (b) traction (blue) and generator (red) torque limits of the EM model.

In the vehicle model, the effective driving resistance force F is calculated as a function of the vehicle velocity x according to Equation (11). f and f are driving resistance coefficients which cause constant and velocity proportional driving resistances. f causes a resistance force which is proportional to the square of the vehicle velocity. The values of the coefficients are listed in Table A3. F =f +f ⋅x +f ⋅x (11) The forces and torques described above act on a torsional vibration model of the mechanical drivetrain (Figure 6). Individual inertias are represented as cylinders connected by spring-damper combinations or clutches. External torques act through the ICE, EM, MB and driving resistance. For the recommended [19] consideration of the lower natural frequencies of the drivetrain, the dual mass flywheel (DMF), the side shafts and the tires were modeled elastically [34]. The wheel-to-ground Appl.contact Sci. 2020is modelled, 10, 7086 ideally (i.e., ideal rolling), since special events such as loss of tire grip will not8 of be 19 investigated in this paper.

Figure 6. Structure of the torsional vibration model used for the hybrid drivetrain. For the conventional

drivetrain the torque Tact,EM is omitted and the inertia at the transmission input is added to the inertia of the clutch drum.

Since the vehicle motion is calculated in the torsional vibration model and the driving resistances are calculated in the translational degree of freedom, a transfer of the corresponding quantities is carried out with Equations (12) and (13). Tdrag is the equivalent driving resistance torque for a driving resistance force acting on the tire radius rtire. The vehicle velocity is calculated from the angular velocity . of the equivalent inertia of the vehicle φveh in the torsional vibration model.

T = F r (12) drag drag · . . x = φ r (13) veh veh · tire The motion of the individual drivetrain inertias is described in the torsional degree of freedom and is shown as an example for the vehicle inertia in Equation (14). ..    . . .  J φ = c φ + φ 2φ + d φ + φ 2φ T (14) veh · veh tire wheel,l wheel,r − veh tire wheel,l wheel,r − veh − drag .. . φ, φ and φ thus describe angular acceleration, angular velocity and angle of the corresponding inertia, respectively. ctire and dtire are stiffness and damping of the tires, respectively. The equivalent inertia of the vehicle Jveh is calculated using Equation (15). The parameters of the torsional vibration model are summarized in Table A4. 1 J = m r2 (15) veh 2 · veh · tire A rigid body model of the transmission (Figure2), including a transmission control unit (TCU), is embedded in the drivetrain model. The overall purpose of the transmission model is to calculate the motion and load condition of each bearing. The rotational speeds of the bearings directly result from the torsional motion of the model, which is calculated for each individual shaft using Equation (16). .. Jshaft is an individual shaft inertia in and φshaft its corresponding angular acceleration. Ti are the torques acting on the shaft and consider driving torques and torque losses [35].

.. X J φ = Ti (16) shaft · shaft i

The bearing loads result from the support of the tangential (Ft), radial (Fr) and axial (Fa) gear forces. To determine the bearing forces, static equilibria are evaluated for the individual transmission shafts (Equations (17) and (18)) at each time step. However, in Equation (18) only the bending moments are considered, since the torsional behavior is calculated in the rigid body model. X Fi = 0 (17) i X Mi = 0 (18) i Appl. Sci. 2020, 10, 7086 9 of 19

The forces are strongly dependent on the geometry of the shaft and gears, as well as the arrangement of the gears on the shaft. Figure7 shows an exemplary force diagram of intermediate shaft 1 for the calculation of bearing forces Fb15 and Fb16. For illustration purposes, all forces are rotated into the section plane, since the tooth meshes are not arranged identically around the circumference for all gears. The locations (gear diameters dG and shaft section lengths l) of the gear forces are considered in theAppl. model. Sci. 2020 FD, 10, isx FOR a gear PEER of REVIEW the final drive. 9 of 19

Figure 7. Force diagram of the first intermediate shaft. All forces are rotated into the section plane. Figure 7. Force diagram of the first intermediate shaft. All forces are rotated into the section plane. An acting tangential force Ft is calculated using Equation (19). TG is the acting torque on a gear wheel.An It acting is read tangential from the transmissionforce F is calculated model at eachusing time Equation step,taking (19). T into is accountthe acting the torque meshing on a losses. gear Thewheel. direction It is read of the from tangential the transmission force depends model on at the each torque time direction. step, taking into account the meshing losses. The direction of the tangential force depends on the torque direction. 2 TG Ft = ·2⋅T (19) F =dG (19) d The acting radial force Fr is calculated using Equation (20). αn is the normal pressure angle and The acting radial force F is calculated using Equation (20). α is the normal pressure angle and β β the the helixhelix angleangle ofof thethe correspondingcorresponding gear.gear. The amplitudeamplitude of the radial force is dependent on the amplitude of the tangential force. Since Since the the radial radial force force is is always or orientediented to to the the center center of of the the gear, the radialradial forceforce ofof anan idleridler correspondscorresponds toto itsits needleneedle bearingbearing load.load.

tantan(α(nα)) FrF= =F|tF| ⋅ (20) | | · coscos(β()β) The axial force F is given by Equation (21). Its direction depends on the direction of the The axial force Fa is given by Equation (21). Its direction depends on the direction of the tangential force,tangential as well force, as onas thewell helix as on direction the helix of direction the gear. of the gear.

F =F ⋅tan(β) (21) Fa = Ft tan(β) (21) In addition to the calculation of the motion and· load of the individual bearings, the influences of gearIn additionshifts, synchronizations to the calculation and of losses the motion are taken and into load account. of the individual In gear shifts bearings, and synchronizations the influences of gearof gears, shifts, additional synchronizations synchronization and losses torques are taken act in into addition account. to the In geardriving shifts resistances, and synchronizations which cause ofhigher gears, gear additional forces and synchronization must ultimately torques be support act ined addition by the tobearings. the driving For resistances,this purpose, which both causeinput higherclutches gear and forces the seven and synchronizers must ultimately were be modeled supported, which by the are bearings. operated For by thisan implemented purpose, both TCU input on clutchesthe basis andof a theshift seven map. synchronizersThis enables launching were modeled, with the which ICE and are gear operated shifts by without an implemented interrupting TCU the ontractive the basis force. of These a shift friction map. This elements enables have launching identical with models the ICE that and distingu gear shiftsish between without interruptinga slip and a thestick tractive state. In force. the slip These state, friction the torque elements capacity have identicalset by the models TCU acts that on distinguish the two neighboring between a inertias slip and of a stickthe friction state. Inelement the slip as state, a synchronizing the torque capacity torque. setIf both by the speeds TCU actsare synchronized, on the two neighboring the friction inertias element of thechanges friction to the element stick state as a synchronizingand the torque torque.transmitted If both by the speeds friction are synchronized,element is limited the by friction the set element torque changescapacity. to For the reasons stick state of andsimplification, the torque transmittedthe set torque by the of frictionthe TCU element is applied is limited without by the delay set torqueto the capacity.friction elements. For reasons To evaluate of simplification, transmission the losses set torque, modules of the are TCUimplemented is applied which without take delayinto account to the load-dependent and -independent bearing and gear losses, drag losses of the wet input clutches and seal losses. The loss calculation was validated in an earlier publication in large parts of the operating range of the transmission so that the calculated power flows were plausibly represented. For further information on the validation, transmission controls and loss calculation, we refer the reader to [35].

Appl. Sci. 2020, 10, 7086 10 of 19 friction elements. To evaluate transmission losses, modules are implemented which take into account load-dependent and -independent bearing and gear losses, drag losses of the wet input clutches and seal losses. The loss calculation was validated in an earlier publication in large parts of the operating range of the transmission so that the calculated power flows were plausibly represented. For further information on the validation, transmission controls and loss calculation, we refer the reader to [35].

3. Results Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 19 In this section, the results of the evaluation of the simulation model and the subsequent lifetime calculation3. Results are presented. First, in Section 3.1, the load and the resulting bearing lifetime of the conventionalIn this drivetrain section, the are results evaluated. of the evaluation The results of the of simulation the hybrid model drivetrain and the aresubsequent compared lifetime to the conventionalcalculation drivetrain are presented. in Section First, 3.2 in. InSection Section 3.1, 3.3 the, a load sensitivity and the analysis resulting is bearing performed lifetime to determine of the the influenceconventional of the drivetrain maximum are regeneration evaluated. The torque result ons theof the bearing hybrid lifetime. drivetrain are compared to the Forconventional all investigations, drivetrain the in Section Worldwide 3.2. In HarmonizedSection 3.3, a sensitivity Light Vehicles analysis Test is performed Cycle (WLTC) to determine [36] was chosenthe as influence the representative of the maximum cycle. regeneration Details of its torque properties on the bearing are summarized lifetime. in Table A5. The WLTC was chosenFor because all investigations, it has combined the Worldwide cycle components Harmonized of Light low, medium,Vehicles Test high Cycle and (WLTC) extra high [36] vehiclewas chosen as the representative cycle. Details of its properties are summarized in Table A5. The WLTC velocities. In addition, vehicle launches, gear shifts and decelerations occur. Special conditions such was chosen because it has combined cycle components of low, medium, high and extra high vehicle as lossvelocities. of tire adhesion In addition, or misusevehicle launches, do not occur gear shift in thes and WLTC, decelerations and are occur. not consideredSpecial conditions in this such paper. With theas choiceloss of oftire the adhesion WLTC, or the misuse driver’s do not influence occur in on the the WLTC, calculated and are loads not isconsidered also deliberately in this paper. omitted. In general,With representativethe choice of the load WLTC, cycles the aredriver’s subject influe tonce extensive on the boundarycalculated loads conditions. is also deliberately Load spectra are experimentallyomitted. In general, measured representative or generated load manufacturer-specifically,cycles are subject to extensive taking boundary into accountconditions. the Load driver, environmentspectra andare experimentally vehicle [14]. measured or generated manufacturer-specifically, taking into account the driver, environment and vehicle [14]. 3.1. Conventional Drivetrain 3.1. Conventional Drivetrain The load spectrum at the transmission input resulting from the applied driving cycle is shown in Figure8. Here,The theload transmission spectrum at the input transmission torque is showninput re versussulting from the transmission the applied driving input speed.cycle is Theshown color in Figure 8. Here, the transmission input torque is shown versus the transmission input speed. The indicates the frequency of occurrence of the respective operating point. The torque at the transmission color indicates the frequency of occurrence of the respective operating point. The torque at the input istransmission to be considered input is hereto be asconsidered a representation here as a representation of the bearing of load,the bearing since theload, gear since forces the gear to be supportedforces by to the be bearingssupported are by proportionalthe bearings are to proporti the torque.onal Theto the range torque. of The positive range torques of positive arises torques during drive operation.arises during Negative drive operation. torques occur Negative during torques coast o operationccur during of coast the ICE. operation Negative of the torques ICE. Negative beyond the thrust characteristictorques beyond curve the thrust of the characteristic ICE occur duringcurve of downshiftsthe ICE occur in during coasting. downshifts In this in case, coasting. an overtorque In this is appliedcase, by an the overtorque clutches is to applied synchronize by the clutches the ICE to to synchronize the transmission the ICE to input the transmission speed. input speed.

FigureFigure 8. Load 8. Load spectrum spectrum of the of the transmission transmission input input torquetorque for for the the conventional conventional drivetrain drivetrain in the in WLTC. the WLTC.

The failureThe failure probabilities probabilities of theof the individual individual rolling rolling bearingsbearings of the the transmission transmission and and their their superpositionsuperposition over over the travelthe travel distance distance are are shown shown in double-logarithmic double-logarithmic form form in Figure in Figure 9. In 9[37],. In the [ 37 ], the transmissiontransmissionB10 B-lifetime-lifetime is is given given as as 150,000–250,000 150,000–250,000 km, km, dep dependingending on on the the vehicle vehicle type. type. In [38] In [and12,38 ], the lifetime[12], ofthe 300,000 lifetime km of for300,000 the vehiclekm forand the transmissionvehicle and transmission is given respectively is given respectively without specifying without the probabilityspecifying of failure. the probability Therefore, of the failure. superposed Therefore, lifetime the superposed of the bearings lifetime with of the about bearings 195,000 with km about in the 195,000 km in the WLTC is considered plausible by the authors. The failure distribution is dominated by bearings 16 and 18. They are used as locating bearings for the two intermediate shafts, and are particularly frequently present in the power flow. In addition, as locating bearings, they are subjected to the combined load of axial and radial bearing forces resulting from the gears. The high scattering of bearing lifetimes of two orders of magnitude according to the authors’ assessment can be attributed Appl. Sci. 2020, 10, 7086 11 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 19

WLTC isto consideredthe fact that plausible different byload the assumptions authors. The than failure thosedistribution caused in the is WLTC dominated and the by vehicle bearings under 16 and 18. Theyconsideration are used as were locating used bearingsfor the design for the of the two transmission. intermediate shafts, and are particularly frequently present in theAccording power flow.to the In calculation addition, approach as locating presented, bearings, the they needle are subjectedroller bearings to the used combined are fatigue- load of axial andresistant, radial since bearing they forces only experience resulting fromsignificant the gears. operating The loads high in scattering the synchronized of bearing case, lifetimes i.e., without of two orders ofrelative magnitude motion of according the bearing to raceways. the authors’ Loads assessment that occur canduring be synchronization attributed to the and fact due that to bearing different friction do not contribute to damage in this case. In fact, wear in the form of indentations in the raceway load assumptions than those caused in the WLTC and the vehicle under consideration were used for under high stationary loads is mentioned in the literature as a frequent cause of failure [38,39]. If a lifetime the designmodel of becomes the transmission. available, the influence of the needle bearings can be included in the approach. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 19

to the fact that different load assumptions than those caused in the WLTC and the vehicle under consideration were used for the design of the transmission. According to the calculation approach presented, the needle roller bearings used are fatigue- resistant, since they only experience significant operating loads in the synchronized case, i.e., without relative motion of the bearing raceways. Loads that occur during synchronization and due to bearing friction do not contribute to damage in this case. In fact, wear in the form of indentations in the raceway under high stationary loads is mentioned in the literature as a frequent cause of failure [38,39]. If a lifetime model becomes available, the influence of the needle bearings can be included in the approach.

Figure 9.FigureFailure 9. probabilitiesFailure probabilities of the individual of the individual bearings bear andings the bearingsand the asbearings a group as for a thegroup conventional for the drivetrain.conventional The position drivetrain. numbers The position of the bearingsnumbers of are th showne bearings framed are shown (see framed Figure 2(see). The Figure dotted 2). The line

indicatesdotted the Bline10-lifetime indicates with the B 195,000-lifetime km. with 195,000 km.

According3.2. Hybrid toDrivetrain the calculation approach presented, the needle roller bearings used are fatigue-resistant,The calculated since they load only spectrum experience of the transmission significant input operating torque loads for the in HEV the synchronizedis shown in Figure case, 10. i.e., withoutThe relative positive motion torque range of the remains bearing essentially raceways. unchange Loadsd, that since occur propulsion during is still synchronization performed by the and ICE. due to bearingHowever, friction operating do not po contributeints with a tohigh damage negative in torque this case.occur Inmore fact, frequently wear in due the to form regeneration. of indentations in the raceway under high stationary loads is mentioned in the literature as a frequent cause of failure [38,39Figure]. If 9. a Failure lifetime probabilities model becomes of the individual available, bear theings influence and the bearings of the as needle a group bearings for the can be included inconventional the approach. drivetrain. The position numbers of the bearings are shown framed (see Figure 2). The dotted line indicates the B-lifetime with 195,000 km. 3.2. Hybrid Drivetrain 3.2. Hybrid Drivetrain The calculated load spectrum of the transmission input torque for the HEV is shown in Figure 10. The calculated load spectrum of the transmission input torque for the HEV is shown in Figure 10. The positive torque range remains essentially unchanged, since propulsion is still performed by the The positive torque range remains essentially unchanged, since propulsion is still performed by the ICE. ICE. However,However, operating operating po pointsints with with a ahigh high negative negative torque torque occu occurr more morefrequently frequently due to regeneration. due to regeneration.

Figure 10. Load spectrum of the transmission input torque for the hybrid drivetrain without limitation of the regeneration torque in the WLTC. Scaling of the color bar identical to Figure 8.

The resulting failure probabilities of the bearings for the HEV are shown in Figure 11. The load spectrum of the hybrid drivetrain causes a shift in the bearing lifetimes towards lower driving distances. In total, the superposed lifetime of all bearings is 123,000 km, which corresponds to a reduction of 36.9% compared to the conventional drivetrain. In addition to the lower overall lifetime, the curves scatter more strongly compared to the conventional drivetrain.

FigureFigure 10. Load 10. Load spectrum spectrum of the of transmissionthe transmission input input torque torque for forthe the hybrid drivetrain drivetrain without without limitation limitation of the regenerationof the regeneration torque torque in the in WLTC.the WLTC. Scaling Scaling of theof the color color bar bar identical identical to Figure 8.8.

The resulting failure probabilities of the bearings for the HEV are shown in Figure 11. The load spectrum of the hybrid drivetrain causes a shift in the bearing lifetimes towards lower driving distances. In total, the superposed lifetime of all bearings is 123,000 km, which corresponds to a reduction of 36.9% compared to the conventional drivetrain. In addition to the lower overall lifetime, the curves scatter more strongly compared to the conventional drivetrain. Appl. Sci. 2020, 10, 7086 12 of 19

The resulting failure probabilities of the bearings for the HEV are shown in Figure 11. The load spectrum of the hybrid drivetrain causes a shift in the bearing lifetimes towards lower driving distances. In total, the superposed lifetime of all bearings is 123,000 km, which corresponds to a reduction of 36.9% compared to the conventional drivetrain. In addition to the lower overall lifetime, the curves scatterAppl. Sci. more 2020, strongly10, x FOR PEER compared REVIEW to the conventional drivetrain. 12 of 19

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 19

FigureFigure 11. 11.Failure Failure probabilities probabilities of of the the individual individual bearings bearings and and the the bearings bearings as as a a group group for for the the hybrid hybrid drivetrain.drivetrain. TheThe positionposition numbersnumbers ofof thethe bearingsbearings areare shown shown framed framed (see (see Figure Figure2 ).2). The The dotted dotted line line Figure 11. Failure probabilities of the individual bearings and the bearings as a group for the hybrid indicatesindicates the the B 10B-lifetime-lifetime of of the the bearing bearing group group with with 123,000 123,000 km. km. drivetrain. The position numbers of the bearings are shown framed (see Figure 2). The dotted line 3.3. Sensitivity Analysis 3.3. Sensitivityindicates Analysis the B -lifetime of the bearing group with 123,000 km. In this section, a parameter variation of the maximum regeneration torque is performed. In reality, In3.3. this Sensitivity section, Analysis a parameter variation of the maximum regeneration torque is performed. In the limitation of the regeneration torque may have its cause on the electrical, mechanical and human reality, theIn limitation this section, of thea parameter regeneration variation torque of themay maximum have its causeregeneration on the torque electrical, is performed. mechanical In and side, and is not unusual in practice. Boundary conditions such as efficiency, component limits and humanreality, side, theand limitation is not unusual of the regeneration in practice. torqueBoundary may conditionshave its cause such on theas efficiency,electrical, mechanical component and limits driving comfort influence the control of the regeneration process [40]. The load cycles presented and drivinghuman comfortside, and influenceis not unusual the incontrol practice. of Boundarythe regeneration conditions process such as [40]. efficiency, The load component cycles presentedlimits in [12] show a clear limitation of the regeneration torque. This may be due to a limitation of the in [12]and show driving a clear comfort limitation influence of the the control regeneration of the rege torque.neration Thisprocess may [40]. be The due load to cyclesa limitation presented of the charging power when the battery is close to fully charged [41]. On the mechanical side, an overload chargingin [12] power show when a clear the limitation battery isof closethe regeneration to fully charged torque. [41]. This On may the be mechanical due to a limitation side, an of overload the of the transmission can always be avoided by limiting the recuperation torque by controls. As an of thecharging transmission power canwhen always the battery be avoided is close to by fully limiti chargedng the [41]. recuperation On the mechanical torque side, by controls.an overload As an alternative,of the atransmission load-related can operating always be strategy avoided forby thelimiti vehicleng the canrecuperation be implemented, torque by whichcontrols. adjusts As an the alternative, a load-related operating strategy for the vehicle can be implemented, which adjusts the possiblealternative, maximum a load-related torque on theoperating basis of strategy the so for far the experienced vehicle can component be implemented, damage. which adjusts the possible maximum torque on the basis of the so far experienced component damage. Figurepossible 12 maximumshows an torque exemplary on the basis load of spectrum the so far of experienced the transmission component input damage. torque for the hybrid Figure 12 shows an exemplary load spectrum of the transmission input torque for the hybrid drivetrain withFigure a 12 regenerative shows an exemplary torque limit load of spectrum 100 Nm of in the the transmission WLTC. The input limitation torque can for bethe seen hybrid by an drivetraindrivetrain with with a regenerative a regenerative torque torque limit limit of of 100 100 Nm Nm in the WLTC.WLTC. The The limita limitationtion can can be seenbe seen by an by an increase of the operating points at 100 Nm, whereby operating points smaller than 100 Nm are increaseincrease of the of operatingthe operating points points at− at−100 −100 Nm, Nm, whereby whereby operating points points smaller smaller than than −100− − 100Nm Nmare are caused by the overtorques of the clutches in powershifting. causedcaused by the by overtorques the overtorques of theof the clutches clutches in in powershifting. powershifting.

FigureFigure 12. Load 12. spectrumLoad spectrum of the transmissionof the transm inputission torque input fortorque thehybrid for the drivetrain hybrid drivetrain with a regenerative with a torqueregenerative limit of the torque electric limit machine of the electric of -100 machine Nm in of the -100 WLTC. Nm in Scaling the WLTC. of theScaling color of barthe identicalcolor bar to Figure 12. Load spectrum of the transmission input torque for the hybrid drivetrain with a Figure8identical. to Figure 8. regenerative torque limit of the electric machine of -100 Nm in the WLTC. Scaling of the color bar identicalFigure to Figure 13 depicts 8. the results of the parameter variation of the maximum regenerative torque of the EM. It can be seen that with increasing maximum regenerative torque, the life of the bearings Figuredecreases. 13 depictsThe increasing the results gradient of the of parameter the bearing va liferiation with ofincreasing the maximum recuperation regenerative torque can torque be of the EM.explained It can bybe theseen decreasing that with number increasing of operat maximuming points regenerative of high regenerative torque, torques. the life of the bearings decreases. The increasing gradient of the bearing life with increasing recuperation torque can be explained by the decreasing number of operating points of high regenerative torques. Appl. Sci. 2020, 10, 7086 13 of 19

Figure 13 depicts the results of the parameter variation of the maximum regenerative torque of the EM. It can be seen that with increasing maximum regenerative torque, the life of the bearings decreases. The increasing gradient of the bearing life with increasing recuperation torque can be explained by the Appl.decreasing Sci. 2020, number 10, x FOR ofPEER operating REVIEW points of high regenerative torques. 13 of 19

Figure 13. Results of the sensitivity analysis. Bearing lifetime over maximum regenerative torque in Figure 13. Results of the sensitivity analysis. Bearing lifetime over maximum regenerative torque in the WLTC. the WLTC. 4. Summary, Conclusions and Outlook 4. Summary, Conclusions and Outlook In this paper, the bearing lifetime of a transmission in a passenger application was investigated and theIn this influence paper, of the regenerative bearing brakinglifetime onof thisa transmission lifetime was in quantified. a passenger Here, car both application a conventional was investigatedand a parallel and hybrid the drivetraininfluence of were regenerative investigated. brak Theing bearing on this lifetimeslifetime werewas quantified. calculated usingHere,ISO both 281 a conventionalas part of a method and fora parallel estimating hybrid the lifetime drivetrain of technical were systems.investigated. The loadsThe onbearing the rolling lifetimes bearings were in calculatedthe transmission using wereISO 281 determined as part of with a method the help for of estimating a detailed drivetrainthe lifetime model of technical consisting systems. of ICE, The EM, loadsmechanical on the drivetrain, rolling bearings vehicle, in driver the andtransmission HCU. In thewere conventional determined drivetrain, with the help the EM of modela detailed was drivetrainomitted. To model take into consisting account of the ICE, additional EM, mechanical loads from drivetrain, the eigenbehavior vehicle, ofdriver the drivetrain, and HCU. the In DMF, the conventionalside shafts and drivetrain, tires were the considered EM model to bewas elastic. omitted. The To WLTC take was into chosen account as the the additional representative loads driving from thecycle. eigenbehavior Additional loads of the from drivetrain, the actuation the DMF, of the side clutches shafts inand gear tires shifts were were considered taken into to accountbe elastic. by The the WLTCimplemented was chosen power as shifts. the representative A simple approach driving was cy consideredcle. Additional as the loads operating from strategy,the actuation which of splits the clutchesthe driver’s in gear braking shifts torque were ta demandken into between account the by dragthe implemented torque of the power ICE, the shifts. EM andA simple the MB. approach was consideredThe results as show the thatoperating in the conventionalstrategy, which drivetrain, splits the the driver’s bearing braking lifetime torque of the demand locating between bearings theof the drag intermediate torque of the shafts ICE, dominates the EM and the the failure MB. distribution. The lifetime of the bearings as a group wasThe evaluated results asshow plausible that in onthe theconventional basis of literature drivetrain, references. the bearing In lifetime the hybrid of the drivetrain locating bearings without oftorque the intermediate limitation, a shafts 36.9% dominates reduction inthe bearing failure lifetimedistribution. was determined.The lifetime of The the lifetime bearings in as the a hybridgroup wasdrivetrain evaluated is shorter, as plausible because on high the negative basis of input literatu torquesre references. occur more In frequentlythe hybrid duringdrivetrain regenerative without torquebraking. limitation, These torques a 36.9% cause reduction additional in bearing bearing lifetime loads, depending was determined. on the gearThe engaged.lifetime in A the reduction hybrid drivetrainof the regenerative is shorter, torque, because as high can negative be justified input by thetorques numerous occur more boundaries frequently of regenerative during regenerative braking, braking.was investigated These torques in a sensitivity cause additional analysis. bearing The bearing loads, lifetime depending decreases on the with gear falling engaged. gradient A reduction up to the ofmaximum the regenerative regenerative torque, torque as can of thebe EM.justified by the numerous boundaries of regenerative braking, was investigatedThe findings in reported a sensitivity here analysis. demonstrate The bear thating the lifetime additional decreases loads occurringwith falling in gradient HEV must up beto thetaken maximum into account regenerative in transmission torque of development the EM. and operation. Design possibilities are a stronger dimensioningThe findings of the reported components, here demonstrate a design adaptation that th ofe additional the transmission loads layoutoccurring or the in useHEV of must a hybrid be takentopology into with account a di ffinerent transmission position of development the EM. If the an regenerationd operation. canDesign be realizedpossibilities without are additionala stronger dimensioningpower flow through of the thecomponents, transmission a design (e.g., topologies adaptation P3 orof P4),the thistransmission influence layout on the lifetimeor the use remains of a hybridnegligible. topology However, with these a different topologies position require of athe higher EM. speedIf the spreadregeneration of the EM.can be realized without additionalIn addition, power the flow loads through can be takenthe transmission into account on(e.g the., operatingtopologies strategy P3 or P4), side. this During influence development, on the lifetimethe operating remains strategy negligible. can be However, adapted to these the load topologies capacity require of the transmissiona higher speed by spread limiting of the the maximum EM. regenerativeIn addition, torque the or byloads a load-related can be taken gear shiftinginto acco strategyunt on that the takes operating into account strategy uniform side. loadsDuring on development,the bearings. During the operating operation, strategy the regenerative can be adapted torque canto the be adjustedload capacity online of to thethe damagetransmission observed by limitingin the components. the maximum The regenerative simultaneous torque consideration or by a lo ofad-related operating gear strategy shifting and strategy transmission that takes lifetime into account uniform loads on the bearings. During operation, the regenerative torque can be adjusted online to the damage observed in the components. The simultaneous consideration of operating strategy and transmission lifetime leads to a multiobjective optimization problem, the solution of which can yield a suitable tradeoff between energy saving and transmission lifetime. Although the results are considered plausible, and a transmission model validated with regard to losses was used, the calculation approaches should nevertheless be validated in future work. This includes the physical behavior of the torsional vibration model, as well as the bearing force Appl. Sci. 2020, 10, 7086 14 of 19 leads to a multiobjective optimization problem, the solution of which can yield a suitable tradeoff between energy saving and transmission lifetime. Although the results are considered plausible, and a transmission model validated with regard to losses was used, the calculation approaches should nevertheless be validated in future work. This includes the physical behavior of the torsional vibration model, as well as the bearing force calculation. The torsional vibration model can be validated by means of a modal analysis. A validation of the bearing force calculation is possible by measuring the bearing forces. However, the necessary application of strain gauges in the proximity of the bearing is costly. In addition to the physics, the results of the lifetime calculation should be validated. For this purpose, condensed durability tests mentioned in the introduction can be carried out on a test bench, or the manufacturer’s return data can be evaluated. Furthermore, the control influences of the modeled TCU and HCU (e.g., calibration) should be validated. Specific use cases should be investigated through more detailed and usage-specific, experimental cycles, taking special events into account. In the future, both the method and the model can be used for more detailed investigations. For example, the influences of a variation in operating strategy, transmission layout or components can be systematically quantified.

Author Contributions: Conceptualization, C.H., S.N. and K.W.; methodology, C.H., G.J., S.N. and K.W.; formal analysis, C.H.; writing—original draft preparation, C.H. and K.W.; writing—review and editing, G.J. and S.N.; supervision, G.J. and S.N. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the European Regional Development Fund (ERDF) as part of the research project DUETT (Diesel hybrid vehicles for environmentally conscious mobility: networked system development in a physical and virtual environment). Reference number ERDF-0800852. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

BEV Battery Electric Vehicle DMF Dual Mass Flywheel EM Electric Motor FD Final Drive HCU Hybrid Control Unit HEV Hybrid Electric Vehicle ICE Internal Combustion Engine ISO International Organization for Standardization MB Mechanical Brake PMSM Permanent Magnet Synchronous Motor TCU Transmission Control Unit WLTC Worldwide Harmonized Light Vehicles Test Cycle

Nomenclature

αn Normal pressure angle (◦) β Helix angle (◦) φ Angle (1) . 1 φ Angular velocity (s− ) .. 2 φ Angular acceleration (s− ) φ Vehicle equivalent inertia angle (1) . veh 1 φ Vehicle equivalent inertia angular velocity (s− ) .. veh 2 φveh Vehicle equivalent inertia angular acceleration (s− ) φ Left wheel angle (1) . wheel,l 1 φwheel,l Left wheel angular velocity (s− ) φwheel,r Right wheel angle (1) Appl. Sci. 2020, 10, 7086 15 of 19

. 1 φwheel,r Right wheel angular velocity (s− ) aISO Life modification factor (1) b Shape parameter (1) B10 Bearing lifetime with a failure probability of 10% (h) c Stiffness (Nm rad 1) · − C Dynamic load rating (N) d Damping (N m s rad 1) · · · − dG Gear diameter (mm) dm Mean bearing diameter (mm) D Damage (1) fd0 Driving resistance parameter (N) f Driving resistance parameter (N s m 1) d1 · · − f Driving resistance parameter (N s2 m 2) d2 · · − Fa Axial gear force (N) Fb Bearing force (N) FA(t) Axial force (N) FB(t) Failure probability of all bearings over time (1) Fdrag Driving resistance force (N) Fi Force i (N) Fj(t) Failure probability of a single bearing j over time (1) Fr Radial gear force (N) Ft Tangential gear force (N) FR(t) Radial force (N) i Index (1) iG Gear ratio (1) j Index (1) J Inertia (kg m2) · k Number of load classes (1) l Length of a shaft section (mm) 6 Li Bearable number of rotations of a load class i (10 ) m Vehicle mass (kg) M Moment i (N m) i · ni Number of bearing revolutions in a load class i (1) p Lifetime exponent (1) P(t) Dynamically equivalent bearing load (N) Pi Dynamically equivalent bearing load of a load class i (N) rtire Tire radius (m) t time (s) t0 Failure free time (h) tEM EM time constant (s) tICE ICE time constant (s) Tcycle Cycle duration (s) T Acting EM torque (N m) act,EM · T Acting ICE torque (N m) act,ICE · T Acting MB torque (N m) act,MB · T Demanded deceleration torque (N m) dem,brk · T Desired EM torque (N m) des,EM · T Desired ICE torque (N m) des, ICE · T Desired MB torque (N m) des,MB · T Equivalent driving resistance torque (N m) drag · T Gear torque (N m) G · Appl. Sci. 2020, 10, 7086 16 of 19

T Torque i (N m) i · T Sum of maximum EM generator and ICE drag torque (N m) max,EM,ICE · T Maximum ICE drag torque (N m) max, ICE · Tj Characteristic lifetime of a single bearing j (h) 1 vact Actual velocity (km h ) · − v Desired velocity (km h 1) des · − xbrk Brake pedal position (1) xthr Throttle pedal position (1) . x Vehicle velocity (km h 1) veh · − X dynamic radial load factor (1) Y dynamic axial load factor (1)

Appendix A

Table A1. Bearing parameters. Position according to Figure2. Dynamic load rating C and mean diameter dm.

Position Type C (N) dm (mm) 1, 2 tapered roller bearing 64,000 65 3 axial needle roller bearing 30,000 65 4 needle roller bearing 26,500 30 5 needle roller bearing 27,500 36 6, 15, 17 ball bearing 32,000 49.5 7 cylindrical roller bearing 33,500 53.3 8 needle roller bearing 33,500 45.5 9 needle roller bearing 33,500 45.5 10 needle roller bearing 33,500 45.5 11 needle roller bearing 33,500 45.5 12 needle roller bearing 33,500 45.5 13 needle roller bearing 33,500 45.5 14 needle roller bearing 33,500 37 16 cylindrical roller bearing 59,000 53.9 18 cylindrical roller bearing 33,500 56 19, 20 cylindrical roller bearing 28,500 29.5

Table A2. Total gear ratios iG of the transmission.

Gear Value 1 14.07 2 7.66 3 4.68 4 3.26 5 2.52 6 2.04

Table A3. Coefficients of driving resistance.

Parameter Value and Unit

fd0 160 N f 4 N s m 1 d1 · · − f 0.37 N s2 m 2 d2 · · − Appl. Sci. 2020, 10, 7086 17 of 19

Table A4. Parameters of the torsional vibration model.

Parameter Value and Unit

rtire 0.302 m J 72.96 kg m2 Veh · J 0.3 kg m2 ICE · J 0.3 kg m2 DMF · J 0.2 kg m2 EE · J 0.5 kg m2 wheel · c 580 N m rad 1 DMF · · − d 50 N m s rad 1 DMF · · · − c 1.1E4 N m rad 1 shaft,l · · − c 9.5E3 N m rad 1 shaft,r · · − d 6 N m s rad 1 shaft,l · · · − d 6 N m s rad 1 shaft,r · · · − c 2.3E4 N m rad 1 tire · · − d 1.15 N m s rad 1 tire · · · −

Table A5. Properties of the WLTC Class 3 cycle.

Parameter Value and Unit

Duration Tcycle 1800 s Distance 23,266 m Average velocity 46.5 km h 1 · − Maximum velocity 131 km h 1 · −

References

1. Liu, Z.; Ivanco, A.; Filipi, Z.S. Impacts of real-world driving and driver aggressiveness on fuel consumption of 48V mild hybrid vehicle. SAE Int. J. Altern. Power. 2016, 5, 249–258. [CrossRef] 2. Joud, L.; Da Silva, R.; Chrenko, D.; Kéromnès, A.; Le Moyne, L. Smart energy management for series hybrid electric vehicles based on driver habits recognition and prediction. Energies 2020, 13, 2954. [CrossRef] 3. Mock, P. European vehicle market statistics: Pocketbook 2018/2019. Available online: https://theicct.org/sites/ default/files/publications/ICCT_Pocketbook_2018_Final_20190408.pdf (accessed on 18 August 2020). 4. Ehsani, M.; Gao, Y.; Gay, S.E.; Emadi, A. Modern Electric, Hybrid Electric, and Fuel Cell Vehicles. Fundamentals, Theory, and Design; CRC Press: Boca Raton, FL, USA, 2005; ISBN 0-8493-3154-4. 5. Guercioni, G.R.; Vigliani, A. Gearshift control strategies for hybrid electric vehicles: A comparison of powertrains equipped with automated manual transmissions and dual-clutch transmissions. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 2019, 233, 2761–2779. [CrossRef] 6. Fischer, R.; Küçükay, F.; Jürgens, G.; Najork, R.; Pollak, B. The Automotive Transmission Book; Springer International Publishing: Cham, Germany, 2015; ISBN 978-3-319-05262-5. 7. Sieg, C.; Küçükay, F. Benchmarking of dedicated hybrid transmissions. Vehicles 2020, 2, 6. [CrossRef] 8. Foulard, S. Online and Real-Time Load Monitoring for Remaining Service Life Prediction of Automotive Transmissions: Damage Level Estimation of Transmission Components Based on a Torque Acquisition; Darmstadt University of Technology: Darmstadt, Germany; Shaker: Aachen, Germany, 2015; ISBN 9783844039504. 9. Winner, H. Challenges of automotive systems engineering for industry and academia. In Automotive Systems Engineering; Maurer, M., Winner, H., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; pp. 3–15. 10. Bertsche, B.; Göhner, P.; Jensen, U.; Schinköthe, W.; Wunderlich, H.-J. Zuverlässigkeit mechatronischer Systeme. Grundlagen und Bewertung in Frühen Entwicklungsphasen; Springer: Berlin/Heidelberg, Germany, 2009; ISBN 978-3-540-85089-2. 11. Leopold, T. Ganzheitliche Datenerfassung für verbesserte Zuverlässigkeitsanalysen. Ph.D. Thesis, University of Stuttgart, Stuttgart, Germany, 2012. 12. Xue, X.; Guo, R.; He Esq, J.; Hong, Z. A road load data processing method for transmission durability optimization development. In Proceedings of the WCX SAE World Congress Experience, Washington, DC, USA, 21–23 April 2020. [CrossRef] Appl. Sci. 2020, 10, 7086 18 of 19

13. Müller-Kose, J.-P. Repräsentative Lastkollektive für Fahrzeuggetriebe; Technical University of Braunschweig: Braunschweig, Germany; Shaker: Aachen, Germany, 2002; ISBN 3832210032. 14. Kücükay, F.; Kassel, T.; Eghtessad, M.; Kollmer, H. Requirement Engineering Using the 3D Method; SAE International: Warrendale, PA, USA, 2011. 15. Belingardi, G.; Cuffaro, V.; Curà, F. Dynamic additional loads influencing the fatigue life of gears in an electric vehicle transmission. Frat. Integrità Strutt. 2014, 8, 469–477. [CrossRef] 16. Kamper, T.; Hwang, D.H.; Juretzki, B.; Neumann, S.; Wöll, L. Comprehensive reliability model of a passenger car gearbox. In Proceedings of the Tagungsband Antriebstechnisches Kolloquium ATK 2017, Aachen, Germany, 7–8 March 2017; ISBN 978-3-7431-4897-0. 17. Foulard, S.; Rinderknecht, S.; Ichchou, M.; Perret-Liaudet, J. Automotive drivetrain model for transmission damage prediction. Mechatron 2015, 30, 27–54. [CrossRef] 18. Foulard, S.; Ichchou, M.; Rinderknecht, S.; Perret-Liaudet, J. Online and real-time monitoring system for remaining service life estimation of automotive transmissions—Application to a . Mechatronics 2015, 30, 140–157. [CrossRef] 19. Foulard, S.; Rinderknecht, S.; Fietzek, R. Lightweight design of automotive transmissions through online and real-time lifetime monitoring. ATZ Worldw. 2016, 118, 72–77. [CrossRef] 20. Rinderknecht, S.; Fietzek, R.; Foulard, S. Online and real-time condition prediction for transmissions based on CAN-signals. In Proceedings of the WCX™ 17: SAE World Congress Experience, Detroit, MI, USA, 4 April 2017. 21. Haq, S.; Joseph, B.; Lee, Y.-L.; Taylor, D.; Attibele, P. Vehicle powertrain loading simulation and variability. J. Mater. Manuf. 2004, 113, 751–756. [CrossRef] 22. Friedmann, M.; Kollmeier, H.-P.; Gindele, J.; Schmid, J.M. Synthetic driving cycles in the area of powertrain testing. ATZ Worldw. 2015, 117, 40–45. [CrossRef] 23. Fugel, M.; Scholz, N.; Kücükay, F. Anforderungen an die Getriebe in Hybridantrieben. In Proceedings of the Getriebe in Fahrzeugen 2006, Friedrichshafen, Germany, 27–28 June 2006; ISBN 3-18-091943-4. 24. Lavall, T. The “Hybrid Effect”: Influence of hybridisation on the durability of automatic transmissions. In Proceedings of the Getriebe in Fahrzeugen 2009, Friedrichshafen, Germany, 30–31 July 2009; pp. 661–672, ISBN 9783180920719. 25. Kurtzke, A.; Hierlwimmer, P. CAE-basierte abstimmung bezüglich des fahrzeug-leistungsverhaltens und der getriebelebensdauer. In Proceedings of the 7. Fachtagung Dynamisches Gesamtsystemverhalten von Fahrzeugantrieben, Munich, Germany, 10–11 March 2009; ISBN 9783816928447. 26. Naunheimer, H.; Bertsche, B.; Ryborz, J.; Novak, W. Automotive Transmissions. Fundamentals, Selection, Design and Application; Springer: Berlin/Heidelberg, Germany, 2011; ISBN 978-3-642-16213-8. 27. Commission of the European Communities. Regulation (EEC) No 4064/89 Merger Procedure. 1999. Available online: https://ec.europa.eu/competition/mergers/cases/decisions/m1406_en.pdf (accessed on 18 August 2020). 28. Wöll, L.; Feldermann, A.; Jacobs, G. Sensitivity analysis on the reliability of an offshore winch regarding selected gearbox parameters. MIC 2017, 38, 51–58. [CrossRef] 29. Neumann, S.; Wöll, L.; Feldermann, A.; Strassburger, F.; Jacobs, G. Modular system modeling for quantitative reliability evaluation of technical systems. MIC 2016, 37, 19–29. [CrossRef] 30. ISO International Organization for Standardization. Rolling Bearings—Dynamic Load Ratings and Rating Life (ISO 281); ISO International Organization for Standardization: Zurich, Switzerland, 2007. 31. Köhler, M.; Jenne, S.; Pötter, K.; Zenner, H. Load Assumption for Fatigue Design of Structures and Components. Counting Methods, Safety Aspects, Practical Application; Springer: Berlin/Heidelberg, Germany, 2017; ISBN 978-3-642-55248-9. 32. Bertsche, B.; Lechner, G. Reliability in Automotive and Mechanical Engineering; Springer: Berlin/Heidelberg, Germany, 2008; ISBN 978-3-540-33969-4. 33. Keller, M.; Schmitt, L.; Abel, D. Nonlinear hierarchical model predictive control for the energy management of a hybrid electric vehicle. In Proceedings of the 2019 27th Mediterranean Conference on Control and Automation (MED), Akko, Israel, 1–4 July 2019; pp. 451–456. [CrossRef] 34. Habermehl, C.; Kramer, A.; Jacobs, G. Interconnected drivetrain development in a physical and virtual environment. ATZ Worldw. 2019, 121, 78–83. [CrossRef] 35. Habermehl, C.; Jacobs, G.; Neumann, S. A modeling method for gear transmission efficiency in transient operating conditions. Mech. Mach. Theory 2020, 153, 103996. [CrossRef] Appl. Sci. 2020, 10, 7086 19 of 19

36. United Nations Economic Commission for Europe. Global Technical Regulation No. 15. Worldwide harmonized Light vehicles Test Procedure. ECE/TRANS/180, 2014. Available online: https://www.unece.org/ trans/main/wp29/wp29wgs/wp29gen/wp29glob_registry.html (accessed on 18 August 2020). 37. Naunheimer, H.; Bertsche, B.; Ryborz, J.; Novak, W.; Fietkau, P. Fahrzeuggetriebe; Springer: Berlin/Heidelberg, Germany, 2019; ISBN 978-3-662-58882-6. 38. Malik, R.; Masur, E.; Schick, A. Bearings and bearing design for transmissions. Encyclopedia Automot. Eng. 2014, 1, 1–15. [CrossRef] 39. Chen, Y.; Li, K.; Zang, L.; Zheng, Y.; Jia, S.; Zhou, H.; Yu, M.; Xue, B. Analysis on contact strength of needle roller bearing of transmission and effect of surface modification. In Proceedings of China SAE Congress 2018: Selected Papers; Springer: Singapore, 2020; pp. 879–891; ISBN 978-981-13-9717-2. 40. Kubaisi, R.; Herold, K.; Gauterin, F.; Giessler, M. Regenerative braking systems for electric driven vehicles: Potential analysis and concept of an adaptive system. In Proceedings of the SAE 2013 Brake Colloquium & Exhibition—31st Annual; SAE International: Warrendale, PA, USA, 2013. 41. Solberg, G. The Magic of Tesla Roadster Regenerative Braking. Available online: https://www.tesla.com/ blog/magic-tesla-roadster-regenerative-braking (accessed on 18 August 2020).

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