FinalProjectReport

Winter2017

MECH393–MachineElementDesign

April11th,2017

GroupNumber:2

GroupMembers:GeorgesMatta 260608769RiadHaissamElCharif 260631084

StanislavNemirovsky 260660024

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TableofContentsExecutiveSummary................................................................................................................4

IndividualContribution.........................................................................................................4

IntroductiontotheDesignProblem.................................................................................5

DetailedDesignSolution.......................................................................................................8I. GearDevelopment...................................................................................................................9i. GearboxLayout.....................................................................................................................................................9ii. DeterminingtheGearRatio.............................................................................................................................9iii. PowerandTorqueRequirements..............................................................................................................11iv. Gear1SampleCalculationsforSafetyFactors.....................................................................................11v. CalculationsforGears2,3,4,5,6...............................................................................................................18

II. ShaftDevelopment...............................................................................................................19i. FreeBodyDiagramsoftheShaftswithGearsandBearings:.................................21ii. Shaft1-InputShaft...............................................................................................................24iii. Shafts2-4-ReductionShafts............................................................................................27iv. Shaft3-OutputShafts.........................................................................................................28

Bearings...................................................................................................................................30

TheCparameterisobtainedfromthemanufacturerspecificationforeachbearing,andsotoallowforsuccessfuliteration,alargeamountbearingtableswereinputintoourexcelsheetstoallowforeasyselection.Thetableswould“iterate”theseformulasformanydifferenttypesofbearings,untilasuitablelifelimitwasfound.Weshouldnotethatwasnotcompareddirectlywithour,butratheritwasmultipliedbyareliabilityfactorofKr=0.33(forareliabilityof98%)formostofourbearingchoices.Wenowusedthisnewtocomparewithourideallife.31

Afteranumberofbearingsofacceptablelifewerefound,theparticularbearingwasselectedbasedoffgeometricalconstraintsofourshaft.Sincewehadaverylowstressstate,wecouldaffordtoaddalargeshoulderonthebearingportionofourshaft,andsodoutwasnotafactor.However,sinceweusedahollowshaft,dinwasthelimitingparameter.Theselectedbearingsarepresentedinlaterinthisreport.................................31

Note:Duetoourverylowloading,thelifeexpectanciesofthebearingsonshaft1aresignificantlyhigherthannecessary,howeverthesebearingswerefoundtobesuitableforourgeometry,andsatisfiedourneeds,and

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thustheywerechosen.ThesameappliesforShaft2,4,howeverthelifeexpectanciesarenotmuchhigherthantheidealrequirement..................31

ThefinalsphericalthrustbearingwaschosenfromSKF,anditwaschosenforasuitableitssuitablegeometryandlifeexpectancies.ThebearingisaSphericalRollerBearing...........................................................................................31

DesignResults.......................................................................................................................32

ModifiedGoodmanDiagrams...........................................................................................36

Conclusions.............................................................................................................................39

AppendixA–AllCalculations...........................................................................................40

AppendixB–FiguresandTables....................................................................................44

AppendixC–MechanicalDrawingsofProposedDesign........................................48

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ExecutiveSummary

NikolaDrive Team The design report presented here was commissioned by the Solar Impulse initiative for the design of a gearbox for the titular aircraft. NikolaDrive is a collective of highly motivated innovative aeronautical engineers, who form a vital subdivision within the Solar Impulse family. Headed by our chief engineerMarkDriscoll,theteamembarkedontheproposeddesignforadoublebranchdoublereductiongearbox,intended for use on the final aircraft. The team had 3 main design goals: Minimize weight, maximize efficiency and endure the aircraft’s lifetime. Integrated design principles were used for the design of Gears, Shafts and Bearings in our system. A targeted safety factor of 1.5 was chosen for our general design allowing for static, dynamic, and fatigue failure analysis to be performed on each component. The designs were iterated until satisfactory results were obtained. All components fall within a safety factor of1.5.Thewhole systemoperatesatpower lossesbelow5%asdesired.Theweightofoursystemis14.4kg,whichdidnotmeetthetargetcriterionof5.5kg.Furtherrefinementinfuturedesignscouldbeconsidered. The NikolaDrive Team is proud to present its first and most innovative design: Solar Impulse Double Branch Double Reduction Gearbox.

IndividualContributionThe NikolaDrive team consists of three engineers, Georges Matta, Stanislav Nemirovsky and Riad Haissam El Charif, working under the supervision of Mr. Driscoll. With such a tightly knit and well-functioning unit, all members had significant contributions on each aspect of design. However, a rough division of individual contribution can nonetheless be made. Georges Matta, a U3 Mechanical Engineering Student at McGill University, oversaw spreadsheet production, gear design, as well as material and bearing selection. Riad Haissam El Charif, also a U3 Mechanical Engineering Student at McGill University worked on shaft and bearing design, layout, and optimization. Finally, Stanislav Nemirovsky, U3 Mechanical Engineering, optimized and realized the design for the bearings, shafts, as well as the gear parameters. The apparent division of labor provided a rough structure of the team’s organization; however, it should be reemphasized that NikolaDrive operates on a highly collaborative structure, in which team members share a significant amount of responsibilities

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IntroductiontotheDesignProblemSolar Impulse’s design objective is simple, yet awe inspiring: Fly around the globe with no onboard fuel. To achieve this unique challenge, every engineering sub team involved in the aircraft design highly optimized parts for maximum efficiency and minimum weight, while maintaining reasonable reliability constraints. Conversely, the uniqueness of this project has given rise to some unusual liberties. First, as an experimental aircraft, no specific certifications are mandated upon our designs, which gave us incredible freedom to pursue our design goals. Second, as a well-funded experimental aircraft team, Solar Impulse has managed to secure enough funds that cost is never taken as a constraint. Such freedom allowed for more open-ended innovation. The gearbox to be designed is of the Double Branched Double Reduction Gearbox type, a sketch of which is shown in Figure 1.

Figure 1- Double Branched Design

The design allows for the reduction on speed, while increasing torque along two stages.

The double branch design allows for the distribution of loads from the input shaft, to reduce stresses on the reduction shaft. The input shaft is directly connected to a 5000-rpm brushless motor, which gets stepdown to a maximum of 525 rpm imparted to the propeller. The propeller shaft, being the heart of our propulsion system, withstands a 1500lb axial load produced by the propeller rotation.

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The design constraints are presented in table 1. The table presents the specifications given to us by the Solar Impulse team, as well as the values we calculated and chose for our final gearbox in bold red.

Powertrain Specification Gearbox Specifications

Motor Brushless + Sensor less Gear Ratio 7.84

Maximum [rpm] 5000 Total Weight [kg] 14.4

Fuel Consumed [L] 0 L of Fossil Fuel Endurance Life-Gears and Output Shaft [hrs] 2000

MaximumMotor Temperature [°C] 135 Temperature Range [°C] -40 to

+40

Propeller Twin Blade Composite Size limitations [cm] 30 X45

X 45

Propeller Thrust [lbf] 1500 Safety Factor 1.5

Propeller Weight [kg] 160

Propeller-Max RPM [rpm] 525

Table 1- Design Requirments

It should alsobenoted that theGears considered in thisdesign areAGMAspur gearsofcoursediametralpitch,duetotheirsimplicityandversatility.Thegearsaretobelubricatedwith SAE 30W. The team’s target was a fictional loss of less than 5%, and this was successfully achieved. Finally, the operating conditions considered in our design are shown below.

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Operating Conditions

Operation Electric power to Motor Driver

[hp]

Electric motor Shaft Torque

[Nm] RPM

Take- Off 40 70.2 4000

Slow Climb 7 16.4 3000

Steep Climb 13.4 29.6 3180

Descent Glide 0.7 2.2 2225

Horizontal

Flight 4.7 14.8 2225

Table2OperatingConditions

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DetailedDesignSolution

The generalmethodology followed in our design process was “assume and iterate”.Westarted process followed for our given points, began with an assumption and tried outmultipledifferentvaluesuntilweconvergedtoourdesiredsafetyfactorsandgeometry.Onefundamentaldesignissuewewantedtoavoidwasthe“DesignParadox”;aswelearnedmore about our design and progressed through it, we became more capable ofunderstandingourdesign, and its specific requirements.However, aswe learnmore,webecameincreasinglyincapableofeditinganyofourvaluesduetoadeepdesigninvestment.To keep a versatile and dynamic system and avoid this crutch,we parametrized all ourvaluesthroughamasterExcelWorkbook.Thisworkbookcontainsallourdesign,diagramsandresultsandallowedustomodifyourdesigndrasticallymultipletimeswithease.ThisworkbookwasprovidedalongwiththeProjectReportandwillbesubsequentlyreferencedmultipletimesinthesectionsbelow.Anotherimportantdesignconsiderationforanyshaftbasedsystemistheshaftdeflection.This variable typically pushes designers to minimize shaft length ‘L’.However, as ourdesignrequirementsdonotincludeashaftdeflectionanalysis,wedecidedtouseupthefullhorizontallengthof30cmforourlayoutasaninitialguess.Thislayoutprovideduswithacceptableresults,andsoitwaschosenforourdesignwithlittlemodification.OurgearboxlayoutcanbeseeninFigure2,aswellasinourdesigndrawings:

Figure2

Thedesignprocessthenprogressedwithalogicalorder,wedesignedthegearsbasedoffthegiveninputsandoutputs,usedthegeardata-FaceWidthandDiameter-toproduceapreliminary design layout for our whole gearbox. The layout was then used along with

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torquedatafromtheGearstoproduceTorqueandMomentdiagrams.Thesewereusedtosizepreliminarydiametersfortheshaft,whichwasthencompletedwiththeselectionofanappropriateBall/RollerBearingwhichfitsourloadandlifetimerequirements.Finally,thedesignwasoptimizedforweightconsiderationsandsuitableSafetyFactors.

I. GearDevelopment

i. GearboxLayout

As shown in Figure 2, a rough sketch of the gearboxwas developed in order tohave a general idea about the dimensions of all the components. In particular,estimationsweremadefortheoveralllengthofthegearbox,aswellasthedistancebetween intermediate gears, and mounting requirements for the shafts andbearings.Allfourshaftswererepresentedintheabovesketch,withanindicationofhowthesixgearsandeightbearingswouldbemountedinthisgearbox.

ii. DeterminingtheGearRatio

As a first step in the design process, an ideal gear ratiomust be determined inordertosatisfy theconditionsandconstraintsprovided.Maximuminputangularvelocities were provided as well as a maximum output RPM. As a result, aminimum gear ratio will be considered a reference for evaluating the severaliterationsstudied.

Now, an accepted combination of gear and pinion teethmust result in a gear ratio that is at least equal to thisvalue.Anotheralternativetothisselectioncriterion is tocompare the output RPM generated by the gear ratioselected; this value cannot exceed the given maximumvalue of 525 rpm. In addition, the fact that a pressureangleof20°isgivenmeansthatthereisalsoaminimumnumber of pinion teeth that could be used in order toavoid interference, the value of which is 18 teeth, asshowninTable12-4.Oneofthefinalconstraintswhenitcomestoselectingthenumberofpinionandgearteethischeckingthatthegeometrysatisfiestherequirementsgiven,asthesumofdiametersoftwoconsecutivegearsinthefirststagemustbeequaltothatinthesecondstage.Andsincethemodulusistakenasconstantforallgears,thismeansthatthesumofteethbetweenadjacentpinionsandgearsmustbeequal

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betweenthetwostages.ThisisexpressedbytheequationsbelowandcanbeseeninmoredetailintheExcelsheetprovided.

Now,gearratioscanbecalculatedforseveralvaluesofN1throughN6,basedontheequationforcompoundgears:

whereN2=N6andN3=N5

The table below includes several iterationsperformedbeforesettlingonthevaluesshownonthe right, which satisfy all the constraintsprovided.

N1 N2 N3 N4 N5 N6ActualGearRatio

Satisfies

Geometry?

OutputRPM

18 55 18 45 18 557.638888889

523.6363636

36 110 36 90 36 1107.638888889

523.6363636

18 40 18 60 18 407.407407407 540

20 50 21 64 21 507.619047619

525.00000

18 56 18 44 18 567.604938272

525.974026

18 54 18 46 18 547.666666667

521.7391304

18 60 18 41 18 607.5925925

93 526.8292683

35 105 36 91 36 1057.5833333

33 527.4725275

20 56 20 56 20 56 7.84 510.2040816

GEARTEETHN1 20N2 56N3 20N4 56N5 20N6 56

RATIO 7.84OutputRPM

510.2040816

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iii. PowerandTorqueRequirementsByconsideringoursysteminanidealscenario,thepowergeneratedbythemotorwill equal the power provided to the propeller. But because our gearbox is adouble-branchdouble-reductiontype,wecansuggestthatthepowerprovidedbythemotor,whichpassesthroughGear1,getsdividedequallybetweenGears2and6,remainsconstantalongGears3and5(sameshaftas2and6respectively),onlyto return to approximately the same initial value through Gear 4, powering thepropeller.Inrealityhowever, lossesinpowerexistduetothepresenceoffactorssuch as friction; the fact that ball bearings were selected played a role in theassumption that such losses are considered negligible, as will be explained incomingsections.Now,afterfindingasuitablegearratio,wecandeterminetheangularvelocitiesandtorquesateachofthegears:

iv. Gear1SampleCalculationsforSafetyFactors

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Themodulusm for all the gearswas estimated to be 3mm as it was themostreasonable value that provides relatively good safety factors in bending andcontact,aswillbeshown.Thus, thepitchdiameterd, thepitchradiusr, aswellas theaddendumaanddedendumbforGear1canallbefoundfromthevalueofm:

ThepitchlinevelocityVTandthetangentialcomponentofloadWTcanalsobefound:

Inaddition, the lifeforinputGear1 (aswellasoutputGear4) isdoublethatoftheothergearsbecauseitisasingledrivingpiniondrivingtwoindependentgearscausingtwofatiguecyclesperrevolution:

Now, theAGMAapproach(SIform) forbothbendingandcontactstresswillbe

appliedtodeterminesuitablegearparametersandsafetyfactors.

TheAGMAbendingstressequationisgivenby:

Thevaluesoftheseconstantsandunknownswillnowbecalculated.

TheApplicationFactorKAischosenas1.25sincethetransmittedloadcannotbeconsidereduniformas it fluctuateswith time, at least for thedrivenmachine asopposedtothedrivingmachine(electricmotor/turbine).Moderateshockisthuschosen.(RefertoTable12-17)

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TheRimThicknessFactorKBis1sinceweareanalyzingasolid-diskpinion.ThisisnotthecaseforGears2,4,and6sincerimswithspokesareused.TheIdlerFactorKIis1sinceanon-idlergearisbeinganalyzed.TheSizeFactorKsis1sincenodrasticchangesinsizearepresent.

TheDynamic factor KV attempts to account for internally generated vibrationloads from tooth-tooth impacts induced by non-conjugate meshing of the gearteeth.Thesevibrationloadsarecalledtransmissionerrorandwillbeworsewithlow-accuracygears.Thisfactorisgivenby:

where:

ThesevaluesofAandBweredeterminedforaqualityindexQV=10.Thus,

The AGMA Bending Geometry Factor J is determined from Table 12-9 for apinionbyrelatingthenumberofpinionteeth(20)tothenumberofgearteeth(56)

ThisvalueisdifferentwhenanalyzingGears2,4,and6

TheFaceWidthFischosenastheminimuminthesuitablerangeinordertogetthebestvaluesforsafetyfactorsforthisgear.

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SincethisvalueofFisbelow100mm,theLoadDistributionFactorKmistakenfromTable12-16:

Thus,thevalueoftheAGMABendingStressis:

TheAGMAContactStressequationisgivenby:

ThefactorsCa,Cm,Cv,andCsareequal,respectively,toKa,Km,Kv,andKsasdefinedforthebendingstressequation

TheSurfaceGeometryFactorIisgivenby:

TheSurfaceFinishFactorCFisusedtoaccountforunusuallyroughsurfacefinishesongears,soitcanbesetas1forgearsmadebyconventionalmethods.

TheElasticCoefficientCPisgivenby:

whereEp,Eg,vp,andvgaregiven:

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Thus,

Now,thevalueoftheAGMAContactStressis:

TheAGMABendingFatigueStrengthisgivenby:

where the uncorrected bending strength was chosen for Steel AISI A1-A5ThroughHardened330HB(Figure12-25)

ReferringtoFigure12-24,TheLifeFactorKLisgivenby:

TheTemperatureFactorKTis1forsteelmaterialinoiltemperaturesupto250°F

TheReliabilityFactorKRistakenfromTable12-19forareliabilityof99%

Thus,thevalueoftheAGMABendingFatigueStrengthis:

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TheAGMAContactFatigueStrengthisgivenby:

where the uncorrected contact strength was chosen for Steel AISI A1-A5ThroughHardened330HB(Figure12-27)

ReferringtoFigure12-26,TheSurface-LifeFactorCLisgivenby:

TheHardnessRatioFactorCFis1forthepinion(gear1)

TheTemperatureFactorCTisidenticaltoKT

TheReliabilityFactorCRisisidenticaltoKR

Thus,thevalueoftheAGMAContactFatigueStrengthis:

Now,wecancalculateoursafetyfactors:

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v. CalculationsforGears2,3,4,5,6

Gears2and6areidentical,andsoareGears3and5WhencomparedtoGear1,theonlyfactorsthatchangewhencalculatingthesafetyfactorsfortherestofthegearsare:

• ThelifeforGears2,3,5,6ishalfthatofGears1and4(480000000cyclesinsteadof960000000cycles)

• The angular velocities, torques, and power differ as shown previously inSection(iii)

• Thefacewidthchangedfromstage1tostage2(from24mmto45mm)• Thegeometry factor J becomes0.4 insteadof0.34whenanalyzinga gear

insteadofapinion.

• ArimthicknessfactorKbexistsforgears2,4,and6duetothepresenceofspokes. Its value is determined for a certain assumed rim thickness asfollows:

• TheHardnessRatioFactorCHisnot1whenanalyzinggearsbecauseofachangeinmaterial.BetweenGear2andGear1,itiscalculatedasfollows:

AthoroughcalculationofsafetyfactorsforGear2(andGear6byassociation)canbefoundintheAppendix,andvaluesfortheothergearscanbefoundintheresultssectionofthisreportaswellasintheExcelsheetaccompanyingit.

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II. ShaftDevelopmentThe4shaftsweredesignedbasedoffthedataproducedbythegeardesignprocess.Sincethegearsdesignshave littledependenceontheir innerdiameters, theshaftsdiametershadmuchsizingfreedom.Itshouldbenotedthattheinputshaft,Shaft1,wasthesimplesttodesignduetoits lowloadingstate,andhadasimilarlayouttoour output shaft, Shaft 3. The reduction shafts, Shaft 2 and 4,were completelyidentical(albeitoperatingatadifferentdirection),andassuchweredesignedonlyonce.Webeganwith the input shaft.Asdescribedpreviously, the shaft layout thefirststepinshaftdesign.As for the loading conditionson the shafts, all shafts experienceda fully reversedalternating bendingmoment, caused by either radial and tangential forces or theweightof the shaft.The final shaft alsoexperienceda significantaxial load,whichwasnottransferredovertotheothershafts.Finally,thetorsionalloading,duetothetorque,wasconsideredtobeaconstantmeantorsionatamagnitudeequal to theTake-Off conditions of 40 HP and 4000 rpm. The reasoning for this designconsiderationwasthatthissteadytorquewasthelargesttorquevaluesthatwouldbe applied onto the aircraft under regular flight conditions, and so designing forthoseoperatingconditionsprovidesuswithamoreconservativeandencompassingfailurecriterion.OurteamcouldhavealternativelychosentheTake-OffconditionsandDescentGlideconditionsasaMaxandMinforanalternatingtorsionformuchmore conservative results, but we felt that this would be an inappropriateassumptionforouraircraft.Ouraircraftistobeflownunderhighlycontrolledflightpaths, with a known and limited number of landings and take-offs, whichmakesassumingacompletelycyclicalternatingtorsionanunnecessaryoverlyconservativeestimate.Duetothenatureofthesymmetricdoublebranchactingupontheinnergears(Gear1andGear4), radial and tangential forces cancel out and no moments diagramsneedbeproducedinthoseplanes,onthoseshafts.

This is shown in the free body diagram below:

Figure3

WrWt

WrWt

Tin Tin=2(WT)(r)

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Duetothefactthatnootherradialortransverseloadingsexist,weusedthegearweightasaforceintheanalysisofShaft1.However,duetothesmallmagnitude,itseffectwasnegligiblerelativetothemuchlargertorque.ThisisveryevidentinthenumberspresentedinourEXCELcalculations.Weightwassubsequentlyignoredinallothershafts. Thefinaldesignconsiderationthatweappliedtoalltheshaftswastheuseofahollowshaft.Thiswasdoneprimarilytoconserveonweight,butalsogivesusabetterstiffness/massratio,whichimprovesourdesign’sdeflectionresistance(asdeflectionwasnotaprimaryconstraint,weightwastheprimarypurpose).Thehollowshaftchoicemeantallourequationshadanouterdiametersubtractedbyaninnerdiameter.

𝑑N = 𝑑OPQN − 𝑑SNN , 𝑛 = 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑛𝑎𝑛𝑦𝑔𝑖𝑣𝑒𝑛𝑒𝑢𝑞𝑎𝑡𝑖𝑜𝑛Theinnerdiameterofeveryshaftwaschosenbasedonaniterativeprocessontheanalysispreformedatthesmallestdiameterofanygivenshaft.Thisinsurednoneofourshaftshoulderswouldclashwithourboredinnershaftdiameter.Furthermore,the𝑑SNwaschosentobeafullheightHofthekeychosenforourshaft.Thisinsuredthatnosignificantstressconcentrationwouldarisebetweentheendofthekeyway𝑑SN.TheseheightswerechosenbasedoffthestandardrecommendationsprovidedinNorton-MachineDesign-Table10-2.

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i. FreeBodyDiagramsoftheShaftswithGearsandBearings:

Figure4–Shaft1

22

Figure5–Shaft2

23

Figure6–Shaft3

Figure7–Shaft4

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Thenextsectioncontainsananalysisforeachshaft.However,asthereissignificantoverlapbetweentheanalysismethods,Shaft1willcontainmostofthesamplecalculationsandequations.FurthercalculatedreferencesforeachshaftcanbefoundincompleteexpansivedetailinourappendixandExcelfile.

ii. Shaft1-InputShaft Shaft 1 was designed to contain gear 1 as well as bearing 1 and 2. Bearing 2 is to be press fit onto the end, while Bearing 1 is locked onto the shaft using a clamp and spacer mechanism (the spacer extending to the motor). The shaft diameter in the bearings is identical, but thinner than that of the rest of the shaft. 1 mm clearance was added between the shoulder and the bearings to allow for thermal expansion during regular operation. The gear is supported axially by a shoulder, and fixed to the shaft using a key. All this is evident in the shaft layout presented in Figure 4, as well as in our machine drawings in the appendix Our moment diagrams, considering weight and torque are presented:

A more detailed calculation rundown is provided in our excel file. We can immediately note that the point of highest Bending Moment and Torque is the same, which is at the center of our gear. Given the fact that this point will contain our keyway, it is safe to assume our highest stress concentration will be occurring there, which we named Point B. The Gear shoulder will have a larger d our gear, and so no further stress calculations are needed there. We took the shoulder to be 3mmlarger than our dB . Finally, our shaft was fit into our bearings with a d < dB however, due to the low moment and lack of a significant stress concentration, we do not expect a failure point to exist there. Furthermore, we designed the dB with a safety factor range of 1.3-1.5, insuring that we are comfortable away from any potential failure points at the bearings. The d at bearings, henceforth denoted as dbr was sized based off an appropriate design choice of roller bearing. Point B: Critical Stress location. For this shaft, since we have no axial forces and constant Torsion was assumed, the formula dB was:

Figure8

25

𝑑bO =cd∗fgh

i∗

jg∗klmno.pq∗ jgr∗sl

m

tu+

jgw∗kwmno.pq∗ jgrw∗sw

m

txy− 𝑑bS

c

z{

For our loading, Ta and Mm are zero. This formula was then solved for Nab, our safety fatigue safety factor for point B.

𝑁}~ =i∗(�h�{��h�

{ )cd

∗jg∗kl

m

tu+

o.pq∗ jgrw∗swm

txy

��

Note, both these equations consider din . The first formula provides us with an estimate for dB , based off NfBDESIRED , which is an initial target safety factor used to provide us with our first guess. This was set to a conservative 2.25 initially. Next, this value was chosen, rounded to allow for ease of manufacturability, and then corresponding din was chosen based off the process outlined in the first section above. These choices caused equation 2 to produce our new NfB . At this point we would repeat the process again iteratively until the results were satisfying, and we have a NfB = [1.3,1.5]. Excel was an excellent tool for this iteration. A sample calculation is presented here detailing all the parameters and factors chosen for design. These results were the ones chosen for our Shaft 1. For further reference, please see the appendix and the excel file. Plug in with Desired Safety Factor of 2.25

𝑑bO =cd∗(d.dq)

i∗ �.�p∗cc.p[f��] m

�q�[k��]+

o.pq∗ d.d�∗podcq.� [f��] m

���[k��]− 10[𝑚𝑚]c

z{

𝑑bO = 17.8325[𝑚𝑚]

Choose a 𝑑bO that is appropriate for machining: 𝑑bO = 17 𝑚𝑚

Plug into Safety Factor equation:

𝑁}~ =𝜋 ∗ (17c − 10c)

32 ∗1.67 ∗ 33.7[𝑁𝑚𝑚] d

154[𝑀𝑃𝑎] +0.75 ∗ 2.24 ∗ 70235.1 [𝑁𝑚𝑚] d

469[𝑀𝑃𝑎]

��

𝑁}~ = 1.32

which is acceptable as a target safety factor. Note: The development shown above contained many iterations, but brevity only the final stage is presented.

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Material:Forourshaft,wearrivedatourchosen𝑁}~byselectingSAE1020ColdRolledSteel(TableA-9,Norton).Subsequently,duetoourLifetimeof2000hour,ournumberofcyclesforshaft1isintherangeof𝑁 = 10�𝐶𝑦𝑐𝑙𝑒𝑠.Basedonthis,weshalldesignforanendurancelimitof𝑆�using:

𝑆� = 𝐶�S��𝐶�O��𝐶�P�}𝐶Q���𝐶���S�~𝑆��where

𝑆�� ≅ 0.5𝑆PQforsteels.• 𝐶�S�� wascalculatedusing𝐶�S�� = 1.189𝑑�o.o�p,andremainedconsistentwithexcel

parametrization.• 𝐶�O��=1forloadinginbendingloads,asthisshaftasnoaxialloads.Shaft3isthe

onlyshaftinwhich𝐶�O�� = 0.70.

• 𝐶�P�}wasobtainedusingtherelationships𝐶�P�} ≅ 𝐴 𝑆PQ ~,withourparametersAandbobtainedfromNortonTable6-3,forMachinedsteel.𝐶�P�} ≅ 0.798.

• 𝐶Q���=1,asouroperatingconditionsforourgearboxistobe−40℃ ≤ 𝑇 ≤ 40℃.

• 𝐶���S�~=0.814forareliabilityof99%whichseemedlikeanacceptablereliability

rangeforthecriticalapplicationneededforSolarImpulse.Itshouldbenotedthatallpartsoftheaircraftundergoextensivequalitytesting,andsoahigherreliabilityisunnecessary.

Finally,our𝑘}and𝑘}��valuesforouranalysisatpointB(keyway)wereobtainedfromthisdevelopment:Obtain𝐾Qand𝐾Q�fromNortonFigure10-16,anestimateof2.2wastakenforanr/dratioof

0.021forthefirstiteration.Thiswaslatercorrectedasdwasobtained.𝐾Q = 2.2𝐾Q� = 3.0

ThesewasconvertedafatiguesafetyfactorusingtheNeuberequation:𝐾} = 1 + 𝑞 𝐾Q − 1

𝑞 =1

1 + 𝑎𝑟

where 𝑎isobtainedfromTable6-6.Finally,Test𝐾} 𝜎¦§¨NO� < 𝑆ª

iftruethen𝐾}� = 𝐾}and𝐾}�� = 𝐾}�.Thiswasthecaseforalltheshafts.

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iii. Shafts2-4-ReductionShafts

Shaft 2 and 4 are the reduction shafts, and they both have identical designs while rotating in opposite directions. Unlike Shaft 1, these two shafts have significant bending loads due to the tangential and radial forces applied at the gears. Bearing 3,4,7 and 8 are to be press fit onto the ends of both shafts. The shaft diameter in the bearings are identical, but thinner than the rest of the shaft. Just as Shaft 1, a 1 mm clearance was added between the shoulder and the bearings to allow for thermal expansion during regular operation. The gears are supported axially by a shoulder, and both are fixed to the shaft using keys. All this is evident in the shaft layout presented in Figure 5, as well as in our machine drawings in the appendix. Due to the 3-dimensional nature of the loading, we produced moment diagrams in two planes, then composed them into a total Moment diagram.

Ourmomentdiagrams,consideringweightandtorque:

Amoredetailedcalculationrundownisprovidedinourexcelfile.WecanimmediatelynotethatthepointofhighestBendingMomentandTorqueisthesame,whichisatthecenterofourGear3.Giventhefactthatthispointwillcontainourkeyway,itissafetoassumeourhigheststressconcentrationwillbeoccurringthere,whichwenamedPointB.SimilarreasoningforwhythispointwillhavethehigheststressconcentrationisfollowedasShaft1.Sincethisdiameterwillbedesignedforfailure,weshalluseanidenticaldiameterforthelocationatGear2.WefollowedanidenticalanalysisprocedureforfatiguefailureasShaft1.Thediametricresultsareshownintheresultssection,aswellasourExcelfile.Note,tosizeourbearings,wehadtochoosefromalistofstandardrollerbearings,whichforcedustotailorourbearingdiametertothisshaft.Tomakesurethesediametersdonotcausefailure,asafetyfactoranalysiswasdoneusingshearstressfromshearforceastheonlyloading(asnootherloadingexistsonthosepoints).Theywereallcomfortablywithinacceptablerange.

Figure9

28

Material:Forourshaft,wearrivedatourchosen𝑁}~byselectingSAE1050ColdRolledSteel(TableA-9,Norton).Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range of 𝑁 = 10�𝐶𝑦𝑐𝑙𝑒𝑠. Based on this, we shall design for an endurance limit of 𝑆�using:

𝑆� = 𝐶�S��𝐶�O��𝐶�P�}𝐶Q���𝐶���S�~𝑆�� All developments beyond this point are identical to Shaft 1.

iv. Shaft3-OutputShafts Shaft 3 has slightly different considerations than the last two shafts, since it contains a cantilevered propeller weight, as well as a significant thrust load. The design methodology for the shaft itself was identical; iterate different values of din and dout until an acceptable safety factor is achieved. Bearing 5 and 6 are carefully selected to account for the thrust load. This is further expanded upon in the bearing section. Bearing 5 is to be press fit, while bearing 6 is attached using a clamp and nut arrangement. No clearance was given between the shaft shoulder and bearing 6, due to the axial thrust consideration. The shaft diameter in bearing 5 is thinner than that of the rest of the shaft, while bearing 6’s diameter will be our design diameter. A 1 mm clearance was added between the shoulder and bearing 5 to allow for thermal expansion during regular operation. The gear is supported axially by a shoulder, and fixed to the shaft using a key. Keys are typically non-ideal for attachments were axial loading is present, but our bearing 6 was designed to absorb all axial loads from the propeller. All this is evident in the shaft layout presented in Figure 6, as well as in our machine drawings in the appendix Our moment diagrams, considering weight and torque are presented:

Unlike previous shafts, the point of highest stress is not so immediately clear. Two points are likely contenders, the keyway at the gear attachment, due to a high torque and moment, and the second point is at Bearing 6 which has a high torque and maximum moment. Due to this, the design analysis was repeated on those two points. It is safe to assume that the rest of the shaft

Figure10

29

will have lower stresses than those two points, just as we did in previous shafts. We started with point B at our bearing. This formula :

𝑑bO =cd∗fgh

i∗

jg∗klmno.pq∗ jgr∗sl

m

tu+

jgw∗kwmno.pq∗ jgrw∗sw

m

txy− 𝑑bS

c

z{

Is not applicable for this shaft, as it was derived from the Case 3 loading of the modified Goodman diagram, with the assumption of no axial load. To remedy this, the original equations were used for the calculations instead. Namely:

1𝑁𝑓 =

𝜎�𝑆�+𝜎��

𝑆PQ

where

𝜎� = 𝑘}32𝑀�

𝜋(𝑑bOc − 𝑑bSc )

𝜎�� = 𝜎��¬S��d + 3𝜏�d o.q

𝜏� = 𝑘}��16𝑇�

𝜋(𝑑bOc − 𝑑bSc )

𝜎��¬S�� = 𝑘}�34𝐹�¬S��

𝜋(𝑑bOd − 𝑑bSd )

We then proceeded in a fashion similar to the previous shafts, iterating diameters until our target safety factor was achieved. We then repeated this process for the keyway point, and found it to be less critical. We chose a diameter 5mm larger than that of the bearing point. Material: For our shaft, we arrived at our chosen 𝑁}~ by selecting SAE 1050 Cold Rolled Steel (Table A-9, Norton). Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range of 𝑁 = 10�𝐶𝑦𝑐𝑙𝑒𝑠. Based on this, we shall design for an endurance limit of 𝑆�using:

𝑆� = 𝐶�S��𝐶�O��𝐶�P�}𝐶Q���𝐶���S�~𝑆��where 𝐶�O�� = 0.70, as opposed to 1 in our other shafts. Note, our 𝐾Q and 𝐾Q� were obtained from Norton Tables C-1 – C-3 for our bearing analysis , as it has a shoulder stress concentration point at that area. Otherwise, 𝑘}and𝑘}��wereobtainedidenticallytoShaft1.

30

BearingsEarlyon in theprocessofbearingselection the teamdecided itwouldbebest to chooseroller/ballbearingsfortheentiresystemratherthanjournalbearings.Ballbearingshavemany desirable attributes, primarily an very operating friction, which reduces frictionallosesinouroverallsystem,thatwouldhaveotherwisebeengeneratedbytheviscosityinajournalbearing.Rollingbearingsalsohavenotransientstart-upspeeds,whichmakesthemidealforacriticalapplicationsuchasanaircraft.Finally,theycanhandleaxialandradialloads,which isdesirable if theaircrafthappens tooperateunderunexpected conditions.Rollerbearingsarealotlesssensitivetoanypotentialinterruptionswiththeirlubricationaswell.We used ball bearings formost of our system,where axial loadswere not present. Theselection of these bearingswas fairly straight forward. For our axial load on the outputshaft,asphericalrollerbearingwaschosentoabsorballtheaxialloadsfromthepropeller.ThischoicewasmadesothatnostressconcentrationsmayariseintheremainderofShaft3, shielding the gear and preventing the transfer of any axial loads to the rest of thegearboxinthecaseofanunexpectedfluctuationinthepropeller.TheBallbearingselectionprocesscannowbeoutlined.ForeveryoneofourshaftsweusedtheFreeBodyDiagramanalysisoutlinedintheShaftsection(Figure4,5,6and7)toobtainthereactionforcesnecessaryforourbearings.ThisallowedustoobtainreactionforcesFRandFA,forradialandaxialrespectively.Wethencombinedthoseloadsusing:

𝑃 = 𝑋𝑉𝐹± + 𝑌𝐹³

whereX,VandYarebearingloadfactorsobtainedfromNortonFigure11-24.

Now,using𝐿�o =µ�

c

wemayobtaintheexpected𝐿�olifeinmillionsofrevolutions.

This𝐿�olifecanthenbecomparedtotheidealnumberofcyclesprovidedasadesignconstraint,toappropriatelyselectabearingthatwillexceedthislimit.

Our𝐿S���� =�ooo ¶� ∗�o∗·¸

�o¹

Where𝜔Nistheangularvelocityofeachshaft.

31

TheCparameterisobtainedfromthemanufacturerspecificationforeachbearing,andsotoallowforsuccessful iteration,a largeamountbearingtables wereinput intoourexcelsheets to allow for easy selection. The tables would “iterate” these formulas for manydifferenttypesofbearings,untilasuitable life limitwasfound.Weshouldnotethat wasnotcompareddirectlywithour,butratheritwasmultipliedbyareliabilityfactorofKr=0.33(forareliabilityof98%)formostofourbearingchoices. Wenowusedthisnewtocomparewithourideallife.

After a number of bearings of acceptable life were found, the particular bearing wasselectedbasedoffgeometricalconstraintsofourshaft.Sincewehadaverylowstressstate,wecouldaffordtoaddalargeshoulderonthebearingportionofourshaft,andsodoutwasnotafactor.However,sinceweusedahollowshaft,dinwasthelimitingparameter.Theselectedbearingsarepresentedinlaterinthisreport.

Note: Due to our very low loading, the life expectancies of the bearings on shaft 1 aresignificantlyhigherthannecessary,howeverthesebearingswerefoundtobesuitableforourgeometry ,andsatisfiedourneeds,andthus theywerechosen.Thesameapplies forShaft2,4,howeverthelifeexpectanciesarenotmuchhigherthantheidealrequirement.

ThefinalsphericalthrustbearingwaschosenfromSKF,anditwaschosenforasuitableitssuitablegeometryandlifeexpectancies.ThebearingisaSphericalRollerBearing.

32

DesignResultsInsummary,theresultinggearspecificationsare:

33

Asforshafts,thespecificationsareasfollows:

34

35

Theresultssummarizedabovearetheoutputofalongiterativedesignprocess.Aswecanseefromthetablesabove,thechosendesignprovidessatisfactorysafetyfactors,inconcordancewithourinitialestimations.Forthegears,thefactorsofsafetyforbothbendingandpittingareacceptable.Also,forshafts,thesafetyfactorsforbothfailureandyieldareacceptable,andfinally,forkeys,thesafetyfactorsforbearingstressandthoseforshaftstressaregood.Thisensurestheviabilityofthedesign.Moreover,thelifeconstraintsarealsorespectedforthebearings.Thelifecalculatedforthebearingchosenishigherthantheliferequirementsgiven,whichprovesthatthedesignisenduringaswell.Unfortunately,themassrequirementcouldnotberespectedwiththecurrentdesignchoices.Alternativescouldbediscussedintheconclusion.Adetaileddrawingofeachpartispresentintheappendixandsupplementarycalculationsareprovidedintheexcelfile.

36

ModifiedGoodmanDiagramsAfter compiling all the data in the tables above, we can now illustrate the stress and safety factors using modified Goodman Diagrams. The equations below will help us to determine the different values that construct the axis and lines.

Toobtainthesafetyfactor,wealsohavetochoose

for

whichfailurecasewearesolving.Astandardchoicewouldbecase3whereweassume

thatƠmandƠawillincreaseinaconstantratio.

37

Shaft 1

Shaft 2/4

SafetyFactorCase31.418797553

SafetyFactorCase31.31817201

38

Shaft 3

SafetyFactorCase31.48390892

39

Conclusions

NikolaDrive'svisionforthefutureandpassionforinnovationdrovetheteamtogiveit'sbestinthedesignofthisgearbox.,Usingthespecificationsoftheplansuchasthepowerinput,themassofthepropellerandrespectingthedimensionalandmassconstraints.

Thedesignprocessforthedouble-brunchdouble-reductiongearboxquicklyproveditselfasanincrediblychallengingone.Astheteamtackledthevarietyofproblemsandconstraints,oftentimes,therewasnovisiblesolution,asdifferentvariablesandfactorsaredependentoneachother.Toovercometheseemergingdifficulties,conceptualunderstandingofmachineelementdesignwasrequiredforlayingtrueandqualityassumptionsthathelpednarrowingdowntherangeofpossiblesolutions.

Thegearboxdesignpresentedinthisreportisaresultofconsiderableamountofexperimentaliterationsbasedonstressanalysisknowledgeandtheteam'sabilitytomanipulatetheprovideddataintoafeasiblewell-performingsolution.Thedesignachievesmostofitsinitialgoalsexceptfortheweightreductionrequirement.Thisparametercouldalsobeimprovedwithmoreiterativeexperimentation.Specifically,withtheuseoflighterandstrongermetalssuchastitanium,magnesiumandothers.Inaddition,thereisstillroomforreducingthespacingbetweenthecomponentswhichmayimprovetheperformanceintermsofloweringthebendingstressesaswellascuttingonweight.

Forconclusions,therigorousprocessthatledtheteamtoitsfinaldesignwasanextremelychallengingandfruitfulprocess.Allteammembershadtojoinforces,thoughtsandsacrificeprecioussleepinghourstoachievethisimpressivesolution.Itisthesekindofchallenges,thatencouragesustoconstantlyimproveandmakesusbetterfutureengineers.

40

AppendixA–AllCalculations

• Full Calculations for Gear 2 (and Gear 6 by Association)

41

42

43

44

AppendixB–FiguresandTables

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00 20 40 60 80 100 120 140

T1Nmm

-1-0.8-0.6-0.4-0.2

00.20.40.60.81

0 20 40 60 80 100 120 140

V1(yzplane)

05

10152025303540

0 20 40 60 80 100 120 140

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48

AppendixC–MechanicalDrawingsofProposedDesign

IsometricView

49

TopView

50

Courtesy of Engineering edge website. http://www.engineersedge.com/mechanical,045tolerances/general_iso_tolerance_.htm