the design of formula sae half shafts for optimum vehicle acceleration

ABSTRACTMany Formula SAE teams choose to design half shaftsinstead of purchase them. Commercial half shafts are usuallyover-designed, so teams make custom shafts to reduce themass and rotational inertia. Half shafts are commonlydesigned by predicting the applied torsional loads andselecting inner and outer diameters to not exceed thematerial's yield strength. Various combinations of inner andouter diameters will support the loads, and the finaldimensions may be chosen arbitrarily based on the designer'sattempt to minimize both mass and rotational inertia.However, the mass and rotational inertia of a hollow shaft areinversely related and both quantities cannot be minimizedsimultaneously. Designers must therefore compromisebetween mass and rotational inertia reductions to maximizevehicle performance.

This paper will present the derivation of an equation whichcalculates the optimum inner and outer half shaft diameter tomaximize vehicle acceleration. Graphical explanations andpredictions of vehicle acceleration improvements will beprovided. The design, manufacturing, and testing procedureof Cal Poly Pomona's Formula SAE half shafts will beexplained as an example for other teams.

INTRODUCTIONHalf shafts are critical components for any Formula SAEvehicle and require detailed analysis to ensure a sufficientfactor of safety. Due to the complications involved withdesign and manufacturing, many teams are rightfully hesitantto create their own half shafts. However, vehicle performancecan be significantly improved by designing optimum halfshafts tailored to a specific vehicle. Strictly consideringvehicle performance, the benefits far outweigh the risks if aproper design procedure is followed. Additionally, design

scores at SAE competitions can be improved with customdesigned half shafts.

Vehicle acceleration is greatly affected by half shaftproperties. Wheel torque is decreased by half shaft rotationalinertia and overall vehicle mass is increased by half shaftmass. Both quantities (shaft rotational inertia and shaft mass)reduce power limited vehicle acceleration. While thesenegative impacts can never be removed, they can beminimized by proper design.

This paper will begin by deriving1. an equation relating vehicle acceleration to half shaftdimensions, and2. an equation relating inner and outer half shaft diametersbased on applied loads and fatigue failure criteria

Then, the paper will explain how to use these equations toselect optimum shaft diameters and verify the factor of safety.A summary of how Cal Poly Pomona utilized the aboveequations will be provided. Additionally, the testingprocedure used to validate the design will be explained.

RELATING VEHICLEACCELERATION TO HALF SHAFTDIMENSIONSNewton's Second law will be used to calculate power limitedacceleration. The section will demonstrate that the vehicle'snet force and total mass are functions of shaft diameters.Substituting the force and mass functions into Newton'sSecond law will give a relationship between vehicleacceleration and shaft dimensions.

The following analysis is valid only when the vehicle'swheels are not slipping (i.e. power limited). This may seem

The Design of Formula SAE Half Shafts forOptimum Vehicle Acceleration

2013-01-1772Published

04/08/2013

James P. ParsonsCalifornia State Polytechnic Univ-Pomona

Copyright 2013 SAE Internationaldoi:10.4271/2013-01-1772

Downloaded from SAE International by Clemson University Libraries, Sunday, March 22, 2015

like a substantial limitation to the analysis, but furtherthought indicates that no limitation exists at all. Half shaftmass and rotational inertia should only be considered whenthe wheels have traction, because neither quantity reducesvehicle acceleration when the wheels are slipping. Seereference [4] for more details on this topic.

Beginning with Newton's Second law, longitudinalacceleration is

(A.1)

To relate net force to half shaft diameters, we define the netlongitudinal force as

(A.2)

The vehicle weight can be neglected assuming the car is onlevel ground, and air drag will be neglected to simplify thefinal equation. (Gravity and air drag can be included ifdesired.) This gives

(A.3)

Before continuing, realize FWheelX is the net force acting inthe longitudinal direction by the tires. FWheelX ordinarilywould include both thrust and rolling resistance forces.Although rolling resistance is not negligible, it will beneglected in this analysis. Neglecting rolling resistanceimplies that predictions of vehicle acceleration will havesome error, but this error eventually cancels itself out duringthe optimization procedure.

The vehicle mass is given by

(A.4)

The force output on the wheels can be calculated by

(A.5)

Torque loss from the half shafts is given by

(A.6)

The rotational inertia is given by reference [3] as

(A.7)

The angular acceleration of the half shaft is

(A.8)where r is the wheel radius.

Substituting Equations A.8 and A.7 into A.6, thensubstituting A.6 into A.5 gives

(A.9)

Before continuing, it's important to understand what equationA.9 represents. The force exerted by the tires on the vehicledue to engine torque is equal to the wheel force that would beexerted if all drivetrain losses except for half shaft rotational


inertia were considered, subtracted by the force lost by halfshaft rotational inertia.

Substituting equation A.9 and A.4 into A.1 gives

(A.10)

The longitudinal vehicle acceleration can now be expressedby rearranging equation A. 10

(A.11)

Equation A.11 will be referred to as the acceleration equationand it provides one of relationships necessary to findoptimum shaft dimensions. A.11 shows how power limitedvehicle acceleration depends on the outer diameter and innerdiameter of the half shafts. The next step is to determine whatID and OD combinations are capable of withstanding appliedloads. Then, the ID and OD that yields the highest vehicleacceleration can be selected.

RELATIONSHIP BETWEEN INNERAND OUTER DIAMETERSWe will now use stress analysis techniques to derive arelationship between ID and OD that will ensure the shaft canavoid a fatigue failure under the applied loads. The followingprocedure will be utilized:1. Determine the applied loads acting on the half shafts2. Relate the applied loads to internal stresses3. Substitute the internal stress relations into the ASMEElliptic failure criterion. This will relate ID to OD and ensurean infinite fatigue life.

The shafts will be analyzed for static failure later.

Determine Applied LoadsIdeally, half shafts for a Formula SAE vehicle should only besubjected to torsional loads. Assuming the shaft joints arefunctioning properly, bending loads should be zero, and axialloads are negligible in comparison to torsion. Finite elementanalysis techniques can be used to validate the assumption ofnegligible axial loads.

The maximum torsional load applied to the half shaft can becalculated using tire data. The half shaft is under maximumtorsional stress when the tire has reached the traction limit.Maximum torque is given by

where FMax represents the maximum tire tractive force. It'simportant to note that dynamic effects acting on the half shaftmost likely produce additional torsional forces which makesTMax an under prediction. A large static factor of safetyshould be used to account for additional dynamic loads.

For fatigue considerations, the variation of applied torsionalloads must be determined. In worst case conditions, the halfshafts would be transitioned from engine braking to tractionlimited acceleration. Therefore the largest possible variationin torque is

(AP.1)where TEngine Brake is the engine braking torque. A negativevalue for engine braking torque should be substituted intoequation AP.1 because it acts opposite in direction tomaximum torque.

Also, the mean torque under cyclic loading is the average ofmaximum and minimum applied torque.

(AP.2)A negative value for engine braking torque should be used forequation AP.2.

Determine Internal StressesBecause we're neglecting axial loads, and no bending loadsare present, the shaft is in pure torsion. The internal shearstress is highest at the outer fibers and is given by

Substituting and into theabove equation gives

At the traction limit, maximum shear stress is

(S.1)


The maximum variation in shear stress is

(S.2)

Finally, the mean shear stress is

(S.3)

Fatigue Failure CriterionTo ensure an infinite fatigue life, the ASME Elliptic failurecriterion will be used. ASME Elliptic has been chosenbecause it is less conservative than other fatigue criteria andwill result in a lighter half shaft. More conservative failurecriteria can be used, such as the Modified Goodman, at thediscretion of the designer. However, the following derivationwould have to be changed. In case the reader is unfamiliarwith fatigue failure criteria, a comprehensive review can befound in reference [1].

To clarify, Formula SAE half shafts probably do not requirean infinite fatigue life. However, specifying an infinitefatigue life provides higher reliability and allows designers toavoid a crack propagation analysis which may require non-destructive testing to verify internal flaw size.

The ASME Elliptic equation is

(S.4)

The equations for Amp and Mean are given by reference[1] to be

(S.5)

(S.6)

Substituting zero values for all bending stresses, and insertingequation S.2 into S.5 gives

(S.7)

Substituting equation S.3 into S.6 gives

(S.8)

Substituting Equations S.7 and S.8 into S.4 gives

(S.9)

Solving equation S.9 for ID and neglecting imaginary andnegative solutions gives

(S.10)

Equation S.10 now can be used to calculate the required shaftID (based on a given OD) to withstand the applied loadingcycles without fatigue failure.

SELECTING THE OPTIMUM HALFSHAFT DIAMETERSUsing the vehicle acceleration equation A.11 and the stressanalysis equation S.10, optimum shaft diameters can beselected. A graphical approach will be used to helpconceptualize the procedure. Vehicle acceleration will becompared to various combinations of ID and OD dimensions.The combinations of inner and outer diameters that canwithstand the applied loads will be graphed. The combinationof ID and OD resulting in the largest vehicle acceleration thatstill supports torsional loads will be chosen. At the end of the


section, an algebraic approach will be explained to obtainmore accurate results.

Figure 1 is a graph of equation A.11 and equation S.10created by Mathematica 8.0. Vehicle acceleration is plottedon the vertical axis, ID on the horizontal x-axis, and OD onthe horizontal y-axis. The curved surface represents thevehicle acceleration equation, and demonstrates variations inacceleration with half shaft dimensions.1 Notice when OD isheld constant, acceleration increases with increasing ID.When ID is held constant, acceleration decreases withincreasing OD. The curved line is the stress analysisequation, which indicates all combinations of OD and ID thatare strong enough to withstand applied loads.

At this point, the designer simply selects the combination ofID and OD that yields maximum vehicle acceleration.

It's possible to select optimum diameters directly off thegraph. Also, a designer can substitute equation S.10 into A.11and use calculus to find the ID and OD corresponding tomaximum acceleration. However, both of those approacheswould be difficult to do by hand accurately.

A more accurate, simpler approach utilizing an equationsolver should be conducted. Using an equation solver,equation S.10 can be substituted into equation A.11. Then,the OD that gives maximum acceleration can be determinedfrom a numerical approach. Back substituting this optimumOD into equation S.10 gives the optimum ID. Now that half

shaft dimensions have been selected, the final task thatremains is to verify the static factor of safety.

VERIFY THE STATIC FACTOR OFSAFETYTo check the static factor of safety, simply divide thematerial's yield strength by the maximum Von Mises stress

(FS.1)

The maximum Von Mises stress, after neglecting bending, isgiven by

(FS.2)where Max is given by equation S.1. However, we now mustinclude a stress concentration factor KT to account for anystress risers. This gives equation FS.2 in simplified form as

and equation FS.1 in simplified form as

(FS.3)

1Numerical values required in Equations A.11 and S.9 were obtained from Cal Poly Pomona's FSAE vehicle.

Figure 1. Graph of vehicle acceleration versus shaft OD and ID. (The numerical values on this graph apply to Cal PolyPomona's FSAE vehicle only. Every vehicle will have slightly different constants substituted into Equations A.11 and S.9 which

will result in different vehicle accelerations.)


The stress concentration factor can be determined using anydesign handbook. For half shafts, there should not be anystress risers on the cylindrical portion between the splines,and the stress concentration factor should be one. Accountingfor stress risers at the connection between the shaft andsplines should be done using another stress analysis.

CAL POLY POMONA DESIGNPROCESSAfter deriving equations A.11, S.10, and FS.3, the half shaftdesign process was started.

Initial Design DecisionsThe shaft material was selected first. 4340 steel was chosenbecause a large yield strength can be obtained after heattreatment. Titanium was considered, but seemed unrealisticdue to the cost. Additionally, titanium would require largerdiameters to withstand the applied loads.

The length of each half shaft was determined usingdimensions between the drivetrain and rear hubs. Cal PolyPomona's half shafts are connected to tripod bearings insidethe hub and differential housing. It was important to verifythat the shafts were long enough to reach both CV joints, butshort enough to allow relative motion between the wheel anddifferential when the wheel bumps.

Estimating Vehicle ParametersTo approximate the wheel torque outputted to each wheel,data from an engine dynamometer test was used. Themaximum engine torque output multiplied by the vehicle'sgear ratio provided an estimation of the wheel torquedelivered to the wheels. Wheel radius was calculated from thenominal tire diameter. The shaftless mass of the vehiclewas approximated by subtracting both half shaft masses fromtotal vehicle mass. Even though wheel torque, wheel radius,and shaftless mass were only approximated, their accuracywas still sufficient to provide a close estimation of theoptimum shaft dimensions.

In hindsight, more testing should have been conducted to findaccurate values of wheel torque, shaftless mass, andeffective wheel radius. Wheel torque can be measured bytesting a vehicle on a dynamometer and subtracting estimatedhalf shaft losses. Additionally, the vehicle should have beenweighed without half shafts to obtain shaftless mass.Finally, the effective wheel radius should be used instead ofthe nominal. This introduces new problems though becausethe effective wheel radius varies in different drivingconditions, and no better technique to approximate the actualtire's radius for this application has been thought of.

After the material, length, wheel torque, shaftless mass, andwheel radius was determined, numerical values weresubstituted into the acceleration equation A.11.

(A.11)Estimating Stress Analysis ParametersTo begin the stress analysis, it was necessary to determinenumerical values for each constant in equation S.10.

The factor of safety against fatigue was initially set at 1.5, butafter design iterations, it reduced to 1.06. This may seem alittle aggressive considering the risks involved. However,caution was taken throughout the entire design process toensure the shafts wouldn't fail during operation.

The endurance strength of the material was determined usinga procedure presented by reference [1]. A specimen underfully reversed loading has an endurance strength of

To adjust the endurance strength for actual loadingconditions, SE was multiplied by a series of correctionfactors. However, many of the correction factors required theshaft ID and OD to be known, so the shaft endurance strengthwas approximated as thirty percent of the ultimate strength.Design iterations were then used to adjust the endurancestrength of the material by appropriate factors after selectinginitial ID and OD values.

The fatigue stress concentration factor was set to a value ofone because no stress risers were present on the hollow shaft.However, the connection between the splines and shaft didintroduce a stress riser which was analyzed by handcalculations and FEA later.

The amplitude and mean torque was calculated usingequations AP.1 and AP.2. Determining both of these valuesrequires knowledge of engine braking torque. It's wasdifficult to determine this value accurately, but it can beapproximated using data recorded by the vehicle's dataacquisition system. The vehicle's deceleration under enginebraking provides an estimate of engine braking torque. Abetter approach should be used in the future to moreaccurately determine the engine braking torque. Due to theinaccuracies involved, the engine braking torque wasintentionally overestimated to be thirty percent of maximumengine torque. With a more accurate test, the half shafts couldbe lightened because less conservative figures would be used.

After determining the fatigue factor of safety, endurancestrength, ultimate strength, amplitude torque, mean torque,


and stress concentration factor, the numerical values weresubstituted into equation S.10.

(S.10)Selecting Half Shaft DiametersOnce equations A.11 and S.10 were completely defined withnumerical values, they were inputted into Mathematica 8.0.The equations were graphed together and the result is shownin Figure 1.

Based on Figure 1, it was obvious that acceleration improvedwhen the wall of the shaft became thinner. This is becauseshaft mass reduces as the wall becomes thinner. However,there were also improvements in vehicle acceleration whenthe rotational inertia decreased.

The optimum OD was found using the procedure describedpreviously. Equation S.10 was substituted into A.11 usingMathematica 8.0, and then the OD that corresponded tomaximum vehicle acceleration was outputted. The result wasoutstanding. Apparently, maximum vehicle accelerationoccurs when the OD is approximately twenty inches.

Obviously the optimum OD is not practical, so one inch wasselected instead. While it may seem that the optimum OD isnot being used, there are not excessively large differences invehicle acceleration between an OD that is one inch or twentyinches.

Next, the ID of the shaft was selected using equation S.10with the OD equal to 1 inch. The result was 0.872in. Thestatic load factor of safety was then checked using equationFS.3 shown below.

(FS.3)The static stress concentration factor was set to one becauseno stress risers existed in the hollow portion of the shaft.

Final Remaining TasksThe half shafts have not yet been manufactured or tested, butthe following plans are in place.

To manufacture the half shafts, one of two options exist: 1.)tubes of the proper inner and outer diameters will bepurchased and proper splines will be welded onto the tubes,or 2.) half shafts will be machined from round stock andsplines will be cut onto the ends. Option one is the most risky

because welds are difficult to analyze and can behaveunpredictably. However, option one allows the tubes to besized to optimum dimensions. Also, detailed welding analysiscan be done as explained by reference [1] and reference [2].If option two is used, the inner diameter of the shaft will bedetermined by the spline stresses. Option two therefore doesnot allow the inner diameter of the half shafts to be optimum.The goal is to use option one, but testing will ultimately bedecide which option is more realistic.

To complete testing, the half shafts need to be staticallyloaded and fatigue loaded. Static testing will be used to verifydesign calculations and check the stiffness. Currently,calculations predict that the half shafts will deflect about sixdegrees under maximum load. Fatigue testing will beconducted on the Cal Poly Pomona Formula SAE vehicleduring practices.

CONCLUSIONWhile the half shaft design process is lengthy, following theillustrated procedure will result in maximum power limitedvehicle acceleration. Cal Poly Pomona will be usingoptimized half shafts in their upcoming competition for 2013,and they will be designed according to the procedurespecified in this technical paper.

As a final note, the theory behind this process does not haveto be limited to shaft design. In fact, any rotating componenton a vehicle can be optimized to balance rotational inertia andmass. Possible examples include hubs, differential housings,and wheel centers. Because components like these are morecomplicated geometrically, it would be difficult to deriveclosed form equations that would result in optimum designs.However, design software, FEA, and an iterative approachcan be used to balance rotational inertia and mass tomaximize acceleration.

REFERENCES1. Budynas, Richard, and Nisbett Keith. Shigley'sMechanical Engineering Design. New York: McGraw-Hill,2011.2. WEAVER, M. Determination of Weld Loads andThroat. Welding Journal. (1999): 1-11. http://www.aws.org/wj/supplement/Weaver/ARTICLE2.pdf(accessed October 24, 2012).3. Meriam, J., and Kraige L.. Engineering MechanicsDynamics. Hoboken: John Wiley & Sons, Inc., 2010.4. Rajamani, Rajesh. Vehicle Dynamics and Control. NewYork: Springer, 2006.

CONTACT INFORMATIONJames ParsonsCalifornia State Polytechnic University, Pomona


[email protected]

ACKNOWLEDGEMENTSI would like to thank Cal Poly Pomona's Formula SAE teamfor the opportunities they have provided me to conduct thisresearch. The ideas for this design approach were formed bydiscussions held with various team members. ProfessorClifford Stover deserves to be thanked as well for the time hededicates as the team's advisor. Go CPP FSAE!

The Engineering Meetings Board has approved this paper for publication. It hassuccessfully completed SAE's peer review process under the supervision of the sessionorganizer. This process requires a minimum of three (3) reviews by industry experts.All rights reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the prior written permission of SAE.ISSN 0148-7191

Positions and opinions advanced in this paper are those of the author(s) and notnecessarily those of SAE. The author is solely responsible for the content of the paper.SAE Customer Service:Tel: 877-606-7323 (inside USA and Canada)Tel: 724-776-4970 (outside USA)Fax: 724-776-0790Email: [email protected] Web Address: http://www.sae.orgPrinted in USA


the design of formula sae half shafts for optimum vehicle acceleration

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