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The Design of Formula SAE Half Shafts for Optimum Vehicle Acceleration
2013-01-1772 Published 04/08/2013
James P. Parsons California State Polytechnic Univ-Pomona Copyright © 2013 SAE International doi:10.4271/2013-01-1772
ABSTRACT Many Formula SAE teams choose to design half shafts instead of purchase them. Commercial half shafts are usually over-designed, so teams make custom shafts to reduce the mass and rotational inertia. Half shafts are commonly designed by predicting the applied torsional loads and selecting inner and outer diameters to not exceed the material's yield strength. Various combinations of inner and outer diameters will support the loads, and the final dimensions may be chosen arbitrarily based on the designer's attempt to minimize both mass and rotational inertia. However, the mass and rotational inertia of a hollow shaft are inversely related and both quantities cannot be minimized simultaneously. Designers must therefore compromise between mass and rotational inertia reductions to maximize vehicle performance. This paper will present the derivation of an equation which calculates the optimum inner and outer half shaft diameter to maximize vehicle acceleration. Graphical explanations and predictions of vehicle acceleration improvements will be provided. The design, manufacturing, and testing procedure of Cal Poly Pomona's Formula SAE half shafts will be explained as an example for other teams.
INTRODUCTION Half shafts are critical components for any Formula SAE vehicle and require detailed analysis to ensure a sufficient factor of safety. Due to the complications involved with design and manufacturing, many teams are rightfully hesitant to create their own half shafts. However, vehicle performance can be significantly improved by designing optimum half shafts tailored to a specific vehicle. Strictly considering vehicle performance, the benefits far outweigh the risks if a proper design procedure is followed. Additionally, design
scores at SAE competitions can be improved with custom designed half shafts. Vehicle acceleration is greatly affected by half shaft properties. Wheel torque is decreased by half shaft rotational inertia and overall vehicle mass is increased by half shaft mass. Both quantities (shaft rotational inertia and shaft mass) reduce power limited vehicle acceleration. While these negative impacts can never be removed, they can be minimized by proper design. This paper will begin by deriving 1. an equation relating vehicle acceleration to half shaft dimensions, and 2. an equation relating inner and outer half shaft diameters based on applied loads and fatigue failure criteria Then, the paper will explain how to use these equations to select optimum shaft diameters and verify the factor of safety. A summary of how Cal Poly Pomona utilized the above equations will be provided. Additionally, the testing procedure used to validate the design will be explained.
RELATING VEHICLE ACCELERATION TO HALF SHAFT DIMENSIONS Newton's Second law will be used to calculate power limited acceleration. The section will demonstrate that the vehicle's net force and total mass are functions of shaft diameters. Substituting the force and mass functions into Newton's Second law will give a relationship between vehicle acceleration and shaft dimensions. The following analysis is valid only when the vehicle's wheels are not slipping (i.e. “power limited”). This may seem
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like a substantial limitation to the analysis, but further thought indicates that no limitation exists at all. Half shaft mass and rotational inertia should only be considered when the wheels have traction, because neither quantity reduces vehicle acceleration when the wheels are slipping. See reference [4] for more details on this topic. Beginning with acceleration is
Newton's
Second
law,
longitudinal The force output on the wheels can be calculated by (A.1) (A.5)
To relate net force to half shaft diameters, we define the net longitudinal force as Torque loss from the half shafts is given by (A.2) (A.6)
The vehicle weight can be neglected assuming the car is on level ground, and air drag will be neglected to simplify the final equation. (Gravity and air drag can be included if desired.) This gives
The rotational inertia is given by reference [3] as
(A.7) (A.3)
The angular acceleration of the half shaft is
Before continuing, realize FWheelX is the net force acting in the longitudinal direction by the tires. FWheelX ordinarily would include both thrust and rolling resistance forces. Although rolling resistance is not negligible, it will be neglected in this analysis. Neglecting rolling resistance implies that predictions of vehicle acceleration will have some error, but this error eventually cancels itself out during the optimization procedure.
(A.8) where r is the wheel radius. Substituting Equations A.8 and A.7 into A.6, then substituting A.6 into A.5 gives
The vehicle mass is given by
(A.9) (A.4)
Before continuing, it's important to understand what equation A.9 represents. The force exerted by the tires on the vehicle due to engine torque is equal to the wheel force that would be exerted if all drivetrain losses except for half shaft rotational
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inertia were considered, subtracted by the force lost by half shaft rotational inertia. Substituting equation A.9 and A.4 into A.1 gives
(A.10) The longitudinal vehicle acceleration can now be expressed by rearranging equation A. 10
(A.11) Equation A.11 will be referred to as the acceleration equation and it provides one of relationships necessary to find optimum shaft dimensions. A.11 shows how power limited vehicle acceleration depends on the outer diameter and inner diameter of the half shafts. The next step is to determine what ID and OD combinations are capable of withstanding applied loads. Then, the ID and OD that yields the highest vehicle acceleration can be selected.
RELATIONSHIP BETWEEN INNER AND OUTER DIAMETERS We will now use stress analysis techniques to derive a relationship between ID and OD that will ensure the shaft can avoid a fatigue failure under the applied loads. The following procedure will be utilized: 1. Determine the applied loads acting on the half shafts
where FMax represents the maximum tire tractive force. It's important to note that dynamic effects acting on the half shaft most likely produce additional torsional forces which makes TMax an under prediction. A large static factor of safety should be used to account for additional dynamic loads. For fatigue considerations, the variation of applied torsional loads must be determined. In worst case conditions, the half shafts would be transitioned from engine braking to traction limited acceleration. Therefore the largest possible variation in torque is
(AP.1) where TEngine Brake is the engine braking torque. A negative value for engine braking torque should be substituted into equation AP.1 because it acts opposite in direction to maximum torque. Also, the mean torque under cyclic loading is the average of maximum and minimum applied torque.
(AP.2) A negative value for engine braking torque should be used for equation AP.2.
Determine Internal Stresses Because we're neglecting axial loads, and no bending loads are present, the shaft is in pure torsion. The internal shear stress is highest at the outer fibers and is given by
2. Relate the applied loads to internal stresses 3. Substitute the internal stress relations into the ASME Elliptic failure criterion. This will relate ID to OD and ensure an infinite fatigue life. The shafts will be analyzed for static failure later.
Determine Applied Loads Ideally, half shafts for a Formula SAE vehicle should only be subjected to torsional loads. Assuming the shaft joints are functioning properly, bending loads should be zero, and axial loads are negligible in comparison to torsion. Finite element analysis techniques can be used to validate the assumption of negligible axial loads. The maximum torsional load applied to the half shaft can be calculated using tire data. The half shaft is under maximum torsional stress when the tire has reached the traction limit. Maximum torque is given by
Substituting above equation gives
and
into the
At the traction limit, maximum shear stress is
(S.1)
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The maximum variation in shear stress is
(S.2) Finally, the mean shear stress is Substituting zero values for all bending stresses, and inserting equation S.2 into S.5 gives (S.3)
Fatigue Failure Criterion
(S.7)
To ensure an infinite fatigue life, the ASME Elliptic failure criterion will be used. ASME Elliptic has been chosen because it is less conservative than other fatigue criteria and will result in a lighter half shaft. More conservative failure criteria can be used, such as the Modified Goodman, at the discretion of the designer. However, the following derivation would have to be changed. In case the reader is unfamiliar with fatigue failure criteria, a comprehensive review can be found in reference [1].
Substituting equation S.3 into S.6 gives
(S.8) Substituting Equations S.7 and S.8 into S.4 gives
To clarify, Formula SAE half shafts probably do not require an infinite fatigue life. However, specifying an infinite fatigue life provides higher reliability and allows designers to avoid a crack propagation analysis which may require nondestructive testing to verify internal flaw size. The ASME Elliptic equation is
(S.9)
(S.4)
Solving equation S.9 for ID and neglecting imaginary and negative solutions gives
(S.10) The equations for σAmp’ and σMean’ are given by reference [1] to be
Equation S.10 now can be used to calculate the required shaft ID (based on a given OD) to withstand the applied loading cycles without fatigue failure.
SELECTING THE OPTIMUM HALF SHAFT DIAMETERS
(S.5)
(S.6)
Using the vehicle acceleration equation A.11 and the stress analysis equation S.10, optimum shaft diameters can be selected. A graphical approach will be used to help conceptualize the procedure. Vehicle acceleration will be compared to various combinations of ID and OD dimensions. The combinations of inner and outer diameters that can withstand the applied loads will be graphed. The combination of ID and OD resulting in the largest vehicle acceleration that still supports torsional loads will be chosen. At the end of the
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Figure 1. Graph of vehicle acceleration versus shaft OD and ID. (The numerical values on this graph apply to Cal Poly Pomona'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.) section, an algebraic approach will be explained to obtain more accurate results.
shaft dimensions have been selected, the final task that remains is to verify the static factor of safety.
Figure 1 is a graph of equation A.11 and equation S.10 created by Mathematica 8.0. Vehicle acceleration is plotted on the vertical axis, ID on the horizontal x-axis, and OD on the horizontal y-axis. The curved surface represents the vehicle acceleration equation, and demonstrates variations in acceleration with half shaft dimensions.1 Notice when OD is held constant, acceleration increases with increasing ID. When ID is held constant, acceleration decreases with increasing OD. The curved line is the stress analysis equation, which indicates all combinations of OD and ID that are strong enough to withstand applied loads.
VERIFY THE STATIC FACTOR OF SAFETY To check the static factor of safety, simply divide the material's yield strength by the maximum Von Mises stress
(FS.1) The maximum Von Mises stress, after neglecting bending, is given by
At this point, the designer simply selects the combination of ID and OD that yields maximum vehicle acceleration. It's possible to select optimum diameters directly off the graph. Also, a designer can substitute equation S.10 into A.11 and use calculus to find the ID and OD corresponding to maximum acceleration. However, both of those approaches would be difficult to do by hand accurately. A more accurate, simpler approach utilizing an equation solver 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 determined from a numerical approach. Back substituting this optimum OD into equation S.10 gives the optimum ID. Now that half 1Numerical values required in Equations
(FS.2) where τMax is given by equation S.1. However, we now must include a stress concentration factor KT to account for any stress risers. This gives equation FS.2 in simplified form as
and equation FS.1 in simplified form as
A.11 and S.9 were obtained from Cal Poly Pomona's FSAE vehicle.
(FS.3)
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The stress concentration factor can be determined using any design handbook. For half shafts, there should not be any stress risers on the cylindrical portion between the splines, and the stress concentration factor should be one. Accounting for stress risers at the connection between the shaft and splines should be done using another stress analysis.
CAL POLY POMONA DESIGN PROCESS After deriving equations A.11, S.10, and FS.3, the half shaft design process was started.
Initial Design Decisions The shaft material was selected first. 4340 steel was chosen because a large yield strength can be obtained after heat treatment. Titanium was considered, but seemed unrealistic due to the cost. Additionally, titanium would require larger diameters to withstand the applied loads. The length of each half shaft was determined using dimensions between the drivetrain and rear hubs. Cal Poly Pomona's half shafts are connected to tripod bearings inside the hub and differential housing. It was important to verify that the shafts were long enough to reach both CV joints, but short enough to allow relative motion between the wheel and differential when the wheel bumps.
Estimating Vehicle Parameters To approximate the wheel torque outputted to each wheel, data from an engine dynamometer test was used. The maximum engine torque output multiplied by the vehicle's gear ratio provided an estimation of the wheel torque delivered to the wheels. Wheel radius was calculated from the nominal tire diameter. The “shaftless” mass of the vehicle was approximated by subtracting both half shaft masses from total vehicle mass. Even though wheel torque, wheel radius, and “shaftless” mass were only approximated, their accuracy was still sufficient to provide a close estimation of the optimum shaft dimensions. In hindsight, more testing should have been conducted to find accurate values of wheel torque, “shaftless” mass, and effective wheel radius. Wheel torque can be measured by testing a vehicle on a dynamometer and subtracting estimated half shaft losses. Additionally, the vehicle should have been weighed without half shafts to obtain “shaftless” mass. Finally, the effective wheel radius should be used instead of the nominal. This introduces new problems though because the effective wheel radius varies in different driving conditions, and no better technique to approximate the actual tire's radius for this application has been thought of.
After the material, length, wheel torque, “shaftless” mass, and wheel radius was determined, numerical values were substituted into the acceleration equation A.11.
(A.11)
Estimating Stress Analysis Parameters To begin the stress analysis, it was necessary to determine numerical values for each constant in equation S.10. The factor of safety against fatigue was initially set at 1.5, but after design iterations, it reduced to 1.06. This may seem a little aggressive considering the risks involved. However, caution was taken throughout the entire design process to ensure the shafts wouldn't fail during operation. The endurance strength of the material was determined using a procedure presented by reference [1]. A specimen under fully reversed loading has an endurance strength of
To adjust the endurance strength for actual loading conditions, SE was multiplied by a series of correction factors. However, many of the correction factors required the shaft ID and OD to be known, so the shaft endurance strength was approximated as thirty percent of the ultimate strength. Design iterations were then used to adjust the endurance strength of the material by appropriate factors after selecting initial ID and OD values. The fatigue stress concentration factor was set to a value of one because no stress risers were present on the hollow shaft. However, the connection between the splines and shaft did introduce a stress riser which was analyzed by hand calculations and FEA later. The amplitude and mean torque was calculated using equations AP.1 and AP.2. Determining both of these values requires knowledge of engine braking torque. It's was difficult to determine this value accurately, but it can be approximated using data recorded by the vehicle's data acquisition system. The vehicle's deceleration under engine braking provides an estimate of engine braking torque. A better approach should be used in the future to more accurately determine the engine braking torque. Due to the inaccuracies involved, the engine braking torque was intentionally overestimated to be thirty percent of maximum engine torque. With a more accurate test, the half shafts could be lightened because less conservative figures would be used. After determining the fatigue factor of safety, endurance strength, ultimate strength, amplitude torque, mean torque,
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and stress concentration factor, the numerical values were substituted into equation S.10.
(S.10)
Selecting Half Shaft Diameters Once equations A.11 and S.10 were completely defined with numerical values, they were inputted into Mathematica 8.0. The equations were graphed together and the result is shown in Figure 1. Based on Figure 1, it was obvious that acceleration improved when the wall of the shaft became thinner. This is because shaft mass reduces as the wall becomes thinner. However, there were also improvements in vehicle acceleration when the rotational inertia decreased. The optimum OD was found using the procedure described previously. Equation S.10 was substituted into A.11 using Mathematica 8.0, and then the OD that corresponded to maximum vehicle acceleration was outputted. The result was outstanding. Apparently, maximum vehicle acceleration occurs when the OD is approximately twenty inches. Obviously the optimum OD is not practical, so one inch was selected instead. While it may seem that the optimum OD is not being used, there are not excessively large differences in vehicle acceleration between an OD that is one inch or twenty inches. Next, the ID of the shaft was selected using equation S.10 with the OD equal to 1 inch. The result was 0.872in. The static load factor of safety was then checked using equation FS.3 shown below.
(FS.3) The static stress concentration factor was set to one because no stress risers existed in the hollow portion of the shaft.
Final Remaining Tasks The half shafts have not yet been manufactured or tested, but the 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 be purchased and proper splines will be welded onto the tubes, or 2.) half shafts will be machined from round stock and splines will be cut onto the ends. Option one is the most risky
because welds are difficult to analyze and can behave unpredictably. However, option one allows the tubes to be sized to optimum dimensions. Also, detailed welding analysis can be done as explained by reference [1] and reference [2]. If option two is used, the inner diameter of the shaft will be determined by the spline stresses. Option two therefore does not allow the inner diameter of the half shafts to be optimum. The goal is to use option one, but testing will ultimately be decide which option is more realistic. To complete testing, the half shafts need to be statically loaded and fatigue loaded. Static testing will be used to verify design calculations and check the stiffness. Currently, calculations predict that the half shafts will deflect about six degrees under maximum load. Fatigue testing will be conducted on the Cal Poly Pomona Formula SAE vehicle during practices.
CONCLUSION While the half shaft design process is lengthy, following the illustrated procedure will result in maximum power limited vehicle acceleration. Cal Poly Pomona will be using optimized half shafts in their upcoming competition for 2013, and they will be designed according to the procedure specified in this technical paper. As a final note, the theory behind this process does not have to be limited to shaft design. In fact, any rotating component on a vehicle can be optimized to balance rotational inertia and mass. Possible examples include hubs, differential housings, and wheel centers. Because components like these are more complicated geometrically, it would be difficult to derive closed form equations that would result in optimum designs. However, design software, FEA, and an iterative approach can be used to balance rotational inertia and mass to maximize acceleration.
REFERENCES 1. Budynas, Richard, and Nisbett Keith. Shigley's Mechanical Engineering Design. New York: McGraw-Hill, 2011. 2. WEAVER, M. “Determination of Weld Loads and Throat.” 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 Mechanics Dynamics. Hoboken: John Wiley & Sons, Inc., 2010. 4. Rajamani, Rajesh. Vehicle Dynamics and Control. New York: Springer, 2006.
CONTACT INFORMATION James Parsons California State Polytechnic University, Pomona
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[email protected]
ACKNOWLEDGEMENTS I would like to thank Cal Poly Pomona's Formula SAE team for the opportunities they have provided me to conduct this research. The ideas for this design approach were formed by discussions held with various team members. Professor Clifford Stover deserves to be thanked as well for the time he dedicates as the team's advisor. Go CPP FSAE!
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