The Benefits of Four-Wheel Drive for a High-Performance FSAE Electric Racecar by Elliot Douglas Owen Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Bachelor of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 c Elliot Douglas Owen, MMXVIII. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Author...... Department of Mechanical Engineering May 18, 2018

Certified by ...... David L. Trumper Professor Thesis Supervisor

Accepted by ...... Rohit Karnik Associate Professor of Mechanical Engineering Undergraduate Officer 2 The Benefits of Four-Wheel Drive for a High-Performance FSAE Electric Racecar by Elliot Douglas Owen

Submitted to the Department of Mechanical Engineering on May 18, 2018, in partial fulfillment of the requirements for the degree of Bachelor of Science in Mechanical Engineering

Abstract This thesis explores the performance of Rear-Wheel Drive (RWD) and Four-Wheel Drive (4WD) FSAE Electric racecars with regards to acceleration and regenerative braking. The benefits of a 4WD architecture are presented along with the tools for further optimization and understanding. The goal is to provide real, actionable information to teams deciding to pursue 4WD vehicles and quantify the results of difficult engineering tradeoffs. Analytical bicycle models are used to discuss the effect of the Center of Gravity location on vehicle performance, and Acceleration-Velocity Phase Space (AVPS) is introduced as a useful tool for optimization. Lap-time Simulation is used to determine the regenerative braking energy available for recovery during a race for RWD and 4WD vehicles.

Thesis Supervisor: David L. Trumper Title: Professor

3 4 Acknowledgments

This thesis is dedicated to MIT Motorsports, because racecar.

I have many people to thank for my current state in life. My family has enabled me to attend MIT and apply myself whole-heartedly to mechanical engineering. My parents have always been extremely supportive of my projects even when it means they cannot walk into my room due to unorganized piles of hardware on the floor. It is a small miracle no one ever impaled their foot on a 6-32 tap. My brother has always served as an example of academic discipline and I have learned much from him.

MIT, the mechanical engineering department, its professors, the GEL program, and the Edgerton Center have all provided incredible opportunities to learn why things break, make yo-yos, lead projects, and turn big pieces of metal into smaller pieces of metal.

My teammates and peers have created the amazing environment that allows FSAE to flourish, and allowed me to push myself on large projects like the MY2017 battery pack. No one person can build a racecar alone, and I feel very fortunate to have seen the early years of the team’s struggle to get a vehicle to turn on. We have come so far since the dark days of MY2014 and MY2015 and still have very far to go. Special thanks to Orlando Ward, Kevin Chan, Brian Sennett, Nelson Brock, Andrew Carlson, Roberto Melendez, Sammi Bray, Luis Mora, Cheyenne Hua, Henry Merrow, and Ethan Perrin for teaching me and helping me with the battery. A final shoutout to a chance encounter with Adam from the CTU Prague Team at the 2016 competition who destroyed all the American teams. Your advice has proved very helpful. “I will share another, maybe the most important experience: The only thing that can stop you from achieving your goal is you. Literally.” - Adam Podhrazsky

Thanks to all my friends at MIT and around the country for their surprising acceptance of “that racecar thing” I do. Thanks to Susan Nitta, Kayla Rajsky, Katherine Paseman, and Becky Steinmeyer for all their mentorship. Finally, thanks to my amazing girlfriend, Ka-Yen Yau, for her unending kindness, support, and impeccable comma usage. Roll Tech!

5 Story of this Thesis

This thesis was born from the desire to make MIT Motorsport’s MY2019 vehicle better than MY2018. During December of 2017, I thought about improving car performance in quantifiable terms such as power, mass, drag, and efficiency. I determined that the Power to Mass Ratio (PTMR) of a vehicle was critically important, but I still knew very little about vehicle design. I tried analyzing a simple bicycle model of a rear-wheel drive racecar and realized that something as trivial as Center of Mass location played an enormous role in the maximum acceleration of a vehicle. I tried to separately analyze improved power to mass ratio and improved traction, but was looking for a single way to characterize the entire performance of the vehicle. In late December, I stumbled upon a way to visualize all the limits of the vehicle simultaneously and started my exploration of Acceleration-Velocity Phase Space (AVPS)-which will be discussed thoroughly. In the beginning of January, I traveled through Europe to visit six of the top FSAE Electric and Combustion teams. The trip was eye-opening. I left convinced that the Euro- pean teams did everything better than their counterparts in the States. They had lighter and more powerful batteries, Four Wheel Drive (4WD), carbon fiber monocoque frames and custom made everything. After that trip, my focus turned to 4WD. How important was it? Could MIt even do it? I started looking for motors, but realized I didn’t even know what to look for. This pushed me to develop more complicated analytical bicycle models to estimate vehicle acceleration. From there, I started to integrate aerodynamic effects, tire models and bicycle models into AVPS. With an analytical framework to estimate vehicle performance, I then set to learn as much as I could about the benefits of 4WD. I realized that a vehicle doesn’t need a very powerful front powertrain to achieve high acceleration. However, my experience in Europe showed that many teams used high power motors on the front. To figure out what sort of power was really necessary, I started to analyze regenerative braking. My investigation led me to lap-time simulation and to the realization of a strong and weak 4WD architecture. Taking this knowledge back to the AVPS models, I realized that the power-limited region offered even more opportunities for vehicle-level optimization.

6 Contents

1 FSAE Background ...... 15 1.1 FSAE Electric Competition ...... 15 1.2 MIT Motorsports Recent History ...... 15 1.3 State of the Art in Formula Student Electric ...... 17 2 Four-Wheel Drive Background ...... 18 2.1 Preliminary Appeal of 4WD ...... 18 2.2 Scope of this Thesis ...... 19 2.3 Common Models: Point Mass, Bicycle, and Two-Track ...... 19 3 Acceleration Analysis ...... 21 3.1 Summary of Benefits and Approaches ...... 21 3.2 Bicycle Models ...... 22 3.3 Tire Load Sensitivity ...... 28 3.4 Load Sensitive Rear-Wheel Drive (LSRWD) ...... 29 3.5 Load Sensitive Four-Wheel Drive (LS4WD) ...... 32 3.6 Aerodynamic Forces ...... 36 3.7 Preliminary Front-Rear Power Split ...... 40 3.8 Practical Limits to Acceleration Curves ...... 42 3.9 Acceleration-Velocity Phase Space ...... 43 3.10 Fine Tuning AVPS Models ...... 50 3.11 Preliminary Acceleration Takeaways ...... 55 3.12 Power Split in the Power Limited Region ...... 55 3.13 Weak and Strong 4WD ...... 56 3.14 Acceleration Takeaways ...... 64

7 3.15 Sensitivities and Final Comments ...... 64 4 Endurance Analysis ...... 65 4.1 Limitations of the Point Mass Model ...... 66 4.2 The Importance of Regenerative Braking ...... 67 4.3 Outputs of OptimumLap ...... 69 4.4 Preliminary Endurance Takeaways ...... 72 4.5 Power Split During Braking ...... 73 4.6 Braking Power Distributions ...... 76 4.7 Final Comments ...... 79 5 Next Steps ...... 80

8 List of Figures

1.1 MIT Motorsports MY2017 Vehicle ...... 16 1.2 AMZ Racing’s Pilatus Vehicle ...... 17 2.1 Point Mass Model ...... 20 2.2 Bicycle Model Sketch ...... 20 2.3 Two Track Model ...... 21 3.1 Bicycle Model FBD ...... 23 3.2 RWD G’s by CG Location ...... 25 3.3 RWD G’s with Reasonable CG ...... 26 3.4 Bicycle Model FBD ...... 27 3.5 Load Sensitivity Plot ...... 28 3.6 LSRWD G’s by CG ...... 31 3.7 LSRWD G’s with Reasonable CG ...... 32 3.8 LS4WD G’s by CG Location ...... 34 3.9 LS4WD G’s with Reasonable CG Location ...... 35 3.10 Comparison Between Models by CGx ...... 36 3.11 Aerodynamic Forces ...... 37 3.12 Aerodynamic FBD ...... 38 3.13 Model Acceleration with Aero ...... 39 3.14 4WD Acceleration by Front and Rear ...... 41 3.15 LS4WD Power Split ...... 42 3.16 AVPS Acceleration Limited ...... 44 3.17 AVPS Acceleration Limited Simulation ...... 45 3.18 AVPS Acceleration and Speed Limited ...... 46

9 3.19 AVPS Acceleration and Speed Limited Simulation ...... 47 3.20 AVPS Acceleration, Speed and Power Limited ...... 48 3.21 AVPS Acceleration, Speed, and Power Limited Simulation ...... 49 3.22 AVPS Model for LS4WD Standard Car ...... 51 3.23 AVPS LS4WD Simulation ...... 52 3.24 AVPS Model for LSRWD Standard Car ...... 53 3.25 AVPS LSRWD Simulation ...... 54 3.26 LS4WD AB Plot V = 0 ...... 58 3.27 LS4WD AB Plot V = 20 ...... 59 3.28 Constant Force Contour at V = 20 ...... 60 3.29 Strong and Weak Power Split ...... 61 3.30 Strong and Weak Force Split ...... 62 3.31 Strong and Weak Torque Split ...... 63 4.1 OptimumLap Simulation ...... 66 4.2 MY2017 Battery Discharge Curve ...... 68 4.3 Longitudinal Braking Forces ...... 70 4.4 Normalized Braking Power ...... 71 4.5 Braking Energy ...... 72 4.6 Vehicle Normal Forces ...... 74 4.7 Vehicle Traction Forces ...... 75 4.8 Braking Power Split ...... 76 4.9 Braking Power Distribution ...... 77 4.10 Regenerative Energy by Motor Power ...... 78

10 List of Tables

1 Standard MY2018 Car Parameters ...... 22 2 RWD Performance of the Standard Car ...... 24 3 RWD and LSRWD Comparison ...... 30 4 4WD and LS4WD Comparison ...... 33

11 12 Abbreviations and Nomenclature

4WD ...... Four-Wheel Drive RWD...... Rear-Wheel Drive LS4WD ...... Load Sensitive Four-Wheel Drive LSRWD...... Load Sensitive Rear-Wheel Drive FSAE ...... Formula Society of Automotive Engineers MY2017...... Model Year 2017 Vehicle AVPS ...... Acceleration-Velocity Phase Space Weak 4WD...... Weak Four-Wheel Drive Architecture Strong 4WD...... Strong Four-Wheel Drive Architecture CG...... Center of Gravity PTMR ...... Power-to-Mass Ratio RPM...... Revolutions Per Minute kWh ...... Kilo-Watt-hour

Rbz ...... Vertical Reaction Force on Rear Wheel, N

Rbx ...... Horizontal Reaction Force on Rear Wheel, N

Rfz ...... Vertical Reaction Force on Front Wheel, N

Rfx ...... Horizontal Reaction Force on Front Wheel, N

Fx ...... Resultant Force in the X Direction, N m ax ...... Resultant Acceleration in the X Direction, s2

Gx ...... Resultant Scaled Acceleration in the X Direction, unitless g m...... Mass of Vehicle, kg

m g...... Gravitational Acceleration of Earth, s2 P...... Power, W

13 X...... Horizontal Distance from CG Location to Front Wheel, m W...... Wheelbase Length, m Z...... Vertical Height of the CG location Above the Ground, m µ...... Tire Longitudinal Ideal Coefficient of Friction, unitless

µ1 ...... Tire Longitudinal Ideal Coefficient of Friction at No Load, unitless 1 µ2 ...... Tire Longitudinal Load Sensitivity Parameter, N

CGx ...... Horizontal Center of Gravity Position, X, m

Fd ...... Aerodynamic Drag Force, N

Fl ...... Aerodynamic Down Force (Lift), N kg ρair ...... Density of Air, m3 CLA ...... Coefficient of Lift Times Area, m2 CLA ...... Coefficient of Drag Times Area, m2

Fz ...... Vertical Force on Car Including Weight and Aero, N

µeff ...... Effective Tire Longitudinal Coefficient of Friction, unitless m V (t)...... Velocity as a Function of Time From Simulation, s X(t)...... Position as a Function of Time From Simulation, m A...... Fraction of Available Front Traction, unitless B...... Fraction of Available Rear Traction, unitless

δt ...... Delta Timestep, s

14 1 FSAE Background

1.1 FSAE Electric Competition

The Formula Society of Automotive Engineer’s (FSAE) Electric competition in the United States is part of the Society for Automotive Engineering’s Collegiate Design Series FSAE. The competition involves University teams from across the nation and around the world who design, build, present, and drive a custom, single-seat, high-performance electric vehicle. The official description of the event is that your team is a small company with a prototype vehicle looking for funding in order to go into production and sell approximately 1,000 vehicles a year of this type. FSAE provides a long and rigorous rule book describing the competition and vehicle requirements and safety rules. The rules typically specify that a certain feature must be present but do not describe how it must be done. Thus, FSAE Electric is a very innovative and dynamic competition and vehicle performance has improved drastically since the start of the US competition in 2013.

1.2 MIT Motorsports Recent History

MIT Motorsports is a student team within the Edgerton Center at MIT which has been involved in FSAE since 2001. From 2001 to 2012, the team was small and competed in the FSAE Internal Combustion competition reaching a peak of 8th place in 2010. After 2012, the team spent two years transitioning to the electric competition to compete in 2014.

15 During this two-year build cycle, the team collapsed as senior members left and a collection of sophomores led an almost failed team in 2015. The 2015 and 2016 year were very difficult, and our car barely passed the rules inspection in 2016. By 2017, the founding group of sophomores were seniors and a revolution in car performance was achieved. The vehicle passed rules inspection and competed in all dynamic events to win second place overall behind the University of Pennsylvania. The MY2018 vehicle represents an iteration on the MY2017 vehicle (shown in figure 1.1) to maximize the potential performance of the Rear Wheel Drive (RWD) architecture. As the team has grown and matured, we are looking towards the next step in vehicle performance for a radical new MY2019 vehicle. This thesis fits into this vision by providing quantitative and useful metrics to predict the performance of a Four Wheel Drive vehicle architecture.

Figure 1.1: The MY2017 vehicle at the Lincoln Nebraska competition June 2017.

16 1.3 State of the Art in Formula Student Electric

The Formula Student competition in Europe started much earlier than FSAE Electric in the US. The combination of experience and educational differences has enabled European Electric teams to reach awe-inspiring levels of performance. The world record for the fastest

km 0 − 100 h time in an electric vehicle (of any type) is held by the student team AMZ Racing at ETH Zurich in Switzerland with their 2014 vehicle Grimsel. The Europeans teams are truly operating on the limits of performance. Team TU FAST from TU Munich bragged that this year they would do all sorts of unholy things to their suspension to save 5kg of mass on a 155kg car. Considering that their competition from TU Delft, AMZ, and GreenTeam are all in the 160-170kg range, this is actually an amazing advantage. These teams fight gram by gram to improve performance and go to great lengths to save weight by using titanium fasteners. These teams have reached the extremes of customization and make their own brake cylinders, suspension springs, steering racks, and even custom video cameras “because GoPro’s are too heavy and add too much aerodynamic drag”. Their rules allow a 600-volt battery with minimal fuses which allows them to minimize resistive losses and complexity. The bar of engineering in Formula Student makes the US FSAE Electric competition look like a dinosaur race. In the US, vehicles range in the 200 to 250kg range, use welded steel frames, RWD architectures, off-the shelf steering wheels, and are stuck with a 300-volt battery limit. The state of the art in Europe is perhaps five years away in the United States. By developing core technologies like Four Wheel Drive, we hope to reach their level of performance and eventually compete with them.

Figure 1.2: ETH Zurich’s 2017 vehicle Pilatus [1] features custom made motors, a carbon fiber monocoque, and innovative hydraulic suspension. Shown with over 20 awards.

17 2 Four-Wheel Drive Background

2.1 Preliminary Appeal of 4WD

The appeal of an electric 4WD vehicle is hard to deny. In its simplest terms, having 4WD allows the normal force on each tire to create forward traction to propel the vehicle. Instead of using the two front tires for braking (negative longitudinal acceleration) and cornering (lateral acceleration), they can also add additional positive longitudinal acceleration. In na¨ıve terms, the car is dominated by Fx = max. With the same mass, more longitudinal force from tire traction will increase acceleration. If you squint at a car and assume the weight distribution front and back is approximately 50%, you could argue that having 4WD should in theory double the acceleration of the vehicle. While this is a simplification, the theoretical promise of double acceleration is enough to justify further analysis to find the actual realizable amount of acceleration. While producing positive longitudinal acceleration clearly improves vehicle performance, 4WD also offers a new take on longitudinal deceleration. A traditional vehicle has brakes on all four wheels. When the vehicle decelerates, the normal forces on the tires enable a certain maximum traction to be applied to slow the vehicle down. While adding motors will not change this amount of traction, the power that is dissipated as P = FV while slowing the car down can be recovered as regenerative braking rather than thermal dissipation in the brake rotors. To first order, one may guess that having 4WD will enable approximately twice as much regenerative braking capacity as a RWD vehicle. Again, this assumption is a gross simplification of complex system, but the promise is so great, that it is worth investigating further into the exact details of how much extra energy can be regenerated with a 4WD architecture. There are many other advantages to 4WD that are more difficult to quantify. A vehicle with actuation available on all four wheels may control traction and limit slip on all four wheels at once. This can prevent wheel slip which can result in inefficiency, tire damage (burn out), and loss of control of the vehicle (spinning out). Beyond traction control, 4WD can enable faster yaw-rates and allow a car to turn faster than with a RWD drivetrain. Fundamentally, 4WD gives much more freedom to the vehicle designer and software control

18 developers to maximize the performance of the vehicle. Since all the available normal force on all the tires can be used to its greatest advantage, 4WD unlocks a new regime of vehicle optimization.

2.2 Scope of this Thesis

The study of vehicle dynamics, performance, and optimization is a large and active field of research in industry and to some extent academia. It is impossible and undesirable to capture the entire field as it related to 4WD so I will limit my discussion to the single-seat racecars that compete in the FSAE Electric Collegiate Design Series. Within that field, I will focus on the promise of improved acceleration and regenerative braking as discussed above. My goal is to quantify the benefits and limits of a 4WD architecture as it relates to those two metrics and discuss the optimization and simulation of a 4WD vehicle. This information is intended for use by MIT Motorsports and other FSAE Electric teams that are considering switching to 4WD. This thesis seeks to inform a team that is debating the advantages of 4WD against the cost and complexity of its implementation by giving them concrete principles to evaluate these vehicle architectures against.

2.3 Common Models: Point Mass, Bicycle, and Two-Track

Vehicle modeling can range from first principles point mass kinematics to incredibly com- plicated full vehicle simulations. For the purposes of this thesis, I will use three models: the point-mass model, the bicycle model, and the two-track model. The point mass model is simple. It abstracts away the details of the tires, suspension, rigid body motions, and deformations and treats the car as a lumped mass at a single point. This point is then accelerated longitudinally and laterally by applied forces. When considering races on a flat surface, the forces on this point consist of gravity, aerodynamic loads, the tire normal force and traction forces. A schematic drawing of the point-mass model is shown in figure 2.1.

19 Figure 2.1: The simplified point mass model from the OptimumLap manual[2] removes lots of modeling complexity.

The second model is the bicycle model which will be developed more fully when discussing acceleration of a 4WD vehicle. While the point mass has one force input from the road, the bicycle model like its namesake has two. The bicycle model presented below in figure 2.2 is a description of a car as seen from the side. Now the vehicle must be defined by the separation of the two tires, as well as the position of the center of gravity.

Figure 2.2: This model is advanced enough to describe a vehicle that is subjected to arbitrary longitudinal accelerations, but ignores any out of plane or lateral forces. For an acceleration simulation, this model is sufficient to describe the vehicles performance with the assumption that the car is driven in a straight line.

The final model that will be briefly used to process braking simulations is the two-track model. Where the bicycle model ignores lateral forces, the two-track model accounts for them, resulting in a system with four wheels and a CG position described in three dimensions. This model allows analysis of cornering loads, steering geometry, and lateral acceleration. A

20 schematic of the two-track model from Wielitzka et al.[3] is shown below in figure 2.3.

Figure 2.3: The two-track model allows for forces at each tire in all directions as well as steering and slip angles. The two-track model can be used to predict the impact of vehicle suspension properties and transient cornering.

The two-track model will be used to decompose the results of lap-time simulation into estimated normal forces and traction forces on each wheel. This data will be used for regenerative braking estimates. Due to the complexity of a full two-track model, I will not use it to estimate cornering and lateral acceleration performance of a 4WD vehicle.

3 Acceleration Analysis

3.1 Summary of Benefits and Approaches

At the beginning of this thesis, I guessed that with simple intuition one could reason that the acceleration of a 4WD vehicle could be as much as twice that of a RWD vehicle. While this is untrue, it is a statement worthy of further investigation. In the following sections, I will develop a bicycle model of a RWD and 4WD vehicle with and without the presence of tire load sensitivity (also called saturation). These four models will then be compared to see how much acceleration they can sustain at zero starting velocity. The impact of the CG position on this maximal acceleration will be discussed. The effects of aerodynamic

21 loading on a vehicle will then be introduced to show how the acceleration of these four models vary with increasing velocity. This data will be represented in Acceleration-Velocity Phase Space (AVPS) which will be introduced as a useful visualization and optimization technique. The application of AVPS will introduce the concept of Weak and Strong 4WD architectures as well as define the desired torque-speed curve for the front and rear motors. Finally, acceleration simulations will be carried out with the use of AVPS models to identify the benefit of 4WD with regards to the Acceleration competition in FSAE Electric. During this investigation, many car parameters will vary. Unless otherwise stated, the following table will be used as the standard vehicle to calculate performance. These values represent the current MY2018 RWD car and are a reasonable starting point for future vehicles. Table 1: Standard MY2018 Car Parameters

Parameter Symbol Value Unit Mass (vehicle + driver) m 288 kg Tire Peak Longitudinal µ 1.83 Unitless Coefficient of Lift times Area CLA 3.83 m2 Coefficient of Drag times Area CDA 1.18 m2 Wheelbase W 1.524 m Horizontal CG position from front tire X 0.838 m Vertical CG position from ground Z 0.254 m

3.2 Bicycle Models

Rear-Wheel Drive

The beginning of an acceleration analysis starts with the development of the simplest bicycle model; that of a RWD vehicle with ideal tires that exhibit a constant friction factor regardless of normal force. This model does not include any aerodynamic forces, rolling resistance, or powertrain details. It would not be out of place on a first-year statics blackboard. The free-body diagram figure 3.1 below will guide the solution to the acceleration of such a vehicle.

22 Figure 3.1: The above free-body diagram shows a bicycle model with rear reaction forces Rbz and Rbx, and front reaction forces Rfz and Rfx. The weight of the car mg is applied at the CG location which is X meters rearward of the front tire and Z meters above the ground.

This system can be described by the following equations of motion (1), (2), and (3).

X Fx = Rfx + Rbx = max (1)

X Fz = Rfz + Rbz = mg (2)

X Ty,cg = RfzX + RfxZ − Rbz(W − X) + RbxZ = 0 (3)

The above equations assume that the system is quasi-static in the Z and θZ directions or that the suspension is infinitely stiff and exhibits no dynamics (finite stiffness, inertia, or damping). In order to solve the above system of 3 equations and 5 unknowns, we must introduce two constitutive equations for RWD to govern the relationship between the tire normal force and traction force. These relationships are given below by equation (4).

Rfx = 0,Rbx = µRbz (4)

Solving the above equations (1) through (4) for Rbz yields the following expression (5).

23 mgX R = (5) bz W − µZ

Given this value of Rbz, the other unknowns Rbx,Rfz, and ax can be solved with the equations (6), and (7).

Rfz = mg − Rbz (6)

µmgX R = µR = (7) bx bz W − µZ

Defining Fx to be the total applied force equal to Rbx + Rfx, we see that in this case

Fx Fx = Rbx. The acceleration achievable (ax) with this total force is simply m . To put this

ax acceleration in everyday terms, the metric Gx = g will be used to describe how many times Earth’s gravity the vehicle can accelerate at. Carrying this out, the RWD vehicle results in the following analytical expressions (8).

µgX µX a = ,G = (8) x W − µZ x W − µZ The following table 2 shows the results of using this model to evaluate the standard car described above in table 1. All values are rounded to three significant figures for clarity. Table 2: RWD Performance of the Standard Car

Variable Value Unit

Rbz 2240 Newtons

Rfz 589 Newtons

Rbx 4090 Newtons

Rfx 0 Newtons

Fx 4090 Newtons m Ax 14.2 s2

Gx 1.45 unitless g

In this model, a staggering 72% of the weight of the vehicle ends up on the rear wheels,

24 and produces a massive amount of traction. This vehicle is capable of accelerating at 1.45G from a standstill which is very impressive. However, while this specific example is interesting, the general case solution offers much more intuition. Using the analytical solution for Rbz and the other parameters, it is possible to determine immediately how the vehicle’s performance varies with µ, X, Z, and m. However, this solution does have its limits. If the CG position is too far backward or forward, the vehicle will tip. Given the stability criteria of Rbz ≥ 0 and Rfz ≥ 0, we can plot all possible accelerations over the a wide range of CG positions all the way from fully forward (X = 0) to fully backward (X = W ). The acceleration of the vehicle is shown by CG position in figure 3.2.

Figure 3.2: Contour plot of Gx as a function of CG location normalized to W . The peak acceleration increases with X and Z and reaches a maximum of about 1.6g on the tipping line.

However, the requirements of proper handling and lateral stability (going around a corner without tipping) restrict the reasonable range of CG locations to ≈40% and ≈60% of the wheelbase length (W). Values of X close to 0 or W can result in poor braking performance, bad cornering, and instability. Extremely high Z values stress the suspension, causing large

25 vehicle motions, and could result in tipping during tight cornering. The available acceleration in the reasonable range of CG locations is shown below in figure 3.3.

Figure 3.3: Contour plot of Gx as a function of CG location normalized to W . The achievable accelerations X within this range are very sensitive to changes in CGx = W and increase with Z.

These plots contradict the common wisdom that the CG location of a racecar should be as low as possible. While for cornering and braking, high CG location present problems, when only considering acceleration, a high CG location can be desirable.

Four-Wheel Drive

A similar bicycle model can be developed for a 4WD vehicle with the same assumptions used above. Revisiting the above free body diagram in figure 3.4, it is clear that the equations of motion are the same, but the constitutive tire model must change.

26 Figure 3.4: The above bicycle model can predict 4WD performance.

The three equations of motion that describe this system are the same as (1), (2), and (3) but the constitutive relations of the tires must be changed to equation (9).

Rfx = µRfz,Rbx = µRbz (9)

With the assumption that the tires on the front and rear have the same frictional prop- erties. This new set of constitutive equations results in a new forumula for Rbz in equation (10).

mg(X + µZ) R = (10) bz W

After applying equation (6) and (9) we can solve for Rfz, Rbx, and Rfx. Then Fx can be

calculated to find ax and Gx. The algebra results in the surprisingly simple expression (11).

ax = µg, Gx = µ (11)

However, this trivial answer is to be expected. In the ideal 4WD model presented, all of the weight of the vehicle goes into normal forces on the tires which directly produce traction. This is just as simple as a point-mass with the expression F = µmg and A = µg. In this ideal limit, any value of Z and X that does not tip will result in the highest possible acceleration

27 available at the tires. While the complete independence of CG location will prove to be false, the notion that a 4WD vehicle will be less sensitive to CG position than a RWD vehicle is firmly established.

3.3 Tire Load Sensitivity

Unfortunately, our model is rather idealistic as it ignores one of the most critical deviations from reality. Tires do not act as constant µ devices. They exhibit lots of non-linear behaviors depending on their orientation, temperature, pressure, and slip ratio relative to the ground. Abstracting all of the tire modeling away and assuming that the suspension is well setup to maintain proper tire contact patch, the tire doesn’t overheat or become underinflated, and that the optimal slip ratio is maintained at all times, we can simply say the tire exhibits load sensitivity. Essentially, the friction factor of the tire is dependent on normal force, and in fact decreases with increasing load. Using a Pacejka tire model, my fellow teammates Luis and Cheyenne extracted the following tire load sensitivity for our tires (Hoosier 18x6R25b). The original test data for this tire was collected at the Calspan TIRF and supplied to the team by the Tire Testing Consortium.

Figure 3.5: Available friction in longitudinal (x) and lateral (y) directions given applied normal force. The curve is roughly linear in this region. The Pacejka tire model is fit to data that goes up to 1100 Newtons. The −4 longitudinal load sensitivity is well described by µ(N) = µ1 − µ2N with µ1 = 1.831 and µ2 = 3.141 × 10 .

28 While it is very convenient that the tires display linear behavior in this region, a linear fit to a more complicated function can also be used. If the function displays significant non-linear behavior a quadratic fit is still usable.

This deviation from the ideal tire will then govern the development of two new models, the Load Sensitive RWD and 4WD bicycle models (LSRWD and LS4WD respectively). The dependence on loading will cause the model to result in a quadratic expression for Rbz and other parameters but is still very analytically tractable. However, since the bicycle model assumes one tire, the load sensitivity parameter µ2 must be divided by 2 to account for the performance of two tires, each of which sees half the applied load. Thus the value of mu2 used will be 1.571 × 10−4.

3.4 Load Sensitive Rear-Wheel Drive (LSRWD)

Returning to the equations of motion for the RWD bicycle model, the constitutive model of the tires must be adjusted again. The result is given below as equation (12).

Rfx = 0,Rbx = Rbz(µ1 − µ2Rbz) (12)

When Rbz is calculated, the load appears on both sides of the equation and further algebraic manipulation results in the following quadratic equation (13).

2 Rbz(µ2Z) + Rbz(W − µ1Z) + (−mgX) = 0 (13)

As a quick check, equation (13) collapses down to (5) when µ2 = 0. Once the value of Rbz is established and the vehicle is confirmed not to tip, the rest of the parameters can be calculated as before with equations (6) and (12). The resulting Fx, ax, and Gx are not visually tractable because they rely on the solution to quadratic equation above. The following table 3 shows a comparison between the ideal RWD model and the new LSRWD model in terms of forces on the standard car. All of the variables are rounded to three significant figures except for the percent differences.

29 Table 3: RWD and LSRWD Comparison

LSRW D−RWD Variable RWD model LSRWD model Percent Difference RWD

Rbz 2240N 2080N −7.0%

Rfz 589N 746N +27%

Rbx 4090N 3160N −23%

Rfx 0N 0N -

Fx 4090N 3160N −23% m m Ax 14.2 s2 11.0 s2 −23%

Gx 1.45g 1.12g −23%

The reduction in µ has a dramatic effect on the vehicle performance, resulting in a −23%

change in Gx for the same standard car. In the first row of the table, it can be seen that the reduced traction changes the torque balance established in equation (1) and reduces the normal force available at the rear tires. While the normal force Rbz decreases by only −7.0%,

Rbx the effective coefficient of friction (µeff = ) drops as well resulting in a −23% change in Rbz

Rbx. In this case µeff is equal to 1.52 instead of 1.831.

The performance of a LSRWD vehicle is also heavily dependent on CG location. The

following figure 3.6 shows the available acceleration ax over a wide range of CG positions:

30 Figure 3.6: A contour plot of available acceleration Gx as it relates to normalized X and Z values. As the CG position moves upward and to the rear, the available acceleration increases until a value of about 1.3g before the vehicle tips.

In the useful range of X = 0.4 − 0.6W the following figure 3.7 shows what realistic accelerations can be attained:

31 Figure 3.7: A contour plot of available acceleration Gx as it relates to normalized X and Z values. To achieve high accelerations in this range, the CG needs to be as far back as possible and as high as allowable. The acceleration is very sensitive to changes in CG position and 0.05 change in CGx can have ≈ 10% in Gx.

3.5 Load Sensitive Four-Wheel Drive (LS4WD)

The next logical step is the development of the LS4WD model which includes a final variation of the constitutive friction equations given below in equation (14).

Rfx = Rfz(µ1 − µ2Rfz),Rbx = Rbz(µ1 − µ2Rbz) (14)

Armed with this new set of equations, we return to (1), (2), and (3) to solve for Rbz. The result is another quadratic equation (15).

2 2 Rbz[2µ2Z] + Rbz[W − 2µ2Zmg] + [−Xmg − µ1Zmg + µ2Z(mg) ] = 0 (15)

A quick check indicated that when µ2 is set to zero, this equation collapses down to (10). This can be solved and used to determine the other variables. The comparison between the

32 LS4WD and 4WD models is shown in table 4.

Table 4: 4WD and LS4WD Comparison

LS4WD−4WD Variable 4WD model LS4WD model Percent Difference 4WD

Rbz 2420N 2280N −5.7%

Rfz 410N 547N +34%

Rbx 4420N 3390N −23%

Rfx 750N 957N +28%

Fx 5170N 4350N −16% m m Ax 18.0 s2 15.1 s2 −16%

Gx 1.83g 1.54g −16%

The LS4WD model provides 16% less acceleration than the 4WD model, but still main- tains a higher acceleration than both the RWD and LSRWD models. Interestingly, the load sensitivity greatly changes the weight distribution of the vehicle with a large percent increase

in Rfz, but little change in Rbz. Since the front wheels aren’t highly loaded compared to

Rfx the rears, they produce a high µeff = of 1.75 and contribute approximately 22% of the Rfz traction with only 19% of the normal force. This suggests that LS4WD vehicles have large advantages over LSRWD vehicles: they can efficiently use the small amount of normal force available in the front to produce traction.

In section 3.2 when I modeled the idealistic 4WD case, I claimed that a 4WD vehicle should exhibit low sensitivity to CG placement compared to a RWD vehicle. With the final LS4WD model above, we can see how the non-ideal load sensitivity impacts CG placement.

The following figure 3.8 indicates the available acceleration Gx over a wide range of CG locations.

33 Figure 3.8: A contour plot of available acceleration Gx as it relates to normalized X and Z values. The LS4WD exhibits substantially higher acceleration than the LSRWD model at a wide variety of CG positions. The plot indicates an optimal region of maximum acceleration that is stable and far from the tipping line. The LS4WD model is less sensitive to Z height and small changes in X than the LSRWD model.

Focusing on the reasonable range of X = 0.4W − 0.6W , it is easy to see the benefits of a LS4WD vehicle compared to a LSRWD vehicle in figure 3.9.

34 Figure 3.9: A contour plot of available acceleration Gx as it relates to normalized X and Z values. The lowest available acceleration in the reasonable range of the LS4WD model (1.425g) is higher than largest possible acceleration for the LSRWD model (1.3g). The LS4WD model also favors CGx locations that are further forward (X closer to 0) which gives more packaging freedom to the designer. In this region, decreasing the CG height increases acceleration, contrary to the LSRWD model.

The advantages of a 4WD architecture over RWD architecture are substantial when it comes to CG placement. In the ideal case of 4WD, CG location simply doesn’t matter, but in the ideal RWD case it must be kept within a narrow region to get peak performance without tipping. These fundamentals influence the more complicated LSRWD and LS4WD models which offer a more realistic description of vehicle performance. To visualize the differences between the four models, the following figure 3.10 takes a constant Z = 0.25m and sweeps across X = (0 − 1)W to reveal the underlying behavior of the models:

35 Figure 3.10: The available acceleration Gx is plotted at a constant Z = 0.25m while X is swept across the feasible range. The two RWD models show almost linear increases in acceleration until the vehicle tips, while the two 4WD models show almost constant acceleration until the vehicle tips. The superiority of a 4WD system is very clear with a CGx ≈ 0.5. The stable maximum region where 1.6g is attainable is also clearly visible in the LS4WD model.

3.6 Aerodynamic Forces

The four models above have been generated with the assumption that the only load on the vehicle is the force of gravity acting on the vehicles mass. On a real vehicle operating at speeds up to 70 mph, aerodynamic loading plays a critical role in the acceleration of a vehicle. Racecars are typically designed with inverted wings to apply additional force downwards on the vehicle while adding a small amount of aerodynamic drag. The figure 3.11 below shows the MY2017 vehicle with its aero package with force vectors added for clarity.

36 Figure 3.11: MY2017’s front and rear wing both generate lift and drag forces on the vehicle. The vehicle body provides the majority of the drag for the system. Photo Credit John Burchardt.

While it is clear that extra aerodynamic force will be applied downwards and backwards to the vehicle, it is not obvious where these forces end up being applied. With Computational Fluid Dynamics, the center of pressure for a wing element can be located. Each airfoil can then be considered as applying a drag force and down force at a specific point in space usually around 0.25-0.5 chord lengths from the front of the wing. Abstracting this idea further, multiple airfoils can be considered to act on a single total center of pressure at a point. The location of the vehicle-wide center of pressure plays a similar role to the CG location as it determines how the aerodynamic forces are split between the tires and how aerodynamic drag causes the vehicle to pitch backwards. Unfortunately, detailed discussion of racecar aerodynamic design is not within the scope of this thesis nor the author’s expertise. When aerodynamic forces are considered in the next section, I will assume that the center of pressure for the vehicle is located at the CG location of the vehicle. Optimal and controllable Center of Pressure location is definitely worth further consideration for top vehicle performance. Top European teams like TU Stuttgart’s Rennteam use active aerodynamic elements to dynamically change the aero center of pressure as well as downforce and drag during the race.

With the Center of Pressure discussed, the following figure 3.12 introduces aerodynamic forces into the bicycle model.

37 Figure 3.12: The bicycle model is modified to include Fl and Fd.

The following modified equations (16) through (18) dictate quasi-static mechanical equi- librium for the model above.

X Fx = Rfx + Rbx − Fd = max (16)

X Fz = Rfz + Rbz = mg + Fl = Fz (17)

X Ty,cg = RfzX + RfxZ − Rbz(W − X) + RbxZ = 0 (18)

Given the specific model being used (RWD, LSRWD, 4WD, or LS4WD) the additional aerodynamic forces will provide different amounts of additional acceleration as a function of velocity. While the exact values of down force and drag are also a target for detailed analysis and optimization, I will used the following equations (19) and (20) and values for further discussion.

38 1 F = ρ CDAV 2 (19) d 2 air

1 F = ρ CLA V 2 (20) l 2 air

For the values of the standard car in table 1, CLA = 3.83, CDA = 1.18 and ρair = 1.2 in the temperature range the car operates in.

In order to appreciate the impact of aerodynamic forces, we must compare the accelera- tion of the four models as a function of velocity. The idealized RWD and 4WD models should exhibit the most benefit from an aero package, while the LSRWD and LS4WD models will still improve despite the load sensitivity of the tires. The four models response to increased velocity is plotted below in figure 3.13.

Figure 3.13: The acceleration available at the standard CG location is plotted as a function of velocity V for the four models. All of the models show increased acceleration at increased velocities, but the load sensitive models lag behind the ideal models. The difference between the LS4WD and LSRWD models is quite large at high speeds when load sensitivity greatly reduces the µeff on the rear wheels.

39 Aerodynamic packages on racecars are very effective at producing large amounts of down force at high speeds. Highly load-sensitive tires are not able to take advantage of the increased down force, so spreading the normal force evenly over the front and rear of the vehicle allows

higher µeff to be reached with a 4WD architecture. 4WD vehicles are able to take advantage of the additional normal force on the front tires to make large increases in total acceleration.

3.7 Preliminary Front-Rear Power Split

m mi The MY2017 vehicle attained a top speed of =32 s or ≈ 70 h during testing and it is assumed that the MY2018 vehicle will be able to do the same. It is obvious that the traction-limited acceleration of the vehicle only increases with speed, offering great benefits to a straight-line acceleration competition. In the case of the 4WD and LS4WD vehicle, we can determine what portion of the available acceleration comes from the front and rear wheels. In this case, the wheels share the drag force ratiometrically and create traction corresponding to

the forces Rbx and Rfx on each wheel. The decomposition of the contributions of the front and rear wheels allow the designer to gauge the relative motor size required on the front and rear. The total and separate contributions for the 4WD and LS4WD models are shown below in figure 3.14.

40 Figure 3.14: The available acceleration ax from 4WD and LS4WD models is plotted against velocity. Both models use the standard CG location. The individual components of the acceleration are broken out to see how they change with increasing downforce. The LS4WD model predicts that the front wheels play a large role in the generation of traction.

The importance of this curve is incredible. This curve dictates the available acceleration delivered by the front and rear of the vehicle as a function of velocity and is therefore a key input to proper powertrain sizing. Since acceleration can be related to force and torque, this curve is almost a scaled version of the torque-speed curve required at the wheels for a LS4WD vehicle. The detailed torque-speed curves will be developed later after power limits are introduced, but they can be guessed at now. Since both curves give acceleration by velocity, the ratio of accelerations is approximately the same as the ratio of torque needed, and power required. The ratio between front to rear power is given in figure 3.15 below as a function of velocity.

41 Figure 3.15: The fraction of the total power that is on the front and rear as a function of velocity. The unitless ratio of the front fraction to the rear fraction shows the relative size of the front powertrain to the rear.

1 1 This demonstrates that for peak acceleration, the front axle only needs to provide ≈ 3 − 2 the power and torque of the rear axle. This is an extremely important number to have when selecting a drivetrain. Motor weight and cost both grow quickly with peak power ratings, and choosing the smallest motor required for the performance needed is a key strategic design decision. The above discussion has been couched with “almost” and “approximately” because the acceleration curves do not show the extra force required to overcome aerodynamic drag forces. The actual forces Rbx and Rfx are the proper curves to use to evaluate motors and they will be shown later.

3.8 Practical Limits to Acceleration Curves

Returning back to the four curves in the Acceleration Velocity plane, it is clear these curves cannot allow arbitrarily high acceleration at arbitrarily high velocities without encountering

42 other physical vehicle limits. At very high speeds, the drag force becomes so large, that the drag power matches or exceeds the available motor power. The expression for top drag- limited speed is given below in equation (21) with the assumption that aerodynamic drag dominates rolling resistance, bearing viscosity and other sources of loss.

1 P = F V = ρ CDAV 3 (21) drag drag 2 air

Since the maximum rules allowed electrical power Pmax = 80kW, the drag limited velocity q 3 2Pmax m for the standard car should be Vmax = = 48.3 . ρairCDA s

In the limit of Vmax, all of the available electrical power is perfectly converted into me- chanical energy which then solely goes into overcoming aerodynamic drag. However, the vehicle is limited to 80kW peak at all times, not just at Vmax, and the acceleration available will be affected before the top speed is reached. In order to visualize and understand the ef- fect of the power limit on vehicle performance I must introduce Acceleration-Velocity Phase Space (AVPS) more formally.

3.9 Acceleration-Velocity Phase Space

Let us consider a simple vehicle RWD, 4WD or otherwise that exhibits no aerodynamic effects. This vehicle has an infinitely capable powertrain, no power limit or sources of loss. It is purely governed by tire traction and is always capable of sustaining an acceleration of

m ax = 9.81 s2 . This vehicle’s range of possible behaviors can be displayed in AVPS as the following figure 3.16

43 m Figure 3.16: This simple vehicle is able to exhibit a constant acceleration of ax = 9.81 s2 . It can reach arbitrarily high velocities given enough time. This model is purely traction limited.

This model is very simple, but still allows us to predict the vehicle’s performance at the FSAE Acceleration event. The acceleration event metric is the time it takes to get from 0-75 meters. Integrating the constant acceleration produces linearly increasing velocity of the form V (t) = 9.81t. Integrating that velocity provides the position of the vehicle over time X(t) = 4.905t2. While the math is very simple in this case and can be integrated by inspection, more complicated curves in AVPS will require numerical simulation in order to be solved in a reasonable amount of time. I have numerically integrated ax to plot V (t) and X(t) for the above case to give the results in figure 3.17.

44 Figure 3.17: The acceleration limited vehicle√ model in AVPS is numerically integrated to get V (t) and X(t). This vehicle should reach 75 meters in 15.29 = 3.910 seconds.

In this model, the only way to improve performance is to increase the value of the q constant acceleration. The time to 75 meters will be given as T = 2∗75 and thus finish ax

further increases in ax result in diminishing returns.

However, this model is clearly simplistic and allows arbitrarily high velocities to be at- tained. The next realistic limit to consider is a maximum velocity as limited by a motor, gear ratio, and tire radius. Electric motors cannot spin up to arbitrarily high speeds and therefore regardless of aerodynamic drag, an electric vehicle’s top speed will always have a

m powertrain limit. Assuming that Vmax = 30 s , the description of the car’s performance is

now ax = 9.81 for V < Vmax and ax = 0 for V = Vmax. The new plot in AVPS now encloses an operating region with respect to the A and V axes shown in figure 3.18.

45 Figure 3.18: The acceleration and speed limited vehicle model in AVPS represents a rectangular operating region with the assumption that ax and V are both positive. Every possible longitudinal motion of the car is limited to within the bounded region. The highest performance of the car is defined by the limiting curve which rides along the edge of the boundary region. The limiting curve (black dashed line) does not perfectly follow the motor speed limit because discrete quantities of V are used.

This vehicle will now accelerate at a constant rate until it hits V = Vmax. After that point the acceleration drops to zero and the vehicle continues at constant speed. Integrating this function will reveal a linearly increasing velocity until the limit is hit. The position as a function of time for this vehicle will begin quadratically and then remain at constant slope once the vehicle is speed limited. The results of the numerical integration are shown in the figure 3.19 below.

46 Figure 3.19: The acceleration and speed limited vehicle model in AVPS is numerically integrated to get V (t) and X(t). This vehicle will become speed limited at T = 3.058 seconds and will reach the finish line in 4.029 seconds, 0.119 seconds slower than an acceleration limited vehicle.

It is clear that the acceleration and speed limited vehicle will perform worse than the acceleration limited vehicle. However, in the case that Vmax is higher than V (t = tfinish) for the acceleration only case, the speed limit doesn’t matter. In the above example, if the

m limiting speed were greater than 38.73 s , its value would be irrelevant over the course of 75 meters. This discussion assumes that the powertrain is capable of any acceleration up until it reaches its maximum speed. Even in the idealistic case of a speed-limit powertrain, the importance of selecting the proper gear ratio becomes readily apparent. If the gear ratio is too high, the maximum speed of the car will be too low, and acceleration performance will suffer. However, even this acceleration and speed limited model leaves much to be desired. In our competition, we must obey a rules mandated (and empirically measured) 80kW electrical power limit. Assuming 100% electrical-mechanical conversion efficiency (drivetrain efficiency is actually closer to 95% for the motor, 99% for the inverter, and 99% for the drivetrain) the

47 mechanical power exerted on the vehicle must always be less than 80kW. Mechanical power

use can be described as P = FV , or with F = max, P = maxV . Converting this into AVPS,

Pmax a constant power limit, given a car weight appears as an acceleration limit of ax = mV this

Pmax can be simplified with the knowledge that m is simply the vehicle’s peak power to mass PTMR ratio (PTMR). Such that ax = V . This new bound on the vehicle performance will show

up as a hyperbola through AVPS. The acceleration of the car will now be ax = 9.81 for PTMR V < Vmax, and subject to the limit that ax ≤ V . This new car with its three limits is shown in AVPS below:

Figure 3.20: The acceleration, speed, and power limited vehicle model in AVPS represents a rectangular operating region with a hyperbolic notch removed from the top corner. Every possible motion of the car is limited to within the bounded region described by the dashed limiting curve. This vehicle has a very w low PTMR of 178 kg representing a vehicle capable of 80kW with a mass of 450kg. The standard car has a w PTMR of 278 kg which is so high that the hyperbola would not intersect the rectangular region.

In this situation, the car now follows three separate limits during three modes of oper- ation. At low velocities, the acceleration is traction limited by the tires, and is well below the power limit. At higher velocities, the vehicle is power limited, but has excess traction it could use. During this region acceleration starts to decrease hyperbolicly with increasing

48 velocity. Finally, at top motor speed, the vehicle cannot sustain any more acceleration. To visualize the velocity profile of the above AVPS limits requires a numerical simulation, the output of which is given in figure 3.21 below.

Figure 3.21: The acceleration, speed, and power limited vehicle model in AVPS is numerically integrated to get V (t) and X(t). The velocity first increases linearly, then along a constant power curve, and finally reaches a constant maximum value. This vehicle reaches the finish line in 4.078 seconds: 0.049 seconds slower than the previous model.

As the operating region defined by the limiting curve gets smaller and smaller, the ve- hicle’s performance decreases and the time to 75 meters increases. AVPS and numerical simulations based on it offer a very simple way to visualize and quantify the limits to vehicle performance in the acceleration event. The usefulness of AVPS is that it lends itself easily to numerical simulation of straight-line acceleration events with minimum effort. Any number of limits of arbitrary complexity in AVPS can be then used to generate a limiting curve (the minimum available acceleration at a given velocity) which can then be fed to a numerical solver. Instead of building an acceleration simulator from numerical timesteps and Newton’s equations as done by many other teams (well documented by University of Wisconsin cour-

49 tesy of Max Liben), AVPS simulations allow vehicle limits to be handled at the high level. The utility of AVPS is only limited by the models that generate the various limiting curves and can even be fit to empirical data. In the three-limit model above, it is clear that the vehicle reaches maximum velocity before crossing the finish line. This suggests that the gear ratio should change such that the vehicle just barely hits maximum speed as it crosses the finish line, assuming the powertrain has enough torque to provide the necessary acceleration at all other speeds. Since the above

m vehicle is power-limited above V ≈ 17 s it is in the power limited regime for about half of the acceleration run. Taking efforts to reduce mass would help move the PTMR curve up and to the right and enable higher performance. The next section will focus on fine tuning of the AVPS models for the LS4WD standard car to see what the limits of performance are for such a vehicle and its LSRWD counterpart. This will allow us to determine the increased performance of the 4WD architecture in the acceleration event.

3.10 Fine Tuning AVPS Models

The first major change I will make to the simple three-limit AVPS model described in the previous section, is replace the simple constant acceleration assumption with the aerody- namic LS4WD traction model developed in section 3.5. The increased acceleration available at low velocities means that a car with aerodynamics will accelerate faster and therefore reach the power limit earlier and at a lower velocity. The limit of this argument is that a vehicle with arbitrarily high traction will spend all of its time in the power limited regime. It will also be seen that extremely high CLAs which allow monstrous amounts of down force at high speeds do not offer increased vehicle performance once the vehicle is power limited. Furthermore, the simple 80kW mechanical power limit must also be augmented to account for the aerodynamic drag power discussed above in equation (19). The ideal 80kW limit will drive the vehicles acceleration asymptotically to zero at large velocities, while the addition of aerodynamic drag causes the acceleration to reach zero at finite velocity when all 80kW of mechanical power are used to resist aero-drag at Vmax. The new combined power and aero drag limit is given below as equation (22).

50 P P − P 80000 − 1 ρ CDAV 3 a = mech = max drag = 2 air (22) x, power limit mV mV mV Assuming the standard car as described in table 1 and using a speed limited powertrain

m (that can achieve any acceleration up to a higher and more realistic Vmax = 38.8 s ), we get the following curve in AVPS, shown below in figure 3.22.

Figure 3.22: The LS4WD, power, drag, and speed limited model in AVPS is defined by an oddly shaped boundary region. The available acceleration quickly increases almost quadratically (due to aerodynamics) at low velocities until the power limit is hit. Above this speed, the vehicle rides down an almost pure power limit before the drag contribution to the power limit starts to dominate and drives the acceleration down almost linearly. Finally, the vehicle hits a maximum motor speed limit before reaching the drag speed limit. The ideal power limit is shown to emphasize the massive effect of aero drag at high speeds.

The more complex AVPS model presented above offers strong intuition about vehicle limits. The extremely high traction limit achievable at high speeds simply doesn’t matter

m since the power limit cuts in at V ≈ 16 s . While the power limiting curve starts out steep, it tapers off hyperbolically. Unfortunately, the drag contribution to the power limiting drastically reduces available acceleration at close to top speed. If the vehicle had a drag

m coefficient (CDA) of zero, the vehicle would be accelerating at ≈ 7 s2 when it reached its

51 motor speed limit, but due to aero drag it only reaches about half that acceleration. My current model assumes that the vehicle has an unlimited powertrain that is characterized by a maximum speed. If this is not true, than another limiting curve representing the power-train will replace the speed limiting curve and add even more complexity. The torque-speed curve of a motor with the appropriate adjustments for inertia and gear ratio can be translated into AVPS to aid the designer. This graph suggests that increasing the PTMR of the vehicle and reducing drag at high speeds while retaining downforce at low speeds will do the most to increase the operating envelope of the vehicle. The results of this AVPS model were numerically simulated and are plotted below in figure 3.23.

Figure 3.23: The LS4WD, power, drag, and speed limited vehicle model in AVPS is numerically integrated to get V (t) and X(t). The acceleration run of this vehicle is characterized by high initial acceleration which peaks at t ≈ 1 second. From there, the vehicle is power and drag limited and accelerates at a slower and decreasing rate for the rest of the run. The vehicle does not hit its motor limited speed during this run. The finish time of this vehicle is a blistering 3.409 seconds.

While this model seems encouraging on its own, it must be compared to the LSRWD model which represents the current potential of the MY2018 vehicle. The AVPS model for the LSRWD standard car is shown below in figure 3.24.

52 Figure 3.24: The LSRWD, power, drag, and speed limited model in AVPS is defined by an oddly shaped boundary region. The available acceleration begins at a point lower than the LS4WD model and increases almost quadratically at low velocities but at a reduced rate compared to LS4WD. The LSRWD model enters the power limited region at a higher velocity than the LS4WD model and therefore spends more time in the traction limited region. The power and speed limits are the same for the LSRWD model and it achieves the same acceleration as the LS4WD model as it follows the power and drag curve towards its maximum motor speed.

The LSRWD model is expected to exhibit reduced acceleration in the beginning of the run and enter the power limited region later in time and at a higher velocity. From there, it will travel down the same velocity profile that the LS4WD model does, but with the disadvantage of getting there later in time. The AVPS simulation results for this model are shown below in figure 3.25.

53 Figure 3.25: The LSRWD, power, drag, and speed limited vehicle model in AVPS is numerically integrated to get V (t) and X(t). The acceleration run of this vehicle is characterized by low initial acceleration which peaks at t ≈ 1.75 seconds. From there, the vehicle is power and drag limited and accelerates at a slower and decreasing rate for the rest of the run. The vehicle does not hit its motor limited speed during this run. The finish time of this vehicle is increased to 3.748 seconds, 0.339 seconds longer than the LS4WD model.

While the original bicycle model analysis suggested that the LS4WD model offered sub- stantial benefits over the LSRWD model, the simulation results may seem disappointing. While the LS4WD models do offer 1.54g of acceleration at a standstill instead of 1.12g, this 37.5% increase in acceleration only results in a 0.339 second or 9.0% reduction in accelera- tion time. This startling discovery shows that while the initial assumption that the car is dominated by Fx = max is true for the first 1 or 2 seconds of the acceleration run, PTMR and drag limits play a major role in the vehicle’s performance and are not influenced by a 4WD vs RWD architecture. However, this does not mean that a 4WD architecture isn’t useful. A 9.0% reduction in acceleration time is an enormous gain and is equivalent to mov- ing multiple ranking positions in the stiff competition in Europe where acceleration trophies are decided by 1% margins. At the 2017 Formula Student Germany (FSG) competition, the winning acceleration time was 3.528 seconds. A 9% longer time would put you in 6th place.

54 While this AVPS simulation is a best case scenario that ignores many sources of loss, the differences predicted between models are still worth considering.

3.11 Preliminary Acceleration Takeaways

Given the above analysis, I can argue that the ideal LS4WD and LSRWD models demon- strate that a 4WD architecture can accelerate as much as 37.5% faster than a RWD vehicle. The 4WD architecture also gives much more flexibility in CG location, and better use of aerodynamic downforce. To achieve maximum acceleration, the front motors need to be

1 1 sized for ≈ 2 − 3 the peak torque and power of the rears. The result of AVPS simulations indicate that the same 0-75 meter time can be had with a 288 kg LSRWD vehicle or a 450kg LS4WD car, meaning that as long as the front wheel powertrain and associated costs in vehi- cle mass are lower than 162kg, the system will perform better. This staggering improvement in timing suggests that acceleration time is very hard to improve with weight reductions alone. More in depth analysis of the tradeoff between weight and 0-75 meter time will be presented later.

3.12 Power Split in the Power Limited Region

When the acceleration as a function of velocity curves were generated for LS4WD and LSRWD bicycle models, we had yet to introduce the power limiting curves in AVPS. Clearly, once a 4WD vehicle enters the power limited region, it has excess traction and is no longer required to operate the front and rear powertrains at their traction limit. In the traction limited region, there was only one solution for the maximum acceleration, but in the power limit regime, there are now multiple solutions for the front-rear power split that each give the maximum acceleration determined by the power limit. Even though there are multiple solutions, how should one choose the power split in the power limited regime? In the University of Wisconsin’s acceleration modeling document[4], Max Liben proposes a maximum efficiency power split. The proposed split optimizes for vehicle efficiency by trying to minimize the combined joule losses of the front and rear drivetrain. This splitting criteria was very important for the University of Wisconsin because

55 their vehicle operated at maximum bus voltage of 140 volts and therefore would draw up 571 amps to reach 80kW at full charge and ≈ 1000 amps at end of race conditions resulting in massive I2R losses. With a 300-volt pack, our currents are approximately half that of Wisconsin and our I2R losses are one quarter of theirs. While there are many criteria one could use to choose a power split in this region, it would useful to know the maximum and minimum limits are.

3.13 Weak and Strong 4WD

In the above section I argued that multiple solutions exist to the power split within the power limited region. The goal is to determine the most extreme power splits that favor the highest use of the rear or front powertrain respectively. With the limit of the minimum front powertrain, a designer could source the smallest possible front motors that would still enable the full AVPS operating region to be used. In the other extreme, knowing the minimum rear powertrain would allow the designer to minimize the requirements of the rear motors in favor of the front. Either way, knowing the extremes of the rear and front powertrain provide valuable design insight and intuition. To understand the differences between the extremes, I will introduce the concept of the Strong and Weak 4WD architecture. The Strong 4WD architecture follows the most extreme power split that results in the maximal use of the front motors. When the Strong 4WD vehicle enters the power-limited regime, it will try to keep the front motors at the highest level of traction possible while slowly ramping down the power and torque going to the rear of the vehicle. The Weak 4WD architecture does the opposite. A Weak 4WD vehicle is essentially a RWD vehicle with the minimum possible front powertrain to achieve the higher acceleration promised by the LS4WD AVPS curves. When the Weak 4WD vehicle enters the power limited region, the front powertrain is quickly throttled down to draw less power while the rear powertrain is kept operating at the highest traction it can within the power limit. Since the majority of the traction is produced on the rear wheels, there will be a point that the Weak 4WD architecture can switch over entirely to RWD and simply let the front powertrain coast at zero net power. The same happens for the Strong solution where at some point, the vehicle can be entirely front-wheel drive. While both the Strong

56 and Weak 4WD system allow the same level of acceleration, in the traction limited and power limited regime, the Weak system has the minimum required front powertrain and the Strong system has the minimum required rear powertrain. Since motor cost, size, and weight increase with increasing peak power and torque requirements, the continuum between a weak and a strong architecture allows further optimization for the ideal powertrain vehicle-wide. When the development of a gear-reduction for outboard motors is considered, selecting a Weak architecture allows the minimum effort to reach the promise of higher acceleration. Now we must determine quantitatively what the limits on front and rear traction are in the power limit regime. At every speed, there is a given traction available on the front and rear of the vehicle. If the traction available at the front is scaled down by a quantity A such that 0 ≤ A ≤ 1 and the traction available at the rear is scaled down by a quantity B such that 0 ≤ A ≤ 1, we can introduce A and B as a modification to the constitutive equations for the LS4WD bicycle model with the result:

Rfx = ARfz(µ1 − µ2Rfz),Rbx = BRbz(µ1 − µ2Rbz) (23)

The equation (24) below solves for Rbz much like in equation (15) but with the addition

of A and B, and Fz from equation (17) representing weight and downforce instead of just mg.

2 2 Rbz[µ2Z(A+B)]+Rbz[W +µ1Z(A−B)−2µ2ZFzA]+[−XFz −µ1ZFzA+µ2ZFz A] = 0 (24)

Equation (24) is similar to (15), but has extra terms where previously A and B would

cancel, and has more complexity where A and B add. Now Rbz is a function of A and B which must be specified before the performance of the vehicle can be calculated. The plot of acceleration at zero velocity for a given A and B reveals a lot about the performance of a vehicle. At the point A = B = 0, no acceleration can be achieved since the traction has been “turned off” at both the front and rear. This would be equivalent to the car spinning out on ice or an infinitely slippery surface. The corner A = 1 and B = 0, describes the performance of a purely front-wheel drive vehicle. From intuition, it is clear that a front

57 wheel drive car will exhibit lower accelerations than a RWD car because the normal force on the front wheels is lower than the rear wheels when accelerating forward. The corner A = 0 and B = 1 describes a LSRWD vehicle. Finally, the corner A = B = 1 describes a purely traction limited LS4WD vehicle which unsurprisingly achieves maximal acceleration. However, the parameters A and B do not need to take on integer values. At the point A = 0.5 and B = 0.75, the vehicle is using 50% of the available traction on the front and 75% of the available traction on the rear. In this configuration it is not immediately clear how this system would compare to the four corner cases described above. At zero velocity, a contour

plot of traction force Fx versus A and B is given below in figure 3.26.

Figure 3.26: The LS4WD model is evaluated as a function of A and B at V = 0. Contours of constant Fx indicate the combinations of A and B that are possible to attain the same traction and acceleration. This shows the continuum between a front-wheel, rear-wheel, and four-wheel vehicle and will be used to identify the Strong and Weak limits.

As we can see, the contours are almost all linear, but the slope and intercept change rapidly. Since the vehicle is not power limited at zero velocity, the corner A = B = 1 will be used in the AVPS simulation. However, if the velocity is high enough such that the vehicle

58 is power limited, the Fx contour through AB space will trace out a set of valid A and B combinations that are not equal to 1. The end-points of the contour represent the extreme values of maximum and minimum A and B. Those extreme values define the limits of the Strong and Weak architectures. It is also clear that the requirements for the front wheel drive system fall faster than that of the RWD system, validating my claim that a Weak LS4WD system will end up with zero power requirement on the front and fully switch to a

m LSRWD system at some velocity. Figure 3.27 below is a contour at V = 20 s .

m Figure 3.27: The LS4WD model is evaluated as a function of A and B at V = 20 s . Contours of constant Fx indicate the combinations of A and B that are possible to attain the same traction and acceleration. Higher values of Fx are attainable at higher velocities and the shape of the contours changes.

m At V = 20 s the LS4WD standard car is power limited and therefore cannot operate at the corner point A = B = 1. Given the power limit, the maximum traction that can be generated is 3716N. The following Fx contour shows the limits of A and B that can match the limited traction allowed.

59 m Figure 3.28: The LS4WD model is evaluated as a function of A and B at V = 20 s . The contour of Fx = 3716N is shown as it represents the highest traction force available without exceeding the power limit.

At this speed, the Weak architecture would require that A ≈ 0.1 and B = 1 while the Strong architecture would require A = 1 and B ≈ 0.6. Thus the claim that the requirements of the front powertrain drop off quickly is validated. By extracting the maximum and minimum A and B values at every velocity, the Strong and Weak powertrain curves can be plotted throughout the AVPS region.

60 Figure 3.29: The Strong (solid) front (red) and rear (blue) lines describe the acceleration that results from a vehicle with the maximum front powertrain and the minimum rear powertrain. The Weak (dashed) front and rear lines describe the contributions from a vehicle with the minimum possible front powertrain. The plot must be read by looking the Strong pair or Weak pair of curves and evaluating the available options between those extremes.

This plot reveals the extreme divide between the Strong and Weak architectures. Tracing out the Strong curves, it is seen that the Strong Rear powertrain is heavily stressed up until

m the power limit, and then falls to 0 by 30 s . During this transition, the Strong Front starts to ramp up until it represents a completely front-wheel drive system and delivers all of the power required. In contrast, the Weak architecture requires very little of the front powertrain, and

m the front is effectively turned off at V = 20 s . The Weak Rear powertrain generates a large amount of traction before jumping to the boundary line. At this point, the Weak 4WD vehicle becomes rear-wheel drive all the way until its aero-limited top speed. While this plot demonstrates the difference between the Strong and Weak architectures, it is not ready for engineering use yet. Since this is the resulting acceleration, blindly taking these curves and multiplying by mass would give the inertial force required to propel the vehicle. However, this curve also includes losses due to drag, which requires additional force

61 to overcome. The actual forces developed, Rbx and Rfx, account for both inertial and aero terms.

Figure 3.30: The forces developed on the front of the vehicle and the rear of the vehicle are shown as split between the Weak and the Strong architectures. The Weak architecture (dashes) requires large forces to be developed on the rear of the vehicle while the Strong architecture (solid) requires comparatively higher forces to be developed on the front.

The shape of the two previous curves is similar, except that the boundary line in figure 3.30 follows the shape of the pure power limit instead of the combined power and drag curve. The most important takeaway is that the Weak Architecture only requires a little over 1000 Newtons to be delivered by the front motors to attain peak acceleration performance. In contrast, the Strong Architecture requires over 2500 Newtons of force from the front of the vehicle. To make this useful for motor selection, the torque-speed curves must be extracted from the force-velocity curve. Assuming 18 inch diameter tires, the following graph describes the total torque required on the front and rear of the vehicle as a function of wheel RPM. Whether this total amount of torque is applied by one or two motors is up to the designer.

62 Figure 3.31: The torques developed on the front of the vehicle and the rear of the vehicle are shown as split between the Weak and the Strong architectures. The Weak architecture (dashes) requires large torques to be developed on the rear of the vehicle while the Strong architecture (solid) requires comparatively higher torques to be developed on the front.

The torque-speed curves presented above make for a very confusing motor selection. The typical Surface Permanent Magnet motor exhibits almost constant torque up to a certain peak-load speed before dropping off rapidly. The Strong Rear and Weak Front curves match that description well and could be well suited to motors (or combinations) that can develop up to 900 and 250 NM respectively. However, the Weak Rear and Strong Front curves are rather strange to fit to this model. The Weak Rear powertrain spends almost the entire range at constant power rather than constant torque and would be better matched to an Internal Permanent Magnet (IPM) motor which tend to exhibit this effect. The Strong Front curve is extremely odd and ranges from constant torque, to rising torque, and finally to a constant power curve. This could be met with a constant torque or constant power motor, but would only use the maximum torque or power for a small range of operation. Perhaps the Strong architecture isn’t particularly useful as an extreme, and a split somewhere between the Strong

63 and Weak would offer a more reasonable torque-speed curve. There exists an infinite amount of solutions between these two extremes, and if a front motor curve is known, the matching rear requirements can be generated to complement it.

3.14 Acceleration Takeaways

The Strong versus Weak architecture decision is very important and strongly affects what type of powertrain is required in the front. A team that pursues the Weak LS4WD architec- ture will minimize the development cost of the front-wheel drive system at the expense of having a larger rear powertrain. The Weak architecture produces a vehicle with a large power split between the front and rear powertrains, while the Strong architecture recommends a vehicle with more similar power and torque requirements for the front and rear.

3.15 Sensitivities and Final Comments

AVPS boundary curves directly inform what physical process is limiting the vehicle at all velocities. To improve the vehicle performance, one must simply raise the limiting curves up and to the right. Assuming an infinite powertrain, the only real variables that define the curves are mass, CG location, tire properties, aero properties, and the power limit. Assuming that the power limit will not change and the CG location has been optimized, it is useful to know what changes to aerodynamics, tires, and vehicle mass will do to the acceleration performance. With all other things equal, decreasing the mass of the car will increase the PTMR that governs the power limit at low speeds (before aero drag power dominates). Reduced mass will also produce a small increase in traction since the tires will have slightly less load on them and therefore will exhibit a friction coefficient closer to ideal, but this is a second order effect. With all other things equal, increasing tire friction coefficient (assuming the same LS coefficient µ2) should raise the traction limit linearly. This means the vehicle will accelerate faster at lower speeds and will reach the power limit faster. With all other things equal, increasing down force without decreasing drag will make the

64 traction limit more quadratic in appearance and will increase the peak traction force and will cause the vehicle to enter the power limited regime at a lower velocity. With all other things equal, decreasing drag force without changing down force will make traction limit slightly move to the right but will greatly affect the power limit at high speeds and will increase the maximum velocity of the vehicle. AVPS can be used to identify the sensitivity of the vehicle’s acceleration performance to various parameters. Once the linear or quadratic sensitivity from a starting point has been established, it is easy to compare performance enhancements to the vehicle. If the aero team proposes a new wing with a CDA of 4.0 and CLA of 1.5 and a mass of 5kg, you can easily calculate the anticipated first-order benefit without even running the simulation. Since the system is very non-linear, it would be best to run the AVPS acceleration simulation afterwards to confirm, but the sensitivity tables give you a good place to start and evaluate small changes and where to put engineering effort.

4 Endurance Analysis

The Endurance event is the single largest source of points at competition and simultane- ously tests the vehicle’s acceleration, cornering, braking, and reliability. Endurance success is measured by total time to complete 16 laps (22km) with penalties for hitting cones, going off-track or other driver errors. Since the vehicle operates within its AVPS region during endurance, the factors that improve acceleration will also aid the vehicle as it completes the Endurance competition. However, one of the major differences between the Endurance race and the Acceleration competition is the heavy braking and cornering involved. A complete description of Endurance performance would require a detailed two-track simulation to eval- uate the vehicle’s longitudinal and lateral acceleration capabilities at all times. Sparing that, even a bicycle model based Endurance simulation would quantify the differences between the LSRWD and LS4WD models with regards to lap time, top speed, and other metrics. How- ever, even the development of a bicycle model Endurance simulator is beyond the scope of this thesis. Fortunately, a free lap-time simulation program called OptimumLap exists, but it uses the simple point-mass model to evaluate vehicle performance.

65 Figure 4.1: A LS4WD vehicle with the standard car parameters simulated around the FSAE Lincoln 2012 track in OptimumLap.

4.1 Limitations of the Point Mass Model

The point mass simulator is very easy to use and fast to simulate, but leaves out many of the desirable features of the more complex models. First and foremost, in the point mass model, CG location and all bicycle-model geometric relations simply do not exist, so it is impossible to see the difference between LSRWD and an LS4WD model with a point-mass. The only way to mimic the difference between the models is to tweak the tire parameters in the simulator to try and match the AVPS limiting curves. This sort of spoofing would give the difference in lap-time for a LSRWD and LS4WD model, but would not provide much intuition or deeper understanding-the goal of this thesis. The AVPS curves were loosely matched to the LS4WD standard car model to get an idea of what such a vehicle could do around the track. No attempt was made to compare this to the LSRWD model. The detailed outputs of the lap-time simulation should be considered with a healthy dose of suspicion. In the current model, the battery, tractive system, and transmission are all assumed to be 100% efficient and this serves as an ideal upper bound.

66 4.2 The Importance of Regenerative Braking

Regenerative braking is a capability unique to electric vehicles to recover mechanical energy from motion and reduce the requirements of the mechanical brake system. The importance of regenerative braking cannot be understated. The battery pack of an FSAE electric racecar

1 1 can easily be 4 to 3 of the vehicle by mass. Our MY2017 vehicle had a 72 kg battery pack in a 231 kg vehicle representing 31% of the vehicle mass. The battery pack is by far the heaviest single system on an FSAE electric racecar excluding the driver. Since battery mass is so large and the energy contained within the battery is typically on the order of 7kWh, the ability to regenerate even one additional kWh could enable a battery to be 10kg lighter, or the vehicle to sustain 14% higher average electrical power. The promise of reduced mass or increased average power both increase the PTMR of the vehicle which has a direct impact on the vehicle’s AVPS operating region. The potential benefits of aggressive regenerative braking have been demonstrated by European Electric teams like AMZ from ETH Zurich. AMZ claims that with their 7.0kWh battery pack, their energy meter measures a total of 10kWh of energy usage, equal to 3kWh of regenerated energy-almost half of their pack’s capacity.

Beyond energy recovery, regenerative braking also enables the battery pack to sustain a higher average bus voltage during the race. Lithium-ion batteries exhibit energy dependent voltage profiles, and decrease in voltage as they run out of charge. This effect can be quite dramatic at low states of charge when the cell voltage can plummet from 3 to 2.5 volts very quickly. This dependence is illustrated by figure 4.2 which shows a pack-level discharge of the MY2017 battery with voltage as a function of energy.

67 Figure 4.2: The MY2017 battery was discharged with Megger Torkel programmable load at an almost constant 12kW power draw. The battery voltage drops as energy is removed. In this test, a single bad cell caused an undervoltage fault while the remainder of the pack had ≈ 0.2kW h of energy left.

The curve starts at high voltage but then falls quickly, stabilizes, and then falls precip- itously again. The addition of regenerative braking allows the battery to recharge slightly when the vehicle brakes, and move back up the voltage-energy curve. Falling bus voltage causes increased current draw at the same power, as well as decreases the maximum speed of the electric powertrain which further limits the vehicle’s AVPS operating region. Finally, the FSAE rules require the vehicle to disable itself if any single cell drops below the safe 2.5 volt operating limit. Regenerative braking helps improve all of these factors in addition to increasing the system wide PTMR.

To determine the regenerative braking energy available, we must evaluate the braking force applied on the vehicle as it completes the endurance race. The braking force multiplied by the vehicle’s forward velocity represents the braking power dissipation which can then be integrated versus time to evaluate the braking energy recovered. The total mechanical energy dissipated is the maximum theoretical braking energy that can be recovered assuming 100% efficient conversion and regenerative capability across all combinations of torque, speed, and

68 power levels.

4.3 Outputs of OptimumLap

The output acceleration and velocity of an OptimumLap simulation as a function of time was recorded. OptimumLap was used instead of empirical data from MY2017 because it represents an optimal case of perfect driving in perfect conditions. This will give an over- estimate of the regenerative energy potential but is still useful to compare RWD and 4WD vehicle architectures.

First, the longitudinal force on the vehicle is extracted by Newton’s law Fx = max. However, this represents positive forces generated by the powertrain as well as negative forces generated by aerodynamic drag. To prevent drag forces from being counted as regenerative energy, the aero-drag as modeled by equation (19) is added to Fx. Whenever the force on the vehicle (with drag removed) goes below zero, the brakes are being applied and slowing down the vehicle. The comparison between the different forces is shown below for a representative portion of a lap in figure 4.3.

69 Figure 4.3: The force output of OptimumLap is shown as a function of time in red. The force on the vehicle without aerodynamic drag is shown in blue. The braking force in black dashes is non-zero whenever the force without drag goes negative.

We can then multiply the braking force by the vehicle’s velocity as a function of time. When the vehicle is traveling quickly in a straight line, the large aerodynamic downforce allows large braking forces to be generated. The large force at high velocity results in a very high peak braking power. However, as the brakes work to slow down the car, the vehicle’s speed and downforce both decrease causing the braking power to decay super-linearly over time. When the vehicle corners, the system gets more complicated. Since tires cannot provide maximum braking and lateral forces at the same time, the braking force can actually increase as the vehicle transitions between two corners. The power, force, and velocity curves are shown below in figure 4.4.

70 Figure 4.4: The values of the braking force, velocity, and braking power are scaled for viewing on the same axes. The brake force in red jumps and decays almost linearly as braking continues. The velocity changes slope from positive to negative when braking force is applied. The braking power shown as a black dash decays super-linearly as both of its components decrease almost linearly. When the vehicle brakes as it transitions between two corners, the braking force can increase as more traction is available for braking.

The curve exhibits the expected behavior, but the braking power is startling high. The peak vehicle electrical output power is limited to 80kW, but hard braking can dissipate over twice that with values in excess of 200kW vehicle wide. Since peak braking power is only traction limited, power levels higher than 80kW are not unreasonable.

Now, with a plot of braking power as a function of time, the entire curve can be integrated to provide an estimate of the total mechanical energy available during braking.

71 Figure 4.5: The braking power is integrated to reveal the absorbed braking energy as a function of time in blue. The normalized plot of braking power is provided to show that increases in energy tend to come during short dramatic braking periods rather than continuously.

The resulting mechanical energy is 0.276 kWh per lap. Since this track is only 1170 meters long. There are 18.8 of these laps during a full 22km endurance run. Multiplying by 18.8, the total energy absorbed over the course of endurance could be as high as 5.19 kWh.

4.4 Preliminary Endurance Takeaways

Braking force is only traction limited and can quickly slow down the vehicle. The peak braking forces are in excess of 80kW and actually exceed 200kW for the entire vehicle. The braking power comes in sharp peaks with super-linear decays or rounded peaks. Overall, there are ≈5.0 kWh of mechanical energy available to recover in the ideal case, and it is reasonable that a good fraction of it can be recovered according to AMZ.

72 4.5 Power Split During Braking

While the total recoverable energy is large, the important question is how much of that energy is available on the front wheels versus the rear wheels. The answer to this question determines the impact of a 4WD versus RWD architecture on recoverable braking energy. If a very high performance LSRWD vehicle had the same AVPS boundary as a LS4WD vehicle, then the only difference would be the energy available on the front wheels. Unfortunately, as mentioned above, the point-mass model doesn’t contain the information required to differ- entiate between forces on the front or rear of the vehicle. Furthermore, while OptimumLap generally brakes heavily on the straights before a corner, a small amount of braking is done along a finite turning radius. This means that the lateral acceleration of the car is non-zero and therefore the normal forces on the vehicle are not equally shared between the left and right tires. In order to determine the distribution of traction on the front and rear we need the normal forces on all four tires. Once we have the traction forces, we can find the power at each wheel.

A two-track normal force model developed by Luis Mora on MIT Motorsports outputs the normal forces on each wheel of the vehicle as a function of longitudinal and lateral acceleration as well as vehicle velocity. This model calculates the front-rear weight balance by geometry, and the left-right balance by suspension stiffness and roll-center height. Feeding the output of the OptimumLap software into the model, with the suspension and aero values for MY2018 results in the predicted normal forces of a two-track model that experiences the same accelerations and velocities as the point-mass vehicle.

The normal forces on the two front tires can be combined into a single front force and the forces on the two rear tires are combined into a single rear force to visualize the weight balance of the vehicle while braking. The front and rear combined normal forces are shown below in figure 4.6for a representative portion of the Endurance race.

73 Figure 4.6: The total normal force of the vehicle (black) is broken down into its components on the front (red) and rear (blue). The braking force is shown for context as a black dashed line. As the braking force is applied, the vehicle slows down and the aerodynamic downforce decreases. Thus, the slope of the total normal force is negative when the braking force is applied. The normal force on the front of the car rises dramatically as the braking force is applied. When the vehicle accelerates, the opposite effect is observed.

With the normal forces at each tire, we can estimate the traction available at each wheel by multiplying by our LS4WD constitutive tire model from equation (14). This gives an idea of how much forward traction could be generated on each wheel as a function of time. While this model uses two-track normal forces to evaluate load sensitivity, it still assumes that the tire can provide the required lateral and longitudinal acceleration at the same time. This is not true since a tire has a traction ellipse which defines the possible combinations of longitudinal and lateral traction available at all times. This could be accounted for with a more complex two-track model with known steering geometry, but that is beyond the scope of this thesis. The predicted traction forces available on the front and rear of the vehicle are compared in figure 4.7.

74 Figure 4.7: The available traction force in Newtons on the front (red) and the rear (blue) are shown in comparison to the braking force along the dashed black line. These curves are very similar to the normal force plots above, except they are scaled by a load-sensitive friction coefficient.

When braking, the front of the car has lots of available traction, and when accelerating, the rear of the vehicle has lots of available traction. At every point in time, the front and rear of the vehicle each provide a certain fraction of the total traction. Those fractions can be multiplied by the vehicle’s braking power to determine the fraction of braking power that originated from the front and the rear of the vehicle. The braking power balance between the front and rear of the vehicle is shown in figure 4.8.

75 Figure 4.8: The braking power available at the front (red) and the rear (blue) of the vehicle is compared to the total braking power available (black dashed line). The majority of the braking power is absorbed by the front wheels.

It is immediately clear that the front wheels carry the majority of the braking power for the vehicle. The front wheels develop about twice as much braking power as the rears and

2 therefore contribute ≈ 3 of the total available braking energy. While the fronts deliver more energy, they also do so at a higher power level than the rear. If the front powertrain cannot absorb all of the incoming electrical power, the excess will be converted into heat by the mechanical braking system. Simple integration of the front and rear braking power shows that the fronts can recover a total of 3.38kWh of energy and the rears can recover a total of 1.82 kWh which matches the total predicted value of 5.19kWh presented in section 4.3.

4.6 Braking Power Distributions

If the front and rear recover mechanical braking energy at different power levels, then the front and rear powertrain must be sized differently to make the most out of the available brak-

76 ing power. This means that there will be an optimal power split for maximum regeneration during braking, as there was a power split between the front and rear during acceleration. Since motors with high peak power ratings are expensive, heavy, and difficult to source, it is valuable to know how much energy can be recovered by a motor of a given power level. This knowledge will help the designer size motors that allow a balance between engineering cost and regenerative energy benefit. A first attempt at understanding the power split is the probability distribution of the front and rear power shown below:

Figure 4.9: The probability distribution of the braking power absorbed by the front and rear of the car is shown in 5kW bins. The front develops power across a wide range up to 155kW while the rear develops all of its power at less than 85 kW.

The staggering difference in power level is not surprising. As discussed above, when both front and rear brakes are applied, they exert different forces at the same velocity. If the front brakes exert twice the force as the rear at the same velocity, they will absorb the available energy at a power level twice that of the rears. It is also clear that even if the front powertrain could develop a full 80kW, it will not be able to recover all of the available braking energy. However, it will recover the majority

77 of the available energy because even as the vehicle brakes at 155kW on the front, 80kW will be recovered and the remaining 75kW will be burned off as heat. Because a powertrain rated for X kW can recover all the energy available at power levels ≤ X and a fraction of the energy available at power level > X. The integral of the probability distribution or the cumulative probability distribution does not accurately describe the total recoverable energy. To determine the energy recoverable with a powertrain capable of X kW, the power available at each time step must be analyzed. If the instantaneous power P is ≤ X, all the energy in the timestep (P δt) will be absorbed. If the instantaneous power P is ≥ X than only the energy (Xδt) will be absorbed. Applying this criteria to the front and rear powertrain produces the desired plot of energy available as a function of powertrain power rating. This model assumes that the powertrain is only power limited, 100% efficient, and is able to provide any torque at any speed within the power limit. Thus this plot represents an upper bound in an idealistic case, but is nonetheless useful.

Figure 4.10: The amount of braking energy recoverable during an endurance run is broken down by the peak power rating of the front and rear powertrain. The front powertrain must reach almost 130kW before most of the energy is captured, while the rear powertrain only needs to develop about 70kW.

78 It is clear that with infinite power available on the front and rear powertrain, the total of 3.38kWh from the front and 1.82kWh from the rear discussed earlier can be attained. It is also clear that a staggering 160kW would be required on the front powertrain (two 80kW peak motors) to absorb all the available energy. However, it takes as little as 32kW on the front powertrain (two 16kW motors) to recover half the total energy on the front. This plot adds another dimension to the Strong versus Weak 4WD architecture discussed in section

1 2 3.13 above. Consider an 80kW peak vehicle with 3 of its power on the front and 3 of it on the rear. This vehicle could develop 27kW on the front and 53kW on the rear, capturing ≈ 1.47 kWh from the front and ≈ 1.76 kWh from the rear for a total of ≈ 3.23 kWh. However, an 80kW vehicle with a 40kW even split on the front and rear would recover ≈ 2.0 kWh on the front and ≈ 1.61 kWh on the rear for a total of ≈ 3.61kWh. Clearly, a stronger 4WD architecture with more power on the front of the vehicle enables higher regenerative energy recovery than a weak architecture. Taking this to the extreme, a very strong 4WD would have even more power ability on the front then on the rear to take advantage of the very high peak power regeneration available. However, designing a vehicle for 160kW operation on the front and 80kW on the rear is a losing game. The vehicle carries the weight of a 240kW capable system, yet it will rarely use a full 80kW for acceleration, and only for short periods of time will it use all 240kW for regen. Furthermore, the losses involved in absorbing 240kW of regenerative power even at a full 300-volt bus are staggering. That would amount to 800 amps which even over 0.1 ohm tractive system loss would result in a monstrous 64kW of heat generation. The importance of this curve is that it allows the designer to choose their power level on the front and rear carefully to get the most regenerative potential with the minimum system cost. This curve is definitely an idealistic upper bound, and makes many simplifying assumptions, but while the exact values may be doubtful, the underlying concepts are valid.

4.7 Final Comments

The Endurance simulation and the regenerative braking results suggest an alternate defini- tion to the Strong and Weak 4WD architectures. These simulations are idealistic, but still offer an upper bound for what a real vehicle could achieve. Instead of taking these models

79 with a grain of salt, empirical data for traction, efficiency, and other sources of loss can be measured and incorporated into the model. The evaluation of regenerative braking potential can be done by an empirical torque-speed curve which would account for inefficiency, as well as limits on torque, speed, and power. A more intense model with drivetrain, motor, controller, and battery efficiency data would offer an idea of the losses involved in the regen- erative braking process. This analysis offers a starting point for other teams to continue to develop regenerative braking models with their own constraints.

5 Next Steps

The information presented in the above sections should be enough to guide a team towards their own AVPS models and regeneration simulations. While the exact values I present may not be useful to most teams, the underlying methods should be helpful. I believe that there is much more to do with regards to AVPS modeling such as fitting empirical data, handling changes in curves such as active aerodynamics actuation, and system level multi-parameter optimization to achieve the highest performance car. Developing detailed sensitivity plots and determining where the largest benefits lie should help teams prioritize new projects and make engineering tradeoffs. Finally, I hope this thesis has provided the reader with some intuition for how vehicles accelerate and brake. Good luck!

80 Bibliography

[1] AMZ News. Eiger - on to new hieghts, May 2018.

[2] OptimumG. Optimumlap v1.4, May 2018.

[3] Mark Wielitzka, Alexander Busch, Matthias Dagen, and Tobias Ortmaier. Unscented kalman filter for state and parameter estimation in vehicle dynamics. 02 2018.

[4] Adam Liben. Powertrain architecture, simulation, and controls, May 2018.

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