Limited Slip Differential Characterization
Please enjoy the 2-minute summary gallery below, or scroll further to explore my work in greater detail
I believe that change is not something to be simply dealt with; rather, change is something to take charge of and make into your own opportunity. The 2020 COVID-19 pandemic shook the world, and I went into quarantine with my family for over 5 months. I consider myself lucky to have been blessed with all that time alone in a position free of responsibility. This time was mine to make the most of, and this is one of the significant projects I undertook during that period.
Project Background
Though I did spend a lot of time on the suspension/aero characterization and optimization for the ChassisSim competition, this page represents the majority of the work that I did to keep pushing my skills while quarantined at home this Summer.
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The goal of this project, broadly speaking, was to explore and evaluate the effects of a limited slip differential on the handling, control, and stability of a Formula SAE vehicle. This project is not yet completed to my satisfaction, but since I am beginning grad school at the time of this writing, and I don't know how much more time I will have to commit to this, I wanted to document my current progress with the hope of returning soon. I had always been familiar with the common talking points regarding the effects of a differential on vehicle handling, but this represents my efforts to better quantify and understand these effects on the Clemson FSAE vehicle.
Since 2017, the Clemson FSAE team has utilized a Drexler 2-way clutch limited slip differential.
Modelling Approach
As is the case with the majority of the projects on this website, I carried out this project entirely in MATLAB. I recognize that there are a multitude of well-established vehicle simulation programs on the market, many of which already incorporate differential models that would enable a straightforward exploration process. Despite this, I prefer to code in MATLAB for two primary reasons. Firstly, I prefer having complete control over what parameters are or are not included in the model, and I can build the detail exactly to my liking. But most importantly, I have learned that when I take the time to learn the math, solve the equations, and code the program, I come away with a much greater intuitive and mathematical understanding of the concepts than if I had just used a ready-built program and analyzed the results. My ultimate goal is to learn, and I have found that this is the best way for me to do that.
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I represented the vehicle by building a two-track vehicle model with seven degrees of freedom: forward velocity, sideslip velocity, yaw velocity, and individual wheel speeds. Roll is omitted, but lateral load transfer distribution is included as an adjustable parameter. In addition, the vehicle has three inputs: steer angle, braking torque, and engine throttle. Braking torque is applied to the wheels and split by the brake bias ratio. Engine torque is determined from a lookup table based on throttle percentage and vehicle velocity, and applied to the differential.
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The Drexler FSAE Limited Slip Differential is available with 3 reversible ramp angle combinations, to adjust the locking characteristics in coast and drive, respectively. For certain models, pre-load torque is adjustable as well. The lockup torque represents the proportion of applied torque that can be sent from one wheel to the other. For an open differential, this would be zero, and for a solid axle this is 100%. The figure below describes the representation used in my model. The preload torque pre-dominates the maximum differential torque possible up until enough engine torque is applied to linearly increase the locking further. In reality, temperature and wheel speed difference affect the friction forces between the clutch packs, but for a pure comparative study I considered a linear model to be adequate. If the torque difference between the rear wheels is below the differential torque, they are fixed to have the same rotational velocity. Once the torque difference overcomes the differential torque, the wheel speed difference is allowed to increase until it is no longer exceeded.
Other vehicle characteristics included are a MF5.2 tire model, fixed lift/drag/CoP coefficients, and the ability to adjust static camber and ackermann percentage. I decided to start simple and work my way up to more complex vehicle operating scenarios, checking each step along the way to ensure my results still make intuitive sense. In the following sections, I will discuss these steps and provide more information on the computational decisions involved.
Steady State Cornering
The simplest starting goal was to find maximum steady state cornering available at a single operating point. Previously, for programs like the lap simulation, the fastest way to find an equilibrium solution was using iterative searching loops. For this problem, the amount of variables is too large for this method to be computationally effective. Instead, I used MATLAB's fmincon function to numerically find a global minimum. This solver had to find the combination of steer, torque demand, yaw rate, and sideslip angle achieve maximum cornering.
To constrain the problem, the following equality conditions were imposed:
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1) The lateral acceleration produced by the tires must equal the yaw rate times forward velocity
2) The sum of longitudinal forces (aero+tire drag, applied power) must equal zero
3) Sum of moments about the yaw axis must equal zero
4) Within each function evaluation, rear wheel speeds were solved to match the torque demand and differential torque requirements
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Additional inequality constraints were added so that the tire slip ratios and slip angles did not exceed a maximum value.
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First, I ran a single velocity of 30 feet per second, and solved for a range of differential lockup torques from fully open to fully locked: