Limited Slip Differential Characterization
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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.
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.
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.
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.
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.
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:
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
Additional inequality constraints were added so that the tire slip ratios and slip angles did not exceed a maximum value.
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:
First, a caveat on the understeer definition used above: Understeer is defined as an instantaneous slope describing the deviation between the ackermann steer angle and the actual required steer angle required for an increase in lateral acceleration. However, to illustrate an easy comparison, the understeer here is defined as a "total understeer", or the absolute difference between the actual/ackermann steer angles divided by the total lateral acceleration. Keeping that in mind, the results so far make intuitive sense: as the differential locking increases, the yaw-resisting moment induced by the rear wheels increases the total understeer, and decreases the maximum possible lateral acceleration. So far, so good. The next step was to evaluate a range of speeds, and see how the behavior changes. Those results are shown below:
Some of the current patterns continue, with a little more nuance. As speed increases, the effects of the differential on the handling balance are similar. It is worth noting, however, that the difference between fully locked and fully unlocked become less and less drastic with higher speeds. The handling performance sees similar trends, except at low speeds and high locking ratios, where performance takes a sudden downturn. Intuitively, this is reasonable, because the slower and tighter the turn, the more drastic the effects of dragging the tires around the corner when they can't spin at differential speeds. This is reflected by a similar sharp uptick in the understeer plot at the same point. It took some troubleshooting on the solver parameters, such as the penalty function weights, to get it to run consistently and robustly, but at this point it seemed in a strong enough state to move on.
Combined Acceleration Conditions
The next step was to evaluate the vehicle across the full maximum performance envelope, and quantify the effects of the differential on conditions where braking or acceleration was taking place as well. In other words, it was time to construct the full vehicle traction envelope. In order to do this, I had to modify some of the equality constraints, and introduce some new ones as well:
1) Now, the sum of longitudinal forces must equal the target longitudinal acceleration as well.
2) Rotational inertia of the spinning components (engine, drivetrain, wheels) comes into play in the relevant equilibrium equations.
3) Longitudinal slip ratio is constant. In order for this condition to be true, the following equation must be satisfied:
(derivative of forward velocity)*(rotational velocity) = (forward velocity)*(derivative of rotational velocity)
The full traction circle was constructed in a collection of 18 points, and I evaluated an open diff, locked diff, and all 6 available ramp angle combinations: