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.

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.

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.

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:

A few key insights are apparent from the figure. Firstly, the lower the ramp angle (aka more locking), the worse the differential performs in pure cornering, but that was already confirmed with the previous figures. Another interesting detail is that the maximum cornering capacity typically increases slightly with a little bit of braking. This is likely because the vehicle is set up to understeer, and the normally saturated front tires are benefiting from the extra normal load without being overwhelmed by large braking torques. This begs the question: what is happening to the stability of the vehicle at this point? That will be answered further below.

In the more extreme braking zones, it can be seen that increasing the locking on the differential actually improves maximum cornering capacity. This might be counter-intuitive at first, because the differential provides an understeer torque. But it is likely that the understeer moment generated is actually allowing the vehicle to achieve higher lateral acceleration by increasing the total yaw stability. This turns into a complex problem, because the suspension balance and the brake bias play a significant role in this as well. 

To interrogate this effect, I replicated the diagram (with fewer differential settings) where vehicle parameters were adjusted to induce increased understeer. My hypothesis was that the increased understeer would diminish the benefit of the locked differential under trail-braking scenarios. The first example is below, where I shifted the lateral load distribution forward by 15%, to increase understeer in steady-state cornering:

At 1g of braking, the cornering performance gain by switching from fully open to fully locked decreased by 20% when I shifted the LLTD forward.  This result was reassuring and lined up with my hypothesis, but I wanted to dig deeper to understand why this was happening. The answer came in the form of the rear brake torque distribution. When the load transfers to the outside tires under lateral acceleration, the inside tires are left with less grip to distribute between braking and acceleration. For the front tires this is not an issue because of the forward load transfer that comes with braking, but for the rear tires, the unloaded inside tire comes dangerously close to locking up. By locking the rear axle to a single fixed velocity, the loaded outside tire can generate the majority of the braking torque, allowing the inside tire to generate more cornering force.

To reinforce this explanation, I generated another data set, this time shifting the brake bias 20% rearward to increase the demand on the rear tires. Once again the outputs were consistent with the rest of my results, and this time the benefit of a locked differential under braking was even more dramatic:

Looking back to the original plot, I was surprised to see that under forward acceleration + cornering conditions, the open differential retained an advantage over a locking one. This would be the scenario where it would be beneficial to lock the rear wheels to put the power down effectively. It is important to keep in mind, however, that modern FSAE cars are high downforce and low-power, making them power-limited in acceleration for the majority of operating conditions. I decided to re-create the friction diagram at a lower speed, to hopefully place the car closer to a traction-limited scenario:

Even at lower speeds, the locked differential was still detrimental during combined acceleration. What I forgot was another key characteristic of Formula SAE, and that is the tight autocross course dimensions. At these lower speeds, the cornering radii and subsequent rear wheel slip difference was serving to further highlight the downsides of a locked differential. So I tried again, this time going back to my original velocity, removing the aero, and doubling the engine torque curve. The following vehicle was entirely traction limited in acceleration, and finally the locked differential had it's time to shine:

At this point, I was satisfied with the quality of my model at this stage. Most results aligned with my original intuition about the effect of the differential on cornering and acceleration. The results that initially didn't make sense were easily explained once the vehicle states were examined and compared, and I managed to learn some new insights along the way. The next step was to move beyond pure acceleration performance, and to explore the stability and maneuverability of the vehicle.

Characterizing Stability and Control

My next goal was to evaluate the effects of the differential on the vehicle behavior throughout its entire operating envelope. To do this, I used my solver from the previous section as a starting point, but instead of the objective being to maximize lateral acceleration, it was to achieve a target lateral and longitudinal acceleration value. The same solver constraints apply here as well. This enabled me to determine the vehicle state at any point along or inside the traction envelopes created earlier.

Once I had access to key vehicle conditions, such as loads, wheel speeds and tire slips, I used Liapunov's indirect method to linearize the vehicle at the point of interest. Using traits such as cornering stiffness and load sensitivity, I calculated partial differentials across the vehicle model before condensing to an equivalent 2 degree of freedom bicycle model. This simplified model retains the effective axle compliances and performance traits as the original, but makes stability analysis much easier.

 

For the rest of this section, you will see my results in the form of 3D contour plots, with relevant discussion for how I defined each metric and its significance. I will display plots for a locked and open differential, as well as the configuration we currently run on the car as an in-between comparison. Most of my discussion will be between the locked and open examples, however, as they represent the extremes of the spectrum.

The first metric I will explore is static vehicle balance and stability. It is hard to use understeer gradient as a comparison tool, because of how sharply it saturates near the cornering limit. A much more stable variable to explore, that describes the same essential characteristic, the static directional stability gradient. It describes the vehicle yaw response to a sideslip perturbation (nicknamed the "weathercock effect" by Milliken), and the sign indicates which end of the vehicle is dominant. The locked differential in the third figure, in this case, shows greater directional stability. However, this is not the complete picture because the differential primarily creates a correcting moment in yaw, which is not accounted for here. 

Since yaw motion imparts a differential slip ratio between the inside and outside tires, any locking in the differential will generate a higher yaw-resisting moment, and this is easily seen on the final figure with the locked rear axle. This is the effect that was not captured in the previous figures, and further increases the understeer of the vehicle.

 

Another benefit of increased yaw damping is higher stability in maneuvering changes, and helping eliminate overshoot in cornering response.

A more complete picture of stability can be found by looking at the system eigenvalues. The following plots display the real component of the dominant eigenvalue. Anything above zero (orange) indicates an unstable system, and the total stability of a locking differential is made clear.

An alternate metric is the stability index (pictured below), defined as the normalized returning moment slope to an increase in lateral acceleration. Again, the higher the degree of locking in the differential, the more stability is imparted to the system.

The other important metric to explore is the maneuverability of the vehicle. Since I was working with a linearized representation, I used the frequency response of the yaw:steer transfer function to find the gain bandwidth. This is classically defined as the input frequency at which the gain response falls below 3 dB, and essentially communicates the total range of maneuverability of the vehicle. In the plots below, the locked differential exhibits the largest bandwidth, but again this is not the full picture. Rather than communicating the raw amount of "maneuverability", the bandwidth describes the accessibility of said agility.

To find the absolute agility, I just found the steady state gain of the transfer function, and took its inverse to filter ourt extreme values. Those results are picture below, where a lower number indicates higher agility. This shows that even though the locked differential has a wider bandwidth of agility, the vehicle with the open differential is more maneuverable overall.