Mathematical race model

The premise

The F1 in Schools car design process is similar among most teams; a set of rules of thumb gathered from experience or from looking at past successful teams, with a lot of prototypes and CFD runs. Validation of aerodynamics and all other performance variables is condensed into physical track testing campaigns. I was skeptical of this engineering approach for two main reasons:

Although CFD colours and streamlines look pretty for marketing purposes, high school student CFD among small, 50 gram, fast accelerating, highly turbulent open-wheel cars would be highly inaccurate at best (in a field where even the best aerodynamicists in the highest levels of motorsports can easily see ~30% mismatches between simulations and real-world data). I also question how much aerodynamics work actually makes a difference in such a high-turbulence environment with mandated minimum open wheel sizes.

There is high variance in physical track testing; alignment, tension, and CO2 cartridges all vary between runs to an extent that makes any statistically defensible data outcomes very time and cost intensive. For instance, in the CO2 cartridges we were able to obtain for our own testing, we measured up to a 20% difference in peak thrust.

Thus, I decided the most effective way to determine car performance would be to break all the engineering variables down into isolated measurable, or derivable, components, then piece them back together in a numerical simulation.

The model

I wrote a numerical simulation in Python that integrates the car's motion in millisecond steps, time-stepping SUVAT-based equations (and somehow managing to figure out this approach independently before learning about numerical analysis formally in university). At each step it resolves the net force and moments, and updates acceleration, velocity, yaw, and forwards and sideways displacement. It took in measured/derived data for:

Thrust from the CO₂ canister, taken from force-sensor data in a rig I created, captured at a 0.001s timestep, including a sidewards component for non-centred canister punctures.

Drag forces as the sum of wheel-bearing friction, aerodynamic drag, and tether-line friction — the last of which also depends on sideways displacement and yaw.

Restoring forces and moments from the tether line, wheels, and aerodynamic moment.

The output is a representative model of car behaviour across the track length, showing comparative differences between different prototypes. A further "refine mode" swept any single variable across a range to isolate its effect on lap time.

Engineering priorities

Feeding the model's outputs into a sensitivity analysis (run by a teammate) produced a ranked weighting of how much each variable drove performance:

Car mass - ~42%

Aerodynamic drag - ~29%

Wheel friction - ~21%

Tether-line friction - ~6%

Wheelbase, guide height, yaw stability - the remainder

This is a key output resulting from the model: it validated our engineering priorities and proved mechanical losses are on-par with aerodynamic losses.

Result

The model was ultimately a comparative tool, used to determine relative performance for our designs. It was only through luck that the modelled race time sat within 3 milliseconds of our actual average race time. The model itself is also not perfect; by the end it still had a lot of hard-coded variables (such as centre of mass height and position, tether line guide longitudinal position, aerodynamic moment coefficient) that was representative of our design style but cannot said to be general across teams.

Nevertheless, it was useful within the scope of our project, and it assisted our team in winning both Fastest Car and Best Engineered Car awards at the World Finals. The thesis held up on the against the best teams in the world.