Power, grip, and aero in theory

Six months ago I did some simulator tests where I used Assetto Corsa to answer questions about the relative contributions of power, grip, and drag. I wanted to follow that up a little with something a bit more rigorous. So I took my driving inconsistencies out of the equation and had the AI drive the car. I did a bunch of experiments on a lazy Sunday using the original rFactor. That was fun and informative, but I’m not reporting on that today because I decided to write a program that simulates a car driving around a track. Why? Well, honestly it’s because I wanted to implement the various equations myself. Most of the math is pretty easy in isolation. Equations for acceleration, lift, drag, etc. aren’t too complicated. Putting them together sometimes is though. For example, as you increase speed, you increase drag. So acceleration gets worse the faster you go.

The Model

Track

The track is modeled as a series of alternating straights and corners. The simplest description would look something like this.

S:2000
C:200:60

This means a 2000 foot straight followed by a 200 foot radius corner with a 60 degree arc. You can chain together any number of straights and corners to create whatever track you like. The sections don’t actually need to connect in a closed shape. I decided to use Thunderhill as that’s one of the most popular tracks in the region. I used Google satellite images and scale bar to rough out the track. It comes out as 2.94 miles, which is pretty close to the actual length. Note that my track model doesn’t take into account elevation or camber (yet).

Corners

Under the assumption that the driving line is circular, corner speed depends only on the radius of the corner and the grip of the tires. This is made a little complicated by aero modifications that increase grip and speed by adding downforce, but only a little. Because the corner speed is constant, it’s trivial to determine how much time was spent in the corner.

Straights

Straights are somewhat complex to model because the vehicle increases speed for some time, and then brakes to arrive at the correct speed for the next corner. This calculation depends on initial speed, engine power, gearing, aerodynamic drag, frontal area, and grip of the tires. A simple way of thinking about it is that the total time is the sum of the accelerating time plus the decelerating time. The way I solve it is by binary searching the transition from throttle to brake. At some number of seconds the distance covered and the final speed will be correct: it’s just a matter of making refined guesses.

Vehicle

Since Miata Is Always The Answer, I decided to do experiments with a virtual Miata. People sometimes say “the answer is always Miata” but that would spell out TAIAM, which doesn’t mean shit. Let’s give some parameters on the typical Miata that we will vary to see how the lap times change.

  • 2300 lbs with all fluids including the driver. We’ll strip some weight out of this to see what happens. We’ll also add a little.
  • 100 HP. My model assumes an engine of constant output. I don’t take into account the torque curve or gearing yet. It’s best to imagine “100” as a placeholder for both torque and horsepower, and the value of 100 is not a very healthy example of the breed.
  • 0.40 Coefficient of Drag. A hardtop Miata with windows down has a drag of something like 0.4. But topless it’s worse, and you could always add theme and make it terrible. For reference, a Prius is below 0.3 but it would be hard to get a Miata that low. However, a Prius has a larger frontal area.
  • 0.0 Coefficient of Lift. I abstract the various aero components into a single item rather than wing, splitter, diffuser, etc. A wing can be flat surface made from plywood with a CoL of 0.75  or something wing-shaped with a CoL of 1.0-1.6. The default value is 0.0 but a base Miata probably has some lift.
  • 0.0 sq-ft wing area. I’m not sure how to convert the various aero surfaces into wing area, but 0-12 feet in 4 foot increments seem like a reasonable range. The default value is 0.0.

Results

Power

It’s probably no surprise that more power reduces lap times. This is especially true if you have an anemic engine. Adding 20 HP sees lap times dropping by 3.08 sec. Another 20 HP is 2.57 sec. While there are diminishing returns, there are significant benefits to 200 HP and beyond. What’s amazing about engines is that you can realistically have them vary over a huge range. A turbo or supercharged Miata can make 300 HP. It might not make a good endurance racer at that point. However one of the most successful Lemons cars is the turbocharged Miata from Eyesore racing. In a race situation, high HP is doubly useful because it’s much easier and safer to pass under acceleration than braking or cornering.

Grip

The more corners a track has, the more grip becomes the key factor in performance. On a circular track, grip would be the only factor (assuming you have enough power to drive a given speed). Even a small change in grip can make a large difference in lap time. For example, a change from 1.00g to 1.05g drops lap time by 2.77 seconds. If you look at the telemetry of different drivers in the same car, you’ll see some people can extract more grip than others, and I think this is largely why some drivers are a couple seconds faster than others. In this model, an all season tire is about 0.90 grip. Summer tires 0.95, 200TW 1.0, Semi-slicks 1.1, Slicks 1.2. While these figures may not be correct, it’s the relative difference that’s important. If you want to win, get the grippiest tire allowed by the rules. The UTQG rating is only a rough indicator of the grip. In the crowded 200 treadwear class, I’ll bet there’s more than 0.05g of variation, especially when you consider differences in rim widths and tire pressures.

Aero

There are two components to aerodynamics, drag and lift (three if you count aesthetics). Drag has a relatively mild effect on lap times. Slipstreaming the heck out of it won’t see more than 1 second improvement. Similarly, ruining your CoD to a tune of 0.5 won’t see you slower by more than 1 second. Of course, every second counts, but this is the least useful area to tune. However, cosmetically, not much says racecar more than a wing.

Because lift affects grip, and grip is incredibly important, an aero package that increases downforce has a reasonable effect on lap time. Simply adding a wing could see your lap times dropping by 1.3 seconds (this is the Ideal 4 column below). There is some drag associated with wings, however, and on a track that is more straight than corner, a wing may do more harm than good. Note that a splitter can both decrease drag and increase downforce, so not all downforce increases drag. While you won’t see huge improvements in lap time from aero, it’s a one-time cost, unlike tires, and a well made aero package could see you dropping 1-2 seconds.

Weight

Removing 100 lbs will see lap times dropping by only about 0.6 sec. Weight reduction appears to have a relatively minor effect because it varies over such a restricted range. It’s a lot easier to improve your power:weight ratio by adding horsepower than removing pounds. So weight reduction might not seem like it’s worth doing, but it is. Out in the real world, the relationship between load and grip is sub-linear, so dropping weight is better than the model shows. There are also gains to be had in component longevity and fuel economy. The simple weight loss associated with angle grinders is relatively cheap, but when you start replacing structural parts with lightened versions it gets costly.

Some example builds

Let’s close this out with some example builds and lap times. Note that for a variety of reasons, the absolute lap times aren’t exactly as you would see at the track, but they aren’t very far off. It’s more important to think of the relative differences.

  • First day at the track – untuned engine (120 HP), all season tires (0.90g), open top (0.45 CoD, no downforce), and a coach in the right seat (2500 lbs) = 2:33.20
  • Solo – as above, but with Summer tires (0.95g) and no passenger (2300 lbs) = 2:28:91 (4.29 sec faster than above)
  • Sport build – engine is mildly tuned (130 HP), 200 TW tires (1.0g), hard top (0.40) and enough weight reduction to offset the top = 2:24.25 (4.66 sec faster than above)
  • Budget enduro – 100 lbs of weight reduction (2200), an additional 5 hp (135), DIY splitter and wing (0.35 CoD, 0.75 CoL, 8 sq-ft area) = 2:21.14 (3.11 sec faster than above)
  • TT build – as above, but using stickier, wider tires (1.05g) and professionally designed aero (CoL 1.3) = 2:17.36 (3.78 sec faster than above). Now I’m sure you’re wondering if it’s the tires or aero. It’s mostly tires (2:18.44 vs 2:20.09).
  • Eyesore – a famous Lemons car with a ghetto-charged motor that was dyno’d at 197 hp. It’s light (2200), has 200TW tires (1.0g), and theme for aero (0.45) = 2:17.29.

At some point I need to put this theory through some real life testing. I honestly can’t imagine anything more fun than going to a track day with 4 sets of tires, removable aero, and some ballast. It would be a long, hard day of work, but what a day. It costs $2200 to rent Thunderhill West for a 2-car test day. In the off-season, they sometimes cut that in half. I just need to find another car to share the session with and a crew to help out with the pit work.

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