The Grip Equation

Published Tuesday, March 26, 2024

A Model that Made Me a Faster Driver

While on the #RoadtoAPlus, I found myself at a roadblock. I was gaining more driving experience, but my lap times were plateauing. On most turns I do what I’m supposed to. I brake hard and late on straights, trail brake into turns, coast through them, and then get on the power while exiting. Why then was I not getting faster?

The problem was that I was fighting the car. I was asking too much of the tires at every turn, and this overloading limited tire performance in every aspect. I was braking too hard too late, which kept me in a straight line for too long on turns that didn’t need it (and that’s even if I was stopping in time). I was suddenly and harshly turning on corner entry, which asks too much of the tires on one side of the car and becomes especially detrimental when also trail braking. I was putting too much power down too early, which asks too much of the tires receiving power while they also try to turn the car.

Every tire has a limited amount of grip. This amount changes with tread, temperature, wetness, and other factors, but they all have a limited amount at any given time. At all times during high performance driving, you want to be at or under this limit but never exceed it. To understand the where you are in relation to the limit, consider this equation.

G_r = \frac{s_w}{s_{mw}} + |sin(\theta)| + \frac{F_{eb}}{F_{meb}}

This is: the requested grip of a tire ( $G_r$ ) is the sum of current wheel speed ( $s_w$ ) divided by the maximum wheel speed ( $s_{mw}$ ), the steering angle of the tire ( $|sin(\theta)|$ ), and the current effective brake force ( $F_{eb}$ ) divided by the maximum effective brake force ( $F_{meb}$ ).

More simply, you can only ask a tire to do some combination of speed up, slow down, and turn. Asking it to do too much of these at one time will cause the requested grip ( $G_r$ ) to exceed the actual grip ( $G_a$ ) and lower the car's performance. Maintaining $G_r$ above $G_a$ for too much of a lap often signals that the car is being overdriven and that the driver may be fighting the car to get it to do what they want. This fight comes from demanding grip that just isn’t present at the moment.

Driving while keeping in mind that I must keep $G_r$ at or below $G_a$ forced me to pay closer attention to my entry speed into corners, strike better balance between my braking and turning in the middle of turns (i.e. proper trail braking), and keep an eye on how much power I was putting down on corner exits. I stopped fighting the car and started dancing with it. This culminated in smoother driving, faster lines, and lower lap times.

This may sound a little confusing over text alone. I'll have visual or interactive demonstrations of this model in the future for you to see. In the meantime, trust that this mental model is rather effective.

Update

May 30 2024

As expected, this concept already exists. It's known as the Circle of Forces or Kamm Circle. The circle represents the limit of grip of a car's tires. Points outside of the circle exceed this grip. A simple graphic of the concept is below.

Source: https://www.researchgate.net/post/What-is-Circle-of-Friction

As it turns out, my wheel speed and effective braking force terms were my mental models of Longitudinal Slip Ratio, and my steering angle term was my mental model of Slip Angle. These two properties can be used to calculate $F_x$ and $F_y$ in the above graphic via the Magic Formula.