**Brian Beckman’s – The Physics of Racing.**

Introduction to the Physics of Racing article series

### PhOR – Introduction

I have been interested in racing since the age of three, and Physics since I learned about it in high school. There was once upon a time when I was writing a racing simulator and reading up on some heavy duty Physics stuff. Very interesting and exciting stuff. During that time I first read the...

### PhOR – Part 1 – Weight Transfer

Weight Transfer Most autocrossers and race drivers learn early in their careers the importance of balancing a car. Learning to do it consistently and automatically is one essential part of becoming a truly good driver. While the skills for balancing a car are commonly taught in drivers' schools, the rationale behind them is not usually...

### PhOR – Part 2 – Keeping your Tyres Stuck to the Ground

Keeping Your Tyres Stuck to the Ground In last month's article, we explained the physics behind weight transfer. That is, we explained why braking shifts weight to the front of the car, accelerating shifts weight to the rear, and cornering shifts weight to the outside of a curve. Weight transfer is a side-effect of the...

### PhOR – Part 3 – Basic Calculations

Basic Calculations In the last two articles, we plunged right into some relatively complex issues, namely weight transfer and tyre adhesion. This month, we regroup and review some of the basic units and dimensions needed to do dynamical calculations. Eventually, we can work up to equations sufficient for a full-blown computer simulation of car dynamics....

### PhOR – Part 4 – There is no such thing as Centrifugal Force

There Is No Such Thing as Centrifugal Force One often hears of "centrifugal force." This is the apparent force that throws you to the outside of a turn during cornering. If there is anything loose in the car, it will immediately slide to the right in a left hand turn, and vice versa. Perhaps you...

### PhOR – Part 5 – Introduction to the Racing Line

Introduction to the Racing Line This month, we analyse the best way to go through a corner. "Best" means in the least time, at the greatest average speed. We ask "what is the shape of the driving line through the corner that gives the best time?" and "what are the times for some other lines,...

### PhOR – Part 6 – Speed and Horsepower

Speed and Horsepower The title of this month's article consists of two words dear to every racer's heart. This month, we do some "back of the envelope" calculations to investigate the basic physics of speed and horsepower (the "back of the envelope" style of calculating was covered in part 3 of this series). How much...

### PhOR – Part 7 – The Traction Budget

The Traction Budget This month, we introduce the traction budget. This is a way of thinking about the traction available for car control under various conditions. It can help you make decisions about driving style, the right line around a course, and diagnosing handling problems. We introduce a diagramming technique for visualizing the traction budget...

### PhOR – Part 8 – Simulating Car Dynamics with a Computer Program

Simulating Car Dynamics with a Computer Program This month, we begin writing a computer program to simulate the physics of racing. Such a program is quite an ambitious one. A simple racing video game, such as "Pole Position," probably took an expert programmer several months to write. A big, realistic game like "Hard Drivin'" probably...

### PhOR – Part 9 – Straights

Straights We found in part 5 of this series, "Introduction to the Racing Line," that a driver can lose a shocking amount of time by taking a bad line in a corner. With a six-foot-wide car on a ten-foot-wide course, one can lose sixteen hundredths by "blowing" a single right-angle turn. This month, we extend...

### PhOR – Part 10 – Grip Angle

Grip Angle In many ways, tyre mechanics is an unpleasant topic. It is shrouded in uncertainty, controversy, and trade secrecy. Both theoretical and experimental studies are extremely difficult and expensive. It is probably the most uncontrollable variable in racing today. As such, it is the source of many highs and lows. An improvement in modelling...

### PhOR – Part 11 – Braking

Braking I was recently helping to crew Mark Thornton's effort at the Silver State Grand Prix in Nevada. Mark had built a beautiful car with a theoretical top speed of over 200 miles per hour for the 92 mile time trial from Lund to Hiko. Mark had no experience driving at these speeds and asked...

### PhOR – Part 12 – CyberCar

CyberCar, Every Racer's DWIM Car? The cybernetic DWIM car is coming. DWIM stands for "Do What I Mean."2 It is a commonplace term in the field of Human-machine Interfaces, and refers to systems that automatically interpret the user's intent from his or her inputs. Cybernetics (or at least one aspect of it) is the science...

### PhOR – Part 13 – Transients

Transients Obviously, handling is extremely important in any racing car. In an autocross car, it is critical. A poorly handling car with lots of power will not do well at all on the typical autocross course. A Miata or CRX can usually beat a 60's muscle car like a Pontiac GTO even though the Goat...

### PhOR – Part 14 – Why Smoothness?

Why Smoothness? I'm back after a hiatus of nine years. Time does fly, doesn't it? For those counting articles, the last one published was part 12; there is no Part 13. After such a long time away, it might be worthwhile to repeat the motivation and goals of this "Physics of Racing" series. I am...

### PhOR – Part 15 – Bumps in the Road

Bumps In The Road Brian Beckman PhD, and Jerry Kuch This month, we investigate how the effects of road bumps vary with speed. Everyone has experienced that bumps are more punchy as speed increases. A bump that you barely notice at 50 mph can sting at 100 mph. But what about at 200 mph? Will...

### PhOR – Part 16 – RARS A Simple Racing Simulator

RARS, A Simple Racing Simulator If you've been following this series, you know that I've been moving inexorably toward a computer simulation of racing. I've repeatedly debated with myself writing a new one completely from scratch versus starting with someone else's work. Ten years ago, when I started this series, the choice was easy. Since...

### PhOR – Part 17 – Slow In Fast Out

You may remember way back in part 5 that we did some simple calculations by hand to show that the classic racing line through a 90-degree right-hander is better than the either the line that hugs the inside or the line that hugs the outside of the corner. 'Better' means 'has lowest time.' The 'classic...

### PhOR – Part 18 – Slow In, Fast Out Continued

Physics of Racing, Part 18: "Slow In, Fast Out!" or, Advanced Racing Line, Continued In the previous instalment, we did exact calculations for a dummy line down a 650-foot entry straight, a 180-degree left-hander, and a 650-foot exit chute. Cornering radii vary from 150 feet to 200 feet, and the track is 100 feet wide...

### PhOR – Part 19 – Space, Time and Rubber

Space, Time, and Rubber In part 16, we introduced RARS, the Robot Auto Racing Simulator. We talked about the clever and simple tyre-friction model in RARS and gave a terse presentation of its details in the big table in the article. Here, we'll explain in a little more detail why the model is cool. First,...

### PhOR – Part 20 – Four Point Statics

Four-Point Statics In this instalment, we analyse the stability of a single wheel, a bicycle, tricycle, and, finally, of a four-wheeled vehicle. In the offing, we introduce force moments, vector cross products, matrices and linear algebra, and some interesting facts about how the number of wheels on a vehicle relate to the number of dimensions...

### PhOR – Part 21 – The Magic Formula: Longitudinal Version

The Magic Formula: Longitudinal Version Driving a car is a classic problem in control. Here, we mean control in the technical sense of control theory, an established branch of engineering science (once again, I find http://www.britannica.com to have a very nice, brush-up article on that term). In a more-or-less continuous fashion, the driver compares desired...

### PhOR – Part 22 – The Magic Formula: Lateral Version

The Magic Formula: Lateral Version In this installment, we review the other side of the magic formula: the one that computes lateral or cornering forces from slip angles (or grip angles). This formula is sufficiently similar to the longitudinal version of Part 21 that we can skip many preliminaries. But it's sufficiently different as...

### PhOR – Part 23 – Trail Braking

Trail Braking Trail-braking is a subtle driving technique that allows for later braking and increased corner entry speed. The classical technique is to complete braking before turn-in. This is a safer, easier technique for the driver because it separates traction management into two phases, braking and cornering, so the driver doesn't have to chew...

### PhOR – Part 24 – Combination Slip

Combination Slip The goal in this and the next instalment of the Physics of Racing is to combine the magic formulae of parts 21 and 22, so that we have a model of tyre forces when turning and braking or turning and accelerating at the same time. In this part, we figure out combination slip, and...

### PhOR – Part 25 – Combination Grip

Combination Grip In this instalment of the Physics of Racing, we complete the program begun last time to combine the magic formulae of parts 21 and 22, so that we have a model of tire forces when turning and braking or turning and accelerating at the same time. Parts 21 and 22 introduced the...

### PhOR – Part 26 – The Driving Wheel Chapter 1

The Driving Wheel, Chapter I Imagine 400 ft-lbs of torque measured on a chassis dynamometer like a DynoJet (see references at the end). This is a very nice number to have in any car, street or racing. In dyno-speak, however, the interpretation of this number is a little tricky. To start a dyno session,...

### PhOR – Part 27 – Four Wheel Weight Transfer

Four-Wheel Weight Transfer In this installment, we revisit the four-wheel statics of Part 20, solving the statics problem for level ground, which is very common in simulation. The problem is: given lateral and longitudinal forces, find the balancing vertical forces. In so doing, we introduce a conventional coordinate system and a new tool: Mathematica....

### PhOR – Part 28 – Hazards of Integration

Hazards of Integration The equations of motion are differential equations. Such equations tell us how to calculate "what's happening now" from "what happened a little while ago." They're called differential because they have the form of ratios of differences (qua subtraction or differentiation or derivatives) between "what's happening now" and "what happened a little...

### Physics of Racing – Part 29

The Physics of Racing, Part 29: A Magical Trick Brian Beckman, PhD ©Copyright February 2002 The Magic Formulae (Physics of Racing parts 21, 22, 24, 25) for grip versus slip have some disadvantages for the simulation programmer. Chief among them are the complicated mathematical structure and the large number of parameters. In this instalment, we...

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