When I first started in motor sports we didn’t have a data logger and being curious I thought the only way to learn more about how the car worked on the track was to write a simulation. Again, at the time, there wasn’t much information about this sort of thing so I had to start from scratch. My correlation was simply asking the driver to remember RPM in different corners and at the end of different straights!
The tracks were created from aerial photos (not from Google, I had to buy them!) correlated against track distance stated in the program of the race. I then traced the tracks into AutoCAD and created a DXF driving line. The first simulations were simply straights and corners. Why was this accurate enough for the first development stages? The answer lies in simple integration of a parabola. You can use algebra to solve for the area under the curve, or you can do a number of approximations based on numerical methods. Best to look again at High School maths to prove this one to yourself.
The first simulation was based on distance as the integration variable. So each solution step was variable in length: I typically used 1m steps. This dictated the basic formulae that were used for the simulation
V^2 = U^2 + 2as
V=final velocity of each step
U = initial velocity of each step
a = acceleration
s = distance (or step size for this simulation)
F (longitudinal) = m . a
F = force: from engine torque, drag from tyres and drag from aerodynamics. There will have to be a simple gearbox included as well.
m = mass
a = solve for acceleration to put into the first formula
The we simply have some limits on how fast the car can traverse a given corner of radius R. We get that formula from centrifugal acceleration.
F (lateral) = m . V^2 / R
V = the constant maximum speed around a given corner
R = the radius of the corner
m = is the mass of the vehicle
Hold on, we need to solve for the maximum speed around a given corner. So one last formula, one that tells us what the maximum force can be. Force is proportional to the vertical load and the grip.
F(Lateral) = mu . N
N = vertical force. This is proportional to the downforce of the vehicle and its mass. Another important thing you will notice here is the ratio between the downforce and the mass: notice particularly that the downforce is non inertial. What I mean is that you get more corner speed for free! This is a very important fact in motor racing.
mu = friction coefficient. Probably one of the most complicated variables in our simulation and one that would require a PhD and lots of engineers, or perhaps an entire Formula One team to solve. Mu is proportional to road surface, temperature of the tyres, their compounds, constructions and running pressures, just to name a few. Stick with using 1 to start with!
Thats the basic formulae required for writing a simulation, everything else relates to how accurate you can get all of the unknown variables as they effect aerodynamics, tyres etc.
So why did I go to all this trouble to explain how a simple simulation works? It has a direct effect on your understanding of the 1st order performance drivers of motorsports performance:
- Engine (power) is proportional to the acceleration (a) variable, DIRECTLY.
- Aerodynamic downforce is DIRECTLY proportional to the normal force (N)
- Aerodynamic drag is DIRECTLY proportional to the longitudinal force (F) and engine power (F)
- Tyres grip (mu) is DIRECTLY proportional to the maximum lateral force in a corner
How do the second order variables effect the car? Wait for the next blog post coming soon…