How to Slow Down Faster: The Math behind Brakes. Part 2: The Math

brake2

Part 1 is here.

     Last post we established that its the tires that stop a car; not the brakes.  The brakes (in physics terms) converts the kinetic (moving) energy into heat energy.  That is the reason why many racecars require larger than stock rotors.  Its not only to convert that energy quicker, but to be able to withstand all the heat without cracking/burning/smoking.

     We’ll start off with the equation: Brake Force(lb) = (Wheel Torque(lb)) / [(Tire Rolling Radius(in)) / (12)].  This equation says that the Brake Force of one corner of the car is found by using the Wheel Torque and Tire Rolling Radius.  How do you get those?  Read on.

     Now the fun starts.  I’ll first show you the equations, then I’ll explain it all afterwards.  Try to keep up as there are ALOT of equations to connect together:

mathbrake12

      Ok, so now that you’ve scanned the equations, I’ll try to break it down.  Here we have all the main compenents of the brake system, and their specifications.  From the Mu of the brake pads, to the size of the rotors.  These can all be obtained from the original manufacturer.  I’m going to assume you know a decent amount of brakes, so I won’t get into what each and every part is and what they do.

     We’ll start from the bottom this time, and explain our way to our original equation.

     The Inboard Caliper Piston Area is how large the piston(s) of the calipers are that push on  your rotors when you brake.  We find it by using simple math.  The area of a circle = radius^2 times pi.  You do this one time for each piston you have for caliper.  If you have 2 pistons, you need to do this equation twice.

     The Master Cylinder Pressure is how much it multiplies your foot’s force, magnifying your foot’s force so it can slow/stop a 3,000 pound vehicle.  You find that by dividing the Booster Output Force by the Master Cylinder Piston Area (which you use the area of a circle formula to get if you have the Master Cylinder Piston Diameter).

     So we multiply that, and we get the Caliper Clamp Force, or how much force the caliper pushes on the rotor.  Also, for floating calipers, you need to multiply the end result Inboard Caliper Piston Area by 2, which I’ve already added to the equation.  So if you’re calculating for fixed calipers, then don’t figure in the extra “2”.

     Now onto the next step.  We multiply the Caliper Clamp Force by the Mu of the brake pad and we get the Brake Pad Friction Force.  Remember, there are different types of brake pads for different purposes; anywhere from performance, being cost-effective, or endurance.  You can get the approximate Mu of the brake pad from the manufacturer.

     To get the Rotor Effective Radius, we need the rotor radius, and the caliper piston diameter.  You divide each by 2, and subtract the piston diameter from the rotor radius, and we get the Rotor Effective Radius.  Remember if you have two pistons, you divide each by two, then add them up for the total caliper piston diameter.

     Divide the Brake Pad Friction Force by the Rotor Effective Radius divided first by 12, and we get the Wheel Torque.

     Divide the Wheel Torque by the Tire Rolling Radius (which is the distance from the center of the axle to the ground) first divided by 12, and we finally get the total Brake Force for one corner of the car.  Remember, that the front and rear brakes differ, so please use the right specifications and measurements when using this equation.

     So now that we’ve deciphered how to find the Brake Force of one caliper/rotor, how do we apply that?  Well the next post will show you how to figure out the Vehicle Deceleration as well as Stopping Distance based on our findings here.

 

      EDIT: Most of these variables can be found through the manufacturer’s specifications sheets.  The only one I had a problem finding is the Booster Output Force.  There are equations to find that too, but the variables for those equations are also impossible to find.  I will make a seperate post about the Booster Output Force.

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