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SPArKy_Dave

Brake Force Theory - INFO

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Pedal Ratio
The critical component in the braking equation is the pedal ratio. In operation, the brake pedal acts as a lever to increase the force the driver applies to the master cylinder. In turn, the master cylinder forces fluid to the disc brake caliper pistons or drum brake wheel cylinders. If you examine a brake pedal, you'll see the pivot point (where the pedal swivels) and the mounting point for the master cylinder pushrod are usually different. By varying the length of the pedal, and/or the distance between the pushrod mount and the pivot, you can change how much force (from your leg) is required to energize the master cylinder. This is the "mechanical advantage" or pedal ratio.

 

This formula will help you figure it out: Input Force x Pedal Ratio ÷ Brake Piston Area = PSI.

Mathematical babble?

The arithmetic simply equates to the amount of force exerted by your leg times the pedal ratio divided by the area of the brake piston(s). FYI, the typical adult male can exert roughly 300 pounds of force (maximum) with one leg, and that's a bunch. Something in the order of 1/3 or 1/2 that figure is obviously more comfortable, even in a hardcore racecar.

The average manual (non-power boosted) master cylinder requires somewhere between 600-1,000 PSI to be totally effective. Somehow, 100-150 pounds of leg force has to be translated into 600-1,200 PSI.

 

The way it's accomplished is by way of pedal ratio. While changing the overall length of the pedal is possible, it's often easier and far more practical to shorten the distance between the pivot point and the master cylinder pushrod mount location. That's precisely how many racecar chassis shops modify brake pedals.

Brake Line Pressure
Brake line pressure is a different thing than the force you apply to the pedal. Force acts in one direction and is addressed in pounds. Pressure acts in all directions against surrounding surfaces and is addressed in pounds per square inch or PSI. "Levers" (brake pedals) can be used to change the force. Inside the hydraulic system, the surface area of the piston is what is affected by pressure. Decreasing the bore size of the master cylinder increases the pressure it can build. Pistons in master cylinders are specified by bore size. But there's a hitch: The area of a circle (or bore) is Pi–R-Squared. The area of the piston surface increases or decreases as the square of the bore size or diameter. For example, the area of a common 1-1/8-inch master cylinder is approximately 0.994-inch. The area of an equally common 1.00-inch bore master cylinder is approximately 0.785-inch. Switching from the larger master cylinder to the smaller version will increase the line pressure approximately 26.5% assuming that pedal ratio hasn't changed.

As the pedal force or the pedal ratio (or both) is increased, the stroke of the master cylinder is shortened (brake line pressure is unaffected). When the size of the master cylinder piston increases, the output pressure of the master cylinder decreases. A smaller master cylinder piston will exert more line pressure with the same amount of force (pedal ratio) than a master cylinder piston with a larger piston area. There's another catch: Since the brake line fluid pressure is working against the surface of the wheel cylinder (or disc brake piston), increasing the area of the cylinder will increase brake torque.

The bottom line is, if the stopping power of a car needs improvement, or if there’s a need to reduce the pedal effort, several options are available: (1) Decrease the master cylinder bore size; (2) Increase the pedal ratio; (3) Increase the wheel cylinder bore size. If the pedal ratio is increased, there will be more travel at the master cylinder piston. If the master cylinder bore size is decreased, the piston has to travel further to move the same amount of fluid. Typically, a master cylinder has approximately 1-1/2-inch to 1-3/4-inch of stroke (travel). The idea here is coordinate the pedal ratio with the bore size to arrive at approximately half of the stroke (roughly 1-inch) in order to make the brakes feel comfortable, and of course, to bring the car to a grinding halt.

 

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Good write up. Thanks for sharing. 

If I can offer a criticism, the formula, while correct, does not have a question. The answer is PSI, but what question is it answering?

I think the question is: "how to find the force applied for a given braking effort."

 

You could probably, clearly define the relationships here with some statements such as:

1. At a consistent line pressure, an increase in piston area will result in an increase in braking force. 

2. An increase in piston area will require more pedal movement to maintain the same line pressure. 

3. However as braking force is increased due to an increase in piston area, less line pressure is required to acheave the same braking effort.

 

(Knowing this, you can see how rear and fwd braking biases is a consideration to ensure you don't unbalance the vehicle by increasing piston area on one end, vs the other. Coefficient of friction of the brake pad can be specified to correct for bias issues, as well as pressure bias adjustment in the rear braking circuit. )

 

A question I have here is, does that really equal that much more pedal movement? For the purpose of this we can assume the brake pad cannot be moved, it is already contacting the brake rotor and the materials of the pad and rotor are very Ridgid. So I'm interested to know how much an increase in piston area will atcually result in a requirement for more displaced fluid in the master/more pedal movement. I would expect in most cases you can increase piston size/count dramatically before there is a requirement for a change in master bore size or travel?

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