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BRAKE CALCULATIONS

How do you calculate line pressure and pedal force for a brake system? If the line pressure or pedal force is too high, what can you do about it?

This question comes from the UNC Charlotte Formula SAE team, which I am helping this year after being uninvolved with it for several years. The team is trying to run a legacy design, making the fewest possible changes. The legacy design, however, has a persistent brake problem which caused the car to DNF in the endurance event in 2015 due to the brake overtravel switch being tripped, shutting off the engine. This was also a problem in the brake test that is part of tech inspection, and has occurred again when testing the car. The brake system does not have a sufficiently firm pedal, and also gets some heat fade in protracted use. This combined with the softness of the pedal results in the pedal traveling far enough to trip the required overtravel switch. The switch can be tripped in limit braking with cool brakes, and it becomes easier and easier to trip it as the brakes start to fade.

The system has some problems that are immediately apparent by a quick visual inspection. The master cylinders, which are under the driver’s heel beneath a removable panel, are mounted on their sides to reduce height. The banjo fittings through which the hoses from the remote reservoirs feed the master cylinders are at 3 o’clock and 9 o’clock, not 12 o’clock. That creates an unbled high spot inside the master cylinder. Two of the calipers are mounted with the bleed screws pointing down. These can be bled, but only by removing the calipers. The reservoirs are very nearly at the same height as the calipers, and there are no residual pressure valves.

In sum, there are a number of ways that air in the system might be causing soft pedal. If we are willing to live with taking calipers off to bleed the brakes, all these potential sources of air in the system can be addressed at the master cylinders, pedal, and reservoirs.

The question then becomes: is that likely to be the whole problem? Can we deal with this by revising the layout of the pedal, master cylinders, and reservoirs? Can we keep the existing uprights,

rotors, and calipers? Or are there further problems relating to the brakes themselves? It is to answer this that we need to see if line pressures and pedal forces are within reasonable limits.

The car is built very light. With the lightest driver in 2015, it weighed about 512 pounds. It uses a single-cylinder engine and ten inch wheels. Tire outside diameter is about 18 inches. Front and rear brakes are identical. Calipers are the very light AP CP4226-2S0. These are popular in FSAE due to their small size and weight, but they are designed for use on the rear of racing motorcycles. In the original application they are intended to actually produce less rear braking force than stock calipers, and also provide a small weight reduction. The reason for wanting less braking is that motorcycles on road courses, at least in the dry, are limited in straight-line braking by rear wheel lift rather than grip. Reducing the power of the rear brake makes it easier for the rider to modulate the rear brake when the rear tire has very little load on it.

AP themselves caution on their website that when these calipers are used in FSAE applications, there is a danger of fade, caliper flex and excessive line pressure. They say that line pressure needs to be kept below 1,000 psi, and that even approaching that places the calipers well outside the range that they see in motorcycle use.

It should be obvious why caliper flex will cause soft pedal. High line pressure will also reduce pedal firmness because the fluid has some compressibility and the hoses have some compliance. AP’s recommended guidelines, per US vendor Essex Parts, are that if line pressure in any brake system exceeds 1,000 psi, the system is undersized, and if it is less than 400 psi the system is oversized. An oversized system works fine; it’s just bigger and heavier than it needs to be. An undersized system is likely to have precisely the problems we are experiencing, especially with calipers optimized for lightness rather than rigidity.

Okay then, what line pressures should we expect our system to generate? First, we have to figure out what force the system has to generate at the contact patches. In our case, we have identical brakes front and rear, and roughly 50% rear statically. Some amount of the car’s weight will transfer forward in braking. So we know that we need to look first at the front brakes.

The team has not measured the car’s c.g. height. It’s probably somewhere in the 10 to 12 inch range. The wheelbase is 60 inches. How much of the car’s weight will transfer forward? We can work this out with free body diagrams, but here’s a shortcut: the portion of the weight that will transfer per g of rearward acceleration is the c.g. height divided by the wheelbase. For a 12” c.g. height, that’s 12/60 = 1/5 = 20% per g. For a 10”c.g. height, it’s 10/60 = 1/6 = 16.67% per g.

Essex Parts suggests assuming that the car brakes at 1.25g for purposes of calculation. That’s perhaps a bit lower than FSAE cars will achieve on a good surface with no downforce. With downforce, the accelerations can be substantially larger. The current car has no wings or other downforce generating devices, but the team is considering adding some.

Normally we might start with a heavy driver and a high assumed c.g. height. But let’s see what we get, given charitable assumptions: 512 pound total weight, 10” c.g. height, 1.25g braking.

Portion of the car’s weight that transfers is a sixth times 1.25. That’s 20.83%. The front wheels have 70.83% of the 512 pound weight on them. That’s 362.6 pounds for the wheel pair, or 181.3 pounds per wheel. For simplicity, we’ll assume that the coefficient of friction at the front contact patches equals the acceleration of the car in g’s: 1.25. Friction force at each front contact patch is then 181.3 times 1.25, or 226.7 pounds.

The radius of the tire is about 9 inches. The brake rotor is 7” diameter (3.5” radius) at the outside, and the pads sweep a surface on it about an inch wide. The acting radius is then about 3 inches. That’s about a third of the tire radius. Therefore the two brake pads combined have to generate a friction force roughly three times the contact patch force, or about 675 pounds.

According to Essex Parts, the pads have a coefficient of friction of about .42 to .40. Using the .42 value, the two pads have to press on the rotor with a combined force of 675 pounds divided by .42, or 1607 pounds, or about 800 pounds each.

The pistons are an inch in diameter. The area of each piston is the diameter squared times π/4, or .7854 square inches each. The hydraulic pressure needed to generate 800 pounds of force from a piston that size is 800 pounds divided by .7854 square inches, or 1,019 pounds per square inch. We’re over the recommended limit, using the most charitable assumptions and rounding of numbers. With a lower pad coefficient of friction due to fade or contamination, a grippier road surface, a heavier driver, a higher c.g., and/or some downforce, we could easily see 1,200psi or more.

We can conclude that the front brakes are seriously undersized. What about the rears? They only need to generate about 25 to 30 percent of the stopping force, so they will see at the most 30/70 or 43% as much pressure as the fronts. Even with a lot of downforce, they are within safe operating limits, provided that the master cylinders and pedal are sending them only as much pressure as is needed to have them lock at about the same pedal force as the fronts.

Although the front brakes are operating at higher than recommended pressure, there have not been any leaks or catastrophic failures. We just have a spongy pedal and some fade. We don’t really know how much of the compliance is due the high hydraulic pressure. We can reasonably predict that we won’t get really good operation as long as the front brakes are undersized, no matter what we do with the pedal and master cylinders. On the other hand, we know that the current configuration doesn’t permit proper bleeding, and we’re bound to get a significant improvement just from correcting that. We just don’t know how significant.

We also know that we can shorten the pedal travel by going to bigger master cylinders and/or a smaller pedal motion ratio. We can’t just let the pedal travel more, without changing the frame design. The overtravel switch is close to the front bulkhead now. Moving that forward means lengthening the frame, which in turn means the current nose won’t fit. Having the pedal further

rearward when the brakes are not applied is not an option either, because our tallest drivers can barely fit in the car now, and the rules require us to accommodate a 95^th percentile male. The only way to shorten the pedal travel via pedal or master cylinder changes is to add to the pedal force required, one way or another.

Well then, where are we now on pedal force? The front master cylinder is 5/8” diameter. That’s .307 square inches of piston area. 1,000psi acting on that piston produces a push rod force of just over 300 pounds. 1,200psi produces just over 360 pounds push rod force. The pedal motion ratio is 6.5:1, and there are two push rods. At mid-adjustment on the balance bar, force on each push rod is 3.25 times pedal force. Pedal force is then about 92 pounds for 1,000psi, or about 111 pounds for 1,200psi. Recommended pedal force for a 1.25g stop is 80 pounds. 100 is somewhat heavy but not unmanageable. More than that may be tolerable, depending on how strong the driver’s leg is and how long the event is.

The rear master cylinder is ¾”. That means that a mid-adjustment on the balance bar, the rear brakes are generating about 41% of the braking, not the 25 to 30% they would require if the front wheels are to lock before the rears. At around 1g, at mid-adjustment on the balance bar, the rear wheels lock. As further pedal force is added, the rear calipers continue to deflect, the fluids in the lines continue to compress, the hoses continue to swell, but none of the added pressure does anything to stop the car because the rear tires are already contributing as much retardation as they can. Probably the car has this much rear brake to help crutch understeer.

In any case, some reduction in deflection can be had by not overpressuring the rear brakes. This can be accomplished by balance bar adjustment up to a point. Beyond that, either a larger rear master cylinder or a proportioning valve in the rear line could be used. Using a proportioning valve along with a balance bar makes sense because it allows us to have less rear brake percentage at high apply pressures, where forward load transfer is greatest, without having premature front lockup in conditions of poorer grip, and still having enough rear brake to free the car up in trail braking.

A rear proportioning valve would not reduce pedal force requirement, and it wouldn’t reduce deflections occurring on the master cylinder side of the valve. However, it would reduce deflections occurring on the caliper side of the valve.

Hopefully, this discussion provides some useful information about how to calculate hydraulic pressures and pedal forces in brake systems, and also provides some insight into the complexities of deciding what remedies to apply to a system that has multiple shortcomings.