| The physics of bigger brakes, wheels, tires |
Subject: The physics of bigger brakes, wheels, tires by 92Polo on 2012/4/17 21:40:38 I recently discovered that with my new tires on a warm day, I actually could not lock up my factory (non heavy duty) brakes except for one brief instant. In general I guess you could say this is a good thing, if the wheels aren't locked you still have control. It just came as somewhat of a shock to learn that for the first time since I've owned the car my I have more traction than braking ability. I got to looking and found that the most economical way for me to improve braking performance is to go with the J55 option. Sure there are better brakes out there, but this is a 99.9% street care these days so it will be a nice upgrade for a nice price. This got me to thinking about what I would call a physics thought experiment. Because the J55 brakes utilize the same pads and therefore have the same braking "area" on the rotor, do they really outperform the standard brakes? Your first thought is "of course" but do they really? The reason I ask is that because the rotor is thicker and a larger diameter, it weighs more. The question then becomes, (if you ignore the larger heat dissipating capability of the larger rotor) does the increase in leverage in going from a 12" to a 13" rotor overcome the increase in rotation inertia? The answer my surprise you! Before I go on, let me say right now that these are "rough" calculations. Meaning when I am calculating inertia I am assuming an evenly distributed mass across the rotor from center to outer edge. This is not 100% accurate but it will suffice to illustrate the point. Braking torque is easy. The braking "force" will be a function of the calipers, pads, rotor surface, etc, etc. Since we are dealing with stock calipers that use the same pads, a lot of this can be ignored. Assume that the standard caliper and the J55 caliper will exert the same force on the pad. If the rotors are the same material we can assume that they will produce a certain amount of friction per unit of pad area. Since the pad area is the same, we can deduce that the "clamping" force will be largely the same (yes the J55 might flex a little more and reduce clamping force but we're playing theoretical here) and therefore the only variable is the location of the clamping force is applied. To arrive at braking torque we simply take a force * a lever arm distance. Since the "force" is the same, the only real variable is lever arm distance. Running the math you figure out that by going from a 12" rotor to a 13" rotor you will receive an 8% increase in braking torque, certainly not bad. But, lets look at the weight.... Rotational inertia is a funny thing. Anybody who has ever done the high school weighted sticks experiment has seen first hand that the farther the weight is distributed from the center of rotation, the more inertia said object will have. This makes the object harder to accelerate and harder to decelerate. I looked up the weights of a stock replacement standard rotor and a factory replacement HD rotor. Using my assumption above that the weight was evenly distributed (I know it's not but if anything the weight is probably concentrated AWAY from the center of rotation rather than towards it making this even worse than my assumption) I was able to calculate that the HD rotor has an 83% increase in it's moment of inertia! That's well more than enough to completely and utterly negate the increase in braking torque! So, this brings up the question, why do bigger brakes work? Simple, the brake rotors aren't the only thing turning. Because your wheels and tires weigh every bit as much as your rotors AND their weight distribution is far from the axis of rotation, they play a huge impact in the total rotational inertia of the system. So much so that (using my own wheels and tires as a rough estimate) going from standard to heavy duty rotors only increases the total moment of inertia of the system by 0.96%! When you run the numbers and solve for angular acceleration for a given force, you find that the 8% increase in braking torque results in a 7% increase in angular acceleration (deceleration really) of the wheel / tire / rotor combo. So... what does it all mean? Wait for it.... it means that larger brakes really do work. Hmm... that's rather anti-climactic isn't it? There has to be some real lesson to be learned from all of this... That is this: If you're one of those people who is racing and wanting to minimize the weight on everything to maximize your performance, you'll get a lot more bang for your buck by getting lighter wheels and tires than you will by getting lighter rotors. Of course, light weight rotors help, EVERY LITTLE BIT HELPS. But if you're looking for bang for your buck, you'll stop shorter and accelerate faster if you save weight on the wheels / tires than the brakes. It's simple physics. Dunno if anybody will get any use out of it at all but I was just pondering it the other day until it bugged me. Once I figured out the real answer I thought I would share. |
