Brakes and Physics. [Jason's Next Lecture]

[jason@scirocco.org (16V Jason)]

At 08:40 PM 3/25/2002, Brian McGarvey wrote:
>Ok Folks...
>I got tired of the argument. So I've whipped out the Physics book. Any
>ME's out there or ppl that are better at ACTUAL REAL non-Relativistic or 
>mumbo jumbo my ass
>tells me... physics

3.5 years of MechE and 18 credits away from graduating with a BS in MechE 
-- couldn't deal with the idea of designing toilet valves for the rest of 
my life; switched majors.  I'm also CC'ing this to a friend of mine with a 
PhD in Physics to make sure he doesn't find any inconsistencies.


>So that means that in order to get a larger negative acceleration a larger
>F is required.

Correct.

>Assumption. Your tires are not crap. Your tires are relatively sticky. ie
>they have a decent coeffienct of friction and maximum force point before 
>they turn to
>dust/liquid/gas.

Okay, I'll one-up you.  More than decent -- let's take a Z-Rated 
Bridgestone Potenza RE-71 in 195/50-15 on a Scirocco, inflated to 42psi on 
a 7x15" wheel.  Or the stickiest road tire you can find.  Either one.


>so now you are trying to increase the force to increase the negative
>accleration. no mind you were are still living in the linear force section 
>of the friction curve.

Right.  Your argument is that you have to increase the force acting on the 
car; thus the force created by the brake; in order to increase the negative 
A and therefore decrease the stopping distance.  (s=v1t + 0.5at^2) where 
s=distance in meters, V1 is the starting speed, a is acceleration in ms^-2, 
and t is the time.


>(FORCE lesson)
>IF the force applied to the disc is constant. the ONLY method for
>increasing the torque applied to the rotating mass is to increase the 
>distance from the center.

Right.

<snip>

>so now you know that to get more force with the same pads, calipers, and
>discs it takes a bigger master clyn to get more force.

Right.


>What does increasing the size of the rotors/discs do?
>by using the SAME size calipers on a larger diameter disc with the same
>CONTACT patch. ie. the Force being applied to the brakes using the same 
>master clyn and
>caliper are the same, but the distance from the center of the disc is now 
>longer. so now with a
>little simplified physics magic. Im sure the ME's here will expand on the 
>physics.
>so there..

So there!?
Uh, Brian, you're neglecting one hugely important part of this equation, 
i.e. the limits of adhesion of the tire.

Let's do a little experiment.  You can do this in real life if you want 
(Kids, don't try this at home).
Take a stock Scirocco 8V with 9.4" brakes on the front of it.  These, 
according to your (correct) logic, put out the lowest amount of braking 
force because they're the smallest.  Now, shod the front wheels with the 
stickiest tires you can think of.  The RE71s I mentioned before will serve 
as a wonderful example.

Accelerate the vehicle to its top speed of approximately 108mph.  Then, 
STOP on the brakes.  What happens?
The wheels lock up immediately, and you hear a loud, screetchy sound 
(technical term).  If you're lucky, you stop in a straight line and don't die.

Now, tell me:  Of what benefit would it be if you increased the braking 
force by 2?  Or 10?  Or 500?

None.  Absolutely nothing.  Why?  Because even the small 9.4" rotors are 
able to put out enough force to lock a turning RE-71 at top 
speed.  Therefore, the bottleneck here -- or the "weakest link" is the 
amount of force that the tires can transmit to the ground.  Once that wheel 
is locked, your brakes have nothing more to do with the equation:  It 
becomes a function of the dynamic coefficient of friction of your tire on 
asphalt.  If your brakes are powerful enough to lock your wheels while 
driving, that means that they are capable of creating more braking force 
than the tires can handle.  Increasing the force from the brakes will do 
nothing at all; the only thing you can do to increase A (and shorten 
braking distances) is to increase the tire's grip or reduce the weight of 
the car.

And if your brakes aren't strong enough to lock a wheel?  Well, then your 
car shouldn't be allowed on the road.  In fact, on a healthy brake system, 
it should be quite easy to do so.

So again, by moving the caliper further out on a larger disc, you are 
increasing the distance out from the axle, and therefore increasing the 
torque acting on the axle, increasing braking force - up to the point where 
the tire locks up.  Since you should be able to lock up your wheels under 
any circumstances, it's a fruitless effort.

What it *does* do is (as you stated), increase braking force for any given 
pedal pressure.  So the brakes will feel more responsive.  But as I said in 
my previous post, if you don't adjust the proportioning of the system, your 
rear wheels will be doing proportionally less work and therefore overall 
braking distance is increased.


>and fade can occur for several basic reasons.
>* the amount of force at the wheel decreases due to losses in the system
>   -- ie. the master clynder hits the stops and has no more volume of fluid
>to press.
>      -- losses can be due to a leaky master clyn, leaky wheel clyn.
>      -- the rubber hoses expanding enough to lower the volume of fluid
>transferred to the wheel clyn there by reducing the force applied.

All of which are mechanical problems that should be addressed before even 
considering to upgrade the system.


>* the coefficient of friction decreases at the disc/lining interface.
>   -- get better pads.
>   -- cool the system.

True, although there is no such thing as a fade-free pad; so there is a 
limitation here.


>           they actually pick up loose dirt.

The RE71s do that dirt thing too. :)  It's cool to have a 150hp vacuum 
cleaner. :)


>does improving the front brakes alone make a difference.
>I suppose its time to try it out.

In theory, as long as your existing front brakes can lock your wheels, it 
cannot help you.  And without ABS or an adjustable brake proportioning 
valve, it can only hurt.   Here's why:

[Larry, this is your cue to pay attention] :)


The amount of weight on a wheel is part of the equation on how much grip 
the tire will have.  The more weight, the more grip.  (Counter-intuitive, I 
know but think about it this way:  Take a tire, stand it up, and drag it 
across the floor.  Now have your friend sit on it and try to drag it across 
the floor.  Your friend's fat ass will make it a lot more difficult to 
drag.  'Nuff said.)  Under braking, there is a large transfer of weight to 
the front axle.  However, in most cars, including ours, there is still a 
significant amount of weight on the rear axle.

As you approach 100% weight transfer, you come to a situation where, under 
full braking, the rear wheels are providing only negligible braking.  Under 
these conditions, a front-only brake upgrade (assuming that the brakes were 
not powerful enough to lock up the wheels throughout the whole stop before) 
would help shorten the braking distances.

If you have any significant weight left on the rear wheels, however, those 
rear wheels are still contributing somewhat to braking.  You can see this 
demonstrated very clearly in the early and mid-1980s leaflets that auto 
manufacturers were required to put in new vehicles showing braking 
distances with fully functional braking systems and comparing them with 
front-only and rear-only functioning systems.  (Don't ask why our stupid 
government required these when braking systems, by their own law, are 
dual-diagonal anyway, not dual front-rear).

So let's use an example to illustrate.  You know that most cars are 
proportioned so that the front wheels will lock up before the rears.  So, 
let's say (for math's sake) that you need 100psi of pressure in the brake 
lines to lock the front wheels.  And let's say you need 20% more pressure 
to lock the rears as well, so 120psi.

So, let's say you want to threshold brake -- so you hold the pedal at a 
system pressure of, say, 95psi.  To get the rears at 95% braking, you'd 
need to provide 95 x 1.2 = 114psi of pressure.  Follow me?

So therefore, at the 95% lockup threshold on the front wheels, the system 
is at 95%, which is (95/114 = 0.83) at 83% of the rear wheels' braking 
ability.  Again, 95psi will put the front wheels at 95% of their ability 
and the backs at 83%.  You want that -- you don't want your rears to lock 
up first, lest you lose directional stability. (We all know what happens 
when you yank the E-Brake and the rear wheels lock).

Okay, so now let's say you upgrade the front rotors and calipers to larger 
units.  As we discussed above, we now need less pressure to perform the 
same amount of braking... let's say 80psi is now enough to lock the front 
wheels.  The rear wheels still need 114psi to lock, mind you.  So now, you 
want to perform 95% threshold braking on the fronts.  That means you want 
to have (80 x 0.95) = 76psi of pressure in the system at front lockup.

Just like last time, you're managing 95% of the front tires' traction.  But 
now where's the back?  At a reduced system pressure of only 76psi (compared 
to 83psi before), you're now working at (76/114) = 66.7%  So now, instead 
of providing 83% of the tire's maximum braking force before locking, you're 
providing less than 67%.  That's a 16% drop!  Granted, the majority of the 
weight is on the front of the car, so that doesn't mean overall braking 
force will be reduced by a full 16%, but by that logic, it _will_ be reduced.

Okay, this simplified illustration assumes a few things; most importantly 
that braking force increases linearly with brake system pressure.  While 
this is not a perfectly linear relationship, it doesn't matter.  As long as 
it's not an inverse relationship (which it can't be -- that would mean 
applying less pressure to the pedal increases system pressure), the exact 
nature of the relationship is unimportant.... the effect can either be 
negligible or negative... not positive.


>we'll have to have everybody with all the diff combinations all get the
>same pads.  change em. and bed them in properly. verify that all brake 
>systems work
>properly. and do panic brake tests on the runway measuring distance. with 
>different
>drivers swapping cars.

If we had ABS (which would be a completely different ballgame), that would 
be a good idea.  But not only is it dangerous, but it's a waste of 
time.  With a properly working system, the only factors involved are the 
tires, the surface, the weight, and the suspension geometry.


>but my prediction based on the number of times ive panic braked on
>motorclcyes where i get to actually modulate the rear brakes seperately 
>from the front is that as
>long as the rear brakes are adjusted properly according to the bently the 
>larger brakes in
>the front will decrease the stopping distanc.

Um, but brake proportioning is not adjustable on our cars...


AND NOW THE CONFESSION:
---------------------------------------
There is one argument that no one has brought up yet -- and that is the 
real reason why manufacturers actually DO put in larger brakes for better 
braking.  And that is for better heat dissipation -- for all the reasons we 
discussed yesterday.  The problem is that, while all braking systems (when 
working properly) are more than capable of locking up all 4 wheels under 
any conditions, they heat up quickly on full-braking stops from high 
speeds.  So much so that in fact, on a full stop from 120mph in a Toyota 
Tercel, you're likely to have so much fade that you can't lock up the 
wheels at all by the time you hit 40.  That calls for brakes with better 
heat dissipating abilities.

Remember, German cars can't get away with that.  They need to be able to 
stop from their top speeds on Autobahnen without experiencing enough fade 
to prevent the brakes from extracting the most grip possible from the 
tires.  So rev your Scirocco 16V up to 123mph and threshold brake back down 
to 20mph.  Chances are, if you then mash the pedal, you'll still have 
enough braking ability left to lock up the front wheels.  If you jumped 
back up to 80 as fast as you could and did it all again, would that still 
be the case?  Probably not.  And if that's what you need to do, then you 
need to get brakes with better heat dissipation abilities... not 
necessarily brakes with a further fulcrum length from the axis of rotation.


Phew.  Sorry for the long post guys, but you know how I get. :)
Jason

PS:  If you're not long on the Scirocco List long enough to remember, this 
brings us full-circle to a debate a few years back over the efficacy of 
Anti-Lock braking.  ABS's ability to modulate each wheel individually 
regardless of weight transfer, split-Mu surfaces (where one side of the car 
is on different pavement than the other, or on ice for that matter), or 
each individual brake pad's ability to grip.  In theory, you could put 4 
different brake pads, rotors, and calipers on an ABS-equipped car, and as 
long as they don't fade to the point where they can't lock up, overall 
stopping distance would be the same.  Scary as shit from a handling 
perspective, but the same.

PPS:  And from this argument, you should also see why cars with more 
rearward weight bias (Rear-wheel drive cars, for example -- or rear-engine 
cars), have a huge braking advantage -- less weight transfer onto the front 
wheels means more braking ability from the rears... and less need for 
uneven brake proportioning and smaller rotors in back.