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#1235219 - 05/17/12 11:49 PM Disc brakes: All the math that you never wanted to know
feets Offline
Senior Management

Registered: 01/20/03
Posts: 26737
Loc: Irving, TX
Some folks liked the numbers that came up in the brake thread. I've thrown the most common stuff out in the open. Now, an engineer would be kicking and screaming about this stuff but it's good enough to get an idea what's going on.
I'll try to keep it simple for those who don't care much for math. These calculations build on one another. It's kind of a step by step thing.

Get a pencil, some paper, and a simple calculator. I use the calculator included in Windows.
Click your start button > All Programs > Accessories > Calculator.

It will be helpful if you write down what the numbers mean. Instead of 7 try writing "pedal ratio = 7" so you will remember what each of the numbers mean.

For this example I will use a common brake setup. Nothing exotic here. 11.75" Cordoba rotors, 2.75" slider calipers, manual 15/16" master cylinder, and a typical brake pedal.
Measure the stuff on your car. Don't use my measurements.


PEDAL RATIO

Before you start with crazy math, go find your brake pedal ratio. Measure from the middle of the pedal pivot to the middle of the pad on the brake pedal. Now measure from the middle of the pivot to the middle of the master cylinder push rod pin.

Divide the total length by the push rod distance. That is your pedal ratio.
14" pedal? 2" to the MC pin? Ratio is 14/2 = 7.
7:1 pedal ratio. Write that down. You will need it later.


AREA OF A CIRCLE (PISTON)

The very first thing you need to do is remember how to find the area of a circle. Your circle can be the master cylinder piston or the brake caliper piston.
It's pretty easy.

Pi R squared

Yes, the pies mom made were round but in physics Pi R squared.

Pi = 3.14 (number goes on forever but this is good enough)
R = radius of the circle (half the width of the piston)
squared = radius times the same radius.

Have a 2.75" piston? The radius is 1.375" (half of 2.75).
So, 3.14 times 1.375 times 1.375
The area of the piston is 5.93". Feel free to call it 5.9 inches.

Wanna cheat? Click here: http://www.calculateme.com/cArea/AreaOfCircle.htm

A note for people with multi-piston calipers: Only count the pistons on ONE side. Don't worry about why. Just count one side. It's the correct thing to do. Your numbers will be accurate and your headache will be smaller. Be sure to measure all the pistons on that side. They may be different sizes. Find the area of each piston on that one side. Once you do that, add the numbers together for a total area. Use that number for your caliper piston measurements.

Most master cylinders for our old cars are measured with fractions. Not good with fractions? Grab your calculator. Divide 1 by 16. Now you know 1/16 of an inch is .0625".
A 15/16 master cylinder is 15 times .0625 or .9375 of an inch. Plug that in to the above formula and the area of the master cylinder piston is .6902 square inches.


MASTER CYLINDER PRESSURE

Want to find the pressure made at the master cylinder?
You need the pedal ratio, the size of the master cylinder, and figure out how hard you want to push on the pedal.
I'm working with a MANUAL master cylinder. Throw a booster (vacuum or hydro) in the equation and you're on your own.

Multiply the pressure you push on the pedal times your pedal ratio. Divide your answer by the surface area of the master cylinder piston.
Using the numbers above, the pedal ratio was 7:1 and the master cylinder was .6902". Let's stomp on the pedal with 100 lbs of push.
100 times 7 equals 700. Divide that by .6902. You get 1014 pounds of force.


EFFECTIVE RADIUS OF A ROTOR

To work with the actual braking ability you really need to find what is called the effective radius of your brake rotor. To do that you need to know the caliper piston diameter (only the biggest one for calipers with more than one piston) and the diameter of your rotor.
Effective radius is the total diameter of the brake rotor plus that same diameter AFTER you subtract the diameter of the piston. Then, divide by 4.
Use a "big" Mopar 11.75" rotor and the 2.75" caliper we used above.
11.75" minus the 2.75" piston is 9 inches.
So, 11.75" plus 9" is 20.75".
Divide by 4 and you get a 5.18" effective radius.


BRAKE TORQUE

Now we're ready to find out what stomping on the pedal does for us.
To find the actual brake torque (stopping force of the brakes) we need the master cylinder pressure, the caliper area, and the effective radius.
Multiply the line pressure by the caliper area.
Take that number and multiply by the effective radius.
The result is your brake torque.
Since we were measuring in inches we need to divide by 12 to make it ft/lbs.
Our line pressure was 1014.
The caliper area was 5.9.
The effective radius was 5.18.
1014 * 5.9 * 5.18 = 30989 in/lbs
30989 in/lbs divided by 12 = 2582 ft/lbs.
Guess what? That's only one side of the rotor. The calipers press on both sides of each rotor so double that number.
You can have 5164 ft/lbs of braking on each front wheel with 100 pounds of pedal effort.
That's serious stuff! Think your tires can handle it?


BRAKE PEDAL MOVEMENT VS BRAKE PAD MOVEMENT

To find out how far your brake pads actually move when you push your brake pedal you need the pedal ratio, the master cylinder area and the caliper piston area. We're going to assume you've already taken up all the slack in the pedal and push rod.
Divide your master cylinder area by the caliper piston area.
Divide that number by the pedal ratio.
Divide that number by the 4 front brake pads (2 pads on the left and 2 pads on the right).

.6902" master cylinder divided by the 5.9" caliper piston area gives you .1169".
.1169" Divided by the pedal ratio of 7 gives you .0167".
Dividing by 4 front brake pads gives you a total of .0041" of movement.

So, every time you move that brake pedal one inch the brake pads each move 4.1 thousandths of an inch.
Brake pads are generally a couple thousandths of an inch off the rotor so they can't move far. If they did your pedal would fall to the floor.



Clear as mud?

That's enough math for tonight. It's late and I probably bumbled some of that info already.

Have a question or an issue? Ask away. I'll answer after I get some sleep.

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#1235220 - 05/17/12 11:56 PM Re: Brakes: All the math that you never wanted to know [Re: feets]
Frankenduster Offline
Moparts proctologist

Registered: 02/15/10
Posts: 14681
Loc: Granite Bay CA
THIS should get a sticky!
Thanks Feetsie!

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#1235221 - 05/18/12 12:01 AM Re: Brakes: All the math that you never wanted to know [Re: Frankenduster]
feets Offline
Senior Management

Registered: 01/20/03
Posts: 26737
Loc: Irving, TX
Thanks.

Let's hope it's all correct. I think it is but my mind is mush after a long day at work.

I'm gonna hit the sack and see if those numbers still make sense in the morning.
I didn't bother with coefficients of friction and some other engineering blah blah blah stuff so I'm sure others will chew it pieces. What do I know? I'm just a parts guy.


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#1235222 - 05/18/12 05:51 AM Re: Disc brakes: All the math that you never wanted to know [Re: feets]
BigBlockMopar Offline
master

Registered: 01/22/03
Posts: 3119
Loc: The Netherlands
Good write-up.


Quote:

Have a question or an issue? Ask away.



Since you asked... Where's the start button?



Attachments
7210944-StartButton.jpg (163 downloads)

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#1235223 - 05/18/12 06:23 AM Re: Disc brakes: All the math that you never wanted to know [Re: BigBlockMopar]
70Cuda383 Offline


Registered: 10/07/03
Posts: 21330
Loc: Marysville, O-H-I-O
WHo uses paper and pencil anymore?

build yourself an excel spreadsheet that does all that calculations for you, so that all you have to enter is the master cylinder bore, Caliper piston(s) bore, and the pedal ratio.
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#1235224 - 05/18/12 06:24 AM Re: Disc brakes: All the math that you never wanted to know [Re: BigBlockMopar]
70Cuda383 Offline


Registered: 10/07/03
Posts: 21330
Loc: Marysville, O-H-I-O
Quote:

Good write-up.


Quote:

Have a question or an issue? Ask away.



Since you asked... Where's the start button?






Oooh, I was gonna say, click on the little black circle icon at the bottom that looks like a tachometer with a red needle, it's called "dashboard" but...you don't have that on your desktop!
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#1235225 - 05/18/12 06:28 AM Re: Disc brakes: All the math that you never wanted to know [Re: BigBlockMopar]
DGS Offline
mopar

Registered: 07/08/05
Posts: 451
Loc: Graz, Austria
Very nice!

but maybe you should mention that this is for manual brakes only! Power brakes have a brake linkage between the pedal and the booster which effectively lowers the pedal ratio and shortens pedal travel! This means that 100lbs of brake force at the pedal results in only 300-400 lbs at the booster (vs. 700 lbs at the MC with manual brakes). The booster then adds a certain amount of force to that number (depends on booster size and if it's a single or a dual diaphragm)! Now the available brake power is greater than with manual brakes which allows you to use a bigger bore MC and pedal effort is lower than with a manual car.

The formula to calculate booster assist is:

diaphragm area (in square inches) times vacuum (in hg) times 1/2

example for a single 10" booster (radius is 5"):
5*5 * Pi (3.14) = 78.5
78.5 * 18hg vacuum = 1413
1413 * 0.5 = 706.6 lbs assist with a 10" single diaphragm booster

with a dual diaphragm area you just have to double the diaphragm area. e.g. dual 8" = 4" * 4" * Pi (3.14) * 2 = 100.48 square inch total diaphragm area = just 25% more than a single 10"!

IMPORTANT: This is for a booster with 100% efficiency - normally boosters work with about 80-85% efficiency. Just multiply the number you get by 0.8 to get a more realistic number!


Edited by DGS (05/18/12 06:52 AM)

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#1235226 - 05/18/12 06:29 AM Re: Disc brakes: All the math that you never wanted to know [Re: 70Cuda383]
70Cuda383 Offline


Registered: 10/07/03
Posts: 21330
Loc: Marysville, O-H-I-O
question kevin. on a dual sided caliper, such as my 4-piston Brembos from Viper or your 8-piston Brembos from Merc, you ignore one side of pistons because of how they push against each other and cancels out some of that line pressure.

Does that mean your brake torque calculation where you double it for the other pad, should not be doubled? or do you still double that number because you already accounted for the "loss" of piston force from the opposing pistons?


And what about the coefficient of friction between the pad/rotor? should that be accounted for somewhere?


---or is that some of the details that you said an engineer would throw a fit over?


while I'm on that subject, don't forget to subtract loss of line pressure from rubber brake lines deforming under pressure


Edited by 70Cuda383 (05/18/12 06:32 AM)
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#1235227 - 05/18/12 08:07 AM Re: Disc brakes: All the math that you never wanted to know [Re: 70Cuda383]
DARTH V8Я Offline
Oh No!! I just had a moron attack!

Registered: 11/01/11
Posts: 2154
Loc: Its a TRAP!
also need to take into account fluid expansion when the brakes get hot
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#1235228 - 05/18/12 08:10 AM Re: Disc brakes: All the math that you never wanted to know [Re: feets]
Uhcoog1 Offline
super stock

Registered: 05/12/05
Posts: 717
Loc: Houston Tx
Great post, Feets!

I just learned all of the math and found a few excel spreadsheets. There is a good excel probably 3/4 of the way down or so. It allows you to determine weight transfer at a specified G force stop, etc. I don't like his power brake calculations, though.

http://www.pirate4x4.com/tech/billavista/Brakes/

A couple things I learned along the way:
My duster has a 6.5:1 manual pedal ratio. When adding the power linkage, it drops to 3.5:1.
A dual master cylinder (side by side masters) split the input force, a tandem master does not.

I still have questions about halving the total piston volume in a fixed caliper, though. The above excel doubles the piston volume in a sliding caliper instead.
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-'02 Dodge Viper Ex-World Challenge racecar
-'73 Duster, 6.1 based 392 hilborn hemi, tko600, full floater rear 9", Hellwig custom bars, viper brakes, built for road course

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#1235229 - 05/18/12 08:21 AM Re: Disc brakes: All the math that you never wanted to know [Re: DARTH V8Я]
DynoDave Offline
master

Registered: 01/20/03
Posts: 2995
Loc: MICHIGAN
Thanks feets. This is super!

And yes, this should be in the tech archive.
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WPC #12304
1970 Plymouth Duster
1972 Dodge Charger Rallye
1977 Chrysler Cordoba

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#1235230 - 05/18/12 08:29 AM Re: Disc brakes: All the math that you never wanted to know [Re: DynoDave]
moparx Online   content
master

Registered: 09/03/03
Posts: 7810
Loc: north of coder
feets, i liked the pedal movement vs pad movement. i'v never seen that equasion before... THANKS !

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#1235231 - 05/18/12 08:51 AM Re: Disc brakes: All the math that you never wanted to know [Re: DynoDave]
runinonmt Offline
mopar

Registered: 05/19/05
Posts: 416
Loc: Franklin Co. Illinois
Quote:

Thanks feets. This is super!

And yes, this should be in the tech archive.





Ron
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#1235232 - 05/18/12 10:18 AM Re: Disc brakes: All the math that you never wanted to know [Re: DGS]
feets Offline
Senior Management

Registered: 01/20/03
Posts: 26737
Loc: Irving, TX
Quote:

but maybe you should mention that this is for manual brakes only!





Look under the part where I discussed master cylinder pressure. I clearly stated this was for MANUAL BRAKES ONLY. If you use a vacuum or hydro booster you are on your own.

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#1235233 - 05/18/12 10:48 AM Re: Disc brakes: All the math that you never wanted to know [Re: 70Cuda383]
feets Offline
Senior Management

Registered: 01/20/03
Posts: 26737
Loc: Irving, TX
Quote:

question kevin. on a dual sided caliper, such as my 4-piston Brembos from Viper or your 8-piston Brembos from Merc, you ignore one side of pistons because of how they push against each other and cancels out some of that line pressure.




The pistons are the same size left and right. That means you have to divide the measurements by two. When you finish the equation you have to add the numbers together to get a total. You just did extra work and cancelled it out in the end.

Quote:

Does that mean your brake torque calculation where you double it for the other pad, should not be doubled? or do you still double that number because you already accounted for the "loss" of piston force from the opposing pistons?




The torque calculation solves for rotor force. The brakes grab both sides of the rotor with the same force due to that whole equal and opposite reaction thing. Since you have drag on both sides it adds together.

Quote:


And what about the coefficient of friction between the pad/rotor? should that be accounted for somewhere?
---or is that some of the details that you said an engineer would throw a fit over?





You are correct. The coefficient of friction divides the brake torque. In the sample above we had the magic brake pads with a coefficient of 1.
This is indeed where engineers begin frothing at the mouth. Brake fluid compresses. Different fluids compress at different rates as temperature changes and water content increases. Rubber brake hoses stretch. Steel brake lines stretch. All kinds of other things come into play.

If you want to get technical, everything I listed is "useless" to a Formula 1 team. Pedal pivot friction, play between the mechanical bits, minute amounts of air trapped in the lines, and all kinds of other stuff changes the numbers.

I was going to build an Excel spreadsheet that would allow people to enter their measurements and get the answers but I have nowhere to host it.

This was meant to be a quick and dirty way to run the calculations.
It sure is dirty but there was nothing quick about it.

Hopefully it will help a few people get a rough idea of what's going on in the brake system.

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#1235234 - 05/18/12 12:20 PM Re: Disc brakes: All the math that you never wanted to know [Re: feets]
Uhcoog1 Offline
super stock

Registered: 05/12/05
Posts: 717
Loc: Houston Tx
Quote:



I was going to build an Excel spreadsheet that would allow people to enter their measurements and get the answers but I have nowhere to host





I'll host. PM sent
_________________________
-'02 Dodge Viper Ex-World Challenge racecar
-'73 Duster, 6.1 based 392 hilborn hemi, tko600, full floater rear 9", Hellwig custom bars, viper brakes, built for road course

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#1235235 - 05/18/12 12:32 PM Re: Disc brakes: All the math that you never wanted to know [Re: Uhcoog1]
feets Offline
Senior Management

Registered: 01/20/03
Posts: 26737
Loc: Irving, TX
Thanks!

I'll put something together.

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#1235236 - 05/18/12 02:30 PM Re: Disc brakes: All the math that you never wanted to know [Re: Uhcoog1]
MuuMuu101 Offline
I got lucky at Woodward!

Registered: 12/12/10
Posts: 12101
Loc: Covina, Ca
Quote:

Quote:



I was going to build an Excel spreadsheet that would allow people to enter their measurements and get the answers but I have nowhere to host





I'll host. PM sent




I want a copy of that program!
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#1235237 - 05/18/12 02:45 PM Re: Disc brakes: All the math that you never wanted to know [Re: MuuMuu101]
hooziewhatsit Offline
master

Registered: 10/05/07
Posts: 2551
Loc: State of Jefferson
I can host it too, if need be.

Now I need to calculate how far my pistons move vs. pedal movement. Also need to get a pressure gauge so I can see if my booster is actually doing anything or not
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#1235238 - 05/18/12 09:14 PM Re: Disc brakes: All the math that you never wanted to know [Re: feets]
Uhcoog1 Offline
super stock

Registered: 05/12/05
Posts: 717
Loc: Houston Tx
Excel file is available for download here:

http://www.3gduster.com/brakes.html

Enjoy!
_________________________
-'02 Dodge Viper Ex-World Challenge racecar
-'73 Duster, 6.1 based 392 hilborn hemi, tko600, full floater rear 9", Hellwig custom bars, viper brakes, built for road course

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