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E46 M3 (20012006) Engine: S54  Max Hp: 333 hp at 7,900 rpm / 262 lb/ft at 4,900 rpm Total Produced: 45,000+  Years Produced: 2001 to 2006. 

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Wed, Jul042012, 04:07:45 PM  #1 
Naturally Agitated

How to Calculate Brake Bias
Almost as touchy a subject as the existence of the God (HiggsBoson) particle, the true winner of a French civil war, or the effectiveness of nuclear proliferation in the middle east, is that of E46 ///M3 brake bias.
I aim to clear the confusion by posting rationale, equations, diagrams, and sample calculations for consumption en masse with respect only to the braking system itself. I do not aim to explain how brakes work except with relation to the formulas themselves as it's been timelessly reiterated in just as many different ways all over the Internet. I also will not factor tire widths and the coefficient of friction between various tire and surface combinations. Though I should really start with rationale and theory, I'm going to get to the core of it all  equations, because that's what everyone wants. Please do not hesitate to contact me by whatever means necessary with corrections to any part of this post  it is meant to serve as a reference!  Dimensions required: (default E46 M3 specifications listed in parentheses)
Constants:
Variables:
The first element necessary to get started is the individual line pressures generated for both the front and rear pistons within the master cylinder, obtained by Equation 1: Equation 1: P = F_{pedal} / A_{piston}
The Force exerted by the pedal is obtained by Equation 2: Equation 2: F_{pedal} = F_{leg} · Ratio_{pedal}
The Area of the master cylinder piston is obtained by Equation 3: Equation 3: A_{piston} = pi · r_{piston}^{2}
Equation 4 is used to determine the radius from a given diameter: Equation 4: r_{piston} = D_{piston} / 2
Brake torque generated at the wheel, individually T_{f} and T_{r}, is given by Equation 5: Equation 5: T_{{f,r}} = P · A_{total} · R_{effective} · c_{F}
The total combined effective area of the caliper pistons is obtained by utilizing Equations 6, 7, 8 and 9 based on their respective caliper configurations (Floating: singlepiston, Floating: 2piston, Fixed: 4piston, Fixed: 6piston, Fixed: 8piston): For a Floating: single piston caliper: Equation 6: A_{total} = 2 · pi · r_{piston}^{2}
For a Floating: 2piston and Fixed: 4piston caliper: Equation 7: Atotal = 2 · pi · (rpiston_1^{2} + rpiston_2^{2})
For a Fixed: 6piston caliper: Equation 8: Atotal = 2 · pi · (r_{piston_1}^{2} + r_{piston_2}^{2} + r_{piston_3}^{2})
For a Fixed: 8piston caliper: Equation 9: Atotal = 2 · pi · (r_{piston_1}^{2} + r_{piston_2}^{2} + r_{piston_3}^{2} + r_{piston_4}^{2})
The effective radius is calculated using Equation 10: Equation 10: R_{effective} = r_{rotor}  (w_{pad} / 2)
Finally, Equation 11 gives the ratio of the front brake torque to that of the rear and is an index reflecting overall front:rear bias: Equation 11: Bias = T_{f} / T_{r}
Equation 12: % Bias_{front} = (T_{f} · 100%) / (T_{f} + T_{r}) % Bias_{rear} = (T_{r} · 100%) / (T_{f} + T_{r})^{*}
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Calculating Brake Bias  Brake Bias Cross Reference Chart  Brake Fluid Specs  eMail contact FREE DIY DME (ECU) TUNING: 6MT shift lights, sport mode memory, VMax deregulation, dyno rev limit delete, DTC suppression, secondary air pump delete, TunerPro checksum correction for E39 M5/E46 M3; modify rev limits, throttle response (sport/comfort), oil temp indicator calibration, SMG Auto fix w/gear swaps, CSL DME rev match functionality, Dynamic DME Tune Switching, Pseudoflash counter software reset, and much more  PDF Howto (Updated 08.28.2014) Last edited by p0lar; Wed, Mar272013 at 01:37:30 PM. 
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Wed, Jul042012, 04:08:14 PM  #2  
Naturally Agitated

Re: How to Calculate Brake Bias
Edit by TerraPhantm 01.09.18
I've unembedded p0lar's pictures (though the attachments are still present) as they were causing too much confusion and misinformation. The master cylinder does *not* impact brake bias. This was mathematically and experimentally proven by e36 332ti You can see further detail in his posts here (reproduced below): Quote:
Quote:
Quote:
This is a quick chart I've constructed using Excel to illustrate the bias generated by various combinations of brake calipers/rotors. Please eMail me any additional caliper/rotor/pad combinations you would like to see crossed, and dimensions of the pad swept area, rotor diameter and piston size and count, if possible. Edit  08.02.2012: Added WilWood SL6R to front calipers with 325mm rotor Edit  08.09.2012: Added C6 Z06 Corvette to front calipers with 355mm rotor Edit  09.23.2012: Added a multitude of calipers. Included FULL bias, inclusive of master cylinder as well as % bias front/rear. Added deviation from standard/ZCP E46 M3 biases in terms of %F (front) or %R (rear) movement  this could give more data to the layman. This was a massive update, please review to let me know if there are any issues with my calculations or data presentation. Edit  09.24.2012: Added 4Piston RacingBrake caliper. Edit  03.27.2013: Removed differentiation from OEM bias charts  they were causing way too much confusion. Added charts for differences in DSC (MK60 (>09/2002) vs DSC3 (<09/2002). Also added a small table to show calculated bias results for OEM E46 M3s and a warning for those who wish to downgrade to ZCP/CSL brakes on a DSC3based system from <09/2002. FULL BIAS INCLUDING MASTER CYLINDER PISTON DIFFERENTIATION E46 M3 with MK60based Master Cylinder (Front: 25mm, Rear: 22mm) (post09/2002) http://www.m3forum.net/m3forum/attac...1&d=1364397539 E46 M3 with DSC3based Master Cylinder (Front: 25mm, Rear: 20mm) (pre09/2002) http://www.m3forum.net/m3forum/attac...1&d=1364397540 Warning: If you replace your pre09/2002 DSC3based OEM E46 M3 brakes with ZCP/CSL brakes, including the rear caliper, you are doing yourself a grand disservice. This causes an almost 5% shift in rear bias from the actual ZCP configuration, which was never designed for the DSC3based master cylinder. Consider upgrading to a MK60based master cylinder at the same time  the MK60based powerbooster may also be required.
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Calculating Brake Bias  Brake Bias Cross Reference Chart  Brake Fluid Specs  eMail contact FREE DIY DME (ECU) TUNING: 6MT shift lights, sport mode memory, VMax deregulation, dyno rev limit delete, DTC suppression, secondary air pump delete, TunerPro checksum correction for E39 M5/E46 M3; modify rev limits, throttle response (sport/comfort), oil temp indicator calibration, SMG Auto fix w/gear swaps, CSL DME rev match functionality, Dynamic DME Tune Switching, Pseudoflash counter software reset, and much more  PDF Howto (Updated 08.28.2014) Last edited by terraphantm; Tue, Jan092018 at 09:31:09 PM. 

Wed, Jul042012, 04:09:03 PM  #3 
Naturally Agitated

Re: How to Calculate Brake Bias
Here is a sample calculation for an E46 M3 with the competition package. I apologize for converting to SAE, but most people here will comprehend ftlbs of torque much easier than metric units.
Front Master Cylinder Pressure Calculation: Equation 4: 25mm / 2 = 12.5 mm = r_{piston} Equation 3: (12.5 mm)^{2} * 3.141592 = 490.87375 mm^{2} = A_{piston} Equation 2: 150 lbs x 3 = 450 lbs = F_{pedal} Equation 1: 450 lbs / 490.87375 mm^{2} = 0.917 lbs/mm^{2} = 591 PSI = Pressure (front lines) Rear Master Cylinder Pressure Calculation: Equation 4: 22 mm / 2 = 11.0 mm = r_{piston} Equation 3: (11.0 mm)^{2} * 3.141592 = 380.1326 mm^{2} = A_{piston} Equation 2: 150 lbs x 3 = 450 lbs = F_{pedal} Equation 1: 450 lbs / 380.1326 mm^{2}= 1.183 lbs/mm^{2}. = 764 PSI = Pressure (rear lines)  Front Caliper Brake Torque Calculation: Equation 4: 60 mm / 2 = 30 mm = r_{piston} Equation 6: 2 * 3.141592 * 30 mm^{2} = 5655 mm^{2} = 56.55 cm^{2} = 8.77 in^{2} = A_{total} Equation 4: 345 mm / 2 = 172.5 mm = r_{rotor} Equation 10: 172.5 mm  (61 mm / 2) = 142 mm = 5.59 in = R_{effective} Equation 5: 591 PSI * 8.77 in^{2} * 5.59 in * 0.36 = 10430 in•lb / 12 in = 869 ft•lbs = T_{{f}} Rear Caliper Brake Torque Calculation: Equation 4: 46 mm / 2 = 23 mm = r_{piston} Equation 6: 2 * 3.141592 * 23 mm^{2} = 3324 mm^{2} = 33.24 cm^{2} = 5.15 in^{2} = A_{total} Equation 4: 328 mm / 2 = 164 mm = r_{rotor} Equation 10: 164 mm  (54 mm / 2) = 137 mm = 5.39 in = R_{effective} Equation 5: 764 PSI * 5.15 in^{2} * 5.39 in * 0.36 = 7633 in•lb / 12 in = 636 ft•lbs = T_{{r}}  Ratio of Front to Rear Torque Calculation: Equation 11: 869 ft.lbs. / 636 ft.lbs. = 1.37:1 F:R Ratio If you neglect line pressure, since that's usually constant unless someone changes their master cylinder, and simply look at the variables that are likely to change (rotor diameter, pad width, # & size of pistons, pad compound, etc), you would arrive at Equation 11 (modified): (8.77 in^{2} * 5.59 in * 0.36)/(5.15 in^{2} * 5.39 in * 0.36) = 1.77:1 F:R Ratio Theoretically, if the c_{F} of the pads is also not changing, or is equivalent, rather, then that could also be dropped from the equation to give a relevant base point for comparison.
__________________
Calculating Brake Bias  Brake Bias Cross Reference Chart  Brake Fluid Specs  eMail contact FREE DIY DME (ECU) TUNING: 6MT shift lights, sport mode memory, VMax deregulation, dyno rev limit delete, DTC suppression, secondary air pump delete, TunerPro checksum correction for E39 M5/E46 M3; modify rev limits, throttle response (sport/comfort), oil temp indicator calibration, SMG Auto fix w/gear swaps, CSL DME rev match functionality, Dynamic DME Tune Switching, Pseudoflash counter software reset, and much more  PDF Howto (Updated 08.28.2014) Last edited by p0lar; Wed, Mar272013 at 01:51:12 PM. 
Wed, Jul042012, 04:25:14 PM  #4 
If it ain't lightweight, it ain't the right weight.
Join Date: Jul 2011
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Re: How to Calculate Brake Bias
Very cool.

Wed, Jul042012, 04:34:48 PM  #5 
Registered User

This is all Chinese to me, pretty cool though
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Fri, Jul062012, 09:17:48 PM  #7 
Mechanical? Engineer?
Join Date: Nov 2003
Age: 45
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Location: Houston TX

Re: How to Calculate Brake Bias
Hear! Hear! Thanks for putting this up.
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Fri, Jul062012, 11:03:02 PM  #8 
Naturally Agitated

Re: How to Calculate Brake Bias
Well, someone had to do it. There's more coming  much, much more.
I wish this forum code would let you add sub/superscripts...
__________________
Calculating Brake Bias  Brake Bias Cross Reference Chart  Brake Fluid Specs  eMail contact FREE DIY DME (ECU) TUNING: 6MT shift lights, sport mode memory, VMax deregulation, dyno rev limit delete, DTC suppression, secondary air pump delete, TunerPro checksum correction for E39 M5/E46 M3; modify rev limits, throttle response (sport/comfort), oil temp indicator calibration, SMG Auto fix w/gear swaps, CSL DME rev match functionality, Dynamic DME Tune Switching, Pseudoflash counter software reset, and much more  PDF Howto (Updated 08.28.2014) Last edited by p0lar; Sat, Jul072012 at 12:04:36 AM. 
Fri, Aug032012, 02:31:26 AM  #9 
Registered User
Join Date: Mar 2008
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Location: Baltimore

Re: How to Calculate Brake Bias
Wow, thanks for maintaining this chart! The calculations shown here is great for figuring out your brake bias.
I found out the Wilwood setup I'm thinking of running will put me into 1.57:1 ratio, which is VERY rear biased Thankfully, if needed, I have a set of nonzcp rear calipers sitting around, and that will put me back into the 1.8:1 range. 
Fri, Aug032012, 02:49:20 AM  #10 
user formerly known as bmwbob
Join Date: Apr 2005
Age: 50
Posts: 13,436
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Location: New York #7/Fedic

Re: How to Calculate Brake Bias
welcome to 2002. this isnt the first brake bias chart! great work though!

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