Brake Bias And Stability
This article grew out of two Ask Us Anything questions.
The following can be proved with a bicycle model:
Braking only on the rear axle has the lowest stopping forces, and lateral stability is reduced, especially when the rear tires are locked.
Braking both the front and rear axle at their maximum potential requires careful management of braking and lateral forces.
Braking only the front axle will maintain the majority of stopping forces, and steering ability is reduced, especially when the front tires are locked.
All of the above can also be experienced by riding a bicycle, though the risk of locking a bicycle front wheel is flying over the handlebars.
Mathematically, start by separating the forces on a tire so we can study them independently.
Vertical forces
1a. Vertical forces come from static mass, from longitudinal and/or lateral load transfer, from downforce, and from vertical acceleration.
Most physics classes teach the concept of friction as vertical force * mu.
That gives the total friction or grip.
This is a very good starting point.
More vertical force = more friction = more grip.
1b. Vertical force sensitivity:
For a tire, grip continues going up with vertical force.
As forces get higher, the grip eventually starts going up more slowly than the vertical force.
It's correct to say mu decreases with vertical force.
This is a second order effect.
I am not aware of any practical situation where more vertical force results in less overall grip.
https://en.wikipedia.org/wiki/Tire_load_sensitivity
Lateral Forces
2a. Lateral forces occur during cornering, or from gravity on banking.
Turn the front tires while traveling straight.
This creates a slip angle and a lateral force on the front tires.
The lateral load creates lateral acceleration and a yaw moment that begins rotating the car.
2b. The rotation creates a slip angle and lateral force at the rear tires.
The slip angle and lateral force of the rear tires increases until it balances the yaw moment of the front tires.
If the rear tires cannot generate enough lateral force to balance the yaw moment, the car will spin.
Braking forces * center of mass height create a moment about the center of mass, which is reacted by vertical forces at the tires.
The front tires have higher vertical force and lateral friction capability, and the rear tires have lower vertical force and lateral friction capability.
This is why it is easy to spin while braking.
2c. Slip angle is a second order effect on tire friction.
There is a point where increasing slip angle will not increase lateral load.
If the rear tires reach and go past the slip angle with peak lateral load, the car will start and continue to spin.
If the front tires pass the slip angle with peak lateral load, increase the steering lock, but the lateral load stays the same, or goes down.
Typically, in an understeer situation like that, the driver reduces the amount of steering back to the peak to regain control.
2d. Rear initiated steering is inherently unstable at the beginning of any change.
Turn the rear wheels toward the outside of a corner while traveling straight.
Rear slip angle and lateral force goes negative.
The yaw moment begins rotating the car, and creates a slip angle and lateral force at the front tires.
The rotation reduces the negative slip angle and lateral forces at the rear, bringing them past zero and going positive for balanced cornering.
If the rear tires reach and go past the slip angle with peak lateral load, the car will start and continue to spin.
2e. Turn the rear wheels toward the outside of a corner while already cornering.
Rear slip angle and lateral forces are reduced, or even go negative.
Car yaw increases, and the slip angle and lateral force of the rear tires increases until it balances the yaw moment of the front tires.
If the rear tires reach and go past the slip angle with peak lateral load, the car will start and continue to spin.
Longitudinal Forces
3a. Torque on the wheels from the drivetrain or brakes, including regeneration.
This requires the friction circle or ellipse concept.
If the tires are at their maximum lateral capacity, there is no more friction available for acceleration or braking.
If acceleration or braking torque is added to the tire, the amount of lateral force available for cornering goes down.
https://www.youtube.com/watch?v=JjCcFsGLpaM
3b. The harder the rear tires are braking, the less lateral force they have available.
If one rear tire locks under straightline braking, it loses braking force due to high slip ratio.
The higher braking force of the other rear tire creates a yaw moment.
The locked tire has very little lateral capability to balance the yaw moment.
The still rolling rear tire is probably close to locking, and also has very little lateral capability to balance the yaw moment.
Steering the front tires will be required.
3c. A typical tuning fix for entry oversteer is reducing the rear brake bias.
Reducing the amount of braking force on the rear tires increases the amount of rear lateral capability.
Increasing the amount of braking force on the front tires reduces the amount of front lateral capability.
If one front tire locks under braking, the higher braking force of the other front creates a yaw moment, but the still rolling rears have a better chance of having enough lateral capability to not spin.
Result: It is slightly more difficult for the front tires to generate enough of a yaw moment to exceed the rear tire lateral limits.
Other Second order factors affecting friction ellipses:
4a. Front tires having better camber for grip than rear tires.
4b. Rear tires have too much pressure or too little pressure.
Pressure also changes the shape of the contact patch, and potentially the location of the pneumatic trail (tire center of pressure).
Long, thin contact patches drive differently than short, wide contact patches with the same area.
4c. Rear tires are too hot or too cold.
Combined braking and turning is also known as “Trail Braking”.
5a. The very best drivers can go up to but not over the maximum lateral capability during braking and corner entry.
Under braking, the lateral limit starts very small.
This article shows the lateral limit is usually defined by the rear tires.
As they smoothly come off the brakes, the lateral limit increases due to both the friction circle, and more balanced loads front/rear.
They increase the steering to follow the lateral limit as it rises.
5b. When legal, anti-lock brake systems (ABS) and electronic braking distribution (EBD) are regarded as significant advantages in motorsport.
Both can be used to do more braking and trail braking closer to the friction limits of all four tires.
The systems can be mapped to pedal force, speed, steering, yaw, lateral acceleration, estimated mu, and more.
In Formula 1, LMP1, and Formula E, EBD is referred to as brake-by-wire (BBW) or bias migration (BMIG), which most importantly adjusts the level of friction braking depending on the level of regenerative braking from the electric motor.
Brake bias, regeneration map, and differential locking are the changes you can watch F1 drivers make on the steering wheel between each corner.
Asterisks
A car with extreme rear-weight bias like the Deltawing has higher relative stopping forces on the rear axle than the front axle.
Please, please wear a helmet when cycling.
Yaw should be thought of in terms of moment, acceleration, and velocity.
Yaw velocity is required for the car to change orientation as it goes around a corner.
Appendix: History of four wheel braking Theory
I prefer to go beyond claims of who was first, and get into all the messy details.
Today, many of these are unintentionally funny.
There is a lot of testing and a lot of detail design that goes into early brakes on moving suspension parts.
There are a lot of theories, but very little math, which should always be a red flag.
How much of my own “knowledge” have I failed to check the math on?
Years are approximate, and indicate the start of production rather than the start of development.
Pre-1885: For horse drawn vehicles, brakes (if any) were the easiest to create for the rear, non-steering wheels, often rubbing on the outside circumference.
1885-1909: Brakes on almost all automobiles are on the rear-axle, or even the transmission only.
Now disproved theory: Braking behind the center of mass is more stable than braking in front of the center of mass.
In practice: Cars spin through multiple rotations on wet roads at 30mph.
See test on page 358: https://www.jstor.org/stable/pdf/44723710.pdf
Combined with terrible tires, high centers of gravity, and suspension failures, cars are incredibly dangerous on a wide open track at modern highway speeds.
Images: https://en.wikipedia.org/wiki/1912_Indianapolis_500
1910-1928: Four-wheel braking is introduced and standardized throughout the automotive industry.
Initially, hand and foot controls activate the front and rear brakes separately. Don’t forget to downshift the sliding gear transmission.
Four-wheel brakes are first raced at Indianapolis by Isotta-Fraschini in 1913.
Multiple French manufacturers race four-wheel brakes in 1914.
The Duesenberg team uses hydraulic brakes, but must copy their competitor’s smaller rear brakes to stop spinning and crashing under braking.
https://en.wikipedia.org/wiki/1921_Grand_Prix_season
In 1924, the rising number of models with four wheel brakes are attacked with many now-disproved theories:
https://gahistoricnewspapers.galileo.usg.edu/lccn/sn88054099/1923-10-07/ed-1/seq-5/
- Four wheel brakes stop too hard, and will cause more accidents than they prevent.
- The components are too expensive, inefficient, and hard to maintain to be worthwhile.
- Locked front wheels make the car uncontrollable, but locked rear wheels are controllable.
Two sympathetic factors on control:
Potentially lower chance of a spin on roads of soft gravel, soft dirt, or soft snow.
Spins can happen faster with double the deceleration.
But equating weak braking with control is blatantly missing the point.
Frequent discussion in SAE Journals:
https://books.google.com/books?id=S_9DAAAAYAAJ&printsec=frontcover#v=onepage&q&f=false
In 1926, even early adopters are still figuring out the front brakes should be more powerful than the rear brakes.
https://steemit.com/cars/@icepack/eddie-rickenbacker
1928-1970: Power assisted braking is introduced, typically accomplished using intake manifold vacuum.
Vacuum pumps are necessary when no throttle is present.
In motorsport, independent front and rear brake circuits with bias adjustment begin replacing diagonal circuits with proportioning valves.
Example: 1955 Lotus Mk IX https://www.flickr.com/photos/16168315@N03/7108859541/
Mechanical anti-lock braking (ABS) is implemented in aircraft.
https://www.pistonheads.com/news/general/ph-origins-anti-lock-brakes/37433
1970-1995: Electronic ABS is introduced and standardized throughout the automotive industry.
https://www.roadandtrack.com/car-culture/car-accessories/a22811340/anti-lock-brakes-the-first-technology-to-help-you-avoid-a-crash-turn-40/
Often, to reduce cost, ABS is initially offered on the rear axle only.
Rear-only ABS becomes prevalent in pickup trucks with a wide range between empty and maximum payload,
Some now disproved theories are repeated: Ignoring the risk of a spin with locked rear tires, the front wheels should have ABS first to allow steering.
Complaints are repeated about ABS causing more rear-end crashes for skidding non-ABS cars.
The same components used for ABS are used to implement automatic bias adjustment (electronic braking distribution EBD).
By using ABS components to brake individual wheels outside of braking, stability control is developed and implemented.
1995-2020:
Brake-by-wire (BBW) is developed out of electronic ABS and EBD.
Sometimes, the term BBW is used interchangeably with EBD.
BBW is used for significant pressure reductions in a friction braking system to compensate for regenerative braking.
GM EV1: http://www.tzev.com/files/10_GM_EV1_Brake_System.pdf
BBW can also describe a 100% electronic, 0% mechanical connection between the brake pedal and the friction braking system.
Apocryphally, the Mercedes SLR McLaren retracts its airbrake spoiler below 100km/h to see and give space to following cars that cannot stop at 1.3+g.
