Brake Calculations
There are many books on brake systems but if you need to find a formula for something in particular,
you never can. This page pulls them together with just a little explanation. They should work for any
two axle vehicle but it's YOUR RESPONSIBILITY to verify them. Use them at your risk.....
VEHICLE DYNAMICS
Static Axle Load Distribution
Relative Centre of Gravity Height
Dynamic Axle Loads (Two Axle Vehicles Only)
STOPPING THE VEHICLE
Braking Force
Wheel Lock
Brake Torque
FOUNDATION BRAKE
Disc Effective Radius
Clamp Load
Brake Factor
GENERATING BRAKING
System Pressure
Servo Booster
Pedal Force
REAL LIFE DECELERATION & STOPPING DISTANCE
BRAKE HEATING
Stop Energy
Kinetic Energy
Rotational Energy
Potential Energy
Braking Power
Dry Disc Temperature Rise
Single Stop Temperature Rise
Fade Stop Temperature Rise
PARKING ON AN INCLINE
Axle Loads
Traction Force
LOSSES FROM CABLE OPERATED BRAKES
HYDRAULIC BRAKES
Brake Fluid Volume Requirements
Foundation Brake Requirements
Pads Compressibility
Rubber Hose Expansion
Steel Pipe Expansion
Master Cylinders Losses
Fluid Compression
DYNAMOMETER INERTIA
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VEHICLE DYNAMICS
Static Axle Load Distribution
Note: this changes with the loading of the vehicle so laden and unladen figures are often different.
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Relative Centre of Gravity Height
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Dynamic Axle Loads (Two Axle Vehicles Only)
The changes in axle loads during braking bears no relationship to which axles are braked. They only
depend on the static laden conditions and the deceleration.
Note: The front axle load cannot be greater than the total vehicle mass. The rear axle load is the
difference between the vehicle mass and the front axle load and cannot be negative. It can lift off the
ground though. (Motorcyclists beware)!
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STOPPING THE VEHICLE
Braking Force
The total braking force required can simply be calculated using Newton’s Second Law.
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Wheel Lock
The braking force can only be generated if the wheel does not lock because the friction of a sliding
wheel is much lower than a rotating one. The maximum braking force possible on any particular axle
before wheel lock is given by:
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Brake Torque
Having decided which wheels will need braking to generate sufficient braking force the torque
requirements of each wheel need to be determined. For some legislation the distribution between front
and rear brakes is laid down. This may be achieved by varying the brake size or more likely using a
valve to reduce the actuation pressure.
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FOUNDATION BRAKE
Disc Effective Radius
The effective radius (torque radius) of a brake disc is the centre of the brake pads by area.
For dry discs it is assumed to be:
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For full circle brakes it is:
Note: the difference is because full circle brakes contact on the full face but caliper pads are not usually
a quadrant but have square sides (Given the variability of friction the difference is not important in
practice).
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Clamp Load
The clamping load is assumed to act on all friction surfaces equally. For dry disc brakes it doesn’t matter
whether the brake is of the sliding type or opposed piston. Newton’s Third Law state every force has an
equal and opposite reaction and a reaction force from a sliding caliper is the same as an opposed piston
one.
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Brake Factor
Ball ramp brakes have a self servoing effect rather like a drum brake. The brake factor multiplies the
output torque.
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Brake Sensitivity
High factor brakes become very sensitive to manufacturing tolerances and lining friction variations. A
measure of sensitivity is the amount the brake factor varies for a change in lining friction. It can be
calculated:
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GENERATING BRAKING
System Pressure
Pressure is a function of the required clamp load and the piston area. Remember on an opposed piston
disc brake it’s only the area on one side of the disc.
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Servo Booster
Servo characteristics are defined graphically. The output will have at least two slopes but will also have
a dead band at the bottom.
Pedal Force
The pedal ratio is calculated to the centre of the foot pad. The pedal return springs may make a
significant contribution to the overall pedal force. Especially at full travel.
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REAL LIFE DECELERATION & STOPPING DISTANCE
The deceleration used in calculations is a steady state one called MFDD (mean fully developed
deceleration). It assumes the vehicle is either braking or not. In practice it takes a time for the system
pressure to rise and the friction to build up. This is not the driver reaction time but the system reaction
time. Where a calculation requires a stopping distance or an average stop deceleration then this delay
must be taken into account. For calculations a linear build up over 0.6 second is used ie 0.3 second
delay.
For testing the following graph show the requirements for 71/320/EEC and ECE R13.
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BRAKE HEATING
Stop Energy
The energy dissipated in a stop is the sum of energy from three sources, kinetic, rotational and
potential.
Kinetic Energy
Assuming the stop is from the test speed down to zero then the kinetic energy is given by:-
Rotational Energy
The rotational energy is the energy needed to slow rotating parts. It varies for different vehicles and
which gear is selected however taking 3% of the kinetic energy is a reasonable assumption.
Potential Energy
The potential energy is the energy gained or lost by stopping on a hill.
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Braking Power
Only when the brake is applied (but rotating) is energy being dissipated in the brake system. Some of
the stop energy is dissipated in the tyre as wheel slip. Managing the ideal wheel slip is the ultimate goal
of ABS development but here assume 8%. The energy to each brake depend on the number of brakes
and the proportion of braking on each axle.
In order to calculate the power we need to know the brake on time:
The power is then given by:
This is the average power, the peak power at the onset of braking is double this.
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Dry Disc Temperature Rise
These calculation are based on that given in the following reference:
Brake Design and Safety 2nd edition by Ruldolf Limpert
Single Stop Temperature Rise
In order to approximate the temperature rise of the disc an assumption as to where the energy is going
has to be made. Initially most of the heating takes place in the disc, however this can then be rapidly
cooled by surrounding components and the air stream. The calculation assumes 80% goes to the disc.
Heat flux into one side of the disc:
Single stop temperature rise is:
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Fade Stop Temperature Rise
The temperature rise after repeated stopping can also be approximated, although so many variables
exist it is suggested this is only used for basic optimisation work.
After a number of stops:
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PARKING ON AN INCLINE
Axle Loads
When parking on an incline the lower axle has a higher load than it does on the level.
The rear axle load is the difference between the vehicle mass and the front axle load.
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Traction Force
If the braked wheel is very light on an incline then it is possible the tyre will slip before the brake. Hill
hold is usually required with the vehicle facing both up and down the hill. The traction force required to
park the vehicle is:
Where only one of the two axles is braked the limiting slope is:
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LOSSES FROM CABLE OPERATED BRAKES
Cable losses are not inconsiderable and vary depending on the number and angle of bends. A typical
cable supplier uses the following calculation to calculate cable efficiency:
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HYDRAULIC BRAKES
Brake Fluid Volume Requirements
When an hydraulic brake is applied fluid is required to move through the pipes. If the fluid source is a
master cylinder it has a finite capacity. The following components need fluid:Foundation Brake Requirements
Brake fluid is required to take up running clearance.
It is also needed to compensate for lack of stiffness of the brake housing. For a disc brake the following
approximation can be used:
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Pads Compressibility
Pad compressibility varies between hot and cold conditions. Worst case figures are 2% cold and 5% hot
at a pressure of 16MPa. The fluid required is given by:
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Rubber Hose Expansion
The rubber hose expansion coefficient is usually taken as
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Steel Pipe Expansion
Pipe expansion is very small and unlikely to be of interest however it should be noted that it is
proportional to the cube of the diameter, so using bigger pipe than necessary on a system with a fixed
fluid volume will cause longer travel for two reasons, the stiffness of the pipe and more importantly the
additional fluid compression losses.
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Master Cylinders Losses
Fluid losses in master cylinders increase with bore size and pressure. A reasonable assumption can be
found by using the following:
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Fluid Compression
Fluid compression varies with temperature and the type of fluid used.
The fluid needed to take account of compression is calculated:
It is usual to allow about 3% for trapped air in the circuits that can’t be removed by bleeding. This air is
squashed totally flat during braking.
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DYNAMOMETER INERTIA
When testing Brakes on a dynamometer it is important to calculate the inertia requirements.
Many brakes do not run at the same speed as the wheels so it is important to understand how the
brake
will
be
mounted
on
the
rig.
Ignoring the inertia of the wheels the required dynamometer inertia is given by