Physics of Braking Systems

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  • Topic: Force, Drum brake, Friction
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The Physics of Braking Systems
By James Walker, Jr. of scR motorsports
Copyright © 2005 StopTech LLC
Author’s disclaimer: mechanical systems operating in the physical world are neither 100% efficient nor are they capable of instantaneous changes in state. Consequently, the equations and relationships presented herein are approximations of these braking system components as best as we understand their mechanizations and physical attributes. Where appropriate, several examples of limiting conditions and primary inefficiencies have been identified, but please do not assume these partial lists to be all-encompassing or definitive in their qualifications.

The Conservation of Energy
The braking system exists to convert the energy of a vehicle in motion into thermal energy, more commonly referred to as heat. From basic physics, the kinetic energy of a body in motion is defined as:

Kinetic Energy =



1
2
× mv × vv
2

where mv = the mass (commonly thought of as weight) of the vehicle in motion where vv = the velocity (commonly known as speed) of the vehicle in motion

Ideally, this energy is completely absorbed by the braking system. While this is not entirely the case, for a stopping event at maximum deceleration most of the vehicle’s kinetic energy is converted into thermal energy as defined by: 1

2
× m v × v v ⇒ mb × C p × ∆Tb
2




where mb = the mass of the braking system components which absorb energy where Cp = the specific heat of the braking system components which absorb energy (a constant based on material properties)

where ∆Tb = the temperature rise experienced by the braking system components which absorb energy

Note that for most single-stop events, the rotors serve as the primary energy absorbing components.

It follows then that the temperature rise of the braking system is directly proportional to the mass of the vehicle in motion. More importantly perhaps, the temperature rise of the braking system is directly proportional to the square of the velocity of the vehicle in motion. In other words, doubling speed will theoretically quadruple brake temperatures: In practical application, tire rolling resistance, aerodynamic drag, grade resistance, and other mechanical losses will also play an energy-absorbing role, but value is still placed in establishing this fundamental relationship as a limiting condition. The Brake Pedal

The brake pedal exists to multiply the force exerted by the driver’s foot. From elementary statics, the force increase will be equal to the driver’s applied force multiplied by the lever ratio of the brake pedal assembly:

Fbp = Fd × {L2 ÷ L1}





where Fbp = the force output of the brake pedal assembly
where Fd = the force applied to the pedal pad by the driver
where L1 = the distance from the brake pedal arm pivot to the output rod clevis attachment
where L2 = the distance from the brake pedal arm pivot to the brake pedal pad

Note that this relationship assumes 100% mechanical efficiency of all components in the brake pedal assembly. In practical application, the mechanical deflection of components and friction present in physical interfaces prevents this condition. The Master Cylinder

It is the functional responsibility of the master cylinder to translate the force from the brake pedal assembly into hydraulic fluid pressure. Assuming incompressible liquids and infinitely rigid hydraulic vessels, the pressure generated by the master cylinder will be equal to:

Pmc =



Fbp
Amc

where Pmc = the hydraulic pressure generated by the master cylinder where Amc = the effective area of the master cylinder hydraulic piston

Note that this relationship assumes 100% hydraulic efficiency of all components in the master cylinder assembly. In practical application, fluid properties, seal friction, and compliance the physical components prevents this condition.

Brake Fluid, Brake Pipes, and Hoses
It is the functional responsibility of...
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