Introduction to
Dynamic Systems
Special Assignment 1
Automotive Brake System Analysis
1
Austin Kaiser & Michael Ney
Date: February 21, 2011
Professor John J. Moskwa
Mechanical Engineering
Automotive Brake System Analysis
The purpose of this project was to analyze the dynamics of an idealized automotive disk
brake system in relation to overall vehicle speed. We made various assumptions about the brake
system schematic to simplify the free-body diagram and the analysis. Along with studying the
physics behind the brake system we also researched and explored it in terms of theoretical and
operational analysis.
Observations
The disk brake system operates with the help of a piston and a caliper. When the brake
pedal is activated, a brake pad attached to the piston shifts forward to apply a force to one side of
the rotor. The reaction force on the other side of the rotor comes from another brake pad attached
to the caliper housing that moves as the piston moves. More pressure on the brake pedal equals
more piston and caliper force, and thus faster deceleration of the vehicle.2
Modeling, Free Body Diagrams, and Mathematical Representation
We had to make a few assumptions and disregard certain elements of the disk brake
system in order to come up with a schematic that we were able to analyze for an idealized model.
The assumptions we made to accompany our free-body diagram were:

No wear and tear of the brake pads

Disc brake system is in perfect alignment

The rotor isn’t warped

The friction force from the brake pads is constant

Neglect static friction forces

There is no slip between the road and the tires

The same braking force is applied to all four wheels and all components act identically

Neglect drag from the air and the road

The braking force applied to the pedal is constant, which makes the deceleration constant
Included in Appendix A is a schematic of the actual braking system, our free body diagram
after assumptions were taken into consideration, and the equations of motion for the idealized
system model.
Our equation of motion with respect to the wheel and rotor is as follows:
𝐼
𝑑2 𝜃
− 𝑅𝜇𝑘 𝐹𝑁 = 0
𝑑𝜃 2
Where 𝐹𝑁 is the normal force of the brake pad on the rotor, R is the distance from center of
rotor to brake pad, 𝜇𝑘 is the coefficient of kinetic friction between the brake pad and rotor, and
𝑑2 𝜃
𝑑𝜃2
is the angular deceleration of the rotating wheel and rotor components. When the driver
applies a higher force on the pedal, 𝐹𝑁 will be larger. From this equation, the angular
deceleration will have to increase.
The angular deceleration of the wheel will result in the car slowing down based on the
following equation:
∑ 𝐹𝑥 = 𝑚𝑎 = 𝜇𝑘 𝐹𝑁 − 𝐹𝑎
Where 𝐹𝑁 is the normal force due to the weight of the car, 𝜇𝑘 is the coefficient of kinetic friction
between the road and tire, and 𝐹𝑎 is the force of the axle on the tire. The force of the axle has an
equal and opposite force on the chassis of the vehicle, thus slowing the entire car down.
Analysis
We simulated the system by using the driver’s brake pedal force as the input to the
system, and the vehicle speed during braking as the output, with it starting at any given coasting
speed. Although all four wheels would play a role in an actual vehicle braking maneuver, we
simplified the system by only considering the front wheels with no vacuum assist and simplified
that even more to a single wheel model. Because we considered constant braking force applied to
the system, the result of the analysis was a constant vehicle deceleration.
Answers to Specific Questions
1. When the driver depresses the brake pedal he is moving pistons within the vehicle’s
master cylinder. The master cylinder contains two pistons in one housing; one for
braking the front wheels, and the other for braking the rear.4 Compressing these pistons
moves brake fluid through a series of steel and rubber hoses to the car’s caliper system.
Brake fluid is incompressible, thus transferring the force from the master cylinder to
another piston system in the caliper.3 This system pinches the brake pads on both sides
of the rotor together causing a friction force used to decelerate the vehicle. The brake
force is increased by using the brake pedal as a lever and controlling the piston sizes in
the hydraulic system.
2. After the driver releases the brake pedal, springs in the master cylinder return the pistons
back to their original positions. As the pistons return, ports in the cylinders housing are
uncovered relieving the pressure that had been built up in the wheel cylinders. A check
valve in the system maintains a minimum pressure, thus preventing the entrance of air in
the lines.3 When the pressure in the wheel cylinders is relieved, the normal force on the
brake pads drops significantly, allowing the pads to ride on the rotor with a very small
friction force.
3. Most cars utilize a floating caliper system that allows the caliper to adjust and center
itself over the rotor. The floating caliper system uses a single (or sometimes multiple)
piston on one side of the rotor to apply the braking force. When brake pressure is
applied, the piston in the wheel cylinder pushes the inside brake pad against the rotor.2
The pressure in the cylinder also forces the rigid caliper to slide in its tracks towards the
inside of the vehicle. This results in the caliper pushing the outside brake pad against the
outside part of the rotor. The final result is both pads pinching the rotor resulting in a
braking force. Warped discs cause a much more dynamic system result as a constant
pressure on the brake pedal may produce a brake force that isn’t exactly constant. Our
model assumes that the rotors are not warped and thus the brake system is aligned
perfectly.
4. Since brake systems rely on the incompressibility of the hydraulic brake fluid,
introducing air (or any other compressible fluid) can have detrimental effects on the
braking reliability and efficiency. Having air bubbles in the system can make the brakes
feel spongy.3 This is due to the fact that when the brakes are applied and the pressure in
the system increases, the air pockets will compress. This results in less of the brake force
being transferred to the wheel cylinder, and thus a less efficient braking system. If air has
entered the brake system, it must be bled out to prevent these problems.
Conclusions
Modeling this system resulted in a dynamic equation where the rotational deceleration of
the wheel and rotor were based on the torque caused by the force of friction between the brake
pad and rotor. This friction force is a function of the force the driver applies to the pedal. The
summation of forces acting on the wheel show that the vehicles velocity decreases as the force of
the road on the tire overcomes the force of the axel on the wheel.
This project helped us to learn how disc brake systems operate in automobiles and how
the dynamic equations we’ve learned about in class can relate to real world applications. We now
have a better understanding of how Newton’s laws of motion can be applied to the complex
systems we come across every day.
References
1. Automotive.com Enthusiast Central: Dialing in Discs. (2011). Retrieved from
http://4wheeloffroad.automotive.com/1666/131-0708-4x4-disc-brakes/index.html
2. Bosch automotive handbook. (2007). (7th ed. ed.). Hoboken, N.J. Chichester : Wiley ;
John Wiley [distributor].
3. Marks’ standard handbook for mechanical engineers /(c2007). (11th ed. ed.). New York ;
London: McGraw-Hill.
4. How Stuff Works: How Brakes Work. (1998-2011). Retrieved from
http://auto.howstuffworks.com/auto-parts/brakes/brake-types/brake3.htm
Appendix A
Schematic of Braking System 4
Free Body Diagram
Equations of Motion
∑ 𝑇 = 𝐼𝛼
𝑑2 𝜃
𝐼 𝑑𝜃2 − 𝑅𝜇𝑘 𝐹𝑁 = 0
𝐹𝑁 = normal force of brake pad on rotor; R=distance from center of rotor
to brake pad; 𝜇𝑘 = coefficient of kinetic friction between rotor and brake
pad
∑ 𝐹𝑥 = 𝑚𝑎 = 𝜇𝑘 𝐹𝑁 − 𝐹𝑎
𝐹𝑁 = normal force between road and tire; 𝜇𝑘 = coefficient of kinetic
friction between road and tire; 𝐹𝑎 = force from axel