Braking Fundamentals a Study of the Dynamics of Disc Brakes and
Pad/Rotor Contact
by
Jared Feist
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miraete, Thesis Adviser
Rensselaer Polytechnic Institute
Troy, New York
December, 2014
CONTENTS
Braking Fundamentals a Study of the Dynamics of Disc Brakes and Pad/Rotor Contact . i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ........................................................................................................ vi
ACRONYMS AND DEFINITIONS ............................................................................... vii
KEYWORDS .................................................................................................................. viii
ACKNOWLEDGMENT .................................................................................................. ix
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.1.1
1.2
Principle of Disc Brakes......................................................................... 1
Problem Description........................................................................................... 2
2. Theory and Methodology ............................................................................................ 4
2.1
Theoretical Background ..................................................................................... 4
2.2
Finite Element Model Definition ....................................................................... 5
2.2.1
Part Geometry and Mesh ........................................................................ 5
2.2.2
Materials ................................................................................................. 7
2.2.3
Assembly, Interaction ............................................................................ 7
2.2.4
Loads and Boundary Conditions ............................................................ 9
3. Results........................................................................................................................ 10
3.1
10
4. Conclusion ................................................................................................................. 12
4.1
12
5. References.................................................................................................................. 13
6. Appendices ................................................................................................................ 15
6.1
Vehicle Dynamics ............................................................................................ 15
6.1.1
Aerodynamic Drag ............................................................................... 15
ii
6.1.2
Driveline Drag...................................................................................... 15
6.1.3
Grade .................................................................................................... 15
6.1.4
Brake Proportioning ............................................................................. 15
iii
LIST OF TABLES
iv
LIST OF FIGURES
Figure 1 – Brake Rotor ...................................................................................................... 2
Figure 2 – Brake Pads ........................................................................................................ 2
Figure 4 – 3D Disc Geometry ............................................................................................ 5
Figure 5 – 3D Pad Geometry ............................................................................................. 6
Figure 6 – Disc Mesh ......................................................................................................... 6
Figure 7 – Pad Mesh .......................................................................................................... 6
Figure 8 – Brake FEM Assembly ...................................................................................... 8
Figure 9 – Initial Gap ......................................................................................................... 8
Figure 10 – Tie Constraint ................................................................................................. 9
Figure 3 – Simple Disk Brake Schematic [2] .................................................................. 11
v
LIST OF SYMBOLS
vi
ACRONYMS AND DEFINITIONS
vii
KEYWORDS
viii
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
ix
1. Introduction
1.1 Background
A moving vehicle possesses kinetic energy whose value depends on the weight
and speed of the vehicle. This energy must be partially or completely dissipated when
the vehicle is slowed down or brought to a stop. The focus of this study will be on the
mechanisms of braking, primarily the physical interaction between pads and rotors.
Automotive brakes dissipate energy by converting a vehicles kinetic energy at
any time into heat energy by means of friction. This energy conversion is a highly
complex study in which all three modes of heat transfer come into play; conduction,
convection, and radiation. The topic of head dissipation is beyond the scope of this
work, however, a more thorough study of heat transfer within automotive brake systems
can be found in Reference [1].
1.1.1
Principle of Disc Brakes
When a vehicle is in motion, it has kinetic energy based on its mass and velocity.
To completely stop a car, the braking system will use friction force to oppose the rotational forces of the wheels. An automotive brake system is a hydro-mechanical system
used by the operator to control speed. This system can be broken down into many
different hydraulic and mechanical components; this project will focus on the primary
friction components a disc brake system.
1.1.1.1 Brake Disc
The brake disc, also called the rotor, is connected to the wheel hub which makes
up the rotating component within the brake system. The rotor also provides the friction
surface for the pads, which generates braking torque. Rotors usually are vented or cross
drilled to aid in the dissipation of heat. A typical rotor can be seen in Figure 1.
1
Figure 1 – Brake Rotor
1.1.1.2 Brake Pads
The brake pads consist of a stamped steel baking plate to which the friction material is attached. Brake pads are housed within a brake caliper, make up the non-rotating
component, and provide the interface to the rest of the brake system. The material, also
called lining is what actually engages the rotor surface creating friction to generate
braking torque. Typical pads can be seen in Figure 2.
Figure 2 – Brake Pads
1.2 Problem Description
This project will analyze a typical domestic passenger car with a mass (m) of
1650 kg, a tire radius (R) of 330.2 mm, and a brake pad surface area (Ap) of 10,700 mm2.
Assuming the vehicle has an initial linear velocity (v1) of 40 km/h and has to come to
complete stop (v2) 0 km/m.
The Society of Automotive Engineers (SAE) Handbook contains requirements
and testing procedures for automotive braking systems. Without going in the details of
every test, the deceleration requirements for in-service brake performance of passenger
car and light duty trucks up to 4500 kg is between 2.4 m/s2 and 5 m/s2 not to exceed 6
m/s2.
A simplified finite element disc brake system will be modeled using the physical
characteristics outline above. Explicit finite element analysis (FEA) will be performed
to determine the inputs of the independent system variables needed to meet the SAE
requirements for passenger vehicle braking performance. These results will also be
compared to the theoretical background typically used in the commercial automotive
industry to design passenger vehicle brake systems.
The contact pressure will depend on the size and style of brake pad used in application. Load variation is known to affect coefficients of friction; studies show typical
2
coefficients of friction fluctuate between 0.3 and 0.5. The relationship between output
torque and sliding coefficients of friction is known as the shoe factor. Shoe factors for
various brake types are provided in Table 1.
Table 1 – Shoe Factor for Various Brake Types [Excerpt from Reference xx, Table 11.3]
Type of Brake
Single trailing shoe
Two trailing shoes
Disc and pad
Single leading shoe
Shoe Factor
0.55
1.15
1.2
1.65
Knowing that disc and pad brakes generally have a shoe factor of 1.2; Figure 3 is used to
determine for this study of disc brakes a coefficient of friction of 0.4 will be used
throughout both the analytical and FEA evaluations.
Figure 3 – Relationship of Shoe Factors and the Coefficient of Friction for Various Brake Types
[Ref xx]
3
2. Theory and Methodology
2.1 Theoretical Background
For simplicity it will be assumed that stopping is be completely attributed to the
brakes on a level grade all other forces associated with vehicle deceleration will be
ignored, such as aerodynamic drag, rolling resistance and any driveline drag that may be
associated a vehicle’s powertrain. The total kinetic energy of the vehicle may be given
by Equation [1].
𝐾𝐸𝑣𝑒ℎ𝑖𝑐𝑙𝑒 =
1
𝑚∆𝑣 2
2
[1]
Assuming the total braking force (FBrake) times distance traveled (x) is equal to kinetic
energy; braking force can be determined by Equation [2].
𝐹𝐵𝑟𝑎𝑘𝑒 ∙ 𝑥 = 𝐾𝐸𝑉𝑒ℎ𝑖𝑐𝑙𝑒
𝐹𝐵𝑟𝑎𝑘𝑒
𝑚𝑣1 2
=
2𝑥
[2]
The friction force necessary to stop the vehicle will be equal to the total braking force
divided by the number of brakes in the system (nbrakes) as shown by Equation [3].
𝑓𝑓 =
𝐹𝐵𝑟𝑎𝑙𝑒
𝑛𝑏𝑟𝑎𝑘𝑒𝑠
[3]
If it is assumed that the friction force acts through the center of the pad, then the braking
torque (TB) can be calculated using the mean distance between the center of the pad and
the center of the disc as shown by Equation [4] being sure to note that braking torque is
dependent on the number of pads per brake (npad).
𝑇𝐵 = 𝑛𝑝𝑎𝑑 (𝑓𝑓 )
(𝑅2 − 𝑅1 )
2
[4]
4
In order to achieve the above braking torque hydraulic pressure (p) within the braking
system must be applied over the pad area. Equation [5] can be used to calculate the
hydraulic pressure required to develop the appropriate braking torque.
𝑝=
𝐹𝐵𝑟𝑎𝑘𝑒
𝑛𝑝𝑎𝑑 𝐴𝑝𝑎𝑑 𝜇
[5]
2.2 Finite Element Model Definition
There are two distinct methods to solve finite element problems, implicit and explicit. The implicit method solves for a static or dynamic equilibrium, while the explicit
method solves transient dynamic response problems using an explicit direct-integration
procedure. The finite element code used to model the brake pad rotor system herein will
utilize ABAQUS/Explicit, Abaqus 6.13-EF2 to solve a dynamic simulation of brake pad
to rotor contact and ABAQUS/CAE, Abaqus 6.13-EF2, for modeling the brake system.
2.2.1
Part Geometry and Mesh
The brake FEM used is a simplified version of a typical disc brake system used
in domestic passenger vehicles. Figure 4 shows a simplified 3D solid disc, which allows
the model to use coarser meshes than would be required to model the details of a typical
brake disc with complicated geometrical features such as cooling ducts and bolt holes.
Figure 5 shows simplified 3D pad geometry and is modeled such that it can only
contact part of the circumference of the disc.
5
Figure 5 – 3D Pad Geometry
Figure 4 – 3D Disc Geometry
The rotor has an outside diameter of 355.6 mm, an inside diameter of 203.2 mm, and a
thickness of 31.75 mm. The pad surface area is 10,700 mm2. The FEM is further
simplified by making it symmetrical about a plane normal to the z-axis. Therefore, only
half of the disc thickness and one pad is modeled, and symmetric boundary conditions
are applied.
Each model was meshed dependently on the part. Figure 6 shows the disc mesh
with a global seed size of 0.01 and an edge seed of five to obtain an acceptable number
of elements through the thickness. Figure 7 shows the pad mesh with a global seed size
of 0.0055 and a similar edge seed of five to get an acceptable number of elements though
the thickness.
6
Figure 7 – Pad Mesh
Figure 6 – Disc Mesh
Both pad and rotor utilize eight-node linear brick (C3D8) fully integrated elements in
order to give more resolution at the surface than reduced integration (C3D8R). Artificial
incompressible bending or locking is not a concern in this analysis. Since the FEM was
simplified by using symmetric boundary conditions bending of the disc is not an issue.
Both pad and rotor contact surfaces were modeled as flat, so a quadratic or curved
surface definition is not necessary. The C3D8 element used throughout the analysis is
the most computational efficient and will provide accurate analysis results.
2.2.2
Materials
Elastic material properties were used for the analysis; the values are summarized
below.
2.2.3
Assembly, Interaction
A reference point was established as the center of rotation at the global origin (0,
0, 0). The FEM assembly was created based on the pad and rotor simplifications outlined in section 2.2.1 and the center of the rotor was arranged coincident to the center of
rotation as seen in Figure 8.
7
Figure 8 – Brake FEM Assembly
Figure 9 – Initial Gap
Figure 9 shows the Pad is initially spaced 1.5mm away from the disc to assure the disc
can properly rotate and no friction forces are present prior to prior to pad-disc contact.
Define and explain why General contact was used between the pad and rotor will
be used. Figure xx shows the general contact interactions applied between the pad and
rotor. General contact was used to simulate braking and so the proper contact interactions can be applied.
8
The center of rotation reference point was connected to the interior surface of the
disc using a rigid body tie constraint. A tie constraint was chosen over a coupling
because the rigid body tie is transmitted directly to the solid elements and they are
computational more efficient in this application.
Figure 10 – Tie Constraint
Define and explain why predetermined rotation was applied to center of rotation.
Expand on how it represents vehicle linear velocity of xx.
2.2.4
Loads and Boundary Conditions
Two boundary conditions were used. The first being at the center of rotation all
the pad rotor assembly to only rotate in about the z axis. The second BC was applied to
the entire surface of the disc to represent the symmetry about the z-axis as described in
section 2.2.1 above.
9
3. Results
3.1 Analytical Results
The brake system and operating scenario outlined in Section 1.2 provides all required information to calculate the vehicle’s kinetic energy using Equation xx.
𝐾𝐸 =
1
(1650 𝑘𝑔)(11.11 𝑚⁄𝑠)2 = 1.019𝑥105 𝐽
2
This braking scenario is consistent with the, Baseline Check Stop outlined in Reference
xx, which requires the stopping deceleration to be equal to 2.4 m/s2. Assuming this is an
average deceleration Equation xx can be used to determine the average braking force
necessary to bring the vehicle to a stop.
𝐹𝐴𝑣𝑔 =
[1650𝑘𝑔(11.11 𝑚⁄𝑠)2 ]
= 1979.62 𝑁
2(51.44𝑚)
In general the interaction between brake pads and rotors involve dry sliding contact at varying speeds and contact forces. Dry friction is the resistance of relative motion
between two solid surfaces in contact at rest or in motion. Friction is often quantified as
a ratio of normal force (FN) to lateral or tangential force (Ft), by the friction coefficient
(μ). Typical clamping forces (FN) for normal braking are between 6kN and 10kN.
These variables will be manipulated within the finite element model (FEM) to accurately
represent and study the braking scenario.
Disc brakes slow the rotational motion of automobile wheels with friction caused
by brake pads pushing against the rotors. This force is typically applied by a piston and
cylinder which is hydraulically powered and controlled by the vehicle operator. This
system is mechanically linked to the wheel or drive train and used to stop the vehicle;
friction causes the rotor and attached wheel to slow and/or stop.
10
Figure 11 – Simple Disk Brake Schematic [2]
11
4. Conclusion
4.1
12
5. References
[1] Limpert, R. (1999). Brake design and safety (2nd ed.). Warrendale, Pa.: Society of
Automotive Engineers.
[2] Deaton, Jamie P. "How Brake Rotors Work." howstuffworks, n.d. Web. 25 Oct.
2012.<http://auto.howstuffworks.com/auto-parts/brakes/brake-parts/brakerotors1.htm>
[3] N.M. Kinkaid, O.M. O'Reilly, P. Papadopoulos, Automotive disc brake squeal,
Journal of Sound and Vibration, Volume 267, Issue 1, 9 October 2003, Pages 105166, ISSN 0022-460X, 10.1016/S0022-460X(02)01573-0.
<http://www.sciencedirect.com/science/article/pii/S0022460X02015730>
[4] SAE International J843, Brake System Road Test Code – Passenger Car and Light –
Duty Truck, Surface Vehicle Recommended Practice, dated March 2013
[5] Shigley’s, Mechanical Engineering Design, Eigth Edition, 2008
[6] Wear and Contact Conditions of Brake Pads, M. Eriksson, J. Lord, and S. Jacobson.
s.1: VTT, 2000, Vol 2.
[7] Two Body Abrasive Behavior of Brake Pad Dry Sliding Against Interpenetrating
Network Ceramics. SY. Zhang, SG. Qu, and YY. Li. Guangzhou, China: Wear,
2009, Vol. 268.
[8] On the Dry and Wet Sliding Performance of Potentially New Frictional Brake Pad
Material for the Automotive Industry. EL-Tayeb, NSM and KW. Liew. s.1.: Wear,
2009, Vol. 266.
[9] The Effect of Metal Fibers in Friction Performance of Automobile Brake Friction
Materials. H. Jang. et al. s.1.: Wear, 2004, Vol. 259.
[10]
Load, Speed, and Temperature Sensitivities of a Carbon Fiber Reinforced
Phenolic Friction Material. P. Gopal, LR. Dharani, and D. Frank. S.1.: Wear, 1995,
Vol. 181.
[11]
Dry Sliding Wear of AL-alloy. M. Narayan, MK. Surappa, and PK. Pramilla Bai.
s.1.: Wear, 1995, Vol. 181.
13
[12]
Tribological Properties of Automotive Disc Brakes with Solid Lubricants. L.
Gudmand-Hoyer and A. Bach. s.1.: Wear, 1999, Vol. 232.
[13]
The Effects of Antimony Trisulfide Sb S and Zirconium Silicate in the Automo-
tive Brake Friction Material on Friction. H. Jang and S. Kim. s.1.: Wear, 2000, Vol.
239.
[14]
Principles of Automotive Vehicles, Technical Manual TM 9-8000, Department
of the Army, dated 25 October 1985
[15]
Antimony in Brake Pads a Carcinogenic Component? O. Uexkull, et al. 1, s.1.:
Journal of Cleaner Production, 2005, Vol. 13.
14
6. Appendices
6.1 Vehicle Dynamics
This project makes a few assumptions to simplify disc braking analysis of a passenger
vehicle, however, the following sections will outline why it is important to consider all
aspects of vehicle dynamics when designing a brake system which is really a subsystem
of the overall automotive system.
6.1.1
Aerodynamic Drag
Aerodynamic drag will aid deceleration due to wind resistance. Aerodynamic
drag force is dependent on the vehicles drag factor and velocity squared. These forces
can be ignored at normal passenger vehicle operating speeds because they are small
compared to braking force.
6.1.2
Driveline Drag
The engine, transmission, and final drive components contribute to both drag and
inertia effect during braking. The inertia of these comments adds to the effective mass
of the vehicle, and warrants consideration in braking.
6.1.3
Grade
Road grade contributes directly to the braking efforts in a positive sense when
going uphill and negatively when traveling downhill. The additional force acting the
vehicle due to grade is given by Equation xx. For small angles θ typical for most grades
small angle theory can be applied.
6.1.4
Brake Proportioning
Balancing the brake efforts on both the front and rear axles is achieved by pro-
portioning the application pressure on the front and rear brakes and in accordance with
their loads. Additionally, brake systems typically have physically larger brake pads and
rotors on the front axle in comparison to the real axle.
During braking load transfers dynamically from the real to the front axle of a vehicle; the maximum brake force is dependent on the deceleration, varying differently at
15
each axle. The maximum braking force of on an axle is dependent on that present on the
other because deceleration affects transfer load. An attempt to brake the front axle to a
level above the front brake force boundary will cause front wheel lockup to occur in
which steering control will be lost. Braking the real axle beyond its boundary will cause
real axle lockup, which places a vehicle in an unstable condition. The challenge in brake
system design is selecting a proportioning ratio that will satisfy all design goals despite
the variability in: road conditions, vehicle weight distribution and center of gravity, and
braking conditions.
Anti-lock Brake systems (ABS) consist of an electronic control unit to effectively
proportion brakes by releasing individual wheel brakes momentarily when it senses
wheel lockup. ABS systems are not designed to control poorly designed brake systems,
they can only assist a vehicle’s operator in braking when being it is outside the limits of
proportional braking.
16