Brake design and safety

By Rudolf Limpert

Brake Design

and Safety

Third Edition

®

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ISBN 978-0-7680-3438-7

SAE Order No. R-398

DOI 10.4271/R-398

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Limpert, Rudolf.

Brake design and safety / Rudolf Limpert. — 3rd ed.

p. cm.

Includes index.

ISBN 978-0-7680-3438-7

1. Automobiles—Brakes—Design and construction. I. Title.

TL269.L56 2011

629.2’46—dc22

2011011904

Library of Congress Cataloging-in-Publication Data

v

Dedication

To Dr. Hans Strien,

at Alfred Teves, Co. in Frankfurt, Germany,

who took into his brake department a young engineering trainee.

16. April 1963.

415

About the Author

Dr. Rudolf Limpert is retiring from a long career as consulting engineer on motor

vehicle and traffic safety. He continues to publish and teach motor vehicle accident

reconstruction and design of braking systems. The author of many publications

and four other books related to automotive safety, Dr. Limpert received his Ph.D.

in mechanical engineering from the University of Michigan, his M.S. and B.E.S.

from Brigham Young University, and his B.S. from the Engineering School of

Wolfenbuettel.

xv

The purpose of this book is to provide a systems approach to designing safer brakes.

Much of the material presented was developed during my work as a brake design

engineer, conducting automotive research, consulting as a brake expert, and teaching

brake design.

The book is written for automotive engineers, technical consultants, accident

reconstruction experts, and lawyers involved with the design of brake systems, the

analysis of braking performance, and product liability issues. Junior engineers will

benefit from the book by finding in one single source all essential concepts, guidelines,

and design checks required for designing safer brakes.

Chapter 1 reviews basic stopping distance performance, design rules, and product

liability factors.

In Chapter 2, drum and disc brakes are discussed. Brake torque computations are

shown for different drum and disc brake designs.

Temperature and thermal stresses are analyzed in Chapter 3. Practical temperature

equations are shown whenever possible.

Chapter 4 briefly reviews basic concepts involved in analyzing mechanical brake

systems.

The operation and design of hydraulic brakes are discussed in Chapter 5.

Air brake systems and their components are discussed and analyzed in Chapter 6.

Brake force distribution, braking efficiency, optimum brake force distribution, and

vehicle stability during braking for the single vehicle are analyzed in Chapter 7.

Car-trailer and commercial truck-trailer braking is discussed in Chapter 8.

Important elements of antilock braking performance and design are introduced in

Chapter 9.

Brake failures are discussed in Chapter 10.

Preface to the First Edition

xiii

The Second Edition continues to provide a systems approach to designing safer brakes.

Consulting experts will find it a single reference in determining the involvement of

brakes in accident causation.

Brake system technology has attained a high standard of quality over the last two

decades. Nearly all automobiles are now equipped with antilock brakes. Federal

braking standards require commercial vehicles to use antilock brakes. Revolutionary

innovative brake designs are not expected. Improvements in brake systems will only

be achieved through basic research, the application of sound engineering concepts, and

testing, resulting in small, yet important, design changes.

The objective of the Second Edition is to assist the brake engineer in accomplishing his

task to design safer brakes that can be operated and maintained safely. The brake expert

will find all the analytical tools to study and determine the potential causes of brake

failures. The Second Edition is expanded to cover all essential subjects, including the

mechanical and thermal analysis of disk brakes. Mistakes found in the First Edition

were corrected.

I thank all those who have made valuable suggestions and comments and helped me to

understand brakes better, in particular the many individuals who attended my Brake

Design and Safety seminars.

Preface to the Second Edition

xi

While writing the third edition, I have carefully considered the comments received from

readers all over the world. One engineer remarked that whenever he has new trainees

in his brake department, they must read Limpert’s brake book. Following that mandate

I have added explanations and examples to the theoretical analysis of braking and brake

temperature while retaining the practical aspects of brake system design.

Electronic system controls have significantly increased the potential of braking systems.

Notwithstanding the advances made in applying brakes by mechanical, hydraulic, or

electrical means, vehicles are slowed and stopped by friction between pad and rotor.

Only when the underlying brake system is properly engineered will automatic controls

perform effectively and vehicles brake safely under all foreseeable operating conditions.

The third edition provides the fundamental tools necessary to design efficient braking

systems that will comply with safety standards, minimize consumer complaints, and

perform safely and efficiently long before and while electronic brake controls become

active. New to the readers is the brake design software, developed by the author as

an effective companion tool to this edition. The efficient design of automotive brake

systems, including trucks and trailers, with PC-BRAKE software is demonstrated with

detailed examples. Automotive engineering students, brake engineers, and forensic

experts will benefit greatly from the third edition in conjunction with the computer

programs and brake design workshop available from the author’s website www.

pcbrakeinc.com.

Rudy Limpert

Preface to the Third Edition

vii

Table of Contents

Chapter 1 Fundamentals of Braking Performance, Design,

and Safety ..........................................................................................1

1.1 The Functions of a Brake System........................................................1

1.2 Vehicle Deceleration and Stopping Distance .................................2

1.3 Elements of Automotive Brake System Design.............................. 10

1.4 Pedal Force and Pedal Travel .......................................................... 17

1.5 Design Solution Selection Process................................................... 18

1.6 Braking System Involvement in Accidents..................................... 20

Chapter 2 Design and Analysis of Friction Brakes....................... 27

2.1 Brake Torque ...................................................................................... 27

2.2 Brake Factor........................................................................................ 27

2.3 Brake Factor of Drum Brakes .......................................................... 29

2.4 Disc Brakes ......................................................................................... 48

Chapter 3 Thermal Analysis of Automotive Brakes ..................... 65

3.1 Temperature Analysis........................................................................ 65

3.2 Thermal Stress Analysis .................................................................. 107

3.3 Thermal Design Measures.............................................................. 112

Chapter 4 Analysis of Mechanical Brake Systems .................... 119

4.1 General Observations...................................................................... 119

4.2 Wheel Brakes.................................................................................... 120

4.3 Driveshaft-Mounted Brakes ........................................................... 122

Chapter 5 Analysis of Hydraulic Brake Systems........................ 125

5.1 Manual Hydraulic Brakes ............................................................... 125

5.2 Boost System Analysis..................................................................... 127

5.3 Brake Line Pressure Control Devices............................................ 141

5.4 Brake Fluid Volume Analysis......................................................... 150

5.5 Dynamic Response of Hydraulic Brake Systems......................... 175

Brake Design and Safety

viii

Chapter 6 Analysis of Air Brake Systems .......................................... 183

6.1 Basic Concepts ................................................................................. 183

6.2 Foundation Brakes........................................................................... 184

6.3 Brake Torque .................................................................................... 190

6.4 Vehicle Deceleration........................................................................ 194

6.5 ABS Modulating Valves .................................................................. 196

6.6 PC-BRAKE AIR Multi-Axle Software Application .................... 199

6.7 Response Time of Air Brake Systems............................................ 200

6.8 Electronic Brake Control (Braking by Wire) ............................... 209

Chapter 7 Single Vehicle Braking Dynamics ............................. 213

7.1 Static Axle Loads.............................................................................. 213

7.2 Dynamic Axle Loads....................................................................... 214

7.3 Optimum Braking Forces............................................................... 216

7.4 Actual Braking Forces Developed by Brakes................................ 224

7.5 Comparison of Optimum and Actual Braking Forces...................225

7.6 Tire-Road Friction Utilization ....................................................... 228

7.7 Braking Efficiency............................................................................ 230

7.8 Fixed Brake Force Distribution Analysis...................................... 232

7.9 Variable Brake Force Distribution Analysis................................. 238

7.10 Braking Dynamics of Two-Axle Truck Equipped

with Air Brakes................................................................................. 249

7.11 Three-Axle Straight Truck – Air Brakes ...................................... 253

7.12 Vehicle Stability Analysis................................................................ 258

7.13 Braking Dynamics While Turning ................................................ 267

Chapter 8 Braking Dynamics of Combination Vehicles ............ 275

8.1 Tow Vehicle-Trailer Combination................................................. 275

8.2 Electronic Stability Control and Trailer Swing ............................ 278

8.3 Braking of Tractor-Trailer Combinations..................................... 279

8.4 Braking of 2-S1 Combination ....................................................... 281

8.5 2-S1 Tractor-Trailer Combination – PC-BRAKE AIR

Software............................................................................................. 302

8.6 Braking of 3-S2 Tractor-Semitrailer Combination...................... 312

8.7 2-S1–2 Combination: Two-Axle Tractor, Single-Axle Semitrailer,

and Double-Axle Trailer ................................................................ 318

8.8 2-S2 Tractor-Semitrailer ................................................................ 320

8.9 2-S3 Tractor-Semitrailer – Triple-Axle Trailer with

Leaf Springs ..................................................................................... 321

8.10 Test Results ....................................................................................... 325

Table of Contents

ix

Chapter 9 Automatic Brake Control ............................................ 327

9.1 Basic Considerations ...................................................................... 327

9.2 Wheel-Lockup Analysis ................................................................. 328

9.3 Basic Performance Requirements of ABS Systems...................... 343

9.4 Hydraulic ABS Systems................................................................... 353

9.5 ABS System Components............................................................... 360

9.6 Drivetrain Influence on ABS.......................................................... 364

9.7 ABS Systems for Air Brakes............................................................ 364

Chapter 10 Analysis of Brake Failure ........................................... 373

10.1 Basic Considerations....................................................................... 373

10.2 Development of Brake Failure........................................................ 374

10.3 Analysis of Partial Brake Failure.................................................... 376

10.4 Comparison of Dual Brake Systems.............................................. 389

10.5 Vacuum Assist Failure..................................................................... 391

10.6 Full Power Brake Failure................................................................. 392

10.7 Degraded Braking Due to Air Inclusion....................................... 393

10.8 Brake Fluid Considerations in Design and Failure Analysis...... 394

10.9 Seal and Rubber Materials.............................................................. 396

10.10 Data Collection in Brake System Failures..................................... 396

10.11 Failure of Air Brake Systems........................................................... 403

Index .................................................................................................405

About the Author ............................................................................. 415

1

Fundamentals of Braking Performance, Design, and Safety

Chapter 1

Fundamentals of

Braking Performance,

Design, and Safety

1.1 The Functions of a Brake System

A vehicle is connected to the roadway by the normal and traction forces

produced by the tires. Braking, steering, or accelerating forces must be

generated by the small tire tread area contacting the ground. Only forces equal

to or less than the product of tire normal force and tire-road coefficient of

friction can be transmitted between vehicle and ground. Even the ideal braking

and stability control system cannot utilize more traction than provided by the

tires and road.

The safe operation of a motor vehicle requires continuous adjustment of its

speed to changing traffic conditions. The brakes and tires along with the

steering system are the most safety-critical accident avoidance components of

a motor vehicle. Brakes must perform safely under all reasonably foreseeable

operating conditions, including slippery, wet, and dry roads; with a lightly or

fully laden vehicle; when braking straight or while turning; with new or worn

brakes; when applied by the novice or experienced driver; on smooth or rough

roads; or when pulling a trailer.

The basic functions of a brake system must be provided under foreseeable

circumstances, at reasonable cost and brake wear life, while providing

directional stability and acceptable tire-road friction utilization.

The braking system must comply will all applicable safety standards. Under

most conditions, safety standards are considered minimum performance

requirements.

1.1.1 Slowing and/or Stopping

Decelerating a vehicle to a lower speed or to a complete stop is the function

most often performed by the service brakes of a vehicle. Safety standards

and industry practices place stringent requirements on effectiveness of stops

including repeated braking under a variety of operating conditions. Critical

design parameters include proper brake balance front-to-rear to ensure

directional stability while braking at, or near, the limit of tire-road friction,

2

Brake Design and Safety

and optimum brake rotor geometry to minimize brake temperature rise and

thermal stresses. Statistically speaking, most domestic drivers rarely exceed

0.1 to 0.2 g braking severity, and may approach dry-road brake lockup or

antilock braking system (ABS) modulation only twice a year. Consequently,

optimizing brake designs with respect to maximum-effectiveness braking

may not yield optimum braking performance in terms of brake wear life for

low-level braking effectiveness or continued braking. Large values of thermal

conductivity, specific heat, and density for brake rotors yield lower swept

surface temperatures.

1.1.2 Maintaining Speed on a Downgrade

In most downgrade driving situations, service brake systems perform

adequately when properly used by the driver. During constant-speed

downgrade operation, the potential energy is converted into thermal energy by

the brakes, resulting in increased brake temperature. As long as the operating

conditions in terms of the potential energy rate (weight, slope, and speed) are

such that the steady-state brake temperature reached is less than a brake-specific

critical temperature, the vehicle will be able to safely descend the downgrade.

Important design parameters are that each brake produces an optimum share

of the overall low-level braking force over an extended time, which is often

different from the share required for maximum-effectiveness braking; that the

convective cooling is optimized through effective ambient airflow and large

cooling areas; and that, in the case of interstate buses and similar vehicles,

cooling airflow obstructions are eliminated or minimized.

1.1.3 Holding Stationary on a Downgrade

Holding a vehicle stationary is primarily a function of the force transmission

or mechanical gain between the application lever and braked tires. Safety

standards generally require a specified hill-holding capacity. However, because

a parking brake may be used in an emergency situation when the service brake

has failed, both thermal and vehicle dynamic factors must be considered by

the brake engineer. In addition, in such an emergency, human factors may be

of critical importance in terms of parking brake control location, hand or foot

application, and apply modulation.

1.2 Vehicle Deceleration and Stopping Distance

1.2.1 Basic Measures of Motion

The motion of a decelerating vehicle can be described by four measures of

physics, namely distance, time, velocity, and deceleration. Distance and time are

fundamental measures; that is, they cannot be broken down into submeasures.

Velocity and deceleration are measures derived from distance and time; i.e.,

they can be broken down into fundamental measures.

The velocity V of a vehicle is computed by the ratio of distance S and time t:

3

Fundamentals of Braking Performance, Design, and Safety

(1-1)

where S = distance, m (ft)

t = time, s

The term “speed,” often used to describe velocity, only refers to the magnitude

of the velocity, and does not indicate angular orientation and direction of the

moving vehicle.

The deceleration, a, of a vehicle is computed by dividing the velocity decrease by

the time interval during which the velocity change has occurred:

(1-2)

With the basic motion parameters defined, we can now compute the stopping

distance and other related factors of a moving vehicle.

1.2.2 Simplified Stopping Distance Analysis

The motion of a vehicle as it changes over time can be shown graphically in

the velocity-time diagram (V-t diagram) (Refs. 1.1, 1.2). In the case of constant

velocity, the velocity curve is a straight line, as illustrated in Fig. 1-1. The

rectangular area under the V-line is given by the product of height and length,

or velocity multiplied by time. Inspection of Eq. (1-1) reveals that the distance S

traveled is also equal to velocity multiplied by time, i.e., equal to the area under

the V-curve.

V=S/t , m/s (ft/s)

a

V

t

V V

t t

, m/s (ft/s ) 2 1

2 1

= ∆

∆ = −

−

2 2

where t = time at start of deceleration, s

t = time at end

1

2 of deceleration, s

V 1 = velocity at start of deceleration, m/s (ft/s)

V = velocity at end of deceleration, m/s (ft 2 /s)

Figure 1-1. Constant velocity-time diagram.

4

Brake Design and Safety

This observation can be expressed as

The distance traveled by a vehicle is equal to the area under the

velocity-time curve.

In a simple analysis, the vehicle motion for an emergency braking maneuver

with constant deceleration can be approximated as shown in Fig. 1-2. After the

driver’s reaction time tr

and the brakes are applied, the vehicle begins to slow

at constant deceleration from its travel speed Vtr, and the vehicle stops after the

braking time ts

.

Figure 1-2. Velocity-time diagram for stopping process.

The V-t diagram shown in Fig. 1-2 consists of a rectangle under the constant

speed portion, and a triangle under the decreasing speed portion of the

maneuver. The total distance Stotal is equal to the area of the rectangle plus the

area of the triangle, or

(1-3)

The last term of Eq. (1-3) can be rewritten using t2

– t1

= ts

or ts

= Vtr /a from Eq.

(1-2) as

(1-4)

Eq. (1-4) is the basic equation used for simple speed and stopping distance

calculations in accident reconstruction.

1.2.3 Expanded Stopping Distance Analysis

In braking maneuvers where the maximum sustained vehicle deceleration is not

achieved quickly and the deceleration rise cannot be ignored, a more detailed

stopping distance analysis must be carried out.

Consider the basic braking parameters illustrated in Fig. 1-3. The idealized

pedal force as a function of time is shown in Fig. 1-3a. At time zero (0) the

driver recognizes the danger. After the reaction time tr

has elapsed, the driver

begins to apply pedal force. After the brake system application time ta

has

S V t V t m(ft) total = + tr r tr s / , 2

S V t V a m(ft) total = + tr r tr

2 / , 2

5

Fundamentals of Braking Performance, Design, and Safety

passed, the brake shoes contact the drums and vehicle deceleration begins. The

linear rise of pedal force is an approximation and occurs over the time tp

. In

critical emergency situations, unskilled drivers tend to reduce their pedal forces

somewhat after 0.1 to 0.2 s of brake initiation in an attempt to modulate the

braking process (Ref. 1.3). When the obstacle comes closer, pedal forces rise

again. Skilled drivers generally have pedal forces that more closely resemble

the idealization. At higher speeds the pedal force rise characteristics that are

actually present may be of lesser importance because their influence on overall

stopping distance is small. Special designs such as the brake assist compensate

for driver factors by rapidly and forcefully applying brake line pressure

automatically to minimize response time.

Figure 1-3. Stopping distance analysis.