Brake Design and Safety Third Edition By Rudolf Limpert ® Warrendale, PA USA Copyright © 2011 SAE International eISBN: 978-0-7680-5789-8 ® 400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: CustomerService@sae.org Phone: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-0790 Copyright © 2011 SAE International All rights reserved No part of this publication may be reproduced, stored in a retrieval system, distributed, or transmitted, in any form or by any means without the prior written permission of SAE For permission and licensing requests, contact SAE Permissions, 400 Commonwealth Drive, Warrendale, PA 15096-0001 USA; e-mail: copyright@sae.org; phone: 724-772-4028; fax: 724-772-9765 ISBN 978-0-7680-3438-7 SAE Order No R-398 DOI 10.4271/R-398 Library of Congress Cataloging-in-Publication Data Limpert, Rudolf Brake design and safety / Rudolf Limpert — 3rd ed p cm Includes index ISBN 978-0-7680-3438-7 Automobiles—Brakes—Design and construction I Title TL269.L56 2011 629.2’46—dc22 2011011904 Information contained in this work has been obtained by SAE International from sources believed to be reliable However, neither SAE International nor its authors guarantee the accuracy or completeness of any information published herein and neither SAE International nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information This work is published with the understanding that SAE International and its authors are supplying information, but are not attempting to render engineering or other professional services If such services are required, the assistance of an appropriate professional should be sought To purchase bulk quantities, please contact: SAE Customer Service E-mail: CustomerService@sae.org Phone: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-0790 Visit the SAE Bookstore at http://books.sae.org 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 v 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 415 Preface to the First Edition 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 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 Practical temperature equations are shown whenever possible Chapter briefly reviews basic concepts involved in analyzing mechanical brake systems The operation and design of hydraulic brakes are discussed in Chapter Air brake systems and their components are discussed and analyzed in Chapter Brake force distribution, braking efficiency, optimum brake force distribution, and vehicle stability during braking for the single vehicle are analyzed in Chapter Car-trailer and commercial truck-trailer braking is discussed in Chapter Important elements of antilock braking performance and design are introduced in Chapter Brake failures are discussed in Chapter 10 xv Preface to the Second Edition 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 xiii Preface to the Third Edition 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 xi Table of Contents Chapter Fundamentals of Braking Performance, Design, and Safety 1.1 1.2 1.3 1.4 1.5 1.6 The Functions of a Brake System .1 Vehicle Deceleration and Stopping Distance Elements of Automotive Brake System Design 10 Pedal Force and Pedal Travel 17 Design Solution Selection Process 18 Braking System Involvement in Accidents 20 Chapter Design and Analysis of Friction Brakes 27 2.1 2.2 2.3 2.4 Brake Torque 27 Brake Factor 27 Brake Factor of Drum Brakes 29 Disc Brakes 48 Chapter Thermal Analysis of Automotive Brakes 65 3.1 3.2 3.3 Temperature Analysis 65 Thermal Stress Analysis 107 Thermal Design Measures 112 Chapter 4 Analysis of Mechanical Brake Systems 119 4.1 4.2 4.3 General Observations 119 Wheel Brakes 120 Driveshaft-Mounted Brakes 122 Chapter Analysis of Hydraulic Brake Systems 125 5.1 5.2 5.3 5.4 5.5 Manual Hydraulic Brakes 125 Boost System Analysis 127 Brake Line Pressure Control Devices 141 Brake Fluid Volume Analysis 150 Dynamic Response of Hydraulic Brake Systems 175 vii Brake Design and Safety Chapter Analysis of Air Brake Systems 183 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Basic Concepts 183 Foundation Brakes 184 Brake Torque 190 Vehicle Deceleration 194 ABS Modulating Valves 196 PC-BRAKE AIR Multi-Axle Software Application 199 Response Time of Air Brake Systems 200 Electronic Brake Control (Braking by Wire) 209 Chapter Single Vehicle Braking Dynamics 213 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 Static Axle Loads 213 Dynamic Axle Loads 214 Optimum Braking Forces 216 Actual Braking Forces Developed by Brakes 224 Comparison of Optimum and Actual Braking Forces 225 Tire-Road Friction Utilization 228 Braking Efficiency 230 Fixed Brake Force Distribution Analysis 232 Variable Brake Force Distribution Analysis 238 Braking Dynamics of Two-Axle Truck Equipped with Air Brakes 249 Three-Axle Straight Truck – Air Brakes 253 Vehicle Stability Analysis 258 Braking Dynamics While Turning 267 Chapter 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 viii Table of Contents Chapter Automatic Brake Control 327 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Basic Considerations 327 Wheel-Lockup Analysis 328 Basic Performance Requirements of ABS Systems 343 Hydraulic ABS Systems 353 ABS System Components 360 Drivetrain Influence on ABS 364 ABS Systems for Air Brakes 364 Chapter 10 Analysis of Brake Failure 373 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 Basic Considerations 373 Development of Brake Failure 374 Analysis of Partial Brake Failure 376 Comparison of Dual Brake Systems 389 Vacuum Assist Failure 391 Full Power Brake Failure 392 Degraded Braking Due to Air Inclusion 393 Brake Fluid Considerations in Design and Failure Analysis 394 Seal and Rubber Materials 396 Data Collection in Brake System Failures 396 Failure of Air Brake Systems 403 Index 405 About the Author 415 ix Index Terms Links Calipers (Cont.) floating 49 installation of 57 CamLaster drum brake Clamp ring brake chambers Combination valves 50 186 191–192 145 Combination vehicles 2-S1–2 two-axle tractor, single-axle semitrailer, double-axle trailer 2-S2 tractor-semitrailer 318–320 320 2-S3 triple-axle trailer with leaf springs 321–324 3-S2 tractor-semitrailer 312–318 3-S2-AA, air springs on tractor and trailer tandem axles 313 3-S2-ALS, tractor air springs and trailer leaf springs 315–318 3-S2-AWB, air springs on tractor with trailer walking beam 314 tow vehicle-trailer, braking dynamics of 275–278 tractor-trailer, braking dynamics of 279–281 Component sizing, checkpoints for 17 Compression of brake fluid 164–165 of brake pad 161–162 of brake shoe and lining 162–163 Compressive stress, formula for Computer-based temperature analysis Constant deceleration time, formula for Convective cooling 107 100–107 88–99 Convective heat transfer coefficient 88 Crashworthiness Data System (CDS) 22 Critical deceleration Customer complaints and accident data 227 12 This page has been reformatted by Knovel to provide easier navigation Index Terms Links D Danger, hazard, and risk 21–22 Deceleration for air brake systems 194–196 with automatic front brake limiting valve 195–196 average, formula for 10 and brake line failure 378 brakes-unlocked critical 232–233 227 formula for idealized 5f measurement of 10 with proportioning valves 196 rise time, formula for and stopping distance 2–10 Deceleration-sensitive reducer valves Disc brakes 145–147 243 48–61 adjustment of air 49–50 187 caliper installation 57 compared to drum brakes 57–58 complete temperature analysis in a single stop 75–82 design values for 74t pad pressure and wear in 51–57 self-energizing 60–61 solid, heat transfer coefficient for thermal stress in rotors 90 107–109 ventilated heat transfer coefficient for temperature analysis of Downgrade 91–99 104–107 Driver reaction time four phases of This page has been reformatted by Knovel to provide easier navigation Index Terms Links Driver reaction time (Cont.) in emergency braking Driveshaf-mounted brakes, analysis of 8–9 122–123 Drivetrain effect on brake force distribution influence on ABS 236–237 364 Drum brakes air in hydraulics brake factor of brake shoe and lining compression in braking power absorbed by drums CamLaster compared to disc brakes 163 34–46 162–163 70–72 186 57–58 design values for 74t drum deformation in 162 drum expansion in 163 heat transfer coefficient for 89–90 lining pressure distribution and wear in 29–32 self-energizing and self-locking 32–34 temperature fade correction factor for thermal stress in Drum expansion, thermal pushrod travel increase due to Drum stiffness, effects on brake factor Dual brake systems, comparison of 193 192–193 110 163 193 47–48 389–391 Duo-servo brakes with pivot abutment 40–44 with sliding abutment 39–40 Dynamic axle loads for tow vehicle-trailer combination 276–277 for 2-S1 tractor-trailer combination 303–304 for single vehicles 214–216 Dynamic braking forces 217–218 This page has been reformatted by Knovel to provide easier navigation Index Terms Links E Economics 12 Electric trailer brakes 277 Electronic brake control 209–210 Electronic control unit (ECU) 361–362 Electronic stability control (ESC) 278–279 function of major components of 352 352–353 malfunction of 353 operation of 353 Electronic traction control 327 351–353 Emergency brake, see Parking brake Example problems air brake design 250–253 brake factor 121–122 brake line pressure 173–175 and pedal travel 167–173 braking energy and braking power 67–69 braking forces, optimum 218–220 braking performance, brake lockup sequence 305–312 braking stability 264–267 deceleration and braking efficiencies 244–248 maximum 222–224 at peak friction and stopping distance duo-servo drum brake forensic brake system analysis optimum braking forces 338 199–200 41–44 253 218–220 thermal analysis of front disc brake 74–75 of heavy truck brake 87–88 of linearly decreasing heat flux 81–82 This page has been reformatted by Knovel to provide easier navigation 351–353 Index Terms Links Example problems thermal analysis (Cont.) of pickup truck brake 78–80 of rear brake 84–85 of tractor-semitrailer brake 85–86 of ventilated disc brake 93–99 time required to lock front brakes 340–342 wheel peak and sliding friction parameters 332–333 F Failure analyses Fatality Analysis Reporting System (FARS) Federal Motor Vehicle Safety Standard (FMVSS) 12 24–25 105 123 FMVSS 105 84 120 FMVSS 121 280 287 FMVSS 126 328 FMVSS 135 84 Finite difference method temperature analysis of ventilated disc Fixed brake force distribution analysis 119 100–104 104–107 232–237 Fixed caliper 48 49f Floating caliper 49 50 Fourier’s conduction law 100 Friction braking from high- to low-friction surface 338–342 lines of constant friction coefficient 220–224 of lining/pad peak and sliding, in braking forces diagram tire braking, empirical equations for 58–59 336–338 336 tire-road friction utilization 2-S1 tractor-trailer combination and brake line failure single-vehicle Friction brakes, design and analysis of 287–302 379 228–230 27–61 This page has been reformatted by Knovel to provide easier navigation 123 Index Terms Links Full-power hydraulic brakes analysis of 139 failure of 392 G General Estimates System (GES) 22 H Heat distribution, formula for Heat flux, allowable Heat penetration time, formula for Heat transfer, radiative 72 112–114 72 99–100 Heat transfer coefficient for drum brakes, formula for for solid discs, formula for of ventilated disc brakes, formula for Horsepower, into lining or pad Humidity, effects on brake factor Hydraulic ABS systems 89–90 90 91–92 114 46 353–360 Hydraulic boost systems Hydraulic brake systems 125–179 boost systems for 127–141 analysis of 136–139 dynamic response of 177–179 vacuum 176–177 brake fluid viscosity 175–176 brake fluid volume analysis for 150–179 brake line pressure control devices for 177 141–150 brake pedal linkage 176 dynamic response of 175 manual 125–127 master cylinder in 177 This page has been reformatted by Knovel to provide easier navigation Index Terms Hydraulic modulator Links 362–363 Hydrovac analysis of 134–135 fluid loss in 166 Ideal braking force 218 I Inclined abutment Inspection and maintenance Integrated hydraulic ABS systems In-use factors, checkpoints for 38–39 12 358–360 16–17 J Jackknifing 279–280 L Leaf spring tandem axle, rear 257–258 Lines of constant friction coefficient 220–224 Loading-brake force distribution analysis 233–235 M Manual brakes hydraulic, analysis of 125–127 PC-BRAKE HYDRAULIC for 125 pedal force and pedal travel for 17 Master cylinder dual, improved design dynamic response of fluid requirements for sizes of 387–389 177 160–161 169t step bore adjustable volume analysis of 147–148 148–150 244–248 156–159 This page has been reformatted by Knovel to provide easier navigation Index Terms Mastervac Links 127 Materials selection 12 Mechanical brake systems analysis of 119–123 driveshaft-mounted brakes 122–123 general observations 119–120 wheel brakes 120–122 Mechanical gain, formula for 120 N National Accident Statistical Sampling (NASS) New-versus-used 22–24 12 O Offset piston design Optimum brake line pressures 53–54 238–241 Optimum braking forces for 2-S1 combination 281–286 braking traction coefficient for 216–217 dynamic 217–218 lines of constant friction coefficient for 220–224 parabola analysis of 224 peak and sliding friction in 336–338 for single vehicles 216–224 P Packaging, labeling, and shipping Parabola analysis of optimum braking forces Parallel sliding abutment 12 224 36–38 Parking brake analysis of 120–122 checkpoints for 16 design example 19–20 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Parking brake (Cont.) dynamic response of pedal force and pedal travel for torque in 177 18 194 PC-BRAKE AIR for 2-S1 tractor-trailer combination 302–312 for 3-S2 tractor-semitrailer 316–318 brake system design with 249–253 for leaf spring suspensions multi-axle pushrod travels in 257 199–203 191f 192 PC-BRAKE FACTOR for duo-servo brake with pivot support for LT-shoe brake with parallel sliding abutment for S-cam brake PC-BRAKE HYDRAULIC, for manual brakes PC-BRAKE STABILITY 43 37–38 45 125 264–267 PC-BRAKE TEMPERATURE, for continuous braking Peak and sliding friction in braking forces diagram 86 336–338 Pedal force and brake line failure idealized 378–379 4–5 and pedal travel 17–18 Pedal travel and brake line failure 379–383 computation of 167–173 and pedal force 17–18 Piston four-piston caliper design 55–57 offset 53–54 Pivot abutment Pressure control devices, brake line 40 141–150 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Pressure distribution on brake lining 29–32 on brake pad 52–53 Pressure limiter valves for 2-S1 combination automatic, for front brake Pressure reducer valves 142–143 286–287 195 143–144 for 2-S1 combination 286–287 deceleration-sensitive 243 Production approval 12 Production methods 12 Pulled pad design 241–242 196 54–55 Pushrods adjustments for 188t travel adjustment factor for clamp ring chambers 191–192 travel increase due to thermal drum expansion 193 travel measurement 193–194 travels and limits of 192t R Radiation heat transfer coefficient, formula for Reaction and application time, formula for 99 Rear leaf spring tandem axle 257–258 Rear walking beam tandem axle 254–256 Relay quick release valve Reliability 200 204f 11 20–21 Response time of air brake systems checkpoints for Road surface, split-coefficient, braking on Road test, comparison of theoretical and test results Roll stiffness, formula for 200–209 15 267 235–236 270 This page has been reformatted by Knovel to provide easier navigation 242–243 Index Terms Links Rotors allowable heat flux into design considerations for 112–114 115 thermal failure of 110–111 thermal stress in 107–109 Rubber materials, failure of 396 S Safety and product liability 11 Safety regulations, checkpoints for 17 Safety standards 12 S-cam brakes PC-BRAKE FACTOR for 184–186 45 Seal, failure of 396 Select-high control 345 Select-low control 345 Self-energizing brakes disc 60–61 drum 32–34 Self-locking drum brakes Single-vehicle braking dynamics Single-wheel control Sliding abutment Slowing 32–34 213–271 345 39–40 1–2 Specific design measures, checkpoints for Speed, definition of Stability analysis 16 258–267 and brake line failure 383 braking on a split-coefficient road surface 267 expanded 260–267 general considerations for 258–259 simplified 259–260 Static axle loads 213–214 This page has been reformatted by Knovel to provide easier navigation Index Terms Step bore master cylinder adjustable Stopping Links 147–148 148–150 244–248 1–2 Stopping distance and deceleration expanded analysis of lightly and fully laden, checkpoints for 2–10 3–4 15 simplified analysis of 3–4 total 6–7 Strain 29 Stress, thermal, analysis of Surface finish Surge brakes System-based design methods 107–111 12 277–278 11 T Temperature effects on brake factor 46 surface, formula for 73 Temperature analysis 65–107 checkpoints for 16 computer-based 100–107 for continuous braking 85–88 disc, complete in a single stop 75–82 finite difference method general considerations in 100–107 65 for repeated braking 82–85 simplified in a single stop 72–75 Temperature fade correction factor for drum brakes 192–193 Temperature response for constant heat flux, formula for 77 for time-varying heat flux, formula for 80 Test results, for combination vehicles Thermal design measures 325 112–115 This page has been reformatted by Knovel to provide easier navigation Index Terms Thermal drum expansion pushrod travel increase due to Thermal resistance, formula for Thermal rotor failure Links 163 193 71 110–111 Thermal stress analysis of in brake drums in disc brake rotors 107–111 110 107–109 Tree-axle straight truck air springs rear tandem axle 253–254 braking dynamics of 253–267 rear leaf spring tandem axle 257–258 rear walking beam tandem axle 254–256 Threshold angular deceleration 332 Tire-road friction utilization effects of brake balance on and brake line failure single-vehicle Tire/wheel braking analysis Total stopping distance 287–302 379 228–230 328–335 6–7 Tow vehicle-trailer combination braking dynamics of 275–278 trailer with brakes 276–278 trailer without brakes 275–276 Traction coefficient for braking 216–217 Tractor-semitrailer braking instability 279–281 jackknifing trailer swing Tractor-trailer combinations, braking of 279–280 280 279–281 stabile 280–281 Trailer swing 278–279 280 Trailers with brakes braking dynamics of 276–278 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Trailers with brakes (Cont.) dynamic axle loads electric brakes surge brakes without brakes Transcendental equation 276–277 277 277–278 275–276 77 Trucks three-axle straight, braking dynamics of 253–267 two-axle, with air brakes, braking dynamics of 249–253 basic considerations 267–268 braking dynamics while 267–271 optimum brake line pressures for 268–271 Turning Two-axle truck, with air brakes, braking dynamics of 249–253 V Vacuum-assisted brake booster 127–135 analysis of 128–134 dynamic response of 176–177 failure of 381–392 Valves combination 145 modulating 196–199 pressure limiter 241–242 for 2-S1 tractor-trailer 286–287 automatic front brake 195–196 for hydraulic brakes 142–143 pressure reducer for 2-S1 tractor-trailer for air brakes for hydraulic brakes 242–243 286–287 196 143–147 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Valves (Cont.) relay quick release 200 volume loss in 166 Variable brake force distribution analysis Velocity, formula for 204f 238–248 2–3 Velocity-time (V-t) diagram constant 3f for stopping process 4f Ventilated disc brakes heat transfer coefficient for temperature analysis of 91–99 104–107 W Walking beam tandem axle, rear Warnings 254–256 12 Wear relationship, formula for Wedge brakes 30–31 186 Wheel brakes, see Parking brakes Wheel speed sensors signal analysis for Wheel-lockup analysis braking from high- to low-friction surface empirical equations for tire braking friction 360–361 342–343 328–343 338–342 336 peak and sliding friction in braking forces diagram 336–338 tire/wheel braking analysis 328–335 wheel speed sensor signal analysis 342–343 This page has been reformatted by Knovel to provide easier navigation 5f